%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : SEU529^1 : 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 : n014.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 02:43:27 PM UTC 2025
% Result : Theorem 24.13s 24.33s
% Output : Proof 24.13s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 174
% Syntax : Number of formulae : 180 ( 58 unt; 0 typ; 50 def)
% Number of atoms : 683 ( 95 equ; 0 cnn)
% Maximal formula atoms : 31 ( 3 avg)
% Number of connectives : 850 ( 133 ~; 63 |; 0 &; 343 @)
% ( 62 <=>; 249 =>; 0 <=; 0 <~>)
% Maximal formula depth : 36 ( 3 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 15 ( 15 >; 0 *; 0 +; 0 <<)
% Number of symbols : 123 ( 121 usr; 120 con; 0-2 aty)
% Number of variables : 156 ( 7 ^; 149 !; 0 ?; 156 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_nonemptyImpWitness,type,
nonemptyImpWitness: $o ).
thf(ty_setadjoinE,type,
setadjoinE: $o ).
thf(ty_foundationAx,type,
foundationAx: $o ).
thf(ty_wellorderingAx,type,
wellorderingAx: $o ).
thf(ty_setunionE,type,
setunionE: $o ).
thf(ty_setextAx,type,
setextAx: $o ).
thf(ty_setext,type,
setext: $o ).
thf(ty_emptysetE,type,
emptysetE: $o ).
thf(ty_setunionAx,type,
setunionAx: $o ).
thf(ty_exuE3e,type,
exuE3e: $o ).
thf(ty_powersetI,type,
powersetI: $o ).
thf(ty_emptyInPowerset,type,
emptyInPowerset: $o ).
thf(ty_eigen__1,type,
eigen__1: $i ).
thf(ty_eigen__0,type,
eigen__0: $i ).
thf(ty_emptysetimpfalse,type,
emptysetimpfalse: $o ).
thf(ty_exuE1,type,
exuE1: $o ).
thf(ty_powersetAx,type,
powersetAx: $o ).
thf(ty_dsetconstrI,type,
dsetconstrI: $o ).
thf(ty_setbeta,type,
setbeta: $o ).
thf(ty_emptyset,type,
emptyset: $i ).
thf(ty_replAx,type,
replAx: $o ).
thf(ty_prop2setE,type,
prop2setE: $o ).
thf(ty_setadjoinIL,type,
setadjoinIL: $o ).
thf(ty_setadjoinAx,type,
setadjoinAx: $o ).
thf(ty_setoftrueEq,type,
setoftrueEq: $o ).
thf(ty_notinsingleton,type,
notinsingleton: $o ).
thf(ty_exuE2,type,
exuE2: $o ).
thf(ty_omegaSAx,type,
omegaSAx: $o ).
thf(ty_omegaIndAx,type,
omegaIndAx: $o ).
thf(ty_noeltsimpempty,type,
noeltsimpempty: $o ).
thf(ty_powersetE,type,
powersetE: $o ).
thf(ty_emptyI,type,
emptyI: $o ).
thf(ty_subPowSU,type,
subPowSU: $o ).
thf(ty_nonemptyE1,type,
nonemptyE1: $o ).
thf(ty_emptyinunitempty,type,
emptyinunitempty: $o ).
thf(ty_emptyinPowerset,type,
emptyinPowerset: $o ).
thf(ty_setunionI,type,
setunionI: $o ).
thf(ty_dsetconstrER,type,
dsetconstrER: $o ).
thf(ty_nonemptyI1,type,
nonemptyI1: $o ).
thf(ty_emptysetAx,type,
emptysetAx: $o ).
thf(ty_descrp,type,
descrp: $o ).
thf(ty_setadjoinIR,type,
setadjoinIR: $o ).
thf(ty_nonemptyI,type,
nonemptyI: $o ).
thf(ty_in,type,
in: $i > $i > $o ).
thf(ty_setadjoinOr,type,
setadjoinOr: $o ).
thf(ty_uniqinunit,type,
uniqinunit: $o ).
thf(ty_dsetconstrEL,type,
dsetconstrEL: $o ).
thf(ty_notinemptyset,type,
notinemptyset: $o ).
thf(ty_omega0Ax,type,
omega0Ax: $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] :
~ ( ( eigen__0 = X1 )
=> ( in @ eigen__0 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__1])]) ).
thf(eigendef_eigen__0,definition,
( eigen__0
= ( eps__0
@ ^ [X1: $i] :
~ ! [X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__0])]) ).
thf(sP1,plain,
( sP1
<=> ! [X1: $i,X2: $i] :
( ( in @ X2 @ ( setadjoin @ eigen__1 @ X1 ) )
= ( ( X2 != eigen__1 )
=> ( in @ X2 @ X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X3 @ ( setadjoin @ X1 @ X2 ) )
= ( ( X3 != X1 )
=> ( in @ X3 @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> sP4 ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( ( ( in @ eigen__0 @ ( setadjoin @ eigen__1 @ emptyset ) )
= ( ( eigen__0 != eigen__1 )
=> ( in @ eigen__0 @ emptyset ) ) )
=> ~ ( ( eigen__0 != eigen__1 )
=> ( in @ eigen__0 @ emptyset ) ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ! [X1: $o] :
( ( ( in @ eigen__0 @ ( setadjoin @ eigen__1 @ emptyset ) )
= X1 )
=> ~ X1 ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ( emptyI
=> sP5 ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> sP4 ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( ( eigen__0 != eigen__1 )
=> ( in @ eigen__0 @ emptyset ) ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ( setbeta
=> sP10 ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( setext
=> sP8 ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( setadjoinAx = sP3 ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> sP2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> sP15 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> sP16 ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ( ~ ( in @ eigen__0 @ ( setadjoin @ eigen__1 @ emptyset ) )
=> sP7 ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ( descrp
=> sP15 ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(sP22,plain,
( sP22
<=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> sP2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP22])]) ).
thf(sP23,plain,
( sP23
<=> ( powersetE
=> ( setunionI
=> sP20 ) ) ),
introduced(definition,[new_symbols(definition,[sP23])]) ).
thf(sP24,plain,
( sP24
<=> ( setadjoinOr
=> sP4 ) ),
introduced(definition,[new_symbols(definition,[sP24])]) ).
thf(sP25,plain,
( sP25
<=> ( notinemptyset
=> ( exuE3e
=> sP13 ) ) ),
introduced(definition,[new_symbols(definition,[sP25])]) ).
thf(sP26,plain,
( sP26
<=> ( omegaSAx
=> ( omegaIndAx
=> sP16 ) ) ),
introduced(definition,[new_symbols(definition,[sP26])]) ).
thf(sP27,plain,
( sP27
<=> ( notinsingleton
=> sP21 ) ),
introduced(definition,[new_symbols(definition,[sP27])]) ).
thf(sP28,plain,
( sP28
<=> ( eigen__0 = eigen__1 ) ),
introduced(definition,[new_symbols(definition,[sP28])]) ).
thf(sP29,plain,
( sP29
<=> ( omega0Ax
=> sP26 ) ),
introduced(definition,[new_symbols(definition,[sP29])]) ).
thf(sP30,plain,
( sP30
<=> ( powersetAx
=> ( setunionAx
=> sP29 ) ) ),
introduced(definition,[new_symbols(definition,[sP30])]) ).
thf(sP31,plain,
( sP31
<=> ( powersetI
=> sP9 ) ),
introduced(definition,[new_symbols(definition,[sP31])]) ).
thf(sP32,plain,
( sP32
<=> ( foundationAx
=> ( wellorderingAx
=> sP19 ) ) ),
introduced(definition,[new_symbols(definition,[sP32])]) ).
thf(sP33,plain,
( sP33
<=> ( in @ eigen__0 @ ( setadjoin @ eigen__1 @ emptyset ) ) ),
introduced(definition,[new_symbols(definition,[sP33])]) ).
thf(sP34,plain,
( sP34
<=> ( setunionI
=> sP20 ) ),
introduced(definition,[new_symbols(definition,[sP34])]) ).
thf(sP35,plain,
( sP35
<=> ( emptysetAx
=> sP17 ) ),
introduced(definition,[new_symbols(definition,[sP35])]) ).
thf(sP36,plain,
( sP36
<=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> sP24 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP36])]) ).
thf(sP37,plain,
( sP37
<=> ( omegaIndAx
=> sP16 ) ),
introduced(definition,[new_symbols(definition,[sP37])]) ).
thf(sP38,plain,
( sP38
<=> ( uniqinunit
=> sP27 ) ),
introduced(definition,[new_symbols(definition,[sP38])]) ).
thf(sP39,plain,
( sP39
<=> ( setadjoinIL
=> sP36 ) ),
introduced(definition,[new_symbols(definition,[sP39])]) ).
thf(sP40,plain,
( sP40
<=> ( emptysetimpfalse
=> sP25 ) ),
introduced(definition,[new_symbols(definition,[sP40])]) ).
thf(sP41,plain,
( sP41
<=> ! [X1: $i] :
( ( in @ X1 @ ( setadjoin @ eigen__1 @ emptyset ) )
= ( ( X1 != eigen__1 )
=> ( in @ X1 @ emptyset ) ) ) ),
introduced(definition,[new_symbols(definition,[sP41])]) ).
thf(sP42,plain,
( sP42
<=> ( setunionAx
=> sP29 ) ),
introduced(definition,[new_symbols(definition,[sP42])]) ).
thf(sP43,plain,
( sP43
<=> ( exuE2
=> ( nonemptyImpWitness
=> sP38 ) ) ),
introduced(definition,[new_symbols(definition,[sP43])]) ).
thf(sP44,plain,
( sP44
<=> ( sP33 = sP11 ) ),
introduced(definition,[new_symbols(definition,[sP44])]) ).
thf(sP45,plain,
( sP45
<=> ( setadjoinIR
=> ( setadjoinE
=> sP24 ) ) ),
introduced(definition,[new_symbols(definition,[sP45])]) ).
thf(sP46,plain,
( sP46
<=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> sP2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP46])]) ).
thf(sP47,plain,
( sP47
<=> ( setextAx
=> sP35 ) ),
introduced(definition,[new_symbols(definition,[sP47])]) ).
thf(sP48,plain,
( sP48
<=> ( prop2setE
=> sP2 ) ),
introduced(definition,[new_symbols(definition,[sP48])]) ).
thf(sP49,plain,
( sP49
<=> ! [X1: $o,X2: $o > $o] :
( ( X2 @ X1 )
=> ! [X3: $o] :
( ( X1 = X3 )
=> ( X2 @ X3 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP49])]) ).
thf(sP50,plain,
( sP50
<=> ! [X1: $i] :
( ( eigen__0 = X1 )
=> ( in @ eigen__0 @ ( setadjoin @ X1 @ emptyset ) ) ) ),
introduced(definition,[new_symbols(definition,[sP50])]) ).
thf(sP51,plain,
( sP51
<=> setadjoinAx ),
introduced(definition,[new_symbols(definition,[sP51])]) ).
thf(sP52,plain,
( sP52
<=> ! [X1: $o > $o] :
( ( X1 @ sP33 )
=> ! [X2: $o] :
( ( sP33 = X2 )
=> ( X1 @ X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP52])]) ).
thf(sP53,plain,
( sP53
<=> ( wellorderingAx
=> sP19 ) ),
introduced(definition,[new_symbols(definition,[sP53])]) ).
thf(sP54,plain,
( sP54
<=> ( nonemptyI
=> ( nonemptyI1
=> sP39 ) ) ),
introduced(definition,[new_symbols(definition,[sP54])]) ).
thf(sP55,plain,
( sP55
<=> ( sP28
=> sP33 ) ),
introduced(definition,[new_symbols(definition,[sP55])]) ).
thf(sP56,plain,
( sP56
<=> ( emptyInPowerset
=> sP23 ) ),
introduced(definition,[new_symbols(definition,[sP56])]) ).
thf(sP57,plain,
( sP57
<=> ( exuE1
=> sP48 ) ),
introduced(definition,[new_symbols(definition,[sP57])]) ).
thf(sP58,plain,
( sP58
<=> ( exuE3e
=> sP13 ) ),
introduced(definition,[new_symbols(definition,[sP58])]) ).
thf(sP59,plain,
( sP59
<=> ( nonemptyImpWitness
=> sP38 ) ),
introduced(definition,[new_symbols(definition,[sP59])]) ).
thf(sP60,plain,
( sP60
<=> ( setadjoinE
=> sP24 ) ),
introduced(definition,[new_symbols(definition,[sP60])]) ).
thf(sP61,plain,
( sP61
<=> ( nonemptyI1
=> sP39 ) ),
introduced(definition,[new_symbols(definition,[sP61])]) ).
thf(sP62,plain,
( sP62
<=> ( subPowSU
=> sP43 ) ),
introduced(definition,[new_symbols(definition,[sP62])]) ).
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 = sP3 ).
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
= ( ! [X1: $i,X2: $i] :
( ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ X3 @ X1 ) )
=> ( in @ X2 @ ( powerset @ X1 ) ) ) ) ) ).
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
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ X3 )
=> ( ( in @ X3 @ X1 )
=> ( in @ X2 @ ( setunion @ X1 ) ) ) ) ) ) ).
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(def_subPowSU,definition,
( subPowSU
= ( ! [X1: $i,X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ ( powerset @ ( setunion @ X1 ) ) ) ) ) ) ).
thf(def_exuE2,definition,
( exuE2
= ( ! [X1: $i > $o] :
( ( exu @ X1 )
=> ~ ! [X2: $i] :
~ ! [X3: $i] :
( ( X1 @ X3 )
= ( X3 = X2 ) ) ) ) ) ).
thf(def_nonemptyImpWitness,definition,
( nonemptyImpWitness
= ( ! [X1: $i] :
( ( nonempty @ X1 )
=> ~ ! [X2: $i] :
~ ( in @ X2 @ X1 ) ) ) ) ).
thf(def_uniqinunit,definition,
( uniqinunit
= ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) ) ) ) ).
thf(def_notinsingleton,definition,
( notinsingleton
= ( ! [X1: $i,X2: $i] :
( ( X1 != X2 )
=> ~ ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ).
thf(eqinunit,conjecture,
sP47 ).
thf(h1,negated_conjecture,
~ sP47,
inference(assume_negation,[status(cth)],[eqinunit]) ).
thf(1,plain,
( sP11
| ~ sP28 ),
inference(prop_rule,[status(thm)],[]) ).
thf(2,plain,
( ~ sP3
| sP1 ),
inference(all_rule,[status(thm)],[]) ).
thf(3,plain,
( ~ sP1
| sP41 ),
inference(all_rule,[status(thm)],[]) ).
thf(4,plain,
( ~ sP41
| sP44 ),
inference(all_rule,[status(thm)],[]) ).
thf(5,plain,
( ~ sP6
| ~ sP44
| ~ sP11 ),
inference(prop_rule,[status(thm)],[]) ).
thf(6,plain,
( ~ sP7
| sP6 ),
inference(all_rule,[status(thm)],[]) ).
thf(7,plain,
( ~ sP18
| sP33
| sP7 ),
inference(prop_rule,[status(thm)],[]) ).
thf(8,plain,
( ~ sP52
| sP18 ),
inference(all_rule,[status(thm)],[]) ).
thf(9,plain,
( ~ sP49
| sP52 ),
inference(all_rule,[status(thm)],[]) ).
thf(10,plain,
sP49,
inference(eq_ind,[status(thm)],[]) ).
thf(11,plain,
( ~ sP14
| ~ sP51
| sP3 ),
inference(prop_rule,[status(thm)],[]) ).
thf(12,plain,
( sP55
| ~ sP33 ),
inference(prop_rule,[status(thm)],[]) ).
thf(13,plain,
( sP55
| sP28 ),
inference(prop_rule,[status(thm)],[]) ).
thf(14,plain,
( sP50
| ~ sP55 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1]) ).
thf(15,plain,
( sP21
| ~ sP50 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__0]) ).
thf(16,plain,
( sP27
| ~ sP21 ),
inference(prop_rule,[status(thm)],[]) ).
thf(17,plain,
( sP38
| ~ sP27 ),
inference(prop_rule,[status(thm)],[]) ).
thf(18,plain,
( sP59
| ~ sP38 ),
inference(prop_rule,[status(thm)],[]) ).
thf(19,plain,
( sP43
| ~ sP59 ),
inference(prop_rule,[status(thm)],[]) ).
thf(20,plain,
( sP62
| ~ sP43 ),
inference(prop_rule,[status(thm)],[]) ).
thf(21,plain,
( sP20
| ~ sP62 ),
inference(prop_rule,[status(thm)],[]) ).
thf(22,plain,
( sP34
| ~ sP20 ),
inference(prop_rule,[status(thm)],[]) ).
thf(23,plain,
( sP23
| ~ sP34 ),
inference(prop_rule,[status(thm)],[]) ).
thf(24,plain,
( sP56
| ~ sP23 ),
inference(prop_rule,[status(thm)],[]) ).
thf(25,plain,
( sP9
| ~ sP56 ),
inference(prop_rule,[status(thm)],[]) ).
thf(26,plain,
( sP31
| ~ sP9 ),
inference(prop_rule,[status(thm)],[]) ).
thf(27,plain,
( sP4
| ~ sP31 ),
inference(prop_rule,[status(thm)],[]) ).
thf(28,plain,
( sP24
| ~ sP4 ),
inference(prop_rule,[status(thm)],[]) ).
thf(29,plain,
( sP60
| ~ sP24 ),
inference(prop_rule,[status(thm)],[]) ).
thf(30,plain,
( sP45
| ~ sP60 ),
inference(prop_rule,[status(thm)],[]) ).
thf(31,plain,
( sP36
| ~ sP45 ),
inference(prop_rule,[status(thm)],[]) ).
thf(32,plain,
( sP39
| ~ sP36 ),
inference(prop_rule,[status(thm)],[]) ).
thf(33,plain,
( sP61
| ~ sP39 ),
inference(prop_rule,[status(thm)],[]) ).
thf(34,plain,
( sP54
| ~ sP61 ),
inference(prop_rule,[status(thm)],[]) ).
thf(35,plain,
( sP10
| ~ sP54 ),
inference(prop_rule,[status(thm)],[]) ).
thf(36,plain,
( sP12
| ~ sP10 ),
inference(prop_rule,[status(thm)],[]) ).
thf(37,plain,
( sP5
| ~ sP12 ),
inference(prop_rule,[status(thm)],[]) ).
thf(38,plain,
( sP8
| ~ sP5 ),
inference(prop_rule,[status(thm)],[]) ).
thf(39,plain,
( sP13
| ~ sP8 ),
inference(prop_rule,[status(thm)],[]) ).
thf(40,plain,
( sP58
| ~ sP13 ),
inference(prop_rule,[status(thm)],[]) ).
thf(41,plain,
( sP25
| ~ sP58 ),
inference(prop_rule,[status(thm)],[]) ).
thf(42,plain,
( sP40
| ~ sP25 ),
inference(prop_rule,[status(thm)],[]) ).
thf(43,plain,
( sP2
| ~ sP40 ),
inference(prop_rule,[status(thm)],[]) ).
thf(44,plain,
( sP48
| ~ sP2 ),
inference(prop_rule,[status(thm)],[]) ).
thf(45,plain,
( sP57
| ~ sP48 ),
inference(prop_rule,[status(thm)],[]) ).
thf(46,plain,
( sP46
| ~ sP57 ),
inference(prop_rule,[status(thm)],[]) ).
thf(47,plain,
( sP22
| ~ sP46 ),
inference(prop_rule,[status(thm)],[]) ).
thf(48,plain,
( sP15
| ~ sP22 ),
inference(prop_rule,[status(thm)],[]) ).
thf(49,plain,
( sP19
| ~ sP15 ),
inference(prop_rule,[status(thm)],[]) ).
thf(50,plain,
( sP53
| ~ sP19 ),
inference(prop_rule,[status(thm)],[]) ).
thf(51,plain,
( sP32
| ~ sP53 ),
inference(prop_rule,[status(thm)],[]) ).
thf(52,plain,
( sP16
| ~ sP32 ),
inference(prop_rule,[status(thm)],[]) ).
thf(53,plain,
( sP37
| ~ sP16 ),
inference(prop_rule,[status(thm)],[]) ).
thf(54,plain,
( sP26
| ~ sP37 ),
inference(prop_rule,[status(thm)],[]) ).
thf(55,plain,
( sP29
| ~ sP26 ),
inference(prop_rule,[status(thm)],[]) ).
thf(56,plain,
( sP42
| ~ sP29 ),
inference(prop_rule,[status(thm)],[]) ).
thf(57,plain,
( sP30
| ~ sP42 ),
inference(prop_rule,[status(thm)],[]) ).
thf(58,plain,
( sP17
| ~ sP30 ),
inference(prop_rule,[status(thm)],[]) ).
thf(59,plain,
( sP17
| sP51 ),
inference(prop_rule,[status(thm)],[]) ).
thf(60,plain,
( sP35
| ~ sP17 ),
inference(prop_rule,[status(thm)],[]) ).
thf(61,plain,
( sP47
| ~ sP35 ),
inference(prop_rule,[status(thm)],[]) ).
thf(setadjoinAx,axiom,
sP14 ).
thf(62,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,setadjoinAx,h1]) ).
thf(63,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[62,h0]) ).
thf(0,theorem,
sP47,
inference(contra,[status(thm),contra(discharge,[h1])],[62,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU529^1 : TPTP v9.2.0. Released v3.7.0.
% 0.12/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.13/0.34 % Computer : n014.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
% 0.19/0.34 % WCLimit : 300
% 0.19/0.34 % DateTime : Wed Oct 8 21:16:53 EDT 2025
% 0.19/0.34 % CPUTime :
% 24.13/24.33 % SZS status Theorem
% 24.13/24.33 % Mode: mode371
% 24.13/24.33 % Inferences: 3962
% 24.13/24.33 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------