%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : NUM671^4 : TPTP v9.2.0. Released v7.1.0.
% Transfm : none
% Format : tptp:raw
% Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Oct 9 02:11:44 PM UTC 2025
% Result : Theorem 145.56s 145.52s
% Output : Proof 145.56s
% Verified :
% SZS Type : Refutation
% Derivation depth : 10
% Number of leaves : 66
% Syntax : Number of formulae : 80 ( 31 unt; 0 typ; 14 def)
% Number of atoms : 1463 ( 261 equ; 1 cnn)
% Maximal formula atoms : 65 ( 18 avg)
% Number of connectives : 6199 (1061 ~; 26 |; 0 &;4569 @)
% ( 21 <=>; 522 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 6 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 5 ( 5 >; 0 *; 0 +; 0 <<)
% Number of symbols : 57 ( 54 usr; 44 con; 0-3 aty)
% Number of variables : 840 ( 546 ^; 294 !; 0 ?; 840 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_univof,type,
univof: $i > $i ).
thf(ty_eigen__6,type,
eigen__6: $i ).
thf(ty_if,type,
if: $o > $i > $i > $i ).
thf(ty_eigen__2,type,
eigen__2: $i ).
thf(ty_eps,type,
eps: ( $i > $o ) > $i ).
thf(ty_d_Pi,type,
d_Pi: $i > ( $i > $i ) > $i ).
thf(ty_d_ReplSep,type,
d_ReplSep: $i > ( $i > $o ) > ( $i > $i ) > $i ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_eigen__4,type,
eigen__4: $i ).
thf(ty_emptyset,type,
emptyset: $i ).
thf(ty_ind,type,
ind: $i > ( $i > $o ) > $i ).
thf(ty_eigen__3,type,
eigen__3: $i ).
thf(ty_pair,type,
pair: $i > $i > $i ).
thf(ty_proj1,type,
proj1: $i > $i ).
thf(ty_d_24_prop2,type,
d_24_prop2: $i > $i > $o ).
thf(ty_repl,type,
repl: $i > ( $i > $i ) > $i ).
thf(ty_in,type,
in: $i > $i > $o ).
thf(ty_nat_p,type,
nat_p: $i > $o ).
thf(h0,assumption,
! [X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__3,definition,
( eigen__3
= ( eps__0
@ ^ [X1: $i] :
( ( proj1 @ X1 )
!= ( proj1 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__3])]) ).
thf(eigendef_eigen__6,definition,
( eigen__6
= ( eps__0
@ ^ [X1: $i] :
( ( d_24_prop2 @ eigen__0 @ X1 )
!= ( d_24_prop2 @ eigen__1 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__6])]) ).
thf(eigendef_eigen__4,definition,
( eigen__4
= ( eps__0
@ ^ [X1: $i] :
( ( ~ ! [X2: $i] :
( X1
!= ( pair @ eigen__2 @ X2 ) ) )
!= ( ~ ! [X2: $i] :
( X1
!= ( pair @ eigen__2 @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__4])]) ).
thf(sP1,plain,
( sP1
<=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( 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
@ ~ ! [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 ) )
@ ( d_24_prop2 @ eigen__0 ) )
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair @ eigen__2 @ X2 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( 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
@ ~ ! [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 ) )
@ ( d_24_prop2 @ eigen__1 ) )
@ ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair @ eigen__2 @ X2 ) )
@ proj1 ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( X2 = X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( d_24_prop2 @ eigen__1 @ eigen__6 ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( ( ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair @ eigen__2 @ X2 ) ) )
= ( ^ [X1: $i] :
~ ! [X2: $i] :
( X1
!= ( pair @ eigen__2 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( ( eigen__0 = eigen__1 )
=> ( eigen__1 = eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( ( d_24_prop2 @ eigen__0 )
= ( d_24_prop2 @ eigen__1 ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ( proj1 = proj1 ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( ( d_Pi
@ ( 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
@ ~ ! [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 ) )
= ( d_Pi
@ ( 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
@ ~ ! [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 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ! [X1: $i] :
( ( proj1 @ X1 )
= ( proj1 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( eigen__1 = eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( d_24_prop2 @ eigen__0 @ eigen__6 ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ! [X1: $i] :
( ( ~ ! [X2: $i] :
( X1
!= ( pair @ eigen__2 @ X2 ) ) )
= ( ~ ! [X2: $i] :
( X1
!= ( pair @ eigen__2 @ X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ! [X1: $i] :
( ( eigen__0 = X1 )
=> ( X1 = eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( ( ind
@ ( d_Pi
@ ( 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
@ ~ ! [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 ) )
@ ( d_24_prop2 @ eigen__0 ) )
= ( ind
@ ( d_Pi
@ ( 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
@ ~ ! [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 ) )
@ ( d_24_prop2 @ eigen__1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ! [X1: $i] :
( ( d_24_prop2 @ eigen__0 @ X1 )
= ( d_24_prop2 @ eigen__1 @ X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( eigen__6 = eigen__6 ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( eigen__0 = eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ( ( proj1 @ eigen__3 )
= ( proj1 @ eigen__3 ) ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ( ( ~ ! [X1: $i] :
( eigen__4
!= ( pair @ eigen__2 @ X1 ) ) )
= ( ~ ! [X1: $i] :
( eigen__4
!= ( pair @ eigen__2 @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ( eigen__3 = eigen__3 ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ( sP11 = sP3 ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(def_is_of,definition,
( is_of
= ( ^ [X1: $i,X2: $i > $o] : ( X2 @ X1 ) ) ) ).
thf(def_all_of,definition,
( all_of
= ( ^ [X1: $i > $o,X2: $i > $o] :
! [X3: $i] :
( ( is_of @ X3 @ X1 )
=> ( X2 @ X3 ) ) ) ) ).
thf(def_d_Sep,definition,
( d_Sep
= ( ^ [X1: $i,X2: $i > $o] :
( if
@ ~ ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ~ ( X2 @ X3 ) )
@ ( repl @ X1
@ ^ [X3: $i] :
( if @ ( X2 @ X3 ) @ X3
@ ( eps
@ ^ [X4: $i] :
~ ( ( in @ X4 @ X1 )
=> ~ ( X2 @ X4 ) ) ) ) )
@ emptyset ) ) ) ).
thf(def_ordsucc,definition,
( ordsucc
= ( ^ [X1: $i] : ( binunion @ X1 @ ( d_Sing @ X1 ) ) ) ) ).
thf(def_omega,definition,
( omega
= ( d_Sep @ ( univof @ emptyset ) @ nat_p ) ) ).
thf(def_ap,definition,
( ap
= ( ^ [X1: $i,X2: $i] :
( d_ReplSep @ X1
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 ) ) ) ).
thf(def_e_is,definition,
( e_is
= ( ^ [X1: $i] : (=) ) ) ).
thf(def_nat,definition,
( nat
= ( d_Sep @ omega
@ ^ [X1: $i] : ( X1 != emptyset ) ) ) ).
thf(def_n_is,definition,
( n_is
= ( e_is @ nat ) ) ).
thf(def_plus,definition,
( plus
= ( ^ [X1: $i] :
( ind
@ ( d_Pi @ nat
@ ^ [X2: $i] : nat )
@ ( d_24_prop2 @ X1 ) ) ) ) ).
thf(def_n_pl,definition,
( n_pl
= ( ^ [X1: $i] : ( ap @ ( plus @ X1 ) ) ) ) ).
thf(satz19b,conjecture,
! [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 ) )
=> ! [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 ) )
=> ! [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 ) )
=> ( ( X1 = X2 )
=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( 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
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X1 ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( 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
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X2 ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 ) ) ) ) ) ) ).
thf(h1,negated_conjecture,
~ ! [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 ) )
=> ! [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 ) )
=> ! [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 ) )
=> ( ( X1 = X2 )
=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( 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
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X1 ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( 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
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ( X6 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X2 ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 ) ) ) ) ) ),
inference(assume_negation,[status(cth)],[satz19b]) ).
thf(h2,assumption,
~ ( ( in @ eigen__0
@ ( 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] :
( ( 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 ) )
=> ! [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 ) )
=> ( ( eigen__0 = X1 )
=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( 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
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ eigen__0 ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( 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
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X1 ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 ) ) ) ) ) ),
introduced(assumption,[]) ).
thf(h3,assumption,
( in @ eigen__0
@ ( 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 ) ),
introduced(assumption,[]) ).
thf(h4,assumption,
~ ! [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 ) )
=> ! [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 ) )
=> ( ( eigen__0 = X1 )
=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( 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
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ eigen__0 ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( 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
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ( X4 = emptyset ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ( X5 = emptyset ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X1 ) )
@ ^ [X3: $i] :
~ ! [X4: $i] :
( X3
!= ( pair @ X2 @ X4 ) )
@ proj1 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h5,assumption,
~ ( ( in @ eigen__1
@ ( 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] :
( ( 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 ) )
=> ( sP17
=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( 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
@ ~ ! [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 ) )
@ ( d_24_prop2 @ eigen__0 ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ X1 @ X3 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( 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
@ ~ ! [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 ) )
@ ( d_24_prop2 @ eigen__1 ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ X1 @ X3 ) )
@ proj1 ) ) ) ) ),
introduced(assumption,[]) ).
thf(h6,assumption,
( in @ eigen__1
@ ( 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 ) ),
introduced(assumption,[]) ).
thf(h7,assumption,
~ ! [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 ) )
=> ( sP17
=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( 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
@ ~ ! [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 ) )
@ ( d_24_prop2 @ eigen__0 ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ X1 @ X3 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( 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
@ ~ ! [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 ) )
@ ( d_24_prop2 @ eigen__1 ) )
@ ^ [X2: $i] :
~ ! [X3: $i] :
( X2
!= ( pair @ X1 @ X3 ) )
@ proj1 ) ) ) ),
introduced(assumption,[]) ).
thf(h8,assumption,
~ ( ( in @ eigen__2
@ ( 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 ) )
=> ( sP17
=> sP1 ) ),
introduced(assumption,[]) ).
thf(h9,assumption,
( in @ eigen__2
@ ( 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 ) ),
introduced(assumption,[]) ).
thf(h10,assumption,
~ ( sP17
=> sP1 ),
introduced(assumption,[]) ).
thf(h11,assumption,
sP17,
introduced(assumption,[]) ).
thf(h12,assumption,
~ sP1,
introduced(assumption,[]) ).
thf(1,plain,
sP8,
inference(prop_rule,[status(thm)],[]) ).
thf(2,plain,
( ~ sP11
| sP3
| ~ sP17
| ~ sP16 ),
inference(mating_rule,[status(thm)],[]) ).
thf(3,plain,
sP16,
inference(prop_rule,[status(thm)],[]) ).
thf(4,plain,
( ~ sP3
| sP11
| ~ sP10
| ~ sP16 ),
inference(mating_rule,[status(thm)],[]) ).
thf(5,plain,
( sP21
| ~ sP11
| ~ sP3 ),
inference(prop_rule,[status(thm)],[]) ).
thf(6,plain,
( sP21
| sP11
| sP3 ),
inference(prop_rule,[status(thm)],[]) ).
thf(7,plain,
( sP15
| ~ sP21 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__6]) ).
thf(8,plain,
( sP6
| ~ sP15 ),
inference(prop_rule,[status(thm)],[]) ).
thf(9,plain,
( sP14
| ~ sP8
| ~ sP6 ),
inference(prop_rule,[status(thm)],[]) ).
thf(10,plain,
sP19,
inference(prop_rule,[status(thm)],[]) ).
thf(11,plain,
( sP12
| ~ sP19 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__4]) ).
thf(12,plain,
( sP4
| ~ sP12 ),
inference(prop_rule,[status(thm)],[]) ).
thf(13,plain,
sP20,
inference(prop_rule,[status(thm)],[]) ).
thf(14,plain,
( sP18
| ~ sP20 ),
inference(prop_rule,[status(thm)],[]) ).
thf(15,plain,
( sP9
| ~ sP18 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__3]) ).
thf(16,plain,
( sP7
| ~ sP9 ),
inference(prop_rule,[status(thm)],[]) ).
thf(17,plain,
( sP1
| ~ sP14
| ~ sP4
| ~ sP7 ),
inference(prop_rule,[status(thm)],[]) ).
thf(18,plain,
( ~ sP5
| ~ sP17
| sP10 ),
inference(prop_rule,[status(thm)],[]) ).
thf(19,plain,
( ~ sP13
| sP5 ),
inference(all_rule,[status(thm)],[]) ).
thf(20,plain,
( ~ sP2
| sP13 ),
inference(all_rule,[status(thm)],[]) ).
thf(21,plain,
sP2,
inference(eq_sym,[status(thm)],[]) ).
thf(22,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h11,h12,h9,h10,h8,h6,h7,h5,h3,h4,h2,h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,h11,h12]) ).
thf(23,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h9,h10,h8,h6,h7,h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h11,h12])],[h10,22,h11,h12]) ).
thf(24,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h8,h6,h7,h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h9,h10])],[h8,23,h9,h10]) ).
thf(25,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h6,h7,h5,h3,h4,h2,h1,h0]),tab_negall(discharge,[h8]),tab_negall(eigenvar,eigen__2)],[h7,24,h8]) ).
thf(26,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h5,h3,h4,h2,h1,h0]),tab_negimp(discharge,[h6,h7])],[h5,25,h6,h7]) ).
thf(27,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h3,h4,h2,h1,h0]),tab_negall(discharge,[h5]),tab_negall(eigenvar,eigen__1)],[h4,26,h5]) ).
thf(28,plain,
$false,
inference(tab_negimp,[status(thm),assumptions([h2,h1,h0]),tab_negimp(discharge,[h3,h4])],[h2,27,h3,h4]) ).
thf(29,plain,
$false,
inference(tab_negall,[status(thm),assumptions([h1,h0]),tab_negall(discharge,[h2]),tab_negall(eigenvar,eigen__0)],[h1,28,h2]) ).
thf(30,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[29,h0]) ).
thf(0,theorem,
! [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 ) )
=> ! [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 ) )
=> ! [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 ) )
=> ( ( X1 = X2 )
=> ( ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( 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
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X1 ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 )
= ( d_ReplSep
@ ( ind
@ ( d_Pi
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X4 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X4: $i] :
( ( in @ X4 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X4 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X4: $i] :
( if @ ( nat_p @ X4 ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) ) ) ) )
@ emptyset )
@ ^ [X4: $i] :
( if @ ( X4 != emptyset ) @ X4
@ ( eps
@ ^ [X5: $i] :
~ ( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( 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
@ ~ ! [X5: $i] :
( ( in @ X5
@ ( if
@ ~ ! [X6: $i] :
( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X6: $i] :
( if @ ( nat_p @ X6 ) @ X6
@ ( eps
@ ^ [X7: $i] :
~ ( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X5 != emptyset ) ) )
@ ( repl
@ ( if
@ ~ ! [X5: $i] :
( ( in @ X5 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X5 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X5: $i] :
( if @ ( nat_p @ X5 ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X6 ) ) ) ) )
@ emptyset )
@ ^ [X5: $i] :
( if @ ( X5 != emptyset ) @ X5
@ ( eps
@ ^ [X6: $i] :
~ ( ( in @ X6
@ ( if
@ ~ ! [X7: $i] :
( ( in @ X7 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X7 ) )
@ ( repl @ ( univof @ emptyset )
@ ^ [X7: $i] :
( if @ ( nat_p @ X7 ) @ X7
@ ( eps
@ ^ [X8: $i] :
~ ( ( in @ X8 @ ( univof @ emptyset ) )
=> ~ ( nat_p @ X8 ) ) ) ) )
@ emptyset ) )
=> ~ ( ( X6 != emptyset ) ) ) ) ) )
@ emptyset ) )
@ ( d_24_prop2 @ X2 ) )
@ ^ [X4: $i] :
~ ! [X5: $i] :
( X4
!= ( pair @ X3 @ X5 ) )
@ proj1 ) ) ) ) ) ),
inference(contra,[status(thm),contra(discharge,[h1])],[29,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.13/0.13 % Problem : NUM671^4 : TPTP v9.2.0. Released v7.1.0.
% 0.13/0.14 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.14/0.35 % Computer : n016.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Wed Oct 8 22:06:53 EDT 2025
% 0.14/0.36 % CPUTime :
% 145.56/145.52 % SZS status Theorem
% 145.56/145.52 % Mode: mode4a:USE_SINE=true:SINE_TOLERANCE=1.0:SINE_GENERALITY_THRESHOLD=16:SINE_RANK_LIMIT=2.:SINE_DEPTH=0
% 145.56/145.52 % Inferences: 42
% 145.56/145.52 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------