%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU533^1 : 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 09:35:08 AM UTC 2026
% Result : Theorem 2.04s 1.48s
% Output : Refutation 2.04s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 4
% Syntax : Number of formulae : 28 ( 7 unt; 0 typ; 0 def)
% Number of atoms : 1056 ( 300 equ; 0 cnn)
% Maximal formula atoms : 51 ( 37 avg)
% Number of connectives : 721 ( 56 ~; 52 |; 171 &; 240 @)
% ( 4 <=>; 198 =>; 0 <=; 0 <~>)
% Maximal formula depth : 57 ( 17 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 123 ( 120 usr; 57 con; 0-3 aty)
% Number of variables : 66 ( 0 ^; 48 !; 18 ?; 66 :)
% 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_74,type,
sP1: $i > $i > $o ).
thf(func_def_75,type,
sP2: $i > $o ).
thf(func_def_76,type,
sP3: $i > $i > $o ).
thf(func_def_77,type,
sP4: $i > $i > $o ).
thf(func_def_79,type,
sK6: $i > $i ).
thf(func_def_83,type,
sK10: $i > $i > $o ).
thf(func_def_85,type,
sK12: $i > $i ).
thf(func_def_86,type,
sK13: $i > $i ).
thf(func_def_87,type,
sK14: $i > ( $i > $i > $o ) > $i ).
thf(func_def_88,type,
sK15: $i > ( $i > $i > $o ) > $i ).
thf(func_def_89,type,
sK16: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_90,type,
sK17: $i > $o ).
thf(func_def_100,type,
sK27: $i > $i > $i ).
thf(func_def_104,type,
sK31: $i > $o ).
thf(func_def_106,type,
sK33: $i > $o ).
thf(func_def_109,type,
sK36: $i > $i ).
thf(func_def_111,type,
sK38: $i > $i ).
thf(func_def_113,type,
sK40: $i > $i ).
thf(func_def_115,type,
sK42: ( $i > $o ) > $i ).
thf(func_def_116,type,
sK43: $i > $o ).
thf(func_def_117,type,
sK44: $i > $i ).
thf(func_def_120,type,
sK47: $i > $i > $i ).
thf(func_def_121,type,
sK48: $i > $i > $i ).
thf(func_def_122,type,
sK49: $i > $i ).
thf(func_def_126,type,
sK53: $i > $o ).
thf(func_def_138,type,
sK65: $i > $i > $i ).
thf(func_def_139,type,
sK66: $i > $i > $i ).
thf(func_def_140,type,
sK67: $i > $i > $i ).
thf(func_def_141,type,
sK68: $i > $i > $i ).
thf(func_def_142,type,
sK69: $i > $i > $i > $i ).
thf(func_def_143,type,
sK70: $i > $i ).
thf(func_def_144,type,
sK71: $i > $i ).
thf(func_def_145,type,
sK72: $i > $i ).
thf(func_def_146,type,
sK73: $i > $i ).
thf(func_def_147,type,
sK74: $i > $i > $i ).
thf(func_def_148,type,
sK75: $i > $i > $i > $i ).
thf(func_def_149,type,
sK76: $i > $i > $i > $i ).
thf(func_def_150,type,
sK77: $i > $i > $i ).
thf(func_def_151,type,
sK78: $i > $i > $i ).
thf(func_def_152,type,
sK79: $i > $i ).
thf(func_def_153,type,
sK80: $i > $i > $i ).
thf(func_def_155,type,
sK82: $i > $i ).
thf(func_def_156,type,
sK83: $i > $i ).
thf(func_def_157,type,
sK84: $o > $i > $i > $i ).
thf(func_def_163,type,
sK90: $i > $i ).
thf(func_def_167,type,
sK94: $i > $i > $i ).
thf(func_def_173,type,
sK100: $i > $o ).
thf(func_def_174,type,
sK101: ( $i > $o ) > $i ).
thf(func_def_176,type,
sK103: $i > $i ).
thf(func_def_183,type,
sK110: $i > $o ).
thf(func_def_184,type,
sK111: $i > $i ).
thf(func_def_185,type,
sK112: ( $i > $o ) > $i ).
thf(func_def_188,type,
sK115: $i > $i > $i ).
thf(func_def_197,type,
sK124: $i > $i > $i ).
thf(func_def_204,type,
sK131: $i > $o ).
thf(func_def_216,type,
ph143:
!>[X0: $tType] : X0 ).
thf(f1354,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1349,f995]) ).
thf(f995,plain,
! [X4: $i,X5: $i] :
( $true
= ( in @ X4 @ ( setadjoin @ X4 @ X5 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f912]) ).
thf(f912,plain,
! [X4: $i,X5: $i] :
( ( $true != $true )
| ( $true
= ( in @ X4 @ ( setadjoin @ X4 @ X5 ) ) ) ),
inference(equality_resolution,[status(thm)],[f844]) ).
thf(f844,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X4 @ ( setadjoin @ X3 @ X5 ) ) )
| ( X3 != X4 )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f618,f656]) ).
thf(f656,plain,
setadjoinAx = $true,
inference(cnf_transformation,[status(thm)],[f394]) ).
thf(f394,plain,
( ( descrp = $true )
& ( exuE3e = $true )
& ( omegaSAx = $true )
& ( emptysetE = $true )
& ( nonemptyI1 = $true )
& ( setextAx = $true )
& ( subPowSU = $true )
& ( setadjoinIR = $true )
& ( emptysetimpfalse = $true )
& ( emptysetAx = $true )
& ( setunionAx = $true )
& ( wellorderingAx = $true )
& ( emptyinPowerset = $true )
& ( setadjoinE = $true )
& ( noeltsimpempty = $true )
& ( nonemptyImpWitness = $true )
& ( setadjoinAx = $true )
& ( dsetconstrI = $true )
& ( eqinunit = $true )
& ( powersetI = $true )
& ( exuE2 = $true )
& ( setbeta = $true )
& ( setext = $true )
& ( upairsetIL = $true )
& ( dsetconstrER = $true )
& ( setadjoinIL = $true )
& ( setunionE = $true )
& ( prop2setE = $true )
& ( nonemptyI = $true )
& ( setoftrueEq = $true )
& ( foundationAx = $true )
& ( notinsingleton = $true )
& ( replAx = $true )
& ( emptyinunitempty = $true )
& ( setadjoinOr = $true )
& ( omega0Ax = $true )
& ( omegaIndAx = $true )
& ( upairsetE = $true )
& ( exuE1 = $true )
& ( singletonsswitch = $true )
& ( dsetconstrEL = $true )
& ( emptyI = $true )
& ( powersetAx = $true )
& ( powersetE = $true )
& ( notinemptyset = $true )
& ( nonemptyE1 = $true )
& ( uniqinunit = $true )
& ( $true
!= ( in @ sK99 @ ( setadjoin @ sK98 @ ( setadjoin @ sK99 @ emptyset ) ) ) )
& ( setunionI = $true )
& ( emptyInPowerset = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK98,sK99])],[f204,f393]) ).
thf(f393,plain,
( ? [X0: $i,X1: $i] :
( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) )
=> ( $true
!= ( in @ sK99 @ ( setadjoin @ sK98 @ ( setadjoin @ sK99 @ emptyset ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f204,plain,
( ( descrp = $true )
& ( exuE3e = $true )
& ( omegaSAx = $true )
& ( emptysetE = $true )
& ( nonemptyI1 = $true )
& ( setextAx = $true )
& ( subPowSU = $true )
& ( setadjoinIR = $true )
& ( emptysetimpfalse = $true )
& ( emptysetAx = $true )
& ( setunionAx = $true )
& ( wellorderingAx = $true )
& ( emptyinPowerset = $true )
& ( setadjoinE = $true )
& ( noeltsimpempty = $true )
& ( nonemptyImpWitness = $true )
& ( setadjoinAx = $true )
& ( dsetconstrI = $true )
& ( eqinunit = $true )
& ( powersetI = $true )
& ( exuE2 = $true )
& ( setbeta = $true )
& ( setext = $true )
& ( upairsetIL = $true )
& ( dsetconstrER = $true )
& ( setadjoinIL = $true )
& ( setunionE = $true )
& ( prop2setE = $true )
& ( nonemptyI = $true )
& ( setoftrueEq = $true )
& ( foundationAx = $true )
& ( notinsingleton = $true )
& ( replAx = $true )
& ( emptyinunitempty = $true )
& ( setadjoinOr = $true )
& ( omega0Ax = $true )
& ( omegaIndAx = $true )
& ( upairsetE = $true )
& ( exuE1 = $true )
& ( singletonsswitch = $true )
& ( dsetconstrEL = $true )
& ( emptyI = $true )
& ( powersetAx = $true )
& ( powersetE = $true )
& ( notinemptyset = $true )
& ( nonemptyE1 = $true )
& ( uniqinunit = $true )
& ? [X0: $i,X1: $i] :
( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) )
& ( setunionI = $true )
& ( emptyInPowerset = $true ) ),
inference(flattening,[status(thm)],[f203]) ).
thf(f203,plain,
( ? [X0: $i,X1: $i] :
( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( 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)],[f86]) ).
thf(f86,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 )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f85]) ).
thf(f85,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
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ! [X0: $i,X1: $i] : ( in @ X1 @ ( setadjoin @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f54]) ).
thf(f54,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
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f53]) ).
thf(f53,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
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ! [X1: $i,X2: $i] : ( in @ X2 @ ( setadjoin @ X1 @ ( setadjoin @ X2 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.kOEAvEHP3t/DTF2THF_32675.p',upairsetIR) ).
thf(f618,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X4 @ ( setadjoin @ X3 @ X5 ) ) )
| ( X3 != X4 )
| ( setadjoinAx != $true ) ),
inference(cnf_transformation,[status(thm)],[f392]) ).
thf(f392,plain,
( ( ( setadjoinAx = $true )
| ( ( ( ( $true
!= ( in @ sK96 @ sK97 ) )
& ( sK96 != sK95 ) )
| ( $true
!= ( in @ sK96 @ ( setadjoin @ sK95 @ sK97 ) ) ) )
& ( ( $true
= ( in @ sK96 @ sK97 ) )
| ( sK96 = sK95 )
| ( $true
= ( in @ sK96 @ ( setadjoin @ sK95 @ sK97 ) ) ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( $true
= ( in @ X4 @ ( setadjoin @ X3 @ X5 ) ) )
| ( ( $true
!= ( in @ X4 @ X5 ) )
& ( X3 != X4 ) ) )
& ( ( $true
= ( in @ X4 @ X5 ) )
| ( X3 = X4 )
| ( $true
!= ( in @ X4 @ ( setadjoin @ X3 @ X5 ) ) ) ) )
| ( setadjoinAx != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK95,sK96,sK97])],[f390,f391]) ).
thf(f391,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( ( $true
!= ( in @ X1 @ X2 ) )
& ( X0 != X1 ) )
| ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ X2 ) ) ) )
& ( ( $true
= ( in @ X1 @ X2 ) )
| ( X0 = X1 )
| ( $true
= ( in @ X1 @ ( setadjoin @ X0 @ X2 ) ) ) ) )
=> ( ( ( ( $true
!= ( in @ sK96 @ sK97 ) )
& ( sK96 != sK95 ) )
| ( $true
!= ( in @ sK96 @ ( setadjoin @ sK95 @ sK97 ) ) ) )
& ( ( $true
= ( in @ sK96 @ sK97 ) )
| ( sK96 = sK95 )
| ( $true
= ( in @ sK96 @ ( setadjoin @ sK95 @ sK97 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f390,plain,
( ( ( setadjoinAx = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( ( $true
!= ( in @ X1 @ X2 ) )
& ( X0 != X1 ) )
| ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ X2 ) ) ) )
& ( ( $true
= ( in @ X1 @ X2 ) )
| ( X0 = X1 )
| ( $true
= ( in @ X1 @ ( setadjoin @ X0 @ X2 ) ) ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( $true
= ( in @ X4 @ ( setadjoin @ X3 @ X5 ) ) )
| ( ( $true
!= ( in @ X4 @ X5 ) )
& ( X3 != X4 ) ) )
& ( ( $true
= ( in @ X4 @ X5 ) )
| ( X3 = X4 )
| ( $true
!= ( in @ X4 @ ( setadjoin @ X3 @ X5 ) ) ) ) )
| ( setadjoinAx != $true ) ) ),
inference(rectify,[status(thm)],[f389]) ).
thf(f389,plain,
( ( ( setadjoinAx = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( ( $true
!= ( in @ X1 @ X2 ) )
& ( X0 != X1 ) )
| ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ X2 ) ) ) )
& ( ( $true
= ( in @ X1 @ X2 ) )
| ( X0 = X1 )
| ( $true
= ( in @ X1 @ ( setadjoin @ X0 @ X2 ) ) ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( $true
= ( in @ X1 @ ( setadjoin @ X0 @ X2 ) ) )
| ( ( $true
!= ( in @ X1 @ X2 ) )
& ( X0 != X1 ) ) )
& ( ( $true
= ( in @ X1 @ X2 ) )
| ( X0 = X1 )
| ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ X2 ) ) ) ) )
| ( setadjoinAx != $true ) ) ),
inference(flattening,[status(thm)],[f388]) ).
thf(f388,plain,
( ( ( setadjoinAx = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( ( $true
!= ( in @ X1 @ X2 ) )
& ( X0 != X1 ) )
| ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ X2 ) ) ) )
& ( ( $true
= ( in @ X1 @ X2 ) )
| ( X0 = X1 )
| ( $true
= ( in @ X1 @ ( setadjoin @ X0 @ X2 ) ) ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( $true
= ( in @ X1 @ ( setadjoin @ X0 @ X2 ) ) )
| ( ( $true
!= ( in @ X1 @ X2 ) )
& ( X0 != X1 ) ) )
& ( ( $true
= ( in @ X1 @ X2 ) )
| ( X0 = X1 )
| ( $true
!= ( in @ X1 @ ( setadjoin @ X0 @ X2 ) ) ) ) )
| ( setadjoinAx != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f110]) ).
thf(f110,plain,
( ( setadjoinAx = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ ( setadjoin @ X0 @ X2 ) ) )
<=> ( ( $true
= ( in @ X1 @ X2 ) )
| ( X0 = X1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f109]) ).
thf(f109,plain,
( setadjoinAx
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X0 @ X2 ) )
<=> ( ( in @ X1 @ X2 )
| ( X0 = X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f4]) ).
thf(f4,axiom,
( setadjoinAx
= ( ! [X1: $i,X2: $i,X3: $i] :
( ( in @ X2 @ ( setadjoin @ X1 @ X3 ) )
<=> ( ( in @ X2 @ X3 )
| ( X1 = X2 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.kOEAvEHP3t/DTF2THF_32675.p',setadjoinAx) ).
thf(f1349,plain,
( $true
!= ( in @ sK99 @ ( setadjoin @ sK99 @ emptyset ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f1347]) ).
thf(f1347,plain,
( ( $true
!= ( in @ sK99 @ ( setadjoin @ sK99 @ emptyset ) ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f625,f966]) ).
thf(f966,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X4 @ ( setadjoin @ X3 @ X5 ) ) )
| ( $true
!= ( in @ X4 @ X5 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f843]) ).
thf(f843,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( $true != $true )
| ( $true
!= ( in @ X4 @ X5 ) )
| ( $true
= ( in @ X4 @ ( setadjoin @ X3 @ X5 ) ) ) ),
inference(definition_unfolding,[status(thm)],[f619,f656]) ).
thf(f619,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X4 @ ( setadjoin @ X3 @ X5 ) ) )
| ( $true
!= ( in @ X4 @ X5 ) )
| ( setadjoinAx != $true ) ),
inference(cnf_transformation,[status(thm)],[f392]) ).
thf(f625,plain,
( $true
!= ( in @ sK99 @ ( setadjoin @ sK98 @ ( setadjoin @ sK99 @ emptyset ) ) ) ),
inference(cnf_transformation,[status(thm)],[f394]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU533^1 : TPTP v9.2.1. Released v3.7.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n029.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:26:14 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.46 thf(spec,logic,$$dhol).
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % File : SEU533^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.46 % Domain : Set Theory
% 0.19/0.46 % Problem : Preliminary Notions - Equality Laws
% 0.19/0.46 % Version : Especial.
% 0.19/0.46 % English : (! x:i.! y:i.in y (setadjoin x (setadjoin y emptyset)))
% 0.19/0.46
% 0.19/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.46 % Source : [Bro08]
% 0.19/0.46 % Names : ZFC035g [Bro08]
% 0.19/0.46
% 0.19/0.46 % Status : Theorem
% 0.19/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 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, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v3.7.0
% 0.19/0.46 % Syntax : Number of formulae : 113 ( 52 unt; 60 typ; 52 def)
% 0.19/0.46 % Number of atoms : 287 ( 70 equ; 0 cnn)
% 0.19/0.46 % Maximal formula atoms : 50 ( 5 avg)
% 0.19/0.46 % Number of connectives : 468 ( 9 ~; 5 |; 19 &; 309 @)
% 0.19/0.46 % ( 8 <=>; 118 =>; 0 <=; 0 <~>)
% 0.19/0.46 % Maximal formula depth : 56 ( 2 avg)
% 0.19/0.46 % Number of types : 2 ( 0 usr)
% 0.19/0.46 % Number of type conns : 27 ( 27 >; 0 *; 0 +; 0 <<)
% 0.19/0.46 % Number of symbols : 63 ( 60 usr; 53 con; 0-2 aty)
% 0.19/0.46 % Number of variables : 153 ( 16 ^; 119 !; 18 ?; 153 :)
% 0.19/0.46 % SPC : TH0_THM_EQU_NAR
% 0.19/0.46
% 0.19/0.46 % Comments : http://mathgate.info/detsetitem.php?id=506
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 thf(in_type,type,
% 0.19/0.46 in: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(exu_type,type,
% 0.19/0.46 exu: ( $i > $o ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(exu,definition,
% 0.19/0.46 ( exu
% 0.19/0.46 = ( ^ [Xphi: $i > $o] :
% 0.19/0.46 ? [Xx: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 & ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setextAx_type,type,
% 0.19/0.46 setextAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setextAx,definition,
% 0.19/0.46 ( setextAx
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 <=> ( in @ Xx @ B ) )
% 0.19/0.46 => ( A = B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyset_type,type,
% 0.19/0.46 emptyset: $i ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetAx_type,type,
% 0.19/0.46 emptysetAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetAx,definition,
% 0.19/0.46 ( emptysetAx
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoin_type,type,
% 0.19/0.46 setadjoin: $i > $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinAx_type,type,
% 0.19/0.46 setadjoinAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinAx,definition,
% 0.19/0.46 ( setadjoinAx
% 0.19/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.46 <=> ( ( Xy = Xx )
% 0.19/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powerset_type,type,
% 0.19/0.46 powerset: $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(powersetAx_type,type,
% 0.19/0.46 powersetAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetAx,definition,
% 0.19/0.46 ( powersetAx
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.46 <=> ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunion_type,type,
% 0.19/0.46 setunion: $i > $i ).
% 0.19/0.46
% 0.19/0.46 thf(setunionAx_type,type,
% 0.19/0.46 setunionAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionAx,definition,
% 0.19/0.46 ( setunionAx
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.46 <=> ? [B: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(omega_type,type,
% 0.19/0.46 omega: $i ).
% 0.19/0.46
% 0.19/0.46 thf(omega0Ax_type,type,
% 0.19/0.46 omega0Ax: $o ).
% 0.19/0.46
% 0.19/0.46 thf(omega0Ax,definition,
% 0.19/0.46 ( omega0Ax
% 0.19/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(omegaSAx_type,type,
% 0.19/0.46 omegaSAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(omegaSAx,definition,
% 0.19/0.46 ( omegaSAx
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ omega )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(omegaIndAx_type,type,
% 0.19/0.46 omegaIndAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(omegaIndAx,definition,
% 0.19/0.46 ( omegaIndAx
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ( ( in @ emptyset @ A )
% 0.19/0.46 & ! [Xx: $i] :
% 0.19/0.46 ( ( ( in @ Xx @ omega )
% 0.19/0.46 & ( in @ Xx @ A ) )
% 0.19/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ omega )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(replAx_type,type,
% 0.19/0.46 replAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(replAx,definition,
% 0.19/0.46 ( replAx
% 0.19/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( exu
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.46 => ? [B: $i] :
% 0.19/0.46 ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 <=> ? [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(foundationAx_type,type,
% 0.19/0.46 foundationAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(foundationAx,definition,
% 0.19/0.46 ( foundationAx
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.46 => ? [B: $i] :
% 0.19/0.46 ( ( in @ B @ A )
% 0.19/0.46 & ~ ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(wellorderingAx_type,type,
% 0.19/0.46 wellorderingAx: $o ).
% 0.19/0.46
% 0.19/0.46 thf(wellorderingAx,definition,
% 0.19/0.46 ( wellorderingAx
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ? [B: $i] :
% 0.19/0.46 ( ! [C: $i] :
% 0.19/0.46 ( ( in @ C @ B )
% 0.19/0.46 => ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ C )
% 0.19/0.46 => ( in @ Xx @ A ) ) )
% 0.19/0.46 & ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( ( in @ Xx @ A )
% 0.19/0.46 & ( in @ Xy @ A ) )
% 0.19/0.46 => ( ! [C: $i] :
% 0.19/0.46 ( ( in @ C @ B )
% 0.19/0.46 => ( ( in @ Xx @ C )
% 0.19/0.46 <=> ( in @ Xy @ C ) ) )
% 0.19/0.46 => ( Xx = Xy ) ) )
% 0.19/0.46 & ! [C: $i,D: $i] :
% 0.19/0.46 ( ( ( in @ C @ B )
% 0.19/0.46 & ( in @ D @ B ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ C )
% 0.19/0.46 => ( in @ Xx @ D ) )
% 0.19/0.46 | ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ D )
% 0.19/0.46 => ( in @ Xx @ C ) ) ) )
% 0.19/0.46 & ! [C: $i] :
% 0.19/0.46 ( ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ C )
% 0.19/0.46 => ( in @ Xx @ A ) )
% 0.19/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.46 => ? [D: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ D @ B )
% 0.19/0.46 & ( in @ Xx @ C )
% 0.19/0.46 & ~ ? [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ D )
% 0.19/0.46 & ( in @ Xy @ C ) )
% 0.19/0.46 & ! [E: $i] :
% 0.19/0.46 ( ( in @ E @ B )
% 0.19/0.46 => ( ! [Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ E )
% 0.19/0.46 => ( in @ Xy @ D ) )
% 0.19/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(descr_type,type,
% 0.19/0.46 descr: ( $i > $o ) > $i ).
% 0.19/0.46
% 0.19/0.46 thf(descrp_type,type,
% 0.19/0.46 descrp: $o ).
% 0.19/0.46
% 0.19/0.46 thf(descrp,definition,
% 0.19/0.46 ( descrp
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ( Xphi
% 0.19/0.46 @ ( descr
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstr_type,type,
% 0.19/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrI_type,type,
% 0.19/0.46 dsetconstrI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrI,definition,
% 0.19/0.46 ( dsetconstrI
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrEL_type,type,
% 0.19/0.46 dsetconstrEL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrEL,definition,
% 0.19/0.46 ( dsetconstrEL
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrER_type,type,
% 0.19/0.46 dsetconstrER: $o ).
% 0.19/0.46
% 0.19/0.46 thf(dsetconstrER,definition,
% 0.19/0.46 ( dsetconstrER
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuE1_type,type,
% 0.19/0.46 exuE1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuE1,definition,
% 0.19/0.46 ( exuE1
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 & ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(prop2set_type,type,
% 0.19/0.46 prop2set: $o > $i ).
% 0.19/0.46
% 0.19/0.46 thf(prop2set,definition,
% 0.19/0.46 ( prop2set
% 0.19/0.46 = ( ^ [Xphi: $o] :
% 0.19/0.46 ( dsetconstr @ ( powerset @ emptyset )
% 0.19/0.46 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(prop2setE_type,type,
% 0.19/0.46 prop2setE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(prop2setE,definition,
% 0.19/0.46 ( prop2setE
% 0.19/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.46 => Xphi ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetE_type,type,
% 0.19/0.46 emptysetE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetE,definition,
% 0.19/0.46 ( emptysetE
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ emptyset )
% 0.19/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetimpfalse_type,type,
% 0.19/0.46 emptysetimpfalse: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptysetimpfalse,definition,
% 0.19/0.46 ( emptysetimpfalse
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ emptyset )
% 0.19/0.46 => $false ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notinemptyset_type,type,
% 0.19/0.46 notinemptyset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notinemptyset,definition,
% 0.19/0.46 ( notinemptyset
% 0.19/0.46 = ( ! [Xx: $i] :
% 0.19/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuE3e_type,type,
% 0.19/0.46 exuE3e: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuE3e,definition,
% 0.19/0.46 ( exuE3e
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setext_type,type,
% 0.19/0.46 setext: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setext,definition,
% 0.19/0.46 ( setext
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ B ) )
% 0.19/0.46 => ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ A ) )
% 0.19/0.46 => ( A = B ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyI_type,type,
% 0.19/0.46 emptyI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyI,definition,
% 0.19/0.46 ( emptyI
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ~ ( in @ Xx @ A )
% 0.19/0.46 => ( A = emptyset ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(noeltsimpempty_type,type,
% 0.19/0.46 noeltsimpempty: $o ).
% 0.19/0.46
% 0.19/0.46 thf(noeltsimpempty,definition,
% 0.19/0.46 ( noeltsimpempty
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ~ ( in @ Xx @ A )
% 0.19/0.46 => ( A = emptyset ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setbeta_type,type,
% 0.19/0.46 setbeta: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setbeta,definition,
% 0.19/0.46 ( setbeta
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( in @ Xx
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(nonempty_type,type,
% 0.19/0.46 nonempty: $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(nonempty,definition,
% 0.19/0.46 ( nonempty
% 0.19/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyE1_type,type,
% 0.19/0.46 nonemptyE1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyE1,definition,
% 0.19/0.46 ( nonemptyE1
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ( nonempty @ A )
% 0.19/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyI_type,type,
% 0.19/0.46 nonemptyI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyI,definition,
% 0.19/0.46 ( nonemptyI
% 0.19/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( ( Xphi @ Xx )
% 0.19/0.46 => ( nonempty
% 0.19/0.46 @ ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyI1_type,type,
% 0.19/0.46 nonemptyI1: $o ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyI1,definition,
% 0.19/0.46 ( nonemptyI1
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinIL_type,type,
% 0.19/0.46 setadjoinIL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinIL,definition,
% 0.19/0.46 ( setadjoinIL
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyinunitempty_type,type,
% 0.19/0.46 emptyinunitempty: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyinunitempty,definition,
% 0.19/0.46 ( emptyinunitempty
% 0.19/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinIR_type,type,
% 0.19/0.46 setadjoinIR: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinIR,definition,
% 0.19/0.46 ( setadjoinIR
% 0.19/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ A )
% 0.19/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinE_type,type,
% 0.19/0.46 setadjoinE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinE,definition,
% 0.19/0.46 ( setadjoinE
% 0.19/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.46 => ! [Xphi: $o] :
% 0.19/0.46 ( ( ( Xy = Xx )
% 0.19/0.46 => Xphi )
% 0.19/0.46 => ( ( ( in @ Xy @ A )
% 0.19/0.46 => Xphi )
% 0.19/0.46 => Xphi ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinOr_type,type,
% 0.19/0.46 setadjoinOr: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setadjoinOr,definition,
% 0.19/0.46 ( setadjoinOr
% 0.19/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.46 => ( ( Xy = Xx )
% 0.19/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setoftrueEq_type,type,
% 0.19/0.46 setoftrueEq: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setoftrueEq,definition,
% 0.19/0.46 ( setoftrueEq
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ( dsetconstr @ A
% 0.19/0.46 @ ^ [Xx: $i] : $true )
% 0.19/0.46 = A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetI_type,type,
% 0.19/0.46 powersetI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetI,definition,
% 0.19/0.46 ( powersetI
% 0.19/0.46 = ( ! [A: $i,B: $i] :
% 0.19/0.46 ( ! [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ A ) )
% 0.19/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyinPowerset_type,type,
% 0.19/0.46 emptyinPowerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyinPowerset,definition,
% 0.19/0.46 ( emptyinPowerset
% 0.19/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(emptyInPowerset_type,type,
% 0.19/0.46 emptyInPowerset: $o ).
% 0.19/0.46
% 0.19/0.46 thf(emptyInPowerset,definition,
% 0.19/0.46 ( emptyInPowerset
% 0.19/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(powersetE_type,type,
% 0.19/0.46 powersetE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(powersetE,definition,
% 0.19/0.46 ( powersetE
% 0.19/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.46 => ( ( in @ Xx @ B )
% 0.19/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionI_type,type,
% 0.19/0.46 setunionI: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionI,definition,
% 0.19/0.46 ( setunionI
% 0.19/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( ( in @ B @ A )
% 0.19/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setunionE_type,type,
% 0.19/0.46 setunionE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(setunionE,definition,
% 0.19/0.46 ( setunionE
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.46 => ! [Xphi: $o] :
% 0.19/0.46 ( ! [B: $i] :
% 0.19/0.46 ( ( in @ Xx @ B )
% 0.19/0.46 => ( ( in @ B @ A )
% 0.19/0.46 => Xphi ) )
% 0.19/0.46 => Xphi ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(subPowSU_type,type,
% 0.19/0.46 subPowSU: $o ).
% 0.19/0.46
% 0.19/0.46 thf(subPowSU,definition,
% 0.19/0.46 ( subPowSU
% 0.19/0.46 = ( ! [A: $i,Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(exuE2_type,type,
% 0.19/0.46 exuE2: $o ).
% 0.19/0.46
% 0.19/0.46 thf(exuE2,definition,
% 0.19/0.46 ( exuE2
% 0.19/0.46 = ( ! [Xphi: $i > $o] :
% 0.19/0.46 ( ( exu
% 0.19/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyImpWitness_type,type,
% 0.19/0.46 nonemptyImpWitness: $o ).
% 0.19/0.46
% 0.19/0.46 thf(nonemptyImpWitness,definition,
% 0.19/0.46 ( nonemptyImpWitness
% 0.19/0.46 = ( ! [A: $i] :
% 0.19/0.46 ( ( nonempty @ A )
% 0.19/0.46 => ? [Xx: $i] :
% 0.19/0.46 ( ( in @ Xx @ A )
% 0.19/0.46 & $true ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(uniqinunit_type,type,
% 0.19/0.46 uniqinunit: $o ).
% 0.19/0.46
% 0.19/0.46 thf(uniqinunit,definition,
% 0.19/0.46 ( uniqinunit
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(notinsingleton_type,type,
% 0.19/0.46 notinsingleton: $o ).
% 0.19/0.46
% 0.19/0.46 thf(notinsingleton,definition,
% 0.19/0.46 ( notinsingleton
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx != Xy )
% 0.19/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(eqinunit_type,type,
% 0.19/0.46 eqinunit: $o ).
% 0.19/0.46
% 0.19/0.46 thf(eqinunit,definition,
% 0.19/0.46 ( eqinunit
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( Xx = Xy )
% 0.19/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsswitch_type,type,
% 0.19/0.46 singletonsswitch: $o ).
% 0.19/0.46
% 0.19/0.46 thf(singletonsswitch,definition,
% 0.19/0.46 ( singletonsswitch
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetE_type,type,
% 0.19/0.46 upairsetE: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetE,definition,
% 0.19/0.46 ( upairsetE
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.46 => ( ( Xz = Xx )
% 0.19/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetIL_type,type,
% 0.19/0.46 upairsetIL: $o ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetIL,definition,
% 0.19/0.46 ( upairsetIL
% 0.19/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(upairsetIR,conjecture,
% 0.19/0.46 ( setextAx
% 0.19/0.46 => ( emptysetAx
% 0.19/0.46 => ( setadjoinAx
% 0.19/0.46 => ( powersetAx
% 0.19/0.46 => ( setunionAx
% 0.19/0.46 => ( omega0Ax
% 0.19/0.46 => ( omegaSAx
% 0.19/0.46 => ( omegaIndAx
% 0.19/0.46 => ( replAx
% 0.19/0.46 => ( foundationAx
% 0.19/0.46 => ( wellorderingAx
% 0.19/0.46 => ( descrp
% 0.19/0.46 => ( dsetconstrI
% 0.19/0.46 => ( dsetconstrEL
% 0.19/0.46 => ( dsetconstrER
% 0.19/0.46 => ( exuE1
% 0.19/0.46 => ( prop2setE
% 0.19/0.46 => ( emptysetE
% 0.19/0.46 => ( emptysetimpfalse
% 0.19/0.46 => ( notinemptyset
% 0.19/0.46 => ( exuE3e
% 0.19/0.46 => ( setext
% 0.19/0.46 => ( emptyI
% 0.19/0.46 => ( noeltsimpempty
% 0.19/0.46 => ( setbeta
% 0.19/0.46 => ( nonemptyE1
% 0.19/0.46 => ( nonemptyI
% 0.19/0.46 => ( nonemptyI1
% 0.19/0.46 => ( setadjoinIL
% 0.19/0.46 => ( emptyinunitempty
% 0.19/0.46 => ( setadjoinIR
% 0.19/0.46 => ( setadjoinE
% 0.19/0.46 => ( setadjoinOr
% 0.19/0.46 => ( setoftrueEq
% 0.19/0.46 => ( powersetI
% 0.19/0.46 => ( emptyinPowerset
% 0.19/0.46 => ( emptyInPowerset
% 0.19/0.46 => ( powersetE
% 0.19/0.46 => ( setunionI
% 0.19/0.46 => ( setunionE
% 0.19/0.46 => ( subPowSU
% 0.19/0.46 => ( exuE2
% 0.19/0.46 => ( nonemptyImpWitness
% 0.19/0.46 => ( uniqinunit
% 0.19/0.46 => ( notinsingleton
% 0.19/0.46 => ( eqinunit
% 0.19/0.46 => ( singletonsswitch
% 0.19/0.46 => ( upairsetE
% 0.19/0.46 => ( upairsetIL
% 0.19/0.46 => ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 ------------------------
% 1.84/1.39 ---- Embedded in THF ---
% 1.84/1.39 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.84/1.39 %%% Generated on Tue Feb 24 12:26:15 EST 2026
% 1.84/1.39 %%% using '$$dhol' embedding, version 1.3.0.
% 1.84/1.39 %%% Logic specification used:
% 1.84/1.39 %%% thf(spec, logic, $$dhol).
% 1.84/1.39
% 1.84/1.39 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.kOEAvEHP3t/DTF2DTF32675.p
% See solution above
% 1.84/1.39 ------------------------
% 2.04/1.40 ---- Cleaned THF ---
% 2.04/1.40 thf(in_type,type,
% 2.04/1.40 in: $i > $i > $o ).
% 2.04/1.40
% 2.04/1.40 thf(exu_type,type,
% 2.04/1.40 exu: ( $i > $o ) > $o ).
% 2.04/1.40
% 2.04/1.40 thf(exu,definition,
% 2.04/1.40 ( exu
% 2.04/1.40 = ( ^ [Xphi: $i > $o] :
% 2.04/1.40 ? [Xx: $i] :
% 2.04/1.40 ( ( Xphi @ Xx )
% 2.04/1.40 & ! [Xy: $i] :
% 2.04/1.40 ( ( Xphi @ Xy )
% 2.04/1.40 => ( Xx = Xy ) ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(setextAx_type,type,
% 2.04/1.40 setextAx: $o ).
% 2.04/1.40
% 2.04/1.40 thf(setextAx,definition,
% 2.04/1.40 ( setextAx
% 2.04/1.40 = ( ! [A: $i,B: $i] :
% 2.04/1.40 ( ! [Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ A )
% 2.04/1.40 <=> ( in @ Xx @ B ) )
% 2.04/1.40 => ( A = B ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(emptyset_type,type,
% 2.04/1.40 emptyset: $i ).
% 2.04/1.40
% 2.04/1.40 thf(emptysetAx_type,type,
% 2.04/1.40 emptysetAx: $o ).
% 2.04/1.40
% 2.04/1.40 thf(emptysetAx,definition,
% 2.04/1.40 ( emptysetAx
% 2.04/1.40 = ( ! [Xx: $i] :
% 2.04/1.40 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(setadjoin_type,type,
% 2.04/1.40 setadjoin: $i > $i > $i ).
% 2.04/1.40
% 2.04/1.40 thf(setadjoinAx_type,type,
% 2.04/1.40 setadjoinAx: $o ).
% 2.04/1.40
% 2.04/1.40 thf(setadjoinAx,definition,
% 2.04/1.40 ( setadjoinAx
% 2.04/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.04/1.40 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.04/1.40 <=> ( ( Xy = Xx )
% 2.04/1.40 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(powerset_type,type,
% 2.04/1.40 powerset: $i > $i ).
% 2.04/1.40
% 2.04/1.40 thf(powersetAx_type,type,
% 2.04/1.40 powersetAx: $o ).
% 2.04/1.40
% 2.04/1.40 thf(powersetAx,definition,
% 2.04/1.40 ( powersetAx
% 2.04/1.40 = ( ! [A: $i,B: $i] :
% 2.04/1.40 ( ( in @ B @ ( powerset @ A ) )
% 2.04/1.40 <=> ! [Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ B )
% 2.04/1.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(setunion_type,type,
% 2.04/1.40 setunion: $i > $i ).
% 2.04/1.40
% 2.04/1.40 thf(setunionAx_type,type,
% 2.04/1.40 setunionAx: $o ).
% 2.04/1.40
% 2.04/1.40 thf(setunionAx,definition,
% 2.04/1.40 ( setunionAx
% 2.04/1.40 = ( ! [A: $i,Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ ( setunion @ A ) )
% 2.04/1.40 <=> ? [B: $i] :
% 2.04/1.40 ( ( in @ Xx @ B )
% 2.04/1.40 & ( in @ B @ A ) ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(omega_type,type,
% 2.04/1.40 omega: $i ).
% 2.04/1.40
% 2.04/1.40 thf(omega0Ax_type,type,
% 2.04/1.40 omega0Ax: $o ).
% 2.04/1.40
% 2.04/1.40 thf(omega0Ax,definition,
% 2.04/1.40 ( omega0Ax
% 2.04/1.40 = ( in @ emptyset @ omega ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(omegaSAx_type,type,
% 2.04/1.40 omegaSAx: $o ).
% 2.04/1.40
% 2.04/1.40 thf(omegaSAx,definition,
% 2.04/1.40 ( omegaSAx
% 2.04/1.40 = ( ! [Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ omega )
% 2.04/1.40 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(omegaIndAx_type,type,
% 2.04/1.40 omegaIndAx: $o ).
% 2.04/1.40
% 2.04/1.40 thf(omegaIndAx,definition,
% 2.04/1.40 ( omegaIndAx
% 2.04/1.40 = ( ! [A: $i] :
% 2.04/1.40 ( ( ( in @ emptyset @ A )
% 2.04/1.40 & ! [Xx: $i] :
% 2.04/1.40 ( ( ( in @ Xx @ omega )
% 2.04/1.40 & ( in @ Xx @ A ) )
% 2.04/1.40 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.04/1.40 => ! [Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ omega )
% 2.04/1.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(replAx_type,type,
% 2.04/1.40 replAx: $o ).
% 2.04/1.40
% 2.04/1.40 thf(replAx,definition,
% 2.04/1.40 ( replAx
% 2.04/1.40 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.04/1.40 ( ! [Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ A )
% 2.04/1.40 => ( exu
% 2.04/1.40 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.04/1.40 => ? [B: $i] :
% 2.04/1.40 ! [Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ B )
% 2.04/1.40 <=> ? [Xy: $i] :
% 2.04/1.40 ( ( in @ Xy @ A )
% 2.04/1.40 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(foundationAx_type,type,
% 2.04/1.40 foundationAx: $o ).
% 2.04/1.40
% 2.04/1.40 thf(foundationAx,definition,
% 2.04/1.40 ( foundationAx
% 2.04/1.40 = ( ! [A: $i] :
% 2.04/1.40 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.04/1.40 => ? [B: $i] :
% 2.04/1.40 ( ( in @ B @ A )
% 2.04/1.40 & ~ ? [Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ B )
% 2.04/1.40 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(wellorderingAx_type,type,
% 2.04/1.40 wellorderingAx: $o ).
% 2.04/1.40
% 2.04/1.40 thf(wellorderingAx,definition,
% 2.04/1.40 ( wellorderingAx
% 2.04/1.40 = ( ! [A: $i] :
% 2.04/1.40 ? [B: $i] :
% 2.04/1.40 ( ! [C: $i] :
% 2.04/1.40 ( ( in @ C @ B )
% 2.04/1.40 => ! [Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ C )
% 2.04/1.40 => ( in @ Xx @ A ) ) )
% 2.04/1.40 & ! [Xx: $i,Xy: $i] :
% 2.04/1.40 ( ( ( in @ Xx @ A )
% 2.04/1.40 & ( in @ Xy @ A ) )
% 2.04/1.40 => ( ! [C: $i] :
% 2.04/1.40 ( ( in @ C @ B )
% 2.04/1.40 => ( ( in @ Xx @ C )
% 2.04/1.40 <=> ( in @ Xy @ C ) ) )
% 2.04/1.40 => ( Xx = Xy ) ) )
% 2.04/1.40 & ! [C: $i,D: $i] :
% 2.04/1.40 ( ( ( in @ C @ B )
% 2.04/1.40 & ( in @ D @ B ) )
% 2.04/1.40 => ( ! [Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ C )
% 2.04/1.40 => ( in @ Xx @ D ) )
% 2.04/1.40 | ! [Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ D )
% 2.04/1.40 => ( in @ Xx @ C ) ) ) )
% 2.04/1.40 & ! [C: $i] :
% 2.04/1.40 ( ( ! [Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ C )
% 2.04/1.40 => ( in @ Xx @ A ) )
% 2.04/1.40 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.04/1.40 => ? [D: $i,Xx: $i] :
% 2.04/1.40 ( ( in @ D @ B )
% 2.04/1.40 & ( in @ Xx @ C )
% 2.04/1.40 & ~ ? [Xy: $i] :
% 2.04/1.40 ( ( in @ Xy @ D )
% 2.04/1.40 & ( in @ Xy @ C ) )
% 2.04/1.40 & ! [E: $i] :
% 2.04/1.40 ( ( in @ E @ B )
% 2.04/1.40 => ( ! [Xy: $i] :
% 2.04/1.40 ( ( in @ Xy @ E )
% 2.04/1.40 => ( in @ Xy @ D ) )
% 2.04/1.40 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(descr_type,type,
% 2.04/1.40 descr: ( $i > $o ) > $i ).
% 2.04/1.40
% 2.04/1.40 thf(descrp_type,type,
% 2.04/1.40 descrp: $o ).
% 2.04/1.40
% 2.04/1.40 thf(descrp,definition,
% 2.04/1.40 ( descrp
% 2.04/1.40 = ( ! [Xphi: $i > $o] :
% 2.04/1.40 ( ( exu
% 2.04/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.04/1.40 => ( Xphi
% 2.04/1.40 @ ( descr
% 2.04/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(dsetconstr_type,type,
% 2.04/1.40 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.04/1.40
% 2.04/1.40 thf(dsetconstrI_type,type,
% 2.04/1.40 dsetconstrI: $o ).
% 2.04/1.40
% 2.04/1.40 thf(dsetconstrI,definition,
% 2.04/1.40 ( dsetconstrI
% 2.04/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ A )
% 2.04/1.40 => ( ( Xphi @ Xx )
% 2.04/1.40 => ( in @ Xx
% 2.04/1.40 @ ( dsetconstr @ A
% 2.04/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(dsetconstrEL_type,type,
% 2.04/1.40 dsetconstrEL: $o ).
% 2.04/1.40
% 2.04/1.40 thf(dsetconstrEL,definition,
% 2.04/1.40 ( dsetconstrEL
% 2.04/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.04/1.40 ( ( in @ Xx
% 2.04/1.40 @ ( dsetconstr @ A
% 2.04/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.04/1.40 => ( in @ Xx @ A ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(dsetconstrER_type,type,
% 2.04/1.40 dsetconstrER: $o ).
% 2.04/1.40
% 2.04/1.40 thf(dsetconstrER,definition,
% 2.04/1.40 ( dsetconstrER
% 2.04/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.04/1.40 ( ( in @ Xx
% 2.04/1.40 @ ( dsetconstr @ A
% 2.04/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.04/1.40 => ( Xphi @ Xx ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(exuE1_type,type,
% 2.04/1.40 exuE1: $o ).
% 2.04/1.40
% 2.04/1.40 thf(exuE1,definition,
% 2.04/1.40 ( exuE1
% 2.04/1.40 = ( ! [Xphi: $i > $o] :
% 2.04/1.40 ( ( exu
% 2.04/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.04/1.40 => ? [Xx: $i] :
% 2.04/1.40 ( ( Xphi @ Xx )
% 2.04/1.40 & ! [Xy: $i] :
% 2.04/1.40 ( ( Xphi @ Xy )
% 2.04/1.40 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(prop2set_type,type,
% 2.04/1.40 prop2set: $o > $i ).
% 2.04/1.40
% 2.04/1.40 thf(prop2set,definition,
% 2.04/1.40 ( prop2set
% 2.04/1.40 = ( ^ [Xphi: $o] :
% 2.04/1.40 ( dsetconstr @ ( powerset @ emptyset )
% 2.04/1.40 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(prop2setE_type,type,
% 2.04/1.40 prop2setE: $o ).
% 2.04/1.40
% 2.04/1.40 thf(prop2setE,definition,
% 2.04/1.40 ( prop2setE
% 2.04/1.40 = ( ! [Xphi: $o,Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.04/1.40 => Xphi ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(emptysetE_type,type,
% 2.04/1.40 emptysetE: $o ).
% 2.04/1.40
% 2.04/1.40 thf(emptysetE,definition,
% 2.04/1.40 ( emptysetE
% 2.04/1.40 = ( ! [Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ emptyset )
% 2.04/1.40 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(emptysetimpfalse_type,type,
% 2.04/1.40 emptysetimpfalse: $o ).
% 2.04/1.40
% 2.04/1.40 thf(emptysetimpfalse,definition,
% 2.04/1.40 ( emptysetimpfalse
% 2.04/1.40 = ( ! [Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ emptyset )
% 2.04/1.40 => $false ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(notinemptyset_type,type,
% 2.04/1.40 notinemptyset: $o ).
% 2.04/1.40
% 2.04/1.40 thf(notinemptyset,definition,
% 2.04/1.40 ( notinemptyset
% 2.04/1.40 = ( ! [Xx: $i] :
% 2.04/1.40 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(exuE3e_type,type,
% 2.04/1.40 exuE3e: $o ).
% 2.04/1.40
% 2.04/1.40 thf(exuE3e,definition,
% 2.04/1.40 ( exuE3e
% 2.04/1.40 = ( ! [Xphi: $i > $o] :
% 2.04/1.40 ( ( exu
% 2.04/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.04/1.40 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(setext_type,type,
% 2.04/1.40 setext: $o ).
% 2.04/1.40
% 2.04/1.40 thf(setext,definition,
% 2.04/1.40 ( setext
% 2.04/1.40 = ( ! [A: $i,B: $i] :
% 2.04/1.40 ( ! [Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ A )
% 2.04/1.40 => ( in @ Xx @ B ) )
% 2.04/1.40 => ( ! [Xx: $i] :
% 2.04/1.40 ( ( in @ Xx @ B )
% 2.04/1.40 => ( in @ Xx @ A ) )
% 2.04/1.40 => ( A = B ) ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(emptyI_type,type,
% 2.04/1.40 emptyI: $o ).
% 2.04/1.40
% 2.04/1.40 thf(emptyI,definition,
% 2.04/1.40 ( emptyI
% 2.04/1.40 = ( ! [A: $i] :
% 2.04/1.40 ( ! [Xx: $i] :
% 2.04/1.40 ~ ( in @ Xx @ A )
% 2.04/1.40 => ( A = emptyset ) ) ) ) ).
% 2.04/1.40
% 2.04/1.40 thf(noeltsimpempty_type,type,
% 2.04/1.40 noeltsimpempty: $o ).
% 2.04/1.40
% 2.04/1.40 thf(noeltsimpempty,definition,
% 2.04/1.40 ( noeltsimpempty
% 2.04/1.40 = ( ! [A: $i] :
% 2.04/1.41 ( ! [Xx: $i] :
% 2.04/1.41 ~ ( in @ Xx @ A )
% 2.04/1.41 => ( A = emptyset ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setbeta_type,type,
% 2.04/1.41 setbeta: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setbeta,definition,
% 2.04/1.41 ( setbeta
% 2.04/1.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 => ( ( in @ Xx
% 2.04/1.41 @ ( dsetconstr @ A
% 2.04/1.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.04/1.41 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(nonempty_type,type,
% 2.04/1.41 nonempty: $i > $o ).
% 2.04/1.41
% 2.04/1.41 thf(nonempty,definition,
% 2.04/1.41 ( nonempty
% 2.04/1.41 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(nonemptyE1_type,type,
% 2.04/1.41 nonemptyE1: $o ).
% 2.04/1.41
% 2.04/1.41 thf(nonemptyE1,definition,
% 2.04/1.41 ( nonemptyE1
% 2.04/1.41 = ( ! [A: $i] :
% 2.04/1.41 ( ( nonempty @ A )
% 2.04/1.41 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(nonemptyI_type,type,
% 2.04/1.41 nonemptyI: $o ).
% 2.04/1.41
% 2.04/1.41 thf(nonemptyI,definition,
% 2.04/1.41 ( nonemptyI
% 2.04/1.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 => ( ( Xphi @ Xx )
% 2.04/1.41 => ( nonempty
% 2.04/1.41 @ ( dsetconstr @ A
% 2.04/1.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(nonemptyI1_type,type,
% 2.04/1.41 nonemptyI1: $o ).
% 2.04/1.41
% 2.04/1.41 thf(nonemptyI1,definition,
% 2.04/1.41 ( nonemptyI1
% 2.04/1.41 = ( ! [A: $i] :
% 2.04/1.41 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.04/1.41 => ( nonempty @ A ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinIL_type,type,
% 2.04/1.41 setadjoinIL: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinIL,definition,
% 2.04/1.41 ( setadjoinIL
% 2.04/1.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(emptyinunitempty_type,type,
% 2.04/1.41 emptyinunitempty: $o ).
% 2.04/1.41
% 2.04/1.41 thf(emptyinunitempty,definition,
% 2.04/1.41 ( emptyinunitempty
% 2.04/1.41 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinIR_type,type,
% 2.04/1.41 setadjoinIR: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinIR,definition,
% 2.04/1.41 ( setadjoinIR
% 2.04/1.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.04/1.41 ( ( in @ Xy @ A )
% 2.04/1.41 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinE_type,type,
% 2.04/1.41 setadjoinE: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinE,definition,
% 2.04/1.41 ( setadjoinE
% 2.04/1.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.04/1.41 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.04/1.41 => ! [Xphi: $o] :
% 2.04/1.41 ( ( ( Xy = Xx )
% 2.04/1.41 => Xphi )
% 2.04/1.41 => ( ( ( in @ Xy @ A )
% 2.04/1.41 => Xphi )
% 2.04/1.41 => Xphi ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinOr_type,type,
% 2.04/1.41 setadjoinOr: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinOr,definition,
% 2.04/1.41 ( setadjoinOr
% 2.04/1.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.04/1.41 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.04/1.41 => ( ( Xy = Xx )
% 2.04/1.41 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setoftrueEq_type,type,
% 2.04/1.41 setoftrueEq: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setoftrueEq,definition,
% 2.04/1.41 ( setoftrueEq
% 2.04/1.41 = ( ! [A: $i] :
% 2.04/1.41 ( ( dsetconstr @ A
% 2.04/1.41 @ ^ [Xx: $i] : $true )
% 2.04/1.41 = A ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(powersetI_type,type,
% 2.04/1.41 powersetI: $o ).
% 2.04/1.41
% 2.04/1.41 thf(powersetI,definition,
% 2.04/1.41 ( powersetI
% 2.04/1.41 = ( ! [A: $i,B: $i] :
% 2.04/1.41 ( ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ B )
% 2.04/1.41 => ( in @ Xx @ A ) )
% 2.04/1.41 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(emptyinPowerset_type,type,
% 2.04/1.41 emptyinPowerset: $o ).
% 2.04/1.41
% 2.04/1.41 thf(emptyinPowerset,definition,
% 2.04/1.41 ( emptyinPowerset
% 2.04/1.41 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(emptyInPowerset_type,type,
% 2.04/1.41 emptyInPowerset: $o ).
% 2.04/1.41
% 2.04/1.41 thf(emptyInPowerset,definition,
% 2.04/1.41 ( emptyInPowerset
% 2.04/1.41 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(powersetE_type,type,
% 2.04/1.41 powersetE: $o ).
% 2.04/1.41
% 2.04/1.41 thf(powersetE,definition,
% 2.04/1.41 ( powersetE
% 2.04/1.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.04/1.41 ( ( in @ B @ ( powerset @ A ) )
% 2.04/1.41 => ( ( in @ Xx @ B )
% 2.04/1.41 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setunionI_type,type,
% 2.04/1.41 setunionI: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setunionI,definition,
% 2.04/1.41 ( setunionI
% 2.04/1.41 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.04/1.41 ( ( in @ Xx @ B )
% 2.04/1.41 => ( ( in @ B @ A )
% 2.04/1.41 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setunionE_type,type,
% 2.04/1.41 setunionE: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setunionE,definition,
% 2.04/1.41 ( setunionE
% 2.04/1.41 = ( ! [A: $i,Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ ( setunion @ A ) )
% 2.04/1.41 => ! [Xphi: $o] :
% 2.04/1.41 ( ! [B: $i] :
% 2.04/1.41 ( ( in @ Xx @ B )
% 2.04/1.41 => ( ( in @ B @ A )
% 2.04/1.41 => Xphi ) )
% 2.04/1.41 => Xphi ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(subPowSU_type,type,
% 2.04/1.41 subPowSU: $o ).
% 2.04/1.41
% 2.04/1.41 thf(subPowSU,definition,
% 2.04/1.41 ( subPowSU
% 2.04/1.41 = ( ! [A: $i,Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(exuE2_type,type,
% 2.04/1.41 exuE2: $o ).
% 2.04/1.41
% 2.04/1.41 thf(exuE2,definition,
% 2.04/1.41 ( exuE2
% 2.04/1.41 = ( ! [Xphi: $i > $o] :
% 2.04/1.41 ( ( exu
% 2.04/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.04/1.41 => ? [Xx: $i] :
% 2.04/1.41 ! [Xy: $i] :
% 2.04/1.41 ( ( Xphi @ Xy )
% 2.04/1.41 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(nonemptyImpWitness_type,type,
% 2.04/1.41 nonemptyImpWitness: $o ).
% 2.04/1.41
% 2.04/1.41 thf(nonemptyImpWitness,definition,
% 2.04/1.41 ( nonemptyImpWitness
% 2.04/1.41 = ( ! [A: $i] :
% 2.04/1.41 ( ( nonempty @ A )
% 2.04/1.41 => ? [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 & $true ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(uniqinunit_type,type,
% 2.04/1.41 uniqinunit: $o ).
% 2.04/1.41
% 2.04/1.41 thf(uniqinunit,definition,
% 2.04/1.41 ( uniqinunit
% 2.04/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.04/1.41 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.04/1.41 => ( Xx = Xy ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(notinsingleton_type,type,
% 2.04/1.41 notinsingleton: $o ).
% 2.04/1.41
% 2.04/1.41 thf(notinsingleton,definition,
% 2.04/1.41 ( notinsingleton
% 2.04/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.04/1.41 ( ( Xx != Xy )
% 2.04/1.41 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(eqinunit_type,type,
% 2.04/1.41 eqinunit: $o ).
% 2.04/1.41
% 2.04/1.41 thf(eqinunit,definition,
% 2.04/1.41 ( eqinunit
% 2.04/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.04/1.41 ( ( Xx = Xy )
% 2.04/1.41 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(singletonsswitch_type,type,
% 2.04/1.41 singletonsswitch: $o ).
% 2.04/1.41
% 2.04/1.41 thf(singletonsswitch,definition,
% 2.04/1.41 ( singletonsswitch
% 2.04/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.04/1.41 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.04/1.41 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(upairsetE_type,type,
% 2.04/1.41 upairsetE: $o ).
% 2.04/1.41
% 2.04/1.41 thf(upairsetE,definition,
% 2.04/1.41 ( upairsetE
% 2.04/1.41 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.04/1.41 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.04/1.41 => ( ( Xz = Xx )
% 2.04/1.41 | ( Xz = Xy ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(upairsetIL_type,type,
% 2.04/1.41 upairsetIL: $o ).
% 2.04/1.41
% 2.04/1.41 thf(upairsetIL,definition,
% 2.04/1.41 ( upairsetIL
% 2.04/1.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(upairsetIR,conjecture,
% 2.04/1.41 ( setextAx
% 2.04/1.41 => ( emptysetAx
% 2.04/1.41 => ( setadjoinAx
% 2.04/1.41 => ( powersetAx
% 2.04/1.41 => ( setunionAx
% 2.04/1.41 => ( omega0Ax
% 2.04/1.41 => ( omegaSAx
% 2.04/1.41 => ( omegaIndAx
% 2.04/1.41 => ( replAx
% 2.04/1.41 => ( foundationAx
% 2.04/1.41 => ( wellorderingAx
% 2.04/1.41 => ( descrp
% 2.04/1.41 => ( dsetconstrI
% 2.04/1.41 => ( dsetconstrEL
% 2.04/1.41 => ( dsetconstrER
% 2.04/1.41 => ( exuE1
% 2.04/1.41 => ( prop2setE
% 2.04/1.41 => ( emptysetE
% 2.04/1.41 => ( emptysetimpfalse
% 2.04/1.41 => ( notinemptyset
% 2.04/1.41 => ( exuE3e
% 2.04/1.41 => ( setext
% 2.04/1.41 => ( emptyI
% 2.04/1.41 => ( noeltsimpempty
% 2.04/1.41 => ( setbeta
% 2.04/1.41 => ( nonemptyE1
% 2.04/1.41 => ( nonemptyI
% 2.04/1.41 => ( nonemptyI1
% 2.04/1.41 => ( setadjoinIL
% 2.04/1.41 => ( emptyinunitempty
% 2.04/1.41 => ( setadjoinIR
% 2.04/1.41 => ( setadjoinE
% 2.04/1.41 => ( setadjoinOr
% 2.04/1.41 => ( setoftrueEq
% 2.04/1.41 => ( powersetI
% 2.04/1.41 => ( emptyinPowerset
% 2.04/1.41 => ( emptyInPowerset
% 2.04/1.41 => ( powersetE
% 2.04/1.41 => ( setunionI
% 2.04/1.41 => ( setunionE
% 2.04/1.41 => ( subPowSU
% 2.04/1.41 => ( exuE2
% 2.04/1.41 => ( nonemptyImpWitness
% 2.04/1.41 => ( uniqinunit
% 2.04/1.41 => ( notinsingleton
% 2.04/1.41 => ( eqinunit
% 2.04/1.41 => ( singletonsswitch
% 2.04/1.41 => ( upairsetE
% 2.04/1.41 => ( upairsetIL
% 2.04/1.41 => ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.04/1.41 ------------------------
% 2.04/1.42 % (419)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_32675 for (2999ds/275Mi)
% 2.04/1.42 % (418)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_32675 for (2999ds/2Mi)
% 2.04/1.42 % (418)Instruction limit reached!
% 2.04/1.42 % (418)------------------------------
% 2.04/1.42 % (418)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.42 % (418)Termination reason: Unknown
% 2.04/1.42 % (418)Termination phase: shuffling
% 2.04/1.42
% 2.04/1.42 % (418)Memory used [KB]: 1023
% 2.04/1.42 % (418)Time elapsed: 0.002 s
% 2.04/1.42 % (418)Instructions burned: 3 (million)
% 2.04/1.42 % (418)------------------------------
% 2.04/1.42 % (418)------------------------------
% 2.04/1.43 % (415)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_32675 for (2999ds/4Mi)
% 2.04/1.43 % (414)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_32675 for (2999ds/183Mi)
% 2.04/1.43 % (416)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_32675 for (2999ds/27Mi)
% 2.04/1.43 % (417)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_32675 for (2999ds/2Mi)
% 2.04/1.43 % (417)Instruction limit reached!
% 2.04/1.43 % (417)------------------------------
% 2.04/1.43 % (417)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.43 % (417)Termination reason: Unknown
% 2.04/1.43 % (417)Termination phase: shuffling
% 2.04/1.43
% 2.04/1.43 % (417)Memory used [KB]: 1023
% 2.04/1.43 % (417)Time elapsed: 0.003 s
% 2.04/1.43 % (417)Instructions burned: 2 (million)
% 2.04/1.43 % (417)------------------------------
% 2.04/1.43 % (417)------------------------------
% 2.04/1.43 % (415)Instruction limit reached!
% 2.04/1.43 % (415)------------------------------
% 2.04/1.43 % (415)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.43 % (415)Termination reason: Unknown
% 2.04/1.43 % (415)Termination phase: shuffling
% 2.04/1.43
% 2.04/1.43 % (415)Memory used [KB]: 1151
% 2.04/1.43 % (415)Time elapsed: 0.004 s
% 2.04/1.43 % (415)Instructions burned: 4 (million)
% 2.04/1.43 % (415)------------------------------
% 2.04/1.43 % (415)------------------------------
% 2.04/1.43 % (421)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32675 for (2999ds/3Mi)
% 2.04/1.43 % (421)Instruction limit reached!
% 2.04/1.43 % (421)------------------------------
% 2.04/1.43 % (421)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.43 % (421)Termination reason: Unknown
% 2.04/1.43 % (421)Termination phase: shuffling
% 2.04/1.43
% 2.04/1.43 % (421)Memory used [KB]: 1151
% 2.04/1.43 % (421)Time elapsed: 0.003 s
% 2.04/1.43 % (421)Instructions burned: 4 (million)
% 2.04/1.43 % (421)------------------------------
% 2.04/1.43 % (421)------------------------------
% 2.04/1.44 % (420)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_32675 for (2999ds/18Mi)
% 2.04/1.44 % (416)Instruction limit reached!
% 2.04/1.44 % (416)------------------------------
% 2.04/1.44 % (416)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.44 % (416)Termination reason: Unknown
% 2.04/1.44 % (416)Termination phase: Saturation
% 2.04/1.44
% 2.04/1.44 % (416)Memory used [KB]: 5756
% 2.04/1.44 % (416)Time elapsed: 0.016 s
% 2.04/1.44 % (416)Instructions burned: 28 (million)
% 2.04/1.44 % (416)------------------------------
% 2.04/1.44 % (416)------------------------------
% 2.04/1.44 % (424)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32675 for (2999ds/3Mi)
% 2.04/1.45 % (424)Instruction limit reached!
% 2.04/1.45 % (424)------------------------------
% 2.04/1.45 % (424)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.45 % (424)Termination reason: Unknown
% 2.04/1.45 % (424)Termination phase: shuffling
% 2.04/1.45
% 2.04/1.45 % (424)Memory used [KB]: 1151
% 2.04/1.45 % (424)Time elapsed: 0.003 s
% 2.04/1.45 % (424)Instructions burned: 4 (million)
% 2.04/1.45 % (424)------------------------------
% 2.04/1.45 % (424)------------------------------
% 2.04/1.45 % (422)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_32675 for (2999ds/37Mi)
% 2.04/1.45 % (423)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_32675 for (2999ds/15Mi)
% 2.04/1.45 % (420)Instruction limit reached!
% 2.04/1.45 % (420)------------------------------
% 2.04/1.45 % (420)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.45 % (420)Termination reason: Unknown
% 2.04/1.45 % (420)Termination phase: Preprocessing 3
% 2.04/1.45
% 2.04/1.45 % (420)Memory used [KB]: 1535
% 2.04/1.45 % (420)Time elapsed: 0.019 s
% 2.04/1.45 % (420)Instructions burned: 18 (million)
% 2.04/1.45 % (420)------------------------------
% 2.04/1.45 % (420)------------------------------
% 2.04/1.46 % (426)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_32675 for (2999ds/7Mi)
% 2.04/1.46 % (423)Instruction limit reached!
% 2.04/1.46 % (423)------------------------------
% 2.04/1.46 % (423)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.46 % (423)Termination reason: Unknown
% 2.04/1.46 % (423)Termination phase: Preprocessing 3
% 2.04/1.46
% 2.04/1.46 % (423)Memory used [KB]: 1407
% 2.04/1.46 % (423)Time elapsed: 0.011 s
% 2.04/1.46 % (423)Instructions burned: 16 (million)
% 2.04/1.46 % (423)------------------------------
% 2.04/1.46 % (423)------------------------------
% 2.04/1.46 % (426)Instruction limit reached!
% 2.04/1.46 % (426)------------------------------
% 2.04/1.46 % (426)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.46 % (426)Termination reason: Unknown
% 2.04/1.46 % (426)Termination phase: shuffling
% 2.04/1.46
% 2.04/1.46 % (426)Memory used [KB]: 1151
% 2.04/1.46 % (426)Time elapsed: 0.003 s
% 2.04/1.46 % (426)Instructions burned: 7 (million)
% 2.04/1.46 % (426)------------------------------
% 2.04/1.46 % (426)------------------------------
% 2.04/1.46 % (425)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_32675 for (2999ds/1041Mi)
% 2.04/1.47 % (422)Instruction limit reached!
% 2.04/1.47 % (422)------------------------------
% 2.04/1.47 % (422)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.47 % (422)Termination reason: Unknown
% 2.04/1.47 % (422)Termination phase: Saturation
% 2.04/1.47
% 2.04/1.47 % (422)Memory used [KB]: 5756
% 2.04/1.47 % (422)Time elapsed: 0.020 s
% 2.04/1.47 % (422)Instructions burned: 38 (million)
% 2.04/1.47 % (422)------------------------------
% 2.04/1.47 % (422)------------------------------
% 2.04/1.47 % (429)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_32675 for (2999ds/3Mi)
% 2.04/1.47 % (429)Instruction limit reached!
% 2.04/1.47 % (429)------------------------------
% 2.04/1.47 % (429)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.47 % (429)Termination reason: Unknown
% 2.04/1.47 % (429)Termination phase: shuffling
% 2.04/1.47
% 2.04/1.47 % (429)Memory used [KB]: 1151
% 2.04/1.47 % (429)Time elapsed: 0.003 s
% 2.04/1.47 % (429)Instructions burned: 4 (million)
% 2.04/1.47 % (429)------------------------------
% 2.04/1.47 % (429)------------------------------
% 2.04/1.47 % (427)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_32675 for (2999ds/16Mi)
% 2.04/1.47 % (428)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_32675 for (2999ds/3Mi)
% 2.04/1.48 % (428)Instruction limit reached!
% 2.04/1.48 % (428)------------------------------
% 2.04/1.48 % (428)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.48 % (428)Termination reason: Unknown
% 2.04/1.48 % (428)Termination phase: shuffling
% 2.04/1.48
% 2.04/1.48 % (428)Memory used [KB]: 1151
% 2.04/1.48 % (428)Time elapsed: 0.004 s
% 2.04/1.48 % (428)Instructions burned: 4 (million)
% 2.04/1.48 % (428)------------------------------
% 2.04/1.48 % (428)------------------------------
% 2.04/1.48 % (427)Instruction limit reached!
% 2.04/1.48 % (427)------------------------------
% 2.04/1.48 % (427)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.48 % (427)Termination reason: Unknown
% 2.04/1.48 % (427)Termination phase: Property scanning
% 2.04/1.48
% 2.04/1.48 % (427)Memory used [KB]: 1279
% 2.04/1.48 % (427)Time elapsed: 0.010 s
% 2.04/1.48 % (427)Instructions burned: 16 (million)
% 2.04/1.48 % (427)------------------------------
% 2.04/1.48 % (427)------------------------------
% 2.04/1.48 % (419)First to succeed.
% 2.04/1.48 % (434)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_32675 for (2999ds/3Mi)
% 2.04/1.48 % (434)Instruction limit reached!
% 2.04/1.48 % (434)------------------------------
% 2.04/1.48 % (434)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.48 % (434)Termination reason: Unknown
% 2.04/1.48 % (434)Termination phase: shuffling
% 2.04/1.48
% 2.04/1.48 % (434)Memory used [KB]: 1151
% 2.04/1.48 % (434)Time elapsed: 0.003 s
% 2.04/1.48 % (434)Instructions burned: 4 (million)
% 2.04/1.48 % (434)------------------------------
% 2.04/1.48 % (434)------------------------------
% 2.04/1.48 % (419)Refutation found. Thanks to Tanya!
% 2.04/1.48 % SZS status Theorem for DTF2THF_32675
% 2.04/1.48 % SZS output start Proof for DTF2THF_32675
% 2.04/1.48 thf(func_def_0, type, in: $i > $i > $o).
% 2.04/1.48 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.04/1.48 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.04/1.48 thf(func_def_8, type, powerset: $i > $i).
% 2.04/1.48 thf(func_def_10, type, setunion: $i > $i).
% 2.04/1.48 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.04/1.48 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.04/1.48 thf(func_def_26, type, prop2set: $o > $i).
% 2.04/1.48 thf(func_def_36, type, nonempty: $i > $o).
% 2.04/1.48 thf(func_def_74, type, sP1: $i > $i > $o).
% 2.04/1.48 thf(func_def_75, type, sP2: $i > $o).
% 2.04/1.48 thf(func_def_76, type, sP3: $i > $i > $o).
% 2.04/1.48 thf(func_def_77, type, sP4: $i > $i > $o).
% 2.04/1.48 thf(func_def_79, type, sK6: $i > $i).
% 2.04/1.48 thf(func_def_83, type, sK10: $i > $i > $o).
% 2.04/1.48 thf(func_def_85, type, sK12: $i > $i).
% 2.04/1.48 thf(func_def_86, type, sK13: $i > $i).
% 2.04/1.48 thf(func_def_87, type, sK14: $i > ($i > $i > $o) > $i).
% 2.04/1.48 thf(func_def_88, type, sK15: $i > ($i > $i > $o) > $i).
% 2.04/1.48 thf(func_def_89, type, sK16: $i > $i > ($i > $i > $o) > $i).
% 2.04/1.48 thf(func_def_90, type, sK17: $i > $o).
% 2.04/1.48 thf(func_def_100, type, sK27: $i > $i > $i).
% 2.04/1.48 thf(func_def_104, type, sK31: $i > $o).
% 2.04/1.48 thf(func_def_106, type, sK33: $i > $o).
% 2.04/1.48 thf(func_def_109, type, sK36: $i > $i).
% 2.04/1.48 thf(func_def_111, type, sK38: $i > $i).
% 2.04/1.48 thf(func_def_113, type, sK40: $i > $i).
% 2.04/1.48 thf(func_def_115, type, sK42: ($i > $o) > $i).
% 2.04/1.48 thf(func_def_116, type, sK43: $i > $o).
% 2.04/1.48 thf(func_def_117, type, sK44: $i > $i).
% 2.04/1.48 thf(func_def_120, type, sK47: $i > $i > $i).
% 2.04/1.48 thf(func_def_121, type, sK48: $i > $i > $i).
% 2.04/1.48 thf(func_def_122, type, sK49: $i > $i).
% 2.04/1.48 thf(func_def_126, type, sK53: $i > $o).
% 2.04/1.48 thf(func_def_138, type, sK65: $i > $i > $i).
% 2.04/1.48 thf(func_def_139, type, sK66: $i > $i > $i).
% 2.04/1.48 thf(func_def_140, type, sK67: $i > $i > $i).
% 2.04/1.48 thf(func_def_141, type, sK68: $i > $i > $i).
% 2.04/1.48 thf(func_def_142, type, sK69: $i > $i > $i > $i).
% 2.04/1.48 thf(func_def_143, type, sK70: $i > $i).
% 2.04/1.48 thf(func_def_144, type, sK71: $i > $i).
% 2.04/1.48 thf(func_def_145, type, sK72: $i > $i).
% 2.04/1.48 thf(func_def_146, type, sK73: $i > $i).
% 2.04/1.48 thf(func_def_147, type, sK74: $i > $i > $i).
% 2.04/1.48 thf(func_def_148, type, sK75: $i > $i > $i > $i).
% 2.04/1.48 thf(func_def_149, type, sK76: $i > $i > $i > $i).
% 2.04/1.48 thf(func_def_150, type, sK77: $i > $i > $i).
% 2.04/1.48 thf(func_def_151, type, sK78: $i > $i > $i).
% 2.04/1.48 thf(func_def_152, type, sK79: $i > $i).
% 2.04/1.48 thf(func_def_153, type, sK80: $i > $i > $i).
% 2.04/1.48 thf(func_def_155, type, sK82: $i > $i).
% 2.04/1.48 thf(func_def_156, type, sK83: $i > $i).
% 2.04/1.48 thf(func_def_157, type, sK84: $o > $i > $i > $i).
% 2.04/1.48 thf(func_def_163, type, sK90: $i > $i).
% 2.04/1.48 thf(func_def_167, type, sK94: $i > $i > $i).
% 2.04/1.48 thf(func_def_173, type, sK100: $i > $o).
% 2.04/1.48 thf(func_def_174, type, sK101: ($i > $o) > $i).
% 2.04/1.48 thf(func_def_176, type, sK103: $i > $i).
% 2.04/1.48 thf(func_def_183, type, sK110: $i > $o).
% 2.04/1.48 thf(func_def_184, type, sK111: $i > $i).
% 2.04/1.48 thf(func_def_185, type, sK112: ($i > $o) > $i).
% 2.04/1.48 thf(func_def_188, type, sK115: $i > $i > $i).
% 2.04/1.48 thf(func_def_197, type, sK124: $i > $i > $i).
% 2.04/1.48 thf(func_def_204, type, sK131: $i > $o).
% 2.04/1.48 thf(func_def_216, type, ph143: !>[X0: $tType]:(X0)).
% 2.04/1.48 thf(f1354,plain,(
% 2.04/1.48 $false),
% 2.04/1.48 inference(subsumption_resolution,[status(thm)],[f1349,f995])).
% 2.04/1.48 thf(f995,plain,(
% 2.04/1.48 ( ! [X4 : $i,X5 : $i] : (($true = (in @ X4 @ (setadjoin @ X4 @ X5)))) )),
% 2.04/1.48 inference(trivial_inequality_removal,[status(thm)],[f912])).
% 2.04/1.48 thf(f912,plain,(
% 2.04/1.48 ( ! [X4 : $i,X5 : $i] : (($true != $true) | ($true = (in @ X4 @ (setadjoin @ X4 @ X5)))) )),
% 2.04/1.48 inference(equality_resolution,[status(thm)],[f844])).
% 2.04/1.48 thf(f844,plain,(
% 2.04/1.48 ( ! [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X4 @ (setadjoin @ X3 @ X5))) | (X3 != X4) | ($true != $true)) )),
% 2.04/1.48 inference(definition_unfolding,[status(thm)],[f618,f656])).
% 2.04/1.48 thf(f656,plain,(
% 2.04/1.48 (setadjoinAx = $true)),
% 2.04/1.48 inference(cnf_transformation,[status(thm)],[f394])).
% 2.04/1.48 thf(f394,plain,(
% 2.04/1.48 (descrp = $true) & (exuE3e = $true) & (omegaSAx = $true) & (emptysetE = $true) & (nonemptyI1 = $true) & (setextAx = $true) & (subPowSU = $true) & (setadjoinIR = $true) & (emptysetimpfalse = $true) & (emptysetAx = $true) & (setunionAx = $true) & (wellorderingAx = $true) & (emptyinPowerset = $true) & (setadjoinE = $true) & (noeltsimpempty = $true) & (nonemptyImpWitness = $true) & (setadjoinAx = $true) & (dsetconstrI = $true) & (eqinunit = $true) & (powersetI = $true) & (exuE2 = $true) & (setbeta = $true) & (setext = $true) & (upairsetIL = $true) & (dsetconstrER = $true) & (setadjoinIL = $true) & (setunionE = $true) & (prop2setE = $true) & (nonemptyI = $true) & (setoftrueEq = $true) & (foundationAx = $true) & (notinsingleton = $true) & (replAx = $true) & (emptyinunitempty = $true) & (setadjoinOr = $true) & (omega0Ax = $true) & (omegaIndAx = $true) & (upairsetE = $true) & (exuE1 = $true) & (singletonsswitch = $true) & (dsetconstrEL = $true) & (emptyI = $true) & (powersetAx = $true) & (powersetE = $true) & (notinemptyset = $true) & (nonemptyE1 = $true) & (uniqinunit = $true) & ($true != (in @ sK99 @ (setadjoin @ sK98 @ (setadjoin @ sK99 @ emptyset)))) & (setunionI = $true) & (emptyInPowerset = $true)),
% 2.04/1.48 inference(skolemisation,[status(esa),new_symbols(skolem,[sK98,sK99])],[f204,f393])).
% 2.04/1.48 thf(f393,plain,(
% 2.04/1.48 ? [X0 : $i,X1 : $i] : ($true != (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)))) => ($true != (in @ sK99 @ (setadjoin @ sK98 @ (setadjoin @ sK99 @ emptyset))))),
% 2.04/1.48 introduced(definition,[],[choice_axiom])).
% 2.04/1.48 thf(f204,plain,(
% 2.04/1.48 (descrp = $true) & (exuE3e = $true) & (omegaSAx = $true) & (emptysetE = $true) & (nonemptyI1 = $true) & (setextAx = $true) & (subPowSU = $true) & (setadjoinIR = $true) & (emptysetimpfalse = $true) & (emptysetAx = $true) & (setunionAx = $true) & (wellorderingAx = $true) & (emptyinPowerset = $true) & (setadjoinE = $true) & (noeltsimpempty = $true) & (nonemptyImpWitness = $true) & (setadjoinAx = $true) & (dsetconstrI = $true) & (eqinunit = $true) & (powersetI = $true) & (exuE2 = $true) & (setbeta = $true) & (setext = $true) & (upairsetIL = $true) & (dsetconstrER = $true) & (setadjoinIL = $true) & (setunionE = $true) & (prop2setE = $true) & (nonemptyI = $true) & (setoftrueEq = $true) & (foundationAx = $true) & (notinsingleton = $true) & (replAx = $true) & (emptyinunitempty = $true) & (setadjoinOr = $true) & (omega0Ax = $true) & (omegaIndAx = $true) & (upairsetE = $true) & (exuE1 = $true) & (singletonsswitch = $true) & (dsetconstrEL = $true) & (emptyI = $true) & (powersetAx = $true) & (powersetE = $true) & (notinemptyset = $true) & (nonemptyE1 = $true) & (uniqinunit = $true) & ? [X0 : $i,X1 : $i] : ($true != (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)))) & (setunionI = $true) & (emptyInPowerset = $true)),
% 2.04/1.48 inference(flattening,[status(thm)],[f203])).
% 2.04/1.48 thf(f203,plain,(
% 2.04/1.48 ((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : ($true != (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)))) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (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)),
% 2.04/1.48 inference(ennf_transformation,[status(thm)],[f86])).
% 2.04/1.48 thf(f86,plain,(
% 2.04/1.48 ~((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) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.04/1.48 inference(fool_elimination,[status(thm)],[f85])).
% 2.04/1.48 thf(f85,plain,(
% 2.04/1.48 ~(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 => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => ! [X0 : $i,X1 : $i] : (in @ X1 @ (setadjoin @ X0 @ (setadjoin @ X1 @ emptyset))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.04/1.48 inference(rectify,[status(thm)],[f54])).
% 2.04/1.48 thf(f54,negated_conjecture,(
% 2.04/1.48 ~(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 => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => ! [X1 : $i,X2 : $i] : (in @ X2 @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.04/1.48 inference(negated_conjecture,[status(cth)],[f53])).
% 2.04/1.48 thf(f53,conjecture,(
% 2.04/1.48 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 => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => ! [X1 : $i,X2 : $i] : (in @ X2 @ (setadjoin @ X1 @ (setadjoin @ X2 @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.04/1.48 file('/export/starexec/sandbox/tmp/tmp.kOEAvEHP3t/DTF2THF_32675.p',upairsetIR)).
% 2.04/1.48 thf(f618,plain,(
% 2.04/1.48 ( ! [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X4 @ (setadjoin @ X3 @ X5))) | (X3 != X4) | (setadjoinAx != $true)) )),
% 2.04/1.48 inference(cnf_transformation,[status(thm)],[f392])).
% 2.04/1.48 thf(f392,plain,(
% 2.04/1.48 ((setadjoinAx = $true) | (((($true != (in @ sK96 @ sK97)) & (sK96 != sK95)) | ($true != (in @ sK96 @ (setadjoin @ sK95 @ sK97)))) & (($true = (in @ sK96 @ sK97)) | (sK96 = sK95) | ($true = (in @ sK96 @ (setadjoin @ sK95 @ sK97)))))) & (! [X3 : $i,X4 : $i,X5 : $i] : ((($true = (in @ X4 @ (setadjoin @ X3 @ X5))) | (($true != (in @ X4 @ X5)) & (X3 != X4))) & (($true = (in @ X4 @ X5)) | (X3 = X4) | ($true != (in @ X4 @ (setadjoin @ X3 @ X5))))) | (setadjoinAx != $true))),
% 2.04/1.48 inference(skolemisation,[status(esa),new_symbols(skolem,[sK95,sK96,sK97])],[f390,f391])).
% 2.04/1.48 thf(f391,plain,(
% 2.04/1.48 ? [X0 : $i,X1 : $i,X2 : $i] : (((($true != (in @ X1 @ X2)) & (X0 != X1)) | ($true != (in @ X1 @ (setadjoin @ X0 @ X2)))) & (($true = (in @ X1 @ X2)) | (X0 = X1) | ($true = (in @ X1 @ (setadjoin @ X0 @ X2))))) => (((($true != (in @ sK96 @ sK97)) & (sK96 != sK95)) | ($true != (in @ sK96 @ (setadjoin @ sK95 @ sK97)))) & (($true = (in @ sK96 @ sK97)) | (sK96 = sK95) | ($true = (in @ sK96 @ (setadjoin @ sK95 @ sK97)))))),
% 2.04/1.48 introduced(definition,[],[choice_axiom])).
% 2.04/1.48 thf(f390,plain,(
% 2.04/1.48 ((setadjoinAx = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((($true != (in @ X1 @ X2)) & (X0 != X1)) | ($true != (in @ X1 @ (setadjoin @ X0 @ X2)))) & (($true = (in @ X1 @ X2)) | (X0 = X1) | ($true = (in @ X1 @ (setadjoin @ X0 @ X2)))))) & (! [X3 : $i,X4 : $i,X5 : $i] : ((($true = (in @ X4 @ (setadjoin @ X3 @ X5))) | (($true != (in @ X4 @ X5)) & (X3 != X4))) & (($true = (in @ X4 @ X5)) | (X3 = X4) | ($true != (in @ X4 @ (setadjoin @ X3 @ X5))))) | (setadjoinAx != $true))),
% 2.04/1.48 inference(rectify,[status(thm)],[f389])).
% 2.04/1.48 thf(f389,plain,(
% 2.04/1.48 ((setadjoinAx = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((($true != (in @ X1 @ X2)) & (X0 != X1)) | ($true != (in @ X1 @ (setadjoin @ X0 @ X2)))) & (($true = (in @ X1 @ X2)) | (X0 = X1) | ($true = (in @ X1 @ (setadjoin @ X0 @ X2)))))) & (! [X0 : $i,X1 : $i,X2 : $i] : ((($true = (in @ X1 @ (setadjoin @ X0 @ X2))) | (($true != (in @ X1 @ X2)) & (X0 != X1))) & (($true = (in @ X1 @ X2)) | (X0 = X1) | ($true != (in @ X1 @ (setadjoin @ X0 @ X2))))) | (setadjoinAx != $true))),
% 2.04/1.48 inference(flattening,[status(thm)],[f388])).
% 2.04/1.48 thf(f388,plain,(
% 2.04/1.48 ((setadjoinAx = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((($true != (in @ X1 @ X2)) & (X0 != X1)) | ($true != (in @ X1 @ (setadjoin @ X0 @ X2)))) & ((($true = (in @ X1 @ X2)) | (X0 = X1)) | ($true = (in @ X1 @ (setadjoin @ X0 @ X2)))))) & (! [X0 : $i,X1 : $i,X2 : $i] : ((($true = (in @ X1 @ (setadjoin @ X0 @ X2))) | (($true != (in @ X1 @ X2)) & (X0 != X1))) & ((($true = (in @ X1 @ X2)) | (X0 = X1)) | ($true != (in @ X1 @ (setadjoin @ X0 @ X2))))) | (setadjoinAx != $true))),
% 2.04/1.48 inference(nnf_transformation,[status(thm)],[f110])).
% 2.04/1.48 thf(f110,plain,(
% 2.04/1.48 (setadjoinAx = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (setadjoin @ X0 @ X2))) <=> (($true = (in @ X1 @ X2)) | (X0 = X1)))),
% 2.04/1.48 inference(fool_elimination,[status(thm)],[f109])).
% 2.04/1.48 thf(f109,plain,(
% 2.04/1.48 (setadjoinAx = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ (setadjoin @ X0 @ X2)) <=> ((in @ X1 @ X2) | (X0 = X1))))),
% 2.04/1.48 inference(rectify,[status(thm)],[f4])).
% 2.04/1.48 thf(f4,axiom,(
% 2.04/1.48 (setadjoinAx = ! [X1 : $i,X2 : $i,X3 : $i] : ((in @ X2 @ (setadjoin @ X1 @ X3)) <=> ((in @ X2 @ X3) | (X1 = X2))))),
% 2.04/1.48 file('/export/starexec/sandbox/tmp/tmp.kOEAvEHP3t/DTF2THF_32675.p',setadjoinAx)).
% 2.04/1.48 thf(f1349,plain,(
% 2.04/1.48 ($true != (in @ sK99 @ (setadjoin @ sK99 @ emptyset)))),
% 2.04/1.48 inference(trivial_inequality_removal,[status(thm)],[f1347])).
% 2.04/1.48 thf(f1347,plain,(
% 2.04/1.48 ($true != (in @ sK99 @ (setadjoin @ sK99 @ emptyset))) | ($true != $true)),
% 2.04/1.48 inference(superposition,[status(thm)],[f625,f966])).
% 2.04/1.48 thf(f966,plain,(
% 2.04/1.48 ( ! [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X4 @ (setadjoin @ X3 @ X5))) | ($true != (in @ X4 @ X5))) )),
% 2.04/1.48 inference(trivial_inequality_removal,[status(thm)],[f843])).
% 2.04/1.48 thf(f843,plain,(
% 2.04/1.48 ( ! [X3 : $i,X4 : $i,X5 : $i] : (($true != $true) | ($true != (in @ X4 @ X5)) | ($true = (in @ X4 @ (setadjoin @ X3 @ X5)))) )),
% 2.04/1.48 inference(definition_unfolding,[status(thm)],[f619,f656])).
% 2.04/1.48 thf(f619,plain,(
% 2.04/1.48 ( ! [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X4 @ (setadjoin @ X3 @ X5))) | ($true != (in @ X4 @ X5)) | (setadjoinAx != $true)) )),
% 2.04/1.48 inference(cnf_transformation,[status(thm)],[f392])).
% 2.04/1.48 thf(f625,plain,(
% 2.04/1.48 ($true != (in @ sK99 @ (setadjoin @ sK98 @ (setadjoin @ sK99 @ emptyset))))),
% 2.04/1.48 inference(cnf_transformation,[status(thm)],[f394])).
% 2.04/1.48 % SZS output end Proof for DTF2THF_32675
% 2.04/1.48 % (419)------------------------------
% 2.04/1.48 % (419)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.48 % (419)Termination reason: Refutation
% 2.04/1.48
% 2.04/1.48 % (419)Memory used [KB]: 6652
% 2.04/1.48 % (419)Time elapsed: 0.060 s
% 2.04/1.48 % (419)Instructions burned: 180 (million)
% 2.04/1.48 % (419)------------------------------
% 2.04/1.48 % (419)------------------------------
% 2.04/1.48 % (413)Success in time 0.065 s
%------------------------------------------------------------------------------