%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU529^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n029.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 : Wed Feb 25 09:35:06 AM UTC 2026
% Result : Theorem 1.98s 1.47s
% Output : Refutation 1.98s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 4
% Syntax : Number of formulae : 25 ( 8 unt; 0 typ; 0 def)
% Number of atoms : 976 ( 284 equ; 0 cnn)
% Maximal formula atoms : 48 ( 39 avg)
% Number of connectives : 647 ( 49 ~; 46 |; 164 &; 198 @)
% ( 4 <=>; 186 =>; 0 <=; 0 <~>)
% Maximal formula depth : 53 ( 18 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 119 ( 116 usr; 53 con; 0-3 aty)
% Number of variables : 57 ( 0 ^; 39 !; 18 ?; 57 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_70,type,
sP1: $i > $i > $o ).
thf(func_def_71,type,
sP2: $i > $o ).
thf(func_def_72,type,
sP3: $i > $i > $o ).
thf(func_def_73,type,
sP4: $i > $i > $o ).
thf(func_def_86,type,
sK17: $i > $o ).
thf(func_def_95,type,
sK26: $i > $i ).
thf(func_def_97,type,
sK28: ( $i > $o ) > $i ).
thf(func_def_98,type,
sK29: $i > $o ).
thf(func_def_99,type,
sK30: $i > $i ).
thf(func_def_102,type,
sK33: $i > $o ).
thf(func_def_106,type,
sK37: $i > $i > $i ).
thf(func_def_115,type,
sK46: $i > $i > $i ).
thf(func_def_116,type,
sK47: $i > $i > $i ).
thf(func_def_117,type,
sK48: $o > $i > $i > $i ).
thf(func_def_121,type,
sK52: $i > $i ).
thf(func_def_123,type,
sK54: $i > $o ).
thf(func_def_124,type,
sK55: ( $i > $o ) > $i ).
thf(func_def_125,type,
sK56: $i > $o ).
thf(func_def_133,type,
sK64: $i > $i ).
thf(func_def_137,type,
sK68: $i > $i ).
thf(func_def_141,type,
sK72: $i > $i > $o ).
thf(func_def_143,type,
sK74: $i > $i ).
thf(func_def_144,type,
sK75: $i > $i ).
thf(func_def_145,type,
sK76: $i > ( $i > $i > $o ) > $i ).
thf(func_def_146,type,
sK77: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_147,type,
sK78: $i > ( $i > $i > $o ) > $i ).
thf(func_def_148,type,
sK79: ( $i > $o ) > $i ).
thf(func_def_149,type,
sK80: $i > $o ).
thf(func_def_150,type,
sK81: $i > $i ).
thf(func_def_157,type,
sK88: $i > $i ).
thf(func_def_159,type,
sK90: $i > $i > $i ).
thf(func_def_163,type,
sK94: $i > $i > $i ).
thf(func_def_164,type,
sK95: $i > $i > $i ).
thf(func_def_165,type,
sK96: $i > $i > $i ).
thf(func_def_166,type,
sK97: $i > $i > $i ).
thf(func_def_167,type,
sK98: $i > $i > $i ).
thf(func_def_168,type,
sK99: $i > $i > $i > $i ).
thf(func_def_169,type,
sK100: $i > $i ).
thf(func_def_170,type,
sK101: $i > $i ).
thf(func_def_171,type,
sK102: $i > $i ).
thf(func_def_172,type,
sK103: $i > $i ).
thf(func_def_173,type,
sK104: $i > $i > $i ).
thf(func_def_174,type,
sK105: $i > $i > $i > $i ).
thf(func_def_175,type,
sK106: $i > $i > $i > $i ).
thf(func_def_176,type,
sK107: $i > $i > $i ).
thf(func_def_177,type,
sK108: $i > $i > $i ).
thf(func_def_179,type,
sK110: $i > $i ).
thf(func_def_180,type,
sK111: $i > $i ).
thf(func_def_181,type,
sK112: $i > $i ).
thf(func_def_182,type,
sK113: $i > $o ).
thf(func_def_186,type,
sK117: $i > $i ).
thf(func_def_188,type,
sK119: $i > $i ).
thf(func_def_191,type,
sK122: $i > $i > $i ).
thf(func_def_196,type,
sK127: $i > $i > $i ).
thf(func_def_199,type,
sK130: $i > $o ).
thf(func_def_203,type,
ph134:
!>[X0: $tType] : X0 ).
thf(f938,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f807,f932]) ).
thf(f932,plain,
! [X4: $i,X5: $i] :
( $true
= ( in @ X4 @ ( setadjoin @ X4 @ X5 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f854]) ).
thf(f854,plain,
! [X4: $i,X5: $i] :
( ( $true
= ( in @ X4 @ ( setadjoin @ X4 @ X5 ) ) )
| ( $true != $true ) ),
inference(equality_resolution,[status(thm)],[f747]) ).
thf(f747,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X3 @ ( setadjoin @ X4 @ X5 ) ) )
| ( X3 != X4 )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f499,f592]) ).
thf(f592,plain,
setadjoinAx = $true,
inference(cnf_transformation,[status(thm)],[f364]) ).
thf(f364,plain,
( ( omega0Ax = $true )
& ( nonemptyI = $true )
& ( omegaIndAx = $true )
& ( setadjoinIR = $true )
& ( wellorderingAx = $true )
& ( setbeta = $true )
& ( dsetconstrEL = $true )
& ( setunionI = $true )
& ( powersetI = $true )
& ( exuE2 = $true )
& ( setadjoinIL = $true )
& ( descrp = $true )
& ( $true
!= ( in @ sK86 @ ( setadjoin @ sK87 @ emptyset ) ) )
& ( sK87 = sK86 )
& ( emptyinunitempty = $true )
& ( subPowSU = $true )
& ( prop2setE = $true )
& ( emptyinPowerset = $true )
& ( dsetconstrER = $true )
& ( setoftrueEq = $true )
& ( emptyI = $true )
& ( notinsingleton = $true )
& ( setadjoinOr = $true )
& ( dsetconstrI = $true )
& ( setadjoinAx = $true )
& ( setunionE = $true )
& ( setunionAx = $true )
& ( emptysetAx = $true )
& ( setadjoinE = $true )
& ( nonemptyE1 = $true )
& ( omegaSAx = $true )
& ( exuE1 = $true )
& ( emptyInPowerset = $true )
& ( powersetAx = $true )
& ( foundationAx = $true )
& ( powersetE = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( replAx = $true )
& ( setextAx = $true )
& ( nonemptyI1 = $true )
& ( nonemptyImpWitness = $true )
& ( noeltsimpempty = $true )
& ( emptysetimpfalse = $true )
& ( uniqinunit = $true )
& ( notinemptyset = $true )
& ( emptysetE = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK86,sK87])],[f178,f363]) ).
thf(f363,plain,
( ? [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
& ( X0 = X1 ) )
=> ( ( $true
!= ( in @ sK86 @ ( setadjoin @ sK87 @ emptyset ) ) )
& ( sK87 = sK86 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f178,plain,
( ( omega0Ax = $true )
& ( nonemptyI = $true )
& ( omegaIndAx = $true )
& ( setadjoinIR = $true )
& ( wellorderingAx = $true )
& ( setbeta = $true )
& ( dsetconstrEL = $true )
& ( setunionI = $true )
& ( powersetI = $true )
& ( exuE2 = $true )
& ( setadjoinIL = $true )
& ( descrp = $true )
& ? [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
& ( X0 = X1 ) )
& ( emptyinunitempty = $true )
& ( subPowSU = $true )
& ( prop2setE = $true )
& ( emptyinPowerset = $true )
& ( dsetconstrER = $true )
& ( setoftrueEq = $true )
& ( emptyI = $true )
& ( notinsingleton = $true )
& ( setadjoinOr = $true )
& ( dsetconstrI = $true )
& ( setadjoinAx = $true )
& ( setunionE = $true )
& ( setunionAx = $true )
& ( emptysetAx = $true )
& ( setadjoinE = $true )
& ( nonemptyE1 = $true )
& ( omegaSAx = $true )
& ( exuE1 = $true )
& ( emptyInPowerset = $true )
& ( powersetAx = $true )
& ( foundationAx = $true )
& ( powersetE = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( replAx = $true )
& ( setextAx = $true )
& ( nonemptyI1 = $true )
& ( nonemptyImpWitness = $true )
& ( noeltsimpempty = $true )
& ( emptysetimpfalse = $true )
& ( uniqinunit = $true )
& ( notinemptyset = $true )
& ( emptysetE = $true ) ),
inference(flattening,[status(thm)],[f177]) ).
thf(f177,plain,
( ? [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
& ( X0 = X1 ) )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f53]) ).
thf(f53,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f52]) ).
thf(f52,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( 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
=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f50]) ).
thf(f50,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f49]) ).
thf(f49,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.ryFqNTYIaR/DTF2THF_31353.p',eqinunit) ).
thf(f499,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X3 @ ( setadjoin @ X4 @ X5 ) ) )
| ( X3 != X4 )
| ( setadjoinAx != $true ) ),
inference(cnf_transformation,[status(thm)],[f283]) ).
thf(f283,plain,
( ( ( setadjoinAx = $true )
| ( ( ( ( sK42 != sK41 )
& ( ( in @ sK41 @ sK43 )
!= $true ) )
| ( ( in @ sK41 @ ( setadjoin @ sK42 @ sK43 ) )
!= $true ) )
& ( ( sK42 = sK41 )
| ( ( in @ sK41 @ sK43 )
= $true )
| ( ( in @ sK41 @ ( setadjoin @ sK42 @ sK43 ) )
= $true ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( $true
= ( in @ X3 @ ( setadjoin @ X4 @ X5 ) ) )
| ( ( X3 != X4 )
& ( $true
!= ( in @ X3 @ X5 ) ) ) )
& ( ( X3 = X4 )
| ( $true
= ( in @ X3 @ X5 ) )
| ( $true
!= ( in @ X3 @ ( setadjoin @ X4 @ X5 ) ) ) ) )
| ( setadjoinAx != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK41,sK42,sK43])],[f281,f282]) ).
thf(f282,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( ( X0 != X1 )
& ( $true
!= ( in @ X0 @ X2 ) ) )
| ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ X2 ) ) ) )
& ( ( X0 = X1 )
| ( $true
= ( in @ X0 @ X2 ) )
| ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ X2 ) ) ) ) )
=> ( ( ( ( sK42 != sK41 )
& ( ( in @ sK41 @ sK43 )
!= $true ) )
| ( ( in @ sK41 @ ( setadjoin @ sK42 @ sK43 ) )
!= $true ) )
& ( ( sK42 = sK41 )
| ( ( in @ sK41 @ sK43 )
= $true )
| ( ( in @ sK41 @ ( setadjoin @ sK42 @ sK43 ) )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f281,plain,
( ( ( setadjoinAx = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( ( X0 != X1 )
& ( $true
!= ( in @ X0 @ X2 ) ) )
| ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ X2 ) ) ) )
& ( ( X0 = X1 )
| ( $true
= ( in @ X0 @ X2 ) )
| ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ X2 ) ) ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( $true
= ( in @ X3 @ ( setadjoin @ X4 @ X5 ) ) )
| ( ( X3 != X4 )
& ( $true
!= ( in @ X3 @ X5 ) ) ) )
& ( ( X3 = X4 )
| ( $true
= ( in @ X3 @ X5 ) )
| ( $true
!= ( in @ X3 @ ( setadjoin @ X4 @ X5 ) ) ) ) )
| ( setadjoinAx != $true ) ) ),
inference(rectify,[status(thm)],[f280]) ).
thf(f280,plain,
( ( ( setadjoinAx = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( ( X0 != X1 )
& ( $true
!= ( in @ X0 @ X2 ) ) )
| ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ X2 ) ) ) )
& ( ( X0 = X1 )
| ( $true
= ( in @ X0 @ X2 ) )
| ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ X2 ) ) ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ X2 ) ) )
| ( ( X0 != X1 )
& ( $true
!= ( in @ X0 @ X2 ) ) ) )
& ( ( X0 = X1 )
| ( $true
= ( in @ X0 @ X2 ) )
| ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ X2 ) ) ) ) )
| ( setadjoinAx != $true ) ) ),
inference(flattening,[status(thm)],[f279]) ).
thf(f279,plain,
( ( ( setadjoinAx = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( ( X0 != X1 )
& ( $true
!= ( in @ X0 @ X2 ) ) )
| ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ X2 ) ) ) )
& ( ( X0 = X1 )
| ( $true
= ( in @ X0 @ X2 ) )
| ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ X2 ) ) ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ X2 ) ) )
| ( ( X0 != X1 )
& ( $true
!= ( in @ X0 @ X2 ) ) ) )
& ( ( X0 = X1 )
| ( $true
= ( in @ X0 @ X2 ) )
| ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ X2 ) ) ) ) )
| ( setadjoinAx != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f85]) ).
thf(f85,plain,
( ( setadjoinAx = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ X2 ) ) )
<=> ( ( X0 = X1 )
| ( $true
= ( in @ X0 @ X2 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f84]) ).
thf(f84,plain,
( setadjoinAx
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ X2 ) )
<=> ( ( X0 = X1 )
| ( in @ X0 @ X2 ) ) ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
( setadjoinAx
= ( ! [X2: $i,X1: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X1 @ X3 ) )
<=> ( ( X1 = X2 )
| ( in @ X2 @ X3 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.ryFqNTYIaR/DTF2THF_31353.p',setadjoinAx) ).
thf(f807,plain,
( $true
!= ( in @ sK86 @ ( setadjoin @ sK86 @ emptyset ) ) ),
inference(definition_unfolding,[status(thm)],[f604,f603]) ).
thf(f603,plain,
sK87 = sK86,
inference(cnf_transformation,[status(thm)],[f364]) ).
thf(f604,plain,
( $true
!= ( in @ sK86 @ ( setadjoin @ sK87 @ emptyset ) ) ),
inference(cnf_transformation,[status(thm)],[f364]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU529^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.10 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.12/0.29 % Computer : n029.cluster.edu
% 0.12/0.29 % Model : x86_64 x86_64
% 0.12/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.29 % Memory : 8042.1875MB
% 0.12/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.29 % CPULimit : 300
% 0.12/0.29 % WCLimit : 300
% 0.12/0.29 % DateTime : Tue Feb 24 12:25:59 EST 2026
% 0.12/0.30 % CPUTime :
% 0.12/0.30 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.39 ---- Original DTF file ---
% 0.16/0.39 thf(spec,logic,$$dhol).
% 0.16/0.39 %------------------------------------------------------------------------------
% 0.16/0.39 % File : SEU529^1 : TPTP v9.2.1. Released v3.7.0.
% 0.16/0.39 % Domain : Set Theory
% 0.16/0.39 % Problem : Preliminary Notions - Equality Laws
% 0.16/0.39 % Version : Especial.
% 0.16/0.39 % English : (! x:i.! y:i.x = y -> in x (setadjoin y emptyset))
% 0.16/0.39
% 0.16/0.39 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.16/0.39 % Source : [Bro08]
% 0.16/0.39 % Names : ZFC031g [Bro08]
% 0.16/0.39
% 0.16/0.39 % Status : Theorem
% 0.16/0.39 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.60 v5.2.0, 0.20 v5.1.0, 0.40 v5.0.0, 0.20 v4.1.0, 0.33 v4.0.0, 0.67 v3.7.0
% 0.16/0.39 % Syntax : Number of formulae : 105 ( 48 unt; 56 typ; 48 def)
% 0.16/0.39 % Number of atoms : 268 ( 64 equ; 0 cnn)
% 0.16/0.39 % Maximal formula atoms : 47 ( 5 avg)
% 0.16/0.39 % Number of connectives : 435 ( 9 ~; 4 |; 19 &; 283 @)
% 0.16/0.39 % ( 8 <=>; 112 =>; 0 <=; 0 <~>)
% 0.16/0.39 % Maximal formula depth : 52 ( 2 avg)
% 0.16/0.39 % Number of types : 2 ( 0 usr)
% 0.16/0.39 % Number of type conns : 27 ( 27 >; 0 *; 0 +; 0 <<)
% 0.16/0.39 % Number of symbols : 59 ( 56 usr; 49 con; 0-2 aty)
% 0.16/0.39 % Number of variables : 144 ( 16 ^; 110 !; 18 ?; 144 :)
% 0.16/0.39 % SPC : TH0_THM_EQU_NAR
% 0.16/0.39
% 0.16/0.39 % Comments : http://mathgate.info/detsetitem.php?id=94
% 0.16/0.39 %------------------------------------------------------------------------------
% 0.16/0.39 thf(in_type,type,
% 0.16/0.39 in: $i > $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(exu_type,type,
% 0.16/0.39 exu: ( $i > $o ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(exu,definition,
% 0.16/0.39 ( exu
% 0.16/0.39 = ( ^ [Xphi: $i > $o] :
% 0.16/0.39 ? [Xx: $i] :
% 0.16/0.39 ( ( Xphi @ Xx )
% 0.16/0.39 & ! [Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setextAx_type,type,
% 0.16/0.39 setextAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setextAx,definition,
% 0.16/0.39 ( setextAx
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 <=> ( in @ Xx @ B ) )
% 0.16/0.39 => ( A = B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyset_type,type,
% 0.16/0.39 emptyset: $i ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetAx_type,type,
% 0.16/0.39 emptysetAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetAx,definition,
% 0.16/0.39 ( emptysetAx
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoin_type,type,
% 0.16/0.39 setadjoin: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinAx_type,type,
% 0.16/0.39 setadjoinAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinAx,definition,
% 0.16/0.39 ( setadjoinAx
% 0.16/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.39 <=> ( ( Xy = Xx )
% 0.16/0.39 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powerset_type,type,
% 0.16/0.39 powerset: $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(powersetAx_type,type,
% 0.16/0.39 powersetAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powersetAx,definition,
% 0.16/0.39 ( powersetAx
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.39 <=> ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunion_type,type,
% 0.16/0.39 setunion: $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(setunionAx_type,type,
% 0.16/0.39 setunionAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionAx,definition,
% 0.16/0.39 ( setunionAx
% 0.16/0.39 = ( ! [A: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.39 <=> ? [B: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 & ( in @ B @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(omega_type,type,
% 0.16/0.39 omega: $i ).
% 0.16/0.39
% 0.16/0.39 thf(omega0Ax_type,type,
% 0.16/0.39 omega0Ax: $o ).
% 0.16/0.39
% 0.16/0.39 thf(omega0Ax,definition,
% 0.16/0.39 ( omega0Ax
% 0.16/0.39 = ( in @ emptyset @ omega ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(omegaSAx_type,type,
% 0.16/0.39 omegaSAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(omegaSAx,definition,
% 0.16/0.39 ( omegaSAx
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ omega )
% 0.16/0.39 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(omegaIndAx_type,type,
% 0.16/0.39 omegaIndAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(omegaIndAx,definition,
% 0.16/0.39 ( omegaIndAx
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ( ( in @ emptyset @ A )
% 0.16/0.39 & ! [Xx: $i] :
% 0.16/0.39 ( ( ( in @ Xx @ omega )
% 0.16/0.39 & ( in @ Xx @ A ) )
% 0.16/0.39 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ omega )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(replAx_type,type,
% 0.16/0.39 replAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(replAx,definition,
% 0.16/0.39 ( replAx
% 0.16/0.39 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( exu
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.16/0.39 => ? [B: $i] :
% 0.16/0.39 ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 <=> ? [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ A )
% 0.16/0.39 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(foundationAx_type,type,
% 0.16/0.39 foundationAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(foundationAx,definition,
% 0.16/0.39 ( foundationAx
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.39 => ? [B: $i] :
% 0.16/0.39 ( ( in @ B @ A )
% 0.16/0.39 & ~ ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(wellorderingAx_type,type,
% 0.16/0.39 wellorderingAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(wellorderingAx,definition,
% 0.16/0.39 ( wellorderingAx
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ? [B: $i] :
% 0.16/0.39 ( ! [C: $i] :
% 0.16/0.39 ( ( in @ C @ B )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ C )
% 0.16/0.39 => ( in @ Xx @ A ) ) )
% 0.16/0.39 & ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( ( in @ Xx @ A )
% 0.16/0.39 & ( in @ Xy @ A ) )
% 0.16/0.39 => ( ! [C: $i] :
% 0.16/0.39 ( ( in @ C @ B )
% 0.16/0.39 => ( ( in @ Xx @ C )
% 0.16/0.39 <=> ( in @ Xy @ C ) ) )
% 0.16/0.39 => ( Xx = Xy ) ) )
% 0.16/0.39 & ! [C: $i,D: $i] :
% 0.16/0.39 ( ( ( in @ C @ B )
% 0.16/0.39 & ( in @ D @ B ) )
% 0.16/0.39 => ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ C )
% 0.16/0.39 => ( in @ Xx @ D ) )
% 0.16/0.39 | ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ D )
% 0.16/0.39 => ( in @ Xx @ C ) ) ) )
% 0.16/0.39 & ! [C: $i] :
% 0.16/0.39 ( ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ C )
% 0.16/0.39 => ( in @ Xx @ A ) )
% 0.16/0.39 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.16/0.39 => ? [D: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ D @ B )
% 0.16/0.39 & ( in @ Xx @ C )
% 0.16/0.39 & ~ ? [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ D )
% 0.16/0.39 & ( in @ Xy @ C ) )
% 0.16/0.39 & ! [E: $i] :
% 0.16/0.39 ( ( in @ E @ B )
% 0.16/0.39 => ( ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ E )
% 0.16/0.39 => ( in @ Xy @ D ) )
% 0.16/0.39 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(descr_type,type,
% 0.16/0.39 descr: ( $i > $o ) > $i ).
% 0.16/0.39
% 0.16/0.39 thf(descrp_type,type,
% 0.16/0.39 descrp: $o ).
% 0.16/0.39
% 0.16/0.39 thf(descrp,definition,
% 0.16/0.39 ( descrp
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ( Xphi
% 0.16/0.39 @ ( descr
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstr_type,type,
% 0.16/0.39 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrI_type,type,
% 0.16/0.39 dsetconstrI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrI,definition,
% 0.16/0.39 ( dsetconstrI
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 => ( in @ Xx
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrEL_type,type,
% 0.16/0.39 dsetconstrEL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrEL,definition,
% 0.16/0.39 ( dsetconstrEL
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrER_type,type,
% 0.16/0.39 dsetconstrER: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrER,definition,
% 0.16/0.39 ( dsetconstrER
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.39 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuE1_type,type,
% 0.16/0.39 exuE1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuE1,definition,
% 0.16/0.39 ( exuE1
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ( ( Xphi @ Xx )
% 0.16/0.39 & ! [Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(prop2set_type,type,
% 0.16/0.39 prop2set: $o > $i ).
% 0.16/0.39
% 0.16/0.39 thf(prop2set,definition,
% 0.16/0.39 ( prop2set
% 0.16/0.39 = ( ^ [Xphi: $o] :
% 0.16/0.39 ( dsetconstr @ ( powerset @ emptyset )
% 0.16/0.39 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(prop2setE_type,type,
% 0.16/0.39 prop2setE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(prop2setE,definition,
% 0.16/0.39 ( prop2setE
% 0.16/0.39 = ( ! [Xphi: $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.16/0.39 => Xphi ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetE_type,type,
% 0.16/0.39 emptysetE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetE,definition,
% 0.16/0.39 ( emptysetE
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ emptyset )
% 0.16/0.39 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetimpfalse_type,type,
% 0.16/0.39 emptysetimpfalse: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetimpfalse,definition,
% 0.16/0.39 ( emptysetimpfalse
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ emptyset )
% 0.16/0.39 => $false ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(notinemptyset_type,type,
% 0.16/0.39 notinemptyset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(notinemptyset,definition,
% 0.16/0.39 ( notinemptyset
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuE3e_type,type,
% 0.16/0.39 exuE3e: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuE3e,definition,
% 0.16/0.39 ( exuE3e
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setext_type,type,
% 0.16/0.39 setext: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setext,definition,
% 0.16/0.39 ( setext
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xx @ B ) )
% 0.16/0.39 => ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ A ) )
% 0.16/0.39 => ( A = B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyI_type,type,
% 0.16/0.39 emptyI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptyI,definition,
% 0.16/0.39 ( emptyI
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ~ ( in @ Xx @ A )
% 0.16/0.39 => ( A = emptyset ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(noeltsimpempty_type,type,
% 0.16/0.39 noeltsimpempty: $o ).
% 0.16/0.39
% 0.16/0.39 thf(noeltsimpempty,definition,
% 0.16/0.39 ( noeltsimpempty
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ~ ( in @ Xx @ A )
% 0.16/0.39 => ( A = emptyset ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setbeta_type,type,
% 0.16/0.39 setbeta: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setbeta,definition,
% 0.16/0.39 ( setbeta
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( in @ Xx
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.39 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(nonempty_type,type,
% 0.16/0.39 nonempty: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(nonempty,definition,
% 0.16/0.39 ( nonempty
% 0.16/0.39 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyE1_type,type,
% 0.16/0.39 nonemptyE1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyE1,definition,
% 0.16/0.39 ( nonemptyE1
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ( nonempty @ A )
% 0.16/0.39 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyI_type,type,
% 0.16/0.39 nonemptyI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyI,definition,
% 0.16/0.39 ( nonemptyI
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 => ( nonempty
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyI1_type,type,
% 0.16/0.39 nonemptyI1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyI1,definition,
% 0.16/0.39 ( nonemptyI1
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.39 => ( nonempty @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinIL_type,type,
% 0.16/0.39 setadjoinIL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinIL,definition,
% 0.16/0.39 ( setadjoinIL
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyinunitempty_type,type,
% 0.16/0.39 emptyinunitempty: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptyinunitempty,definition,
% 0.16/0.39 ( emptyinunitempty
% 0.16/0.39 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinIR_type,type,
% 0.16/0.39 setadjoinIR: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinIR,definition,
% 0.16/0.39 ( setadjoinIR
% 0.16/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ A )
% 0.16/0.39 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinE_type,type,
% 0.16/0.39 setadjoinE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinE,definition,
% 0.16/0.39 ( setadjoinE
% 0.16/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.39 => ! [Xphi: $o] :
% 0.16/0.39 ( ( ( Xy = Xx )
% 0.16/0.39 => Xphi )
% 0.16/0.39 => ( ( ( in @ Xy @ A )
% 0.16/0.39 => Xphi )
% 0.16/0.39 => Xphi ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinOr_type,type,
% 0.16/0.39 setadjoinOr: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinOr,definition,
% 0.16/0.39 ( setadjoinOr
% 0.16/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.39 => ( ( Xy = Xx )
% 0.16/0.39 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setoftrueEq_type,type,
% 0.16/0.39 setoftrueEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setoftrueEq,definition,
% 0.16/0.39 ( setoftrueEq
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xx: $i] : $true )
% 0.16/0.39 = A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powersetI_type,type,
% 0.16/0.39 powersetI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powersetI,definition,
% 0.16/0.39 ( powersetI
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ A ) )
% 0.16/0.39 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyinPowerset_type,type,
% 0.16/0.39 emptyinPowerset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptyinPowerset,definition,
% 0.16/0.39 ( emptyinPowerset
% 0.16/0.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyInPowerset_type,type,
% 0.16/0.39 emptyInPowerset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptyInPowerset,definition,
% 0.16/0.39 ( emptyInPowerset
% 0.16/0.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powersetE_type,type,
% 0.16/0.39 powersetE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powersetE,definition,
% 0.16/0.39 ( powersetE
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.39 => ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunionI_type,type,
% 0.16/0.39 setunionI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionI,definition,
% 0.16/0.39 ( setunionI
% 0.16/0.39 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( ( in @ B @ A )
% 0.16/0.39 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunionE_type,type,
% 0.16/0.39 setunionE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionE,definition,
% 0.16/0.39 ( setunionE
% 0.16/0.39 = ( ! [A: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.39 => ! [Xphi: $o] :
% 0.16/0.39 ( ! [B: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( ( in @ B @ A )
% 0.16/0.39 => Xphi ) )
% 0.16/0.39 => Xphi ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subPowSU_type,type,
% 0.16/0.39 subPowSU: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subPowSU,definition,
% 0.16/0.39 ( subPowSU
% 0.16/0.39 = ( ! [A: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuE2_type,type,
% 0.16/0.39 exuE2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuE2,definition,
% 0.16/0.39 ( exuE2
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ! [Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xy )
% 0.16/0.39 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyImpWitness_type,type,
% 0.16/0.39 nonemptyImpWitness: $o ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyImpWitness,definition,
% 0.16/0.39 ( nonemptyImpWitness
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ( nonempty @ A )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & $true ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(uniqinunit_type,type,
% 0.16/0.39 uniqinunit: $o ).
% 0.16/0.39
% 0.16/0.39 thf(uniqinunit,definition,
% 0.16/0.39 ( uniqinunit
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(notinsingleton_type,type,
% 0.16/0.39 notinsingleton: $o ).
% 0.16/0.39
% 0.16/0.39 thf(notinsingleton,definition,
% 0.16/0.39 ( notinsingleton
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx != Xy )
% 0.16/0.39 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(eqinunit,conjecture,
% 0.16/0.39 ( setextAx
% 0.16/0.39 => ( emptysetAx
% 0.16/0.39 => ( setadjoinAx
% 0.16/0.39 => ( powersetAx
% 0.16/0.39 => ( setunionAx
% 0.16/0.39 => ( omega0Ax
% 0.16/0.39 => ( omegaSAx
% 0.16/0.39 => ( omegaIndAx
% 0.16/0.39 => ( replAx
% 0.16/0.39 => ( foundationAx
% 0.16/0.39 => ( wellorderingAx
% 0.16/0.39 => ( descrp
% 0.16/0.39 => ( dsetconstrI
% 0.16/0.39 => ( dsetconstrEL
% 0.16/0.39 => ( dsetconstrER
% 0.16/0.39 => ( exuE1
% 0.16/0.39 => ( prop2setE
% 0.16/0.39 => ( emptysetE
% 0.16/0.39 => ( emptysetimpfalse
% 0.16/0.39 => ( notinemptyset
% 0.16/0.39 => ( exuE3e
% 0.16/0.39 => ( setext
% 0.16/0.39 => ( emptyI
% 0.16/0.39 => ( noeltsimpempty
% 0.16/0.39 => ( setbeta
% 0.16/0.39 => ( nonemptyE1
% 0.16/0.39 => ( nonemptyI
% 0.16/0.39 => ( nonemptyI1
% 0.16/0.39 => ( setadjoinIL
% 0.16/0.39 => ( emptyinunitempty
% 0.16/0.39 => ( setadjoinIR
% 0.16/0.39 => ( setadjoinE
% 0.16/0.39 => ( setadjoinOr
% 0.16/0.39 => ( setoftrueEq
% 0.16/0.39 => ( powersetI
% 0.16/0.39 => ( emptyinPowerset
% 0.16/0.39 => ( emptyInPowerset
% 0.16/0.39 => ( powersetE
% 0.16/0.39 => ( setunionI
% 0.16/0.39 => ( setunionE
% 0.16/0.39 => ( subPowSU
% 0.16/0.39 => ( exuE2
% 0.16/0.39 => ( nonemptyImpWitness
% 0.16/0.39 => ( uniqinunit
% 0.16/0.39 => ( notinsingleton
% 0.16/0.39 => ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 %------------------------------------------------------------------------------
% 0.16/0.39 ------------------------
% 1.98/1.41 ---- Embedded in THF ---
% 1.98/1.41 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.98/1.41 %%% Generated on Tue Feb 24 12:26:00 EST 2026
% 1.98/1.41 %%% using '$$dhol' embedding, version 1.3.0.
% 1.98/1.41 %%% Logic specification used:
% 1.98/1.41 %%% thf(spec, logic, $$dhol).
% 1.98/1.41
% 1.98/1.41 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.ryFqNTYIaR/DTF2DTF31353.p
% See solution above
% 1.98/1.41 ------------------------
% 1.98/1.43 ---- Cleaned THF ---
% 1.98/1.43 thf(in_type,type,
% 1.98/1.43 in: $i > $i > $o ).
% 1.98/1.43
% 1.98/1.43 thf(exu_type,type,
% 1.98/1.43 exu: ( $i > $o ) > $o ).
% 1.98/1.43
% 1.98/1.43 thf(exu,definition,
% 1.98/1.43 ( exu
% 1.98/1.43 = ( ^ [Xphi: $i > $o] :
% 1.98/1.43 ? [Xx: $i] :
% 1.98/1.43 ( ( Xphi @ Xx )
% 1.98/1.43 & ! [Xy: $i] :
% 1.98/1.43 ( ( Xphi @ Xy )
% 1.98/1.43 => ( Xx = Xy ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(setextAx_type,type,
% 1.98/1.43 setextAx: $o ).
% 1.98/1.43
% 1.98/1.43 thf(setextAx,definition,
% 1.98/1.43 ( setextAx
% 1.98/1.43 = ( ! [A: $i,B: $i] :
% 1.98/1.43 ( ! [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ A )
% 1.98/1.43 <=> ( in @ Xx @ B ) )
% 1.98/1.43 => ( A = B ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(emptyset_type,type,
% 1.98/1.43 emptyset: $i ).
% 1.98/1.43
% 1.98/1.43 thf(emptysetAx_type,type,
% 1.98/1.43 emptysetAx: $o ).
% 1.98/1.43
% 1.98/1.43 thf(emptysetAx,definition,
% 1.98/1.43 ( emptysetAx
% 1.98/1.43 = ( ! [Xx: $i] :
% 1.98/1.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(setadjoin_type,type,
% 1.98/1.43 setadjoin: $i > $i > $i ).
% 1.98/1.43
% 1.98/1.43 thf(setadjoinAx_type,type,
% 1.98/1.43 setadjoinAx: $o ).
% 1.98/1.43
% 1.98/1.43 thf(setadjoinAx,definition,
% 1.98/1.43 ( setadjoinAx
% 1.98/1.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.98/1.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.98/1.43 <=> ( ( Xy = Xx )
% 1.98/1.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(powerset_type,type,
% 1.98/1.43 powerset: $i > $i ).
% 1.98/1.43
% 1.98/1.43 thf(powersetAx_type,type,
% 1.98/1.43 powersetAx: $o ).
% 1.98/1.43
% 1.98/1.43 thf(powersetAx,definition,
% 1.98/1.43 ( powersetAx
% 1.98/1.43 = ( ! [A: $i,B: $i] :
% 1.98/1.43 ( ( in @ B @ ( powerset @ A ) )
% 1.98/1.43 <=> ! [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ B )
% 1.98/1.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(setunion_type,type,
% 1.98/1.43 setunion: $i > $i ).
% 1.98/1.43
% 1.98/1.43 thf(setunionAx_type,type,
% 1.98/1.43 setunionAx: $o ).
% 1.98/1.43
% 1.98/1.43 thf(setunionAx,definition,
% 1.98/1.43 ( setunionAx
% 1.98/1.43 = ( ! [A: $i,Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ ( setunion @ A ) )
% 1.98/1.43 <=> ? [B: $i] :
% 1.98/1.43 ( ( in @ Xx @ B )
% 1.98/1.43 & ( in @ B @ A ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(omega_type,type,
% 1.98/1.43 omega: $i ).
% 1.98/1.43
% 1.98/1.43 thf(omega0Ax_type,type,
% 1.98/1.43 omega0Ax: $o ).
% 1.98/1.43
% 1.98/1.43 thf(omega0Ax,definition,
% 1.98/1.43 ( omega0Ax
% 1.98/1.43 = ( in @ emptyset @ omega ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(omegaSAx_type,type,
% 1.98/1.43 omegaSAx: $o ).
% 1.98/1.43
% 1.98/1.43 thf(omegaSAx,definition,
% 1.98/1.43 ( omegaSAx
% 1.98/1.43 = ( ! [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ omega )
% 1.98/1.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(omegaIndAx_type,type,
% 1.98/1.43 omegaIndAx: $o ).
% 1.98/1.43
% 1.98/1.43 thf(omegaIndAx,definition,
% 1.98/1.43 ( omegaIndAx
% 1.98/1.43 = ( ! [A: $i] :
% 1.98/1.43 ( ( ( in @ emptyset @ A )
% 1.98/1.43 & ! [Xx: $i] :
% 1.98/1.43 ( ( ( in @ Xx @ omega )
% 1.98/1.43 & ( in @ Xx @ A ) )
% 1.98/1.43 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 1.98/1.43 => ! [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ omega )
% 1.98/1.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(replAx_type,type,
% 1.98/1.43 replAx: $o ).
% 1.98/1.43
% 1.98/1.43 thf(replAx,definition,
% 1.98/1.43 ( replAx
% 1.98/1.43 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 1.98/1.43 ( ! [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ A )
% 1.98/1.43 => ( exu
% 1.98/1.43 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 1.98/1.43 => ? [B: $i] :
% 1.98/1.43 ! [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ B )
% 1.98/1.43 <=> ? [Xy: $i] :
% 1.98/1.43 ( ( in @ Xy @ A )
% 1.98/1.43 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(foundationAx_type,type,
% 1.98/1.43 foundationAx: $o ).
% 1.98/1.43
% 1.98/1.43 thf(foundationAx,definition,
% 1.98/1.43 ( foundationAx
% 1.98/1.43 = ( ! [A: $i] :
% 1.98/1.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.98/1.43 => ? [B: $i] :
% 1.98/1.43 ( ( in @ B @ A )
% 1.98/1.43 & ~ ? [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ B )
% 1.98/1.43 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(wellorderingAx_type,type,
% 1.98/1.43 wellorderingAx: $o ).
% 1.98/1.43
% 1.98/1.43 thf(wellorderingAx,definition,
% 1.98/1.43 ( wellorderingAx
% 1.98/1.43 = ( ! [A: $i] :
% 1.98/1.43 ? [B: $i] :
% 1.98/1.43 ( ! [C: $i] :
% 1.98/1.43 ( ( in @ C @ B )
% 1.98/1.43 => ! [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ C )
% 1.98/1.43 => ( in @ Xx @ A ) ) )
% 1.98/1.43 & ! [Xx: $i,Xy: $i] :
% 1.98/1.43 ( ( ( in @ Xx @ A )
% 1.98/1.43 & ( in @ Xy @ A ) )
% 1.98/1.43 => ( ! [C: $i] :
% 1.98/1.43 ( ( in @ C @ B )
% 1.98/1.43 => ( ( in @ Xx @ C )
% 1.98/1.43 <=> ( in @ Xy @ C ) ) )
% 1.98/1.43 => ( Xx = Xy ) ) )
% 1.98/1.43 & ! [C: $i,D: $i] :
% 1.98/1.43 ( ( ( in @ C @ B )
% 1.98/1.43 & ( in @ D @ B ) )
% 1.98/1.43 => ( ! [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ C )
% 1.98/1.43 => ( in @ Xx @ D ) )
% 1.98/1.43 | ! [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ D )
% 1.98/1.43 => ( in @ Xx @ C ) ) ) )
% 1.98/1.43 & ! [C: $i] :
% 1.98/1.43 ( ( ! [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ C )
% 1.98/1.43 => ( in @ Xx @ A ) )
% 1.98/1.43 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 1.98/1.43 => ? [D: $i,Xx: $i] :
% 1.98/1.43 ( ( in @ D @ B )
% 1.98/1.43 & ( in @ Xx @ C )
% 1.98/1.43 & ~ ? [Xy: $i] :
% 1.98/1.43 ( ( in @ Xy @ D )
% 1.98/1.43 & ( in @ Xy @ C ) )
% 1.98/1.43 & ! [E: $i] :
% 1.98/1.43 ( ( in @ E @ B )
% 1.98/1.43 => ( ! [Xy: $i] :
% 1.98/1.43 ( ( in @ Xy @ E )
% 1.98/1.43 => ( in @ Xy @ D ) )
% 1.98/1.43 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(descr_type,type,
% 1.98/1.43 descr: ( $i > $o ) > $i ).
% 1.98/1.43
% 1.98/1.43 thf(descrp_type,type,
% 1.98/1.43 descrp: $o ).
% 1.98/1.43
% 1.98/1.43 thf(descrp,definition,
% 1.98/1.43 ( descrp
% 1.98/1.43 = ( ! [Xphi: $i > $o] :
% 1.98/1.43 ( ( exu
% 1.98/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.43 => ( Xphi
% 1.98/1.43 @ ( descr
% 1.98/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(dsetconstr_type,type,
% 1.98/1.43 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.98/1.43
% 1.98/1.43 thf(dsetconstrI_type,type,
% 1.98/1.43 dsetconstrI: $o ).
% 1.98/1.43
% 1.98/1.43 thf(dsetconstrI,definition,
% 1.98/1.43 ( dsetconstrI
% 1.98/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ A )
% 1.98/1.43 => ( ( Xphi @ Xx )
% 1.98/1.43 => ( in @ Xx
% 1.98/1.43 @ ( dsetconstr @ A
% 1.98/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(dsetconstrEL_type,type,
% 1.98/1.43 dsetconstrEL: $o ).
% 1.98/1.43
% 1.98/1.43 thf(dsetconstrEL,definition,
% 1.98/1.43 ( dsetconstrEL
% 1.98/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.98/1.43 ( ( in @ Xx
% 1.98/1.43 @ ( dsetconstr @ A
% 1.98/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.98/1.43 => ( in @ Xx @ A ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(dsetconstrER_type,type,
% 1.98/1.43 dsetconstrER: $o ).
% 1.98/1.43
% 1.98/1.43 thf(dsetconstrER,definition,
% 1.98/1.43 ( dsetconstrER
% 1.98/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.98/1.43 ( ( in @ Xx
% 1.98/1.43 @ ( dsetconstr @ A
% 1.98/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.98/1.43 => ( Xphi @ Xx ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(exuE1_type,type,
% 1.98/1.43 exuE1: $o ).
% 1.98/1.43
% 1.98/1.43 thf(exuE1,definition,
% 1.98/1.43 ( exuE1
% 1.98/1.43 = ( ! [Xphi: $i > $o] :
% 1.98/1.43 ( ( exu
% 1.98/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.43 => ? [Xx: $i] :
% 1.98/1.43 ( ( Xphi @ Xx )
% 1.98/1.43 & ! [Xy: $i] :
% 1.98/1.43 ( ( Xphi @ Xy )
% 1.98/1.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(prop2set_type,type,
% 1.98/1.43 prop2set: $o > $i ).
% 1.98/1.43
% 1.98/1.43 thf(prop2set,definition,
% 1.98/1.43 ( prop2set
% 1.98/1.43 = ( ^ [Xphi: $o] :
% 1.98/1.43 ( dsetconstr @ ( powerset @ emptyset )
% 1.98/1.43 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(prop2setE_type,type,
% 1.98/1.43 prop2setE: $o ).
% 1.98/1.43
% 1.98/1.43 thf(prop2setE,definition,
% 1.98/1.43 ( prop2setE
% 1.98/1.43 = ( ! [Xphi: $o,Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 1.98/1.43 => Xphi ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(emptysetE_type,type,
% 1.98/1.43 emptysetE: $o ).
% 1.98/1.43
% 1.98/1.43 thf(emptysetE,definition,
% 1.98/1.43 ( emptysetE
% 1.98/1.43 = ( ! [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ emptyset )
% 1.98/1.43 => ! [Xphi: $o] : Xphi ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(emptysetimpfalse_type,type,
% 1.98/1.43 emptysetimpfalse: $o ).
% 1.98/1.43
% 1.98/1.43 thf(emptysetimpfalse,definition,
% 1.98/1.43 ( emptysetimpfalse
% 1.98/1.43 = ( ! [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ emptyset )
% 1.98/1.43 => $false ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(notinemptyset_type,type,
% 1.98/1.43 notinemptyset: $o ).
% 1.98/1.43
% 1.98/1.43 thf(notinemptyset,definition,
% 1.98/1.43 ( notinemptyset
% 1.98/1.43 = ( ! [Xx: $i] :
% 1.98/1.43 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(exuE3e_type,type,
% 1.98/1.43 exuE3e: $o ).
% 1.98/1.43
% 1.98/1.43 thf(exuE3e,definition,
% 1.98/1.43 ( exuE3e
% 1.98/1.43 = ( ! [Xphi: $i > $o] :
% 1.98/1.43 ( ( exu
% 1.98/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.43 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(setext_type,type,
% 1.98/1.43 setext: $o ).
% 1.98/1.43
% 1.98/1.43 thf(setext,definition,
% 1.98/1.43 ( setext
% 1.98/1.43 = ( ! [A: $i,B: $i] :
% 1.98/1.43 ( ! [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ A )
% 1.98/1.43 => ( in @ Xx @ B ) )
% 1.98/1.43 => ( ! [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ B )
% 1.98/1.43 => ( in @ Xx @ A ) )
% 1.98/1.43 => ( A = B ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(emptyI_type,type,
% 1.98/1.43 emptyI: $o ).
% 1.98/1.43
% 1.98/1.43 thf(emptyI,definition,
% 1.98/1.43 ( emptyI
% 1.98/1.43 = ( ! [A: $i] :
% 1.98/1.43 ( ! [Xx: $i] :
% 1.98/1.43 ~ ( in @ Xx @ A )
% 1.98/1.43 => ( A = emptyset ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(noeltsimpempty_type,type,
% 1.98/1.43 noeltsimpempty: $o ).
% 1.98/1.43
% 1.98/1.43 thf(noeltsimpempty,definition,
% 1.98/1.43 ( noeltsimpempty
% 1.98/1.43 = ( ! [A: $i] :
% 1.98/1.43 ( ! [Xx: $i] :
% 1.98/1.43 ~ ( in @ Xx @ A )
% 1.98/1.43 => ( A = emptyset ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(setbeta_type,type,
% 1.98/1.43 setbeta: $o ).
% 1.98/1.43
% 1.98/1.43 thf(setbeta,definition,
% 1.98/1.43 ( setbeta
% 1.98/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ A )
% 1.98/1.43 => ( ( in @ Xx
% 1.98/1.43 @ ( dsetconstr @ A
% 1.98/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.98/1.43 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(nonempty_type,type,
% 1.98/1.43 nonempty: $i > $o ).
% 1.98/1.43
% 1.98/1.43 thf(nonempty,definition,
% 1.98/1.43 ( nonempty
% 1.98/1.43 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(nonemptyE1_type,type,
% 1.98/1.43 nonemptyE1: $o ).
% 1.98/1.43
% 1.98/1.43 thf(nonemptyE1,definition,
% 1.98/1.43 ( nonemptyE1
% 1.98/1.43 = ( ! [A: $i] :
% 1.98/1.43 ( ( nonempty @ A )
% 1.98/1.43 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(nonemptyI_type,type,
% 1.98/1.43 nonemptyI: $o ).
% 1.98/1.43
% 1.98/1.43 thf(nonemptyI,definition,
% 1.98/1.43 ( nonemptyI
% 1.98/1.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ A )
% 1.98/1.43 => ( ( Xphi @ Xx )
% 1.98/1.43 => ( nonempty
% 1.98/1.43 @ ( dsetconstr @ A
% 1.98/1.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(nonemptyI1_type,type,
% 1.98/1.43 nonemptyI1: $o ).
% 1.98/1.43
% 1.98/1.43 thf(nonemptyI1,definition,
% 1.98/1.43 ( nonemptyI1
% 1.98/1.43 = ( ! [A: $i] :
% 1.98/1.43 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.98/1.43 => ( nonempty @ A ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(setadjoinIL_type,type,
% 1.98/1.43 setadjoinIL: $o ).
% 1.98/1.43
% 1.98/1.43 thf(setadjoinIL,definition,
% 1.98/1.43 ( setadjoinIL
% 1.98/1.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(emptyinunitempty_type,type,
% 1.98/1.43 emptyinunitempty: $o ).
% 1.98/1.43
% 1.98/1.43 thf(emptyinunitempty,definition,
% 1.98/1.43 ( emptyinunitempty
% 1.98/1.43 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(setadjoinIR_type,type,
% 1.98/1.43 setadjoinIR: $o ).
% 1.98/1.43
% 1.98/1.43 thf(setadjoinIR,definition,
% 1.98/1.43 ( setadjoinIR
% 1.98/1.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.98/1.43 ( ( in @ Xy @ A )
% 1.98/1.43 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(setadjoinE_type,type,
% 1.98/1.43 setadjoinE: $o ).
% 1.98/1.43
% 1.98/1.43 thf(setadjoinE,definition,
% 1.98/1.43 ( setadjoinE
% 1.98/1.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.98/1.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.98/1.43 => ! [Xphi: $o] :
% 1.98/1.43 ( ( ( Xy = Xx )
% 1.98/1.43 => Xphi )
% 1.98/1.43 => ( ( ( in @ Xy @ A )
% 1.98/1.43 => Xphi )
% 1.98/1.43 => Xphi ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(setadjoinOr_type,type,
% 1.98/1.43 setadjoinOr: $o ).
% 1.98/1.43
% 1.98/1.43 thf(setadjoinOr,definition,
% 1.98/1.43 ( setadjoinOr
% 1.98/1.43 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.98/1.43 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.98/1.43 => ( ( Xy = Xx )
% 1.98/1.43 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(setoftrueEq_type,type,
% 1.98/1.43 setoftrueEq: $o ).
% 1.98/1.43
% 1.98/1.43 thf(setoftrueEq,definition,
% 1.98/1.43 ( setoftrueEq
% 1.98/1.43 = ( ! [A: $i] :
% 1.98/1.43 ( ( dsetconstr @ A
% 1.98/1.43 @ ^ [Xx: $i] : $true )
% 1.98/1.43 = A ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(powersetI_type,type,
% 1.98/1.43 powersetI: $o ).
% 1.98/1.43
% 1.98/1.43 thf(powersetI,definition,
% 1.98/1.43 ( powersetI
% 1.98/1.43 = ( ! [A: $i,B: $i] :
% 1.98/1.43 ( ! [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ B )
% 1.98/1.43 => ( in @ Xx @ A ) )
% 1.98/1.43 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(emptyinPowerset_type,type,
% 1.98/1.43 emptyinPowerset: $o ).
% 1.98/1.43
% 1.98/1.43 thf(emptyinPowerset,definition,
% 1.98/1.43 ( emptyinPowerset
% 1.98/1.43 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(emptyInPowerset_type,type,
% 1.98/1.43 emptyInPowerset: $o ).
% 1.98/1.43
% 1.98/1.43 thf(emptyInPowerset,definition,
% 1.98/1.43 ( emptyInPowerset
% 1.98/1.43 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(powersetE_type,type,
% 1.98/1.43 powersetE: $o ).
% 1.98/1.43
% 1.98/1.43 thf(powersetE,definition,
% 1.98/1.43 ( powersetE
% 1.98/1.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.98/1.43 ( ( in @ B @ ( powerset @ A ) )
% 1.98/1.43 => ( ( in @ Xx @ B )
% 1.98/1.43 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(setunionI_type,type,
% 1.98/1.43 setunionI: $o ).
% 1.98/1.43
% 1.98/1.43 thf(setunionI,definition,
% 1.98/1.43 ( setunionI
% 1.98/1.43 = ( ! [A: $i,Xx: $i,B: $i] :
% 1.98/1.43 ( ( in @ Xx @ B )
% 1.98/1.43 => ( ( in @ B @ A )
% 1.98/1.43 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(setunionE_type,type,
% 1.98/1.43 setunionE: $o ).
% 1.98/1.43
% 1.98/1.43 thf(setunionE,definition,
% 1.98/1.43 ( setunionE
% 1.98/1.43 = ( ! [A: $i,Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ ( setunion @ A ) )
% 1.98/1.43 => ! [Xphi: $o] :
% 1.98/1.43 ( ! [B: $i] :
% 1.98/1.43 ( ( in @ Xx @ B )
% 1.98/1.43 => ( ( in @ B @ A )
% 1.98/1.43 => Xphi ) )
% 1.98/1.43 => Xphi ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(subPowSU_type,type,
% 1.98/1.43 subPowSU: $o ).
% 1.98/1.43
% 1.98/1.43 thf(subPowSU,definition,
% 1.98/1.43 ( subPowSU
% 1.98/1.43 = ( ! [A: $i,Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ A )
% 1.98/1.43 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(exuE2_type,type,
% 1.98/1.43 exuE2: $o ).
% 1.98/1.43
% 1.98/1.43 thf(exuE2,definition,
% 1.98/1.43 ( exuE2
% 1.98/1.43 = ( ! [Xphi: $i > $o] :
% 1.98/1.43 ( ( exu
% 1.98/1.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.98/1.43 => ? [Xx: $i] :
% 1.98/1.43 ! [Xy: $i] :
% 1.98/1.43 ( ( Xphi @ Xy )
% 1.98/1.43 <=> ( Xy = Xx ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(nonemptyImpWitness_type,type,
% 1.98/1.43 nonemptyImpWitness: $o ).
% 1.98/1.43
% 1.98/1.43 thf(nonemptyImpWitness,definition,
% 1.98/1.43 ( nonemptyImpWitness
% 1.98/1.43 = ( ! [A: $i] :
% 1.98/1.43 ( ( nonempty @ A )
% 1.98/1.43 => ? [Xx: $i] :
% 1.98/1.43 ( ( in @ Xx @ A )
% 1.98/1.43 & $true ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(uniqinunit_type,type,
% 1.98/1.43 uniqinunit: $o ).
% 1.98/1.43
% 1.98/1.43 thf(uniqinunit,definition,
% 1.98/1.43 ( uniqinunit
% 1.98/1.43 = ( ! [Xx: $i,Xy: $i] :
% 1.98/1.43 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 1.98/1.43 => ( Xx = Xy ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(notinsingleton_type,type,
% 1.98/1.43 notinsingleton: $o ).
% 1.98/1.43
% 1.98/1.43 thf(notinsingleton,definition,
% 1.98/1.43 ( notinsingleton
% 1.98/1.43 = ( ! [Xx: $i,Xy: $i] :
% 1.98/1.43 ( ( Xx != Xy )
% 1.98/1.43 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.98/1.43
% 1.98/1.43 thf(eqinunit,conjecture,
% 1.98/1.43 ( setextAx
% 1.98/1.43 => ( emptysetAx
% 1.98/1.43 => ( setadjoinAx
% 1.98/1.43 => ( powersetAx
% 1.98/1.43 => ( setunionAx
% 1.98/1.43 => ( omega0Ax
% 1.98/1.43 => ( omegaSAx
% 1.98/1.43 => ( omegaIndAx
% 1.98/1.43 => ( replAx
% 1.98/1.43 => ( foundationAx
% 1.98/1.43 => ( wellorderingAx
% 1.98/1.43 => ( descrp
% 1.98/1.43 => ( dsetconstrI
% 1.98/1.43 => ( dsetconstrEL
% 1.98/1.43 => ( dsetconstrER
% 1.98/1.43 => ( exuE1
% 1.98/1.43 => ( prop2setE
% 1.98/1.43 => ( emptysetE
% 1.98/1.43 => ( emptysetimpfalse
% 1.98/1.43 => ( notinemptyset
% 1.98/1.43 => ( exuE3e
% 1.98/1.43 => ( setext
% 1.98/1.43 => ( emptyI
% 1.98/1.43 => ( noeltsimpempty
% 1.98/1.43 => ( setbeta
% 1.98/1.43 => ( nonemptyE1
% 1.98/1.43 => ( nonemptyI
% 1.98/1.43 => ( nonemptyI1
% 1.98/1.43 => ( setadjoinIL
% 1.98/1.43 => ( emptyinunitempty
% 1.98/1.43 => ( setadjoinIR
% 1.98/1.43 => ( setadjoinE
% 1.98/1.43 => ( setadjoinOr
% 1.98/1.43 => ( setoftrueEq
% 1.98/1.43 => ( powersetI
% 1.98/1.43 => ( emptyinPowerset
% 1.98/1.43 => ( emptyInPowerset
% 1.98/1.43 => ( powersetE
% 1.98/1.43 => ( setunionI
% 1.98/1.43 => ( setunionE
% 1.98/1.43 => ( subPowSU
% 1.98/1.43 => ( exuE2
% 1.98/1.43 => ( nonemptyImpWitness
% 1.98/1.43 => ( uniqinunit
% 1.98/1.43 => ( notinsingleton
% 1.98/1.43 => ! [Xx: $i,Xy: $i] :
% 1.98/1.43 ( ( Xx = Xy )
% 1.98/1.43 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 1.98/1.43 ------------------------
% 1.98/1.45 % (31452)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_31353 for (2999ds/183Mi)
% 1.98/1.45 % (31454)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_31353 for (2999ds/27Mi)
% 1.98/1.45 % (31457)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_31353 for (2999ds/275Mi)
% 1.98/1.45 % (31455)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_31353 for (2999ds/2Mi)
% 1.98/1.45 % (31458)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_31353 for (2999ds/18Mi)
% 1.98/1.45 % (31453)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_31353 for (2999ds/4Mi)
% 1.98/1.45 % (31456)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_31353 for (2999ds/2Mi)
% 1.98/1.45 % (31455)Instruction limit reached!
% 1.98/1.45 % (31455)------------------------------
% 1.98/1.45 % (31455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.98/1.45 % (31455)Termination reason: Unknown
% 1.98/1.45 % (31455)Termination phase: shuffling
% 1.98/1.45 % (31456)Instruction limit reached!
% 1.98/1.45 % (31456)------------------------------
% 1.98/1.45 % (31456)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.98/1.45
% 1.98/1.45 % (31455)Memory used [KB]: 1023
% 1.98/1.45 % (31455)Time elapsed: 0.003 s
% 1.98/1.45 % (31455)Instructions burned: 3 (million)
% 1.98/1.45 % (31455)------------------------------
% 1.98/1.45 % (31455)------------------------------
% 1.98/1.45 % (31456)Termination reason: Unknown
% 1.98/1.45 % (31456)Termination phase: shuffling
% 1.98/1.45
% 1.98/1.45 % (31456)Memory used [KB]: 1023
% 1.98/1.45 % (31456)Time elapsed: 0.003 s
% 1.98/1.45 % (31456)Instructions burned: 2 (million)
% 1.98/1.45 % (31456)------------------------------
% 1.98/1.45 % (31456)------------------------------
% 1.98/1.45 % (31453)Instruction limit reached!
% 1.98/1.45 % (31453)------------------------------
% 1.98/1.45 % (31453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.98/1.45 % (31453)Termination reason: Unknown
% 1.98/1.45 % (31453)Termination phase: shuffling
% 1.98/1.45
% 1.98/1.45 % (31453)Memory used [KB]: 1151
% 1.98/1.45 % (31453)Time elapsed: 0.003 s
% 1.98/1.45 % (31453)Instructions burned: 4 (million)
% 1.98/1.45 % (31453)------------------------------
% 1.98/1.45 % (31453)------------------------------
% 1.98/1.46 % (31458)Instruction limit reached!
% 1.98/1.46 % (31458)------------------------------
% 1.98/1.46 % (31458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.98/1.46 % (31458)Termination reason: Unknown
% 1.98/1.46 % (31458)Termination phase: Property scanning
% 1.98/1.46
% 1.98/1.46 % (31458)Memory used [KB]: 1535
% 1.98/1.46 % (31458)Time elapsed: 0.011 s
% 1.98/1.46 % (31458)Instructions burned: 19 (million)
% 1.98/1.46 % (31458)------------------------------
% 1.98/1.46 % (31458)------------------------------
% 1.98/1.46 % (31454)Instruction limit reached!
% 1.98/1.46 % (31454)------------------------------
% 1.98/1.46 % (31454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.98/1.46 % (31454)Termination reason: Unknown
% 1.98/1.46 % (31454)Termination phase: Saturation
% 1.98/1.46
% 1.98/1.46 % (31454)Memory used [KB]: 5756
% 1.98/1.46 % (31454)Time elapsed: 0.013 s
% 1.98/1.46 % (31454)Instructions burned: 27 (million)
% 1.98/1.46 % (31454)------------------------------
% 1.98/1.46 % (31454)------------------------------
% 1.98/1.47 % (31460)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_31353 for (2999ds/37Mi)
% 1.98/1.47 % (31459)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_31353 for (2999ds/3Mi)
% 1.98/1.47 % (31457)First to succeed.
% 1.98/1.47 % (31461)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_31353 for (2999ds/15Mi)
% 1.98/1.47 % (31459)Instruction limit reached!
% 1.98/1.47 % (31459)------------------------------
% 1.98/1.47 % (31459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.98/1.47 % (31459)Termination reason: Unknown
% 1.98/1.47 % (31459)Termination phase: shuffling
% 1.98/1.47
% 1.98/1.47 % (31459)Memory used [KB]: 1023
% 1.98/1.47 % (31459)Time elapsed: 0.003 s
% 1.98/1.47 % (31459)Instructions burned: 3 (million)
% 1.98/1.47 % (31459)------------------------------
% 1.98/1.47 % (31459)------------------------------
% 1.98/1.47 % (31457)Refutation found. Thanks to Tanya!
% 1.98/1.47 % SZS status Theorem for DTF2THF_31353
% 1.98/1.47 % SZS output start Proof for DTF2THF_31353
% 1.98/1.47 thf(func_def_0, type, in: $i > $i > $o).
% 1.98/1.47 thf(func_def_1, type, exu: ($i > $o) > $o).
% 1.98/1.47 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 1.98/1.47 thf(func_def_8, type, powerset: $i > $i).
% 1.98/1.47 thf(func_def_10, type, setunion: $i > $i).
% 1.98/1.47 thf(func_def_19, type, descr: ($i > $o) > $i).
% 1.98/1.47 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 1.98/1.47 thf(func_def_26, type, prop2set: $o > $i).
% 1.98/1.47 thf(func_def_36, type, nonempty: $i > $o).
% 1.98/1.47 thf(func_def_70, type, sP1: $i > $i > $o).
% 1.98/1.47 thf(func_def_71, type, sP2: $i > $o).
% 1.98/1.47 thf(func_def_72, type, sP3: $i > $i > $o).
% 1.98/1.47 thf(func_def_73, type, sP4: $i > $i > $o).
% 1.98/1.47 thf(func_def_86, type, sK17: $i > $o).
% 1.98/1.47 thf(func_def_95, type, sK26: $i > $i).
% 1.98/1.47 thf(func_def_97, type, sK28: ($i > $o) > $i).
% 1.98/1.47 thf(func_def_98, type, sK29: $i > $o).
% 1.98/1.47 thf(func_def_99, type, sK30: $i > $i).
% 1.98/1.47 thf(func_def_102, type, sK33: $i > $o).
% 1.98/1.47 thf(func_def_106, type, sK37: $i > $i > $i).
% 1.98/1.47 thf(func_def_115, type, sK46: $i > $i > $i).
% 1.98/1.47 thf(func_def_116, type, sK47: $i > $i > $i).
% 1.98/1.47 thf(func_def_117, type, sK48: $o > $i > $i > $i).
% 1.98/1.47 thf(func_def_121, type, sK52: $i > $i).
% 1.98/1.47 thf(func_def_123, type, sK54: $i > $o).
% 1.98/1.47 thf(func_def_124, type, sK55: ($i > $o) > $i).
% 1.98/1.47 thf(func_def_125, type, sK56: $i > $o).
% 1.98/1.47 thf(func_def_133, type, sK64: $i > $i).
% 1.98/1.47 thf(func_def_137, type, sK68: $i > $i).
% 1.98/1.47 thf(func_def_141, type, sK72: $i > $i > $o).
% 1.98/1.47 thf(func_def_143, type, sK74: $i > $i).
% 1.98/1.47 thf(func_def_144, type, sK75: $i > $i).
% 1.98/1.47 thf(func_def_145, type, sK76: $i > ($i > $i > $o) > $i).
% 1.98/1.47 thf(func_def_146, type, sK77: $i > $i > ($i > $i > $o) > $i).
% 1.98/1.47 thf(func_def_147, type, sK78: $i > ($i > $i > $o) > $i).
% 1.98/1.47 thf(func_def_148, type, sK79: ($i > $o) > $i).
% 1.98/1.47 thf(func_def_149, type, sK80: $i > $o).
% 1.98/1.47 thf(func_def_150, type, sK81: $i > $i).
% 1.98/1.47 thf(func_def_157, type, sK88: $i > $i).
% 1.98/1.47 thf(func_def_159, type, sK90: $i > $i > $i).
% 1.98/1.47 thf(func_def_163, type, sK94: $i > $i > $i).
% 1.98/1.47 thf(func_def_164, type, sK95: $i > $i > $i).
% 1.98/1.47 thf(func_def_165, type, sK96: $i > $i > $i).
% 1.98/1.47 thf(func_def_166, type, sK97: $i > $i > $i).
% 1.98/1.47 thf(func_def_167, type, sK98: $i > $i > $i).
% 1.98/1.47 thf(func_def_168, type, sK99: $i > $i > $i > $i).
% 1.98/1.47 thf(func_def_169, type, sK100: $i > $i).
% 1.98/1.47 thf(func_def_170, type, sK101: $i > $i).
% 1.98/1.47 thf(func_def_171, type, sK102: $i > $i).
% 1.98/1.47 thf(func_def_172, type, sK103: $i > $i).
% 1.98/1.47 thf(func_def_173, type, sK104: $i > $i > $i).
% 1.98/1.47 thf(func_def_174, type, sK105: $i > $i > $i > $i).
% 1.98/1.47 thf(func_def_175, type, sK106: $i > $i > $i > $i).
% 1.98/1.47 thf(func_def_176, type, sK107: $i > $i > $i).
% 1.98/1.47 thf(func_def_177, type, sK108: $i > $i > $i).
% 1.98/1.47 thf(func_def_179, type, sK110: $i > $i).
% 1.98/1.47 thf(func_def_180, type, sK111: $i > $i).
% 1.98/1.47 thf(func_def_181, type, sK112: $i > $i).
% 1.98/1.47 thf(func_def_182, type, sK113: $i > $o).
% 1.98/1.47 thf(func_def_186, type, sK117: $i > $i).
% 1.98/1.47 thf(func_def_188, type, sK119: $i > $i).
% 1.98/1.47 thf(func_def_191, type, sK122: $i > $i > $i).
% 1.98/1.47 thf(func_def_196, type, sK127: $i > $i > $i).
% 1.98/1.47 thf(func_def_199, type, sK130: $i > $o).
% 1.98/1.47 thf(func_def_203, type, ph134: !>[X0: $tType]:(X0)).
% 1.98/1.47 thf(f938,plain,(
% 1.98/1.47 $false),
% 1.98/1.47 inference(subsumption_resolution,[status(thm)],[f807,f932])).
% 1.98/1.47 thf(f932,plain,(
% 1.98/1.47 ( ! [X4 : $i,X5 : $i] : (($true = (in @ X4 @ (setadjoin @ X4 @ X5)))) )),
% 1.98/1.47 inference(trivial_inequality_removal,[status(thm)],[f854])).
% 1.98/1.47 thf(f854,plain,(
% 1.98/1.47 ( ! [X4 : $i,X5 : $i] : (($true = (in @ X4 @ (setadjoin @ X4 @ X5))) | ($true != $true)) )),
% 1.98/1.47 inference(equality_resolution,[status(thm)],[f747])).
% 1.98/1.47 thf(f747,plain,(
% 1.98/1.47 ( ! [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X3 @ (setadjoin @ X4 @ X5))) | (X3 != X4) | ($true != $true)) )),
% 1.98/1.47 inference(definition_unfolding,[status(thm)],[f499,f592])).
% 1.98/1.47 thf(f592,plain,(
% 1.98/1.47 (setadjoinAx = $true)),
% 1.98/1.47 inference(cnf_transformation,[status(thm)],[f364])).
% 1.98/1.47 thf(f364,plain,(
% 1.98/1.47 (omega0Ax = $true) & (nonemptyI = $true) & (omegaIndAx = $true) & (setadjoinIR = $true) & (wellorderingAx = $true) & (setbeta = $true) & (dsetconstrEL = $true) & (setunionI = $true) & (powersetI = $true) & (exuE2 = $true) & (setadjoinIL = $true) & (descrp = $true) & (($true != (in @ sK86 @ (setadjoin @ sK87 @ emptyset))) & (sK87 = sK86)) & (emptyinunitempty = $true) & (subPowSU = $true) & (prop2setE = $true) & (emptyinPowerset = $true) & (dsetconstrER = $true) & (setoftrueEq = $true) & (emptyI = $true) & (notinsingleton = $true) & (setadjoinOr = $true) & (dsetconstrI = $true) & (setadjoinAx = $true) & (setunionE = $true) & (setunionAx = $true) & (emptysetAx = $true) & (setadjoinE = $true) & (nonemptyE1 = $true) & (omegaSAx = $true) & (exuE1 = $true) & (emptyInPowerset = $true) & (powersetAx = $true) & (foundationAx = $true) & (powersetE = $true) & (setext = $true) & (exuE3e = $true) & (replAx = $true) & (setextAx = $true) & (nonemptyI1 = $true) & (nonemptyImpWitness = $true) & (noeltsimpempty = $true) & (emptysetimpfalse = $true) & (uniqinunit = $true) & (notinemptyset = $true) & (emptysetE = $true)),
% 1.98/1.47 inference(skolemisation,[status(esa),new_symbols(skolem,[sK86,sK87])],[f178,f363])).
% 1.98/1.47 thf(f363,plain,(
% 1.98/1.47 ? [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (setadjoin @ X1 @ emptyset))) & (X0 = X1)) => (($true != (in @ sK86 @ (setadjoin @ sK87 @ emptyset))) & (sK87 = sK86))),
% 1.98/1.47 introduced(definition,[],[choice_axiom])).
% 1.98/1.47 thf(f178,plain,(
% 1.98/1.47 (omega0Ax = $true) & (nonemptyI = $true) & (omegaIndAx = $true) & (setadjoinIR = $true) & (wellorderingAx = $true) & (setbeta = $true) & (dsetconstrEL = $true) & (setunionI = $true) & (powersetI = $true) & (exuE2 = $true) & (setadjoinIL = $true) & (descrp = $true) & ? [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (setadjoin @ X1 @ emptyset))) & (X0 = X1)) & (emptyinunitempty = $true) & (subPowSU = $true) & (prop2setE = $true) & (emptyinPowerset = $true) & (dsetconstrER = $true) & (setoftrueEq = $true) & (emptyI = $true) & (notinsingleton = $true) & (setadjoinOr = $true) & (dsetconstrI = $true) & (setadjoinAx = $true) & (setunionE = $true) & (setunionAx = $true) & (emptysetAx = $true) & (setadjoinE = $true) & (nonemptyE1 = $true) & (omegaSAx = $true) & (exuE1 = $true) & (emptyInPowerset = $true) & (powersetAx = $true) & (foundationAx = $true) & (powersetE = $true) & (setext = $true) & (exuE3e = $true) & (replAx = $true) & (setextAx = $true) & (nonemptyI1 = $true) & (nonemptyImpWitness = $true) & (noeltsimpempty = $true) & (emptysetimpfalse = $true) & (uniqinunit = $true) & (notinemptyset = $true) & (emptysetE = $true)),
% 1.98/1.47 inference(flattening,[status(thm)],[f177])).
% 1.98/1.47 thf(f177,plain,(
% 1.98/1.47 ((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (setadjoin @ X1 @ emptyset))) & (X0 = X1)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 1.98/1.47 inference(ennf_transformation,[status(thm)],[f53])).
% 1.98/1.47 thf(f53,plain,(
% 1.98/1.47 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ! [X0 : $i,X1 : $i] : ((X0 = X1) => ($true = (in @ X0 @ (setadjoin @ X1 @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))),
% 1.98/1.47 inference(fool_elimination,[status(thm)],[f52])).
% 1.98/1.47 thf(f52,plain,(
% 1.98/1.47 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (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 => ! [X0 : $i,X1 : $i] : ((X0 = X1) => (in @ X0 @ (setadjoin @ X1 @ emptyset))))))))))))))))))))))))))))))))))))))))))))))))),
% 1.98/1.47 inference(rectify,[status(thm)],[f50])).
% 1.98/1.47 thf(f50,negated_conjecture,(
% 1.98/1.47 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (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))))))))))))))))))))))))))))))))))))))))))))))))),
% 1.98/1.47 inference(negated_conjecture,[status(cth)],[f49])).
% 1.98/1.47 thf(f49,conjecture,(
% 1.98/1.47 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (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)))))))))))))))))))))))))))))))))))))))))))))))),
% 1.98/1.47 file('/export/starexec/sandbox/tmp/tmp.ryFqNTYIaR/DTF2THF_31353.p',eqinunit)).
% 1.98/1.47 thf(f499,plain,(
% 1.98/1.47 ( ! [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X3 @ (setadjoin @ X4 @ X5))) | (X3 != X4) | (setadjoinAx != $true)) )),
% 1.98/1.47 inference(cnf_transformation,[status(thm)],[f283])).
% 1.98/1.47 thf(f283,plain,(
% 1.98/1.47 ((setadjoinAx = $true) | ((((sK42 != sK41) & ((in @ sK41 @ sK43) != $true)) | ((in @ sK41 @ (setadjoin @ sK42 @ sK43)) != $true)) & ((sK42 = sK41) | ((in @ sK41 @ sK43) = $true) | ((in @ sK41 @ (setadjoin @ sK42 @ sK43)) = $true)))) & (! [X3 : $i,X4 : $i,X5 : $i] : ((($true = (in @ X3 @ (setadjoin @ X4 @ X5))) | ((X3 != X4) & ($true != (in @ X3 @ X5)))) & ((X3 = X4) | ($true = (in @ X3 @ X5)) | ($true != (in @ X3 @ (setadjoin @ X4 @ X5))))) | (setadjoinAx != $true))),
% 1.98/1.47 inference(skolemisation,[status(esa),new_symbols(skolem,[sK41,sK42,sK43])],[f281,f282])).
% 1.98/1.47 thf(f282,plain,(
% 1.98/1.47 ? [X0 : $i,X1 : $i,X2 : $i] : ((((X0 != X1) & ($true != (in @ X0 @ X2))) | ($true != (in @ X0 @ (setadjoin @ X1 @ X2)))) & ((X0 = X1) | ($true = (in @ X0 @ X2)) | ($true = (in @ X0 @ (setadjoin @ X1 @ X2))))) => ((((sK42 != sK41) & ((in @ sK41 @ sK43) != $true)) | ((in @ sK41 @ (setadjoin @ sK42 @ sK43)) != $true)) & ((sK42 = sK41) | ((in @ sK41 @ sK43) = $true) | ((in @ sK41 @ (setadjoin @ sK42 @ sK43)) = $true)))),
% 1.98/1.47 introduced(definition,[],[choice_axiom])).
% 1.98/1.47 thf(f281,plain,(
% 1.98/1.47 ((setadjoinAx = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : ((((X0 != X1) & ($true != (in @ X0 @ X2))) | ($true != (in @ X0 @ (setadjoin @ X1 @ X2)))) & ((X0 = X1) | ($true = (in @ X0 @ X2)) | ($true = (in @ X0 @ (setadjoin @ X1 @ X2)))))) & (! [X3 : $i,X4 : $i,X5 : $i] : ((($true = (in @ X3 @ (setadjoin @ X4 @ X5))) | ((X3 != X4) & ($true != (in @ X3 @ X5)))) & ((X3 = X4) | ($true = (in @ X3 @ X5)) | ($true != (in @ X3 @ (setadjoin @ X4 @ X5))))) | (setadjoinAx != $true))),
% 1.98/1.47 inference(rectify,[status(thm)],[f280])).
% 1.98/1.47 thf(f280,plain,(
% 1.98/1.47 ((setadjoinAx = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : ((((X0 != X1) & ($true != (in @ X0 @ X2))) | ($true != (in @ X0 @ (setadjoin @ X1 @ X2)))) & ((X0 = X1) | ($true = (in @ X0 @ X2)) | ($true = (in @ X0 @ (setadjoin @ X1 @ X2)))))) & (! [X0 : $i,X1 : $i,X2 : $i] : ((($true = (in @ X0 @ (setadjoin @ X1 @ X2))) | ((X0 != X1) & ($true != (in @ X0 @ X2)))) & ((X0 = X1) | ($true = (in @ X0 @ X2)) | ($true != (in @ X0 @ (setadjoin @ X1 @ X2))))) | (setadjoinAx != $true))),
% 1.98/1.47 inference(flattening,[status(thm)],[f279])).
% 1.98/1.47 thf(f279,plain,(
% 1.98/1.47 ((setadjoinAx = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : ((((X0 != X1) & ($true != (in @ X0 @ X2))) | ($true != (in @ X0 @ (setadjoin @ X1 @ X2)))) & (((X0 = X1) | ($true = (in @ X0 @ X2))) | ($true = (in @ X0 @ (setadjoin @ X1 @ X2)))))) & (! [X0 : $i,X1 : $i,X2 : $i] : ((($true = (in @ X0 @ (setadjoin @ X1 @ X2))) | ((X0 != X1) & ($true != (in @ X0 @ X2)))) & (((X0 = X1) | ($true = (in @ X0 @ X2))) | ($true != (in @ X0 @ (setadjoin @ X1 @ X2))))) | (setadjoinAx != $true))),
% 1.98/1.47 inference(nnf_transformation,[status(thm)],[f85])).
% 1.98/1.47 thf(f85,plain,(
% 1.98/1.47 (setadjoinAx = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (setadjoin @ X1 @ X2))) <=> ((X0 = X1) | ($true = (in @ X0 @ X2))))),
% 1.98/1.47 inference(fool_elimination,[status(thm)],[f84])).
% 1.98/1.47 thf(f84,plain,(
% 1.98/1.47 (setadjoinAx = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ (setadjoin @ X1 @ X2)) <=> ((X0 = X1) | (in @ X0 @ X2))))),
% 1.98/1.47 inference(rectify,[status(thm)],[f4])).
% 1.98/1.47 thf(f4,axiom,(
% 1.98/1.47 (setadjoinAx = ! [X2 : $i,X1 : $i,X3 : $i] : ((in @ X2 @ (setadjoin @ X1 @ X3)) <=> ((X1 = X2) | (in @ X2 @ X3))))),
% 1.98/1.47 file('/export/starexec/sandbox/tmp/tmp.ryFqNTYIaR/DTF2THF_31353.p',setadjoinAx)).
% 1.98/1.47 thf(f807,plain,(
% 1.98/1.47 ($true != (in @ sK86 @ (setadjoin @ sK86 @ emptyset)))),
% 1.98/1.47 inference(definition_unfolding,[status(thm)],[f604,f603])).
% 1.98/1.47 thf(f603,plain,(
% 1.98/1.47 (sK87 = sK86)),
% 1.98/1.47 inference(cnf_transformation,[status(thm)],[f364])).
% 1.98/1.47 thf(f604,plain,(
% 1.98/1.47 ($true != (in @ sK86 @ (setadjoin @ sK87 @ emptyset)))),
% 1.98/1.47 inference(cnf_transformation,[status(thm)],[f364])).
% 1.98/1.47 % SZS output end Proof for DTF2THF_31353
% 1.98/1.47 % (31457)------------------------------
% 1.98/1.47 % (31457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.98/1.47 % (31457)Termination reason: Refutation
% 1.98/1.47
% 1.98/1.47 % (31457)Memory used [KB]: 6268
% 1.98/1.47 % (31457)Time elapsed: 0.021 s
% 1.98/1.47 % (31457)Instructions burned: 38 (million)
% 1.98/1.47 % (31457)------------------------------
% 1.98/1.47 % (31457)------------------------------
% 1.98/1.47 % (31451)Success in time 0.036 s
%------------------------------------------------------------------------------