%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : NUM726^4 : TPTP v9.2.0. Released v7.1.0.
% Transfm : none
% Format : tptp:raw
% Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% Computer : n001.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Oct 9 02:11:56 PM UTC 2025
% Result : Theorem 61.08s 61.15s
% Output : Proof 61.16s
% Verified :
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)
% Comments :
%------------------------------------------------------------------------------
thf(ty_univof,type,
univof: $i > $i ).
thf(ty_if,type,
if: $o > $i > $i > $i ).
thf(ty_power,type,
power: $i > $i ).
thf(ty_eps,type,
eps: ( $i > $o ) > $i ).
thf(ty_union,type,
union: $i > $i ).
thf(ty_d_ReplSep,type,
d_ReplSep: $i > ( $i > $o ) > ( $i > $i ) > $i ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_l_or,type,
l_or: $o > $o > $o ).
thf(ty_emptyset,type,
emptyset: $i ).
thf(ty_n_1t,type,
n_1t: $i ).
thf(ty_plus,type,
plus: $i > $i ).
thf(ty_pair,type,
pair: $i > $i > $i ).
thf(ty_proj1,type,
proj1: $i > $i ).
thf(ty_n_2t,type,
n_2t: $i ).
thf(ty_d_Sigma,type,
d_Sigma: $i > ( $i > $i ) > $i ).
thf(ty_n_some,type,
n_some: ( $i > $o ) > $o ).
thf(ty_diffprop,type,
diffprop: $i > $i > $i > $o ).
thf(ty_repl,type,
repl: $i > ( $i > $i ) > $i ).
thf(ty_in,type,
in: $i > $i > $o ).
thf(ty_d_428_prop2,type,
d_428_prop2: $i > $i > $o ).
thf(ty_d_Sing,type,
d_Sing: $i > $i ).
thf(ty_nat_p,type,
nat_p: $i > $o ).
thf(ty_binunion,type,
binunion: $i > $i > $i ).
thf(sP1,plain,
( sP1
<=> ( ( ( d_ReplSep
@ ( eps
@ ^ [X1: $i] :
~ ( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X2: $i] :
( if
@ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ eigen__0
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ n_1t @ X3 ) )
@ proj1 )
@ X1 ) ) )
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair
@ ( d_ReplSep @ eigen__1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_2t @ X4 ) )
@ proj1 )
@ X2 ) )
@ proj1 )
= ( d_ReplSep
@ ( eps
@ ^ [X1: $i] :
~ ( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X2: $i] :
( if
@ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ eigen__1
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ n_1t @ X3 ) )
@ proj1 )
@ X1 ) ) )
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair
@ ( d_ReplSep @ eigen__0
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_2t @ X4 ) )
@ proj1 )
@ X2 ) )
@ proj1 ) )
=> ( ( d_ReplSep
@ ( eps
@ ^ [X1: $i] :
~ ( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X2: $i] :
( if
@ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ eigen__1
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ n_1t @ X3 ) )
@ proj1 )
@ X1 ) ) )
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair
@ ( d_ReplSep @ eigen__0
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_2t @ X4 ) )
@ proj1 )
@ X2 ) )
@ proj1 )
= ( d_ReplSep
@ ( eps
@ ^ [X1: $i] :
~ ( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X2: $i] :
( if
@ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ eigen__0
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ n_1t @ X3 ) )
@ proj1 )
@ X1 ) ) )
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair
@ ( d_ReplSep @ eigen__1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_2t @ X4 ) )
@ proj1 )
@ X2 ) )
@ proj1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( ( d_ReplSep
@ ( eps
@ ^ [X1: $i] :
~ ( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X2: $i] :
( if
@ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ eigen__0
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ n_1t @ X3 ) )
@ proj1 )
@ X1 ) ) )
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair
@ ( d_ReplSep @ eigen__1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_2t @ X4 ) )
@ proj1 )
@ X2 ) )
@ proj1 )
= ( d_ReplSep
@ ( eps
@ ^ [X1: $i] :
~ ( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X2: $i] :
( if
@ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ eigen__1
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ n_1t @ X3 ) )
@ proj1 )
@ X1 ) ) )
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair
@ ( d_ReplSep @ eigen__0
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_2t @ X4 ) )
@ proj1 )
@ X2 ) )
@ proj1 ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( ( d_ReplSep
@ ( eps
@ ^ [X1: $i] :
~ ( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X2: $i] :
( if
@ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ eigen__1
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ n_1t @ X3 ) )
@ proj1 )
@ X1 ) ) )
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair
@ ( d_ReplSep @ eigen__0
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_2t @ X4 ) )
@ proj1 )
@ X2 ) )
@ proj1 )
= ( d_ReplSep
@ ( eps
@ ^ [X1: $i] :
~ ( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X2: $i] :
( if
@ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ eigen__0
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ n_1t @ X3 ) )
@ proj1 )
@ X1 ) ) )
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair
@ ( d_ReplSep @ eigen__1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_2t @ X4 ) )
@ proj1 )
@ X2 ) )
@ proj1 ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( X2 = X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ! [X1: $i] :
( ( ( d_ReplSep
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X3: $i] :
( if
@ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ eigen__0
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_1t @ X4 ) )
@ proj1 )
@ X2 ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair
@ ( d_ReplSep @ eigen__1
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_2t @ X5 ) )
@ proj1 )
@ X3 ) )
@ proj1 )
= X1 )
=> ( X1
= ( d_ReplSep
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X3: $i] :
( if
@ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ eigen__0
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_1t @ X4 ) )
@ proj1 )
@ X2 ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair
@ ( d_ReplSep @ eigen__1
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_2t @ X5 ) )
@ proj1 )
@ X3 ) )
@ proj1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(def_is_of,definition,
( is_of
= ( ^ [X1: $i,X2: $i > $o] : ( X2 @ X1 ) ) ) ).
thf(def_all_of,definition,
( all_of
= ( ^ [X1: $i > $o,X2: $i > $o] :
! [X3: $i] :
( ( is_of @ X3 @ X1 )
=> ( X2 @ X3 ) ) ) ) ).
thf(def_d_Sep,definition,
( d_Sep
= ( ^ [X1: $i,X2: $i > $o] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
@ ( repl @ X1
@ ^ [X3: $i] :
( if @ ( X2 @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ X1 )
=> ~ ( X2 @ X4 ) ) ) ) )
@ emptyset ) ) ) ).
thf(def_ordsucc,definition,
( ordsucc
= ( ^ [X1: $i] : ( binunion @ X1 @ ( d_Sing @ X1 ) ) ) ) ).
thf(def_omega,definition,
( omega
= ( d_Sep @ ( univof @ emptyset ) @ nat_p ) ) ).
thf(def_setprod,definition,
( setprod
= ( ^ [X1: $i,X2: $i] :
( d_Sigma @ X1
@ ^ [X3: $i] : X2 ) ) ) ).
thf(def_ap,definition,
( ap
= ( ^ [X1: $i,X2: $i] :
( d_ReplSep @ X1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 ) ) ) ).
thf(def_d_Pi,definition,
( d_Pi
= ( ^ [X1: $i,X2: $i > $i] :
( d_Sep
@ ( power
@ ( d_Sigma @ X1
@ ^ [X3: $i] : ( union @ ( X2 @ X3 ) ) ) )
@ ^ [X3: $i] :
! [X4: $i] :
( ( in @ X4 @ X1 )
=> ( in @ ( ap @ X3 @ X4 ) @ ( X2 @ X4 ) ) ) ) ) ) ).
thf(def_e_is,definition,
( e_is
= ( ^ [X1: $i] : (=) ) ) ).
thf(def_one,definition,
( one
= ( ^ [X1: $i,X2: $i > $o] : ( d_and @ ( amone @ X1 @ X2 ) @ ( l_some @ X1 @ X2 ) ) ) ) ).
thf(def_ind,definition,
( ind
= ( ^ [X1: $i,X2: $i > $o] :
( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) ) ) ) ) ).
thf(def_e_in,definition,
( e_in
= ( ^ [X1: $i,X2: $i > $o,X3: $i] : X3 ) ) ).
thf(def_d_pair,definition,
( d_pair
= ( ^ [X1: $i,X2: $i] : pair ) ) ).
thf(def_first,definition,
( first
= ( ^ [X1: $i,X2: $i] : proj0 ) ) ).
thf(def_second,definition,
( second
= ( ^ [X1: $i,X2: $i] : proj1 ) ) ).
thf(def_nat,definition,
( nat
= ( d_Sep @ omega
@ ^ [X1: $i] : ( X1 != emptyset ) ) ) ).
thf(def_n_is,definition,
( n_is
= ( e_is @ nat ) ) ).
thf(def_n_1,definition,
( n_1
= ( ordsucc @ emptyset ) ) ).
thf(def_n_pl,definition,
( n_pl
= ( ^ [X1: $i] : ( ap @ ( plus @ X1 ) ) ) ) ).
thf(def_d_29_ii,definition,
( d_29_ii
= ( ^ [X1: $i,X2: $i] : ( n_some @ ( diffprop @ X1 @ X2 ) ) ) ) ).
thf(def_iii,definition,
( iii
= ( ^ [X1: $i,X2: $i] : ( n_some @ ( diffprop @ X2 @ X1 ) ) ) ) ).
thf(def_moreis,definition,
( moreis
= ( ^ [X1: $i,X2: $i] : ( l_or @ ( d_29_ii @ X1 @ X2 ) @ ( n_is @ X1 @ X2 ) ) ) ) ).
thf(def_lessis,definition,
( lessis
= ( ^ [X1: $i,X2: $i] : ( l_or @ ( iii @ X1 @ X2 ) @ ( n_is @ X1 @ X2 ) ) ) ) ).
thf(def_times,definition,
( times
= ( ^ [X1: $i] :
( ind
@ ( d_Pi @ nat
@ ^ [X2: $i] : nat )
@ ( d_428_prop2 @ X1 ) ) ) ) ).
thf(def_n_ts,definition,
( n_ts
= ( ^ [X1: $i] : ( ap @ ( times @ X1 ) ) ) ) ).
thf(def_d_1to,definition,
( d_1to
= ( ^ [X1: $i] :
( d_Sep @ nat
@ ^ [X2: $i] : ( lessis @ X2 @ X1 ) ) ) ) ).
thf(def_n_2,definition,
( n_2
= ( n_pl @ n_1 @ n_1 ) ) ).
thf(def_pair1type,definition,
( pair1type
= ( ^ [X1: $i] :
( d_Pi @ ( d_1to @ n_2 )
@ ^ [X2: $i] : X1 ) ) ) ).
thf(def_first1,definition,
( first1
= ( ^ [X1: $i,X2: $i] : ( ap @ X2 @ n_1t ) ) ) ).
thf(def_second1,definition,
( second1
= ( ^ [X1: $i,X2: $i] : ( ap @ X2 @ n_2t ) ) ) ).
thf(def_frac,definition,
( frac
= ( pair1type @ nat ) ) ).
thf(def_num,definition,
( num
= ( first1 @ nat ) ) ).
thf(def_den,definition,
( den
= ( second1 @ nat ) ) ).
thf(def_n_eq,definition,
( n_eq
= ( ^ [X1: $i,X2: $i] : ( n_is @ ( n_ts @ ( num @ X1 ) @ ( den @ X2 ) ) @ ( n_ts @ ( num @ X2 ) @ ( den @ X1 ) ) ) ) ) ).
thf(satz38,conjecture,
! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X3 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X3 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X2 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X3 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X3 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X2: $i] :
( if
@ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) )
@ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) )
@ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X3 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X3: $i] :
( if
@ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) )
@ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) )
@ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) )
@ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset )
@ ^ [X8: $i] :
( if @ ( X8 != emptyset ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9
@ ( if
@ ~ ! [X10: $i] :
( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X10: $i] :
( if @ ( nat_p @ X10 ) @ X10
@ ( eps
@ ^ [X11: $i] :
~ ( ( in @ X11 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X11 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X9 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X8: $i] :
~ ! [X9: $i] :
( X8
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X9 ) )
@ proj1 )
@ X7 ) )
@ ( X7
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X8: $i] :
~ ! [X9: $i] :
( X8
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X9 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ( ( ( d_ReplSep
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X4: $i] :
( if
@ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( union
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X5
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ X6 @ X8 ) )
@ proj1 )
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ X1
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_1t @ X5 ) )
@ proj1 )
@ X3 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X2
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 )
= ( d_ReplSep
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X4: $i] :
( if
@ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( union
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X5
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ X6 @ X8 ) )
@ proj1 )
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_1t @ X5 ) )
@ proj1 )
@ X3 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 ) )
=> ( ( d_ReplSep
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X4: $i] :
( if
@ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( union
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X5
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ X6 @ X8 ) )
@ proj1 )
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_1t @ X5 ) )
@ proj1 )
@ X3 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 )
= ( d_ReplSep
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X4: $i] :
( if
@ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( union
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X5
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ X6 @ X8 ) )
@ proj1 )
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ X1
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_1t @ X5 ) )
@ proj1 )
@ X3 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X2
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 ) ) ) ) ) ).
thf(h0,negated_conjecture,
~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X2: $i] :
( if
@ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) )
@ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) )
@ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X3: $i] :
( if
@ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) )
@ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) )
@ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) )
@ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset )
@ ^ [X8: $i] :
( if @ ( X8 != emptyset ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9
@ ( if
@ ~ ! [X10: $i] :
( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X10: $i] :
( if @ ( nat_p @ X10 ) @ X10
@ ( eps
@ ^ [X11: $i] :
~ ( ( in @ X11 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X11 ) ) ) ) )
@ emptyset ) )
=> ( X9 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X8: $i] :
~ ! [X9: $i] :
( X8
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X9 ) )
@ proj1 )
@ X7 ) )
@ ( X7
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X8: $i] :
~ ! [X9: $i] :
( X8
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X9 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ( ( ( d_ReplSep
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X4: $i] :
( if
@ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( union
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X5
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ X6 @ X8 ) )
@ proj1 )
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ X1
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_1t @ X5 ) )
@ proj1 )
@ X3 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X2
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 )
= ( d_ReplSep
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X4: $i] :
( if
@ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( union
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X5
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ X6 @ X8 ) )
@ proj1 )
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_1t @ X5 ) )
@ proj1 )
@ X3 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 ) )
=> ( ( d_ReplSep
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X4: $i] :
( if
@ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( union
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X5
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ X6 @ X8 ) )
@ proj1 )
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_1t @ X5 ) )
@ proj1 )
@ X3 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 )
= ( d_ReplSep
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X4: $i] :
( if
@ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( union
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X5
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ X6 @ X8 ) )
@ proj1 )
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ X1
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_1t @ X5 ) )
@ proj1 )
@ X3 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X2
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 ) ) ) ) ),
inference(assume_negation,[status(cth)],[satz38]) ).
thf(h1,assumption,
~ ( ( in @ eigen__0
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 )
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X3 ) )
@ proj1 )
@ X1 ) )
@ ( X1
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X3 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset ) )
=> ( X1 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X1 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X1: $i] :
( if @ ( nat_p @ X1 ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if @ ( X1 != emptyset ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X3 ) )
@ proj1 )
@ X1 ) )
@ ( X1
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X3 ) )
@ proj1 ) ) )
@ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( union
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X1: $i] :
( if
@ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 )
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X2: $i] :
( if
@ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) )
@ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ( ( ( d_ReplSep
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X3: $i] :
( if
@ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ eigen__0
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_1t @ X4 ) )
@ proj1 )
@ X2 ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_2t @ X5 ) )
@ proj1 )
@ X3 ) )
@ proj1 )
= ( d_ReplSep
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X3: $i] :
( if
@ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ X1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_1t @ X4 ) )
@ proj1 )
@ X2 ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair
@ ( d_ReplSep @ eigen__0
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_2t @ X5 ) )
@ proj1 )
@ X3 ) )
@ proj1 ) )
=> ( ( d_ReplSep
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X3: $i] :
( if
@ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ X1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_1t @ X4 ) )
@ proj1 )
@ X2 ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair
@ ( d_ReplSep @ eigen__0
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_2t @ X5 ) )
@ proj1 )
@ X3 ) )
@ proj1 )
= ( d_ReplSep
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X3: $i] :
( if
@ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ eigen__0
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_1t @ X4 ) )
@ proj1 )
@ X2 ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_2t @ X5 ) )
@ proj1 )
@ X3 ) )
@ proj1 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h2,assumption,
( in @ eigen__0
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 )
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X3 ) )
@ proj1 )
@ X1 ) )
@ ( X1
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X3 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset ) )
=> ( X1 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X1 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X1: $i] :
( if @ ( nat_p @ X1 ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if @ ( X1 != emptyset ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X3 ) )
@ proj1 )
@ X1 ) )
@ ( X1
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X3 ) )
@ proj1 ) ) )
@ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( union
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X1: $i] :
( if
@ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 )
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) ),
introduced(assumption,[]) ).
thf(h3,assumption,
~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X2: $i] :
( if
@ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) )
@ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ( ( ( d_ReplSep
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X3: $i] :
( if
@ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ eigen__0
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_1t @ X4 ) )
@ proj1 )
@ X2 ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_2t @ X5 ) )
@ proj1 )
@ X3 ) )
@ proj1 )
= ( d_ReplSep
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X3: $i] :
( if
@ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ X1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_1t @ X4 ) )
@ proj1 )
@ X2 ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair
@ ( d_ReplSep @ eigen__0
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_2t @ X5 ) )
@ proj1 )
@ X3 ) )
@ proj1 ) )
=> ( ( d_ReplSep
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X3: $i] :
( if
@ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ X1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_1t @ X4 ) )
@ proj1 )
@ X2 ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair
@ ( d_ReplSep @ eigen__0
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_2t @ X5 ) )
@ proj1 )
@ X3 ) )
@ proj1 )
= ( d_ReplSep
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X3: $i] :
( if
@ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ eigen__0
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_1t @ X4 ) )
@ proj1 )
@ X2 ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_2t @ X5 ) )
@ proj1 )
@ X3 ) )
@ proj1 ) ) ) ),
introduced(assumption,[]) ).
thf(h4,assumption,
~ ( ( in @ eigen__1
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 )
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X3 ) )
@ proj1 )
@ X1 ) )
@ ( X1
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X3 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset ) )
=> ( X1 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X1 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X1: $i] :
( if @ ( nat_p @ X1 ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if @ ( X1 != emptyset ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X3 ) )
@ proj1 )
@ X1 ) )
@ ( X1
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X3 ) )
@ proj1 ) ) )
@ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( union
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X1: $i] :
( if
@ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 )
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> sP1 ),
introduced(assumption,[]) ).
thf(h5,assumption,
( in @ eigen__1
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 )
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X3 ) )
@ proj1 )
@ X1 ) )
@ ( X1
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X3 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset ) )
=> ( X1 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X1 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X1: $i] :
( if @ ( nat_p @ X1 ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if @ ( X1 != emptyset ) @ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X3 ) )
@ proj1 )
@ X1 ) )
@ ( X1
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X3 ) )
@ proj1 ) ) )
@ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X1: $i] :
( union
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X1: $i] :
( if
@ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 )
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) ) )
@ X1
@ ( eps
@ ^ [X2: $i] :
~ ( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) ),
introduced(assumption,[]) ).
thf(h6,assumption,
~ sP1,
introduced(assumption,[]) ).
thf(h7,assumption,
sP2,
introduced(assumption,[]) ).
thf(h8,assumption,
~ sP3,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP1
| ~ sP2
| sP3 ),
inference(prop_rule,[status(thm)],[]) ).
thf(2,plain,
( ~ sP5
| sP1 ),
inference(all_rule,[status(thm)],[]) ).
thf(3,plain,
( ~ sP4
| sP5 ),
inference(all_rule,[status(thm)],[]) ).
thf(4,plain,
sP4,
inference(eq_sym,[status(thm)],[]) ).
thf(5,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h7,h8,h5,h6,h4,h2,h3,h1,h0])],[1,2,3,4,h7,h8]) ).
thf(6,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h7,h8])],[h6,5,h7,h8]) ).
thf(7,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,6,h5,h6]) ).
thf(8,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h2,h3,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,7,h4]) ).
thf(9,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,8,h2,h3]) ).
thf(10,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,9,h1]) ).
thf(0,theorem,
! [X1: $i] :
( ( in @ X1
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X3 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X3 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X2 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X2: $i] :
( if @ ( nat_p @ X2 ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X3 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 )
@ X2 ) )
@ ( X2
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X4 ) )
@ proj1 ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X3 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X2: $i] :
( if
@ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset ) ) )
@ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) )
@ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ! [X2: $i] :
( ( in @ X2
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) )
@ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X3 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 )
@ X3 ) )
@ ( X3
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X5 ) )
@ proj1 ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 )
@ X4 ) )
@ ( X4
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X6 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X3: $i] :
( if
@ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) )
@ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) )
@ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 )
@ X5 ) )
@ ( X5
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X7 ) )
@ proj1 ) ) )
@ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if
@ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 )
@ X6 ) )
@ ( X6
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X8 ) )
@ proj1 ) ) )
@ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset )
@ ^ [X8: $i] :
( if @ ( X8 != emptyset ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9
@ ( if
@ ~ ! [X10: $i] :
( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X10: $i] :
( if @ ( nat_p @ X10 ) @ X10
@ ( eps
@ ^ [X11: $i] :
~ ( ( in @ X11 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X11 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X9 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ~ ( l_or
@ ( n_some
@ ( diffprop
@ ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X8: $i] :
~ ! [X9: $i] :
( X8
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X9 ) )
@ proj1 )
@ X7 ) )
@ ( X7
= ( d_ReplSep @ ( plus @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) )
@ ^ [X8: $i] :
~ ! [X9: $i] :
( X8
!= ( pair @ ( binunion @ emptyset @ ( d_Sing @ emptyset ) ) @ X9 ) )
@ proj1 ) ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ( ( ( d_ReplSep
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X4: $i] :
( if
@ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( union
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X5
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ X6 @ X8 ) )
@ proj1 )
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ X1
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_1t @ X5 ) )
@ proj1 )
@ X3 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X2
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 )
= ( d_ReplSep
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X4: $i] :
( if
@ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( union
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X5
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ X6 @ X8 ) )
@ proj1 )
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_1t @ X5 ) )
@ proj1 )
@ X3 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 ) )
=> ( ( d_ReplSep
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X4: $i] :
( if
@ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( union
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X5
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ X6 @ X8 ) )
@ proj1 )
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_1t @ X5 ) )
@ proj1 )
@ X3 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 )
= ( d_ReplSep
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( union
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X4: $i] :
( if
@ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset ) ) )
@ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( power
@ ( d_Sigma
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( union
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X5
@ ^ [X7: $i] :
~ ! [X8: $i] :
( X7
!= ( pair @ X6 @ X8 ) )
@ proj1 )
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7
@ ( if
@ ~ ! [X8: $i] :
( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X7 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
( if @ ( X7 != emptyset ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
@ ( if
@ ~ ! [X9: $i] :
( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X8 != emptyset ) ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ~ ( d_428_prop2
@ ( d_ReplSep @ X1
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_1t @ X5 ) )
@ proj1 )
@ X3 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X2
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 ) ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h0])],[10,h0]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : NUM726^4 : TPTP v9.2.0. Released v7.1.0.
% 0.12/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.13/0.34 % Computer : n001.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Wed Oct 8 23:00:53 EDT 2025
% 0.13/0.34 % CPUTime :
% 61.08/61.15 % SZS status Theorem
% 61.08/61.15 % Mode: mode8a:USE_SINE=true:SINE_TOLERANCE=1.2:SINE_GENERALITY_THRESHOLD=16:SINE_RANK_LIMIT=3.:SINE_DEPTH=0
% 61.08/61.15 % Inferences: 3
% 61.08/61.15 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------