%------------------------------------------------------------------------------
% File : Lash---1.13
% Problem : NUM726^4 : TPTP v9.2.0. Released v7.1.0.
% Transfm : none
% Format : tptp:raw
% Command : lash -P picomus -M modes -p tstp -t %d %s
% Computer : n031.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 02:09:43 PM UTC 2025
% Result : Theorem 20.28s 20.49s
% Output : Proof 20.32s
% Verified :
% SZS Type : ERROR: Analysing output (MakeTreeStats fails)
% Comments :
%------------------------------------------------------------------------------
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_d_Sigma,type,
d_Sigma: $i > ( $i > $i ) > $i ).
thf(ty_union,type,
union: $i > $i ).
thf(ty_n_2t,type,
n_2t: $i ).
thf(ty_d_428_prop2,type,
d_428_prop2: $i > $i > $o ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_d_ReplSep,type,
d_ReplSep: $i > ( $i > $o ) > ( $i > $i ) > $i ).
thf(ty_proj1,type,
proj1: $i > $i ).
thf(ty_nat_p,type,
nat_p: $i > $o ).
thf(ty_emptyset,type,
emptyset: $i ).
thf(ty_ind,type,
ind: $i > ( $i > $o ) > $i ).
thf(ty_pair,type,
pair: $i > $i > $i ).
thf(ty_repl,type,
repl: $i > ( $i > $i ) > $i ).
thf(ty_lessis,type,
lessis: $i > $i > $o ).
thf(ty_n_1,type,
n_1: $i ).
thf(ty_eps,type,
eps: ( $i > $o ) > $i ).
thf(ty_power,type,
power: $i > $i ).
thf(ty_n_1t,type,
n_1t: $i ).
thf(ty_n_pl,type,
n_pl: $i > $i > $i ).
thf(ty_if,type,
if: $o > $i > $i > $i ).
thf(ty_univof,type,
univof: $i > $i ).
thf(ty_in,type,
in: $i > $i > $o ).
thf(sP1,plain,
( sP1
<=> ( ( d_ReplSep
@ ( ind
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( 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] :
( ( 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 ) )
=> ( 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 @ ( 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] :
( 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 @ ( univof @ emptyset ) )
=> ~ ( 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 ) )
=> ( 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 @ ( univof @ emptyset ) )
=> ~ ( 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 ) ) ) ) ) ) )
@ emptyset )
@ ( d_428_prop2
@ ( d_ReplSep @ eigen__0
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair @ n_1t @ X2 ) )
@ proj1 ) ) )
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair
@ ( d_ReplSep @ eigen__1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_2t @ X4 ) )
@ proj1 )
@ X2 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( 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] :
( ( 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 ) )
=> ( 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 @ ( 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] :
( 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 @ ( univof @ emptyset ) )
=> ~ ( 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 ) )
=> ( 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 @ ( univof @ emptyset ) )
=> ~ ( 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 ) ) ) ) ) ) )
@ emptyset )
@ ( d_428_prop2
@ ( d_ReplSep @ eigen__1
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair @ n_1t @ X2 ) )
@ proj1 ) ) )
@ ^ [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,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( ( d_ReplSep
@ ( ind
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( 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
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( ( in @ X3
@ ( if
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( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( 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 @ ( univof @ emptyset ) )
=> ~ ( 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 ) )
=> ( 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 @ ( 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] :
( 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 @ ( univof @ emptyset ) )
=> ~ ( 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 ) )
=> ( 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 @ ( univof @ emptyset ) )
=> ~ ( 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 ) ) ) ) ) ) )
@ emptyset )
@ ( d_428_prop2
@ ( d_ReplSep @ eigen__1
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair @ n_1t @ X2 ) )
@ proj1 ) ) )
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair
@ ( d_ReplSep @ eigen__0
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_2t @ X4 ) )
@ proj1 )
@ X2 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( if
@ ~ ! [X1: $i] :
( ( in @ X1
@ ( 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] :
( ( 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 ) )
=> ( 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 @ ( 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] :
( 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 @ ( univof @ emptyset ) )
=> ~ ( 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 ) )
=> ( 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 @ ( univof @ emptyset ) )
=> ~ ( 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 ) ) ) ) ) ) )
@ emptyset )
@ ( d_428_prop2
@ ( d_ReplSep @ eigen__0
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair @ n_1t @ X2 ) )
@ proj1 ) ) )
@ ^ [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,[sP2])]) ).
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] :
( ^ [X4: $o,X5: $o] :
( X4
=> X5 )
@ ( 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_omega,definition,
( omega
= ( d_Sep @ ( univof @ emptyset ) @ nat_p ) ) ).
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] :
( ^ [X5: $o,X6: $o] :
( X5
=> X6 )
@ ( in @ X4 @ X1 )
@ ( in @ ( ap @ X3 @ X4 ) @ ( X2 @ X4 ) ) ) ) ) ) ).
thf(def_e_is,definition,
( e_is
= ( ^ [X1: $i,X2: $i,X3: $i] : ( X2 = X3 ) ) ) ).
thf(def_nat,definition,
( nat
= ( d_Sep @ omega
@ ^ [X1: $i] : ( (~) @ ( X1 = emptyset ) ) ) ) ).
thf(def_n_is,definition,
( n_is
= ( e_is @ nat ) ) ).
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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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
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@ ~ ! [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
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~ ( ( in @ X4
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) @ X3
@ ( eps
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~ ( ( in @ X4
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( ( in @ X5
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=> ~ ( nat_p @ 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
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
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@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
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@ ( eps
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~ ( ( in @ X6
@ ( if
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( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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 ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
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( ( 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
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( ( in @ X4
@ ( if
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( ( in @ X5
@ ( if
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( 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 ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
@ ( if
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( ( in @ X4
@ ( if
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( 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
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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 @ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
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( ( in @ X6
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=> ~ ( nat_p @ 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 ) )
=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
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@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( 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
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( 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 ) )
=> ~ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
@ ( if
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( ( in @ X2
@ ( if
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( ( 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
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=> ~ ( 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 @ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
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( ( in @ X4
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=> ~ ( 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] :
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@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
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( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
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=> ~ ( nat_p @ X6 ) )
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@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
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( ( 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
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( ( 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] :
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=> ~ ( nat_p @ X5 ) )
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@ ^ [X5: $i] :
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@ ^ [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
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( ( in @ X5
@ ( if
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
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@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
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( if @ ( nat_p @ X5 ) @ X5
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@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
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@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
@ ( if
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( ( in @ X4
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( ( in @ X5 @ ( univof @ emptyset ) )
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@ ^ [X5: $i] :
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@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
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( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
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@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
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@ ~ ! [X6: $i] :
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=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
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=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
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=> ~ ( nat_p @ X7 ) )
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
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~ ( ( in @ X9 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
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=> ( in
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=> ~ ( nat_p @ X5 ) )
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
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~ ( ( in @ X5 @ ( univof @ emptyset ) )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
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@ X2
@ ( eps
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~ ( ( in @ X3
@ ( power
@ ( d_Sigma
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=> ~ ( nat_p @ X6 ) )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
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=> ( X5 = emptyset ) )
@ ( repl
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=> ~ ( nat_p @ X5 ) )
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=> ( X6 = emptyset ) ) ) ) )
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=> ~ ( nat_p @ X6 ) ) ) ) )
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=> ( X6 = emptyset ) ) ) ) )
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=> ~ ! [X4: $i] :
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@ ( if
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( ( 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 ) )
=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( union
@ ( if
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( ( in @ X5
@ ( if
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( 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
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( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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
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( ( in @ X6
@ ( if
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=> ~ ( nat_p @ 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
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( 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
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( ( 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 ) )
=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
@ ( if
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( ( in @ X5
@ ( if
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
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( if @ ( X5 != emptyset ) @ X5
@ ( eps
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~ ( ( in @ X6
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( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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 @ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
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( ( in @ X7
@ ( if
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=> ~ ( 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
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( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( 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
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=> ~ ( nat_p @ X9 ) )
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@ ^ [X9: $i] :
( if @ ( nat_p @ X9 ) @ X9
@ ( eps
@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( lessis @ X6 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
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( ( in @ X5
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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
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( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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
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( ( in @ X4
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( 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
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
@ ( if
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( ( in @ X3
@ ( if
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( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( 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
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( ( 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
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( 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 @ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
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( ( in @ X5
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=> ~ ( nat_p @ 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
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
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@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
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( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
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=> ~ ( nat_p @ 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 ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
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( ( in @ X4
@ ( if
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( 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
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( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( 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
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~ ( ( in @ X5
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@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ^ [X3: $i] :
( if
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( ( in @ X4
@ ( if
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( ( in @ X5
@ ( if
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( ( in @ X6
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=> ~ ( nat_p @ 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
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
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@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
( if @ ( X6 != emptyset ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7
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=> ~ ( 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 ) )
=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
@ ( if
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( ( in @ X5
@ ( if
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@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
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=> ( X5 = emptyset ) )
@ ( repl
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
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@ ^ [X6: $i] :
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=> ~ ( nat_p @ X6 ) ) ) ) )
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@ ( eps
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@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) @ X5
@ ( eps
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@ ^ [X8: $i] :
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@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
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@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8
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@ ^ [X9: $i] :
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@ ^ [X10: $i] :
~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( lessis @ X6 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X3
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ X4 @ X6 ) )
@ proj1 )
@ ( if
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( ( in @ X5
@ ( if
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
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@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
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@ ^ [X6: $i] :
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@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
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=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
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@ X3
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~ ( ( in @ X4
@ ( power
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=> ~ ( nat_p @ X7 ) )
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
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=> ~ ( nat_p @ X6 ) )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
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@ ( eps
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~ ( ( in @ X7
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~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
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=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
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@ ( eps
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~ ( ( in @ X6
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=> ~ ( nat_p @ X7 ) )
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( if @ ( nat_p @ X7 ) @ X7
@ ( eps
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
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=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
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~ ( ( in @ X9 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
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=> ~ ( nat_p @ X7 ) )
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@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
@ ^ [X7: $i] :
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@ ( eps
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~ ( ( in @ X8
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=> ~ ( nat_p @ X9 ) )
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@ ^ [X9: $i] :
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~ ( ( in @ X10 @ ( univof @ emptyset ) )
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=> ( X8 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X6 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
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@ ^ [X5: $i] :
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( ( in @ X6
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=> ~ ( nat_p @ 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
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( 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
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=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
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@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
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=> ~ ! [X5: $i] :
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=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
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~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
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=> ( X7 = emptyset ) )
@ ( repl
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( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset )
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=> ~ ( nat_p @ X9 ) )
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@ ^ [X9: $i] :
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~ ( ( in @ X10 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X10 ) ) ) ) )
@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X6 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
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( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( 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
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~ ( ( in @ X7
@ ( if
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( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
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~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
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@ ( repl @ ( univof @ emptyset )
@ ^ [X9: $i] :
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~ ( ( in @ X10 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X8 = emptyset ) )
@ ( repl
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@ ( repl @ ( univof @ emptyset )
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@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
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@ ^ [X10: $i] :
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@ ^ [X11: $i] :
~ ( ( in @ X11 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X11 ) ) ) ) )
@ emptyset ) )
=> ( X9 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( lessis @ X7 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X4
@ ^ [X6: $i] :
~ ! [X7: $i] :
( X6
!= ( pair @ X5 @ X7 ) )
@ proj1 )
@ ( if
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( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
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@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
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@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
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@ ^ [X6: $i] :
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~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ( ( ( d_ReplSep
@ ( ind
@ ( if
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=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
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=> ( X4 = emptyset ) )
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=> ( X5 = emptyset ) )
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
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=> ( X6 = emptyset ) ) ) ) )
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=> ( in
@ ( d_ReplSep @ X3
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=> ( X5 = emptyset ) )
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=> ( X6 = emptyset ) ) ) ) )
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@ ( repl
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@ ^ [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 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X2
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( 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 @ X2
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_1t @ X4 ) )
@ proj1 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 ) )
=> ( ( d_ReplSep
@ ( ind
@ ( 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 @ X2
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_1t @ X4 ) )
@ proj1 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( 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
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( ( in @ X5
@ ( if
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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
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( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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
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( ( 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
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( ( 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 ) ) )
@ ^ [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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
=> ~ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
@ ( if
@ ~ ! [X2: $i] :
( ( in @ X2
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=> ~ ( nat_p @ X3 ) )
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@ emptyset ) )
=> ( X2 = emptyset ) )
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@ ^ [X2: $i] :
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=> ~ ( nat_p @ X5 ) )
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=> ( in
@ ( d_ReplSep @ X2
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!= ( pair @ X3 @ X5 ) )
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@ X2
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@ ( d_Sigma
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=> ( X5 = emptyset ) )
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=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
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~ ( ( in @ X9 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X7 = emptyset ) )
@ ( repl
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=> ( X8 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X6 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
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=> ( in
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!= ( pair @ X4 @ X6 ) )
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=> ( X5 = emptyset ) )
@ ( repl
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=> ! [X2: $i] :
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=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
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( if @ ( nat_p @ X6 ) @ X6
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
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=> ~ ( nat_p @ X5 ) )
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@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
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=> ~ ( nat_p @ X5 ) )
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=> ( X4 = emptyset ) )
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=> ~ ! [X4: $i] :
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=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) )
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=> ( in
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@ ( repl
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@ ^ [X3: $i] :
( union
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=> ( ( ( d_ReplSep
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( ( in @ X3
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( ( in @ X4
@ ( if
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( 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
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( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( 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
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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
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( ( in @ X5
@ ( if
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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
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( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X2
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( 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 @ X2
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_1t @ X4 ) )
@ proj1 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 ) )
=> ( ( d_ReplSep
@ ( ind
@ ( 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 @ X2
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_1t @ X4 ) )
@ proj1 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( 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 ) ) )
@ ^ [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 ) )
=> ~ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
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@ ( repl
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=> ~ ( nat_p @ X3 ) )
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~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
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~ ( ( in @ X4
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
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@ ( eps
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
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=> ( X5 = emptyset ) )
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
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@ ^ [X3: $i] :
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
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=> ( X4 = emptyset ) )
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~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
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~ ( ( in @ X5
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=> ~ ( nat_p @ X6 ) )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
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=> ( X5 = emptyset ) ) ) ) )
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=> ~ ! [X3: $i] :
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=> ~ ( nat_p @ X6 ) )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
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=> ~ ( nat_p @ X5 ) )
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
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@ ( eps
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
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~ ( ( in @ X5 @ ( univof @ emptyset ) )
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~ ( ( in @ X5
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
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@ ^ [X4: $i] :
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
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=> ( X7 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
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=> ( in
@ ( d_ReplSep @ X2
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~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
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=> ~ ( nat_p @ X5 ) )
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
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=> ~ ( nat_p @ X4 ) )
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@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
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~ ( ( in @ X5
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=> ~ ( nat_p @ X6 ) )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
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=> ! [X1: $i] :
( ( in @ X1
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@ ( power
@ ( d_Sigma
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=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
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@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
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( ( in @ X4 @ ( univof @ emptyset ) )
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~ ( ( in @ X5 @ ( univof @ emptyset ) )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
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=> ( X5 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
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=> ( X3 = emptyset ) )
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@ ~ ! [X6: $i] :
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=> ~ ( 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
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=> ~ ( nat_p @ X8 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X8: $i] :
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@ ( eps
@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
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@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
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( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
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~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
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~ ( ( in @ X5
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) ) ) )
@ ( repl
@ ( power
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=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
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~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) )
@ ( repl
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( ( 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
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~ ( ( in @ X4
@ ( if
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
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( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
@ ( eps
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~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset ) )
=> ( X2 = emptyset ) )
@ ( repl
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( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
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~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
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@ ( eps
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~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
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=> ~ ( nat_p @ X5 ) )
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@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
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=> ( X4 = emptyset ) )
@ ( repl
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( ( in @ X4 @ ( univof @ emptyset ) )
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~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
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~ ( ( in @ X5
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=> ( X5 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
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@ ^ [X2: $i] :
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( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
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~ ( ( in @ X5 @ ( univof @ emptyset ) )
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=> ( X3 = emptyset ) )
@ ( repl
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~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
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=> ~ ( nat_p @ X6 ) ) ) ) )
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=> ( X4 = emptyset ) ) ) ) )
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@ ^ [X2: $i] :
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=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
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( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
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~ ( ( in @ X6
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=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
@ ( if
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( ( in @ X4
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( if @ ( nat_p @ X5 ) @ X5
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
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=> ( X4 = emptyset ) )
@ ( repl
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( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
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~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
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@ ( eps
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
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@ emptyset ) )
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@ emptyset )
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@ ( eps
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
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=> ( X6 = emptyset ) )
@ ( repl
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( ( in @ X6 @ ( univof @ emptyset ) )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
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@ ^ [X6: $i] :
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@ ( eps
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~ ( ( in @ X7
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@ ^ [X9: $i] :
~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
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=> ( X5 = emptyset ) ) ) ) )
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@ X2
@ ( eps
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~ ( ( in @ X3
@ ( power
@ ( d_Sigma
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@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
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=> ( X5 = emptyset ) )
@ ( repl
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
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=> ( X4 = emptyset ) )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
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@ emptyset ) )
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@ ^ [X4: $i] :
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@ ( eps
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@ ^ [X7: $i] :
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@ ^ [X8: $i] :
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=> ~ ( nat_p @ X8 ) ) ) ) )
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=> ( X6 = emptyset ) )
@ ( repl
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=> ( X7 = emptyset ) ) ) ) )
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@ ^ [X4: $i] :
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@ ( repl
@ ( if
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
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( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
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@ ( eps
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~ ( ( in @ X6
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
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=> ~ ! [X4: $i] :
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
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=> ( X6 = emptyset ) )
@ ( repl
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=> ~ ( nat_p @ X6 ) )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
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@ emptyset )
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~ ( ( in @ X9 @ ( univof @ emptyset ) )
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=> ( X7 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
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=> ~ ( nat_p @ X5 ) )
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
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=> ( X6 = emptyset ) ) ) ) )
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~ ( ( in @ X9 @ ( univof @ emptyset ) )
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=> ( X7 = emptyset ) )
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
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=> ( X8 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X6 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
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=> ( in
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!= ( pair @ X4 @ X6 ) )
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
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=> ~ ( nat_p @ X5 ) )
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
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=> ( ( ( d_ReplSep
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~ ( ( in @ X5 @ ( univof @ emptyset ) )
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
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~ ( ( in @ X5 @ ( univof @ emptyset ) )
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@ ( eps
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~ ( ( in @ X5
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
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=> ~ ! [X3: $i] :
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
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=> ( X4 = emptyset ) )
@ ( repl
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=> ( in
@ ( d_ReplSep @ X2
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@ ( repl
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( 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 ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_2t @ X5 ) )
@ proj1 )
@ X3 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( 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 @ X1
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ n_1t @ X3 ) )
@ proj1 ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair
@ ( d_ReplSep @ eigen__0
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_2t @ X5 ) )
@ proj1 )
@ X3 ) )
@ proj1 ) )
=> ( ( d_ReplSep
@ ( ind
@ ( 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 @ X1
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ n_1t @ X3 ) )
@ proj1 ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair
@ ( d_ReplSep @ eigen__0
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_2t @ X5 ) )
@ proj1 )
@ X3 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( 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 ) ) )
@ ^ [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 ) )
=> ~ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
=> ~ ( lessis @ X1 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X1 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
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( ( in @ X5
@ ( if
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=> ~ ( nat_p @ 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
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( 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
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~ ( ( in @ X6
@ ( if
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=> ~ ( nat_p @ X7 ) )
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@ ^ [X7: $i] :
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@ ( eps
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
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( ( in @ X4
@ ( if
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( 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
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( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( 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
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( ( in @ X4
@ ( if
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( ( in @ X5
@ ( if
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( 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 ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
@ ( if
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( ( in @ X4
@ ( if
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
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@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
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( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
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@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
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=> ~ ( nat_p @ X6 ) )
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@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
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( ( in @ X6
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=> ~ ( nat_p @ 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
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( ( 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 ) )
=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
@ ( if
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( ( in @ X4
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@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( 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 ) )
=> ~ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
@ ( if
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( ( in @ X2
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( ( 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
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( ( in @ X2 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X2 ) )
@ ( repl @ ( univof @ emptyset )
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( if @ ( nat_p @ X2 ) @ X2
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@ ^ [X3: $i] :
~ ( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( X2 != emptyset ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
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@ ^ [X4: $i] :
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@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset ) )
=> ( X3 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( if @ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) @ X2
@ ( eps
@ ^ [X3: $i] :
~ ( ( in @ X3
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( ( in @ X4
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=> ~ ( 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
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( ( in @ X4 @ ( univof @ emptyset ) )
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@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
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@ ( eps
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
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@ ^ [X2: $i] :
( union
@ ( if
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( ( in @ X3
@ ( if
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( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( 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
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( ( in @ X3 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X3 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X3: $i] :
( if @ ( nat_p @ X3 ) @ X3
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@ ^ [X4: $i] :
~ ( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( if @ ( X3 != emptyset ) @ X3
@ ( eps
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=> ~ ( nat_p @ X5 ) )
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@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) ) ) ) )
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@ ^ [X2: $i] :
( if
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( ( in @ X3
@ ( if
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( ( in @ X4
@ ( if
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=> ~ ( nat_p @ X6 ) )
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@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
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@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
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@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
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@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
@ ( if
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( ( in @ X4
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@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
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=> ( X4 = emptyset ) )
@ ( repl
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( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
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@ ^ [X5: $i] :
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@ ( eps
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@ ^ [X6: $i] :
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@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
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@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
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@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) )
@ ( repl
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
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@ ^ [X6: $i] :
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@ ( 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 )
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~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ emptyset ) )
=> ( in
@ ( d_ReplSep @ X2
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!= ( pair @ X3 @ X5 ) )
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
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@ ( eps
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
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=> ~ ( nat_p @ X4 ) )
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@ emptyset )
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@ ( eps
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~ ( ( in @ X5
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
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@ X2
@ ( eps
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~ ( ( in @ X3
@ ( power
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=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
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=> ( X5 = emptyset ) )
@ ( repl
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=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
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=> ~ ( nat_p @ X6 ) ) ) ) )
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
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=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
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~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
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~ ( ( in @ X5
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
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@ ^ [X7: $i] :
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
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=> ( X6 = emptyset ) )
@ ( repl
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
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@ ( eps
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~ ( ( in @ X7
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=> ~ ( nat_p @ X8 ) )
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@ ^ [X8: $i] :
( if @ ( nat_p @ X8 ) @ X8
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~ ( ( in @ X9 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X9 ) ) ) ) )
@ emptyset ) )
=> ( X7 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
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@ ^ [X4: $i] :
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=> ~ ( nat_p @ X6 ) )
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@ ^ [X6: $i] :
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
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=> ~ ( nat_p @ X5 ) )
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@ ^ [X5: $i] :
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
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=> ~ ( nat_p @ X7 ) )
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
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=> ( X6 = emptyset ) ) ) ) )
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=> ~ ! [X4: $i] :
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=> ( X6 = emptyset ) )
@ ( repl
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=> ~ ( nat_p @ X6 ) )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset )
@ ^ [X6: $i] :
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@ ( eps
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=> ~ ( nat_p @ X8 ) )
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~ ( ( in @ X9 @ ( univof @ emptyset ) )
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=> ( X7 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
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=> ~ ( nat_p @ X6 ) )
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( if @ ( nat_p @ X6 ) @ X6
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
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@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
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( ( in @ X5 @ ( univof @ emptyset ) )
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@ ( repl @ ( univof @ emptyset )
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( if @ ( nat_p @ X5 ) @ X5
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@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
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@ ( eps
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@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
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@ ^ [X5: $i] :
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=> ( X7 = emptyset ) )
@ ( repl
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~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
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@ ^ [X10: $i] :
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@ emptyset ) )
=> ( X8 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X6 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
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=> ( in
@ ( d_ReplSep @ X3
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~ ! [X6: $i] :
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@ proj1 )
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=> ~ ( nat_p @ X6 ) )
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
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=> ( X5 = emptyset ) )
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
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=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) ) ) ) ) ) )
@ emptyset ) )
=> ( ( ( d_ReplSep
@ ( ind
@ ( if
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@ ( power
@ ( d_Sigma
@ ( if
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=> ( X5 = emptyset ) ) ) ) )
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=> ( in
@ ( d_ReplSep @ X2
@ ^ [X4: $i] :
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!= ( pair @ X3 @ X5 ) )
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@ ( if
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=> ( X4 = emptyset ) )
@ ( repl
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@ ^ [X5: $i] :
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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 ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_2t @ X5 ) )
@ proj1 )
@ X3 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( 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 @ X1
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ n_1t @ X3 ) )
@ proj1 ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair
@ ( d_ReplSep @ eigen__0
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_2t @ X5 ) )
@ proj1 )
@ X3 ) )
@ proj1 ) )
=> ( ( d_ReplSep
@ ( ind
@ ( 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 @ X1
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ n_1t @ X3 ) )
@ proj1 ) ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair
@ ( d_ReplSep @ eigen__0
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ n_2t @ X5 ) )
@ proj1 )
@ X3 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( 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 ) ) )
@ ^ [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 ) )
=> ~ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
=> ~ ( lessis @ X1 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X1 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
@ ( if
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( ( 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 @ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4
@ ( if
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( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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
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( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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 ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ emptyset )
@ ^ [X3: $i] :
( union
@ ( if
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( ( in @ X4
@ ( if
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( 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
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( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( 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
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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
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( ( in @ X4
@ ( if
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( ( in @ X5
@ ( if
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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
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( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( 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
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( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
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@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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
=> sP2 ) ),
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 ) )
=> ~ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
=> ~ ( lessis @ X1 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X1 @ ( n_pl @ n_1 @ n_1 ) ) @ X1
@ ( eps
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~ ( ( in @ X5 @ ( univof @ emptyset ) )
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=> ( X3 = emptyset ) )
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=> ~ ( nat_p @ X6 ) ) ) ) )
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=> ( X4 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
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=> ( in
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!= ( pair @ X2 @ X4 ) )
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@ emptyset )
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~ ( ( in @ X4
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=> ~ ( nat_p @ X6 ) ) ) ) )
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@ X1
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@ ( d_Sigma
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~ ( ( in @ X6 @ ( univof @ emptyset ) )
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=> ( X4 = emptyset ) )
@ ( repl
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~ ( ( in @ X7 @ ( univof @ emptyset ) )
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=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( repl
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=> ( 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
=> sP2 ),
introduced(assumption,[]) ).
thf(h7,assumption,
sP1,
introduced(assumption,[]) ).
thf(h8,assumption,
~ sP2,
introduced(assumption,[]) ).
thf(1,plain,
( ~ sP1
| sP2 ),
inference(symeq,[status(thm)],[]) ).
thf(2,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h7,h8,h5,h6,h4,h2,h3,h1,h0])],[1,h7,h8]) ).
thf(3,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h5,h6,h4,h2,h3,h1,h0]),tab_negimp(discharge,[h7,h8])],[h6,2,h7,h8]) ).
thf(4,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h4,h2,h3,h1,h0]),tab_negimp(discharge,[h5,h6])],[h4,3,h5,h6]) ).
thf(5,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h2,h3,h1,h0]),tab_negall(discharge,[h4]),tab_negall(eigenvar,eigen__1)],[h3,4,h4]) ).
thf(6,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h1,h0]),tab_negimp(discharge,[h2,h3])],[h1,5,h2,h3]) ).
thf(7,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h0]),tab_negall(discharge,[h1]),tab_negall(eigenvar,eigen__0)],[h0,6,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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ 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 ) )
=> ~ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) )
@ ( 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 @ ( lessis @ X2 @ ( n_pl @ n_1 @ n_1 ) ) @ 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 ) )
=> ~ ( lessis @ X3 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
@ emptyset )
@ ^ [X2: $i] :
( union
@ ( if
@ ~ ! [X3: $i] :
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=> ( X5 = emptyset ) )
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=> ~ ( lessis @ X5 @ ( n_pl @ n_1 @ n_1 ) ) ) ) ) )
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=> ( in
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=> ~ ( nat_p @ X5 ) )
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=> ( X5 = emptyset ) )
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=> ( X6 = emptyset ) ) ) ) )
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=> ~ ( lessis @ X4 @ ( n_pl @ n_1 @ n_1 ) ) )
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=> ( X8 = emptyset ) ) ) ) )
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=> ( in
@ ( d_ReplSep @ X3
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!= ( pair @ X4 @ X6 ) )
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@ ( repl
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=> ( in
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@ X3
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=> ( X6 = emptyset ) )
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=> ~ ! [X5: $i] :
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=> ~ ( nat_p @ X9 ) ) ) ) )
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=> ( X7 = emptyset ) ) ) ) )
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=> ( ( ( d_ReplSep
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( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ 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 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X2
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( 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 @ X2
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_1t @ X4 ) )
@ proj1 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 ) )
=> ( ( d_ReplSep
@ ( ind
@ ( 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 @ X2
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ n_1t @ X4 ) )
@ proj1 ) ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair
@ ( d_ReplSep @ X1
@ ^ [X5: $i] :
~ ! [X6: $i] :
( X5
!= ( pair @ n_2t @ X6 ) )
@ proj1 )
@ X4 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( 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 ) ) )
@ ^ [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])],[7,h0]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : NUM726^4 : TPTP v9.2.0. Released v7.1.0.
% 0.03/0.12 % Command : lash -P picomus -M modes -p tstp -t %d %s
% 0.12/0.33 % Computer : n031.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Wed Oct 8 23:00:53 EDT 2025
% 0.12/0.33 % CPUTime :
% 20.28/20.49 % SZS status Theorem
% 20.28/20.49 % Mode: cade22sinegrackle2xfaf3
% 20.28/20.49 % Steps: 15
% 20.28/20.49 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------