%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU541^1 : 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 : n006.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 09:35:10 AM UTC 2026
% Result : Theorem 2.22s 1.63s
% Output : Refutation 2.22s
% Verified :
% SZS Type : Refutation
% Derivation depth : 11
% Number of leaves : 5
% Syntax : Number of formulae : 30 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 1056 ( 288 equ; 0 cnn)
% Maximal formula atoms : 59 ( 35 avg)
% Number of connectives : 549 ( 28 ~; 15 |; 184 &; 83 @)
% ( 2 <=>; 237 =>; 0 <=; 0 <~>)
% Maximal formula depth : 63 ( 17 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 139 ( 136 usr; 62 con; 0-3 aty)
% Number of variables : 24 ( 5 ^; 13 !; 6 ?; 24 :)
% 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_69,type,
set2prop: $i > $o ).
thf(func_def_82,type,
sP0: $i > $i > $o ).
thf(func_def_83,type,
sP1: $i > $o ).
thf(func_def_84,type,
sP2: $i > $i > $o ).
thf(func_def_85,type,
sP3: $i > $i > $o ).
thf(func_def_89,type,
sK7: $i > $o ).
thf(func_def_94,type,
sK12: $i > $o ).
thf(func_def_98,type,
sK16: $i > $o ).
thf(func_def_99,type,
sK17: ( $i > $o ) > $i ).
thf(func_def_101,type,
sK19: $i > $i > $i ).
thf(func_def_102,type,
sK20: $i > $i > $i ).
thf(func_def_103,type,
sK21: $i > $i > $i ).
thf(func_def_104,type,
sK22: $i > $i > $i ).
thf(func_def_105,type,
sK23: $i > $i > $i > $i ).
thf(func_def_106,type,
sK24: $i > $i ).
thf(func_def_107,type,
sK25: $i > $i ).
thf(func_def_108,type,
sK26: $i > $i ).
thf(func_def_109,type,
sK27: $i > $i ).
thf(func_def_110,type,
sK28: $i > $i > $i > $i ).
thf(func_def_111,type,
sK29: $i > $i > $i > $i ).
thf(func_def_112,type,
sK30: $i > $i > $i ).
thf(func_def_113,type,
sK31: $i > $i > $i ).
thf(func_def_114,type,
sK32: $i > $i > $i ).
thf(func_def_115,type,
sK33: $i > $i ).
thf(func_def_116,type,
sK34: $i > $i > $i ).
thf(func_def_118,type,
sK36: $i > $i ).
thf(func_def_119,type,
sK37: $i > $i ).
thf(func_def_122,type,
sK40: $i > $i > $i ).
thf(func_def_129,type,
sK47: $i > $i ).
thf(func_def_130,type,
sK48: $i > $i ).
thf(func_def_139,type,
sK57: $i > $i ).
thf(func_def_146,type,
sK64: ( $i > $o ) > $i ).
thf(func_def_147,type,
sK65: $i > $o ).
thf(func_def_148,type,
sK66: $i > $i ).
thf(func_def_153,type,
sK71: $i > $o ).
thf(func_def_154,type,
sK72: $i > $i ).
thf(func_def_155,type,
sK73: ( $i > $o ) > $i ).
thf(func_def_156,type,
sK74: $i > $o ).
thf(func_def_159,type,
sK77: $i > $o ).
thf(func_def_164,type,
sK82: $i > $i > $i ).
thf(func_def_165,type,
sK83: $i > $i > $i ).
thf(func_def_168,type,
sK86: $o > $i > $i > $i ).
thf(func_def_173,type,
sK91: $i > $o ).
thf(func_def_176,type,
sK94: $i > $o ).
thf(func_def_178,type,
sK96: ( $i > $o ) > $i > $i ).
thf(func_def_179,type,
sK97: ( $i > $o ) > $i > $i ).
thf(func_def_181,type,
sK99: $i > $o ).
thf(func_def_183,type,
sK101: $i > $o ).
thf(func_def_186,type,
sK104: $i > $i ).
thf(func_def_189,type,
sK107: $i > $i ).
thf(func_def_192,type,
sK110: $i > $i > $i ).
thf(func_def_196,type,
sK114: $i > $i > $i ).
thf(func_def_210,type,
sK128: $i > $i ).
thf(func_def_212,type,
sK130: ( $i > $o ) > $i > $i ).
thf(func_def_214,type,
sK132: $i > $o ).
thf(func_def_217,type,
sK135: $i > $o ).
thf(func_def_220,type,
sK138: $i > ( $i > $o ) > $i ).
thf(func_def_232,type,
sK150: $i > $i > $o ).
thf(func_def_234,type,
sK152: $i > $i ).
thf(func_def_235,type,
sK153: $i > $i ).
thf(func_def_236,type,
sK154: $i > ( $i > $i > $o ) > $i ).
thf(func_def_237,type,
sK155: $i > ( $i > $i > $o ) > $i ).
thf(func_def_238,type,
sK156: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_239,type,
sK157: $i > $i ).
thf(func_def_241,type,
sK159: $i > $i > $i ).
thf(func_def_246,type,
sK164: $i > $o ).
thf(func_def_248,type,
ph166:
!>[X0: $tType] : X0 ).
thf(f1170,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1112,f1118]) ).
thf(f1118,plain,
( ( in @ emptyset @ ( prop2set @ $true ) )
!= $true ),
inference(beta_eta_normalization,[status(thm)],[f906]) ).
thf(f906,plain,
( ( ^ [Y0: $i] : ( in @ emptyset @ Y0 )
@ ( prop2set @ $true ) )
!= $true ),
inference(definition_unfolding,[status(thm)],[f633,f568,f632]) ).
thf(f632,plain,
sK53 = $true,
inference(cnf_transformation,[status(thm)],[f350]) ).
thf(f350,plain,
( ( quantDeMorgan1 = $true )
& ( setadjoinE = $true )
& ( emptysetAx = $true )
& ( nonemptyI = $true )
& ( omega0Ax = $true )
& ( powersetI = $true )
& ( quantDeMorgan4 = $true )
& ( emptysetE = $true )
& ( replAx = $true )
& ( emptyI = $true )
& ( exuE3e = $true )
& ( nonemptyImpWitness = $true )
& ( notinemptyset = $true )
& ( nonemptyI1 = $true )
& ( noeltsimpempty = $true )
& ( upairsetIR = $true )
& ( upairsetE = $true )
& ( setadjoinAx = $true )
& ( setunionE = $true )
& ( setext = $true )
& ( notinsingleton = $true )
& ( prop2setE = $true )
& ( emptyinPowerset = $true )
& ( foundationAx = $true )
& ( omegaIndAx = $true )
& ( dsetconstrEL = $true )
& ( powersetAx = $true )
& ( setadjoinOr = $true )
& ( quantDeMorgan3 = $true )
& ( dsetconstrER = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionAx = $true )
& ( eqinunit = $true )
& ( exuE1 = $true )
& ( vacuousDall = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( setunionI = $true )
& ( emptyInPowerset = $true )
& ( setbeta = $true )
& ( omegaSAx = $true )
& ( singletonsswitch = $true )
& ( emptyE1 = $true )
& ( quantDeMorgan2 = $true )
& ( emptysetimpfalse = $true )
& ( prop2setI = $true )
& ( wellorderingAx = $true )
& ( upairsetIL = $true )
& ( setextAx = $true )
& ( powersetE = $true )
& ( setoftrueEq = $true )
& ( nonemptyE1 = $true )
& ( uniqinunit = $true )
& ( dsetconstrI = $true )
& ( ( set2prop @ ( prop2set @ sK53 ) )
!= $true )
& ( sK53 = $true )
& ( descrp = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK53])],[f237,f349]) ).
thf(f349,plain,
( ? [X0: $o] :
( ( ( set2prop @ ( prop2set @ X0 ) )
!= $true )
& ( $true = X0 ) )
=> ( ( ( set2prop @ ( prop2set @ sK53 ) )
!= $true )
& ( sK53 = $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f237,plain,
( ( quantDeMorgan1 = $true )
& ( setadjoinE = $true )
& ( emptysetAx = $true )
& ( nonemptyI = $true )
& ( omega0Ax = $true )
& ( powersetI = $true )
& ( quantDeMorgan4 = $true )
& ( emptysetE = $true )
& ( replAx = $true )
& ( emptyI = $true )
& ( exuE3e = $true )
& ( nonemptyImpWitness = $true )
& ( notinemptyset = $true )
& ( nonemptyI1 = $true )
& ( noeltsimpempty = $true )
& ( upairsetIR = $true )
& ( upairsetE = $true )
& ( setadjoinAx = $true )
& ( setunionE = $true )
& ( setext = $true )
& ( notinsingleton = $true )
& ( prop2setE = $true )
& ( emptyinPowerset = $true )
& ( foundationAx = $true )
& ( omegaIndAx = $true )
& ( dsetconstrEL = $true )
& ( powersetAx = $true )
& ( setadjoinOr = $true )
& ( quantDeMorgan3 = $true )
& ( dsetconstrER = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionAx = $true )
& ( eqinunit = $true )
& ( exuE1 = $true )
& ( vacuousDall = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( setunionI = $true )
& ( emptyInPowerset = $true )
& ( setbeta = $true )
& ( omegaSAx = $true )
& ( singletonsswitch = $true )
& ( emptyE1 = $true )
& ( quantDeMorgan2 = $true )
& ( emptysetimpfalse = $true )
& ( prop2setI = $true )
& ( wellorderingAx = $true )
& ( upairsetIL = $true )
& ( setextAx = $true )
& ( powersetE = $true )
& ( setoftrueEq = $true )
& ( nonemptyE1 = $true )
& ( uniqinunit = $true )
& ( dsetconstrI = $true )
& ? [X0: $o] :
( ( ( set2prop @ ( prop2set @ X0 ) )
!= $true )
& ( $true = X0 ) )
& ( descrp = $true ) ),
inference(flattening,[status(thm)],[f236]) ).
thf(f236,plain,
( ? [X0: $o] :
( ( ( set2prop @ ( prop2set @ X0 ) )
!= $true )
& ( $true = X0 ) )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( 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)],[f65]) ).
thf(f65,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 )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ! [X0: $o] :
( ( $true = X0 )
=> ( ( set2prop @ ( prop2set @ X0 ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f64]) ).
thf(f64,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
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ! [X0: $o] :
( X0
=> ( set2prop @ ( prop2set @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f62]) ).
thf(f62,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
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ! [X0: $o] :
( X0
=> ( set2prop @ ( prop2set @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f61]) ).
thf(f61,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
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ! [X0: $o] :
( X0
=> ( set2prop @ ( prop2set @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.rXSDAX2vBq/DTF2THF_22746.p',prop2set2propI) ).
thf(f568,plain,
( set2prop
= ( ^ [Y0: $i] : ( in @ emptyset @ Y0 ) ) ),
inference(cnf_transformation,[status(thm)],[f167]) ).
thf(f167,plain,
( set2prop
= ( ^ [Y0: $i] : ( in @ emptyset @ Y0 ) ) ),
inference(fool_elimination,[status(thm)],[f166]) ).
thf(f166,plain,
( set2prop
= ( ^ [X0: $i] : ( in @ emptyset @ X0 ) ) ),
inference(rectify,[status(thm)],[f60]) ).
thf(f60,axiom,
( set2prop
= ( ^ [X3: $i] : ( in @ emptyset @ X3 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.rXSDAX2vBq/DTF2THF_22746.p',set2prop) ).
thf(f633,plain,
( ( set2prop @ ( prop2set @ sK53 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f350]) ).
thf(f1112,plain,
( ( in @ emptyset @ ( prop2set @ $true ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1066]) ).
thf(f1066,plain,
( ( ( in @ emptyset @ ( prop2set @ $true ) )
= $true )
| ( $true != $true ) ),
inference(equality_resolution,[status(thm)],[f1014]) ).
thf(f1014,plain,
! [X1: $o] :
( ( $true
= ( in @ emptyset @ ( prop2set @ X1 ) ) )
| ( $true != X1 )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f795,f642]) ).
thf(f642,plain,
prop2setI = $true,
inference(cnf_transformation,[status(thm)],[f350]) ).
thf(f795,plain,
! [X1: $o] :
( ( $true
= ( in @ emptyset @ ( prop2set @ X1 ) ) )
| ( $true != X1 )
| ( prop2setI != $true ) ),
inference(cnf_transformation,[status(thm)],[f483]) ).
thf(f483,plain,
( ( ( prop2setI = $true )
| ( ( ( in @ emptyset @ ( prop2set @ sK127 ) )
!= $true )
& ( $true = sK127 ) ) )
& ( ! [X1: $o] :
( ( $true
= ( in @ emptyset @ ( prop2set @ X1 ) ) )
| ( $true != X1 ) )
| ( prop2setI != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK127])],[f481,f482]) ).
thf(f482,plain,
( ? [X0: $o] :
( ( ( in @ emptyset @ ( prop2set @ X0 ) )
!= $true )
& ( $true = X0 ) )
=> ( ( ( in @ emptyset @ ( prop2set @ sK127 ) )
!= $true )
& ( $true = sK127 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f481,plain,
( ( ( prop2setI = $true )
| ? [X0: $o] :
( ( ( in @ emptyset @ ( prop2set @ X0 ) )
!= $true )
& ( $true = X0 ) ) )
& ( ! [X1: $o] :
( ( $true
= ( in @ emptyset @ ( prop2set @ X1 ) ) )
| ( $true != X1 ) )
| ( prop2setI != $true ) ) ),
inference(rectify,[status(thm)],[f480]) ).
thf(f480,plain,
( ( ( prop2setI = $true )
| ? [X0: $o] :
( ( ( in @ emptyset @ ( prop2set @ X0 ) )
!= $true )
& ( $true = X0 ) ) )
& ( ! [X0: $o] :
( ( ( in @ emptyset @ ( prop2set @ X0 ) )
= $true )
| ( $true != X0 ) )
| ( prop2setI != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f209]) ).
thf(f209,plain,
( ( prop2setI = $true )
<=> ! [X0: $o] :
( ( ( in @ emptyset @ ( prop2set @ X0 ) )
= $true )
| ( $true != X0 ) ) ),
inference(ennf_transformation,[status(thm)],[f101]) ).
thf(f101,plain,
( ( prop2setI = $true )
<=> ! [X0: $o] :
( ( $true = X0 )
=> ( ( in @ emptyset @ ( prop2set @ X0 ) )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f100]) ).
thf(f100,plain,
! [X0: $o] :
( ( X0
=> ( in @ emptyset @ ( prop2set @ X0 ) ) )
= prop2setI ),
inference(rectify,[status(thm)],[f59]) ).
thf(f59,axiom,
! [X0: $o] :
( ( X0
=> ( in @ emptyset @ ( prop2set @ X0 ) ) )
= prop2setI ),
file('/export/starexec/sandbox/tmp/tmp.rXSDAX2vBq/DTF2THF_22746.p',prop2setI) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU541^1 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.34 % Computer : n006.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 12:26:15 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.34/0.49 ---- Original DTF file ---
% 0.34/0.49 thf(spec,logic,$$dhol).
% 0.34/0.49 %------------------------------------------------------------------------------
% 0.34/0.49 % File : SEU541^1 : TPTP v9.2.1. Released v3.7.0.
% 0.34/0.49 % Domain : Set Theory
% 0.34/0.49 % Problem : Preliminary Notions - Dependent Connective Laws
% 0.34/0.49 % Version : Especial.
% 0.34/0.49 % English : (! phi:o.phi -> set2prop (prop2set phi))
% 0.34/0.49
% 0.34/0.49 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.34/0.49 % Source : [Bro08]
% 0.34/0.49 % Names : ZFC043g [Bro08]
% 0.34/0.49
% 0.34/0.49 % Status : Theorem
% 0.34/0.49 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.12 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, 0.50 v5.4.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v3.7.0
% 0.34/0.49 % Syntax : Number of formulae : 130 ( 60 unt; 69 typ; 60 def)
% 0.34/0.49 % Number of atoms : 327 ( 79 equ; 0 cnn)
% 0.34/0.49 % Maximal formula atoms : 58 ( 5 avg)
% 0.34/0.49 % Number of connectives : 539 ( 17 ~; 5 |; 24 &; 346 @)
% 0.34/0.49 % ( 8 <=>; 139 =>; 0 <=; 0 <~>)
% 0.34/0.49 % Maximal formula depth : 62 ( 2 avg)
% 0.34/0.49 % Number of types : 2 ( 0 usr)
% 0.34/0.49 % Number of type conns : 34 ( 34 >; 0 *; 0 +; 0 <<)
% 0.34/0.49 % Number of symbols : 72 ( 69 usr; 61 con; 0-2 aty)
% 0.34/0.49 % Number of variables : 176 ( 16 ^; 137 !; 23 ?; 176 :)
% 0.34/0.49 % SPC : TH0_THM_EQU_NAR
% 0.34/0.49
% 0.34/0.49 % Comments : http://mathgate.info/detsetitem.php?id=456
% 0.34/0.49 %------------------------------------------------------------------------------
% 0.34/0.49 thf(in_type,type,
% 0.34/0.49 in: $i > $i > $o ).
% 0.34/0.49
% 0.34/0.49 thf(exu_type,type,
% 0.34/0.49 exu: ( $i > $o ) > $o ).
% 0.34/0.49
% 0.34/0.49 thf(exu,definition,
% 0.34/0.49 ( exu
% 0.34/0.49 = ( ^ [Xphi: $i > $o] :
% 0.34/0.49 ? [Xx: $i] :
% 0.34/0.49 ( ( Xphi @ Xx )
% 0.34/0.49 & ! [Xy: $i] :
% 0.34/0.49 ( ( Xphi @ Xy )
% 0.34/0.49 => ( Xx = Xy ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setextAx_type,type,
% 0.34/0.49 setextAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setextAx,definition,
% 0.34/0.49 ( setextAx
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 <=> ( in @ Xx @ B ) )
% 0.34/0.49 => ( A = B ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptyset_type,type,
% 0.34/0.49 emptyset: $i ).
% 0.34/0.49
% 0.34/0.49 thf(emptysetAx_type,type,
% 0.34/0.49 emptysetAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptysetAx,definition,
% 0.34/0.49 ( emptysetAx
% 0.34/0.49 = ( ! [Xx: $i] :
% 0.34/0.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoin_type,type,
% 0.34/0.49 setadjoin: $i > $i > $i ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinAx_type,type,
% 0.34/0.49 setadjoinAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinAx,definition,
% 0.34/0.49 ( setadjoinAx
% 0.34/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.34/0.49 <=> ( ( Xy = Xx )
% 0.34/0.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(powerset_type,type,
% 0.34/0.49 powerset: $i > $i ).
% 0.34/0.49
% 0.34/0.49 thf(powersetAx_type,type,
% 0.34/0.49 powersetAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(powersetAx,definition,
% 0.34/0.49 ( powersetAx
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ( in @ B @ ( powerset @ A ) )
% 0.34/0.49 <=> ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setunion_type,type,
% 0.34/0.49 setunion: $i > $i ).
% 0.34/0.49
% 0.34/0.49 thf(setunionAx_type,type,
% 0.34/0.49 setunionAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setunionAx,definition,
% 0.34/0.49 ( setunionAx
% 0.34/0.49 = ( ! [A: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.34/0.49 <=> ? [B: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 & ( in @ B @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(omega_type,type,
% 0.34/0.49 omega: $i ).
% 0.34/0.49
% 0.34/0.49 thf(omega0Ax_type,type,
% 0.34/0.49 omega0Ax: $o ).
% 0.34/0.49
% 0.34/0.49 thf(omega0Ax,definition,
% 0.34/0.49 ( omega0Ax
% 0.34/0.49 = ( in @ emptyset @ omega ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(omegaSAx_type,type,
% 0.34/0.49 omegaSAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(omegaSAx,definition,
% 0.34/0.49 ( omegaSAx
% 0.34/0.49 = ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ omega )
% 0.34/0.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(omegaIndAx_type,type,
% 0.34/0.49 omegaIndAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(omegaIndAx,definition,
% 0.34/0.49 ( omegaIndAx
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ( ( in @ emptyset @ A )
% 0.34/0.49 & ! [Xx: $i] :
% 0.34/0.49 ( ( ( in @ Xx @ omega )
% 0.34/0.49 & ( in @ Xx @ A ) )
% 0.34/0.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.34/0.49 => ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ omega )
% 0.34/0.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(replAx_type,type,
% 0.34/0.49 replAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(replAx,definition,
% 0.34/0.49 ( replAx
% 0.34/0.49 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( exu
% 0.34/0.49 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.34/0.49 => ? [B: $i] :
% 0.34/0.49 ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 <=> ? [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ A )
% 0.34/0.49 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(foundationAx_type,type,
% 0.34/0.49 foundationAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(foundationAx,definition,
% 0.34/0.49 ( foundationAx
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.34/0.49 => ? [B: $i] :
% 0.34/0.49 ( ( in @ B @ A )
% 0.34/0.49 & ~ ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(wellorderingAx_type,type,
% 0.34/0.49 wellorderingAx: $o ).
% 0.34/0.49
% 0.34/0.49 thf(wellorderingAx,definition,
% 0.34/0.49 ( wellorderingAx
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ? [B: $i] :
% 0.34/0.49 ( ! [C: $i] :
% 0.34/0.49 ( ( in @ C @ B )
% 0.34/0.49 => ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ C )
% 0.34/0.49 => ( in @ Xx @ A ) ) )
% 0.34/0.49 & ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( ( in @ Xx @ A )
% 0.34/0.49 & ( in @ Xy @ A ) )
% 0.34/0.49 => ( ! [C: $i] :
% 0.34/0.49 ( ( in @ C @ B )
% 0.34/0.49 => ( ( in @ Xx @ C )
% 0.34/0.49 <=> ( in @ Xy @ C ) ) )
% 0.34/0.49 => ( Xx = Xy ) ) )
% 0.34/0.49 & ! [C: $i,D: $i] :
% 0.34/0.49 ( ( ( in @ C @ B )
% 0.34/0.49 & ( in @ D @ B ) )
% 0.34/0.49 => ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ C )
% 0.34/0.49 => ( in @ Xx @ D ) )
% 0.34/0.49 | ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ D )
% 0.34/0.49 => ( in @ Xx @ C ) ) ) )
% 0.34/0.49 & ! [C: $i] :
% 0.34/0.49 ( ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ C )
% 0.34/0.49 => ( in @ Xx @ A ) )
% 0.34/0.49 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.34/0.49 => ? [D: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ D @ B )
% 0.34/0.49 & ( in @ Xx @ C )
% 0.34/0.49 & ~ ? [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ D )
% 0.34/0.49 & ( in @ Xy @ C ) )
% 0.34/0.49 & ! [E: $i] :
% 0.34/0.49 ( ( in @ E @ B )
% 0.34/0.49 => ( ! [Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ E )
% 0.34/0.49 => ( in @ Xy @ D ) )
% 0.34/0.49 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(descr_type,type,
% 0.34/0.49 descr: ( $i > $o ) > $i ).
% 0.34/0.49
% 0.34/0.49 thf(descrp_type,type,
% 0.34/0.49 descrp: $o ).
% 0.34/0.49
% 0.34/0.49 thf(descrp,definition,
% 0.34/0.49 ( descrp
% 0.34/0.49 = ( ! [Xphi: $i > $o] :
% 0.34/0.49 ( ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 => ( Xphi
% 0.34/0.49 @ ( descr
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(dsetconstr_type,type,
% 0.34/0.49 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.34/0.49
% 0.34/0.49 thf(dsetconstrI_type,type,
% 0.34/0.49 dsetconstrI: $o ).
% 0.34/0.49
% 0.34/0.49 thf(dsetconstrI,definition,
% 0.34/0.49 ( dsetconstrI
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( ( Xphi @ Xx )
% 0.34/0.49 => ( in @ Xx
% 0.34/0.49 @ ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(dsetconstrEL_type,type,
% 0.34/0.49 dsetconstrEL: $o ).
% 0.34/0.49
% 0.34/0.49 thf(dsetconstrEL,definition,
% 0.34/0.49 ( dsetconstrEL
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx
% 0.34/0.49 @ ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.34/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(dsetconstrER_type,type,
% 0.34/0.49 dsetconstrER: $o ).
% 0.34/0.49
% 0.34/0.49 thf(dsetconstrER,definition,
% 0.34/0.49 ( dsetconstrER
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx
% 0.34/0.49 @ ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.34/0.49 => ( Xphi @ Xx ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(exuE1_type,type,
% 0.34/0.49 exuE1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(exuE1,definition,
% 0.34/0.49 ( exuE1
% 0.34/0.49 = ( ! [Xphi: $i > $o] :
% 0.34/0.49 ( ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 => ? [Xx: $i] :
% 0.34/0.49 ( ( Xphi @ Xx )
% 0.34/0.49 & ! [Xy: $i] :
% 0.34/0.49 ( ( Xphi @ Xy )
% 0.34/0.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(prop2set_type,type,
% 0.34/0.49 prop2set: $o > $i ).
% 0.34/0.49
% 0.34/0.49 thf(prop2setE_type,type,
% 0.34/0.49 prop2setE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(prop2setE,definition,
% 0.34/0.49 ( prop2setE
% 0.34/0.49 = ( ! [Xphi: $o,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.34/0.49 => Xphi ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptysetE_type,type,
% 0.34/0.49 emptysetE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptysetE,definition,
% 0.34/0.49 ( emptysetE
% 0.34/0.49 = ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ emptyset )
% 0.34/0.49 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptysetimpfalse_type,type,
% 0.34/0.49 emptysetimpfalse: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptysetimpfalse,definition,
% 0.34/0.49 ( emptysetimpfalse
% 0.34/0.49 = ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ emptyset )
% 0.34/0.49 => $false ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(notinemptyset_type,type,
% 0.34/0.49 notinemptyset: $o ).
% 0.34/0.49
% 0.34/0.49 thf(notinemptyset,definition,
% 0.34/0.49 ( notinemptyset
% 0.34/0.49 = ( ! [Xx: $i] :
% 0.34/0.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(exuE3e_type,type,
% 0.34/0.49 exuE3e: $o ).
% 0.34/0.49
% 0.34/0.49 thf(exuE3e,definition,
% 0.34/0.49 ( exuE3e
% 0.34/0.49 = ( ! [Xphi: $i > $o] :
% 0.34/0.49 ( ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setext_type,type,
% 0.34/0.49 setext: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setext,definition,
% 0.34/0.49 ( setext
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( in @ Xx @ B ) )
% 0.34/0.49 => ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 => ( in @ Xx @ A ) )
% 0.34/0.49 => ( A = B ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptyI_type,type,
% 0.34/0.49 emptyI: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptyI,definition,
% 0.34/0.49 ( emptyI
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ~ ( in @ Xx @ A )
% 0.34/0.49 => ( A = emptyset ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(noeltsimpempty_type,type,
% 0.34/0.49 noeltsimpempty: $o ).
% 0.34/0.49
% 0.34/0.49 thf(noeltsimpempty,definition,
% 0.34/0.49 ( noeltsimpempty
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ~ ( in @ Xx @ A )
% 0.34/0.49 => ( A = emptyset ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setbeta_type,type,
% 0.34/0.49 setbeta: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setbeta,definition,
% 0.34/0.49 ( setbeta
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( ( in @ Xx
% 0.34/0.49 @ ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.34/0.49 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(nonempty_type,type,
% 0.34/0.49 nonempty: $i > $o ).
% 0.34/0.49
% 0.34/0.49 thf(nonempty,definition,
% 0.34/0.49 ( nonempty
% 0.34/0.49 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(nonemptyE1_type,type,
% 0.34/0.49 nonemptyE1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(nonemptyE1,definition,
% 0.34/0.49 ( nonemptyE1
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ( nonempty @ A )
% 0.34/0.49 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(nonemptyI_type,type,
% 0.34/0.49 nonemptyI: $o ).
% 0.34/0.49
% 0.34/0.49 thf(nonemptyI,definition,
% 0.34/0.49 ( nonemptyI
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( ( Xphi @ Xx )
% 0.34/0.49 => ( nonempty
% 0.34/0.49 @ ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(nonemptyI1_type,type,
% 0.34/0.49 nonemptyI1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(nonemptyI1,definition,
% 0.34/0.49 ( nonemptyI1
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.34/0.49 => ( nonempty @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinIL_type,type,
% 0.34/0.49 setadjoinIL: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinIL,definition,
% 0.34/0.49 ( setadjoinIL
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptyinunitempty_type,type,
% 0.34/0.49 emptyinunitempty: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptyinunitempty,definition,
% 0.34/0.49 ( emptyinunitempty
% 0.34/0.49 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinIR_type,type,
% 0.34/0.49 setadjoinIR: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinIR,definition,
% 0.34/0.49 ( setadjoinIR
% 0.34/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ A )
% 0.34/0.49 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinE_type,type,
% 0.34/0.49 setadjoinE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinE,definition,
% 0.34/0.49 ( setadjoinE
% 0.34/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.34/0.49 => ! [Xphi: $o] :
% 0.34/0.49 ( ( ( Xy = Xx )
% 0.34/0.49 => Xphi )
% 0.34/0.49 => ( ( ( in @ Xy @ A )
% 0.34/0.49 => Xphi )
% 0.34/0.49 => Xphi ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinOr_type,type,
% 0.34/0.49 setadjoinOr: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setadjoinOr,definition,
% 0.34/0.49 ( setadjoinOr
% 0.34/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.34/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.34/0.49 => ( ( Xy = Xx )
% 0.34/0.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setoftrueEq_type,type,
% 0.34/0.49 setoftrueEq: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setoftrueEq,definition,
% 0.34/0.49 ( setoftrueEq
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xx: $i] : $true )
% 0.34/0.49 = A ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(powersetI_type,type,
% 0.34/0.49 powersetI: $o ).
% 0.34/0.49
% 0.34/0.49 thf(powersetI,definition,
% 0.34/0.49 ( powersetI
% 0.34/0.49 = ( ! [A: $i,B: $i] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 => ( in @ Xx @ A ) )
% 0.34/0.49 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptyinPowerset_type,type,
% 0.34/0.49 emptyinPowerset: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptyinPowerset,definition,
% 0.34/0.49 ( emptyinPowerset
% 0.34/0.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptyInPowerset_type,type,
% 0.34/0.49 emptyInPowerset: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptyInPowerset,definition,
% 0.34/0.49 ( emptyInPowerset
% 0.34/0.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(powersetE_type,type,
% 0.34/0.49 powersetE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(powersetE,definition,
% 0.34/0.49 ( powersetE
% 0.34/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ B @ ( powerset @ A ) )
% 0.34/0.49 => ( ( in @ Xx @ B )
% 0.34/0.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setunionI_type,type,
% 0.34/0.49 setunionI: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setunionI,definition,
% 0.34/0.49 ( setunionI
% 0.34/0.49 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 => ( ( in @ B @ A )
% 0.34/0.49 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(setunionE_type,type,
% 0.34/0.49 setunionE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(setunionE,definition,
% 0.34/0.49 ( setunionE
% 0.34/0.49 = ( ! [A: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.34/0.49 => ! [Xphi: $o] :
% 0.34/0.49 ( ! [B: $i] :
% 0.34/0.49 ( ( in @ Xx @ B )
% 0.34/0.49 => ( ( in @ B @ A )
% 0.34/0.49 => Xphi ) )
% 0.34/0.49 => Xphi ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(subPowSU_type,type,
% 0.34/0.49 subPowSU: $o ).
% 0.34/0.49
% 0.34/0.49 thf(subPowSU,definition,
% 0.34/0.49 ( subPowSU
% 0.34/0.49 = ( ! [A: $i,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(exuE2_type,type,
% 0.34/0.49 exuE2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(exuE2,definition,
% 0.34/0.49 ( exuE2
% 0.34/0.49 = ( ! [Xphi: $i > $o] :
% 0.34/0.49 ( ( exu
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 => ? [Xx: $i] :
% 0.34/0.49 ! [Xy: $i] :
% 0.34/0.49 ( ( Xphi @ Xy )
% 0.34/0.49 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(nonemptyImpWitness_type,type,
% 0.34/0.49 nonemptyImpWitness: $o ).
% 0.34/0.49
% 0.34/0.49 thf(nonemptyImpWitness,definition,
% 0.34/0.49 ( nonemptyImpWitness
% 0.34/0.49 = ( ! [A: $i] :
% 0.34/0.49 ( ( nonempty @ A )
% 0.34/0.49 => ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & $true ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(uniqinunit_type,type,
% 0.34/0.49 uniqinunit: $o ).
% 0.34/0.49
% 0.34/0.49 thf(uniqinunit,definition,
% 0.34/0.49 ( uniqinunit
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.34/0.49 => ( Xx = Xy ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(notinsingleton_type,type,
% 0.34/0.49 notinsingleton: $o ).
% 0.34/0.49
% 0.34/0.49 thf(notinsingleton,definition,
% 0.34/0.49 ( notinsingleton
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( Xx != Xy )
% 0.34/0.49 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(eqinunit_type,type,
% 0.34/0.49 eqinunit: $o ).
% 0.34/0.49
% 0.34/0.49 thf(eqinunit,definition,
% 0.34/0.49 ( eqinunit
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( Xx = Xy )
% 0.34/0.49 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(singletonsswitch_type,type,
% 0.34/0.49 singletonsswitch: $o ).
% 0.34/0.49
% 0.34/0.49 thf(singletonsswitch,definition,
% 0.34/0.49 ( singletonsswitch
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.34/0.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.34/0.49 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(upairsetE_type,type,
% 0.34/0.49 upairsetE: $o ).
% 0.34/0.49
% 0.34/0.49 thf(upairsetE,definition,
% 0.34/0.49 ( upairsetE
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.34/0.49 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.34/0.49 => ( ( Xz = Xx )
% 0.34/0.49 | ( Xz = Xy ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(upairsetIL_type,type,
% 0.34/0.49 upairsetIL: $o ).
% 0.34/0.49
% 0.34/0.49 thf(upairsetIL,definition,
% 0.34/0.49 ( upairsetIL
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(upairsetIR_type,type,
% 0.34/0.49 upairsetIR: $o ).
% 0.34/0.49
% 0.34/0.49 thf(upairsetIR,definition,
% 0.34/0.49 ( upairsetIR
% 0.34/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(emptyE1_type,type,
% 0.34/0.49 emptyE1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(emptyE1,definition,
% 0.34/0.49 ( emptyE1
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ( Xphi @ Xx ) )
% 0.34/0.49 => ( ( ( dsetconstr @ A
% 0.34/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.34/0.49 = emptyset )
% 0.34/0.49 => $false ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(vacuousDall_type,type,
% 0.34/0.49 vacuousDall: $o ).
% 0.34/0.49
% 0.34/0.49 thf(vacuousDall,definition,
% 0.34/0.49 ( vacuousDall
% 0.34/0.49 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ emptyset )
% 0.34/0.49 => ( Xphi @ Xx ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(quantDeMorgan1_type,type,
% 0.34/0.49 quantDeMorgan1: $o ).
% 0.34/0.49
% 0.34/0.49 thf(quantDeMorgan1,definition,
% 0.34/0.49 ( quantDeMorgan1
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( ~ ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( Xphi @ Xx ) )
% 0.34/0.49 => ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(quantDeMorgan2_type,type,
% 0.34/0.49 quantDeMorgan2: $o ).
% 0.34/0.49
% 0.34/0.49 thf(quantDeMorgan2,definition,
% 0.34/0.49 ( quantDeMorgan2
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ~ ( Xphi @ Xx ) )
% 0.34/0.49 => ~ ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(quantDeMorgan3_type,type,
% 0.34/0.49 quantDeMorgan3: $o ).
% 0.34/0.49
% 0.34/0.49 thf(quantDeMorgan3,definition,
% 0.34/0.49 ( quantDeMorgan3
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( ~ ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ( Xphi @ Xx ) )
% 0.34/0.49 => ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(quantDeMorgan4_type,type,
% 0.34/0.49 quantDeMorgan4: $o ).
% 0.34/0.49
% 0.34/0.49 thf(quantDeMorgan4,definition,
% 0.34/0.49 ( quantDeMorgan4
% 0.34/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.34/0.49 ( ? [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 & ~ ( Xphi @ Xx ) )
% 0.34/0.49 => ~ ! [Xx: $i] :
% 0.34/0.49 ( ( in @ Xx @ A )
% 0.34/0.49 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(prop2setI_type,type,
% 0.34/0.49 prop2setI: $o ).
% 0.34/0.49
% 0.34/0.49 thf(prop2setI,definition,
% 0.34/0.49 ( prop2setI
% 0.34/0.49 = ( ! [Xphi: $o] :
% 0.34/0.49 ( Xphi
% 0.34/0.49 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(set2prop_type,type,
% 0.34/0.49 set2prop: $i > $o ).
% 0.34/0.49
% 0.34/0.49 thf(set2prop,definition,
% 0.34/0.49 ( set2prop
% 0.34/0.49 = ( ^ [A: $i] : ( in @ emptyset @ A ) ) ) ).
% 0.34/0.49
% 0.34/0.49 thf(prop2set2propI,conjecture,
% 0.34/0.49 ( setextAx
% 0.34/0.49 => ( emptysetAx
% 0.34/0.49 => ( setadjoinAx
% 0.34/0.49 => ( powersetAx
% 0.34/0.49 => ( setunionAx
% 0.34/0.49 => ( omega0Ax
% 0.34/0.49 => ( omegaSAx
% 0.34/0.49 => ( omegaIndAx
% 0.34/0.49 => ( replAx
% 0.34/0.49 => ( foundationAx
% 0.34/0.49 => ( wellorderingAx
% 0.34/0.49 => ( descrp
% 0.34/0.49 => ( dsetconstrI
% 0.34/0.49 => ( dsetconstrEL
% 0.34/0.49 => ( dsetconstrER
% 0.34/0.49 => ( exuE1
% 0.34/0.49 => ( prop2setE
% 0.34/0.49 => ( emptysetE
% 0.34/0.49 => ( emptysetimpfalse
% 0.34/0.49 => ( notinemptyset
% 0.34/0.49 => ( exuE3e
% 0.34/0.49 => ( setext
% 0.34/0.49 => ( emptyI
% 0.34/0.49 => ( noeltsimpempty
% 0.34/0.49 => ( setbeta
% 0.34/0.49 => ( nonemptyE1
% 0.34/0.49 => ( nonemptyI
% 0.34/0.49 => ( nonemptyI1
% 0.34/0.49 => ( setadjoinIL
% 0.34/0.49 => ( emptyinunitempty
% 0.34/0.49 => ( setadjoinIR
% 0.34/0.49 => ( setadjoinE
% 0.34/0.49 => ( setadjoinOr
% 0.34/0.49 => ( setoftrueEq
% 0.34/0.49 => ( powersetI
% 0.34/0.49 => ( emptyinPowerset
% 0.34/0.49 => ( emptyInPowerset
% 0.34/0.49 => ( powersetE
% 0.34/0.49 => ( setunionI
% 0.34/0.49 => ( setunionE
% 0.34/0.49 => ( subPowSU
% 0.34/0.49 => ( exuE2
% 0.34/0.49 => ( nonemptyImpWitness
% 0.34/0.49 => ( uniqinunit
% 0.34/0.49 => ( notinsingleton
% 0.34/0.49 => ( eqinunit
% 0.34/0.49 => ( singletonsswitch
% 0.34/0.49 => ( upairsetE
% 0.34/0.49 => ( upairsetIL
% 0.34/0.49 => ( upairsetIR
% 0.34/0.49 => ( emptyE1
% 0.34/0.49 => ( vacuousDall
% 0.34/0.49 => ( quantDeMorgan1
% 0.34/0.49 => ( quantDeMorgan2
% 0.34/0.49 => ( quantDeMorgan3
% 0.34/0.49 => ( quantDeMorgan4
% 0.34/0.49 => ( prop2setI
% 0.34/0.49 => ! [Xphi: $o] :
% 0.34/0.49 ( Xphi
% 0.34/0.49 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.34/0.49
% 0.34/0.49 %------------------------------------------------------------------------------
% 0.34/0.49 ------------------------
% 2.05/1.56 ---- Embedded in THF ---
% 2.05/1.56 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.05/1.56 %%% Generated on Tue Feb 24 12:26:16 EST 2026
% 2.05/1.56 %%% using '$$dhol' embedding, version 1.3.0.
% 2.05/1.56 %%% Logic specification used:
% 2.05/1.56 %%% thf(spec, logic, $$dhol).
% 2.05/1.56
% 2.05/1.56 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.rXSDAX2vBq/DTF2DTF22746.p
% See solution above
% 2.05/1.56 ------------------------
% 2.05/1.58 ---- Cleaned THF ---
% 2.05/1.58 thf(in_type,type,
% 2.05/1.58 in: $i > $i > $o ).
% 2.05/1.58
% 2.05/1.58 thf(exu_type,type,
% 2.05/1.58 exu: ( $i > $o ) > $o ).
% 2.05/1.58
% 2.05/1.58 thf(exu,definition,
% 2.05/1.58 ( exu
% 2.05/1.58 = ( ^ [Xphi: $i > $o] :
% 2.05/1.58 ? [Xx: $i] :
% 2.05/1.58 ( ( Xphi @ Xx )
% 2.05/1.58 & ! [Xy: $i] :
% 2.05/1.58 ( ( Xphi @ Xy )
% 2.05/1.58 => ( Xx = Xy ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(setextAx_type,type,
% 2.05/1.58 setextAx: $o ).
% 2.05/1.58
% 2.05/1.58 thf(setextAx,definition,
% 2.05/1.58 ( setextAx
% 2.05/1.58 = ( ! [A: $i,B: $i] :
% 2.05/1.58 ( ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 <=> ( in @ Xx @ B ) )
% 2.05/1.58 => ( A = B ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(emptyset_type,type,
% 2.05/1.58 emptyset: $i ).
% 2.05/1.58
% 2.05/1.58 thf(emptysetAx_type,type,
% 2.05/1.58 emptysetAx: $o ).
% 2.05/1.58
% 2.05/1.58 thf(emptysetAx,definition,
% 2.05/1.58 ( emptysetAx
% 2.05/1.58 = ( ! [Xx: $i] :
% 2.05/1.58 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(setadjoin_type,type,
% 2.05/1.58 setadjoin: $i > $i > $i ).
% 2.05/1.58
% 2.05/1.58 thf(setadjoinAx_type,type,
% 2.05/1.58 setadjoinAx: $o ).
% 2.05/1.58
% 2.05/1.58 thf(setadjoinAx,definition,
% 2.05/1.58 ( setadjoinAx
% 2.05/1.58 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.05/1.58 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.05/1.58 <=> ( ( Xy = Xx )
% 2.05/1.58 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(powerset_type,type,
% 2.05/1.58 powerset: $i > $i ).
% 2.05/1.58
% 2.05/1.58 thf(powersetAx_type,type,
% 2.05/1.58 powersetAx: $o ).
% 2.05/1.58
% 2.05/1.58 thf(powersetAx,definition,
% 2.05/1.58 ( powersetAx
% 2.05/1.58 = ( ! [A: $i,B: $i] :
% 2.05/1.58 ( ( in @ B @ ( powerset @ A ) )
% 2.05/1.58 <=> ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ B )
% 2.05/1.58 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(setunion_type,type,
% 2.05/1.58 setunion: $i > $i ).
% 2.05/1.58
% 2.05/1.58 thf(setunionAx_type,type,
% 2.05/1.58 setunionAx: $o ).
% 2.05/1.58
% 2.05/1.58 thf(setunionAx,definition,
% 2.05/1.58 ( setunionAx
% 2.05/1.58 = ( ! [A: $i,Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ ( setunion @ A ) )
% 2.05/1.58 <=> ? [B: $i] :
% 2.05/1.58 ( ( in @ Xx @ B )
% 2.05/1.58 & ( in @ B @ A ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(omega_type,type,
% 2.05/1.58 omega: $i ).
% 2.05/1.58
% 2.05/1.58 thf(omega0Ax_type,type,
% 2.05/1.58 omega0Ax: $o ).
% 2.05/1.58
% 2.05/1.58 thf(omega0Ax,definition,
% 2.05/1.58 ( omega0Ax
% 2.05/1.58 = ( in @ emptyset @ omega ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(omegaSAx_type,type,
% 2.05/1.58 omegaSAx: $o ).
% 2.05/1.58
% 2.05/1.58 thf(omegaSAx,definition,
% 2.05/1.58 ( omegaSAx
% 2.05/1.58 = ( ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ omega )
% 2.05/1.58 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(omegaIndAx_type,type,
% 2.05/1.58 omegaIndAx: $o ).
% 2.05/1.58
% 2.05/1.58 thf(omegaIndAx,definition,
% 2.05/1.58 ( omegaIndAx
% 2.05/1.58 = ( ! [A: $i] :
% 2.05/1.58 ( ( ( in @ emptyset @ A )
% 2.05/1.58 & ! [Xx: $i] :
% 2.05/1.58 ( ( ( in @ Xx @ omega )
% 2.05/1.58 & ( in @ Xx @ A ) )
% 2.05/1.58 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.05/1.58 => ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ omega )
% 2.05/1.58 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(replAx_type,type,
% 2.05/1.58 replAx: $o ).
% 2.05/1.58
% 2.05/1.58 thf(replAx,definition,
% 2.05/1.58 ( replAx
% 2.05/1.58 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.05/1.58 ( ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 => ( exu
% 2.05/1.58 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.05/1.58 => ? [B: $i] :
% 2.05/1.58 ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ B )
% 2.05/1.58 <=> ? [Xy: $i] :
% 2.05/1.58 ( ( in @ Xy @ A )
% 2.05/1.58 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(foundationAx_type,type,
% 2.05/1.58 foundationAx: $o ).
% 2.05/1.58
% 2.05/1.58 thf(foundationAx,definition,
% 2.05/1.58 ( foundationAx
% 2.05/1.58 = ( ! [A: $i] :
% 2.05/1.58 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.05/1.58 => ? [B: $i] :
% 2.05/1.58 ( ( in @ B @ A )
% 2.05/1.58 & ~ ? [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ B )
% 2.05/1.58 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(wellorderingAx_type,type,
% 2.05/1.58 wellorderingAx: $o ).
% 2.05/1.58
% 2.05/1.58 thf(wellorderingAx,definition,
% 2.05/1.58 ( wellorderingAx
% 2.05/1.58 = ( ! [A: $i] :
% 2.05/1.58 ? [B: $i] :
% 2.05/1.58 ( ! [C: $i] :
% 2.05/1.58 ( ( in @ C @ B )
% 2.05/1.58 => ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ C )
% 2.05/1.58 => ( in @ Xx @ A ) ) )
% 2.05/1.58 & ! [Xx: $i,Xy: $i] :
% 2.05/1.58 ( ( ( in @ Xx @ A )
% 2.05/1.58 & ( in @ Xy @ A ) )
% 2.05/1.58 => ( ! [C: $i] :
% 2.05/1.58 ( ( in @ C @ B )
% 2.05/1.58 => ( ( in @ Xx @ C )
% 2.05/1.58 <=> ( in @ Xy @ C ) ) )
% 2.05/1.58 => ( Xx = Xy ) ) )
% 2.05/1.58 & ! [C: $i,D: $i] :
% 2.05/1.58 ( ( ( in @ C @ B )
% 2.05/1.58 & ( in @ D @ B ) )
% 2.05/1.58 => ( ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ C )
% 2.05/1.58 => ( in @ Xx @ D ) )
% 2.05/1.58 | ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ D )
% 2.05/1.58 => ( in @ Xx @ C ) ) ) )
% 2.05/1.58 & ! [C: $i] :
% 2.05/1.58 ( ( ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ C )
% 2.05/1.58 => ( in @ Xx @ A ) )
% 2.05/1.58 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.05/1.58 => ? [D: $i,Xx: $i] :
% 2.05/1.58 ( ( in @ D @ B )
% 2.05/1.58 & ( in @ Xx @ C )
% 2.05/1.58 & ~ ? [Xy: $i] :
% 2.05/1.58 ( ( in @ Xy @ D )
% 2.05/1.58 & ( in @ Xy @ C ) )
% 2.05/1.58 & ! [E: $i] :
% 2.05/1.58 ( ( in @ E @ B )
% 2.05/1.58 => ( ! [Xy: $i] :
% 2.05/1.58 ( ( in @ Xy @ E )
% 2.05/1.58 => ( in @ Xy @ D ) )
% 2.05/1.58 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(descr_type,type,
% 2.05/1.58 descr: ( $i > $o ) > $i ).
% 2.05/1.58
% 2.05/1.58 thf(descrp_type,type,
% 2.05/1.58 descrp: $o ).
% 2.05/1.58
% 2.05/1.58 thf(descrp,definition,
% 2.05/1.58 ( descrp
% 2.05/1.58 = ( ! [Xphi: $i > $o] :
% 2.05/1.58 ( ( exu
% 2.05/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.05/1.58 => ( Xphi
% 2.05/1.58 @ ( descr
% 2.05/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(dsetconstr_type,type,
% 2.05/1.58 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.05/1.58
% 2.05/1.58 thf(dsetconstrI_type,type,
% 2.05/1.58 dsetconstrI: $o ).
% 2.05/1.58
% 2.05/1.58 thf(dsetconstrI,definition,
% 2.05/1.58 ( dsetconstrI
% 2.05/1.58 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 => ( ( Xphi @ Xx )
% 2.05/1.58 => ( in @ Xx
% 2.05/1.58 @ ( dsetconstr @ A
% 2.05/1.58 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(dsetconstrEL_type,type,
% 2.05/1.58 dsetconstrEL: $o ).
% 2.05/1.58
% 2.05/1.58 thf(dsetconstrEL,definition,
% 2.05/1.58 ( dsetconstrEL
% 2.05/1.58 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.05/1.58 ( ( in @ Xx
% 2.05/1.58 @ ( dsetconstr @ A
% 2.05/1.58 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.05/1.58 => ( in @ Xx @ A ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(dsetconstrER_type,type,
% 2.05/1.58 dsetconstrER: $o ).
% 2.05/1.58
% 2.05/1.58 thf(dsetconstrER,definition,
% 2.05/1.58 ( dsetconstrER
% 2.05/1.58 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.05/1.58 ( ( in @ Xx
% 2.05/1.58 @ ( dsetconstr @ A
% 2.05/1.58 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.05/1.58 => ( Xphi @ Xx ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(exuE1_type,type,
% 2.05/1.58 exuE1: $o ).
% 2.05/1.58
% 2.05/1.58 thf(exuE1,definition,
% 2.05/1.58 ( exuE1
% 2.05/1.58 = ( ! [Xphi: $i > $o] :
% 2.05/1.58 ( ( exu
% 2.05/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.05/1.58 => ? [Xx: $i] :
% 2.05/1.58 ( ( Xphi @ Xx )
% 2.05/1.58 & ! [Xy: $i] :
% 2.05/1.58 ( ( Xphi @ Xy )
% 2.05/1.58 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(prop2set_type,type,
% 2.05/1.58 prop2set: $o > $i ).
% 2.05/1.58
% 2.05/1.58 thf(prop2setE_type,type,
% 2.05/1.58 prop2setE: $o ).
% 2.05/1.58
% 2.05/1.58 thf(prop2setE,definition,
% 2.05/1.58 ( prop2setE
% 2.05/1.58 = ( ! [Xphi: $o,Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.05/1.58 => Xphi ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(emptysetE_type,type,
% 2.05/1.58 emptysetE: $o ).
% 2.05/1.58
% 2.05/1.58 thf(emptysetE,definition,
% 2.05/1.58 ( emptysetE
% 2.05/1.58 = ( ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ emptyset )
% 2.05/1.58 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(emptysetimpfalse_type,type,
% 2.05/1.58 emptysetimpfalse: $o ).
% 2.05/1.58
% 2.05/1.58 thf(emptysetimpfalse,definition,
% 2.05/1.58 ( emptysetimpfalse
% 2.05/1.58 = ( ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ emptyset )
% 2.05/1.58 => $false ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(notinemptyset_type,type,
% 2.05/1.58 notinemptyset: $o ).
% 2.05/1.58
% 2.05/1.58 thf(notinemptyset,definition,
% 2.05/1.58 ( notinemptyset
% 2.05/1.58 = ( ! [Xx: $i] :
% 2.05/1.58 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(exuE3e_type,type,
% 2.05/1.58 exuE3e: $o ).
% 2.05/1.58
% 2.05/1.58 thf(exuE3e,definition,
% 2.05/1.58 ( exuE3e
% 2.05/1.58 = ( ! [Xphi: $i > $o] :
% 2.05/1.58 ( ( exu
% 2.05/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.05/1.58 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(setext_type,type,
% 2.05/1.58 setext: $o ).
% 2.05/1.58
% 2.05/1.58 thf(setext,definition,
% 2.05/1.58 ( setext
% 2.05/1.58 = ( ! [A: $i,B: $i] :
% 2.05/1.58 ( ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 => ( in @ Xx @ B ) )
% 2.05/1.58 => ( ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ B )
% 2.05/1.58 => ( in @ Xx @ A ) )
% 2.05/1.58 => ( A = B ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(emptyI_type,type,
% 2.05/1.58 emptyI: $o ).
% 2.05/1.58
% 2.05/1.58 thf(emptyI,definition,
% 2.05/1.58 ( emptyI
% 2.05/1.58 = ( ! [A: $i] :
% 2.05/1.58 ( ! [Xx: $i] :
% 2.05/1.58 ~ ( in @ Xx @ A )
% 2.05/1.58 => ( A = emptyset ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(noeltsimpempty_type,type,
% 2.05/1.58 noeltsimpempty: $o ).
% 2.05/1.58
% 2.05/1.58 thf(noeltsimpempty,definition,
% 2.05/1.58 ( noeltsimpempty
% 2.05/1.58 = ( ! [A: $i] :
% 2.05/1.58 ( ! [Xx: $i] :
% 2.05/1.58 ~ ( in @ Xx @ A )
% 2.05/1.58 => ( A = emptyset ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(setbeta_type,type,
% 2.05/1.58 setbeta: $o ).
% 2.05/1.58
% 2.05/1.58 thf(setbeta,definition,
% 2.05/1.58 ( setbeta
% 2.05/1.58 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 => ( ( in @ Xx
% 2.05/1.58 @ ( dsetconstr @ A
% 2.05/1.58 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.05/1.58 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(nonempty_type,type,
% 2.05/1.58 nonempty: $i > $o ).
% 2.05/1.58
% 2.05/1.58 thf(nonempty,definition,
% 2.05/1.58 ( nonempty
% 2.05/1.58 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(nonemptyE1_type,type,
% 2.05/1.58 nonemptyE1: $o ).
% 2.05/1.58
% 2.05/1.58 thf(nonemptyE1,definition,
% 2.05/1.58 ( nonemptyE1
% 2.05/1.58 = ( ! [A: $i] :
% 2.05/1.58 ( ( nonempty @ A )
% 2.05/1.58 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(nonemptyI_type,type,
% 2.05/1.58 nonemptyI: $o ).
% 2.05/1.58
% 2.05/1.58 thf(nonemptyI,definition,
% 2.05/1.58 ( nonemptyI
% 2.05/1.58 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 => ( ( Xphi @ Xx )
% 2.05/1.58 => ( nonempty
% 2.05/1.58 @ ( dsetconstr @ A
% 2.05/1.58 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(nonemptyI1_type,type,
% 2.05/1.58 nonemptyI1: $o ).
% 2.05/1.58
% 2.05/1.58 thf(nonemptyI1,definition,
% 2.05/1.58 ( nonemptyI1
% 2.05/1.58 = ( ! [A: $i] :
% 2.05/1.58 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.05/1.58 => ( nonempty @ A ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(setadjoinIL_type,type,
% 2.05/1.58 setadjoinIL: $o ).
% 2.05/1.58
% 2.05/1.58 thf(setadjoinIL,definition,
% 2.05/1.58 ( setadjoinIL
% 2.05/1.58 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(emptyinunitempty_type,type,
% 2.05/1.58 emptyinunitempty: $o ).
% 2.05/1.58
% 2.05/1.58 thf(emptyinunitempty,definition,
% 2.05/1.58 ( emptyinunitempty
% 2.05/1.58 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(setadjoinIR_type,type,
% 2.05/1.58 setadjoinIR: $o ).
% 2.05/1.58
% 2.05/1.58 thf(setadjoinIR,definition,
% 2.05/1.58 ( setadjoinIR
% 2.05/1.58 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.05/1.58 ( ( in @ Xy @ A )
% 2.05/1.58 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(setadjoinE_type,type,
% 2.05/1.58 setadjoinE: $o ).
% 2.05/1.58
% 2.05/1.58 thf(setadjoinE,definition,
% 2.05/1.58 ( setadjoinE
% 2.05/1.58 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.05/1.58 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.05/1.58 => ! [Xphi: $o] :
% 2.05/1.58 ( ( ( Xy = Xx )
% 2.05/1.58 => Xphi )
% 2.05/1.58 => ( ( ( in @ Xy @ A )
% 2.05/1.58 => Xphi )
% 2.05/1.58 => Xphi ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(setadjoinOr_type,type,
% 2.05/1.58 setadjoinOr: $o ).
% 2.05/1.58
% 2.05/1.58 thf(setadjoinOr,definition,
% 2.05/1.58 ( setadjoinOr
% 2.05/1.58 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.05/1.58 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.05/1.58 => ( ( Xy = Xx )
% 2.05/1.58 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(setoftrueEq_type,type,
% 2.05/1.58 setoftrueEq: $o ).
% 2.05/1.58
% 2.05/1.58 thf(setoftrueEq,definition,
% 2.05/1.58 ( setoftrueEq
% 2.05/1.58 = ( ! [A: $i] :
% 2.05/1.58 ( ( dsetconstr @ A
% 2.05/1.58 @ ^ [Xx: $i] : $true )
% 2.05/1.58 = A ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(powersetI_type,type,
% 2.05/1.58 powersetI: $o ).
% 2.05/1.58
% 2.05/1.58 thf(powersetI,definition,
% 2.05/1.58 ( powersetI
% 2.05/1.58 = ( ! [A: $i,B: $i] :
% 2.05/1.58 ( ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ B )
% 2.05/1.58 => ( in @ Xx @ A ) )
% 2.05/1.58 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(emptyinPowerset_type,type,
% 2.05/1.58 emptyinPowerset: $o ).
% 2.05/1.58
% 2.05/1.58 thf(emptyinPowerset,definition,
% 2.05/1.58 ( emptyinPowerset
% 2.05/1.58 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(emptyInPowerset_type,type,
% 2.05/1.58 emptyInPowerset: $o ).
% 2.05/1.58
% 2.05/1.58 thf(emptyInPowerset,definition,
% 2.05/1.58 ( emptyInPowerset
% 2.05/1.58 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(powersetE_type,type,
% 2.05/1.58 powersetE: $o ).
% 2.05/1.58
% 2.05/1.58 thf(powersetE,definition,
% 2.05/1.58 ( powersetE
% 2.05/1.58 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.05/1.58 ( ( in @ B @ ( powerset @ A ) )
% 2.05/1.58 => ( ( in @ Xx @ B )
% 2.05/1.58 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(setunionI_type,type,
% 2.05/1.58 setunionI: $o ).
% 2.05/1.58
% 2.05/1.58 thf(setunionI,definition,
% 2.05/1.58 ( setunionI
% 2.05/1.58 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.05/1.58 ( ( in @ Xx @ B )
% 2.05/1.58 => ( ( in @ B @ A )
% 2.05/1.58 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(setunionE_type,type,
% 2.05/1.58 setunionE: $o ).
% 2.05/1.58
% 2.05/1.58 thf(setunionE,definition,
% 2.05/1.58 ( setunionE
% 2.05/1.58 = ( ! [A: $i,Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ ( setunion @ A ) )
% 2.05/1.58 => ! [Xphi: $o] :
% 2.05/1.58 ( ! [B: $i] :
% 2.05/1.58 ( ( in @ Xx @ B )
% 2.05/1.58 => ( ( in @ B @ A )
% 2.05/1.58 => Xphi ) )
% 2.05/1.58 => Xphi ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(subPowSU_type,type,
% 2.05/1.58 subPowSU: $o ).
% 2.05/1.58
% 2.05/1.58 thf(subPowSU,definition,
% 2.05/1.58 ( subPowSU
% 2.05/1.58 = ( ! [A: $i,Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(exuE2_type,type,
% 2.05/1.58 exuE2: $o ).
% 2.05/1.58
% 2.05/1.58 thf(exuE2,definition,
% 2.05/1.58 ( exuE2
% 2.05/1.58 = ( ! [Xphi: $i > $o] :
% 2.05/1.58 ( ( exu
% 2.05/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.05/1.58 => ? [Xx: $i] :
% 2.05/1.58 ! [Xy: $i] :
% 2.05/1.58 ( ( Xphi @ Xy )
% 2.05/1.58 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(nonemptyImpWitness_type,type,
% 2.05/1.58 nonemptyImpWitness: $o ).
% 2.05/1.58
% 2.05/1.58 thf(nonemptyImpWitness,definition,
% 2.05/1.58 ( nonemptyImpWitness
% 2.05/1.58 = ( ! [A: $i] :
% 2.05/1.58 ( ( nonempty @ A )
% 2.05/1.58 => ? [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 & $true ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(uniqinunit_type,type,
% 2.05/1.58 uniqinunit: $o ).
% 2.05/1.58
% 2.05/1.58 thf(uniqinunit,definition,
% 2.05/1.58 ( uniqinunit
% 2.05/1.58 = ( ! [Xx: $i,Xy: $i] :
% 2.05/1.58 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.05/1.58 => ( Xx = Xy ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(notinsingleton_type,type,
% 2.05/1.58 notinsingleton: $o ).
% 2.05/1.58
% 2.05/1.58 thf(notinsingleton,definition,
% 2.05/1.58 ( notinsingleton
% 2.05/1.58 = ( ! [Xx: $i,Xy: $i] :
% 2.05/1.58 ( ( Xx != Xy )
% 2.05/1.58 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(eqinunit_type,type,
% 2.05/1.58 eqinunit: $o ).
% 2.05/1.58
% 2.05/1.58 thf(eqinunit,definition,
% 2.05/1.58 ( eqinunit
% 2.05/1.58 = ( ! [Xx: $i,Xy: $i] :
% 2.05/1.58 ( ( Xx = Xy )
% 2.05/1.58 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(singletonsswitch_type,type,
% 2.05/1.58 singletonsswitch: $o ).
% 2.05/1.58
% 2.05/1.58 thf(singletonsswitch,definition,
% 2.05/1.58 ( singletonsswitch
% 2.05/1.58 = ( ! [Xx: $i,Xy: $i] :
% 2.05/1.58 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.05/1.58 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(upairsetE_type,type,
% 2.05/1.58 upairsetE: $o ).
% 2.05/1.58
% 2.05/1.58 thf(upairsetE,definition,
% 2.05/1.58 ( upairsetE
% 2.05/1.58 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.05/1.58 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.05/1.58 => ( ( Xz = Xx )
% 2.05/1.58 | ( Xz = Xy ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(upairsetIL_type,type,
% 2.05/1.58 upairsetIL: $o ).
% 2.05/1.58
% 2.05/1.58 thf(upairsetIL,definition,
% 2.05/1.58 ( upairsetIL
% 2.05/1.58 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(upairsetIR_type,type,
% 2.05/1.58 upairsetIR: $o ).
% 2.05/1.58
% 2.05/1.58 thf(upairsetIR,definition,
% 2.05/1.58 ( upairsetIR
% 2.05/1.58 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(emptyE1_type,type,
% 2.05/1.58 emptyE1: $o ).
% 2.05/1.58
% 2.05/1.58 thf(emptyE1,definition,
% 2.05/1.58 ( emptyE1
% 2.05/1.58 = ( ! [A: $i,Xphi: $i > $o] :
% 2.05/1.58 ( ? [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 & ( Xphi @ Xx ) )
% 2.05/1.58 => ( ( ( dsetconstr @ A
% 2.05/1.58 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.05/1.58 = emptyset )
% 2.05/1.58 => $false ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(vacuousDall_type,type,
% 2.05/1.58 vacuousDall: $o ).
% 2.05/1.58
% 2.05/1.58 thf(vacuousDall,definition,
% 2.05/1.58 ( vacuousDall
% 2.05/1.58 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ emptyset )
% 2.05/1.58 => ( Xphi @ Xx ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(quantDeMorgan1_type,type,
% 2.05/1.58 quantDeMorgan1: $o ).
% 2.05/1.58
% 2.05/1.58 thf(quantDeMorgan1,definition,
% 2.05/1.58 ( quantDeMorgan1
% 2.05/1.58 = ( ! [A: $i,Xphi: $i > $o] :
% 2.05/1.58 ( ~ ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 => ( Xphi @ Xx ) )
% 2.05/1.58 => ? [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(quantDeMorgan2_type,type,
% 2.05/1.58 quantDeMorgan2: $o ).
% 2.05/1.58
% 2.05/1.58 thf(quantDeMorgan2,definition,
% 2.05/1.58 ( quantDeMorgan2
% 2.05/1.58 = ( ! [A: $i,Xphi: $i > $o] :
% 2.05/1.58 ( ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 => ~ ( Xphi @ Xx ) )
% 2.05/1.58 => ~ ? [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(quantDeMorgan3_type,type,
% 2.05/1.58 quantDeMorgan3: $o ).
% 2.05/1.58
% 2.05/1.58 thf(quantDeMorgan3,definition,
% 2.05/1.58 ( quantDeMorgan3
% 2.05/1.58 = ( ! [A: $i,Xphi: $i > $o] :
% 2.05/1.58 ( ~ ? [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 & ( Xphi @ Xx ) )
% 2.05/1.58 => ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(quantDeMorgan4_type,type,
% 2.05/1.58 quantDeMorgan4: $o ).
% 2.05/1.58
% 2.05/1.58 thf(quantDeMorgan4,definition,
% 2.05/1.58 ( quantDeMorgan4
% 2.05/1.58 = ( ! [A: $i,Xphi: $i > $o] :
% 2.05/1.58 ( ? [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 & ~ ( Xphi @ Xx ) )
% 2.05/1.58 => ~ ! [Xx: $i] :
% 2.05/1.58 ( ( in @ Xx @ A )
% 2.05/1.58 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(prop2setI_type,type,
% 2.05/1.58 prop2setI: $o ).
% 2.05/1.58
% 2.05/1.58 thf(prop2setI,definition,
% 2.05/1.58 ( prop2setI
% 2.05/1.58 = ( ! [Xphi: $o] :
% 2.05/1.58 ( Xphi
% 2.05/1.58 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(set2prop_type,type,
% 2.05/1.58 set2prop: $i > $o ).
% 2.05/1.58
% 2.05/1.58 thf(set2prop,definition,
% 2.05/1.58 ( set2prop
% 2.05/1.58 = ( ^ [A: $i] : ( in @ emptyset @ A ) ) ) ).
% 2.05/1.58
% 2.05/1.58 thf(prop2set2propI,conjecture,
% 2.05/1.58 ( setextAx
% 2.05/1.58 => ( emptysetAx
% 2.05/1.58 => ( setadjoinAx
% 2.05/1.58 => ( powersetAx
% 2.05/1.58 => ( setunionAx
% 2.05/1.58 => ( omega0Ax
% 2.05/1.58 => ( omegaSAx
% 2.05/1.58 => ( omegaIndAx
% 2.05/1.58 => ( replAx
% 2.05/1.58 => ( foundationAx
% 2.05/1.58 => ( wellorderingAx
% 2.05/1.58 => ( descrp
% 2.05/1.58 => ( dsetconstrI
% 2.05/1.58 => ( dsetconstrEL
% 2.05/1.58 => ( dsetconstrER
% 2.05/1.58 => ( exuE1
% 2.05/1.58 => ( prop2setE
% 2.05/1.58 => ( emptysetE
% 2.05/1.58 => ( emptysetimpfalse
% 2.05/1.58 => ( notinemptyset
% 2.05/1.58 => ( exuE3e
% 2.05/1.58 => ( setext
% 2.05/1.58 => ( emptyI
% 2.05/1.58 => ( noeltsimpempty
% 2.05/1.58 => ( setbeta
% 2.05/1.58 => ( nonemptyE1
% 2.05/1.58 => ( nonemptyI
% 2.05/1.58 => ( nonemptyI1
% 2.05/1.58 => ( setadjoinIL
% 2.05/1.58 => ( emptyinunitempty
% 2.05/1.58 => ( setadjoinIR
% 2.05/1.58 => ( setadjoinE
% 2.05/1.58 => ( setadjoinOr
% 2.05/1.58 => ( setoftrueEq
% 2.05/1.58 => ( powersetI
% 2.05/1.58 => ( emptyinPowerset
% 2.05/1.58 => ( emptyInPowerset
% 2.05/1.58 => ( powersetE
% 2.05/1.58 => ( setunionI
% 2.05/1.58 => ( setunionE
% 2.05/1.58 => ( subPowSU
% 2.05/1.58 => ( exuE2
% 2.05/1.58 => ( nonemptyImpWitness
% 2.05/1.58 => ( uniqinunit
% 2.05/1.58 => ( notinsingleton
% 2.05/1.58 => ( eqinunit
% 2.05/1.58 => ( singletonsswitch
% 2.05/1.58 => ( upairsetE
% 2.05/1.58 => ( upairsetIL
% 2.05/1.58 => ( upairsetIR
% 2.05/1.58 => ( emptyE1
% 2.05/1.58 => ( vacuousDall
% 2.05/1.58 => ( quantDeMorgan1
% 2.05/1.58 => ( quantDeMorgan2
% 2.05/1.58 => ( quantDeMorgan3
% 2.05/1.58 => ( quantDeMorgan4
% 2.05/1.58 => ( prop2setI
% 2.05/1.58 => ! [Xphi: $o] :
% 2.05/1.58 ( Xphi
% 2.05/1.58 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.05/1.58 ------------------------
% 2.22/1.60 % (22862)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22746 for (2999ds/2Mi)
% 2.22/1.60 % (22859)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22746 for (2999ds/183Mi)
% 2.22/1.60 % (22860)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_22746 for (2999ds/4Mi)
% 2.22/1.60 % (22861)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_22746 for (2999ds/27Mi)
% 2.22/1.60 % (22863)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_22746 for (2999ds/2Mi)
% 2.22/1.60 % (22864)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_22746 for (2999ds/275Mi)
% 2.22/1.60 % (22865)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22746 for (2999ds/18Mi)
% 2.22/1.60 % (22862)Instruction limit reached!
% 2.22/1.60 % (22862)------------------------------
% 2.22/1.60 % (22862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.22/1.60 % (22862)Termination reason: Unknown
% 2.22/1.60 % (22862)Termination phase: shuffling
% 2.22/1.60
% 2.22/1.60 % (22863)Instruction limit reached!
% 2.22/1.60 % (22863)------------------------------
% 2.22/1.60 % (22863)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.22/1.60 % (22863)Termination reason: Unknown
% 2.22/1.60 % (22863)Termination phase: shuffling
% 2.22/1.60
% 2.22/1.60 % (22863)Memory used [KB]: 1151
% 2.22/1.60 % (22863)Time elapsed: 0.003 s
% 2.22/1.60 % (22863)Instructions burned: 3 (million)
% 2.22/1.60 % (22863)------------------------------
% 2.22/1.60 % (22863)------------------------------
% 2.22/1.60 % (22862)Memory used [KB]: 1151
% 2.22/1.60 % (22862)Time elapsed: 0.003 s
% 2.22/1.60 % (22862)Instructions burned: 3 (million)
% 2.22/1.60 % (22862)------------------------------
% 2.22/1.60 % (22862)------------------------------
% 2.22/1.60 % (22860)Instruction limit reached!
% 2.22/1.60 % (22860)------------------------------
% 2.22/1.60 % (22860)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.22/1.60 % (22860)Termination reason: Unknown
% 2.22/1.60 % (22860)Termination phase: shuffling
% 2.22/1.60
% 2.22/1.60 % (22860)Memory used [KB]: 1151
% 2.22/1.60 % (22860)Time elapsed: 0.004 s
% 2.22/1.60 % (22860)Instructions burned: 5 (million)
% 2.22/1.60 % (22860)------------------------------
% 2.22/1.60 % (22860)------------------------------
% 2.22/1.61 % (22865)Instruction limit reached!
% 2.22/1.61 % (22865)------------------------------
% 2.22/1.61 % (22865)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.22/1.61 % (22865)Termination reason: Unknown
% 2.22/1.61 % (22865)Termination phase: Preprocessing 3
% 2.22/1.61
% 2.22/1.61 % (22865)Memory used [KB]: 1663
% 2.22/1.61 % (22865)Time elapsed: 0.011 s
% 2.22/1.61 % (22865)Instructions burned: 19 (million)
% 2.22/1.61 % (22865)------------------------------
% 2.22/1.61 % (22865)------------------------------
% 2.22/1.61 % (22861)Instruction limit reached!
% 2.22/1.61 % (22861)------------------------------
% 2.22/1.61 % (22861)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.22/1.61 % (22861)Termination reason: Unknown
% 2.22/1.61 % (22861)Termination phase: Property scanning
% 2.22/1.61
% 2.22/1.61 % (22861)Memory used [KB]: 1407
% 2.22/1.61 % (22861)Time elapsed: 0.014 s
% 2.22/1.61 % (22861)Instructions burned: 27 (million)
% 2.22/1.61 % (22861)------------------------------
% 2.22/1.61 % (22861)------------------------------
% 2.22/1.62 % (22866)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22746 for (2999ds/3Mi)
% 2.22/1.62 % (22867)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_22746 for (2999ds/37Mi)
% 2.22/1.62 % (22868)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_22746 for (2999ds/15Mi)
% 2.22/1.62 % (22866)Instruction limit reached!
% 2.22/1.62 % (22866)------------------------------
% 2.22/1.62 % (22866)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.22/1.62 % (22866)Termination reason: Unknown
% 2.22/1.62 % (22866)Termination phase: shuffling
% 2.22/1.62
% 2.22/1.62 % (22866)Memory used [KB]: 1151
% 2.22/1.62 % (22866)Time elapsed: 0.003 s
% 2.22/1.62 % (22866)Instructions burned: 4 (million)
% 2.22/1.62 % (22866)------------------------------
% 2.22/1.62 % (22866)------------------------------
% 2.22/1.63 % (22868)Instruction limit reached!
% 2.22/1.63 % (22868)------------------------------
% 2.22/1.63 % (22868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.22/1.63 % (22869)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_22746 for (2999ds/3Mi)
% 2.22/1.63 % (22868)Termination reason: Unknown
% 2.22/1.63 % (22868)Termination phase: Preprocessing 3
% 2.22/1.63
% 2.22/1.63 % (22868)Memory used [KB]: 1407
% 2.22/1.63 % (22868)Time elapsed: 0.009 s
% 2.22/1.63 % (22868)Instructions burned: 16 (million)
% 2.22/1.63 % (22868)------------------------------
% 2.22/1.63 % (22868)------------------------------
% 2.22/1.63 % (22864)First to succeed.
% 2.22/1.63 % (22869)Instruction limit reached!
% 2.22/1.63 % (22869)------------------------------
% 2.22/1.63 % (22869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.22/1.63 % (22869)Termination reason: Unknown
% 2.22/1.63 % (22869)Termination phase: shuffling
% 2.22/1.63
% 2.22/1.63 % (22869)Memory used [KB]: 1151
% 2.22/1.63 % (22869)Time elapsed: 0.003 s
% 2.22/1.63 % (22869)Instructions burned: 4 (million)
% 2.22/1.63 % (22869)------------------------------
% 2.22/1.63 % (22869)------------------------------
% 2.22/1.63 % (22870)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_22746 for (2999ds/1041Mi)
% 2.22/1.63 % (22864)Refutation found. Thanks to Tanya!
% 2.22/1.63 % SZS status Theorem for DTF2THF_22746
% 2.22/1.63 % SZS output start Proof for DTF2THF_22746
% 2.22/1.63 thf(func_def_0, type, in: $i > $i > $o).
% 2.22/1.63 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.22/1.63 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.22/1.63 thf(func_def_8, type, powerset: $i > $i).
% 2.22/1.63 thf(func_def_10, type, setunion: $i > $i).
% 2.22/1.63 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.22/1.63 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.22/1.63 thf(func_def_26, type, prop2set: $o > $i).
% 2.22/1.63 thf(func_def_36, type, nonempty: $i > $o).
% 2.22/1.63 thf(func_def_69, type, set2prop: $i > $o).
% 2.22/1.63 thf(func_def_82, type, sP0: $i > $i > $o).
% 2.22/1.63 thf(func_def_83, type, sP1: $i > $o).
% 2.22/1.63 thf(func_def_84, type, sP2: $i > $i > $o).
% 2.22/1.63 thf(func_def_85, type, sP3: $i > $i > $o).
% 2.22/1.63 thf(func_def_89, type, sK7: $i > $o).
% 2.22/1.63 thf(func_def_94, type, sK12: $i > $o).
% 2.22/1.63 thf(func_def_98, type, sK16: $i > $o).
% 2.22/1.63 thf(func_def_99, type, sK17: ($i > $o) > $i).
% 2.22/1.63 thf(func_def_101, type, sK19: $i > $i > $i).
% 2.22/1.63 thf(func_def_102, type, sK20: $i > $i > $i).
% 2.22/1.63 thf(func_def_103, type, sK21: $i > $i > $i).
% 2.22/1.63 thf(func_def_104, type, sK22: $i > $i > $i).
% 2.22/1.63 thf(func_def_105, type, sK23: $i > $i > $i > $i).
% 2.22/1.63 thf(func_def_106, type, sK24: $i > $i).
% 2.22/1.63 thf(func_def_107, type, sK25: $i > $i).
% 2.22/1.63 thf(func_def_108, type, sK26: $i > $i).
% 2.22/1.63 thf(func_def_109, type, sK27: $i > $i).
% 2.22/1.63 thf(func_def_110, type, sK28: $i > $i > $i > $i).
% 2.22/1.63 thf(func_def_111, type, sK29: $i > $i > $i > $i).
% 2.22/1.63 thf(func_def_112, type, sK30: $i > $i > $i).
% 2.22/1.63 thf(func_def_113, type, sK31: $i > $i > $i).
% 2.22/1.63 thf(func_def_114, type, sK32: $i > $i > $i).
% 2.22/1.63 thf(func_def_115, type, sK33: $i > $i).
% 2.22/1.63 thf(func_def_116, type, sK34: $i > $i > $i).
% 2.22/1.63 thf(func_def_118, type, sK36: $i > $i).
% 2.22/1.63 thf(func_def_119, type, sK37: $i > $i).
% 2.22/1.63 thf(func_def_122, type, sK40: $i > $i > $i).
% 2.22/1.63 thf(func_def_129, type, sK47: $i > $i).
% 2.22/1.63 thf(func_def_130, type, sK48: $i > $i).
% 2.22/1.63 thf(func_def_139, type, sK57: $i > $i).
% 2.22/1.63 thf(func_def_146, type, sK64: ($i > $o) > $i).
% 2.22/1.63 thf(func_def_147, type, sK65: $i > $o).
% 2.22/1.63 thf(func_def_148, type, sK66: $i > $i).
% 2.22/1.63 thf(func_def_153, type, sK71: $i > $o).
% 2.22/1.63 thf(func_def_154, type, sK72: $i > $i).
% 2.22/1.63 thf(func_def_155, type, sK73: ($i > $o) > $i).
% 2.22/1.63 thf(func_def_156, type, sK74: $i > $o).
% 2.22/1.63 thf(func_def_159, type, sK77: $i > $o).
% 2.22/1.63 thf(func_def_164, type, sK82: $i > $i > $i).
% 2.22/1.63 thf(func_def_165, type, sK83: $i > $i > $i).
% 2.22/1.63 thf(func_def_168, type, sK86: $o > $i > $i > $i).
% 2.22/1.63 thf(func_def_173, type, sK91: $i > $o).
% 2.22/1.63 thf(func_def_176, type, sK94: $i > $o).
% 2.22/1.63 thf(func_def_178, type, sK96: ($i > $o) > $i > $i).
% 2.22/1.63 thf(func_def_179, type, sK97: ($i > $o) > $i > $i).
% 2.22/1.63 thf(func_def_181, type, sK99: $i > $o).
% 2.22/1.63 thf(func_def_183, type, sK101: $i > $o).
% 2.22/1.63 thf(func_def_186, type, sK104: $i > $i).
% 2.22/1.63 thf(func_def_189, type, sK107: $i > $i).
% 2.22/1.63 thf(func_def_192, type, sK110: $i > $i > $i).
% 2.22/1.63 thf(func_def_196, type, sK114: $i > $i > $i).
% 2.22/1.63 thf(func_def_210, type, sK128: $i > $i).
% 2.22/1.63 thf(func_def_212, type, sK130: ($i > $o) > $i > $i).
% 2.22/1.63 thf(func_def_214, type, sK132: $i > $o).
% 2.22/1.63 thf(func_def_217, type, sK135: $i > $o).
% 2.22/1.63 thf(func_def_220, type, sK138: $i > ($i > $o) > $i).
% 2.22/1.63 thf(func_def_232, type, sK150: $i > $i > $o).
% 2.22/1.63 thf(func_def_234, type, sK152: $i > $i).
% 2.22/1.63 thf(func_def_235, type, sK153: $i > $i).
% 2.22/1.63 thf(func_def_236, type, sK154: $i > ($i > $i > $o) > $i).
% 2.22/1.63 thf(func_def_237, type, sK155: $i > ($i > $i > $o) > $i).
% 2.22/1.63 thf(func_def_238, type, sK156: $i > $i > ($i > $i > $o) > $i).
% 2.22/1.63 thf(func_def_239, type, sK157: $i > $i).
% 2.22/1.63 thf(func_def_241, type, sK159: $i > $i > $i).
% 2.22/1.63 thf(func_def_246, type, sK164: $i > $o).
% 2.22/1.63 thf(func_def_248, type, ph166: !>[X0: $tType]:(X0)).
% 2.22/1.63 thf(f1170,plain,(
% 2.22/1.63 $false),
% 2.22/1.63 inference(subsumption_resolution,[status(thm)],[f1112,f1118])).
% 2.22/1.63 thf(f1118,plain,(
% 2.22/1.63 ((in @ emptyset @ (prop2set @ $true)) != $true)),
% 2.22/1.63 inference(beta_eta_normalization,[status(thm)],[f906])).
% 2.22/1.63 thf(f906,plain,(
% 2.22/1.63 (((^[Y0 : $i]: (in @ emptyset @ Y0)) @ (prop2set @ $true)) != $true)),
% 2.22/1.63 inference(definition_unfolding,[status(thm)],[f633,f568,f632])).
% 2.22/1.63 thf(f632,plain,(
% 2.22/1.63 (sK53 = $true)),
% 2.22/1.63 inference(cnf_transformation,[status(thm)],[f350])).
% 2.22/1.63 thf(f350,plain,(
% 2.22/1.63 (quantDeMorgan1 = $true) & (setadjoinE = $true) & (emptysetAx = $true) & (nonemptyI = $true) & (omega0Ax = $true) & (powersetI = $true) & (quantDeMorgan4 = $true) & (emptysetE = $true) & (replAx = $true) & (emptyI = $true) & (exuE3e = $true) & (nonemptyImpWitness = $true) & (notinemptyset = $true) & (nonemptyI1 = $true) & (noeltsimpempty = $true) & (upairsetIR = $true) & (upairsetE = $true) & (setadjoinAx = $true) & (setunionE = $true) & (setext = $true) & (notinsingleton = $true) & (prop2setE = $true) & (emptyinPowerset = $true) & (foundationAx = $true) & (omegaIndAx = $true) & (dsetconstrEL = $true) & (powersetAx = $true) & (setadjoinOr = $true) & (quantDeMorgan3 = $true) & (dsetconstrER = $true) & (exuE2 = $true) & (subPowSU = $true) & (setunionAx = $true) & (eqinunit = $true) & (exuE1 = $true) & (vacuousDall = $true) & (setadjoinIR = $true) & (emptyinunitempty = $true) & (setadjoinIL = $true) & (setunionI = $true) & (emptyInPowerset = $true) & (setbeta = $true) & (omegaSAx = $true) & (singletonsswitch = $true) & (emptyE1 = $true) & (quantDeMorgan2 = $true) & (emptysetimpfalse = $true) & (prop2setI = $true) & (wellorderingAx = $true) & (upairsetIL = $true) & (setextAx = $true) & (powersetE = $true) & (setoftrueEq = $true) & (nonemptyE1 = $true) & (uniqinunit = $true) & (dsetconstrI = $true) & (((set2prop @ (prop2set @ sK53)) != $true) & (sK53 = $true)) & (descrp = $true)),
% 2.22/1.63 inference(skolemisation,[status(esa),new_symbols(skolem,[sK53])],[f237,f349])).
% 2.22/1.63 thf(f349,plain,(
% 2.22/1.63 ? [X0 : $o] : (((set2prop @ (prop2set @ X0)) != $true) & ($true = X0)) => (((set2prop @ (prop2set @ sK53)) != $true) & (sK53 = $true))),
% 2.22/1.63 introduced(definition,[],[choice_axiom])).
% 2.22/1.63 thf(f237,plain,(
% 2.22/1.63 (quantDeMorgan1 = $true) & (setadjoinE = $true) & (emptysetAx = $true) & (nonemptyI = $true) & (omega0Ax = $true) & (powersetI = $true) & (quantDeMorgan4 = $true) & (emptysetE = $true) & (replAx = $true) & (emptyI = $true) & (exuE3e = $true) & (nonemptyImpWitness = $true) & (notinemptyset = $true) & (nonemptyI1 = $true) & (noeltsimpempty = $true) & (upairsetIR = $true) & (upairsetE = $true) & (setadjoinAx = $true) & (setunionE = $true) & (setext = $true) & (notinsingleton = $true) & (prop2setE = $true) & (emptyinPowerset = $true) & (foundationAx = $true) & (omegaIndAx = $true) & (dsetconstrEL = $true) & (powersetAx = $true) & (setadjoinOr = $true) & (quantDeMorgan3 = $true) & (dsetconstrER = $true) & (exuE2 = $true) & (subPowSU = $true) & (setunionAx = $true) & (eqinunit = $true) & (exuE1 = $true) & (vacuousDall = $true) & (setadjoinIR = $true) & (emptyinunitempty = $true) & (setadjoinIL = $true) & (setunionI = $true) & (emptyInPowerset = $true) & (setbeta = $true) & (omegaSAx = $true) & (singletonsswitch = $true) & (emptyE1 = $true) & (quantDeMorgan2 = $true) & (emptysetimpfalse = $true) & (prop2setI = $true) & (wellorderingAx = $true) & (upairsetIL = $true) & (setextAx = $true) & (powersetE = $true) & (setoftrueEq = $true) & (nonemptyE1 = $true) & (uniqinunit = $true) & (dsetconstrI = $true) & ? [X0 : $o] : (((set2prop @ (prop2set @ X0)) != $true) & ($true = X0)) & (descrp = $true)),
% 2.22/1.63 inference(flattening,[status(thm)],[f236])).
% 2.22/1.63 thf(f236,plain,(
% 2.22/1.63 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $o] : (((set2prop @ (prop2set @ X0)) != $true) & ($true = X0)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (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)),
% 2.22/1.63 inference(ennf_transformation,[status(thm)],[f65])).
% 2.22/1.63 thf(f65,plain,(
% 2.22/1.63 ~((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) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ! [X0 : $o] : (($true = X0) => ((set2prop @ (prop2set @ X0)) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.22/1.63 inference(fool_elimination,[status(thm)],[f64])).
% 2.22/1.63 thf(f64,plain,(
% 2.22/1.63 ~(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 => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => ! [X0 : $o] : (X0 => (set2prop @ (prop2set @ X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.22/1.63 inference(rectify,[status(thm)],[f62])).
% 2.22/1.63 thf(f62,negated_conjecture,(
% 2.22/1.63 ~(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 => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => ! [X0 : $o] : (X0 => (set2prop @ (prop2set @ X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.22/1.63 inference(negated_conjecture,[status(cth)],[f61])).
% 2.22/1.63 thf(f61,conjecture,(
% 2.22/1.63 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 => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => ! [X0 : $o] : (X0 => (set2prop @ (prop2set @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.22/1.63 file('/export/starexec/sandbox/tmp/tmp.rXSDAX2vBq/DTF2THF_22746.p',prop2set2propI)).
% 2.22/1.63 thf(f568,plain,(
% 2.22/1.63 (set2prop = (^[Y0 : $i]: (in @ emptyset @ Y0)))),
% 2.22/1.63 inference(cnf_transformation,[status(thm)],[f167])).
% 2.22/1.63 thf(f167,plain,(
% 2.22/1.63 (set2prop = (^[Y0 : $i]: (in @ emptyset @ Y0)))),
% 2.22/1.63 inference(fool_elimination,[status(thm)],[f166])).
% 2.22/1.63 thf(f166,plain,(
% 2.22/1.63 (set2prop = (^[X0 : $i] : (in @ emptyset @ X0)))),
% 2.22/1.63 inference(rectify,[status(thm)],[f60])).
% 2.22/1.63 thf(f60,axiom,(
% 2.22/1.63 (set2prop = (^[X3 : $i] : (in @ emptyset @ X3)))),
% 2.22/1.63 file('/export/starexec/sandbox/tmp/tmp.rXSDAX2vBq/DTF2THF_22746.p',set2prop)).
% 2.22/1.63 thf(f633,plain,(
% 2.22/1.63 ((set2prop @ (prop2set @ sK53)) != $true)),
% 2.22/1.63 inference(cnf_transformation,[status(thm)],[f350])).
% 2.22/1.63 thf(f1112,plain,(
% 2.22/1.63 ((in @ emptyset @ (prop2set @ $true)) = $true)),
% 2.22/1.63 inference(trivial_inequality_removal,[status(thm)],[f1066])).
% 2.22/1.63 thf(f1066,plain,(
% 2.22/1.63 ((in @ emptyset @ (prop2set @ $true)) = $true) | ($true != $true)),
% 2.22/1.63 inference(equality_resolution,[status(thm)],[f1014])).
% 2.22/1.63 thf(f1014,plain,(
% 2.22/1.63 ( ! [X1 : $o] : (($true = (in @ emptyset @ (prop2set @ X1))) | ($true != X1) | ($true != $true)) )),
% 2.22/1.63 inference(definition_unfolding,[status(thm)],[f795,f642])).
% 2.22/1.63 thf(f642,plain,(
% 2.22/1.63 (prop2setI = $true)),
% 2.22/1.63 inference(cnf_transformation,[status(thm)],[f350])).
% 2.22/1.63 thf(f795,plain,(
% 2.22/1.63 ( ! [X1 : $o] : (($true = (in @ emptyset @ (prop2set @ X1))) | ($true != X1) | (prop2setI != $true)) )),
% 2.22/1.63 inference(cnf_transformation,[status(thm)],[f483])).
% 2.22/1.63 thf(f483,plain,(
% 2.22/1.63 ((prop2setI = $true) | (((in @ emptyset @ (prop2set @ sK127)) != $true) & ($true = sK127))) & (! [X1 : $o] : (($true = (in @ emptyset @ (prop2set @ X1))) | ($true != X1)) | (prop2setI != $true))),
% 2.22/1.63 inference(skolemisation,[status(esa),new_symbols(skolem,[sK127])],[f481,f482])).
% 2.22/1.63 thf(f482,plain,(
% 2.22/1.63 ? [X0 : $o] : (((in @ emptyset @ (prop2set @ X0)) != $true) & ($true = X0)) => (((in @ emptyset @ (prop2set @ sK127)) != $true) & ($true = sK127))),
% 2.22/1.63 introduced(definition,[],[choice_axiom])).
% 2.22/1.63 thf(f481,plain,(
% 2.22/1.63 ((prop2setI = $true) | ? [X0 : $o] : (((in @ emptyset @ (prop2set @ X0)) != $true) & ($true = X0))) & (! [X1 : $o] : (($true = (in @ emptyset @ (prop2set @ X1))) | ($true != X1)) | (prop2setI != $true))),
% 2.22/1.63 inference(rectify,[status(thm)],[f480])).
% 2.22/1.63 thf(f480,plain,(
% 2.22/1.63 ((prop2setI = $true) | ? [X0 : $o] : (((in @ emptyset @ (prop2set @ X0)) != $true) & ($true = X0))) & (! [X0 : $o] : (((in @ emptyset @ (prop2set @ X0)) = $true) | ($true != X0)) | (prop2setI != $true))),
% 2.22/1.63 inference(nnf_transformation,[status(thm)],[f209])).
% 2.22/1.63 thf(f209,plain,(
% 2.22/1.63 (prop2setI = $true) <=> ! [X0 : $o] : (((in @ emptyset @ (prop2set @ X0)) = $true) | ($true != X0))),
% 2.22/1.63 inference(ennf_transformation,[status(thm)],[f101])).
% 2.22/1.63 thf(f101,plain,(
% 2.22/1.63 (prop2setI = $true) <=> ! [X0 : $o] : (($true = X0) => ((in @ emptyset @ (prop2set @ X0)) = $true))),
% 2.22/1.63 inference(fool_elimination,[status(thm)],[f100])).
% 2.22/1.63 thf(f100,plain,(
% 2.22/1.63 (! [X0 : $o] : (X0 => (in @ emptyset @ (prop2set @ X0))) = prop2setI)),
% 2.22/1.63 inference(rectify,[status(thm)],[f59])).
% 2.22/1.63 thf(f59,axiom,(
% 2.22/1.63 (! [X0 : $o] : (X0 => (in @ emptyset @ (prop2set @ X0))) = prop2setI)),
% 2.22/1.63 file('/export/starexec/sandbox/tmp/tmp.rXSDAX2vBq/DTF2THF_22746.p',prop2setI)).
% 2.22/1.63 % SZS output end Proof for DTF2THF_22746
% 2.22/1.63 % (22864)------------------------------
% 2.22/1.63 % (22864)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.22/1.63 % (22864)Termination reason: Refutation
% 2.22/1.63
% 2.22/1.63 % (22864)Memory used [KB]: 6396
% 2.22/1.63 % (22864)Time elapsed: 0.029 s
% 2.22/1.63 % (22864)Instructions burned: 55 (million)
% 2.22/1.63 % (22864)------------------------------
% 2.22/1.63 % (22864)------------------------------
% 2.22/1.63 % (22858)Success in time 0.045 s
%------------------------------------------------------------------------------