%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU530^1 : 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 : n005.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 2.23s 1.57s
% Output : Refutation 2.23s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 6
% Syntax : Number of formulae : 37 ( 12 unt; 0 typ; 0 def)
% Number of atoms : 980 ( 267 equ; 0 cnn)
% Maximal formula atoms : 50 ( 26 avg)
% Number of connectives : 613 ( 38 ~; 24 |; 154 &; 200 @)
% ( 3 <=>; 194 =>; 0 <=; 0 <~>)
% Maximal formula depth : 54 ( 13 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 121 ( 118 usr; 55 con; 0-3 aty)
% Number of variables : 64 ( 0 ^; 46 !; 18 ?; 64 :)
% 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_71,type,
sP1: $i > $i > $o ).
thf(func_def_72,type,
sP2: $i > $o ).
thf(func_def_73,type,
sP3: $i > $i > $o ).
thf(func_def_74,type,
sP4: $i > $i > $o ).
thf(func_def_77,type,
sK7: $i > $o ).
thf(func_def_82,type,
sK12: $i > $i ).
thf(func_def_86,type,
sK16: $i > $i > $i ).
thf(func_def_92,type,
sK22: $i > $i ).
thf(func_def_99,type,
sK29: $i > $i > $i ).
thf(func_def_103,type,
sK33: ( $i > $o ) > $i ).
thf(func_def_104,type,
sK34: $i > $o ).
thf(func_def_112,type,
sK42: $i > $i > $o ).
thf(func_def_113,type,
sK43: $i > $i ).
thf(func_def_114,type,
sK44: $i > $i ).
thf(func_def_115,type,
sK45: ( $i > $i > $o ) > $i > $i ).
thf(func_def_116,type,
sK46: ( $i > $i > $o ) > $i > $i ).
thf(func_def_117,type,
sK47: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_122,type,
sK52: $i > $i ).
thf(func_def_127,type,
sK57: $i > $i ).
thf(func_def_129,type,
sK59: $i > $o ).
thf(func_def_130,type,
sK60: $i > $i ).
thf(func_def_131,type,
sK61: ( $i > $o ) > $i ).
thf(func_def_137,type,
sK67: $i > $o ).
thf(func_def_140,type,
sK70: $i > $i ).
thf(func_def_152,type,
sK82: $i > $i ).
thf(func_def_155,type,
sK85: $i > $i ).
thf(func_def_157,type,
sK87: $i > $o ).
thf(func_def_160,type,
sK90: $i > $o ).
thf(func_def_164,type,
sK94: $i > $i > $i ).
thf(func_def_165,type,
sK95: $i > $i > $i ).
thf(func_def_166,type,
sK96: $i > $i > $i ).
thf(func_def_167,type,
sK97: $i > $i > $i ).
thf(func_def_168,type,
sK98: $i > $i > $i ).
thf(func_def_169,type,
sK99: $i > $i > $i ).
thf(func_def_170,type,
sK100: $i > $i > $i > $i ).
thf(func_def_171,type,
sK101: $i > $i ).
thf(func_def_172,type,
sK102: $i > $i ).
thf(func_def_173,type,
sK103: $i > $i ).
thf(func_def_174,type,
sK104: $i > $i ).
thf(func_def_175,type,
sK105: $i > $i > $i ).
thf(func_def_176,type,
sK106: $i > $i > $i > $i ).
thf(func_def_177,type,
sK107: $i > $i > $i > $i ).
thf(func_def_178,type,
sK108: $i > $i > $i ).
thf(func_def_179,type,
sK109: $i > $i > $i ).
thf(func_def_180,type,
sK110: $i > $i ).
thf(func_def_182,type,
sK112: $i > $i ).
thf(func_def_183,type,
sK113: $i > $i ).
thf(func_def_187,type,
sK117: $i > $o ).
thf(func_def_191,type,
sK121: $o > $i > $i > $i ).
thf(func_def_193,type,
sK123: $i > $o ).
thf(func_def_194,type,
sK124: $i > $i ).
thf(func_def_195,type,
sK125: ( $i > $o ) > $i ).
thf(func_def_197,type,
sK127: $i > $i > $i ).
thf(func_def_203,type,
sK133: $i > $i > $i ).
thf(func_def_204,type,
sK134: $i > $i > $i ).
thf(func_def_206,type,
ph136:
!>[X0: $tType] : X0 ).
thf(f1222,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1221,f932]) ).
thf(f932,plain,
! [X2: $i,X3: $i] :
( $true
= ( in @ X3 @ ( setadjoin @ X3 @ X2 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f754]) ).
thf(f754,plain,
! [X2: $i,X3: $i] :
( ( $true
= ( in @ X3 @ ( setadjoin @ X3 @ X2 ) ) )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f558,f479]) ).
thf(f479,plain,
setadjoinIL = $true,
inference(cnf_transformation,[status(thm)],[f222]) ).
thf(f222,plain,
( ( uniqinunit = $true )
& ( emptysetimpfalse = $true )
& ( replAx = $true )
& ( setunionE = $true )
& ( powersetAx = $true )
& ( setunionI = $true )
& ( nonemptyI = $true )
& ( setadjoinAx = $true )
& ( dsetconstrEL = $true )
& ( emptysetE = $true )
& ( wellorderingAx = $true )
& ( setextAx = $true )
& ( setadjoinE = $true )
& ( setunionAx = $true )
& ( nonemptyE1 = $true )
& ( emptyinPowerset = $true )
& ( emptyinunitempty = $true )
& ( powersetI = $true )
& ( setbeta = $true )
& ( notinsingleton = $true )
& ( omegaIndAx = $true )
& ( setadjoinOr = $true )
& ( setadjoinIL = $true )
& ( foundationAx = $true )
& ( emptysetAx = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( nonemptyI1 = $true )
& ( powersetE = $true )
& ( exuE2 = $true )
& ( notinemptyset = $true )
& ( $true
= ( in @ sK10 @ ( setadjoin @ sK11 @ emptyset ) ) )
& ( $true
!= ( in @ sK11 @ ( setadjoin @ sK10 @ emptyset ) ) )
& ( emptyInPowerset = $true )
& ( eqinunit = $true )
& ( descrp = $true )
& ( exuE3e = $true )
& ( dsetconstrI = $true )
& ( subPowSU = $true )
& ( emptyI = $true )
& ( setadjoinIR = $true )
& ( setoftrueEq = $true )
& ( omegaSAx = $true )
& ( prop2setE = $true )
& ( omega0Ax = $true )
& ( setext = $true )
& ( noeltsimpempty = $true )
& ( nonemptyImpWitness = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK10,sK11])],[f191,f221]) ).
thf(f221,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
& ( ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) )
!= $true ) )
=> ( ( $true
= ( in @ sK10 @ ( setadjoin @ sK11 @ emptyset ) ) )
& ( $true
!= ( in @ sK11 @ ( setadjoin @ sK10 @ emptyset ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f191,plain,
( ( uniqinunit = $true )
& ( emptysetimpfalse = $true )
& ( replAx = $true )
& ( setunionE = $true )
& ( powersetAx = $true )
& ( setunionI = $true )
& ( nonemptyI = $true )
& ( setadjoinAx = $true )
& ( dsetconstrEL = $true )
& ( emptysetE = $true )
& ( wellorderingAx = $true )
& ( setextAx = $true )
& ( setadjoinE = $true )
& ( setunionAx = $true )
& ( nonemptyE1 = $true )
& ( emptyinPowerset = $true )
& ( emptyinunitempty = $true )
& ( powersetI = $true )
& ( setbeta = $true )
& ( notinsingleton = $true )
& ( omegaIndAx = $true )
& ( setadjoinOr = $true )
& ( setadjoinIL = $true )
& ( foundationAx = $true )
& ( emptysetAx = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( nonemptyI1 = $true )
& ( powersetE = $true )
& ( exuE2 = $true )
& ( notinemptyset = $true )
& ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
& ( ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) )
!= $true ) )
& ( emptyInPowerset = $true )
& ( eqinunit = $true )
& ( descrp = $true )
& ( exuE3e = $true )
& ( dsetconstrI = $true )
& ( subPowSU = $true )
& ( emptyI = $true )
& ( setadjoinIR = $true )
& ( setoftrueEq = $true )
& ( omegaSAx = $true )
& ( prop2setE = $true )
& ( omega0Ax = $true )
& ( setext = $true )
& ( noeltsimpempty = $true )
& ( nonemptyImpWitness = $true ) ),
inference(flattening,[status(thm)],[f190]) ).
thf(f190,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
& ( ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) )
!= $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)],[f98]) ).
thf(f98,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 )
=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f97]) ).
thf(f97,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
=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) )
=> ( in @ X1 @ ( setadjoin @ X0 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f51]) ).
thf(f51,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
=> ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f50]) ).
thf(f50,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
=> ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( in @ X2 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.xT3Ehs1D5u/DTF2THF_25397.p',singletonsswitch) ).
thf(f558,plain,
! [X2: $i,X3: $i] :
( ( $true
= ( in @ X3 @ ( setadjoin @ X3 @ X2 ) ) )
| ( setadjoinIL != $true ) ),
inference(cnf_transformation,[status(thm)],[f295]) ).
thf(f295,plain,
( ( ( setadjoinIL = $true )
| ( $true
!= ( in @ sK51 @ ( setadjoin @ sK51 @ sK50 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( $true
= ( in @ X3 @ ( setadjoin @ X3 @ X2 ) ) )
| ( setadjoinIL != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK50,sK51])],[f293,f294]) ).
thf(f294,plain,
( ? [X0: $i,X1: $i] :
( $true
!= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) )
=> ( $true
!= ( in @ sK51 @ ( setadjoin @ sK51 @ sK50 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f293,plain,
( ( ( setadjoinIL = $true )
| ? [X0: $i,X1: $i] :
( $true
!= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) ) )
& ( ! [X2: $i,X3: $i] :
( $true
= ( in @ X3 @ ( setadjoin @ X3 @ X2 ) ) )
| ( setadjoinIL != $true ) ) ),
inference(rectify,[status(thm)],[f292]) ).
thf(f292,plain,
( ( ( setadjoinIL = $true )
| ? [X0: $i,X1: $i] :
( $true
!= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) ) )
& ( ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) )
| ( setadjoinIL != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f80]) ).
thf(f80,plain,
( ( setadjoinIL = $true )
<=> ! [X0: $i,X1: $i] :
( $true
= ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) ) ),
inference(fool_elimination,[status(thm)],[f79]) ).
thf(f79,plain,
( setadjoinIL
= ( ! [X0: $i,X1: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f32]) ).
thf(f32,axiom,
( setadjoinIL
= ( ! [X2: $i,X1: $i] : ( in @ X1 @ ( setadjoin @ X1 @ X2 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.xT3Ehs1D5u/DTF2THF_25397.p',setadjoinIL) ).
thf(f1221,plain,
( $true
!= ( in @ sK10 @ ( setadjoin @ sK10 @ emptyset ) ) ),
inference(superposition,[status(thm)],[f469,f1198]) ).
thf(f1198,plain,
sK10 = sK11,
inference(trivial_inequality_removal,[status(thm)],[f1187]) ).
thf(f1187,plain,
( ( $true != $true )
| ( sK10 = sK11 ) ),
inference(superposition,[status(thm)],[f924,f470]) ).
thf(f470,plain,
( $true
= ( in @ sK10 @ ( setadjoin @ sK11 @ emptyset ) ) ),
inference(cnf_transformation,[status(thm)],[f222]) ).
thf(f924,plain,
! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
| ( X0 = X1 ) ),
inference(trivial_inequality_removal,[status(thm)],[f779]) ).
thf(f779,plain,
! [X0: $i,X1: $i] :
( ( $true != $true )
| ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
| ( X0 = X1 ) ),
inference(definition_unfolding,[status(thm)],[f587,f501]) ).
thf(f501,plain,
uniqinunit = $true,
inference(cnf_transformation,[status(thm)],[f222]) ).
thf(f587,plain,
! [X0: $i,X1: $i] :
( ( X0 = X1 )
| ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
| ( uniqinunit != $true ) ),
inference(cnf_transformation,[status(thm)],[f332]) ).
thf(f332,plain,
( ( ! [X0: $i,X1: $i] :
( ( X0 = X1 )
| ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) )
| ( uniqinunit != $true ) )
& ( ( uniqinunit = $true )
| ( ( sK68 != sK69 )
& ( $true
= ( in @ sK68 @ ( setadjoin @ sK69 @ emptyset ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK68,sK69])],[f330,f331]) ).
thf(f331,plain,
( ? [X2: $i,X3: $i] :
( ( X2 != X3 )
& ( ( in @ X2 @ ( setadjoin @ X3 @ emptyset ) )
= $true ) )
=> ( ( sK68 != sK69 )
& ( $true
= ( in @ sK68 @ ( setadjoin @ sK69 @ emptyset ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f330,plain,
( ( ! [X0: $i,X1: $i] :
( ( X0 = X1 )
| ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) )
| ( uniqinunit != $true ) )
& ( ( uniqinunit = $true )
| ? [X2: $i,X3: $i] :
( ( X2 != X3 )
& ( ( in @ X2 @ ( setadjoin @ X3 @ emptyset ) )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f329]) ).
thf(f329,plain,
( ( ! [X0: $i,X1: $i] :
( ( X0 = X1 )
| ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) )
| ( uniqinunit != $true ) )
& ( ( uniqinunit = $true )
| ? [X0: $i,X1: $i] :
( ( X0 != X1 )
& ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f200]) ).
thf(f200,plain,
( ! [X0: $i,X1: $i] :
( ( X0 = X1 )
| ( $true
!= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) ) )
<=> ( uniqinunit = $true ) ),
inference(ennf_transformation,[status(thm)],[f94]) ).
thf(f94,plain,
( ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) ) )
=> ( X0 = X1 ) )
<=> ( uniqinunit = $true ) ),
inference(fool_elimination,[status(thm)],[f93]) ).
thf(f93,plain,
( uniqinunit
= ( ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( setadjoin @ X1 @ emptyset ) )
=> ( X0 = X1 ) ) ) ),
inference(rectify,[status(thm)],[f47]) ).
thf(f47,axiom,
( uniqinunit
= ( ! [X1: $i,X2: $i] :
( ( in @ X1 @ ( setadjoin @ X2 @ emptyset ) )
=> ( X1 = X2 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.xT3Ehs1D5u/DTF2THF_25397.p',uniqinunit) ).
thf(f469,plain,
( $true
!= ( in @ sK11 @ ( setadjoin @ sK10 @ emptyset ) ) ),
inference(cnf_transformation,[status(thm)],[f222]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU530^1 : TPTP v9.2.1. Released v3.7.0.
% 0.13/0.14 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.35 % Computer : n005.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Tue Feb 24 12:25:44 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.47 ---- Original DTF file ---
% 0.29/0.47 thf(spec,logic,$$dhol).
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 % File : SEU530^1 : TPTP v9.2.1. Released v3.7.0.
% 0.29/0.47 % Domain : Set Theory
% 0.29/0.47 % Problem : Preliminary Notions - Equality Laws
% 0.29/0.47 % Version : Especial.
% 0.29/0.47 % English : (! x:i.! y:i.in x (setadjoin y emptyset) -> in y
% 0.29/0.47 % (setadjoin x emptyset))
% 0.29/0.47
% 0.29/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.29/0.47 % Source : [Bro08]
% 0.29/0.47 % Names : ZFC032g [Bro08]
% 0.29/0.47
% 0.29/0.47 % Status : Theorem
% 0.29/0.47 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.31 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, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v3.7.0
% 0.29/0.47 % Syntax : Number of formulae : 107 ( 49 unt; 57 typ; 49 def)
% 0.29/0.47 % Number of atoms : 273 ( 65 equ; 0 cnn)
% 0.29/0.47 % Maximal formula atoms : 48 ( 5 avg)
% 0.29/0.47 % Number of connectives : 445 ( 9 ~; 4 |; 19 &; 291 @)
% 0.29/0.47 % ( 8 <=>; 114 =>; 0 <=; 0 <~>)
% 0.29/0.47 % Maximal formula depth : 53 ( 2 avg)
% 0.29/0.47 % Number of types : 2 ( 0 usr)
% 0.29/0.47 % Number of type conns : 27 ( 27 >; 0 *; 0 +; 0 <<)
% 0.29/0.47 % Number of symbols : 60 ( 57 usr; 50 con; 0-2 aty)
% 0.29/0.47 % Number of variables : 146 ( 16 ^; 112 !; 18 ?; 146 :)
% 0.29/0.47 % SPC : TH0_THM_EQU_NAR
% 0.29/0.47
% 0.29/0.47 % Comments : http://mathgate.info/detsetitem.php?id=471
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 thf(in_type,type,
% 0.29/0.47 in: $i > $i > $o ).
% 0.29/0.47
% 0.29/0.47 thf(exu_type,type,
% 0.29/0.47 exu: ( $i > $o ) > $o ).
% 0.29/0.47
% 0.29/0.47 thf(exu,definition,
% 0.29/0.47 ( exu
% 0.29/0.47 = ( ^ [Xphi: $i > $o] :
% 0.29/0.47 ? [Xx: $i] :
% 0.29/0.47 ( ( Xphi @ Xx )
% 0.29/0.47 & ! [Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xy )
% 0.29/0.47 => ( Xx = Xy ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setextAx_type,type,
% 0.29/0.47 setextAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setextAx,definition,
% 0.29/0.47 ( setextAx
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 <=> ( in @ Xx @ B ) )
% 0.29/0.47 => ( A = B ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyset_type,type,
% 0.29/0.47 emptyset: $i ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetAx_type,type,
% 0.29/0.47 emptysetAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetAx,definition,
% 0.29/0.47 ( emptysetAx
% 0.29/0.47 = ( ! [Xx: $i] :
% 0.29/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoin_type,type,
% 0.29/0.47 setadjoin: $i > $i > $i ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinAx_type,type,
% 0.29/0.47 setadjoinAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinAx,definition,
% 0.29/0.47 ( setadjoinAx
% 0.29/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.47 <=> ( ( Xy = Xx )
% 0.29/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(powerset_type,type,
% 0.29/0.47 powerset: $i > $i ).
% 0.29/0.47
% 0.29/0.47 thf(powersetAx_type,type,
% 0.29/0.47 powersetAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(powersetAx,definition,
% 0.29/0.47 ( powersetAx
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.29/0.47 <=> ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setunion_type,type,
% 0.29/0.47 setunion: $i > $i ).
% 0.29/0.47
% 0.29/0.47 thf(setunionAx_type,type,
% 0.29/0.47 setunionAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setunionAx,definition,
% 0.29/0.47 ( setunionAx
% 0.29/0.47 = ( ! [A: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.29/0.47 <=> ? [B: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(omega_type,type,
% 0.29/0.47 omega: $i ).
% 0.29/0.47
% 0.29/0.47 thf(omega0Ax_type,type,
% 0.29/0.47 omega0Ax: $o ).
% 0.29/0.47
% 0.29/0.47 thf(omega0Ax,definition,
% 0.29/0.47 ( omega0Ax
% 0.29/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(omegaSAx_type,type,
% 0.29/0.47 omegaSAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(omegaSAx,definition,
% 0.29/0.47 ( omegaSAx
% 0.29/0.47 = ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ omega )
% 0.29/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(omegaIndAx_type,type,
% 0.29/0.47 omegaIndAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(omegaIndAx,definition,
% 0.29/0.47 ( omegaIndAx
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ( ( in @ emptyset @ A )
% 0.29/0.47 & ! [Xx: $i] :
% 0.29/0.47 ( ( ( in @ Xx @ omega )
% 0.29/0.47 & ( in @ Xx @ A ) )
% 0.29/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.29/0.47 => ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ omega )
% 0.29/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(replAx_type,type,
% 0.29/0.47 replAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(replAx,definition,
% 0.29/0.47 ( replAx
% 0.29/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( exu
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.29/0.47 => ? [B: $i] :
% 0.29/0.47 ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 <=> ? [Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ A )
% 0.29/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(foundationAx_type,type,
% 0.29/0.47 foundationAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(foundationAx,definition,
% 0.29/0.47 ( foundationAx
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.29/0.47 => ? [B: $i] :
% 0.29/0.47 ( ( in @ B @ A )
% 0.29/0.47 & ~ ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(wellorderingAx_type,type,
% 0.29/0.47 wellorderingAx: $o ).
% 0.29/0.47
% 0.29/0.47 thf(wellorderingAx,definition,
% 0.29/0.47 ( wellorderingAx
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ? [B: $i] :
% 0.29/0.47 ( ! [C: $i] :
% 0.29/0.47 ( ( in @ C @ B )
% 0.29/0.47 => ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ C )
% 0.29/0.47 => ( in @ Xx @ A ) ) )
% 0.29/0.47 & ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( ( in @ Xx @ A )
% 0.29/0.47 & ( in @ Xy @ A ) )
% 0.29/0.47 => ( ! [C: $i] :
% 0.29/0.47 ( ( in @ C @ B )
% 0.29/0.47 => ( ( in @ Xx @ C )
% 0.29/0.47 <=> ( in @ Xy @ C ) ) )
% 0.29/0.47 => ( Xx = Xy ) ) )
% 0.29/0.47 & ! [C: $i,D: $i] :
% 0.29/0.47 ( ( ( in @ C @ B )
% 0.29/0.47 & ( in @ D @ B ) )
% 0.29/0.47 => ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ C )
% 0.29/0.47 => ( in @ Xx @ D ) )
% 0.29/0.47 | ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ D )
% 0.29/0.47 => ( in @ Xx @ C ) ) ) )
% 0.29/0.47 & ! [C: $i] :
% 0.29/0.47 ( ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ C )
% 0.29/0.47 => ( in @ Xx @ A ) )
% 0.29/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.29/0.47 => ? [D: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ D @ B )
% 0.29/0.47 & ( in @ Xx @ C )
% 0.29/0.47 & ~ ? [Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ D )
% 0.29/0.47 & ( in @ Xy @ C ) )
% 0.29/0.47 & ! [E: $i] :
% 0.29/0.47 ( ( in @ E @ B )
% 0.29/0.47 => ( ! [Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ E )
% 0.29/0.47 => ( in @ Xy @ D ) )
% 0.29/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(descr_type,type,
% 0.29/0.47 descr: ( $i > $o ) > $i ).
% 0.29/0.47
% 0.29/0.47 thf(descrp_type,type,
% 0.29/0.47 descrp: $o ).
% 0.29/0.47
% 0.29/0.47 thf(descrp,definition,
% 0.29/0.47 ( descrp
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ( Xphi
% 0.29/0.47 @ ( descr
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstr_type,type,
% 0.29/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrI_type,type,
% 0.29/0.47 dsetconstrI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrI,definition,
% 0.29/0.47 ( dsetconstrI
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( ( Xphi @ Xx )
% 0.29/0.47 => ( in @ Xx
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrEL_type,type,
% 0.29/0.47 dsetconstrEL: $o ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrEL,definition,
% 0.29/0.47 ( dsetconstrEL
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrER_type,type,
% 0.29/0.47 dsetconstrER: $o ).
% 0.29/0.47
% 0.29/0.47 thf(dsetconstrER,definition,
% 0.29/0.47 ( dsetconstrER
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuE1_type,type,
% 0.29/0.47 exuE1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuE1,definition,
% 0.29/0.47 ( exuE1
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ? [Xx: $i] :
% 0.29/0.47 ( ( Xphi @ Xx )
% 0.29/0.47 & ! [Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xy )
% 0.29/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(prop2set_type,type,
% 0.29/0.47 prop2set: $o > $i ).
% 0.29/0.47
% 0.29/0.47 thf(prop2set,definition,
% 0.29/0.47 ( prop2set
% 0.29/0.47 = ( ^ [Xphi: $o] :
% 0.29/0.47 ( dsetconstr @ ( powerset @ emptyset )
% 0.29/0.47 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(prop2setE_type,type,
% 0.29/0.47 prop2setE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(prop2setE,definition,
% 0.29/0.47 ( prop2setE
% 0.29/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.29/0.47 => Xphi ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetE_type,type,
% 0.29/0.47 emptysetE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetE,definition,
% 0.29/0.47 ( emptysetE
% 0.29/0.47 = ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ emptyset )
% 0.29/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetimpfalse_type,type,
% 0.29/0.47 emptysetimpfalse: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptysetimpfalse,definition,
% 0.29/0.47 ( emptysetimpfalse
% 0.29/0.47 = ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ emptyset )
% 0.29/0.47 => $false ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(notinemptyset_type,type,
% 0.29/0.47 notinemptyset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(notinemptyset,definition,
% 0.29/0.47 ( notinemptyset
% 0.29/0.47 = ( ! [Xx: $i] :
% 0.29/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuE3e_type,type,
% 0.29/0.47 exuE3e: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuE3e,definition,
% 0.29/0.47 ( exuE3e
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setext_type,type,
% 0.29/0.47 setext: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setext,definition,
% 0.29/0.47 ( setext
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ Xx @ B ) )
% 0.29/0.47 => ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 => ( in @ Xx @ A ) )
% 0.29/0.47 => ( A = B ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyI_type,type,
% 0.29/0.47 emptyI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptyI,definition,
% 0.29/0.47 ( emptyI
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ~ ( in @ Xx @ A )
% 0.29/0.47 => ( A = emptyset ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(noeltsimpempty_type,type,
% 0.29/0.47 noeltsimpempty: $o ).
% 0.29/0.47
% 0.29/0.47 thf(noeltsimpempty,definition,
% 0.29/0.47 ( noeltsimpempty
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ~ ( in @ Xx @ A )
% 0.29/0.47 => ( A = emptyset ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setbeta_type,type,
% 0.29/0.47 setbeta: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setbeta,definition,
% 0.29/0.47 ( setbeta
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( ( in @ Xx
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.29/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(nonempty_type,type,
% 0.29/0.47 nonempty: $i > $o ).
% 0.29/0.47
% 0.29/0.47 thf(nonempty,definition,
% 0.29/0.47 ( nonempty
% 0.29/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyE1_type,type,
% 0.29/0.47 nonemptyE1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyE1,definition,
% 0.29/0.47 ( nonemptyE1
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ( nonempty @ A )
% 0.29/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyI_type,type,
% 0.29/0.47 nonemptyI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyI,definition,
% 0.29/0.47 ( nonemptyI
% 0.29/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( ( Xphi @ Xx )
% 0.29/0.47 => ( nonempty
% 0.29/0.47 @ ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyI1_type,type,
% 0.29/0.47 nonemptyI1: $o ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyI1,definition,
% 0.29/0.47 ( nonemptyI1
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.29/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinIL_type,type,
% 0.29/0.47 setadjoinIL: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinIL,definition,
% 0.29/0.47 ( setadjoinIL
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyinunitempty_type,type,
% 0.29/0.47 emptyinunitempty: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptyinunitempty,definition,
% 0.29/0.47 ( emptyinunitempty
% 0.29/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinIR_type,type,
% 0.29/0.47 setadjoinIR: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinIR,definition,
% 0.29/0.47 ( setadjoinIR
% 0.29/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ A )
% 0.29/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinE_type,type,
% 0.29/0.47 setadjoinE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinE,definition,
% 0.29/0.47 ( setadjoinE
% 0.29/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.47 => ! [Xphi: $o] :
% 0.29/0.47 ( ( ( Xy = Xx )
% 0.29/0.47 => Xphi )
% 0.29/0.47 => ( ( ( in @ Xy @ A )
% 0.29/0.47 => Xphi )
% 0.29/0.47 => Xphi ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinOr_type,type,
% 0.29/0.47 setadjoinOr: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setadjoinOr,definition,
% 0.29/0.47 ( setadjoinOr
% 0.29/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.29/0.47 => ( ( Xy = Xx )
% 0.29/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setoftrueEq_type,type,
% 0.29/0.47 setoftrueEq: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setoftrueEq,definition,
% 0.29/0.47 ( setoftrueEq
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ( dsetconstr @ A
% 0.29/0.47 @ ^ [Xx: $i] : $true )
% 0.29/0.47 = A ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(powersetI_type,type,
% 0.29/0.47 powersetI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(powersetI,definition,
% 0.29/0.47 ( powersetI
% 0.29/0.47 = ( ! [A: $i,B: $i] :
% 0.29/0.47 ( ! [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 => ( in @ Xx @ A ) )
% 0.29/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyinPowerset_type,type,
% 0.29/0.47 emptyinPowerset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptyinPowerset,definition,
% 0.29/0.47 ( emptyinPowerset
% 0.29/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(emptyInPowerset_type,type,
% 0.29/0.47 emptyInPowerset: $o ).
% 0.29/0.47
% 0.29/0.47 thf(emptyInPowerset,definition,
% 0.29/0.47 ( emptyInPowerset
% 0.29/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(powersetE_type,type,
% 0.29/0.47 powersetE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(powersetE,definition,
% 0.29/0.47 ( powersetE
% 0.29/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.29/0.47 => ( ( in @ Xx @ B )
% 0.29/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setunionI_type,type,
% 0.29/0.47 setunionI: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setunionI,definition,
% 0.29/0.47 ( setunionI
% 0.29/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 => ( ( in @ B @ A )
% 0.29/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(setunionE_type,type,
% 0.29/0.47 setunionE: $o ).
% 0.29/0.47
% 0.29/0.47 thf(setunionE,definition,
% 0.29/0.47 ( setunionE
% 0.29/0.47 = ( ! [A: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.29/0.47 => ! [Xphi: $o] :
% 0.29/0.47 ( ! [B: $i] :
% 0.29/0.47 ( ( in @ Xx @ B )
% 0.29/0.47 => ( ( in @ B @ A )
% 0.29/0.47 => Xphi ) )
% 0.29/0.47 => Xphi ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subPowSU_type,type,
% 0.29/0.47 subPowSU: $o ).
% 0.29/0.47
% 0.29/0.47 thf(subPowSU,definition,
% 0.29/0.47 ( subPowSU
% 0.29/0.47 = ( ! [A: $i,Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(exuE2_type,type,
% 0.29/0.47 exuE2: $o ).
% 0.29/0.47
% 0.29/0.47 thf(exuE2,definition,
% 0.29/0.47 ( exuE2
% 0.29/0.47 = ( ! [Xphi: $i > $o] :
% 0.29/0.47 ( ( exu
% 0.29/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.29/0.47 => ? [Xx: $i] :
% 0.29/0.47 ! [Xy: $i] :
% 0.29/0.47 ( ( Xphi @ Xy )
% 0.29/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyImpWitness_type,type,
% 0.29/0.47 nonemptyImpWitness: $o ).
% 0.29/0.47
% 0.29/0.47 thf(nonemptyImpWitness,definition,
% 0.29/0.47 ( nonemptyImpWitness
% 0.29/0.47 = ( ! [A: $i] :
% 0.29/0.47 ( ( nonempty @ A )
% 0.29/0.47 => ? [Xx: $i] :
% 0.29/0.47 ( ( in @ Xx @ A )
% 0.29/0.47 & $true ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(uniqinunit_type,type,
% 0.29/0.47 uniqinunit: $o ).
% 0.29/0.47
% 0.29/0.47 thf(uniqinunit,definition,
% 0.29/0.47 ( uniqinunit
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(notinsingleton_type,type,
% 0.29/0.47 notinsingleton: $o ).
% 0.29/0.47
% 0.29/0.47 thf(notinsingleton,definition,
% 0.29/0.47 ( notinsingleton
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xx != Xy )
% 0.29/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(eqinunit_type,type,
% 0.29/0.47 eqinunit: $o ).
% 0.29/0.47
% 0.29/0.47 thf(eqinunit,definition,
% 0.29/0.47 ( eqinunit
% 0.29/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( Xx = Xy )
% 0.29/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(singletonsswitch,conjecture,
% 0.29/0.47 ( setextAx
% 0.29/0.47 => ( emptysetAx
% 0.29/0.47 => ( setadjoinAx
% 0.29/0.47 => ( powersetAx
% 0.29/0.47 => ( setunionAx
% 0.29/0.47 => ( omega0Ax
% 0.29/0.47 => ( omegaSAx
% 0.29/0.47 => ( omegaIndAx
% 0.29/0.47 => ( replAx
% 0.29/0.47 => ( foundationAx
% 0.29/0.47 => ( wellorderingAx
% 0.29/0.47 => ( descrp
% 0.29/0.47 => ( dsetconstrI
% 0.29/0.47 => ( dsetconstrEL
% 0.29/0.47 => ( dsetconstrER
% 0.29/0.47 => ( exuE1
% 0.29/0.47 => ( prop2setE
% 0.29/0.47 => ( emptysetE
% 0.29/0.47 => ( emptysetimpfalse
% 0.29/0.47 => ( notinemptyset
% 0.29/0.47 => ( exuE3e
% 0.29/0.47 => ( setext
% 0.29/0.47 => ( emptyI
% 0.29/0.47 => ( noeltsimpempty
% 0.29/0.47 => ( setbeta
% 0.29/0.47 => ( nonemptyE1
% 0.29/0.47 => ( nonemptyI
% 0.29/0.47 => ( nonemptyI1
% 0.29/0.47 => ( setadjoinIL
% 0.29/0.47 => ( emptyinunitempty
% 0.29/0.47 => ( setadjoinIR
% 0.29/0.47 => ( setadjoinE
% 0.29/0.47 => ( setadjoinOr
% 0.29/0.47 => ( setoftrueEq
% 0.29/0.47 => ( powersetI
% 0.29/0.47 => ( emptyinPowerset
% 0.29/0.47 => ( emptyInPowerset
% 0.29/0.47 => ( powersetE
% 0.29/0.47 => ( setunionI
% 0.29/0.47 => ( setunionE
% 0.29/0.47 => ( subPowSU
% 0.29/0.47 => ( exuE2
% 0.29/0.47 => ( nonemptyImpWitness
% 0.29/0.47 => ( uniqinunit
% 0.29/0.47 => ( notinsingleton
% 0.29/0.47 => ( eqinunit
% 0.29/0.47 => ! [Xx: $i,Xy: $i] :
% 0.29/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.29/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 ------------------------
% 2.08/1.47 ---- Embedded in THF ---
% 2.08/1.47 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.08/1.47 %%% Generated on Tue Feb 24 12:25:45 EST 2026
% 2.08/1.47 %%% using '$$dhol' embedding, version 1.3.0.
% 2.08/1.47 %%% Logic specification used:
% 2.08/1.47 %%% thf(spec, logic, $$dhol).
% 2.08/1.47
% 2.08/1.47 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.xT3Ehs1D5u/DTF2DTF25397.p
% See solution above
% 2.08/1.47 ------------------------
% 2.08/1.49 ---- Cleaned THF ---
% 2.08/1.49 thf(in_type,type,
% 2.08/1.49 in: $i > $i > $o ).
% 2.08/1.49
% 2.08/1.49 thf(exu_type,type,
% 2.08/1.49 exu: ( $i > $o ) > $o ).
% 2.08/1.49
% 2.08/1.49 thf(exu,definition,
% 2.08/1.49 ( exu
% 2.08/1.49 = ( ^ [Xphi: $i > $o] :
% 2.08/1.49 ? [Xx: $i] :
% 2.08/1.49 ( ( Xphi @ Xx )
% 2.08/1.49 & ! [Xy: $i] :
% 2.08/1.49 ( ( Xphi @ Xy )
% 2.08/1.49 => ( Xx = Xy ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(setextAx_type,type,
% 2.08/1.49 setextAx: $o ).
% 2.08/1.49
% 2.08/1.49 thf(setextAx,definition,
% 2.08/1.49 ( setextAx
% 2.08/1.49 = ( ! [A: $i,B: $i] :
% 2.08/1.49 ( ! [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ A )
% 2.08/1.49 <=> ( in @ Xx @ B ) )
% 2.08/1.49 => ( A = B ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(emptyset_type,type,
% 2.08/1.49 emptyset: $i ).
% 2.08/1.49
% 2.08/1.49 thf(emptysetAx_type,type,
% 2.08/1.49 emptysetAx: $o ).
% 2.08/1.49
% 2.08/1.49 thf(emptysetAx,definition,
% 2.08/1.49 ( emptysetAx
% 2.08/1.49 = ( ! [Xx: $i] :
% 2.08/1.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(setadjoin_type,type,
% 2.08/1.49 setadjoin: $i > $i > $i ).
% 2.08/1.49
% 2.08/1.49 thf(setadjoinAx_type,type,
% 2.08/1.49 setadjoinAx: $o ).
% 2.08/1.49
% 2.08/1.49 thf(setadjoinAx,definition,
% 2.08/1.49 ( setadjoinAx
% 2.08/1.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.08/1.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.08/1.49 <=> ( ( Xy = Xx )
% 2.08/1.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(powerset_type,type,
% 2.08/1.49 powerset: $i > $i ).
% 2.08/1.49
% 2.08/1.49 thf(powersetAx_type,type,
% 2.08/1.49 powersetAx: $o ).
% 2.08/1.49
% 2.08/1.49 thf(powersetAx,definition,
% 2.08/1.49 ( powersetAx
% 2.08/1.49 = ( ! [A: $i,B: $i] :
% 2.08/1.49 ( ( in @ B @ ( powerset @ A ) )
% 2.08/1.49 <=> ! [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ B )
% 2.08/1.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(setunion_type,type,
% 2.08/1.49 setunion: $i > $i ).
% 2.08/1.49
% 2.08/1.49 thf(setunionAx_type,type,
% 2.08/1.49 setunionAx: $o ).
% 2.08/1.49
% 2.08/1.49 thf(setunionAx,definition,
% 2.08/1.49 ( setunionAx
% 2.08/1.49 = ( ! [A: $i,Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ ( setunion @ A ) )
% 2.08/1.49 <=> ? [B: $i] :
% 2.08/1.49 ( ( in @ Xx @ B )
% 2.08/1.49 & ( in @ B @ A ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(omega_type,type,
% 2.08/1.49 omega: $i ).
% 2.08/1.49
% 2.08/1.49 thf(omega0Ax_type,type,
% 2.08/1.49 omega0Ax: $o ).
% 2.08/1.49
% 2.08/1.49 thf(omega0Ax,definition,
% 2.08/1.49 ( omega0Ax
% 2.08/1.49 = ( in @ emptyset @ omega ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(omegaSAx_type,type,
% 2.08/1.49 omegaSAx: $o ).
% 2.08/1.49
% 2.08/1.49 thf(omegaSAx,definition,
% 2.08/1.49 ( omegaSAx
% 2.08/1.49 = ( ! [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ omega )
% 2.08/1.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(omegaIndAx_type,type,
% 2.08/1.49 omegaIndAx: $o ).
% 2.08/1.49
% 2.08/1.49 thf(omegaIndAx,definition,
% 2.08/1.49 ( omegaIndAx
% 2.08/1.49 = ( ! [A: $i] :
% 2.08/1.49 ( ( ( in @ emptyset @ A )
% 2.08/1.49 & ! [Xx: $i] :
% 2.08/1.49 ( ( ( in @ Xx @ omega )
% 2.08/1.49 & ( in @ Xx @ A ) )
% 2.08/1.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.08/1.49 => ! [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ omega )
% 2.08/1.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(replAx_type,type,
% 2.08/1.49 replAx: $o ).
% 2.08/1.49
% 2.08/1.49 thf(replAx,definition,
% 2.08/1.49 ( replAx
% 2.08/1.49 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.08/1.49 ( ! [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ A )
% 2.08/1.49 => ( exu
% 2.08/1.49 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.08/1.49 => ? [B: $i] :
% 2.08/1.49 ! [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ B )
% 2.08/1.49 <=> ? [Xy: $i] :
% 2.08/1.49 ( ( in @ Xy @ A )
% 2.08/1.49 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(foundationAx_type,type,
% 2.08/1.49 foundationAx: $o ).
% 2.08/1.49
% 2.08/1.49 thf(foundationAx,definition,
% 2.08/1.49 ( foundationAx
% 2.08/1.49 = ( ! [A: $i] :
% 2.08/1.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.08/1.49 => ? [B: $i] :
% 2.08/1.49 ( ( in @ B @ A )
% 2.08/1.49 & ~ ? [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ B )
% 2.08/1.49 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(wellorderingAx_type,type,
% 2.08/1.49 wellorderingAx: $o ).
% 2.08/1.49
% 2.08/1.49 thf(wellorderingAx,definition,
% 2.08/1.49 ( wellorderingAx
% 2.08/1.49 = ( ! [A: $i] :
% 2.08/1.49 ? [B: $i] :
% 2.08/1.49 ( ! [C: $i] :
% 2.08/1.49 ( ( in @ C @ B )
% 2.08/1.49 => ! [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ C )
% 2.08/1.49 => ( in @ Xx @ A ) ) )
% 2.08/1.49 & ! [Xx: $i,Xy: $i] :
% 2.08/1.49 ( ( ( in @ Xx @ A )
% 2.08/1.49 & ( in @ Xy @ A ) )
% 2.08/1.49 => ( ! [C: $i] :
% 2.08/1.49 ( ( in @ C @ B )
% 2.08/1.49 => ( ( in @ Xx @ C )
% 2.08/1.49 <=> ( in @ Xy @ C ) ) )
% 2.08/1.49 => ( Xx = Xy ) ) )
% 2.08/1.49 & ! [C: $i,D: $i] :
% 2.08/1.49 ( ( ( in @ C @ B )
% 2.08/1.49 & ( in @ D @ B ) )
% 2.08/1.49 => ( ! [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ C )
% 2.08/1.49 => ( in @ Xx @ D ) )
% 2.08/1.49 | ! [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ D )
% 2.08/1.49 => ( in @ Xx @ C ) ) ) )
% 2.08/1.49 & ! [C: $i] :
% 2.08/1.49 ( ( ! [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ C )
% 2.08/1.49 => ( in @ Xx @ A ) )
% 2.08/1.49 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.08/1.49 => ? [D: $i,Xx: $i] :
% 2.08/1.49 ( ( in @ D @ B )
% 2.08/1.49 & ( in @ Xx @ C )
% 2.08/1.49 & ~ ? [Xy: $i] :
% 2.08/1.49 ( ( in @ Xy @ D )
% 2.08/1.49 & ( in @ Xy @ C ) )
% 2.08/1.49 & ! [E: $i] :
% 2.08/1.49 ( ( in @ E @ B )
% 2.08/1.49 => ( ! [Xy: $i] :
% 2.08/1.49 ( ( in @ Xy @ E )
% 2.08/1.49 => ( in @ Xy @ D ) )
% 2.08/1.49 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(descr_type,type,
% 2.08/1.49 descr: ( $i > $o ) > $i ).
% 2.08/1.49
% 2.08/1.49 thf(descrp_type,type,
% 2.08/1.49 descrp: $o ).
% 2.08/1.49
% 2.08/1.49 thf(descrp,definition,
% 2.08/1.49 ( descrp
% 2.08/1.49 = ( ! [Xphi: $i > $o] :
% 2.08/1.49 ( ( exu
% 2.08/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.08/1.49 => ( Xphi
% 2.08/1.49 @ ( descr
% 2.08/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(dsetconstr_type,type,
% 2.08/1.49 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.08/1.49
% 2.08/1.49 thf(dsetconstrI_type,type,
% 2.08/1.49 dsetconstrI: $o ).
% 2.08/1.49
% 2.08/1.49 thf(dsetconstrI,definition,
% 2.08/1.49 ( dsetconstrI
% 2.08/1.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ A )
% 2.08/1.49 => ( ( Xphi @ Xx )
% 2.08/1.49 => ( in @ Xx
% 2.08/1.49 @ ( dsetconstr @ A
% 2.08/1.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(dsetconstrEL_type,type,
% 2.08/1.49 dsetconstrEL: $o ).
% 2.08/1.49
% 2.08/1.49 thf(dsetconstrEL,definition,
% 2.08/1.49 ( dsetconstrEL
% 2.08/1.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.08/1.49 ( ( in @ Xx
% 2.08/1.49 @ ( dsetconstr @ A
% 2.08/1.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.08/1.49 => ( in @ Xx @ A ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(dsetconstrER_type,type,
% 2.08/1.49 dsetconstrER: $o ).
% 2.08/1.49
% 2.08/1.49 thf(dsetconstrER,definition,
% 2.08/1.49 ( dsetconstrER
% 2.08/1.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.08/1.49 ( ( in @ Xx
% 2.08/1.49 @ ( dsetconstr @ A
% 2.08/1.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.08/1.49 => ( Xphi @ Xx ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(exuE1_type,type,
% 2.08/1.49 exuE1: $o ).
% 2.08/1.49
% 2.08/1.49 thf(exuE1,definition,
% 2.08/1.49 ( exuE1
% 2.08/1.49 = ( ! [Xphi: $i > $o] :
% 2.08/1.49 ( ( exu
% 2.08/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.08/1.49 => ? [Xx: $i] :
% 2.08/1.49 ( ( Xphi @ Xx )
% 2.08/1.49 & ! [Xy: $i] :
% 2.08/1.49 ( ( Xphi @ Xy )
% 2.08/1.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(prop2set_type,type,
% 2.08/1.49 prop2set: $o > $i ).
% 2.08/1.49
% 2.08/1.49 thf(prop2set,definition,
% 2.08/1.49 ( prop2set
% 2.08/1.49 = ( ^ [Xphi: $o] :
% 2.08/1.49 ( dsetconstr @ ( powerset @ emptyset )
% 2.08/1.49 @ ^ [Xx: $i] : Xphi ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(prop2setE_type,type,
% 2.08/1.49 prop2setE: $o ).
% 2.08/1.49
% 2.08/1.49 thf(prop2setE,definition,
% 2.08/1.49 ( prop2setE
% 2.08/1.49 = ( ! [Xphi: $o,Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.08/1.49 => Xphi ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(emptysetE_type,type,
% 2.08/1.49 emptysetE: $o ).
% 2.08/1.49
% 2.08/1.49 thf(emptysetE,definition,
% 2.08/1.49 ( emptysetE
% 2.08/1.49 = ( ! [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ emptyset )
% 2.08/1.49 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(emptysetimpfalse_type,type,
% 2.08/1.49 emptysetimpfalse: $o ).
% 2.08/1.49
% 2.08/1.49 thf(emptysetimpfalse,definition,
% 2.08/1.49 ( emptysetimpfalse
% 2.08/1.49 = ( ! [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ emptyset )
% 2.08/1.49 => $false ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(notinemptyset_type,type,
% 2.08/1.49 notinemptyset: $o ).
% 2.08/1.49
% 2.08/1.49 thf(notinemptyset,definition,
% 2.08/1.49 ( notinemptyset
% 2.08/1.49 = ( ! [Xx: $i] :
% 2.08/1.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(exuE3e_type,type,
% 2.08/1.49 exuE3e: $o ).
% 2.08/1.49
% 2.08/1.49 thf(exuE3e,definition,
% 2.08/1.49 ( exuE3e
% 2.08/1.49 = ( ! [Xphi: $i > $o] :
% 2.08/1.49 ( ( exu
% 2.08/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.08/1.49 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(setext_type,type,
% 2.08/1.49 setext: $o ).
% 2.08/1.49
% 2.08/1.49 thf(setext,definition,
% 2.08/1.49 ( setext
% 2.08/1.49 = ( ! [A: $i,B: $i] :
% 2.08/1.49 ( ! [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ A )
% 2.08/1.49 => ( in @ Xx @ B ) )
% 2.08/1.49 => ( ! [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ B )
% 2.08/1.49 => ( in @ Xx @ A ) )
% 2.08/1.49 => ( A = B ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(emptyI_type,type,
% 2.08/1.49 emptyI: $o ).
% 2.08/1.49
% 2.08/1.49 thf(emptyI,definition,
% 2.08/1.49 ( emptyI
% 2.08/1.49 = ( ! [A: $i] :
% 2.08/1.49 ( ! [Xx: $i] :
% 2.08/1.49 ~ ( in @ Xx @ A )
% 2.08/1.49 => ( A = emptyset ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(noeltsimpempty_type,type,
% 2.08/1.49 noeltsimpempty: $o ).
% 2.08/1.49
% 2.08/1.49 thf(noeltsimpempty,definition,
% 2.08/1.49 ( noeltsimpempty
% 2.08/1.49 = ( ! [A: $i] :
% 2.08/1.49 ( ! [Xx: $i] :
% 2.08/1.49 ~ ( in @ Xx @ A )
% 2.08/1.49 => ( A = emptyset ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(setbeta_type,type,
% 2.08/1.49 setbeta: $o ).
% 2.08/1.49
% 2.08/1.49 thf(setbeta,definition,
% 2.08/1.49 ( setbeta
% 2.08/1.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ A )
% 2.08/1.49 => ( ( in @ Xx
% 2.08/1.49 @ ( dsetconstr @ A
% 2.08/1.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.08/1.49 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(nonempty_type,type,
% 2.08/1.49 nonempty: $i > $o ).
% 2.08/1.49
% 2.08/1.49 thf(nonempty,definition,
% 2.08/1.49 ( nonempty
% 2.08/1.49 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(nonemptyE1_type,type,
% 2.08/1.49 nonemptyE1: $o ).
% 2.08/1.49
% 2.08/1.49 thf(nonemptyE1,definition,
% 2.08/1.49 ( nonemptyE1
% 2.08/1.49 = ( ! [A: $i] :
% 2.08/1.49 ( ( nonempty @ A )
% 2.08/1.49 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(nonemptyI_type,type,
% 2.08/1.49 nonemptyI: $o ).
% 2.08/1.49
% 2.08/1.49 thf(nonemptyI,definition,
% 2.08/1.49 ( nonemptyI
% 2.08/1.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ A )
% 2.08/1.49 => ( ( Xphi @ Xx )
% 2.08/1.49 => ( nonempty
% 2.08/1.49 @ ( dsetconstr @ A
% 2.08/1.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(nonemptyI1_type,type,
% 2.08/1.49 nonemptyI1: $o ).
% 2.08/1.49
% 2.08/1.49 thf(nonemptyI1,definition,
% 2.08/1.49 ( nonemptyI1
% 2.08/1.49 = ( ! [A: $i] :
% 2.08/1.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.08/1.49 => ( nonempty @ A ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(setadjoinIL_type,type,
% 2.08/1.49 setadjoinIL: $o ).
% 2.08/1.49
% 2.08/1.49 thf(setadjoinIL,definition,
% 2.08/1.49 ( setadjoinIL
% 2.08/1.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(emptyinunitempty_type,type,
% 2.08/1.49 emptyinunitempty: $o ).
% 2.08/1.49
% 2.08/1.49 thf(emptyinunitempty,definition,
% 2.08/1.49 ( emptyinunitempty
% 2.08/1.49 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(setadjoinIR_type,type,
% 2.08/1.49 setadjoinIR: $o ).
% 2.08/1.49
% 2.08/1.49 thf(setadjoinIR,definition,
% 2.08/1.49 ( setadjoinIR
% 2.08/1.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.08/1.49 ( ( in @ Xy @ A )
% 2.08/1.49 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(setadjoinE_type,type,
% 2.08/1.49 setadjoinE: $o ).
% 2.08/1.49
% 2.08/1.49 thf(setadjoinE,definition,
% 2.08/1.49 ( setadjoinE
% 2.08/1.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.08/1.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.08/1.49 => ! [Xphi: $o] :
% 2.08/1.49 ( ( ( Xy = Xx )
% 2.08/1.49 => Xphi )
% 2.08/1.49 => ( ( ( in @ Xy @ A )
% 2.08/1.49 => Xphi )
% 2.08/1.49 => Xphi ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(setadjoinOr_type,type,
% 2.08/1.49 setadjoinOr: $o ).
% 2.08/1.49
% 2.08/1.49 thf(setadjoinOr,definition,
% 2.08/1.49 ( setadjoinOr
% 2.08/1.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.08/1.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.08/1.49 => ( ( Xy = Xx )
% 2.08/1.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(setoftrueEq_type,type,
% 2.08/1.49 setoftrueEq: $o ).
% 2.08/1.49
% 2.08/1.49 thf(setoftrueEq,definition,
% 2.08/1.49 ( setoftrueEq
% 2.08/1.49 = ( ! [A: $i] :
% 2.08/1.49 ( ( dsetconstr @ A
% 2.08/1.49 @ ^ [Xx: $i] : $true )
% 2.08/1.49 = A ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(powersetI_type,type,
% 2.08/1.49 powersetI: $o ).
% 2.08/1.49
% 2.08/1.49 thf(powersetI,definition,
% 2.08/1.49 ( powersetI
% 2.08/1.49 = ( ! [A: $i,B: $i] :
% 2.08/1.49 ( ! [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ B )
% 2.08/1.49 => ( in @ Xx @ A ) )
% 2.08/1.49 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(emptyinPowerset_type,type,
% 2.08/1.49 emptyinPowerset: $o ).
% 2.08/1.49
% 2.08/1.49 thf(emptyinPowerset,definition,
% 2.08/1.49 ( emptyinPowerset
% 2.08/1.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(emptyInPowerset_type,type,
% 2.08/1.49 emptyInPowerset: $o ).
% 2.08/1.49
% 2.08/1.49 thf(emptyInPowerset,definition,
% 2.08/1.49 ( emptyInPowerset
% 2.08/1.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(powersetE_type,type,
% 2.08/1.49 powersetE: $o ).
% 2.08/1.49
% 2.08/1.49 thf(powersetE,definition,
% 2.08/1.49 ( powersetE
% 2.08/1.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.08/1.49 ( ( in @ B @ ( powerset @ A ) )
% 2.08/1.49 => ( ( in @ Xx @ B )
% 2.08/1.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(setunionI_type,type,
% 2.08/1.49 setunionI: $o ).
% 2.08/1.49
% 2.08/1.49 thf(setunionI,definition,
% 2.08/1.49 ( setunionI
% 2.08/1.49 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.08/1.49 ( ( in @ Xx @ B )
% 2.08/1.49 => ( ( in @ B @ A )
% 2.08/1.49 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(setunionE_type,type,
% 2.08/1.49 setunionE: $o ).
% 2.08/1.49
% 2.08/1.49 thf(setunionE,definition,
% 2.08/1.49 ( setunionE
% 2.08/1.49 = ( ! [A: $i,Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ ( setunion @ A ) )
% 2.08/1.49 => ! [Xphi: $o] :
% 2.08/1.49 ( ! [B: $i] :
% 2.08/1.49 ( ( in @ Xx @ B )
% 2.08/1.49 => ( ( in @ B @ A )
% 2.08/1.49 => Xphi ) )
% 2.08/1.49 => Xphi ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(subPowSU_type,type,
% 2.08/1.49 subPowSU: $o ).
% 2.08/1.49
% 2.08/1.49 thf(subPowSU,definition,
% 2.08/1.49 ( subPowSU
% 2.08/1.49 = ( ! [A: $i,Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ A )
% 2.08/1.49 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(exuE2_type,type,
% 2.08/1.49 exuE2: $o ).
% 2.08/1.49
% 2.08/1.49 thf(exuE2,definition,
% 2.08/1.49 ( exuE2
% 2.08/1.49 = ( ! [Xphi: $i > $o] :
% 2.08/1.49 ( ( exu
% 2.08/1.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.08/1.49 => ? [Xx: $i] :
% 2.08/1.49 ! [Xy: $i] :
% 2.08/1.49 ( ( Xphi @ Xy )
% 2.08/1.49 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(nonemptyImpWitness_type,type,
% 2.08/1.49 nonemptyImpWitness: $o ).
% 2.08/1.49
% 2.08/1.49 thf(nonemptyImpWitness,definition,
% 2.08/1.49 ( nonemptyImpWitness
% 2.08/1.49 = ( ! [A: $i] :
% 2.08/1.49 ( ( nonempty @ A )
% 2.08/1.49 => ? [Xx: $i] :
% 2.08/1.49 ( ( in @ Xx @ A )
% 2.08/1.49 & $true ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(uniqinunit_type,type,
% 2.08/1.49 uniqinunit: $o ).
% 2.08/1.49
% 2.08/1.49 thf(uniqinunit,definition,
% 2.08/1.49 ( uniqinunit
% 2.08/1.49 = ( ! [Xx: $i,Xy: $i] :
% 2.08/1.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.08/1.49 => ( Xx = Xy ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(notinsingleton_type,type,
% 2.08/1.49 notinsingleton: $o ).
% 2.08/1.49
% 2.08/1.49 thf(notinsingleton,definition,
% 2.08/1.49 ( notinsingleton
% 2.08/1.49 = ( ! [Xx: $i,Xy: $i] :
% 2.08/1.49 ( ( Xx != Xy )
% 2.08/1.49 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(eqinunit_type,type,
% 2.08/1.49 eqinunit: $o ).
% 2.08/1.49
% 2.08/1.49 thf(eqinunit,definition,
% 2.08/1.49 ( eqinunit
% 2.08/1.49 = ( ! [Xx: $i,Xy: $i] :
% 2.08/1.49 ( ( Xx = Xy )
% 2.08/1.49 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.08/1.49
% 2.08/1.49 thf(singletonsswitch,conjecture,
% 2.08/1.49 ( setextAx
% 2.08/1.49 => ( emptysetAx
% 2.08/1.49 => ( setadjoinAx
% 2.08/1.49 => ( powersetAx
% 2.08/1.49 => ( setunionAx
% 2.08/1.49 => ( omega0Ax
% 2.08/1.49 => ( omegaSAx
% 2.08/1.49 => ( omegaIndAx
% 2.08/1.49 => ( replAx
% 2.08/1.49 => ( foundationAx
% 2.08/1.49 => ( wellorderingAx
% 2.08/1.49 => ( descrp
% 2.08/1.49 => ( dsetconstrI
% 2.08/1.49 => ( dsetconstrEL
% 2.08/1.49 => ( dsetconstrER
% 2.08/1.49 => ( exuE1
% 2.08/1.49 => ( prop2setE
% 2.08/1.49 => ( emptysetE
% 2.08/1.49 => ( emptysetimpfalse
% 2.08/1.49 => ( notinemptyset
% 2.08/1.49 => ( exuE3e
% 2.08/1.49 => ( setext
% 2.08/1.49 => ( emptyI
% 2.08/1.49 => ( noeltsimpempty
% 2.08/1.49 => ( setbeta
% 2.08/1.49 => ( nonemptyE1
% 2.08/1.49 => ( nonemptyI
% 2.08/1.49 => ( nonemptyI1
% 2.08/1.49 => ( setadjoinIL
% 2.08/1.49 => ( emptyinunitempty
% 2.08/1.49 => ( setadjoinIR
% 2.08/1.49 => ( setadjoinE
% 2.08/1.49 => ( setadjoinOr
% 2.08/1.49 => ( setoftrueEq
% 2.08/1.49 => ( powersetI
% 2.08/1.49 => ( emptyinPowerset
% 2.08/1.49 => ( emptyInPowerset
% 2.08/1.49 => ( powersetE
% 2.08/1.49 => ( setunionI
% 2.08/1.49 => ( setunionE
% 2.08/1.49 => ( subPowSU
% 2.08/1.49 => ( exuE2
% 2.08/1.49 => ( nonemptyImpWitness
% 2.08/1.49 => ( uniqinunit
% 2.08/1.49 => ( notinsingleton
% 2.08/1.49 => ( eqinunit
% 2.08/1.49 => ! [Xx: $i,Xy: $i] :
% 2.08/1.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.08/1.49 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.08/1.49 ------------------------
% 2.23/1.51 % (25623)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_25397 for (2999ds/18Mi)
% 2.23/1.51 % (25621)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_25397 for (2999ds/2Mi)
% 2.23/1.51 % (25619)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_25397 for (2999ds/27Mi)
% 2.23/1.51 % (25622)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_25397 for (2999ds/275Mi)
% 2.23/1.51 % (25621)Instruction limit reached!
% 2.23/1.51 % (25621)------------------------------
% 2.23/1.51 % (25621)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.51 % (25621)Termination reason: Unknown
% 2.23/1.51 % (25621)Termination phase: shuffling
% 2.23/1.51
% 2.23/1.51 % (25621)Memory used [KB]: 1023
% 2.23/1.51 % (25621)Time elapsed: 0.003 s
% 2.23/1.51 % (25621)Instructions burned: 3 (million)
% 2.23/1.51 % (25621)------------------------------
% 2.23/1.51 % (25621)------------------------------
% 2.23/1.51 % (25617)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_25397 for (2999ds/183Mi)
% 2.23/1.51 % (25618)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_25397 for (2999ds/4Mi)
% 2.23/1.52 % (25618)Instruction limit reached!
% 2.23/1.52 % (25618)------------------------------
% 2.23/1.52 % (25618)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.52 % (25618)Termination reason: Unknown
% 2.23/1.52 % (25618)Termination phase: shuffling
% 2.23/1.52
% 2.23/1.52 % (25618)Memory used [KB]: 1151
% 2.23/1.52 % (25618)Time elapsed: 0.004 s
% 2.23/1.52 % (25618)Instructions burned: 4 (million)
% 2.23/1.52 % (25618)------------------------------
% 2.23/1.52 % (25618)------------------------------
% 2.23/1.52 % (25623)Instruction limit reached!
% 2.23/1.52 % (25623)------------------------------
% 2.23/1.52 % (25623)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.52 % (25623)Termination reason: Unknown
% 2.23/1.52 % (25623)Termination phase: Property scanning
% 2.23/1.52
% 2.23/1.52 % (25623)Memory used [KB]: 1535
% 2.23/1.52 % (25623)Time elapsed: 0.011 s
% 2.23/1.52 % (25623)Instructions burned: 19 (million)
% 2.23/1.52 % (25623)------------------------------
% 2.23/1.52 % (25623)------------------------------
% 2.23/1.52 % (25620)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_25397 for (2999ds/2Mi)
% 2.23/1.52 % (25620)Instruction limit reached!
% 2.23/1.52 % (25620)------------------------------
% 2.23/1.52 % (25620)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.52 % (25620)Termination reason: Unknown
% 2.23/1.52 % (25620)Termination phase: shuffling
% 2.23/1.52
% 2.23/1.52 % (25620)Memory used [KB]: 1023
% 2.23/1.52 % (25620)Time elapsed: 0.003 s
% 2.23/1.52 % (25620)Instructions burned: 2 (million)
% 2.23/1.52 % (25620)------------------------------
% 2.23/1.52 % (25620)------------------------------
% 2.23/1.53 % (25619)Instruction limit reached!
% 2.23/1.53 % (25619)------------------------------
% 2.23/1.53 % (25619)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.53 % (25619)Termination reason: Unknown
% 2.23/1.53 % (25619)Termination phase: Saturation
% 2.23/1.53
% 2.23/1.53 % (25619)Memory used [KB]: 5756
% 2.23/1.53 % (25619)Time elapsed: 0.015 s
% 2.23/1.53 % (25619)Instructions burned: 28 (million)
% 2.23/1.53 % (25619)------------------------------
% 2.23/1.53 % (25619)------------------------------
% 2.23/1.53 % (25627)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_25397 for (2999ds/3Mi)
% 2.23/1.53 % (25627)Instruction limit reached!
% 2.23/1.53 % (25627)------------------------------
% 2.23/1.53 % (25627)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.53 % (25627)Termination reason: Unknown
% 2.23/1.53 % (25627)Termination phase: shuffling
% 2.23/1.53
% 2.23/1.53 % (25627)Memory used [KB]: 1023
% 2.23/1.53 % (25627)Time elapsed: 0.003 s
% 2.23/1.53 % (25627)Instructions burned: 4 (million)
% 2.23/1.53 % (25627)------------------------------
% 2.23/1.53 % (25627)------------------------------
% 2.23/1.54 % (25631)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_25397 for (2999ds/15Mi)
% 2.23/1.54 % (25628)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_25397 for (2999ds/37Mi)
% 2.23/1.54 % (25632)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_25397 for (2999ds/3Mi)
% 2.23/1.54 % (25632)Instruction limit reached!
% 2.23/1.54 % (25632)------------------------------
% 2.23/1.54 % (25632)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.54 % (25632)Termination reason: Unknown
% 2.23/1.54 % (25632)Termination phase: shuffling
% 2.23/1.54
% 2.23/1.54 % (25632)Memory used [KB]: 1151
% 2.23/1.54 % (25632)Time elapsed: 0.004 s
% 2.23/1.54 % (25632)Instructions burned: 5 (million)
% 2.23/1.54 % (25632)------------------------------
% 2.23/1.54 % (25632)------------------------------
% 2.23/1.54 % (25631)Instruction limit reached!
% 2.23/1.54 % (25631)------------------------------
% 2.23/1.54 % (25631)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.54 % (25631)Termination reason: Unknown
% 2.23/1.54 % (25631)Termination phase: Preprocessing 3
% 2.23/1.54
% 2.23/1.54 % (25631)Memory used [KB]: 1407
% 2.23/1.54 % (25631)Time elapsed: 0.008 s
% 2.23/1.54 % (25631)Instructions burned: 16 (million)
% 2.23/1.54 % (25631)------------------------------
% 2.23/1.54 % (25631)------------------------------
% 2.23/1.55 % (25635)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_25397 for (2999ds/7Mi)
% 2.23/1.55 % (25635)Instruction limit reached!
% 2.23/1.55 % (25635)------------------------------
% 2.23/1.55 % (25635)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.55 % (25635)Termination reason: Unknown
% 2.23/1.55 % (25635)Termination phase: Property scanning
% 2.23/1.55
% 2.23/1.55 % (25635)Memory used [KB]: 1151
% 2.23/1.55 % (25635)Time elapsed: 0.004 s
% 2.23/1.55 % (25635)Instructions burned: 9 (million)
% 2.23/1.55 % (25635)------------------------------
% 2.23/1.55 % (25635)------------------------------
% 2.23/1.55 % (25634)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_25397 for (2999ds/1041Mi)
% 2.23/1.55 % (25628)Instruction limit reached!
% 2.23/1.55 % (25628)------------------------------
% 2.23/1.55 % (25628)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.55 % (25628)Termination reason: Unknown
% 2.23/1.55 % (25628)Termination phase: Saturation
% 2.23/1.55
% 2.23/1.55 % (25628)Memory used [KB]: 5884
% 2.23/1.55 % (25628)Time elapsed: 0.013 s
% 2.23/1.55 % (25628)Instructions burned: 38 (million)
% 2.23/1.55 % (25628)------------------------------
% 2.23/1.55 % (25628)------------------------------
% 2.23/1.55 % (25639)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_25397 for (2999ds/3Mi)
% 2.23/1.55 % (25638)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_25397 for (2999ds/16Mi)
% 2.23/1.55 % (25639)Instruction limit reached!
% 2.23/1.55 % (25639)------------------------------
% 2.23/1.55 % (25639)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.55 % (25639)Termination reason: Unknown
% 2.23/1.55 % (25639)Termination phase: shuffling
% 2.23/1.55
% 2.23/1.55 % (25639)Memory used [KB]: 1023
% 2.23/1.55 % (25639)Time elapsed: 0.002 s
% 2.23/1.55 % (25639)Instructions burned: 4 (million)
% 2.23/1.55 % (25639)------------------------------
% 2.23/1.55 % (25639)------------------------------
% 2.23/1.56 % (25638)Instruction limit reached!
% 2.23/1.56 % (25638)------------------------------
% 2.23/1.56 % (25638)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.56 % (25638)Termination reason: Unknown
% 2.23/1.56 % (25638)Termination phase: Function definition elimination
% 2.23/1.56
% 2.23/1.56 % (25638)Memory used [KB]: 1279
% 2.23/1.56 % (25638)Time elapsed: 0.006 s
% 2.23/1.56 % (25638)Instructions burned: 18 (million)
% 2.23/1.56 % (25638)------------------------------
% 2.23/1.56 % (25638)------------------------------
% 2.23/1.56 % (25640)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_25397 for (2999ds/3Mi)
% 2.23/1.56 % (25640)Instruction limit reached!
% 2.23/1.56 % (25640)------------------------------
% 2.23/1.56 % (25640)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.56 % (25640)Termination reason: Unknown
% 2.23/1.56 % (25640)Termination phase: shuffling
% 2.23/1.56
% 2.23/1.56 % (25640)Memory used [KB]: 1151
% 2.23/1.56 % (25640)Time elapsed: 0.003 s
% 2.23/1.56 % (25640)Instructions burned: 4 (million)
% 2.23/1.56 % (25640)------------------------------
% 2.23/1.56 % (25640)------------------------------
% 2.23/1.56 % (25641)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_25397 for (2999ds/7Mi)
% 2.23/1.56 % (25642)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_25397 for (2999ds/3Mi)
% 2.23/1.56 % (25642)Instruction limit reached!
% 2.23/1.56 % (25642)------------------------------
% 2.23/1.56 % (25642)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.56 % (25642)Termination reason: Unknown
% 2.23/1.56 % (25642)Termination phase: shuffling
% 2.23/1.56
% 2.23/1.56 % (25642)Memory used [KB]: 1023
% 2.23/1.56 % (25642)Time elapsed: 0.002 s
% 2.23/1.56 % (25642)Instructions burned: 3 (million)
% 2.23/1.56 % (25642)------------------------------
% 2.23/1.56 % (25642)------------------------------
% 2.23/1.56 % (25641)Instruction limit reached!
% 2.23/1.56 % (25641)------------------------------
% 2.23/1.56 % (25641)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.56 % (25641)Termination reason: Unknown
% 2.23/1.56 % (25641)Termination phase: shuffling
% 2.23/1.56
% 2.23/1.56 % (25641)Memory used [KB]: 1151
% 2.23/1.56 % (25641)Time elapsed: 0.004 s
% 2.23/1.56 % (25641)Instructions burned: 8 (million)
% 2.23/1.56 % (25641)------------------------------
% 2.23/1.56 % (25641)------------------------------
% 2.23/1.57 % (25622)First to succeed.
% 2.23/1.57 % (25643)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_25397 for (2999ds/4Mi)
% 2.23/1.57 % (25622)Refutation found. Thanks to Tanya!
% 2.23/1.57 % SZS status Theorem for DTF2THF_25397
% 2.23/1.57 % SZS output start Proof for DTF2THF_25397
% 2.23/1.57 thf(func_def_0, type, in: $i > $i > $o).
% 2.23/1.57 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.23/1.57 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.23/1.57 thf(func_def_8, type, powerset: $i > $i).
% 2.23/1.57 thf(func_def_10, type, setunion: $i > $i).
% 2.23/1.57 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.23/1.57 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.23/1.57 thf(func_def_26, type, prop2set: $o > $i).
% 2.23/1.57 thf(func_def_36, type, nonempty: $i > $o).
% 2.23/1.57 thf(func_def_71, type, sP1: $i > $i > $o).
% 2.23/1.57 thf(func_def_72, type, sP2: $i > $o).
% 2.23/1.57 thf(func_def_73, type, sP3: $i > $i > $o).
% 2.23/1.57 thf(func_def_74, type, sP4: $i > $i > $o).
% 2.23/1.57 thf(func_def_77, type, sK7: $i > $o).
% 2.23/1.57 thf(func_def_82, type, sK12: $i > $i).
% 2.23/1.57 thf(func_def_86, type, sK16: $i > $i > $i).
% 2.23/1.57 thf(func_def_92, type, sK22: $i > $i).
% 2.23/1.57 thf(func_def_99, type, sK29: $i > $i > $i).
% 2.23/1.57 thf(func_def_103, type, sK33: ($i > $o) > $i).
% 2.23/1.57 thf(func_def_104, type, sK34: $i > $o).
% 2.23/1.57 thf(func_def_112, type, sK42: $i > $i > $o).
% 2.23/1.57 thf(func_def_113, type, sK43: $i > $i).
% 2.23/1.57 thf(func_def_114, type, sK44: $i > $i).
% 2.23/1.57 thf(func_def_115, type, sK45: ($i > $i > $o) > $i > $i).
% 2.23/1.57 thf(func_def_116, type, sK46: ($i > $i > $o) > $i > $i).
% 2.23/1.57 thf(func_def_117, type, sK47: $i > ($i > $i > $o) > $i > $i).
% 2.23/1.57 thf(func_def_122, type, sK52: $i > $i).
% 2.23/1.57 thf(func_def_127, type, sK57: $i > $i).
% 2.23/1.57 thf(func_def_129, type, sK59: $i > $o).
% 2.23/1.57 thf(func_def_130, type, sK60: $i > $i).
% 2.23/1.57 thf(func_def_131, type, sK61: ($i > $o) > $i).
% 2.23/1.57 thf(func_def_137, type, sK67: $i > $o).
% 2.23/1.57 thf(func_def_140, type, sK70: $i > $i).
% 2.23/1.57 thf(func_def_152, type, sK82: $i > $i).
% 2.23/1.57 thf(func_def_155, type, sK85: $i > $i).
% 2.23/1.57 thf(func_def_157, type, sK87: $i > $o).
% 2.23/1.57 thf(func_def_160, type, sK90: $i > $o).
% 2.23/1.57 thf(func_def_164, type, sK94: $i > $i > $i).
% 2.23/1.57 thf(func_def_165, type, sK95: $i > $i > $i).
% 2.23/1.57 thf(func_def_166, type, sK96: $i > $i > $i).
% 2.23/1.57 thf(func_def_167, type, sK97: $i > $i > $i).
% 2.23/1.57 thf(func_def_168, type, sK98: $i > $i > $i).
% 2.23/1.57 thf(func_def_169, type, sK99: $i > $i > $i).
% 2.23/1.57 thf(func_def_170, type, sK100: $i > $i > $i > $i).
% 2.23/1.57 thf(func_def_171, type, sK101: $i > $i).
% 2.23/1.57 thf(func_def_172, type, sK102: $i > $i).
% 2.23/1.57 thf(func_def_173, type, sK103: $i > $i).
% 2.23/1.57 thf(func_def_174, type, sK104: $i > $i).
% 2.23/1.57 thf(func_def_175, type, sK105: $i > $i > $i).
% 2.23/1.57 thf(func_def_176, type, sK106: $i > $i > $i > $i).
% 2.23/1.57 thf(func_def_177, type, sK107: $i > $i > $i > $i).
% 2.23/1.57 thf(func_def_178, type, sK108: $i > $i > $i).
% 2.23/1.57 thf(func_def_179, type, sK109: $i > $i > $i).
% 2.23/1.57 thf(func_def_180, type, sK110: $i > $i).
% 2.23/1.57 thf(func_def_182, type, sK112: $i > $i).
% 2.23/1.57 thf(func_def_183, type, sK113: $i > $i).
% 2.23/1.57 thf(func_def_187, type, sK117: $i > $o).
% 2.23/1.57 thf(func_def_191, type, sK121: $o > $i > $i > $i).
% 2.23/1.57 thf(func_def_193, type, sK123: $i > $o).
% 2.23/1.57 thf(func_def_194, type, sK124: $i > $i).
% 2.23/1.57 thf(func_def_195, type, sK125: ($i > $o) > $i).
% 2.23/1.57 thf(func_def_197, type, sK127: $i > $i > $i).
% 2.23/1.57 thf(func_def_203, type, sK133: $i > $i > $i).
% 2.23/1.57 thf(func_def_204, type, sK134: $i > $i > $i).
% 2.23/1.57 thf(func_def_206, type, ph136: !>[X0: $tType]:(X0)).
% 2.23/1.57 thf(f1222,plain,(
% 2.23/1.57 $false),
% 2.23/1.57 inference(subsumption_resolution,[status(thm)],[f1221,f932])).
% 2.23/1.57 thf(f932,plain,(
% 2.23/1.57 ( ! [X2 : $i,X3 : $i] : (($true = (in @ X3 @ (setadjoin @ X3 @ X2)))) )),
% 2.23/1.57 inference(trivial_inequality_removal,[status(thm)],[f754])).
% 2.23/1.57 thf(f754,plain,(
% 2.23/1.57 ( ! [X2 : $i,X3 : $i] : (($true = (in @ X3 @ (setadjoin @ X3 @ X2))) | ($true != $true)) )),
% 2.23/1.57 inference(definition_unfolding,[status(thm)],[f558,f479])).
% 2.23/1.57 thf(f479,plain,(
% 2.23/1.57 (setadjoinIL = $true)),
% 2.23/1.57 inference(cnf_transformation,[status(thm)],[f222])).
% 2.23/1.57 thf(f222,plain,(
% 2.23/1.57 (uniqinunit = $true) & (emptysetimpfalse = $true) & (replAx = $true) & (setunionE = $true) & (powersetAx = $true) & (setunionI = $true) & (nonemptyI = $true) & (setadjoinAx = $true) & (dsetconstrEL = $true) & (emptysetE = $true) & (wellorderingAx = $true) & (setextAx = $true) & (setadjoinE = $true) & (setunionAx = $true) & (nonemptyE1 = $true) & (emptyinPowerset = $true) & (emptyinunitempty = $true) & (powersetI = $true) & (setbeta = $true) & (notinsingleton = $true) & (omegaIndAx = $true) & (setadjoinOr = $true) & (setadjoinIL = $true) & (foundationAx = $true) & (emptysetAx = $true) & (exuE1 = $true) & (dsetconstrER = $true) & (nonemptyI1 = $true) & (powersetE = $true) & (exuE2 = $true) & (notinemptyset = $true) & (($true = (in @ sK10 @ (setadjoin @ sK11 @ emptyset))) & ($true != (in @ sK11 @ (setadjoin @ sK10 @ emptyset)))) & (emptyInPowerset = $true) & (eqinunit = $true) & (descrp = $true) & (exuE3e = $true) & (dsetconstrI = $true) & (subPowSU = $true) & (emptyI = $true) & (setadjoinIR = $true) & (setoftrueEq = $true) & (omegaSAx = $true) & (prop2setE = $true) & (omega0Ax = $true) & (setext = $true) & (noeltsimpempty = $true) & (nonemptyImpWitness = $true)),
% 2.23/1.57 inference(skolemisation,[status(esa),new_symbols(skolem,[sK10,sK11])],[f191,f221])).
% 2.23/1.57 thf(f221,plain,(
% 2.23/1.57 ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (setadjoin @ X1 @ emptyset))) & ((in @ X1 @ (setadjoin @ X0 @ emptyset)) != $true)) => (($true = (in @ sK10 @ (setadjoin @ sK11 @ emptyset))) & ($true != (in @ sK11 @ (setadjoin @ sK10 @ emptyset))))),
% 2.23/1.57 introduced(definition,[],[choice_axiom])).
% 2.23/1.57 thf(f191,plain,(
% 2.23/1.57 (uniqinunit = $true) & (emptysetimpfalse = $true) & (replAx = $true) & (setunionE = $true) & (powersetAx = $true) & (setunionI = $true) & (nonemptyI = $true) & (setadjoinAx = $true) & (dsetconstrEL = $true) & (emptysetE = $true) & (wellorderingAx = $true) & (setextAx = $true) & (setadjoinE = $true) & (setunionAx = $true) & (nonemptyE1 = $true) & (emptyinPowerset = $true) & (emptyinunitempty = $true) & (powersetI = $true) & (setbeta = $true) & (notinsingleton = $true) & (omegaIndAx = $true) & (setadjoinOr = $true) & (setadjoinIL = $true) & (foundationAx = $true) & (emptysetAx = $true) & (exuE1 = $true) & (dsetconstrER = $true) & (nonemptyI1 = $true) & (powersetE = $true) & (exuE2 = $true) & (notinemptyset = $true) & ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (setadjoin @ X1 @ emptyset))) & ((in @ X1 @ (setadjoin @ X0 @ emptyset)) != $true)) & (emptyInPowerset = $true) & (eqinunit = $true) & (descrp = $true) & (exuE3e = $true) & (dsetconstrI = $true) & (subPowSU = $true) & (emptyI = $true) & (setadjoinIR = $true) & (setoftrueEq = $true) & (omegaSAx = $true) & (prop2setE = $true) & (omega0Ax = $true) & (setext = $true) & (noeltsimpempty = $true) & (nonemptyImpWitness = $true)),
% 2.23/1.57 inference(flattening,[status(thm)],[f190])).
% 2.23/1.57 thf(f190,plain,(
% 2.23/1.57 (((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (setadjoin @ X1 @ emptyset))) & ((in @ X1 @ (setadjoin @ X0 @ emptyset)) != $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)),
% 2.23/1.57 inference(ennf_transformation,[status(thm)],[f98])).
% 2.23/1.57 thf(f98,plain,(
% 2.23/1.57 ~((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) => ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (setadjoin @ X1 @ emptyset))) => ((in @ X1 @ (setadjoin @ X0 @ emptyset)) = $true))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.23/1.57 inference(fool_elimination,[status(thm)],[f97])).
% 2.23/1.57 thf(f97,plain,(
% 2.23/1.57 ~(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 => ! [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ emptyset)) => (in @ X1 @ (setadjoin @ X0 @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.23/1.57 inference(rectify,[status(thm)],[f51])).
% 2.23/1.57 thf(f51,negated_conjecture,(
% 2.23/1.57 ~(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 => ! [X1 : $i,X2 : $i] : ((in @ X1 @ (setadjoin @ X2 @ emptyset)) => (in @ X2 @ (setadjoin @ X1 @ emptyset)))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.23/1.57 inference(negated_conjecture,[status(cth)],[f50])).
% 2.23/1.57 thf(f50,conjecture,(
% 2.23/1.57 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 => ! [X1 : $i,X2 : $i] : ((in @ X1 @ (setadjoin @ X2 @ emptyset)) => (in @ X2 @ (setadjoin @ X1 @ emptyset))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.23/1.57 file('/export/starexec/sandbox/tmp/tmp.xT3Ehs1D5u/DTF2THF_25397.p',singletonsswitch)).
% 2.23/1.57 thf(f558,plain,(
% 2.23/1.57 ( ! [X2 : $i,X3 : $i] : (($true = (in @ X3 @ (setadjoin @ X3 @ X2))) | (setadjoinIL != $true)) )),
% 2.23/1.57 inference(cnf_transformation,[status(thm)],[f295])).
% 2.23/1.57 thf(f295,plain,(
% 2.23/1.57 ((setadjoinIL = $true) | ($true != (in @ sK51 @ (setadjoin @ sK51 @ sK50)))) & (! [X2 : $i,X3 : $i] : ($true = (in @ X3 @ (setadjoin @ X3 @ X2))) | (setadjoinIL != $true))),
% 2.23/1.57 inference(skolemisation,[status(esa),new_symbols(skolem,[sK50,sK51])],[f293,f294])).
% 2.23/1.57 thf(f294,plain,(
% 2.23/1.57 ? [X0 : $i,X1 : $i] : ($true != (in @ X1 @ (setadjoin @ X1 @ X0))) => ($true != (in @ sK51 @ (setadjoin @ sK51 @ sK50)))),
% 2.23/1.57 introduced(definition,[],[choice_axiom])).
% 2.23/1.57 thf(f293,plain,(
% 2.23/1.57 ((setadjoinIL = $true) | ? [X0 : $i,X1 : $i] : ($true != (in @ X1 @ (setadjoin @ X1 @ X0)))) & (! [X2 : $i,X3 : $i] : ($true = (in @ X3 @ (setadjoin @ X3 @ X2))) | (setadjoinIL != $true))),
% 2.23/1.57 inference(rectify,[status(thm)],[f292])).
% 2.23/1.57 thf(f292,plain,(
% 2.23/1.57 ((setadjoinIL = $true) | ? [X0 : $i,X1 : $i] : ($true != (in @ X1 @ (setadjoin @ X1 @ X0)))) & (! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X1 @ X0))) | (setadjoinIL != $true))),
% 2.23/1.57 inference(nnf_transformation,[status(thm)],[f80])).
% 2.23/1.57 thf(f80,plain,(
% 2.23/1.57 (setadjoinIL = $true) <=> ! [X0 : $i,X1 : $i] : ($true = (in @ X1 @ (setadjoin @ X1 @ X0)))),
% 2.23/1.57 inference(fool_elimination,[status(thm)],[f79])).
% 2.23/1.57 thf(f79,plain,(
% 2.23/1.57 (setadjoinIL = ! [X0 : $i,X1 : $i] : (in @ X1 @ (setadjoin @ X1 @ X0)))),
% 2.23/1.57 inference(rectify,[status(thm)],[f32])).
% 2.23/1.57 thf(f32,axiom,(
% 2.23/1.57 (setadjoinIL = ! [X2 : $i,X1 : $i] : (in @ X1 @ (setadjoin @ X1 @ X2)))),
% 2.23/1.57 file('/export/starexec/sandbox/tmp/tmp.xT3Ehs1D5u/DTF2THF_25397.p',setadjoinIL)).
% 2.23/1.57 thf(f1221,plain,(
% 2.23/1.57 ($true != (in @ sK10 @ (setadjoin @ sK10 @ emptyset)))),
% 2.23/1.57 inference(superposition,[status(thm)],[f469,f1198])).
% 2.23/1.57 thf(f1198,plain,(
% 2.23/1.57 (sK10 = sK11)),
% 2.23/1.57 inference(trivial_inequality_removal,[status(thm)],[f1187])).
% 2.23/1.57 thf(f1187,plain,(
% 2.23/1.57 ($true != $true) | (sK10 = sK11)),
% 2.23/1.57 inference(superposition,[status(thm)],[f924,f470])).
% 2.23/1.57 thf(f470,plain,(
% 2.23/1.57 ($true = (in @ sK10 @ (setadjoin @ sK11 @ emptyset)))),
% 2.23/1.57 inference(cnf_transformation,[status(thm)],[f222])).
% 2.23/1.57 thf(f924,plain,(
% 2.23/1.57 ( ! [X0 : $i,X1 : $i] : (($true != (in @ X0 @ (setadjoin @ X1 @ emptyset))) | (X0 = X1)) )),
% 2.23/1.57 inference(trivial_inequality_removal,[status(thm)],[f779])).
% 2.23/1.57 thf(f779,plain,(
% 2.23/1.57 ( ! [X0 : $i,X1 : $i] : (($true != $true) | ($true != (in @ X0 @ (setadjoin @ X1 @ emptyset))) | (X0 = X1)) )),
% 2.23/1.57 inference(definition_unfolding,[status(thm)],[f587,f501])).
% 2.23/1.57 thf(f501,plain,(
% 2.23/1.57 (uniqinunit = $true)),
% 2.23/1.57 inference(cnf_transformation,[status(thm)],[f222])).
% 2.23/1.57 thf(f587,plain,(
% 2.23/1.57 ( ! [X0 : $i,X1 : $i] : ((X0 = X1) | ($true != (in @ X0 @ (setadjoin @ X1 @ emptyset))) | (uniqinunit != $true)) )),
% 2.23/1.57 inference(cnf_transformation,[status(thm)],[f332])).
% 2.23/1.57 thf(f332,plain,(
% 2.23/1.57 (! [X0 : $i,X1 : $i] : ((X0 = X1) | ($true != (in @ X0 @ (setadjoin @ X1 @ emptyset)))) | (uniqinunit != $true)) & ((uniqinunit = $true) | ((sK68 != sK69) & ($true = (in @ sK68 @ (setadjoin @ sK69 @ emptyset)))))),
% 2.23/1.57 inference(skolemisation,[status(esa),new_symbols(skolem,[sK68,sK69])],[f330,f331])).
% 2.23/1.57 thf(f331,plain,(
% 2.23/1.57 ? [X2 : $i,X3 : $i] : ((X2 != X3) & ((in @ X2 @ (setadjoin @ X3 @ emptyset)) = $true)) => ((sK68 != sK69) & ($true = (in @ sK68 @ (setadjoin @ sK69 @ emptyset))))),
% 2.23/1.57 introduced(definition,[],[choice_axiom])).
% 2.23/1.57 thf(f330,plain,(
% 2.23/1.57 (! [X0 : $i,X1 : $i] : ((X0 = X1) | ($true != (in @ X0 @ (setadjoin @ X1 @ emptyset)))) | (uniqinunit != $true)) & ((uniqinunit = $true) | ? [X2 : $i,X3 : $i] : ((X2 != X3) & ((in @ X2 @ (setadjoin @ X3 @ emptyset)) = $true)))),
% 2.23/1.57 inference(rectify,[status(thm)],[f329])).
% 2.23/1.57 thf(f329,plain,(
% 2.23/1.57 (! [X0 : $i,X1 : $i] : ((X0 = X1) | ($true != (in @ X0 @ (setadjoin @ X1 @ emptyset)))) | (uniqinunit != $true)) & ((uniqinunit = $true) | ? [X0 : $i,X1 : $i] : ((X0 != X1) & ($true = (in @ X0 @ (setadjoin @ X1 @ emptyset)))))),
% 2.23/1.57 inference(nnf_transformation,[status(thm)],[f200])).
% 2.23/1.57 thf(f200,plain,(
% 2.23/1.57 ! [X0 : $i,X1 : $i] : ((X0 = X1) | ($true != (in @ X0 @ (setadjoin @ X1 @ emptyset)))) <=> (uniqinunit = $true)),
% 2.23/1.57 inference(ennf_transformation,[status(thm)],[f94])).
% 2.23/1.57 thf(f94,plain,(
% 2.23/1.57 ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (setadjoin @ X1 @ emptyset))) => (X0 = X1)) <=> (uniqinunit = $true)),
% 2.23/1.57 inference(fool_elimination,[status(thm)],[f93])).
% 2.23/1.57 thf(f93,plain,(
% 2.23/1.57 (uniqinunit = ! [X0 : $i,X1 : $i] : ((in @ X0 @ (setadjoin @ X1 @ emptyset)) => (X0 = X1)))),
% 2.23/1.57 inference(rectify,[status(thm)],[f47])).
% 2.23/1.57 thf(f47,axiom,(
% 2.23/1.57 (uniqinunit = ! [X1 : $i,X2 : $i] : ((in @ X1 @ (setadjoin @ X2 @ emptyset)) => (X1 = X2)))),
% 2.23/1.57 file('/export/starexec/sandbox/tmp/tmp.xT3Ehs1D5u/DTF2THF_25397.p',uniqinunit)).
% 2.23/1.57 thf(f469,plain,(
% 2.23/1.57 ($true != (in @ sK11 @ (setadjoin @ sK10 @ emptyset)))),
% 2.23/1.57 inference(cnf_transformation,[status(thm)],[f222])).
% 2.23/1.57 % SZS output end Proof for DTF2THF_25397
% 2.23/1.57 % (25622)------------------------------
% 2.23/1.57 % (25622)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.57 % (25622)Termination reason: Refutation
% 2.23/1.57
% 2.23/1.57 % (25622)Memory used [KB]: 6524
% 2.23/1.57 % (25622)Time elapsed: 0.055 s
% 2.23/1.57 % (25622)Instructions burned: 132 (million)
% 2.23/1.57 % (25622)------------------------------
% 2.23/1.57 % (25622)------------------------------
% 2.23/1.58 % (25613)Success in time 0.069 s
%------------------------------------------------------------------------------