%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : SEU524^1 : TPTP v9.2.0. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% Computer : n027.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 02:43:25 PM UTC 2025
% Result : Theorem 14.69s 14.90s
% Output : Proof 14.69s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 159
% Syntax : Number of formulae : 166 ( 52 unt; 0 typ; 46 def)
% Number of atoms : 811 ( 132 equ; 0 cnn)
% Maximal formula atoms : 101 ( 4 avg)
% Number of connectives : 1967 ( 288 ~; 60 |; 0 &;1068 @)
% ( 57 <=>; 494 =>; 0 <=; 0 <~>)
% Maximal formula depth : 49 ( 5 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 42 ( 42 >; 0 *; 0 +; 0 <<)
% Number of symbols : 114 ( 112 usr; 107 con; 0-2 aty)
% Number of variables : 512 ( 14 ^; 498 !; 0 ?; 512 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_setunion,type,
setunion: $i > $i ).
thf(ty_eigen__7,type,
eigen__7: $i ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_emptyset,type,
emptyset: $i ).
thf(ty_descr,type,
descr: ( $i > $o ) > $i ).
thf(ty_dsetconstr,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(ty_powerset,type,
powerset: $i > $i ).
thf(ty_omega,type,
omega: $i ).
thf(ty_in,type,
in: $i > $i > $o ).
thf(ty_setadjoin,type,
setadjoin: $i > $i > $i ).
thf(h0,assumption,
! [X1: $i > $o,X2: $i] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__1,definition,
( eigen__1
= ( eps__0
@ ^ [X1: $i] :
~ ( ( in @ X1 @ eigen__0 )
=> ( in @ X1 @ ( powerset @ ( setunion @ eigen__0 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__1])]) ).
thf(eigendef_eigen__0,definition,
( eigen__0
= ( eps__0
@ ^ [X1: $i] :
~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__0])]) ).
thf(eigendef_eigen__7,definition,
( eigen__7
= ( eps__0
@ ^ [X1: $i] :
~ ( ( in @ X1 @ eigen__1 )
=> ( in @ X1 @ ( setunion @ eigen__0 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__7])]) ).
thf(sP1,plain,
( sP1
<=> ! [X1: $i] :
( ( in @ X1 @ eigen__1 )
=> ( in @ X1 @ ( setunion @ eigen__0 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> ( ! [X1: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( ! [X1: $i] :
( ( X1 != emptyset )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( ! [X1: $o,X2: $i] :
( ( in @ X2
@ ( dsetconstr @ ( powerset @ emptyset )
@ ^ [X3: $i] : X1 ) )
=> X1 )
=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> sP3 ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( in @ eigen__1 @ eigen__0 ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) )
=> sP3 ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( sP6
=> ( in @ eigen__7 @ ( setunion @ eigen__0 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) )
=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( in @ eigen__7 @ ( setunion @ eigen__0 ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) )
=> sP2 ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ( in @ eigen__1 @ ( powerset @ ( setunion @ eigen__0 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> sP7 ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> sP5 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> sP7 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 != emptyset ) )
=> sP14 ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( sP1
=> sP12 ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> sP15 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> sP4 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) )
=> sP9 ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(sP22,plain,
( sP22
<=> ! [X1: $i] :
( ( in @ eigen__7 @ X1 )
=> ( ( in @ X1 @ eigen__0 )
=> sP10 ) ) ),
introduced(definition,[new_symbols(definition,[sP22])]) ).
thf(sP23,plain,
( sP23
<=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> sP4 ) ) ),
introduced(definition,[new_symbols(definition,[sP23])]) ).
thf(sP24,plain,
( sP24
<=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> sP7 ) ),
introduced(definition,[new_symbols(definition,[sP24])]) ).
thf(sP25,plain,
( sP25
<=> ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) )
=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> sP19 ) ) ),
introduced(definition,[new_symbols(definition,[sP25])]) ).
thf(sP26,plain,
( sP26
<=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP26])]) ).
thf(sP27,plain,
( sP27
<=> ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) )
=> sP24 ) ),
introduced(definition,[new_symbols(definition,[sP27])]) ).
thf(sP28,plain,
( sP28
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) )
=> sP20 ) ),
introduced(definition,[new_symbols(definition,[sP28])]) ).
thf(sP29,plain,
( sP29
<=> ! [X1: $i] :
( ( in @ X1 @ eigen__0 )
=> ( in @ X1 @ ( powerset @ ( setunion @ eigen__0 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP29])]) ).
thf(sP30,plain,
( sP30
<=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) )
=> sP4 ) ),
introduced(definition,[new_symbols(definition,[sP30])]) ).
thf(sP31,plain,
( sP31
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) )
=> sP5 ) ),
introduced(definition,[new_symbols(definition,[sP31])]) ).
thf(sP32,plain,
( sP32
<=> ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ! [X4: $i] :
( ( X1 @ X3 @ X4 )
=> ~ ! [X5: $i] :
( ( X1 @ X3 @ X5 )
=> ( X4 = X5 ) ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) )
=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> sP25 ) ) ),
introduced(definition,[new_symbols(definition,[sP32])]) ).
thf(sP33,plain,
( sP33
<=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ( X1 @ ( descr @ X1 ) ) )
=> sP19 ) ),
introduced(definition,[new_symbols(definition,[sP33])]) ).
thf(sP34,plain,
( sP34
<=> ( ! [X1: $i > $o] :
( ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) )
=> sP15 ) ),
introduced(definition,[new_symbols(definition,[sP34])]) ).
thf(sP35,plain,
( sP35
<=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> sP28 ) ) ),
introduced(definition,[new_symbols(definition,[sP35])]) ).
thf(sP36,plain,
( sP36
<=> ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) )
=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> sP32 ) ) ),
introduced(definition,[new_symbols(definition,[sP36])]) ).
thf(sP37,plain,
( sP37
<=> ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> ( ( in @ emptyset @ omega )
=> sP36 ) ) ),
introduced(definition,[new_symbols(definition,[sP37])]) ).
thf(sP38,plain,
( sP38
<=> ( ( in @ emptyset @ omega )
=> sP36 ) ),
introduced(definition,[new_symbols(definition,[sP38])]) ).
thf(sP39,plain,
( sP39
<=> ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 )
=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> sP35 ) ) ),
introduced(definition,[new_symbols(definition,[sP39])]) ).
thf(sP40,plain,
( sP40
<=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) )
=> sP35 ) ),
introduced(definition,[new_symbols(definition,[sP40])]) ).
thf(sP41,plain,
( sP41
<=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( ( dsetconstr @ X1 @ X2 )
!= emptyset ) ) )
=> sP16 ) ),
introduced(definition,[new_symbols(definition,[sP41])]) ).
thf(sP42,plain,
( sP42
<=> ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) )
=> sP23 ) ),
introduced(definition,[new_symbols(definition,[sP42])]) ).
thf(sP43,plain,
( sP43
<=> ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) )
=> sP28 ) ),
introduced(definition,[new_symbols(definition,[sP43])]) ).
thf(sP44,plain,
( sP44
<=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> sP11 ) ),
introduced(definition,[new_symbols(definition,[sP44])]) ).
thf(sP45,plain,
( sP45
<=> ( ( in @ eigen__7 @ eigen__1 )
=> sP8 ) ),
introduced(definition,[new_symbols(definition,[sP45])]) ).
thf(sP46,plain,
( sP46
<=> ( ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) )
=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> sP31 ) ) ),
introduced(definition,[new_symbols(definition,[sP46])]) ).
thf(sP47,plain,
( sP47
<=> ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) )
=> sP44 ) ),
introduced(definition,[new_symbols(definition,[sP47])]) ).
thf(sP48,plain,
( sP48
<=> ! [X1: $i] :
( ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( setunion @ eigen__0 ) ) )
=> ( in @ X1 @ ( powerset @ ( setunion @ eigen__0 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP48])]) ).
thf(sP49,plain,
( sP49
<=> ( sP6
=> sP12 ) ),
introduced(definition,[new_symbols(definition,[sP49])]) ).
thf(sP50,plain,
( sP50
<=> ! [X1: $i,X2: $i] :
( ( in @ X1 @ X2 )
=> ( ( in @ X2 @ eigen__0 )
=> ( in @ X1 @ ( setunion @ eigen__0 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP50])]) ).
thf(sP51,plain,
( sP51
<=> ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) )
=> sP31 ) ),
introduced(definition,[new_symbols(definition,[sP51])]) ).
thf(sP52,plain,
( sP52
<=> ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) )
=> sP32 ) ),
introduced(definition,[new_symbols(definition,[sP52])]) ).
thf(sP53,plain,
( sP53
<=> ( ( in @ eigen__7 @ eigen__1 )
=> sP10 ) ),
introduced(definition,[new_symbols(definition,[sP53])]) ).
thf(sP54,plain,
( sP54
<=> ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP54])]) ).
thf(sP55,plain,
( sP55
<=> ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) )
=> sP25 ) ),
introduced(definition,[new_symbols(definition,[sP55])]) ).
thf(sP56,plain,
( sP56
<=> ( ! [X1: $i] :
~ ( in @ X1 @ emptyset )
=> sP34 ) ),
introduced(definition,[new_symbols(definition,[sP56])]) ).
thf(sP57,plain,
( sP57
<=> ( in @ eigen__7 @ eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP57])]) ).
thf(def_exu,definition,
( exu
= ( ^ [X1: $i > $o] :
~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ).
thf(def_setextAx,definition,
( setextAx
= ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
= ( in @ X3 @ X2 ) )
=> ( X1 = X2 ) ) ) ) ).
thf(def_emptysetAx,definition,
( emptysetAx
= ( ! [X1: $i] :
~ ( in @ X1 @ emptyset ) ) ) ).
thf(def_setadjoinAx,definition,
( setadjoinAx
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_powersetAx,definition,
( powersetAx
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
= ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) ) ) ) ).
thf(def_setunionAx,definition,
( setunionAx
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
= ( ~ ! [X3: $i] :
( ( in @ X2 @ X3 )
=> ~ ( in @ X3 @ X1 ) ) ) ) ) ) ).
thf(def_omega0Ax,definition,
( omega0Ax
= ( in @ emptyset @ omega ) ) ).
thf(def_omegaSAx,definition,
( omegaSAx
= ( ! [X1: $i] :
( ( in @ X1 @ omega )
=> ( in @ ( setadjoin @ X1 @ X1 ) @ omega ) ) ) ) ).
thf(def_omegaIndAx,definition,
( omegaIndAx
= ( ! [X1: $i] :
( ~ ( ( in @ emptyset @ X1 )
=> ~ ! [X2: $i] :
( ~ ( ( in @ X2 @ omega )
=> ~ ( in @ X2 @ X1 ) )
=> ( in @ ( setadjoin @ X2 @ X2 ) @ X1 ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ omega )
=> ( in @ X2 @ X1 ) ) ) ) ) ).
thf(def_replAx,definition,
( replAx
= ( ! [X1: $i > $i > $o,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( exu @ ( X1 @ X3 ) ) )
=> ~ ! [X3: $i] :
~ ! [X4: $i] :
( ( in @ X4 @ X3 )
= ( ~ ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ~ ( X1 @ X5 @ X4 ) ) ) ) ) ) ) ).
thf(def_foundationAx,definition,
( foundationAx
= ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ~ ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ~ ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X1 ) ) ) ) ) ) ).
thf(def_wellorderingAx,definition,
( wellorderingAx
= ( ! [X1: $i] :
~ ! [X2: $i] :
( ~ ( ~ ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X1 )
=> ~ ( in @ X4 @ X1 ) )
=> ( ! [X5: $i] :
( ( in @ X5 @ X2 )
=> ( ( in @ X3 @ X5 )
= ( in @ X4 @ X5 ) ) )
=> ( X3 = X4 ) ) ) )
=> ~ ! [X3: $i,X4: $i] :
( ~ ( ( in @ X3 @ X2 )
=> ~ ( in @ X4 @ X2 ) )
=> ( ~ ! [X5: $i] :
( ( in @ X5 @ X3 )
=> ( in @ X5 @ X4 ) )
=> ! [X5: $i] :
( ( in @ X5 @ X4 )
=> ( in @ X5 @ X3 ) ) ) ) )
=> ~ ! [X3: $i] :
( ~ ( ! [X4: $i] :
( ( in @ X4 @ X3 )
=> ( in @ X4 @ X1 ) )
=> ! [X4: $i] :
~ ( in @ X4 @ X3 ) )
=> ~ ! [X4: $i,X5: $i] :
( ~ ( ~ ( ( in @ X4 @ X2 )
=> ~ ( in @ X5 @ X3 ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X4 )
=> ~ ( in @ X6 @ X3 ) ) )
=> ~ ! [X6: $i] :
( ( in @ X6 @ X2 )
=> ( ~ ! [X7: $i] :
( ( in @ X7 @ X6 )
=> ( in @ X7 @ X4 ) )
=> ( in @ X5 @ X6 ) ) ) ) ) ) ) ) ).
thf(def_descrp,definition,
( descrp
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ( X1 @ ( descr @ X1 ) ) ) ) ) ).
thf(def_dsetconstrI,definition,
( dsetconstrI
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_dsetconstrEL,definition,
( dsetconstrEL
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( in @ X3 @ X1 ) ) ) ) ).
thf(def_dsetconstrER,definition,
( dsetconstrER
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
=> ( X2 @ X3 ) ) ) ) ).
thf(def_exuE1,definition,
( exuE1
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
( ( X1 @ X2 )
=> ~ ! [X3: $i] :
( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) ) ) ).
thf(def_prop2set,definition,
( prop2set
= ( ^ [X1: $o] :
( dsetconstr @ ( powerset @ emptyset )
@ ^ [X2: $i] : X1 ) ) ) ).
thf(def_prop2setE,definition,
( prop2setE
= ( ! [X1: $o,X2: $i] :
( ( in @ X2 @ ( prop2set @ X1 ) )
=> X1 ) ) ) ).
thf(def_emptysetE,definition,
( emptysetE
= ( ! [X1: $i] :
( ( in @ X1 @ emptyset )
=> ! [X2: $o] : X2 ) ) ) ).
thf(def_emptysetimpfalse,definition,
( emptysetimpfalse
= ( ! [X1: $i] :
~ ( in @ X1 @ emptyset ) ) ) ).
thf(def_notinemptyset,definition,
( notinemptyset
= ( ! [X1: $i] :
~ ( in @ X1 @ emptyset ) ) ) ).
thf(def_exuE3e,definition,
( exuE3e
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
~ ( X1 @ X2 ) ) ) ) ).
thf(def_setext,definition,
( setext
= ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( in @ X3 @ X2 ) )
=> ( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( X1 = X2 ) ) ) ) ) ).
thf(def_emptyI,definition,
( emptyI
= ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) ) ) ) ).
thf(def_noeltsimpempty,definition,
( noeltsimpempty
= ( ! [X1: $i] :
( ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( X1 = emptyset ) ) ) ) ).
thf(def_setbeta,definition,
( setbeta
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( dsetconstr @ X1 @ X2 ) )
= ( X2 @ X3 ) ) ) ) ) ).
thf(def_nonempty,definition,
( nonempty
= ( ^ [X1: $i] : ( X1 != emptyset ) ) ) ).
thf(def_nonemptyE1,definition,
( nonemptyE1
= ( ! [X1: $i] :
( ( nonempty @ X1 )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) ) ) ) ).
thf(def_nonemptyI,definition,
( nonemptyI
= ( ! [X1: $i,X2: $i > $o,X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( X2 @ X3 )
=> ( nonempty @ ( dsetconstr @ X1 @ X2 ) ) ) ) ) ) ).
thf(def_nonemptyI1,definition,
( nonemptyI1
= ( ! [X1: $i] :
( ~ ! [X2: $i] :
~ ( in @ X2 @ X1 )
=> ( nonempty @ X1 ) ) ) ) ).
thf(def_setadjoinIL,definition,
( setadjoinIL
= ( ! [X1: $i,X2: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) ) ) ) ).
thf(def_emptyinunitempty,definition,
( emptyinunitempty
= ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
thf(def_setadjoinIR,definition,
( setadjoinIR
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ ( setadjoin @ X1 @ X2 ) ) ) ) ) ).
thf(def_setadjoinE,definition,
( setadjoinE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ! [X4: $o] :
( ( ( X3 = X1 )
=> X4 )
=> ( ( ( in @ X3 @ X2 )
=> X4 )
=> X4 ) ) ) ) ) ).
thf(def_setadjoinOr,definition,
( setadjoinOr
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
=> ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) ) ) ) ).
thf(def_setoftrueEq,definition,
( setoftrueEq
= ( ! [X1: $i] :
( ( dsetconstr @ X1
@ ^ [X2: $i] : ~ $false )
= X1 ) ) ) ).
thf(def_powersetI,definition,
powersetI = sP54 ).
thf(def_emptyinPowerset,definition,
( emptyinPowerset
= ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) ) ) ) ).
thf(def_emptyInPowerset,definition,
( emptyInPowerset
= ( ! [X1: $i] : ( in @ emptyset @ ( powerset @ X1 ) ) ) ) ).
thf(def_powersetE,definition,
( powersetE
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) ) ) ) ) ).
thf(def_setunionI,definition,
setunionI = sP21 ).
thf(def_setunionE,definition,
( setunionE
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setunion @ X1 ) )
=> ! [X3: $o] :
( ! [X4: $i] :
( ( in @ X2 @ X4 )
=> ( ( in @ X4 @ X1 )
=> X3 ) )
=> X3 ) ) ) ) ).
thf(subPowSU,conjecture,
sP47 ).
thf(h1,negated_conjecture,
~ sP47,
inference(assume_negation,[status(cth)],[subPowSU]) ).
thf(1,plain,
( ~ sP50
| sP22 ),
inference(all_rule,[status(thm)],[]) ).
thf(2,plain,
( ~ sP22
| sP45 ),
inference(all_rule,[status(thm)],[]) ).
thf(3,plain,
( ~ sP45
| ~ sP57
| sP8 ),
inference(prop_rule,[status(thm)],[]) ).
thf(4,plain,
( ~ sP8
| ~ sP6
| sP10 ),
inference(prop_rule,[status(thm)],[]) ).
thf(5,plain,
( sP53
| ~ sP10 ),
inference(prop_rule,[status(thm)],[]) ).
thf(6,plain,
( sP53
| sP57 ),
inference(prop_rule,[status(thm)],[]) ).
thf(7,plain,
( sP1
| ~ sP53 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__7]) ).
thf(8,plain,
( ~ sP21
| sP50 ),
inference(all_rule,[status(thm)],[]) ).
thf(9,plain,
( ~ sP54
| sP48 ),
inference(all_rule,[status(thm)],[]) ).
thf(10,plain,
( ~ sP48
| sP17 ),
inference(all_rule,[status(thm)],[]) ).
thf(11,plain,
( ~ sP17
| ~ sP1
| sP12 ),
inference(prop_rule,[status(thm)],[]) ).
thf(12,plain,
( sP49
| ~ sP12 ),
inference(prop_rule,[status(thm)],[]) ).
thf(13,plain,
( sP49
| sP6 ),
inference(prop_rule,[status(thm)],[]) ).
thf(14,plain,
( sP29
| ~ sP49 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1]) ).
thf(15,plain,
( sP26
| ~ sP29 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__0]) ).
thf(16,plain,
( sP9
| ~ sP26 ),
inference(prop_rule,[status(thm)],[]) ).
thf(17,plain,
( sP20
| ~ sP9 ),
inference(prop_rule,[status(thm)],[]) ).
thf(18,plain,
( sP20
| sP21 ),
inference(prop_rule,[status(thm)],[]) ).
thf(19,plain,
( sP28
| ~ sP20 ),
inference(prop_rule,[status(thm)],[]) ).
thf(20,plain,
( sP43
| ~ sP28 ),
inference(prop_rule,[status(thm)],[]) ).
thf(21,plain,
( sP35
| ~ sP43 ),
inference(prop_rule,[status(thm)],[]) ).
thf(22,plain,
( sP40
| ~ sP35 ),
inference(prop_rule,[status(thm)],[]) ).
thf(23,plain,
( sP40
| sP54 ),
inference(prop_rule,[status(thm)],[]) ).
thf(24,plain,
( sP39
| ~ sP40 ),
inference(prop_rule,[status(thm)],[]) ).
thf(25,plain,
( sP5
| ~ sP39 ),
inference(prop_rule,[status(thm)],[]) ).
thf(26,plain,
( sP31
| ~ sP5 ),
inference(prop_rule,[status(thm)],[]) ).
thf(27,plain,
( sP51
| ~ sP31 ),
inference(prop_rule,[status(thm)],[]) ).
thf(28,plain,
( sP46
| ~ sP51 ),
inference(prop_rule,[status(thm)],[]) ).
thf(29,plain,
( sP14
| ~ sP46 ),
inference(prop_rule,[status(thm)],[]) ).
thf(30,plain,
( sP16
| ~ sP14 ),
inference(prop_rule,[status(thm)],[]) ).
thf(31,plain,
( sP41
| ~ sP16 ),
inference(prop_rule,[status(thm)],[]) ).
thf(32,plain,
( sP3
| ~ sP41 ),
inference(prop_rule,[status(thm)],[]) ).
thf(33,plain,
( sP7
| ~ sP3 ),
inference(prop_rule,[status(thm)],[]) ).
thf(34,plain,
( sP24
| ~ sP7 ),
inference(prop_rule,[status(thm)],[]) ).
thf(35,plain,
( sP27
| ~ sP24 ),
inference(prop_rule,[status(thm)],[]) ).
thf(36,plain,
( sP15
| ~ sP27 ),
inference(prop_rule,[status(thm)],[]) ).
thf(37,plain,
( sP34
| ~ sP15 ),
inference(prop_rule,[status(thm)],[]) ).
thf(38,plain,
( sP56
| ~ sP34 ),
inference(prop_rule,[status(thm)],[]) ).
thf(39,plain,
( sP18
| ~ sP56 ),
inference(prop_rule,[status(thm)],[]) ).
thf(40,plain,
( sP13
| ~ sP18 ),
inference(prop_rule,[status(thm)],[]) ).
thf(41,plain,
( sP4
| ~ sP13 ),
inference(prop_rule,[status(thm)],[]) ).
thf(42,plain,
( sP30
| ~ sP4 ),
inference(prop_rule,[status(thm)],[]) ).
thf(43,plain,
( sP23
| ~ sP30 ),
inference(prop_rule,[status(thm)],[]) ).
thf(44,plain,
( sP42
| ~ sP23 ),
inference(prop_rule,[status(thm)],[]) ).
thf(45,plain,
( sP19
| ~ sP42 ),
inference(prop_rule,[status(thm)],[]) ).
thf(46,plain,
( sP33
| ~ sP19 ),
inference(prop_rule,[status(thm)],[]) ).
thf(47,plain,
( sP25
| ~ sP33 ),
inference(prop_rule,[status(thm)],[]) ).
thf(48,plain,
( sP55
| ~ sP25 ),
inference(prop_rule,[status(thm)],[]) ).
thf(49,plain,
( sP32
| ~ sP55 ),
inference(prop_rule,[status(thm)],[]) ).
thf(50,plain,
( sP52
| ~ sP32 ),
inference(prop_rule,[status(thm)],[]) ).
thf(51,plain,
( sP36
| ~ sP52 ),
inference(prop_rule,[status(thm)],[]) ).
thf(52,plain,
( sP38
| ~ sP36 ),
inference(prop_rule,[status(thm)],[]) ).
thf(53,plain,
( sP37
| ~ sP38 ),
inference(prop_rule,[status(thm)],[]) ).
thf(54,plain,
( sP2
| ~ sP37 ),
inference(prop_rule,[status(thm)],[]) ).
thf(55,plain,
( sP11
| ~ sP2 ),
inference(prop_rule,[status(thm)],[]) ).
thf(56,plain,
( sP44
| ~ sP11 ),
inference(prop_rule,[status(thm)],[]) ).
thf(57,plain,
( sP47
| ~ sP44 ),
inference(prop_rule,[status(thm)],[]) ).
thf(58,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,h1]) ).
thf(59,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[58,h0]) ).
thf(0,theorem,
sP47,
inference(contra,[status(thm),contra(discharge,[h1])],[58,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SEU524^1 : TPTP v9.2.0. Released v3.7.0.
% 0.07/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.13/0.34 % Computer : n027.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Wed Oct 8 21:34:53 EDT 2025
% 0.13/0.34 % CPUTime :
% 14.69/14.90 % SZS status Theorem
% 14.69/14.90 % Mode: mode454
% 14.69/14.90 % Inferences: 705
% 14.69/14.90 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------