%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU567^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n024.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 09:35:19 AM UTC 2026
% Result : Theorem 1.56s 1.06s
% Output : Refutation 1.56s
% Verified :
% SZS Type : Refutation
% Derivation depth : 12
% Number of leaves : 4
% Syntax : Number of formulae : 26 ( 8 unt; 0 typ; 0 def)
% Number of atoms : 1656 ( 477 equ; 0 cnn)
% Maximal formula atoms : 85 ( 63 avg)
% Number of connectives : 798 ( 38 ~; 15 |; 262 &; 56 @)
% ( 2 <=>; 425 =>; 0 <=; 0 <~>)
% Maximal formula depth : 91 ( 30 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 197 ( 194 usr; 89 con; 0-4 aty)
% Number of variables : 42 ( 0 ^; 30 !; 12 ?; 42 :)
% 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_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_112,type,
sP1: $i > $i > $o ).
thf(func_def_113,type,
sP2: $i > $o ).
thf(func_def_114,type,
sP3: $i > $i > $o ).
thf(func_def_115,type,
sP4: $i > $i > $o ).
thf(func_def_118,type,
sK7: $i > $i > $i ).
thf(func_def_119,type,
sK8: $i > $i > $i ).
thf(func_def_120,type,
sK9: $i > $i > $i ).
thf(func_def_123,type,
sK12: $i > $i ).
thf(func_def_128,type,
sK17: $i > $o ).
thf(func_def_130,type,
sK19: $i > ( $i > $o ) > $i ).
thf(func_def_135,type,
sK24: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_136,type,
sK25: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_137,type,
sK26: $i > $o ).
thf(func_def_138,type,
sK27: $i > $o ).
thf(func_def_139,type,
sK28: $i > $o ).
thf(func_def_145,type,
sK34: $i > $o ).
thf(func_def_158,type,
sK47: $i > $o ).
thf(func_def_163,type,
sK52: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_164,type,
sK53: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_165,type,
sK54: $i > $o ).
thf(func_def_166,type,
sK55: $i > $o ).
thf(func_def_167,type,
sK56: $i > ( $i > $o ) > $i ).
thf(func_def_168,type,
sK57: $i > $o ).
thf(func_def_170,type,
sK59: $i > $i > $o ).
thf(func_def_172,type,
sK61: $i > $i ).
thf(func_def_173,type,
sK62: $i > $i ).
thf(func_def_174,type,
sK63: $i > ( $i > $i > $o ) > $i ).
thf(func_def_175,type,
sK64: $i > ( $i > $i > $o ) > $i ).
thf(func_def_176,type,
sK65: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_183,type,
sK72: $i > $i > $i ).
thf(func_def_184,type,
sK73: $i > $i > $i ).
thf(func_def_187,type,
sK76: $i > $i ).
thf(func_def_191,type,
sK80: $i > $i ).
thf(func_def_194,type,
sK83: $i > $o ).
thf(func_def_197,type,
sK86: $i > $o ).
thf(func_def_200,type,
sK89: $i > $o ).
thf(func_def_207,type,
sK96: ( $i > $o ) > $i ).
thf(func_def_208,type,
sK97: ( $i > $o ) > $i ).
thf(func_def_209,type,
sK98: $i > $o ).
thf(func_def_219,type,
sK108: $i > $o ).
thf(func_def_221,type,
sK110: $i > $o ).
thf(func_def_224,type,
sK113: $i > ( $i > $o ) > $i ).
thf(func_def_226,type,
sK115: $i > $o ).
thf(func_def_232,type,
sK121: $i > $o ).
thf(func_def_233,type,
sK122: $i > $i > $i ).
thf(func_def_239,type,
sK128: $i > $o ).
thf(func_def_241,type,
sK130: ( $i > $o ) > $i > $i ).
thf(func_def_243,type,
sK132: $i > $o ).
thf(func_def_245,type,
sK134: ( $i > $o ) > $i > $i ).
thf(func_def_247,type,
sK136: $i > $o ).
thf(func_def_251,type,
sK140: $i > $o ).
thf(func_def_253,type,
sK142: ( $i > $o ) > $i > $i ).
thf(func_def_261,type,
sK150: $o > $i > $i > $i ).
thf(func_def_264,type,
sK153: $i > $o ).
thf(func_def_265,type,
sK154: $i > $o ).
thf(func_def_266,type,
sK155: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_267,type,
sK156: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_276,type,
sK165: $i > $i ).
thf(func_def_279,type,
sK168: $i > ( $i > $o ) > $i ).
thf(func_def_280,type,
sK169: $i > $o ).
thf(func_def_283,type,
sK172: $i > $i > $i ).
thf(func_def_284,type,
sK173: $i > $i > $i ).
thf(func_def_285,type,
sK174: $i > $i > $i ).
thf(func_def_286,type,
sK175: $i > $i > $i ).
thf(func_def_287,type,
sK176: $i > $i > $i > $i ).
thf(func_def_288,type,
sK177: $i > $i ).
thf(func_def_289,type,
sK178: $i > $i ).
thf(func_def_290,type,
sK179: $i > $i ).
thf(func_def_291,type,
sK180: $i > $i ).
thf(func_def_292,type,
sK181: $i > $i > $i > $i ).
thf(func_def_293,type,
sK182: $i > $i > $i > $i ).
thf(func_def_294,type,
sK183: $i > $i > $i ).
thf(func_def_295,type,
sK184: $i > $i > $i ).
thf(func_def_296,type,
sK185: $i > $i > $i ).
thf(func_def_297,type,
sK186: $i > $i ).
thf(func_def_298,type,
sK187: $i > $i > $i ).
thf(func_def_300,type,
sK189: $i > $i ).
thf(func_def_301,type,
sK190: $i > $i ).
thf(func_def_304,type,
sK193: $i > $o ).
thf(func_def_305,type,
sK194: $i > $i ).
thf(func_def_306,type,
sK195: ( $i > $o ) > $i ).
thf(func_def_307,type,
sK196: $i > $i ).
thf(func_def_309,type,
sK198: $i > $i ).
thf(func_def_311,type,
sK200: $i > $i > $i ).
thf(func_def_315,type,
sK204: ( $i > $o ) > $i ).
thf(func_def_316,type,
sK205: $i > $o ).
thf(func_def_317,type,
sK206: $i > $i ).
thf(func_def_323,type,
sK212: ( $i > $o ) > $i > $i ).
thf(func_def_325,type,
sK214: $i > $o ).
thf(func_def_330,type,
sK219: $i > $i > $i ).
thf(func_def_343,type,
sK232: ( $i > $o ) > $i ).
thf(func_def_344,type,
sK233: $i > $o ).
thf(func_def_346,type,
sK235: $i > $i ).
thf(func_def_353,type,
ph242:
!>[X0: $tType] : X0 ).
thf(f1730,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1354,f1719]) ).
thf(f1719,plain,
! [X3: $i] :
( ( subset @ X3 @ X3 )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f1553]) ).
thf(f1553,plain,
! [X3: $i] :
( ( $true != $true )
| ( ( subset @ X3 @ X3 )
= $true ) ),
inference(equality_resolution,[status(thm)],[f1443]) ).
thf(f1443,plain,
! [X2: $i,X3: $i] :
( ( X2 != X3 )
| ( $true
= ( subset @ X2 @ X3 ) )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f1098,f941]) ).
thf(f941,plain,
eqimpsubset2 = $true,
inference(cnf_transformation,[status(thm)],[f543]) ).
thf(f543,plain,
( ( exuEu = $true )
& ( upairsetE = $true )
& ( sK102 = sK103 )
& ( ( subset @ sK102 @ sK103 )
!= $true )
& ( exuE3u = $true )
& ( prop2setI = $true )
& ( emptyinPowerset = $true )
& ( setadjoinE = $true )
& ( in__Cong = $true )
& ( emptyset__Cong = $true )
& ( powersetE = $true )
& ( powerset__Cong = $true )
& ( notdexE = $true )
& ( setadjoin__Cong = $true )
& ( subPowSU = $true )
& ( notdallE = $true )
& ( setext = $true )
& ( exuE2 = $true )
& ( descr__Cong = $true )
& ( setadjoinIR = $true )
& ( setextAx = $true )
& ( vacuousDall = $true )
& ( emptyInPowerset = $true )
& ( setoftrueEq = $true )
& ( omega0Ax = $true )
& ( dsetconstrI = $true )
& ( nonemptyI = $true )
& ( setunionI = $true )
& ( exuE3e = $true )
& ( nonemptyI1 = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( notsubsetI = $true )
& ( descrp = $true )
& ( exuI3 = $true )
& ( setunion__Cong = $true )
& ( omegaSAx = $true )
& ( emptyinunitempty = $true )
& ( uniqinunit = $true )
& ( setadjoinOr = $true )
& ( emptysetimpfalse = $true )
& ( emptysetsubset = $true )
& ( subsetE = $true )
& ( setadjoinAx = $true )
& ( powersetAx = $true )
& ( emptysetAx = $true )
& ( exuI2 = $true )
& ( notinsingleton = $true )
& ( exuE1 = $true )
& ( upairsetIR = $true )
& ( eqimpsubset1 = $true )
& ( notinemptyset = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( prop2setE = $true )
& ( replAx = $true )
& ( setunionE = $true )
& ( setunionAx = $true )
& ( powersetI = $true )
& ( quantDeMorgan1 = $true )
& ( quantDeMorgan3 = $true )
& ( prop2set2propI = $true )
& ( nonemptyImpWitness = $true )
& ( subsetI1 = $true )
& ( subsetI2 = $true )
& ( setadjoinIL = $true )
& ( upairsetIL = $true )
& ( exuI1 = $true )
& ( exu__Cong = $true )
& ( noeltsimpempty = $true )
& ( dsetconstrEL = $true )
& ( eqimpsubset2 = $true )
& ( dsetconstr__Cong = $true )
& ( wellorderingAx = $true )
& ( dsetconstrER = $true )
& ( inCongP = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan4 = $true )
& ( emptysetE = $true )
& ( omega__Cong = $true )
& ( emptyI = $true )
& ( foundationAx = $true )
& ( omegaIndAx = $true )
& ( emptyE1 = $true )
& ( subsetE2 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK102,sK103])],[f364,f542]) ).
thf(f542,plain,
( ? [X0: $i,X1: $i] :
( ( X0 = X1 )
& ( ( subset @ X0 @ X1 )
!= $true ) )
=> ( ( sK102 = sK103 )
& ( ( subset @ sK102 @ sK103 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f364,plain,
( ( exuEu = $true )
& ( upairsetE = $true )
& ? [X0: $i,X1: $i] :
( ( X0 = X1 )
& ( ( subset @ X0 @ X1 )
!= $true ) )
& ( exuE3u = $true )
& ( prop2setI = $true )
& ( emptyinPowerset = $true )
& ( setadjoinE = $true )
& ( in__Cong = $true )
& ( emptyset__Cong = $true )
& ( powersetE = $true )
& ( powerset__Cong = $true )
& ( notdexE = $true )
& ( setadjoin__Cong = $true )
& ( subPowSU = $true )
& ( notdallE = $true )
& ( setext = $true )
& ( exuE2 = $true )
& ( descr__Cong = $true )
& ( setadjoinIR = $true )
& ( setextAx = $true )
& ( vacuousDall = $true )
& ( emptyInPowerset = $true )
& ( setoftrueEq = $true )
& ( omega0Ax = $true )
& ( dsetconstrI = $true )
& ( nonemptyI = $true )
& ( setunionI = $true )
& ( exuE3e = $true )
& ( nonemptyI1 = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( notsubsetI = $true )
& ( descrp = $true )
& ( exuI3 = $true )
& ( setunion__Cong = $true )
& ( omegaSAx = $true )
& ( emptyinunitempty = $true )
& ( uniqinunit = $true )
& ( setadjoinOr = $true )
& ( emptysetimpfalse = $true )
& ( emptysetsubset = $true )
& ( subsetE = $true )
& ( setadjoinAx = $true )
& ( powersetAx = $true )
& ( emptysetAx = $true )
& ( exuI2 = $true )
& ( notinsingleton = $true )
& ( exuE1 = $true )
& ( upairsetIR = $true )
& ( eqimpsubset1 = $true )
& ( notinemptyset = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( prop2setE = $true )
& ( replAx = $true )
& ( setunionE = $true )
& ( setunionAx = $true )
& ( powersetI = $true )
& ( quantDeMorgan1 = $true )
& ( quantDeMorgan3 = $true )
& ( prop2set2propI = $true )
& ( nonemptyImpWitness = $true )
& ( subsetI1 = $true )
& ( subsetI2 = $true )
& ( setadjoinIL = $true )
& ( upairsetIL = $true )
& ( exuI1 = $true )
& ( exu__Cong = $true )
& ( noeltsimpempty = $true )
& ( dsetconstrEL = $true )
& ( eqimpsubset2 = $true )
& ( dsetconstr__Cong = $true )
& ( wellorderingAx = $true )
& ( dsetconstrER = $true )
& ( inCongP = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan4 = $true )
& ( emptysetE = $true )
& ( omega__Cong = $true )
& ( emptyI = $true )
& ( foundationAx = $true )
& ( omegaIndAx = $true )
& ( emptyE1 = $true )
& ( subsetE2 = $true ) ),
inference(flattening,[status(thm)],[f363]) ).
thf(f363,plain,
( ? [X0: $i,X1: $i] :
( ( X0 = X1 )
& ( ( subset @ X0 @ X1 )
!= $true ) )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( 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)],[f255]) ).
thf(f255,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 )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ! [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
!= $true )
=> ( X0 != X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f90]) ).
thf(f90,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 )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ! [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
!= $true )
=> ( X0 != X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f89]) ).
thf(f89,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
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ! [X0: $i,X1: $i] :
( ~ ( subset @ X0 @ X1 )
=> ( X0 != X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f87]) ).
thf(f87,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
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ! [X3: $i,X4: $i] :
( ~ ( subset @ X3 @ X4 )
=> ( X3 != X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f86]) ).
thf(f86,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
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ! [X3: $i,X4: $i] :
( ~ ( subset @ X3 @ X4 )
=> ( X3 != X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.HAHRJ9zahH/DTF2THF_17640.p',notequalI1) ).
thf(f1098,plain,
! [X2: $i,X3: $i] :
( ( X2 != X3 )
| ( $true
= ( subset @ X2 @ X3 ) )
| ( eqimpsubset2 != $true ) ),
inference(cnf_transformation,[status(thm)],[f648]) ).
thf(f648,plain,
( ( ( eqimpsubset2 = $true )
| ( ( sK162 = sK163 )
& ( ( subset @ sK162 @ sK163 )
!= $true ) ) )
& ( ! [X2: $i,X3: $i] :
( ( X2 != X3 )
| ( $true
= ( subset @ X2 @ X3 ) ) )
| ( eqimpsubset2 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK162,sK163])],[f646,f647]) ).
thf(f647,plain,
( ? [X0: $i,X1: $i] :
( ( X0 = X1 )
& ( ( subset @ X0 @ X1 )
!= $true ) )
=> ( ( sK162 = sK163 )
& ( ( subset @ sK162 @ sK163 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f646,plain,
( ( ( eqimpsubset2 = $true )
| ? [X0: $i,X1: $i] :
( ( X0 = X1 )
& ( ( subset @ X0 @ X1 )
!= $true ) ) )
& ( ! [X2: $i,X3: $i] :
( ( X2 != X3 )
| ( $true
= ( subset @ X2 @ X3 ) ) )
| ( eqimpsubset2 != $true ) ) ),
inference(rectify,[status(thm)],[f645]) ).
thf(f645,plain,
( ( ( eqimpsubset2 = $true )
| ? [X0: $i,X1: $i] :
( ( X0 = X1 )
& ( ( subset @ X0 @ X1 )
!= $true ) ) )
& ( ! [X0: $i,X1: $i] :
( ( X0 != X1 )
| ( ( subset @ X0 @ X1 )
= $true ) )
| ( eqimpsubset2 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f306]) ).
thf(f306,plain,
( ( eqimpsubset2 = $true )
<=> ! [X0: $i,X1: $i] :
( ( X0 != X1 )
| ( ( subset @ X0 @ X1 )
= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f129]) ).
thf(f129,plain,
( ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ( ( subset @ X0 @ X1 )
= $true ) )
<=> ( eqimpsubset2 = $true ) ),
inference(fool_elimination,[status(thm)],[f128]) ).
thf(f128,plain,
! [X0: $i,X1: $i] :
( ( ( X0 = X1 )
=> ( subset @ X0 @ X1 ) )
= eqimpsubset2 ),
inference(rectify,[status(thm)],[f79]) ).
thf(f79,axiom,
! [X4: $i,X3: $i] :
( ( ( X3 = X4 )
=> ( subset @ X4 @ X3 ) )
= eqimpsubset2 ),
file('/export/starexec/sandbox2/tmp/tmp.HAHRJ9zahH/DTF2THF_17640.p',eqimpsubset2) ).
thf(f1354,plain,
( ( subset @ sK103 @ sK103 )
!= $true ),
inference(definition_unfolding,[status(thm)],[f1009,f1010]) ).
thf(f1010,plain,
sK102 = sK103,
inference(cnf_transformation,[status(thm)],[f543]) ).
thf(f1009,plain,
( ( subset @ sK102 @ sK103 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f543]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : SEU567^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.06 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.06/0.24 % Computer : n024.cluster.edu
% 0.06/0.24 % Model : x86_64 x86_64
% 0.06/0.24 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.06/0.24 % Memory : 8042.1875MB
% 0.06/0.24 % OS : Linux 3.10.0-693.el7.x86_64
% 0.06/0.24 % CPULimit : 300
% 0.06/0.24 % WCLimit : 300
% 0.06/0.24 % DateTime : Tue Feb 24 07:26:55 EST 2026
% 0.06/0.24 % CPUTime :
% 0.06/0.24 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.06/0.31 ---- Original DTF file ---
% 0.06/0.31 thf(spec,logic,$$dhol).
% 0.06/0.31 %------------------------------------------------------------------------------
% 0.06/0.31 % File : SEU567^1 : TPTP v9.2.1. Released v3.7.0.
% 0.06/0.31 % Domain : Set Theory
% 0.06/0.31 % Problem : Preliminary Notions - Relations on Sets - Subsets
% 0.06/0.31 % Version : Especial.
% 0.06/0.31 % English : (! A:i.! B:i.~(subset A B) -> ~(A = B))
% 0.06/0.31
% 0.06/0.31 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.06/0.31 % Source : [Bro08]
% 0.06/0.31 % Names : ZFC069g [Bro08]
% 0.06/0.31
% 0.06/0.31 % Status : Theorem
% 0.06/0.31 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v3.7.0
% 0.06/0.31 % Syntax : Number of formulae : 184 ( 85 unt; 98 typ; 85 def)
% 0.06/0.31 % Number of atoms : 467 ( 131 equ; 0 cnn)
% 0.06/0.31 % Maximal formula atoms : 85 ( 5 avg)
% 0.06/0.31 % Number of connectives : 746 ( 27 ~; 5 |; 27 &; 458 @)
% 0.06/0.31 % ( 14 <=>; 215 =>; 0 <=; 0 <~>)
% 0.06/0.31 % Maximal formula depth : 90 ( 2 avg)
% 0.06/0.31 % Number of types : 2 ( 0 usr)
% 0.06/0.31 % Number of type conns : 53 ( 53 >; 0 *; 0 +; 0 <<)
% 0.06/0.31 % Number of symbols : 101 ( 98 usr; 87 con; 0-2 aty)
% 0.06/0.31 % Number of variables : 264 ( 28 ^; 208 !; 28 ?; 264 :)
% 0.06/0.31 % SPC : TH0_THM_EQU_NAR
% 0.06/0.31
% 0.06/0.31 % Comments : http://mathgate.info/detsetitem.php?id=432
% 0.06/0.31 %------------------------------------------------------------------------------
% 0.06/0.31 thf(in_type,type,
% 0.06/0.31 in: $i > $i > $o ).
% 0.06/0.31
% 0.06/0.31 thf(exu_type,type,
% 0.06/0.31 exu: ( $i > $o ) > $o ).
% 0.06/0.31
% 0.06/0.31 thf(exu,definition,
% 0.06/0.31 ( exu
% 0.06/0.31 = ( ^ [Xphi: $i > $o] :
% 0.06/0.31 ? [Xx: $i] :
% 0.06/0.31 ( ( Xphi @ Xx )
% 0.06/0.31 & ! [Xy: $i] :
% 0.06/0.31 ( ( Xphi @ Xy )
% 0.06/0.31 => ( Xx = Xy ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(setextAx_type,type,
% 0.06/0.31 setextAx: $o ).
% 0.06/0.31
% 0.06/0.31 thf(setextAx,definition,
% 0.06/0.31 ( setextAx
% 0.06/0.31 = ( ! [A: $i,B: $i] :
% 0.06/0.31 ( ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 <=> ( in @ Xx @ B ) )
% 0.06/0.31 => ( A = B ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(emptyset_type,type,
% 0.06/0.31 emptyset: $i ).
% 0.06/0.31
% 0.06/0.31 thf(emptysetAx_type,type,
% 0.06/0.31 emptysetAx: $o ).
% 0.06/0.31
% 0.06/0.31 thf(emptysetAx,definition,
% 0.06/0.31 ( emptysetAx
% 0.06/0.31 = ( ! [Xx: $i] :
% 0.06/0.31 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(setadjoin_type,type,
% 0.06/0.31 setadjoin: $i > $i > $i ).
% 0.06/0.31
% 0.06/0.31 thf(setadjoinAx_type,type,
% 0.06/0.31 setadjoinAx: $o ).
% 0.06/0.31
% 0.06/0.31 thf(setadjoinAx,definition,
% 0.06/0.31 ( setadjoinAx
% 0.06/0.31 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.06/0.31 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.06/0.31 <=> ( ( Xy = Xx )
% 0.06/0.31 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(powerset_type,type,
% 0.06/0.31 powerset: $i > $i ).
% 0.06/0.31
% 0.06/0.31 thf(powersetAx_type,type,
% 0.06/0.31 powersetAx: $o ).
% 0.06/0.31
% 0.06/0.31 thf(powersetAx,definition,
% 0.06/0.31 ( powersetAx
% 0.06/0.31 = ( ! [A: $i,B: $i] :
% 0.06/0.31 ( ( in @ B @ ( powerset @ A ) )
% 0.06/0.31 <=> ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ B )
% 0.06/0.31 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(setunion_type,type,
% 0.06/0.31 setunion: $i > $i ).
% 0.06/0.31
% 0.06/0.31 thf(setunionAx_type,type,
% 0.06/0.31 setunionAx: $o ).
% 0.06/0.31
% 0.06/0.31 thf(setunionAx,definition,
% 0.06/0.31 ( setunionAx
% 0.06/0.31 = ( ! [A: $i,Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ ( setunion @ A ) )
% 0.06/0.31 <=> ? [B: $i] :
% 0.06/0.31 ( ( in @ Xx @ B )
% 0.06/0.31 & ( in @ B @ A ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(omega_type,type,
% 0.06/0.31 omega: $i ).
% 0.06/0.31
% 0.06/0.31 thf(omega0Ax_type,type,
% 0.06/0.31 omega0Ax: $o ).
% 0.06/0.31
% 0.06/0.31 thf(omega0Ax,definition,
% 0.06/0.31 ( omega0Ax
% 0.06/0.31 = ( in @ emptyset @ omega ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(omegaSAx_type,type,
% 0.06/0.31 omegaSAx: $o ).
% 0.06/0.31
% 0.06/0.31 thf(omegaSAx,definition,
% 0.06/0.31 ( omegaSAx
% 0.06/0.31 = ( ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ omega )
% 0.06/0.31 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(omegaIndAx_type,type,
% 0.06/0.31 omegaIndAx: $o ).
% 0.06/0.31
% 0.06/0.31 thf(omegaIndAx,definition,
% 0.06/0.31 ( omegaIndAx
% 0.06/0.31 = ( ! [A: $i] :
% 0.06/0.31 ( ( ( in @ emptyset @ A )
% 0.06/0.31 & ! [Xx: $i] :
% 0.06/0.31 ( ( ( in @ Xx @ omega )
% 0.06/0.31 & ( in @ Xx @ A ) )
% 0.06/0.31 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.06/0.31 => ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ omega )
% 0.06/0.31 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(replAx_type,type,
% 0.06/0.31 replAx: $o ).
% 0.06/0.31
% 0.06/0.31 thf(replAx,definition,
% 0.06/0.31 ( replAx
% 0.06/0.31 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.06/0.31 ( ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 => ( exu
% 0.06/0.31 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.06/0.31 => ? [B: $i] :
% 0.06/0.31 ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ B )
% 0.06/0.31 <=> ? [Xy: $i] :
% 0.06/0.31 ( ( in @ Xy @ A )
% 0.06/0.31 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(foundationAx_type,type,
% 0.06/0.31 foundationAx: $o ).
% 0.06/0.31
% 0.06/0.31 thf(foundationAx,definition,
% 0.06/0.31 ( foundationAx
% 0.06/0.31 = ( ! [A: $i] :
% 0.06/0.31 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.06/0.31 => ? [B: $i] :
% 0.06/0.31 ( ( in @ B @ A )
% 0.06/0.31 & ~ ? [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ B )
% 0.06/0.31 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(wellorderingAx_type,type,
% 0.06/0.31 wellorderingAx: $o ).
% 0.06/0.31
% 0.06/0.31 thf(wellorderingAx,definition,
% 0.06/0.31 ( wellorderingAx
% 0.06/0.31 = ( ! [A: $i] :
% 0.06/0.31 ? [B: $i] :
% 0.06/0.31 ( ! [C: $i] :
% 0.06/0.31 ( ( in @ C @ B )
% 0.06/0.31 => ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ C )
% 0.06/0.31 => ( in @ Xx @ A ) ) )
% 0.06/0.31 & ! [Xx: $i,Xy: $i] :
% 0.06/0.31 ( ( ( in @ Xx @ A )
% 0.06/0.31 & ( in @ Xy @ A ) )
% 0.06/0.31 => ( ! [C: $i] :
% 0.06/0.31 ( ( in @ C @ B )
% 0.06/0.31 => ( ( in @ Xx @ C )
% 0.06/0.31 <=> ( in @ Xy @ C ) ) )
% 0.06/0.31 => ( Xx = Xy ) ) )
% 0.06/0.31 & ! [C: $i,D: $i] :
% 0.06/0.31 ( ( ( in @ C @ B )
% 0.06/0.31 & ( in @ D @ B ) )
% 0.06/0.31 => ( ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ C )
% 0.06/0.31 => ( in @ Xx @ D ) )
% 0.06/0.31 | ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ D )
% 0.06/0.31 => ( in @ Xx @ C ) ) ) )
% 0.06/0.31 & ! [C: $i] :
% 0.06/0.31 ( ( ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ C )
% 0.06/0.31 => ( in @ Xx @ A ) )
% 0.06/0.31 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.06/0.31 => ? [D: $i,Xx: $i] :
% 0.06/0.31 ( ( in @ D @ B )
% 0.06/0.31 & ( in @ Xx @ C )
% 0.06/0.31 & ~ ? [Xy: $i] :
% 0.06/0.31 ( ( in @ Xy @ D )
% 0.06/0.31 & ( in @ Xy @ C ) )
% 0.06/0.31 & ! [E: $i] :
% 0.06/0.31 ( ( in @ E @ B )
% 0.06/0.31 => ( ! [Xy: $i] :
% 0.06/0.31 ( ( in @ Xy @ E )
% 0.06/0.31 => ( in @ Xy @ D ) )
% 0.06/0.31 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(descr_type,type,
% 0.06/0.31 descr: ( $i > $o ) > $i ).
% 0.06/0.31
% 0.06/0.31 thf(descrp_type,type,
% 0.06/0.31 descrp: $o ).
% 0.06/0.31
% 0.06/0.31 thf(descrp,definition,
% 0.06/0.31 ( descrp
% 0.06/0.31 = ( ! [Xphi: $i > $o] :
% 0.06/0.31 ( ( exu
% 0.06/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.31 => ( Xphi
% 0.06/0.31 @ ( descr
% 0.06/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(dsetconstr_type,type,
% 0.06/0.31 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.06/0.31
% 0.06/0.31 thf(dsetconstrI_type,type,
% 0.06/0.31 dsetconstrI: $o ).
% 0.06/0.31
% 0.06/0.31 thf(dsetconstrI,definition,
% 0.06/0.31 ( dsetconstrI
% 0.06/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 => ( ( Xphi @ Xx )
% 0.06/0.31 => ( in @ Xx
% 0.06/0.31 @ ( dsetconstr @ A
% 0.06/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(dsetconstrEL_type,type,
% 0.06/0.31 dsetconstrEL: $o ).
% 0.06/0.31
% 0.06/0.31 thf(dsetconstrEL,definition,
% 0.06/0.31 ( dsetconstrEL
% 0.06/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.06/0.31 ( ( in @ Xx
% 0.06/0.31 @ ( dsetconstr @ A
% 0.06/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.06/0.31 => ( in @ Xx @ A ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(dsetconstrER_type,type,
% 0.06/0.31 dsetconstrER: $o ).
% 0.06/0.31
% 0.06/0.31 thf(dsetconstrER,definition,
% 0.06/0.31 ( dsetconstrER
% 0.06/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.06/0.31 ( ( in @ Xx
% 0.06/0.31 @ ( dsetconstr @ A
% 0.06/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.06/0.31 => ( Xphi @ Xx ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(exuE1_type,type,
% 0.06/0.31 exuE1: $o ).
% 0.06/0.31
% 0.06/0.31 thf(exuE1,definition,
% 0.06/0.31 ( exuE1
% 0.06/0.31 = ( ! [Xphi: $i > $o] :
% 0.06/0.31 ( ( exu
% 0.06/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.31 => ? [Xx: $i] :
% 0.06/0.31 ( ( Xphi @ Xx )
% 0.06/0.31 & ! [Xy: $i] :
% 0.06/0.31 ( ( Xphi @ Xy )
% 0.06/0.31 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(prop2set_type,type,
% 0.06/0.31 prop2set: $o > $i ).
% 0.06/0.31
% 0.06/0.31 thf(prop2setE_type,type,
% 0.06/0.31 prop2setE: $o ).
% 0.06/0.31
% 0.06/0.31 thf(prop2setE,definition,
% 0.06/0.31 ( prop2setE
% 0.06/0.31 = ( ! [Xphi: $o,Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.06/0.31 => Xphi ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(emptysetE_type,type,
% 0.06/0.31 emptysetE: $o ).
% 0.06/0.31
% 0.06/0.31 thf(emptysetE,definition,
% 0.06/0.31 ( emptysetE
% 0.06/0.31 = ( ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ emptyset )
% 0.06/0.31 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(emptysetimpfalse_type,type,
% 0.06/0.31 emptysetimpfalse: $o ).
% 0.06/0.31
% 0.06/0.31 thf(emptysetimpfalse,definition,
% 0.06/0.31 ( emptysetimpfalse
% 0.06/0.31 = ( ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ emptyset )
% 0.06/0.31 => $false ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(notinemptyset_type,type,
% 0.06/0.31 notinemptyset: $o ).
% 0.06/0.31
% 0.06/0.31 thf(notinemptyset,definition,
% 0.06/0.31 ( notinemptyset
% 0.06/0.31 = ( ! [Xx: $i] :
% 0.06/0.31 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(exuE3e_type,type,
% 0.06/0.31 exuE3e: $o ).
% 0.06/0.31
% 0.06/0.31 thf(exuE3e,definition,
% 0.06/0.31 ( exuE3e
% 0.06/0.31 = ( ! [Xphi: $i > $o] :
% 0.06/0.31 ( ( exu
% 0.06/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.31 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(setext_type,type,
% 0.06/0.31 setext: $o ).
% 0.06/0.31
% 0.06/0.31 thf(setext,definition,
% 0.06/0.31 ( setext
% 0.06/0.31 = ( ! [A: $i,B: $i] :
% 0.06/0.31 ( ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 => ( in @ Xx @ B ) )
% 0.06/0.31 => ( ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ B )
% 0.06/0.31 => ( in @ Xx @ A ) )
% 0.06/0.31 => ( A = B ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(emptyI_type,type,
% 0.06/0.31 emptyI: $o ).
% 0.06/0.31
% 0.06/0.31 thf(emptyI,definition,
% 0.06/0.31 ( emptyI
% 0.06/0.31 = ( ! [A: $i] :
% 0.06/0.31 ( ! [Xx: $i] :
% 0.06/0.31 ~ ( in @ Xx @ A )
% 0.06/0.31 => ( A = emptyset ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(noeltsimpempty_type,type,
% 0.06/0.31 noeltsimpempty: $o ).
% 0.06/0.31
% 0.06/0.31 thf(noeltsimpempty,definition,
% 0.06/0.31 ( noeltsimpempty
% 0.06/0.31 = ( ! [A: $i] :
% 0.06/0.31 ( ! [Xx: $i] :
% 0.06/0.31 ~ ( in @ Xx @ A )
% 0.06/0.31 => ( A = emptyset ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(setbeta_type,type,
% 0.06/0.31 setbeta: $o ).
% 0.06/0.31
% 0.06/0.31 thf(setbeta,definition,
% 0.06/0.31 ( setbeta
% 0.06/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 => ( ( in @ Xx
% 0.06/0.31 @ ( dsetconstr @ A
% 0.06/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.06/0.31 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(nonempty_type,type,
% 0.06/0.31 nonempty: $i > $o ).
% 0.06/0.31
% 0.06/0.31 thf(nonempty,definition,
% 0.06/0.31 ( nonempty
% 0.06/0.31 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(nonemptyE1_type,type,
% 0.06/0.31 nonemptyE1: $o ).
% 0.06/0.31
% 0.06/0.31 thf(nonemptyE1,definition,
% 0.06/0.31 ( nonemptyE1
% 0.06/0.31 = ( ! [A: $i] :
% 0.06/0.31 ( ( nonempty @ A )
% 0.06/0.31 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(nonemptyI_type,type,
% 0.06/0.31 nonemptyI: $o ).
% 0.06/0.31
% 0.06/0.31 thf(nonemptyI,definition,
% 0.06/0.31 ( nonemptyI
% 0.06/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 => ( ( Xphi @ Xx )
% 0.06/0.31 => ( nonempty
% 0.06/0.31 @ ( dsetconstr @ A
% 0.06/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(nonemptyI1_type,type,
% 0.06/0.31 nonemptyI1: $o ).
% 0.06/0.31
% 0.06/0.31 thf(nonemptyI1,definition,
% 0.06/0.31 ( nonemptyI1
% 0.06/0.31 = ( ! [A: $i] :
% 0.06/0.31 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.06/0.31 => ( nonempty @ A ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(setadjoinIL_type,type,
% 0.06/0.31 setadjoinIL: $o ).
% 0.06/0.31
% 0.06/0.31 thf(setadjoinIL,definition,
% 0.06/0.31 ( setadjoinIL
% 0.06/0.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(emptyinunitempty_type,type,
% 0.06/0.31 emptyinunitempty: $o ).
% 0.06/0.31
% 0.06/0.31 thf(emptyinunitempty,definition,
% 0.06/0.31 ( emptyinunitempty
% 0.06/0.31 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(setadjoinIR_type,type,
% 0.06/0.31 setadjoinIR: $o ).
% 0.06/0.31
% 0.06/0.31 thf(setadjoinIR,definition,
% 0.06/0.31 ( setadjoinIR
% 0.06/0.31 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.06/0.31 ( ( in @ Xy @ A )
% 0.06/0.31 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(setadjoinE_type,type,
% 0.06/0.31 setadjoinE: $o ).
% 0.06/0.31
% 0.06/0.31 thf(setadjoinE,definition,
% 0.06/0.31 ( setadjoinE
% 0.06/0.31 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.06/0.31 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.06/0.31 => ! [Xphi: $o] :
% 0.06/0.31 ( ( ( Xy = Xx )
% 0.06/0.31 => Xphi )
% 0.06/0.31 => ( ( ( in @ Xy @ A )
% 0.06/0.31 => Xphi )
% 0.06/0.31 => Xphi ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(setadjoinOr_type,type,
% 0.06/0.31 setadjoinOr: $o ).
% 0.06/0.31
% 0.06/0.31 thf(setadjoinOr,definition,
% 0.06/0.31 ( setadjoinOr
% 0.06/0.31 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.06/0.31 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.06/0.31 => ( ( Xy = Xx )
% 0.06/0.31 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(setoftrueEq_type,type,
% 0.06/0.31 setoftrueEq: $o ).
% 0.06/0.31
% 0.06/0.31 thf(setoftrueEq,definition,
% 0.06/0.31 ( setoftrueEq
% 0.06/0.31 = ( ! [A: $i] :
% 0.06/0.31 ( ( dsetconstr @ A
% 0.06/0.31 @ ^ [Xx: $i] : $true )
% 0.06/0.31 = A ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(powersetI_type,type,
% 0.06/0.31 powersetI: $o ).
% 0.06/0.31
% 0.06/0.31 thf(powersetI,definition,
% 0.06/0.31 ( powersetI
% 0.06/0.31 = ( ! [A: $i,B: $i] :
% 0.06/0.31 ( ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ B )
% 0.06/0.31 => ( in @ Xx @ A ) )
% 0.06/0.31 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(emptyinPowerset_type,type,
% 0.06/0.31 emptyinPowerset: $o ).
% 0.06/0.31
% 0.06/0.31 thf(emptyinPowerset,definition,
% 0.06/0.31 ( emptyinPowerset
% 0.06/0.31 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(emptyInPowerset_type,type,
% 0.06/0.31 emptyInPowerset: $o ).
% 0.06/0.31
% 0.06/0.31 thf(emptyInPowerset,definition,
% 0.06/0.31 ( emptyInPowerset
% 0.06/0.31 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(powersetE_type,type,
% 0.06/0.31 powersetE: $o ).
% 0.06/0.31
% 0.06/0.31 thf(powersetE,definition,
% 0.06/0.31 ( powersetE
% 0.06/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.31 ( ( in @ B @ ( powerset @ A ) )
% 0.06/0.31 => ( ( in @ Xx @ B )
% 0.06/0.31 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(setunionI_type,type,
% 0.06/0.31 setunionI: $o ).
% 0.06/0.31
% 0.06/0.31 thf(setunionI,definition,
% 0.06/0.31 ( setunionI
% 0.06/0.31 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.06/0.31 ( ( in @ Xx @ B )
% 0.06/0.31 => ( ( in @ B @ A )
% 0.06/0.31 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(setunionE_type,type,
% 0.06/0.31 setunionE: $o ).
% 0.06/0.31
% 0.06/0.31 thf(setunionE,definition,
% 0.06/0.31 ( setunionE
% 0.06/0.31 = ( ! [A: $i,Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ ( setunion @ A ) )
% 0.06/0.31 => ! [Xphi: $o] :
% 0.06/0.31 ( ! [B: $i] :
% 0.06/0.31 ( ( in @ Xx @ B )
% 0.06/0.31 => ( ( in @ B @ A )
% 0.06/0.31 => Xphi ) )
% 0.06/0.31 => Xphi ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(subPowSU_type,type,
% 0.06/0.31 subPowSU: $o ).
% 0.06/0.31
% 0.06/0.31 thf(subPowSU,definition,
% 0.06/0.31 ( subPowSU
% 0.06/0.31 = ( ! [A: $i,Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(exuE2_type,type,
% 0.06/0.31 exuE2: $o ).
% 0.06/0.31
% 0.06/0.31 thf(exuE2,definition,
% 0.06/0.31 ( exuE2
% 0.06/0.31 = ( ! [Xphi: $i > $o] :
% 0.06/0.31 ( ( exu
% 0.06/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.31 => ? [Xx: $i] :
% 0.06/0.31 ! [Xy: $i] :
% 0.06/0.31 ( ( Xphi @ Xy )
% 0.06/0.31 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(nonemptyImpWitness_type,type,
% 0.06/0.31 nonemptyImpWitness: $o ).
% 0.06/0.31
% 0.06/0.31 thf(nonemptyImpWitness,definition,
% 0.06/0.31 ( nonemptyImpWitness
% 0.06/0.31 = ( ! [A: $i] :
% 0.06/0.31 ( ( nonempty @ A )
% 0.06/0.31 => ? [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 & $true ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(uniqinunit_type,type,
% 0.06/0.31 uniqinunit: $o ).
% 0.06/0.31
% 0.06/0.31 thf(uniqinunit,definition,
% 0.06/0.31 ( uniqinunit
% 0.06/0.31 = ( ! [Xx: $i,Xy: $i] :
% 0.06/0.31 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.06/0.31 => ( Xx = Xy ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(notinsingleton_type,type,
% 0.06/0.31 notinsingleton: $o ).
% 0.06/0.31
% 0.06/0.31 thf(notinsingleton,definition,
% 0.06/0.31 ( notinsingleton
% 0.06/0.31 = ( ! [Xx: $i,Xy: $i] :
% 0.06/0.31 ( ( Xx != Xy )
% 0.06/0.31 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(eqinunit_type,type,
% 0.06/0.31 eqinunit: $o ).
% 0.06/0.31
% 0.06/0.31 thf(eqinunit,definition,
% 0.06/0.31 ( eqinunit
% 0.06/0.31 = ( ! [Xx: $i,Xy: $i] :
% 0.06/0.31 ( ( Xx = Xy )
% 0.06/0.31 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(singletonsswitch_type,type,
% 0.06/0.31 singletonsswitch: $o ).
% 0.06/0.31
% 0.06/0.31 thf(singletonsswitch,definition,
% 0.06/0.31 ( singletonsswitch
% 0.06/0.31 = ( ! [Xx: $i,Xy: $i] :
% 0.06/0.31 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.06/0.31 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(upairsetE_type,type,
% 0.06/0.31 upairsetE: $o ).
% 0.06/0.31
% 0.06/0.31 thf(upairsetE,definition,
% 0.06/0.31 ( upairsetE
% 0.06/0.31 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.06/0.31 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.06/0.31 => ( ( Xz = Xx )
% 0.06/0.31 | ( Xz = Xy ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(upairsetIL_type,type,
% 0.06/0.31 upairsetIL: $o ).
% 0.06/0.31
% 0.06/0.31 thf(upairsetIL,definition,
% 0.06/0.31 ( upairsetIL
% 0.06/0.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(upairsetIR_type,type,
% 0.06/0.31 upairsetIR: $o ).
% 0.06/0.31
% 0.06/0.31 thf(upairsetIR,definition,
% 0.06/0.31 ( upairsetIR
% 0.06/0.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(emptyE1_type,type,
% 0.06/0.31 emptyE1: $o ).
% 0.06/0.31
% 0.06/0.31 thf(emptyE1,definition,
% 0.06/0.31 ( emptyE1
% 0.06/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.31 ( ? [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 & ( Xphi @ Xx ) )
% 0.06/0.31 => ( ( ( dsetconstr @ A
% 0.06/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.31 = emptyset )
% 0.06/0.31 => $false ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(vacuousDall_type,type,
% 0.06/0.31 vacuousDall: $o ).
% 0.06/0.31
% 0.06/0.31 thf(vacuousDall,definition,
% 0.06/0.31 ( vacuousDall
% 0.06/0.31 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ emptyset )
% 0.06/0.31 => ( Xphi @ Xx ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(quantDeMorgan1_type,type,
% 0.06/0.31 quantDeMorgan1: $o ).
% 0.06/0.31
% 0.06/0.31 thf(quantDeMorgan1,definition,
% 0.06/0.31 ( quantDeMorgan1
% 0.06/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.31 ( ~ ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 => ( Xphi @ Xx ) )
% 0.06/0.31 => ? [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(quantDeMorgan2_type,type,
% 0.06/0.31 quantDeMorgan2: $o ).
% 0.06/0.31
% 0.06/0.31 thf(quantDeMorgan2,definition,
% 0.06/0.31 ( quantDeMorgan2
% 0.06/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.31 ( ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 => ~ ( Xphi @ Xx ) )
% 0.06/0.31 => ~ ? [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(quantDeMorgan3_type,type,
% 0.06/0.31 quantDeMorgan3: $o ).
% 0.06/0.31
% 0.06/0.31 thf(quantDeMorgan3,definition,
% 0.06/0.31 ( quantDeMorgan3
% 0.06/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.31 ( ~ ? [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 & ( Xphi @ Xx ) )
% 0.06/0.31 => ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(quantDeMorgan4_type,type,
% 0.06/0.31 quantDeMorgan4: $o ).
% 0.06/0.31
% 0.06/0.31 thf(quantDeMorgan4,definition,
% 0.06/0.31 ( quantDeMorgan4
% 0.06/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.31 ( ? [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 & ~ ( Xphi @ Xx ) )
% 0.06/0.31 => ~ ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(prop2setI_type,type,
% 0.06/0.31 prop2setI: $o ).
% 0.06/0.31
% 0.06/0.31 thf(prop2setI,definition,
% 0.06/0.31 ( prop2setI
% 0.06/0.31 = ( ! [Xphi: $o] :
% 0.06/0.31 ( Xphi
% 0.06/0.31 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(set2prop_type,type,
% 0.06/0.31 set2prop: $i > $o ).
% 0.06/0.31
% 0.06/0.31 thf(prop2set2propI_type,type,
% 0.06/0.31 prop2set2propI: $o ).
% 0.06/0.31
% 0.06/0.31 thf(prop2set2propI,definition,
% 0.06/0.31 ( prop2set2propI
% 0.06/0.31 = ( ! [Xphi: $o] :
% 0.06/0.31 ( Xphi
% 0.06/0.31 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(notdexE_type,type,
% 0.06/0.31 notdexE: $o ).
% 0.06/0.31
% 0.06/0.31 thf(notdexE,definition,
% 0.06/0.31 ( notdexE
% 0.06/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.31 ( ~ ? [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 & ( Xphi @ Xx ) )
% 0.06/0.31 => ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(notdallE_type,type,
% 0.06/0.31 notdallE: $o ).
% 0.06/0.31
% 0.06/0.31 thf(notdallE,definition,
% 0.06/0.31 ( notdallE
% 0.06/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.06/0.31 ( ~ ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 => ( Xphi @ Xx ) )
% 0.06/0.31 => ? [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(exuI1_type,type,
% 0.06/0.31 exuI1: $o ).
% 0.06/0.31
% 0.06/0.31 thf(exuI1,definition,
% 0.06/0.31 ( exuI1
% 0.06/0.31 = ( ! [Xphi: $i > $o] :
% 0.06/0.31 ( ? [Xx: $i] :
% 0.06/0.31 ( ( Xphi @ Xx )
% 0.06/0.31 & ! [Xy: $i] :
% 0.06/0.31 ( ( Xphi @ Xy )
% 0.06/0.31 => ( Xx = Xy ) ) )
% 0.06/0.31 => ( exu
% 0.06/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(exuI3_type,type,
% 0.06/0.31 exuI3: $o ).
% 0.06/0.31
% 0.06/0.31 thf(exuI3,definition,
% 0.06/0.31 ( exuI3
% 0.06/0.31 = ( ! [Xphi: $i > $o] :
% 0.06/0.31 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.06/0.31 => ( ! [Xx: $i,Xy: $i] :
% 0.06/0.31 ( ( Xphi @ Xx )
% 0.06/0.31 => ( ( Xphi @ Xy )
% 0.06/0.31 => ( Xx = Xy ) ) )
% 0.06/0.31 => ( exu
% 0.06/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(exuI2_type,type,
% 0.06/0.31 exuI2: $o ).
% 0.06/0.31
% 0.06/0.31 thf(exuI2,definition,
% 0.06/0.31 ( exuI2
% 0.06/0.31 = ( ! [Xphi: $i > $o] :
% 0.06/0.31 ( ? [Xx: $i] :
% 0.06/0.31 ! [Xy: $i] :
% 0.06/0.31 ( ( Xphi @ Xy )
% 0.06/0.31 <=> ( Xy = Xx ) )
% 0.06/0.31 => ( exu
% 0.06/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(inCongP_type,type,
% 0.06/0.31 inCongP: $o ).
% 0.06/0.31
% 0.06/0.31 thf(inCongP,definition,
% 0.06/0.31 ( inCongP
% 0.06/0.31 = ( ! [A: $i,B: $i] :
% 0.06/0.31 ( ( A = B )
% 0.06/0.31 => ! [Xx: $i,Xy: $i] :
% 0.06/0.31 ( ( Xx = Xy )
% 0.06/0.31 => ( ( in @ Xx @ A )
% 0.06/0.31 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(in__Cong_type,type,
% 0.06/0.31 in__Cong: $o ).
% 0.06/0.31
% 0.06/0.31 thf(in__Cong,definition,
% 0.06/0.31 ( in__Cong
% 0.06/0.31 = ( ! [A: $i,B: $i] :
% 0.06/0.31 ( ( A = B )
% 0.06/0.31 => ! [Xx: $i,Xy: $i] :
% 0.06/0.31 ( ( Xx = Xy )
% 0.06/0.31 => ( ( in @ Xx @ A )
% 0.06/0.31 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(exuE3u_type,type,
% 0.06/0.31 exuE3u: $o ).
% 0.06/0.31
% 0.06/0.31 thf(exuE3u,definition,
% 0.06/0.31 ( exuE3u
% 0.06/0.31 = ( ! [Xphi: $i > $o] :
% 0.06/0.31 ( ( exu
% 0.06/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.31 => ! [Xx: $i,Xy: $i] :
% 0.06/0.31 ( ( Xphi @ Xx )
% 0.06/0.31 => ( ( Xphi @ Xy )
% 0.06/0.31 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(exu__Cong_type,type,
% 0.06/0.31 exu__Cong: $o ).
% 0.06/0.31
% 0.06/0.31 thf(exu__Cong,definition,
% 0.06/0.31 ( exu__Cong
% 0.06/0.31 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.06/0.31 ( ! [Xx: $i,Xy: $i] :
% 0.06/0.31 ( ( Xx = Xy )
% 0.06/0.31 => ( ( Xphi @ Xx )
% 0.06/0.31 <=> ( Xpsi @ Xy ) ) )
% 0.06/0.31 => ( ( exu
% 0.06/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.31 <=> ( exu
% 0.06/0.31 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(emptyset__Cong_type,type,
% 0.06/0.31 emptyset__Cong: $o ).
% 0.06/0.31
% 0.06/0.31 thf(emptyset__Cong,definition,
% 0.06/0.31 ( emptyset__Cong
% 0.06/0.31 = ( emptyset = emptyset ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(setadjoin__Cong_type,type,
% 0.06/0.31 setadjoin__Cong: $o ).
% 0.06/0.31
% 0.06/0.31 thf(setadjoin__Cong,definition,
% 0.06/0.31 ( setadjoin__Cong
% 0.06/0.31 = ( ! [Xx: $i,Xy: $i] :
% 0.06/0.31 ( ( Xx = Xy )
% 0.06/0.31 => ! [Xz: $i,Xu: $i] :
% 0.06/0.31 ( ( Xz = Xu )
% 0.06/0.31 => ( ( setadjoin @ Xx @ Xz )
% 0.06/0.31 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(powerset__Cong_type,type,
% 0.06/0.31 powerset__Cong: $o ).
% 0.06/0.31
% 0.06/0.31 thf(powerset__Cong,definition,
% 0.06/0.31 ( powerset__Cong
% 0.06/0.31 = ( ! [A: $i,B: $i] :
% 0.06/0.31 ( ( A = B )
% 0.06/0.31 => ( ( powerset @ A )
% 0.06/0.31 = ( powerset @ B ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(setunion__Cong_type,type,
% 0.06/0.31 setunion__Cong: $o ).
% 0.06/0.31
% 0.06/0.31 thf(setunion__Cong,definition,
% 0.06/0.31 ( setunion__Cong
% 0.06/0.31 = ( ! [A: $i,B: $i] :
% 0.06/0.31 ( ( A = B )
% 0.06/0.31 => ( ( setunion @ A )
% 0.06/0.31 = ( setunion @ B ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(omega__Cong_type,type,
% 0.06/0.31 omega__Cong: $o ).
% 0.06/0.31
% 0.06/0.31 thf(omega__Cong,definition,
% 0.06/0.31 ( omega__Cong
% 0.06/0.31 = ( omega = omega ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(exuEu_type,type,
% 0.06/0.31 exuEu: $o ).
% 0.06/0.31
% 0.06/0.31 thf(exuEu,definition,
% 0.06/0.31 ( exuEu
% 0.06/0.31 = ( ! [Xphi: $i > $o] :
% 0.06/0.31 ( ( exu
% 0.06/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.31 => ! [Xx: $i,Xy: $i] :
% 0.06/0.31 ( ( Xphi @ Xx )
% 0.06/0.31 => ( ( Xphi @ Xy )
% 0.06/0.31 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(descr__Cong_type,type,
% 0.06/0.31 descr__Cong: $o ).
% 0.06/0.31
% 0.06/0.31 thf(descr__Cong,definition,
% 0.06/0.31 ( descr__Cong
% 0.06/0.31 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.06/0.31 ( ! [Xx: $i,Xy: $i] :
% 0.06/0.31 ( ( Xx = Xy )
% 0.06/0.31 => ( ( Xphi @ Xx )
% 0.06/0.31 <=> ( Xpsi @ Xy ) ) )
% 0.06/0.31 => ( ( exu
% 0.06/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.31 => ( ( exu
% 0.06/0.31 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.06/0.31 => ( ( descr
% 0.06/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.31 = ( descr
% 0.06/0.31 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(dsetconstr__Cong_type,type,
% 0.06/0.31 dsetconstr__Cong: $o ).
% 0.06/0.31
% 0.06/0.31 thf(dsetconstr__Cong,definition,
% 0.06/0.31 ( dsetconstr__Cong
% 0.06/0.31 = ( ! [A: $i,B: $i] :
% 0.06/0.31 ( ( A = B )
% 0.06/0.31 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.06/0.31 ( ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 => ! [Xy: $i] :
% 0.06/0.31 ( ( in @ Xy @ B )
% 0.06/0.31 => ( ( Xx = Xy )
% 0.06/0.31 => ( ( Xphi @ Xx )
% 0.06/0.31 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.06/0.31 => ( ( dsetconstr @ A
% 0.06/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.06/0.31 = ( dsetconstr @ B
% 0.06/0.31 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(subset_type,type,
% 0.06/0.31 subset: $i > $i > $o ).
% 0.06/0.31
% 0.06/0.31 thf(disjoint_type,type,
% 0.06/0.31 disjoint: $i > $i > $o ).
% 0.06/0.31
% 0.06/0.31 thf(setsmeet_type,type,
% 0.06/0.31 setsmeet: $i > $i > $o ).
% 0.06/0.31
% 0.06/0.31 thf(subsetI1_type,type,
% 0.06/0.31 subsetI1: $o ).
% 0.06/0.31
% 0.06/0.31 thf(subsetI1,definition,
% 0.06/0.31 ( subsetI1
% 0.06/0.31 = ( ! [A: $i,B: $i] :
% 0.06/0.31 ( ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 => ( in @ Xx @ B ) )
% 0.06/0.31 => ( subset @ A @ B ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(eqimpsubset2_type,type,
% 0.06/0.31 eqimpsubset2: $o ).
% 0.06/0.31
% 0.06/0.31 thf(eqimpsubset2,definition,
% 0.06/0.31 ( eqimpsubset2
% 0.06/0.31 = ( ! [A: $i,B: $i] :
% 0.06/0.31 ( ( A = B )
% 0.06/0.31 => ( subset @ B @ A ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(eqimpsubset1_type,type,
% 0.06/0.31 eqimpsubset1: $o ).
% 0.06/0.31
% 0.06/0.31 thf(eqimpsubset1,definition,
% 0.06/0.31 ( eqimpsubset1
% 0.06/0.31 = ( ! [A: $i,B: $i] :
% 0.06/0.31 ( ( A = B )
% 0.06/0.31 => ( subset @ A @ B ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(subsetI2_type,type,
% 0.06/0.31 subsetI2: $o ).
% 0.06/0.31
% 0.06/0.31 thf(subsetI2,definition,
% 0.06/0.31 ( subsetI2
% 0.06/0.31 = ( ! [A: $i,B: $i] :
% 0.06/0.31 ( ! [Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 => ( in @ Xx @ B ) )
% 0.06/0.31 => ( subset @ A @ B ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(emptysetsubset_type,type,
% 0.06/0.31 emptysetsubset: $o ).
% 0.06/0.31
% 0.06/0.31 thf(emptysetsubset,definition,
% 0.06/0.31 ( emptysetsubset
% 0.06/0.31 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(subsetE_type,type,
% 0.06/0.31 subsetE: $o ).
% 0.06/0.31
% 0.06/0.31 thf(subsetE,definition,
% 0.06/0.31 ( subsetE
% 0.06/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.31 ( ( subset @ A @ B )
% 0.06/0.31 => ( ( in @ Xx @ A )
% 0.06/0.31 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(subsetE2_type,type,
% 0.06/0.31 subsetE2: $o ).
% 0.06/0.31
% 0.06/0.31 thf(subsetE2,definition,
% 0.06/0.31 ( subsetE2
% 0.06/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.31 ( ( subset @ A @ B )
% 0.06/0.31 => ( ~ ( in @ Xx @ B )
% 0.06/0.31 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(notsubsetI_type,type,
% 0.06/0.31 notsubsetI: $o ).
% 0.06/0.31
% 0.06/0.31 thf(notsubsetI,definition,
% 0.06/0.31 ( notsubsetI
% 0.06/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.06/0.31 ( ( in @ Xx @ A )
% 0.06/0.31 => ( ~ ( in @ Xx @ B )
% 0.06/0.31 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(notequalI1,conjecture,
% 0.06/0.31 ( setextAx
% 0.06/0.31 => ( emptysetAx
% 0.06/0.31 => ( setadjoinAx
% 0.06/0.31 => ( powersetAx
% 0.06/0.31 => ( setunionAx
% 0.06/0.31 => ( omega0Ax
% 0.06/0.31 => ( omegaSAx
% 0.06/0.31 => ( omegaIndAx
% 0.06/0.31 => ( replAx
% 0.06/0.31 => ( foundationAx
% 0.06/0.31 => ( wellorderingAx
% 0.06/0.31 => ( descrp
% 0.06/0.31 => ( dsetconstrI
% 0.06/0.31 => ( dsetconstrEL
% 0.06/0.31 => ( dsetconstrER
% 0.06/0.31 => ( exuE1
% 0.06/0.31 => ( prop2setE
% 0.06/0.31 => ( emptysetE
% 0.06/0.31 => ( emptysetimpfalse
% 0.06/0.31 => ( notinemptyset
% 0.06/0.31 => ( exuE3e
% 0.06/0.31 => ( setext
% 0.06/0.31 => ( emptyI
% 0.06/0.31 => ( noeltsimpempty
% 0.06/0.31 => ( setbeta
% 0.06/0.31 => ( nonemptyE1
% 0.06/0.31 => ( nonemptyI
% 0.06/0.31 => ( nonemptyI1
% 0.06/0.31 => ( setadjoinIL
% 0.06/0.31 => ( emptyinunitempty
% 0.06/0.31 => ( setadjoinIR
% 0.06/0.31 => ( setadjoinE
% 0.06/0.31 => ( setadjoinOr
% 0.06/0.31 => ( setoftrueEq
% 0.06/0.31 => ( powersetI
% 0.06/0.31 => ( emptyinPowerset
% 0.06/0.31 => ( emptyInPowerset
% 0.06/0.31 => ( powersetE
% 0.06/0.31 => ( setunionI
% 0.06/0.31 => ( setunionE
% 0.06/0.31 => ( subPowSU
% 0.06/0.31 => ( exuE2
% 0.06/0.31 => ( nonemptyImpWitness
% 0.06/0.31 => ( uniqinunit
% 0.06/0.31 => ( notinsingleton
% 0.06/0.31 => ( eqinunit
% 0.06/0.31 => ( singletonsswitch
% 0.06/0.31 => ( upairsetE
% 0.06/0.31 => ( upairsetIL
% 0.06/0.31 => ( upairsetIR
% 0.06/0.31 => ( emptyE1
% 0.06/0.31 => ( vacuousDall
% 0.06/0.31 => ( quantDeMorgan1
% 0.06/0.31 => ( quantDeMorgan2
% 0.06/0.31 => ( quantDeMorgan3
% 0.06/0.31 => ( quantDeMorgan4
% 0.06/0.31 => ( prop2setI
% 0.06/0.31 => ( prop2set2propI
% 0.06/0.31 => ( notdexE
% 0.06/0.31 => ( notdallE
% 0.06/0.31 => ( exuI1
% 0.06/0.31 => ( exuI3
% 0.06/0.31 => ( exuI2
% 0.06/0.31 => ( inCongP
% 0.06/0.31 => ( in__Cong
% 0.06/0.31 => ( exuE3u
% 0.06/0.31 => ( exu__Cong
% 0.06/0.31 => ( emptyset__Cong
% 0.06/0.31 => ( setadjoin__Cong
% 0.06/0.31 => ( powerset__Cong
% 0.06/0.31 => ( setunion__Cong
% 0.06/0.31 => ( omega__Cong
% 0.06/0.31 => ( exuEu
% 0.06/0.31 => ( descr__Cong
% 0.06/0.31 => ( dsetconstr__Cong
% 0.06/0.31 => ( subsetI1
% 0.06/0.31 => ( eqimpsubset2
% 0.06/0.31 => ( eqimpsubset1
% 0.06/0.31 => ( subsetI2
% 0.06/0.31 => ( emptysetsubset
% 0.06/0.31 => ( subsetE
% 0.06/0.31 => ( subsetE2
% 0.06/0.31 => ( notsubsetI
% 0.06/0.31 => ! [A: $i,B: $i] :
% 0.06/0.31 ( ~ ( subset @ A @ B )
% 0.06/0.31 => ( A != B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 %------------------------------------------------------------------------------
% 0.06/0.31 ------------------------
% 1.27/1.00 ---- Embedded in THF ---
% 1.27/1.00 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.27/1.00 %%% Generated on Tue Feb 24 07:26:56 EST 2026
% 1.27/1.00 %%% using '$$dhol' embedding, version 1.3.0.
% 1.27/1.00 %%% Logic specification used:
% 1.27/1.00 %%% thf(spec, logic, $$dhol).
% 1.27/1.00
% 1.27/1.00 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.HAHRJ9zahH/DTF2DTF17640.p
% See solution above
% 1.27/1.00 ------------------------
% 1.56/1.01 ---- Cleaned THF ---
% 1.56/1.01 thf(in_type,type,
% 1.56/1.01 in: $i > $i > $o ).
% 1.56/1.01
% 1.56/1.01 thf(exu_type,type,
% 1.56/1.01 exu: ( $i > $o ) > $o ).
% 1.56/1.01
% 1.56/1.01 thf(exu,definition,
% 1.56/1.01 ( exu
% 1.56/1.01 = ( ^ [Xphi: $i > $o] :
% 1.56/1.01 ? [Xx: $i] :
% 1.56/1.01 ( ( Xphi @ Xx )
% 1.56/1.01 & ! [Xy: $i] :
% 1.56/1.01 ( ( Xphi @ Xy )
% 1.56/1.01 => ( Xx = Xy ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(setextAx_type,type,
% 1.56/1.01 setextAx: $o ).
% 1.56/1.01
% 1.56/1.01 thf(setextAx,definition,
% 1.56/1.01 ( setextAx
% 1.56/1.01 = ( ! [A: $i,B: $i] :
% 1.56/1.01 ( ! [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ A )
% 1.56/1.01 <=> ( in @ Xx @ B ) )
% 1.56/1.01 => ( A = B ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(emptyset_type,type,
% 1.56/1.01 emptyset: $i ).
% 1.56/1.01
% 1.56/1.01 thf(emptysetAx_type,type,
% 1.56/1.01 emptysetAx: $o ).
% 1.56/1.01
% 1.56/1.01 thf(emptysetAx,definition,
% 1.56/1.01 ( emptysetAx
% 1.56/1.01 = ( ! [Xx: $i] :
% 1.56/1.01 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(setadjoin_type,type,
% 1.56/1.01 setadjoin: $i > $i > $i ).
% 1.56/1.01
% 1.56/1.01 thf(setadjoinAx_type,type,
% 1.56/1.01 setadjoinAx: $o ).
% 1.56/1.01
% 1.56/1.01 thf(setadjoinAx,definition,
% 1.56/1.01 ( setadjoinAx
% 1.56/1.01 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.56/1.01 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.56/1.01 <=> ( ( Xy = Xx )
% 1.56/1.01 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(powerset_type,type,
% 1.56/1.01 powerset: $i > $i ).
% 1.56/1.01
% 1.56/1.01 thf(powersetAx_type,type,
% 1.56/1.01 powersetAx: $o ).
% 1.56/1.01
% 1.56/1.01 thf(powersetAx,definition,
% 1.56/1.01 ( powersetAx
% 1.56/1.01 = ( ! [A: $i,B: $i] :
% 1.56/1.01 ( ( in @ B @ ( powerset @ A ) )
% 1.56/1.01 <=> ! [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ B )
% 1.56/1.01 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(setunion_type,type,
% 1.56/1.01 setunion: $i > $i ).
% 1.56/1.01
% 1.56/1.01 thf(setunionAx_type,type,
% 1.56/1.01 setunionAx: $o ).
% 1.56/1.01
% 1.56/1.01 thf(setunionAx,definition,
% 1.56/1.01 ( setunionAx
% 1.56/1.01 = ( ! [A: $i,Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ ( setunion @ A ) )
% 1.56/1.01 <=> ? [B: $i] :
% 1.56/1.01 ( ( in @ Xx @ B )
% 1.56/1.01 & ( in @ B @ A ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(omega_type,type,
% 1.56/1.01 omega: $i ).
% 1.56/1.01
% 1.56/1.01 thf(omega0Ax_type,type,
% 1.56/1.01 omega0Ax: $o ).
% 1.56/1.01
% 1.56/1.01 thf(omega0Ax,definition,
% 1.56/1.01 ( omega0Ax
% 1.56/1.01 = ( in @ emptyset @ omega ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(omegaSAx_type,type,
% 1.56/1.01 omegaSAx: $o ).
% 1.56/1.01
% 1.56/1.01 thf(omegaSAx,definition,
% 1.56/1.01 ( omegaSAx
% 1.56/1.01 = ( ! [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ omega )
% 1.56/1.01 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(omegaIndAx_type,type,
% 1.56/1.01 omegaIndAx: $o ).
% 1.56/1.01
% 1.56/1.01 thf(omegaIndAx,definition,
% 1.56/1.01 ( omegaIndAx
% 1.56/1.01 = ( ! [A: $i] :
% 1.56/1.01 ( ( ( in @ emptyset @ A )
% 1.56/1.01 & ! [Xx: $i] :
% 1.56/1.01 ( ( ( in @ Xx @ omega )
% 1.56/1.01 & ( in @ Xx @ A ) )
% 1.56/1.01 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 1.56/1.01 => ! [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ omega )
% 1.56/1.01 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(replAx_type,type,
% 1.56/1.01 replAx: $o ).
% 1.56/1.01
% 1.56/1.01 thf(replAx,definition,
% 1.56/1.01 ( replAx
% 1.56/1.01 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 1.56/1.01 ( ! [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ A )
% 1.56/1.01 => ( exu
% 1.56/1.01 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 1.56/1.01 => ? [B: $i] :
% 1.56/1.01 ! [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ B )
% 1.56/1.01 <=> ? [Xy: $i] :
% 1.56/1.01 ( ( in @ Xy @ A )
% 1.56/1.01 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(foundationAx_type,type,
% 1.56/1.01 foundationAx: $o ).
% 1.56/1.01
% 1.56/1.01 thf(foundationAx,definition,
% 1.56/1.01 ( foundationAx
% 1.56/1.01 = ( ! [A: $i] :
% 1.56/1.01 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.56/1.01 => ? [B: $i] :
% 1.56/1.01 ( ( in @ B @ A )
% 1.56/1.01 & ~ ? [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ B )
% 1.56/1.01 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(wellorderingAx_type,type,
% 1.56/1.01 wellorderingAx: $o ).
% 1.56/1.01
% 1.56/1.01 thf(wellorderingAx,definition,
% 1.56/1.01 ( wellorderingAx
% 1.56/1.01 = ( ! [A: $i] :
% 1.56/1.01 ? [B: $i] :
% 1.56/1.01 ( ! [C: $i] :
% 1.56/1.01 ( ( in @ C @ B )
% 1.56/1.01 => ! [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ C )
% 1.56/1.01 => ( in @ Xx @ A ) ) )
% 1.56/1.01 & ! [Xx: $i,Xy: $i] :
% 1.56/1.01 ( ( ( in @ Xx @ A )
% 1.56/1.01 & ( in @ Xy @ A ) )
% 1.56/1.01 => ( ! [C: $i] :
% 1.56/1.01 ( ( in @ C @ B )
% 1.56/1.01 => ( ( in @ Xx @ C )
% 1.56/1.01 <=> ( in @ Xy @ C ) ) )
% 1.56/1.01 => ( Xx = Xy ) ) )
% 1.56/1.01 & ! [C: $i,D: $i] :
% 1.56/1.01 ( ( ( in @ C @ B )
% 1.56/1.01 & ( in @ D @ B ) )
% 1.56/1.01 => ( ! [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ C )
% 1.56/1.01 => ( in @ Xx @ D ) )
% 1.56/1.01 | ! [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ D )
% 1.56/1.01 => ( in @ Xx @ C ) ) ) )
% 1.56/1.01 & ! [C: $i] :
% 1.56/1.01 ( ( ! [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ C )
% 1.56/1.01 => ( in @ Xx @ A ) )
% 1.56/1.01 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 1.56/1.01 => ? [D: $i,Xx: $i] :
% 1.56/1.01 ( ( in @ D @ B )
% 1.56/1.01 & ( in @ Xx @ C )
% 1.56/1.01 & ~ ? [Xy: $i] :
% 1.56/1.01 ( ( in @ Xy @ D )
% 1.56/1.01 & ( in @ Xy @ C ) )
% 1.56/1.01 & ! [E: $i] :
% 1.56/1.01 ( ( in @ E @ B )
% 1.56/1.01 => ( ! [Xy: $i] :
% 1.56/1.01 ( ( in @ Xy @ E )
% 1.56/1.01 => ( in @ Xy @ D ) )
% 1.56/1.01 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(descr_type,type,
% 1.56/1.01 descr: ( $i > $o ) > $i ).
% 1.56/1.01
% 1.56/1.01 thf(descrp_type,type,
% 1.56/1.01 descrp: $o ).
% 1.56/1.01
% 1.56/1.01 thf(descrp,definition,
% 1.56/1.01 ( descrp
% 1.56/1.01 = ( ! [Xphi: $i > $o] :
% 1.56/1.01 ( ( exu
% 1.56/1.01 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.56/1.01 => ( Xphi
% 1.56/1.01 @ ( descr
% 1.56/1.01 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(dsetconstr_type,type,
% 1.56/1.01 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.56/1.01
% 1.56/1.01 thf(dsetconstrI_type,type,
% 1.56/1.01 dsetconstrI: $o ).
% 1.56/1.01
% 1.56/1.01 thf(dsetconstrI,definition,
% 1.56/1.01 ( dsetconstrI
% 1.56/1.01 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ A )
% 1.56/1.01 => ( ( Xphi @ Xx )
% 1.56/1.01 => ( in @ Xx
% 1.56/1.01 @ ( dsetconstr @ A
% 1.56/1.01 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(dsetconstrEL_type,type,
% 1.56/1.01 dsetconstrEL: $o ).
% 1.56/1.01
% 1.56/1.01 thf(dsetconstrEL,definition,
% 1.56/1.01 ( dsetconstrEL
% 1.56/1.01 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.56/1.01 ( ( in @ Xx
% 1.56/1.01 @ ( dsetconstr @ A
% 1.56/1.01 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.56/1.01 => ( in @ Xx @ A ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(dsetconstrER_type,type,
% 1.56/1.01 dsetconstrER: $o ).
% 1.56/1.01
% 1.56/1.01 thf(dsetconstrER,definition,
% 1.56/1.01 ( dsetconstrER
% 1.56/1.01 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.56/1.01 ( ( in @ Xx
% 1.56/1.01 @ ( dsetconstr @ A
% 1.56/1.01 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.56/1.01 => ( Xphi @ Xx ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(exuE1_type,type,
% 1.56/1.01 exuE1: $o ).
% 1.56/1.01
% 1.56/1.01 thf(exuE1,definition,
% 1.56/1.01 ( exuE1
% 1.56/1.01 = ( ! [Xphi: $i > $o] :
% 1.56/1.01 ( ( exu
% 1.56/1.01 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.56/1.01 => ? [Xx: $i] :
% 1.56/1.01 ( ( Xphi @ Xx )
% 1.56/1.01 & ! [Xy: $i] :
% 1.56/1.01 ( ( Xphi @ Xy )
% 1.56/1.01 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(prop2set_type,type,
% 1.56/1.01 prop2set: $o > $i ).
% 1.56/1.01
% 1.56/1.01 thf(prop2setE_type,type,
% 1.56/1.01 prop2setE: $o ).
% 1.56/1.01
% 1.56/1.01 thf(prop2setE,definition,
% 1.56/1.01 ( prop2setE
% 1.56/1.01 = ( ! [Xphi: $o,Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 1.56/1.01 => Xphi ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(emptysetE_type,type,
% 1.56/1.01 emptysetE: $o ).
% 1.56/1.01
% 1.56/1.01 thf(emptysetE,definition,
% 1.56/1.01 ( emptysetE
% 1.56/1.01 = ( ! [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ emptyset )
% 1.56/1.01 => ! [Xphi: $o] : Xphi ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(emptysetimpfalse_type,type,
% 1.56/1.01 emptysetimpfalse: $o ).
% 1.56/1.01
% 1.56/1.01 thf(emptysetimpfalse,definition,
% 1.56/1.01 ( emptysetimpfalse
% 1.56/1.01 = ( ! [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ emptyset )
% 1.56/1.01 => $false ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(notinemptyset_type,type,
% 1.56/1.01 notinemptyset: $o ).
% 1.56/1.01
% 1.56/1.01 thf(notinemptyset,definition,
% 1.56/1.01 ( notinemptyset
% 1.56/1.01 = ( ! [Xx: $i] :
% 1.56/1.01 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(exuE3e_type,type,
% 1.56/1.01 exuE3e: $o ).
% 1.56/1.01
% 1.56/1.01 thf(exuE3e,definition,
% 1.56/1.01 ( exuE3e
% 1.56/1.01 = ( ! [Xphi: $i > $o] :
% 1.56/1.01 ( ( exu
% 1.56/1.01 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.56/1.01 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(setext_type,type,
% 1.56/1.01 setext: $o ).
% 1.56/1.01
% 1.56/1.01 thf(setext,definition,
% 1.56/1.01 ( setext
% 1.56/1.01 = ( ! [A: $i,B: $i] :
% 1.56/1.01 ( ! [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ A )
% 1.56/1.01 => ( in @ Xx @ B ) )
% 1.56/1.01 => ( ! [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ B )
% 1.56/1.01 => ( in @ Xx @ A ) )
% 1.56/1.01 => ( A = B ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(emptyI_type,type,
% 1.56/1.01 emptyI: $o ).
% 1.56/1.01
% 1.56/1.01 thf(emptyI,definition,
% 1.56/1.01 ( emptyI
% 1.56/1.01 = ( ! [A: $i] :
% 1.56/1.01 ( ! [Xx: $i] :
% 1.56/1.01 ~ ( in @ Xx @ A )
% 1.56/1.01 => ( A = emptyset ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(noeltsimpempty_type,type,
% 1.56/1.01 noeltsimpempty: $o ).
% 1.56/1.01
% 1.56/1.01 thf(noeltsimpempty,definition,
% 1.56/1.01 ( noeltsimpempty
% 1.56/1.01 = ( ! [A: $i] :
% 1.56/1.01 ( ! [Xx: $i] :
% 1.56/1.01 ~ ( in @ Xx @ A )
% 1.56/1.01 => ( A = emptyset ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(setbeta_type,type,
% 1.56/1.01 setbeta: $o ).
% 1.56/1.01
% 1.56/1.01 thf(setbeta,definition,
% 1.56/1.01 ( setbeta
% 1.56/1.01 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ A )
% 1.56/1.01 => ( ( in @ Xx
% 1.56/1.01 @ ( dsetconstr @ A
% 1.56/1.01 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.56/1.01 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(nonempty_type,type,
% 1.56/1.01 nonempty: $i > $o ).
% 1.56/1.01
% 1.56/1.01 thf(nonempty,definition,
% 1.56/1.01 ( nonempty
% 1.56/1.01 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(nonemptyE1_type,type,
% 1.56/1.01 nonemptyE1: $o ).
% 1.56/1.01
% 1.56/1.01 thf(nonemptyE1,definition,
% 1.56/1.01 ( nonemptyE1
% 1.56/1.01 = ( ! [A: $i] :
% 1.56/1.01 ( ( nonempty @ A )
% 1.56/1.01 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(nonemptyI_type,type,
% 1.56/1.01 nonemptyI: $o ).
% 1.56/1.01
% 1.56/1.01 thf(nonemptyI,definition,
% 1.56/1.01 ( nonemptyI
% 1.56/1.01 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ A )
% 1.56/1.01 => ( ( Xphi @ Xx )
% 1.56/1.01 => ( nonempty
% 1.56/1.01 @ ( dsetconstr @ A
% 1.56/1.01 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(nonemptyI1_type,type,
% 1.56/1.01 nonemptyI1: $o ).
% 1.56/1.01
% 1.56/1.01 thf(nonemptyI1,definition,
% 1.56/1.01 ( nonemptyI1
% 1.56/1.01 = ( ! [A: $i] :
% 1.56/1.01 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.56/1.01 => ( nonempty @ A ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(setadjoinIL_type,type,
% 1.56/1.01 setadjoinIL: $o ).
% 1.56/1.01
% 1.56/1.01 thf(setadjoinIL,definition,
% 1.56/1.01 ( setadjoinIL
% 1.56/1.01 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(emptyinunitempty_type,type,
% 1.56/1.01 emptyinunitempty: $o ).
% 1.56/1.01
% 1.56/1.01 thf(emptyinunitempty,definition,
% 1.56/1.01 ( emptyinunitempty
% 1.56/1.01 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(setadjoinIR_type,type,
% 1.56/1.01 setadjoinIR: $o ).
% 1.56/1.01
% 1.56/1.01 thf(setadjoinIR,definition,
% 1.56/1.01 ( setadjoinIR
% 1.56/1.01 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.56/1.01 ( ( in @ Xy @ A )
% 1.56/1.01 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(setadjoinE_type,type,
% 1.56/1.01 setadjoinE: $o ).
% 1.56/1.01
% 1.56/1.01 thf(setadjoinE,definition,
% 1.56/1.01 ( setadjoinE
% 1.56/1.01 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.56/1.01 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.56/1.01 => ! [Xphi: $o] :
% 1.56/1.01 ( ( ( Xy = Xx )
% 1.56/1.01 => Xphi )
% 1.56/1.01 => ( ( ( in @ Xy @ A )
% 1.56/1.01 => Xphi )
% 1.56/1.01 => Xphi ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(setadjoinOr_type,type,
% 1.56/1.01 setadjoinOr: $o ).
% 1.56/1.01
% 1.56/1.01 thf(setadjoinOr,definition,
% 1.56/1.01 ( setadjoinOr
% 1.56/1.01 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.56/1.01 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.56/1.01 => ( ( Xy = Xx )
% 1.56/1.01 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(setoftrueEq_type,type,
% 1.56/1.01 setoftrueEq: $o ).
% 1.56/1.01
% 1.56/1.01 thf(setoftrueEq,definition,
% 1.56/1.01 ( setoftrueEq
% 1.56/1.01 = ( ! [A: $i] :
% 1.56/1.01 ( ( dsetconstr @ A
% 1.56/1.01 @ ^ [Xx: $i] : $true )
% 1.56/1.01 = A ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(powersetI_type,type,
% 1.56/1.01 powersetI: $o ).
% 1.56/1.01
% 1.56/1.01 thf(powersetI,definition,
% 1.56/1.01 ( powersetI
% 1.56/1.01 = ( ! [A: $i,B: $i] :
% 1.56/1.01 ( ! [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ B )
% 1.56/1.01 => ( in @ Xx @ A ) )
% 1.56/1.01 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(emptyinPowerset_type,type,
% 1.56/1.01 emptyinPowerset: $o ).
% 1.56/1.01
% 1.56/1.01 thf(emptyinPowerset,definition,
% 1.56/1.01 ( emptyinPowerset
% 1.56/1.01 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(emptyInPowerset_type,type,
% 1.56/1.01 emptyInPowerset: $o ).
% 1.56/1.01
% 1.56/1.01 thf(emptyInPowerset,definition,
% 1.56/1.01 ( emptyInPowerset
% 1.56/1.01 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(powersetE_type,type,
% 1.56/1.01 powersetE: $o ).
% 1.56/1.01
% 1.56/1.01 thf(powersetE,definition,
% 1.56/1.01 ( powersetE
% 1.56/1.01 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.56/1.01 ( ( in @ B @ ( powerset @ A ) )
% 1.56/1.01 => ( ( in @ Xx @ B )
% 1.56/1.01 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(setunionI_type,type,
% 1.56/1.01 setunionI: $o ).
% 1.56/1.01
% 1.56/1.01 thf(setunionI,definition,
% 1.56/1.01 ( setunionI
% 1.56/1.01 = ( ! [A: $i,Xx: $i,B: $i] :
% 1.56/1.01 ( ( in @ Xx @ B )
% 1.56/1.01 => ( ( in @ B @ A )
% 1.56/1.01 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(setunionE_type,type,
% 1.56/1.01 setunionE: $o ).
% 1.56/1.01
% 1.56/1.01 thf(setunionE,definition,
% 1.56/1.01 ( setunionE
% 1.56/1.01 = ( ! [A: $i,Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ ( setunion @ A ) )
% 1.56/1.01 => ! [Xphi: $o] :
% 1.56/1.01 ( ! [B: $i] :
% 1.56/1.01 ( ( in @ Xx @ B )
% 1.56/1.01 => ( ( in @ B @ A )
% 1.56/1.01 => Xphi ) )
% 1.56/1.01 => Xphi ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(subPowSU_type,type,
% 1.56/1.01 subPowSU: $o ).
% 1.56/1.01
% 1.56/1.01 thf(subPowSU,definition,
% 1.56/1.01 ( subPowSU
% 1.56/1.01 = ( ! [A: $i,Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ A )
% 1.56/1.01 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(exuE2_type,type,
% 1.56/1.01 exuE2: $o ).
% 1.56/1.01
% 1.56/1.01 thf(exuE2,definition,
% 1.56/1.01 ( exuE2
% 1.56/1.01 = ( ! [Xphi: $i > $o] :
% 1.56/1.01 ( ( exu
% 1.56/1.01 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.56/1.01 => ? [Xx: $i] :
% 1.56/1.01 ! [Xy: $i] :
% 1.56/1.01 ( ( Xphi @ Xy )
% 1.56/1.01 <=> ( Xy = Xx ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(nonemptyImpWitness_type,type,
% 1.56/1.01 nonemptyImpWitness: $o ).
% 1.56/1.01
% 1.56/1.01 thf(nonemptyImpWitness,definition,
% 1.56/1.01 ( nonemptyImpWitness
% 1.56/1.01 = ( ! [A: $i] :
% 1.56/1.01 ( ( nonempty @ A )
% 1.56/1.01 => ? [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ A )
% 1.56/1.01 & $true ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(uniqinunit_type,type,
% 1.56/1.01 uniqinunit: $o ).
% 1.56/1.01
% 1.56/1.01 thf(uniqinunit,definition,
% 1.56/1.01 ( uniqinunit
% 1.56/1.01 = ( ! [Xx: $i,Xy: $i] :
% 1.56/1.01 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 1.56/1.01 => ( Xx = Xy ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(notinsingleton_type,type,
% 1.56/1.01 notinsingleton: $o ).
% 1.56/1.01
% 1.56/1.01 thf(notinsingleton,definition,
% 1.56/1.01 ( notinsingleton
% 1.56/1.01 = ( ! [Xx: $i,Xy: $i] :
% 1.56/1.01 ( ( Xx != Xy )
% 1.56/1.01 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(eqinunit_type,type,
% 1.56/1.01 eqinunit: $o ).
% 1.56/1.01
% 1.56/1.01 thf(eqinunit,definition,
% 1.56/1.01 ( eqinunit
% 1.56/1.01 = ( ! [Xx: $i,Xy: $i] :
% 1.56/1.01 ( ( Xx = Xy )
% 1.56/1.01 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(singletonsswitch_type,type,
% 1.56/1.01 singletonsswitch: $o ).
% 1.56/1.01
% 1.56/1.01 thf(singletonsswitch,definition,
% 1.56/1.01 ( singletonsswitch
% 1.56/1.01 = ( ! [Xx: $i,Xy: $i] :
% 1.56/1.01 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 1.56/1.01 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(upairsetE_type,type,
% 1.56/1.01 upairsetE: $o ).
% 1.56/1.01
% 1.56/1.01 thf(upairsetE,definition,
% 1.56/1.01 ( upairsetE
% 1.56/1.01 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 1.56/1.01 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 1.56/1.01 => ( ( Xz = Xx )
% 1.56/1.01 | ( Xz = Xy ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(upairsetIL_type,type,
% 1.56/1.01 upairsetIL: $o ).
% 1.56/1.01
% 1.56/1.01 thf(upairsetIL,definition,
% 1.56/1.01 ( upairsetIL
% 1.56/1.01 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(upairsetIR_type,type,
% 1.56/1.01 upairsetIR: $o ).
% 1.56/1.01
% 1.56/1.01 thf(upairsetIR,definition,
% 1.56/1.01 ( upairsetIR
% 1.56/1.01 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(emptyE1_type,type,
% 1.56/1.01 emptyE1: $o ).
% 1.56/1.01
% 1.56/1.01 thf(emptyE1,definition,
% 1.56/1.01 ( emptyE1
% 1.56/1.01 = ( ! [A: $i,Xphi: $i > $o] :
% 1.56/1.01 ( ? [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ A )
% 1.56/1.01 & ( Xphi @ Xx ) )
% 1.56/1.01 => ( ( ( dsetconstr @ A
% 1.56/1.01 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.56/1.01 = emptyset )
% 1.56/1.01 => $false ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(vacuousDall_type,type,
% 1.56/1.01 vacuousDall: $o ).
% 1.56/1.01
% 1.56/1.01 thf(vacuousDall,definition,
% 1.56/1.01 ( vacuousDall
% 1.56/1.01 = ( ! [Xphi: $i > $o,Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ emptyset )
% 1.56/1.01 => ( Xphi @ Xx ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(quantDeMorgan1_type,type,
% 1.56/1.01 quantDeMorgan1: $o ).
% 1.56/1.01
% 1.56/1.01 thf(quantDeMorgan1,definition,
% 1.56/1.01 ( quantDeMorgan1
% 1.56/1.01 = ( ! [A: $i,Xphi: $i > $o] :
% 1.56/1.01 ( ~ ! [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ A )
% 1.56/1.01 => ( Xphi @ Xx ) )
% 1.56/1.01 => ? [Xx: $i] :
% 1.56/1.01 ( ( in @ Xx @ A )
% 1.56/1.01 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.56/1.01
% 1.56/1.01 thf(quantDeMorgan2_type,type,
% 1.56/1.01 quantDeMorgan2: $o ).
% 1.56/1.02
% 1.56/1.02 thf(quantDeMorgan2,definition,
% 1.56/1.02 ( quantDeMorgan2
% 1.56/1.02 = ( ! [A: $i,Xphi: $i > $o] :
% 1.56/1.02 ( ! [Xx: $i] :
% 1.56/1.02 ( ( in @ Xx @ A )
% 1.56/1.02 => ~ ( Xphi @ Xx ) )
% 1.56/1.02 => ~ ? [Xx: $i] :
% 1.56/1.02 ( ( in @ Xx @ A )
% 1.56/1.02 & ( Xphi @ Xx ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(quantDeMorgan3_type,type,
% 1.56/1.02 quantDeMorgan3: $o ).
% 1.56/1.02
% 1.56/1.02 thf(quantDeMorgan3,definition,
% 1.56/1.02 ( quantDeMorgan3
% 1.56/1.02 = ( ! [A: $i,Xphi: $i > $o] :
% 1.56/1.02 ( ~ ? [Xx: $i] :
% 1.56/1.02 ( ( in @ Xx @ A )
% 1.56/1.02 & ( Xphi @ Xx ) )
% 1.56/1.02 => ! [Xx: $i] :
% 1.56/1.02 ( ( in @ Xx @ A )
% 1.56/1.02 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(quantDeMorgan4_type,type,
% 1.56/1.02 quantDeMorgan4: $o ).
% 1.56/1.02
% 1.56/1.02 thf(quantDeMorgan4,definition,
% 1.56/1.02 ( quantDeMorgan4
% 1.56/1.02 = ( ! [A: $i,Xphi: $i > $o] :
% 1.56/1.02 ( ? [Xx: $i] :
% 1.56/1.02 ( ( in @ Xx @ A )
% 1.56/1.02 & ~ ( Xphi @ Xx ) )
% 1.56/1.02 => ~ ! [Xx: $i] :
% 1.56/1.02 ( ( in @ Xx @ A )
% 1.56/1.02 => ( Xphi @ Xx ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(prop2setI_type,type,
% 1.56/1.02 prop2setI: $o ).
% 1.56/1.02
% 1.56/1.02 thf(prop2setI,definition,
% 1.56/1.02 ( prop2setI
% 1.56/1.02 = ( ! [Xphi: $o] :
% 1.56/1.02 ( Xphi
% 1.56/1.02 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(set2prop_type,type,
% 1.56/1.02 set2prop: $i > $o ).
% 1.56/1.02
% 1.56/1.02 thf(prop2set2propI_type,type,
% 1.56/1.02 prop2set2propI: $o ).
% 1.56/1.02
% 1.56/1.02 thf(prop2set2propI,definition,
% 1.56/1.02 ( prop2set2propI
% 1.56/1.02 = ( ! [Xphi: $o] :
% 1.56/1.02 ( Xphi
% 1.56/1.02 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(notdexE_type,type,
% 1.56/1.02 notdexE: $o ).
% 1.56/1.02
% 1.56/1.02 thf(notdexE,definition,
% 1.56/1.02 ( notdexE
% 1.56/1.02 = ( ! [A: $i,Xphi: $i > $o] :
% 1.56/1.02 ( ~ ? [Xx: $i] :
% 1.56/1.02 ( ( in @ Xx @ A )
% 1.56/1.02 & ( Xphi @ Xx ) )
% 1.56/1.02 => ! [Xx: $i] :
% 1.56/1.02 ( ( in @ Xx @ A )
% 1.56/1.02 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(notdallE_type,type,
% 1.56/1.02 notdallE: $o ).
% 1.56/1.02
% 1.56/1.02 thf(notdallE,definition,
% 1.56/1.02 ( notdallE
% 1.56/1.02 = ( ! [A: $i,Xphi: $i > $o] :
% 1.56/1.02 ( ~ ! [Xx: $i] :
% 1.56/1.02 ( ( in @ Xx @ A )
% 1.56/1.02 => ( Xphi @ Xx ) )
% 1.56/1.02 => ? [Xx: $i] :
% 1.56/1.02 ( ( in @ Xx @ A )
% 1.56/1.02 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(exuI1_type,type,
% 1.56/1.02 exuI1: $o ).
% 1.56/1.02
% 1.56/1.02 thf(exuI1,definition,
% 1.56/1.02 ( exuI1
% 1.56/1.02 = ( ! [Xphi: $i > $o] :
% 1.56/1.02 ( ? [Xx: $i] :
% 1.56/1.02 ( ( Xphi @ Xx )
% 1.56/1.02 & ! [Xy: $i] :
% 1.56/1.02 ( ( Xphi @ Xy )
% 1.56/1.02 => ( Xx = Xy ) ) )
% 1.56/1.02 => ( exu
% 1.56/1.02 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(exuI3_type,type,
% 1.56/1.02 exuI3: $o ).
% 1.56/1.02
% 1.56/1.02 thf(exuI3,definition,
% 1.56/1.02 ( exuI3
% 1.56/1.02 = ( ! [Xphi: $i > $o] :
% 1.56/1.02 ( ? [Xx: $i] : ( Xphi @ Xx )
% 1.56/1.02 => ( ! [Xx: $i,Xy: $i] :
% 1.56/1.02 ( ( Xphi @ Xx )
% 1.56/1.02 => ( ( Xphi @ Xy )
% 1.56/1.02 => ( Xx = Xy ) ) )
% 1.56/1.02 => ( exu
% 1.56/1.02 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(exuI2_type,type,
% 1.56/1.02 exuI2: $o ).
% 1.56/1.02
% 1.56/1.02 thf(exuI2,definition,
% 1.56/1.02 ( exuI2
% 1.56/1.02 = ( ! [Xphi: $i > $o] :
% 1.56/1.02 ( ? [Xx: $i] :
% 1.56/1.02 ! [Xy: $i] :
% 1.56/1.02 ( ( Xphi @ Xy )
% 1.56/1.02 <=> ( Xy = Xx ) )
% 1.56/1.02 => ( exu
% 1.56/1.02 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(inCongP_type,type,
% 1.56/1.02 inCongP: $o ).
% 1.56/1.02
% 1.56/1.02 thf(inCongP,definition,
% 1.56/1.02 ( inCongP
% 1.56/1.02 = ( ! [A: $i,B: $i] :
% 1.56/1.02 ( ( A = B )
% 1.56/1.02 => ! [Xx: $i,Xy: $i] :
% 1.56/1.02 ( ( Xx = Xy )
% 1.56/1.02 => ( ( in @ Xx @ A )
% 1.56/1.02 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(in__Cong_type,type,
% 1.56/1.02 in__Cong: $o ).
% 1.56/1.02
% 1.56/1.02 thf(in__Cong,definition,
% 1.56/1.02 ( in__Cong
% 1.56/1.02 = ( ! [A: $i,B: $i] :
% 1.56/1.02 ( ( A = B )
% 1.56/1.02 => ! [Xx: $i,Xy: $i] :
% 1.56/1.02 ( ( Xx = Xy )
% 1.56/1.02 => ( ( in @ Xx @ A )
% 1.56/1.02 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(exuE3u_type,type,
% 1.56/1.02 exuE3u: $o ).
% 1.56/1.02
% 1.56/1.02 thf(exuE3u,definition,
% 1.56/1.02 ( exuE3u
% 1.56/1.02 = ( ! [Xphi: $i > $o] :
% 1.56/1.02 ( ( exu
% 1.56/1.02 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.56/1.02 => ! [Xx: $i,Xy: $i] :
% 1.56/1.02 ( ( Xphi @ Xx )
% 1.56/1.02 => ( ( Xphi @ Xy )
% 1.56/1.02 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(exu__Cong_type,type,
% 1.56/1.02 exu__Cong: $o ).
% 1.56/1.02
% 1.56/1.02 thf(exu__Cong,definition,
% 1.56/1.02 ( exu__Cong
% 1.56/1.02 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 1.56/1.02 ( ! [Xx: $i,Xy: $i] :
% 1.56/1.02 ( ( Xx = Xy )
% 1.56/1.02 => ( ( Xphi @ Xx )
% 1.56/1.02 <=> ( Xpsi @ Xy ) ) )
% 1.56/1.02 => ( ( exu
% 1.56/1.02 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.56/1.02 <=> ( exu
% 1.56/1.02 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(emptyset__Cong_type,type,
% 1.56/1.02 emptyset__Cong: $o ).
% 1.56/1.02
% 1.56/1.02 thf(emptyset__Cong,definition,
% 1.56/1.02 ( emptyset__Cong
% 1.56/1.02 = ( emptyset = emptyset ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(setadjoin__Cong_type,type,
% 1.56/1.02 setadjoin__Cong: $o ).
% 1.56/1.02
% 1.56/1.02 thf(setadjoin__Cong,definition,
% 1.56/1.02 ( setadjoin__Cong
% 1.56/1.02 = ( ! [Xx: $i,Xy: $i] :
% 1.56/1.02 ( ( Xx = Xy )
% 1.56/1.02 => ! [Xz: $i,Xu: $i] :
% 1.56/1.02 ( ( Xz = Xu )
% 1.56/1.02 => ( ( setadjoin @ Xx @ Xz )
% 1.56/1.02 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(powerset__Cong_type,type,
% 1.56/1.02 powerset__Cong: $o ).
% 1.56/1.02
% 1.56/1.02 thf(powerset__Cong,definition,
% 1.56/1.02 ( powerset__Cong
% 1.56/1.02 = ( ! [A: $i,B: $i] :
% 1.56/1.02 ( ( A = B )
% 1.56/1.02 => ( ( powerset @ A )
% 1.56/1.02 = ( powerset @ B ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(setunion__Cong_type,type,
% 1.56/1.02 setunion__Cong: $o ).
% 1.56/1.02
% 1.56/1.02 thf(setunion__Cong,definition,
% 1.56/1.02 ( setunion__Cong
% 1.56/1.02 = ( ! [A: $i,B: $i] :
% 1.56/1.02 ( ( A = B )
% 1.56/1.02 => ( ( setunion @ A )
% 1.56/1.02 = ( setunion @ B ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(omega__Cong_type,type,
% 1.56/1.02 omega__Cong: $o ).
% 1.56/1.02
% 1.56/1.02 thf(omega__Cong,definition,
% 1.56/1.02 ( omega__Cong
% 1.56/1.02 = ( omega = omega ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(exuEu_type,type,
% 1.56/1.02 exuEu: $o ).
% 1.56/1.02
% 1.56/1.02 thf(exuEu,definition,
% 1.56/1.02 ( exuEu
% 1.56/1.02 = ( ! [Xphi: $i > $o] :
% 1.56/1.02 ( ( exu
% 1.56/1.02 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.56/1.02 => ! [Xx: $i,Xy: $i] :
% 1.56/1.02 ( ( Xphi @ Xx )
% 1.56/1.02 => ( ( Xphi @ Xy )
% 1.56/1.02 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(descr__Cong_type,type,
% 1.56/1.02 descr__Cong: $o ).
% 1.56/1.02
% 1.56/1.02 thf(descr__Cong,definition,
% 1.56/1.02 ( descr__Cong
% 1.56/1.02 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 1.56/1.02 ( ! [Xx: $i,Xy: $i] :
% 1.56/1.02 ( ( Xx = Xy )
% 1.56/1.02 => ( ( Xphi @ Xx )
% 1.56/1.02 <=> ( Xpsi @ Xy ) ) )
% 1.56/1.02 => ( ( exu
% 1.56/1.02 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.56/1.02 => ( ( exu
% 1.56/1.02 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 1.56/1.02 => ( ( descr
% 1.56/1.02 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.56/1.02 = ( descr
% 1.56/1.02 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(dsetconstr__Cong_type,type,
% 1.56/1.02 dsetconstr__Cong: $o ).
% 1.56/1.02
% 1.56/1.02 thf(dsetconstr__Cong,definition,
% 1.56/1.02 ( dsetconstr__Cong
% 1.56/1.02 = ( ! [A: $i,B: $i] :
% 1.56/1.02 ( ( A = B )
% 1.56/1.02 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 1.56/1.02 ( ! [Xx: $i] :
% 1.56/1.02 ( ( in @ Xx @ A )
% 1.56/1.02 => ! [Xy: $i] :
% 1.56/1.02 ( ( in @ Xy @ B )
% 1.56/1.02 => ( ( Xx = Xy )
% 1.56/1.02 => ( ( Xphi @ Xx )
% 1.56/1.02 <=> ( Xpsi @ Xy ) ) ) ) )
% 1.56/1.02 => ( ( dsetconstr @ A
% 1.56/1.02 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.56/1.02 = ( dsetconstr @ B
% 1.56/1.02 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(subset_type,type,
% 1.56/1.02 subset: $i > $i > $o ).
% 1.56/1.02
% 1.56/1.02 thf(disjoint_type,type,
% 1.56/1.02 disjoint: $i > $i > $o ).
% 1.56/1.02
% 1.56/1.02 thf(setsmeet_type,type,
% 1.56/1.02 setsmeet: $i > $i > $o ).
% 1.56/1.02
% 1.56/1.02 thf(subsetI1_type,type,
% 1.56/1.02 subsetI1: $o ).
% 1.56/1.02
% 1.56/1.02 thf(subsetI1,definition,
% 1.56/1.02 ( subsetI1
% 1.56/1.02 = ( ! [A: $i,B: $i] :
% 1.56/1.02 ( ! [Xx: $i] :
% 1.56/1.02 ( ( in @ Xx @ A )
% 1.56/1.02 => ( in @ Xx @ B ) )
% 1.56/1.02 => ( subset @ A @ B ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(eqimpsubset2_type,type,
% 1.56/1.02 eqimpsubset2: $o ).
% 1.56/1.02
% 1.56/1.02 thf(eqimpsubset2,definition,
% 1.56/1.02 ( eqimpsubset2
% 1.56/1.02 = ( ! [A: $i,B: $i] :
% 1.56/1.02 ( ( A = B )
% 1.56/1.02 => ( subset @ B @ A ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(eqimpsubset1_type,type,
% 1.56/1.02 eqimpsubset1: $o ).
% 1.56/1.02
% 1.56/1.02 thf(eqimpsubset1,definition,
% 1.56/1.02 ( eqimpsubset1
% 1.56/1.02 = ( ! [A: $i,B: $i] :
% 1.56/1.02 ( ( A = B )
% 1.56/1.02 => ( subset @ A @ B ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(subsetI2_type,type,
% 1.56/1.02 subsetI2: $o ).
% 1.56/1.02
% 1.56/1.02 thf(subsetI2,definition,
% 1.56/1.02 ( subsetI2
% 1.56/1.02 = ( ! [A: $i,B: $i] :
% 1.56/1.02 ( ! [Xx: $i] :
% 1.56/1.02 ( ( in @ Xx @ A )
% 1.56/1.02 => ( in @ Xx @ B ) )
% 1.56/1.02 => ( subset @ A @ B ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(emptysetsubset_type,type,
% 1.56/1.02 emptysetsubset: $o ).
% 1.56/1.02
% 1.56/1.02 thf(emptysetsubset,definition,
% 1.56/1.02 ( emptysetsubset
% 1.56/1.02 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(subsetE_type,type,
% 1.56/1.02 subsetE: $o ).
% 1.56/1.02
% 1.56/1.02 thf(subsetE,definition,
% 1.56/1.02 ( subsetE
% 1.56/1.02 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.56/1.02 ( ( subset @ A @ B )
% 1.56/1.02 => ( ( in @ Xx @ A )
% 1.56/1.02 => ( in @ Xx @ B ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(subsetE2_type,type,
% 1.56/1.02 subsetE2: $o ).
% 1.56/1.02
% 1.56/1.02 thf(subsetE2,definition,
% 1.56/1.02 ( subsetE2
% 1.56/1.02 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.56/1.02 ( ( subset @ A @ B )
% 1.56/1.02 => ( ~ ( in @ Xx @ B )
% 1.56/1.02 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(notsubsetI_type,type,
% 1.56/1.02 notsubsetI: $o ).
% 1.56/1.02
% 1.56/1.02 thf(notsubsetI,definition,
% 1.56/1.02 ( notsubsetI
% 1.56/1.02 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.56/1.02 ( ( in @ Xx @ A )
% 1.56/1.02 => ( ~ ( in @ Xx @ B )
% 1.56/1.02 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 1.56/1.02
% 1.56/1.02 thf(notequalI1,conjecture,
% 1.56/1.02 ( setextAx
% 1.56/1.02 => ( emptysetAx
% 1.56/1.02 => ( setadjoinAx
% 1.56/1.02 => ( powersetAx
% 1.56/1.02 => ( setunionAx
% 1.56/1.02 => ( omega0Ax
% 1.56/1.02 => ( omegaSAx
% 1.56/1.02 => ( omegaIndAx
% 1.56/1.02 => ( replAx
% 1.56/1.02 => ( foundationAx
% 1.56/1.02 => ( wellorderingAx
% 1.56/1.02 => ( descrp
% 1.56/1.02 => ( dsetconstrI
% 1.56/1.02 => ( dsetconstrEL
% 1.56/1.02 => ( dsetconstrER
% 1.56/1.02 => ( exuE1
% 1.56/1.02 => ( prop2setE
% 1.56/1.02 => ( emptysetE
% 1.56/1.02 => ( emptysetimpfalse
% 1.56/1.02 => ( notinemptyset
% 1.56/1.02 => ( exuE3e
% 1.56/1.02 => ( setext
% 1.56/1.02 => ( emptyI
% 1.56/1.02 => ( noeltsimpempty
% 1.56/1.02 => ( setbeta
% 1.56/1.02 => ( nonemptyE1
% 1.56/1.02 => ( nonemptyI
% 1.56/1.02 => ( nonemptyI1
% 1.56/1.02 => ( setadjoinIL
% 1.56/1.02 => ( emptyinunitempty
% 1.56/1.02 => ( setadjoinIR
% 1.56/1.02 => ( setadjoinE
% 1.56/1.02 => ( setadjoinOr
% 1.56/1.02 => ( setoftrueEq
% 1.56/1.02 => ( powersetI
% 1.56/1.02 => ( emptyinPowerset
% 1.56/1.02 => ( emptyInPowerset
% 1.56/1.02 => ( powersetE
% 1.56/1.02 => ( setunionI
% 1.56/1.02 => ( setunionE
% 1.56/1.02 => ( subPowSU
% 1.56/1.02 => ( exuE2
% 1.56/1.02 => ( nonemptyImpWitness
% 1.56/1.02 => ( uniqinunit
% 1.56/1.02 => ( notinsingleton
% 1.56/1.02 => ( eqinunit
% 1.56/1.02 => ( singletonsswitch
% 1.56/1.02 => ( upairsetE
% 1.56/1.02 => ( upairsetIL
% 1.56/1.02 => ( upairsetIR
% 1.56/1.02 => ( emptyE1
% 1.56/1.02 => ( vacuousDall
% 1.56/1.02 => ( quantDeMorgan1
% 1.56/1.02 => ( quantDeMorgan2
% 1.56/1.02 => ( quantDeMorgan3
% 1.56/1.02 => ( quantDeMorgan4
% 1.56/1.02 => ( prop2setI
% 1.56/1.02 => ( prop2set2propI
% 1.56/1.02 => ( notdexE
% 1.56/1.02 => ( notdallE
% 1.56/1.02 => ( exuI1
% 1.56/1.02 => ( exuI3
% 1.56/1.02 => ( exuI2
% 1.56/1.02 => ( inCongP
% 1.56/1.02 => ( in__Cong
% 1.56/1.02 => ( exuE3u
% 1.56/1.02 => ( exu__Cong
% 1.56/1.02 => ( emptyset__Cong
% 1.56/1.02 => ( setadjoin__Cong
% 1.56/1.02 => ( powerset__Cong
% 1.56/1.02 => ( setunion__Cong
% 1.56/1.02 => ( omega__Cong
% 1.56/1.02 => ( exuEu
% 1.56/1.02 => ( descr__Cong
% 1.56/1.02 => ( dsetconstr__Cong
% 1.56/1.02 => ( subsetI1
% 1.56/1.02 => ( eqimpsubset2
% 1.56/1.02 => ( eqimpsubset1
% 1.56/1.02 => ( subsetI2
% 1.56/1.02 => ( emptysetsubset
% 1.56/1.02 => ( subsetE
% 1.56/1.02 => ( subsetE2
% 1.56/1.02 => ( notsubsetI
% 1.56/1.02 => ! [A: $i,B: $i] :
% 1.56/1.02 ( ~ ( subset @ A @ B )
% 1.56/1.02 => ( A != B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 1.56/1.02 ------------------------
% 1.56/1.03 % (17827)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_17640 for (2999ds/4Mi)
% 1.56/1.03 % (17829)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_17640 for (2999ds/2Mi)
% 1.56/1.03 % (17830)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_17640 for (2999ds/2Mi)
% 1.56/1.03 % (17828)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_17640 for (2999ds/27Mi)
% 1.56/1.03 % (17826)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_17640 for (2999ds/183Mi)
% 1.56/1.03 % (17831)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_17640 for (2999ds/275Mi)
% 1.56/1.03 % (17832)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_17640 for (2999ds/18Mi)
% 1.56/1.03 % (17829)Instruction limit reached!
% 1.56/1.03 % (17829)------------------------------
% 1.56/1.03 % (17829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.56/1.03 % (17829)Termination reason: Unknown
% 1.56/1.03 % (17829)Termination phase: shuffling
% 1.56/1.03
% 1.56/1.03 % (17829)Memory used [KB]: 1151
% 1.56/1.03 % (17829)Time elapsed: 0.002 s
% 1.56/1.03 % (17829)Instructions burned: 3 (million)
% 1.56/1.03 % (17829)------------------------------
% 1.56/1.03 % (17829)------------------------------
% 1.56/1.03 % (17830)Instruction limit reached!
% 1.56/1.03 % (17830)------------------------------
% 1.56/1.03 % (17830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.56/1.03 % (17830)Termination reason: Unknown
% 1.56/1.03 % (17830)Termination phase: shuffling
% 1.56/1.03
% 1.56/1.03 % (17830)Memory used [KB]: 1151
% 1.56/1.03 % (17830)Time elapsed: 0.002 s
% 1.56/1.03 % (17830)Instructions burned: 3 (million)
% 1.56/1.03 % (17830)------------------------------
% 1.56/1.03 % (17830)------------------------------
% 1.56/1.03 % (17827)Instruction limit reached!
% 1.56/1.03 % (17827)------------------------------
% 1.56/1.03 % (17827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.56/1.03 % (17827)Termination reason: Unknown
% 1.56/1.03 % (17827)Termination phase: shuffling
% 1.56/1.03
% 1.56/1.03 % (17827)Memory used [KB]: 1279
% 1.56/1.03 % (17827)Time elapsed: 0.003 s
% 1.56/1.03 % (17827)Instructions burned: 7 (million)
% 1.56/1.03 % (17827)------------------------------
% 1.56/1.03 % (17827)------------------------------
% 1.56/1.03 % (17832)Instruction limit reached!
% 1.56/1.03 % (17832)------------------------------
% 1.56/1.03 % (17832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.56/1.03 % (17832)Termination reason: Unknown
% 1.56/1.03 % (17832)Termination phase: Preprocessing 3
% 1.56/1.03
% 1.56/1.03 % (17832)Memory used [KB]: 1663
% 1.56/1.03 % (17832)Time elapsed: 0.007 s
% 1.56/1.03 % (17832)Instructions burned: 20 (million)
% 1.56/1.03 % (17832)------------------------------
% 1.56/1.03 % (17832)------------------------------
% 1.56/1.04 % (17828)Instruction limit reached!
% 1.56/1.04 % (17828)------------------------------
% 1.56/1.04 % (17828)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.56/1.04 % (17828)Termination reason: Unknown
% 1.56/1.04 % (17828)Termination phase: Function definition elimination
% 1.56/1.04
% 1.56/1.04 % (17828)Memory used [KB]: 1407
% 1.56/1.04 % (17828)Time elapsed: 0.009 s
% 1.56/1.04 % (17828)Instructions burned: 30 (million)
% 1.56/1.04 % (17828)------------------------------
% 1.56/1.04 % (17828)------------------------------
% 1.56/1.04 % (17833)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_17640 for (2999ds/3Mi)
% 1.56/1.04 % (17834)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_17640 for (2999ds/37Mi)
% 1.56/1.04 % (17835)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_17640 for (2999ds/15Mi)
% 1.56/1.04 % (17833)Instruction limit reached!
% 1.56/1.04 % (17833)------------------------------
% 1.56/1.04 % (17833)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.56/1.04 % (17833)Termination reason: Unknown
% 1.56/1.04 % (17833)Termination phase: shuffling
% 1.56/1.04
% 1.56/1.04 % (17833)Memory used [KB]: 1151
% 1.56/1.04 % (17833)Time elapsed: 0.002 s
% 1.56/1.04 % (17833)Instructions burned: 4 (million)
% 1.56/1.04 % (17833)------------------------------
% 1.56/1.04 % (17833)------------------------------
% 1.56/1.04 % (17836)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_17640 for (2999ds/3Mi)
% 1.56/1.04 % (17835)Instruction limit reached!
% 1.56/1.04 % (17835)------------------------------
% 1.56/1.04 % (17835)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.56/1.04 % (17835)Termination reason: Unknown
% 1.56/1.04 % (17835)Termination phase: Preprocessing 3
% 1.56/1.04
% 1.56/1.04 % (17835)Memory used [KB]: 1535
% 1.56/1.04 % (17835)Time elapsed: 0.006 s
% 1.56/1.04 % (17835)Instructions burned: 17 (million)
% 1.56/1.04 % (17835)------------------------------
% 1.56/1.04 % (17835)------------------------------
% 1.56/1.04 % (17836)Instruction limit reached!
% 1.56/1.04 % (17836)------------------------------
% 1.56/1.04 % (17836)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.56/1.04 % (17836)Termination reason: Unknown
% 1.56/1.04 % (17836)Termination phase: shuffling
% 1.56/1.04
% 1.56/1.04 % (17836)Memory used [KB]: 1151
% 1.56/1.04 % (17836)Time elapsed: 0.002 s
% 1.56/1.04 % (17836)Instructions burned: 3 (million)
% 1.56/1.04 % (17836)------------------------------
% 1.56/1.04 % (17836)------------------------------
% 1.56/1.05 % (17837)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_17640 for (2999ds/1041Mi)
% 1.56/1.05 % (17834)Instruction limit reached!
% 1.56/1.05 % (17834)------------------------------
% 1.56/1.05 % (17834)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.56/1.05 % (17834)Termination reason: Unknown
% 1.56/1.05 % (17834)Termination phase: Property scanning
% 1.56/1.05
% 1.56/1.05 % (17834)Memory used [KB]: 1407
% 1.56/1.05 % (17834)Time elapsed: 0.011 s
% 1.56/1.05 % (17834)Instructions burned: 40 (million)
% 1.56/1.05 % (17834)------------------------------
% 1.56/1.05 % (17834)------------------------------
% 1.56/1.05 % (17838)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_17640 for (2999ds/7Mi)
% 1.56/1.05 % (17838)Instruction limit reached!
% 1.56/1.05 % (17838)------------------------------
% 1.56/1.05 % (17838)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.56/1.05 % (17838)Termination reason: Unknown
% 1.56/1.05 % (17838)Termination phase: shuffling
% 1.56/1.05
% 1.56/1.05 % (17838)Memory used [KB]: 1151
% 1.56/1.05 % (17838)Time elapsed: 0.003 s
% 1.56/1.05 % (17838)Instructions burned: 7 (million)
% 1.56/1.05 % (17838)------------------------------
% 1.56/1.05 % (17838)------------------------------
% 1.56/1.05 % (17839)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_17640 for (2999ds/16Mi)
% 1.56/1.05 % (17840)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_17640 for (2999ds/3Mi)
% 1.56/1.06 % (17840)Instruction limit reached!
% 1.56/1.06 % (17840)------------------------------
% 1.56/1.06 % (17840)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.56/1.06 % (17840)Termination reason: Unknown
% 1.56/1.06 % (17840)Termination phase: shuffling
% 1.56/1.06
% 1.56/1.06 % (17840)Memory used [KB]: 1151
% 1.56/1.06 % (17840)Time elapsed: 0.002 s
% 1.56/1.06 % (17840)Instructions burned: 3 (million)
% 1.56/1.06 % (17840)------------------------------
% 1.56/1.06 % (17840)------------------------------
% 1.56/1.06 % (17831)First to succeed.
% 1.56/1.06 % (17831)Refutation found. Thanks to Tanya!
% 1.56/1.06 % SZS status Theorem for DTF2THF_17640
% 1.56/1.06 % SZS output start Proof for DTF2THF_17640
% 1.56/1.06 thf(func_def_0, type, in: $i > $i > $o).
% 1.56/1.06 thf(func_def_1, type, exu: ($i > $o) > $o).
% 1.56/1.06 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 1.56/1.06 thf(func_def_8, type, powerset: $i > $i).
% 1.56/1.06 thf(func_def_10, type, setunion: $i > $i).
% 1.56/1.06 thf(func_def_19, type, descr: ($i > $o) > $i).
% 1.56/1.06 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 1.56/1.06 thf(func_def_26, type, prop2set: $o > $i).
% 1.56/1.06 thf(func_def_36, type, nonempty: $i > $o).
% 1.56/1.06 thf(func_def_69, type, set2prop: $i > $o).
% 1.56/1.06 thf(func_def_88, type, subset: $i > $i > $o).
% 1.56/1.06 thf(func_def_89, type, disjoint: $i > $i > $o).
% 1.56/1.06 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 1.56/1.06 thf(func_def_112, type, sP1: $i > $i > $o).
% 1.56/1.06 thf(func_def_113, type, sP2: $i > $o).
% 1.56/1.06 thf(func_def_114, type, sP3: $i > $i > $o).
% 1.56/1.06 thf(func_def_115, type, sP4: $i > $i > $o).
% 1.56/1.06 thf(func_def_118, type, sK7: $i > $i > $i).
% 1.56/1.06 thf(func_def_119, type, sK8: $i > $i > $i).
% 1.56/1.06 thf(func_def_120, type, sK9: $i > $i > $i).
% 1.56/1.06 thf(func_def_123, type, sK12: $i > $i).
% 1.56/1.06 thf(func_def_128, type, sK17: $i > $o).
% 1.56/1.06 thf(func_def_130, type, sK19: $i > ($i > $o) > $i).
% 1.56/1.06 thf(func_def_135, type, sK24: ($i > $o) > ($i > $o) > $i).
% 1.56/1.06 thf(func_def_136, type, sK25: ($i > $o) > ($i > $o) > $i).
% 1.56/1.06 thf(func_def_137, type, sK26: $i > $o).
% 1.56/1.06 thf(func_def_138, type, sK27: $i > $o).
% 1.56/1.06 thf(func_def_139, type, sK28: $i > $o).
% 1.56/1.06 thf(func_def_145, type, sK34: $i > $o).
% 1.56/1.06 thf(func_def_158, type, sK47: $i > $o).
% 1.56/1.06 thf(func_def_163, type, sK52: ($i > $o) > ($i > $o) > $i).
% 1.56/1.06 thf(func_def_164, type, sK53: ($i > $o) > ($i > $o) > $i).
% 1.56/1.06 thf(func_def_165, type, sK54: $i > $o).
% 1.56/1.06 thf(func_def_166, type, sK55: $i > $o).
% 1.56/1.06 thf(func_def_167, type, sK56: $i > ($i > $o) > $i).
% 1.56/1.06 thf(func_def_168, type, sK57: $i > $o).
% 1.56/1.06 thf(func_def_170, type, sK59: $i > $i > $o).
% 1.56/1.06 thf(func_def_172, type, sK61: $i > $i).
% 1.56/1.06 thf(func_def_173, type, sK62: $i > $i).
% 1.56/1.06 thf(func_def_174, type, sK63: $i > ($i > $i > $o) > $i).
% 1.56/1.06 thf(func_def_175, type, sK64: $i > ($i > $i > $o) > $i).
% 1.56/1.06 thf(func_def_176, type, sK65: $i > $i > ($i > $i > $o) > $i).
% 1.56/1.06 thf(func_def_183, type, sK72: $i > $i > $i).
% 1.56/1.06 thf(func_def_184, type, sK73: $i > $i > $i).
% 1.56/1.06 thf(func_def_187, type, sK76: $i > $i).
% 1.56/1.06 thf(func_def_191, type, sK80: $i > $i).
% 1.56/1.06 thf(func_def_194, type, sK83: $i > $o).
% 1.56/1.06 thf(func_def_197, type, sK86: $i > $o).
% 1.56/1.06 thf(func_def_200, type, sK89: $i > $o).
% 1.56/1.06 thf(func_def_207, type, sK96: ($i > $o) > $i).
% 1.56/1.06 thf(func_def_208, type, sK97: ($i > $o) > $i).
% 1.56/1.06 thf(func_def_209, type, sK98: $i > $o).
% 1.56/1.06 thf(func_def_219, type, sK108: $i > $o).
% 1.56/1.06 thf(func_def_221, type, sK110: $i > $o).
% 1.56/1.06 thf(func_def_224, type, sK113: $i > ($i > $o) > $i).
% 1.56/1.06 thf(func_def_226, type, sK115: $i > $o).
% 1.56/1.06 thf(func_def_232, type, sK121: $i > $o).
% 1.56/1.06 thf(func_def_233, type, sK122: $i > $i > $i).
% 1.56/1.06 thf(func_def_239, type, sK128: $i > $o).
% 1.56/1.06 thf(func_def_241, type, sK130: ($i > $o) > $i > $i).
% 1.56/1.06 thf(func_def_243, type, sK132: $i > $o).
% 1.56/1.06 thf(func_def_245, type, sK134: ($i > $o) > $i > $i).
% 1.56/1.06 thf(func_def_247, type, sK136: $i > $o).
% 1.56/1.06 thf(func_def_251, type, sK140: $i > $o).
% 1.56/1.06 thf(func_def_253, type, sK142: ($i > $o) > $i > $i).
% 1.56/1.06 thf(func_def_261, type, sK150: $o > $i > $i > $i).
% 1.56/1.06 thf(func_def_264, type, sK153: $i > $o).
% 1.56/1.06 thf(func_def_265, type, sK154: $i > $o).
% 1.56/1.06 thf(func_def_266, type, sK155: ($i > $o) > ($i > $o) > $i > $i > $i).
% 1.56/1.06 thf(func_def_267, type, sK156: ($i > $o) > ($i > $o) > $i > $i > $i).
% 1.56/1.06 thf(func_def_276, type, sK165: $i > $i).
% 1.56/1.06 thf(func_def_279, type, sK168: $i > ($i > $o) > $i).
% 1.56/1.06 thf(func_def_280, type, sK169: $i > $o).
% 1.56/1.06 thf(func_def_283, type, sK172: $i > $i > $i).
% 1.56/1.06 thf(func_def_284, type, sK173: $i > $i > $i).
% 1.56/1.06 thf(func_def_285, type, sK174: $i > $i > $i).
% 1.56/1.06 thf(func_def_286, type, sK175: $i > $i > $i).
% 1.56/1.06 thf(func_def_287, type, sK176: $i > $i > $i > $i).
% 1.56/1.06 thf(func_def_288, type, sK177: $i > $i).
% 1.56/1.06 thf(func_def_289, type, sK178: $i > $i).
% 1.56/1.06 thf(func_def_290, type, sK179: $i > $i).
% 1.56/1.06 thf(func_def_291, type, sK180: $i > $i).
% 1.56/1.06 thf(func_def_292, type, sK181: $i > $i > $i > $i).
% 1.56/1.06 thf(func_def_293, type, sK182: $i > $i > $i > $i).
% 1.56/1.06 thf(func_def_294, type, sK183: $i > $i > $i).
% 1.56/1.06 thf(func_def_295, type, sK184: $i > $i > $i).
% 1.56/1.06 thf(func_def_296, type, sK185: $i > $i > $i).
% 1.56/1.06 thf(func_def_297, type, sK186: $i > $i).
% 1.56/1.06 thf(func_def_298, type, sK187: $i > $i > $i).
% 1.56/1.06 thf(func_def_300, type, sK189: $i > $i).
% 1.56/1.06 thf(func_def_301, type, sK190: $i > $i).
% 1.56/1.06 thf(func_def_304, type, sK193: $i > $o).
% 1.56/1.06 thf(func_def_305, type, sK194: $i > $i).
% 1.56/1.06 thf(func_def_306, type, sK195: ($i > $o) > $i).
% 1.56/1.06 thf(func_def_307, type, sK196: $i > $i).
% 1.56/1.06 thf(func_def_309, type, sK198: $i > $i).
% 1.56/1.06 thf(func_def_311, type, sK200: $i > $i > $i).
% 1.56/1.06 thf(func_def_315, type, sK204: ($i > $o) > $i).
% 1.56/1.06 thf(func_def_316, type, sK205: $i > $o).
% 1.56/1.06 thf(func_def_317, type, sK206: $i > $i).
% 1.56/1.06 thf(func_def_323, type, sK212: ($i > $o) > $i > $i).
% 1.56/1.06 thf(func_def_325, type, sK214: $i > $o).
% 1.56/1.06 thf(func_def_330, type, sK219: $i > $i > $i).
% 1.56/1.06 thf(func_def_343, type, sK232: ($i > $o) > $i).
% 1.56/1.06 thf(func_def_344, type, sK233: $i > $o).
% 1.56/1.06 thf(func_def_346, type, sK235: $i > $i).
% 1.56/1.06 thf(func_def_353, type, ph242: !>[X0: $tType]:(X0)).
% 1.56/1.06 thf(f1730,plain,(
% 1.56/1.06 $false),
% 1.56/1.06 inference(subsumption_resolution,[status(thm)],[f1354,f1719])).
% 1.56/1.06 thf(f1719,plain,(
% 1.56/1.06 ( ! [X3 : $i] : (((subset @ X3 @ X3) = $true)) )),
% 1.56/1.06 inference(trivial_inequality_removal,[status(thm)],[f1553])).
% 1.56/1.06 thf(f1553,plain,(
% 1.56/1.06 ( ! [X3 : $i] : (($true != $true) | ((subset @ X3 @ X3) = $true)) )),
% 1.56/1.06 inference(equality_resolution,[status(thm)],[f1443])).
% 1.56/1.06 thf(f1443,plain,(
% 1.56/1.06 ( ! [X2 : $i,X3 : $i] : ((X2 != X3) | ($true = (subset @ X2 @ X3)) | ($true != $true)) )),
% 1.56/1.06 inference(definition_unfolding,[status(thm)],[f1098,f941])).
% 1.56/1.06 thf(f941,plain,(
% 1.56/1.06 (eqimpsubset2 = $true)),
% 1.56/1.06 inference(cnf_transformation,[status(thm)],[f543])).
% 1.56/1.06 thf(f543,plain,(
% 1.56/1.06 (exuEu = $true) & (upairsetE = $true) & ((sK102 = sK103) & ((subset @ sK102 @ sK103) != $true)) & (exuE3u = $true) & (prop2setI = $true) & (emptyinPowerset = $true) & (setadjoinE = $true) & (in__Cong = $true) & (emptyset__Cong = $true) & (powersetE = $true) & (powerset__Cong = $true) & (notdexE = $true) & (setadjoin__Cong = $true) & (subPowSU = $true) & (notdallE = $true) & (setext = $true) & (exuE2 = $true) & (descr__Cong = $true) & (setadjoinIR = $true) & (setextAx = $true) & (vacuousDall = $true) & (emptyInPowerset = $true) & (setoftrueEq = $true) & (omega0Ax = $true) & (dsetconstrI = $true) & (nonemptyI = $true) & (setunionI = $true) & (exuE3e = $true) & (nonemptyI1 = $true) & (nonemptyE1 = $true) & (setbeta = $true) & (notsubsetI = $true) & (descrp = $true) & (exuI3 = $true) & (setunion__Cong = $true) & (omegaSAx = $true) & (emptyinunitempty = $true) & (uniqinunit = $true) & (setadjoinOr = $true) & (emptysetimpfalse = $true) & (emptysetsubset = $true) & (subsetE = $true) & (setadjoinAx = $true) & (powersetAx = $true) & (emptysetAx = $true) & (exuI2 = $true) & (notinsingleton = $true) & (exuE1 = $true) & (upairsetIR = $true) & (eqimpsubset1 = $true) & (notinemptyset = $true) & (singletonsswitch = $true) & (eqinunit = $true) & (prop2setE = $true) & (replAx = $true) & (setunionE = $true) & (setunionAx = $true) & (powersetI = $true) & (quantDeMorgan1 = $true) & (quantDeMorgan3 = $true) & (prop2set2propI = $true) & (nonemptyImpWitness = $true) & (subsetI1 = $true) & (subsetI2 = $true) & (setadjoinIL = $true) & (upairsetIL = $true) & (exuI1 = $true) & (exu__Cong = $true) & (noeltsimpempty = $true) & (dsetconstrEL = $true) & (eqimpsubset2 = $true) & (dsetconstr__Cong = $true) & (wellorderingAx = $true) & (dsetconstrER = $true) & (inCongP = $true) & (quantDeMorgan2 = $true) & (quantDeMorgan4 = $true) & (emptysetE = $true) & (omega__Cong = $true) & (emptyI = $true) & (foundationAx = $true) & (omegaIndAx = $true) & (emptyE1 = $true) & (subsetE2 = $true)),
% 1.56/1.06 inference(skolemisation,[status(esa),new_symbols(skolem,[sK102,sK103])],[f364,f542])).
% 1.56/1.06 thf(f542,plain,(
% 1.56/1.06 ? [X0 : $i,X1 : $i] : ((X0 = X1) & ((subset @ X0 @ X1) != $true)) => ((sK102 = sK103) & ((subset @ sK102 @ sK103) != $true))),
% 1.56/1.06 introduced(definition,[],[choice_axiom])).
% 1.56/1.06 thf(f364,plain,(
% 1.56/1.06 (exuEu = $true) & (upairsetE = $true) & ? [X0 : $i,X1 : $i] : ((X0 = X1) & ((subset @ X0 @ X1) != $true)) & (exuE3u = $true) & (prop2setI = $true) & (emptyinPowerset = $true) & (setadjoinE = $true) & (in__Cong = $true) & (emptyset__Cong = $true) & (powersetE = $true) & (powerset__Cong = $true) & (notdexE = $true) & (setadjoin__Cong = $true) & (subPowSU = $true) & (notdallE = $true) & (setext = $true) & (exuE2 = $true) & (descr__Cong = $true) & (setadjoinIR = $true) & (setextAx = $true) & (vacuousDall = $true) & (emptyInPowerset = $true) & (setoftrueEq = $true) & (omega0Ax = $true) & (dsetconstrI = $true) & (nonemptyI = $true) & (setunionI = $true) & (exuE3e = $true) & (nonemptyI1 = $true) & (nonemptyE1 = $true) & (setbeta = $true) & (notsubsetI = $true) & (descrp = $true) & (exuI3 = $true) & (setunion__Cong = $true) & (omegaSAx = $true) & (emptyinunitempty = $true) & (uniqinunit = $true) & (setadjoinOr = $true) & (emptysetimpfalse = $true) & (emptysetsubset = $true) & (subsetE = $true) & (setadjoinAx = $true) & (powersetAx = $true) & (emptysetAx = $true) & (exuI2 = $true) & (notinsingleton = $true) & (exuE1 = $true) & (upairsetIR = $true) & (eqimpsubset1 = $true) & (notinemptyset = $true) & (singletonsswitch = $true) & (eqinunit = $true) & (prop2setE = $true) & (replAx = $true) & (setunionE = $true) & (setunionAx = $true) & (powersetI = $true) & (quantDeMorgan1 = $true) & (quantDeMorgan3 = $true) & (prop2set2propI = $true) & (nonemptyImpWitness = $true) & (subsetI1 = $true) & (subsetI2 = $true) & (setadjoinIL = $true) & (upairsetIL = $true) & (exuI1 = $true) & (exu__Cong = $true) & (noeltsimpempty = $true) & (dsetconstrEL = $true) & (eqimpsubset2 = $true) & (dsetconstr__Cong = $true) & (wellorderingAx = $true) & (dsetconstrER = $true) & (inCongP = $true) & (quantDeMorgan2 = $true) & (quantDeMorgan4 = $true) & (emptysetE = $true) & (omega__Cong = $true) & (emptyI = $true) & (foundationAx = $true) & (omegaIndAx = $true) & (emptyE1 = $true) & (subsetE2 = $true)),
% 1.56/1.06 inference(flattening,[status(thm)],[f363])).
% 1.56/1.06 thf(f363,plain,(
% 1.56/1.06 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : ((X0 = X1) & ((subset @ X0 @ X1) != $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (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)),
% 1.56/1.06 inference(ennf_transformation,[status(thm)],[f255])).
% 1.56/1.06 thf(f255,plain,(
% 1.56/1.06 ~((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) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ! [X0 : $i,X1 : $i] : (((subset @ X0 @ X1) != $true) => (X0 != X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 1.56/1.06 inference(flattening,[status(thm)],[f90])).
% 1.56/1.06 thf(f90,plain,(
% 1.56/1.06 ~((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) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ! [X0 : $i,X1 : $i] : (~((subset @ X0 @ X1) = $true) => (X0 != X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 1.56/1.06 inference(fool_elimination,[status(thm)],[f89])).
% 1.56/1.06 thf(f89,plain,(
% 1.56/1.06 ~(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 => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => ! [X0 : $i,X1 : $i] : (~(subset @ X0 @ X1) => (X0 != X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 1.56/1.06 inference(rectify,[status(thm)],[f87])).
% 1.56/1.06 thf(f87,negated_conjecture,(
% 1.56/1.06 ~(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 => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => ! [X3 : $i,X4 : $i] : (~(subset @ X3 @ X4) => (X3 != X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 1.56/1.06 inference(negated_conjecture,[status(cth)],[f86])).
% 1.56/1.06 thf(f86,conjecture,(
% 1.56/1.06 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 => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => ! [X3 : $i,X4 : $i] : (~(subset @ X3 @ X4) => (X3 != X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 1.56/1.06 file('/export/starexec/sandbox2/tmp/tmp.HAHRJ9zahH/DTF2THF_17640.p',notequalI1)).
% 1.56/1.06 thf(f1098,plain,(
% 1.56/1.06 ( ! [X2 : $i,X3 : $i] : ((X2 != X3) | ($true = (subset @ X2 @ X3)) | (eqimpsubset2 != $true)) )),
% 1.56/1.06 inference(cnf_transformation,[status(thm)],[f648])).
% 1.56/1.06 thf(f648,plain,(
% 1.56/1.06 ((eqimpsubset2 = $true) | ((sK162 = sK163) & ((subset @ sK162 @ sK163) != $true))) & (! [X2 : $i,X3 : $i] : ((X2 != X3) | ($true = (subset @ X2 @ X3))) | (eqimpsubset2 != $true))),
% 1.56/1.06 inference(skolemisation,[status(esa),new_symbols(skolem,[sK162,sK163])],[f646,f647])).
% 1.56/1.06 thf(f647,plain,(
% 1.56/1.06 ? [X0 : $i,X1 : $i] : ((X0 = X1) & ((subset @ X0 @ X1) != $true)) => ((sK162 = sK163) & ((subset @ sK162 @ sK163) != $true))),
% 1.56/1.06 introduced(definition,[],[choice_axiom])).
% 1.56/1.06 thf(f646,plain,(
% 1.56/1.06 ((eqimpsubset2 = $true) | ? [X0 : $i,X1 : $i] : ((X0 = X1) & ((subset @ X0 @ X1) != $true))) & (! [X2 : $i,X3 : $i] : ((X2 != X3) | ($true = (subset @ X2 @ X3))) | (eqimpsubset2 != $true))),
% 1.56/1.06 inference(rectify,[status(thm)],[f645])).
% 1.56/1.06 thf(f645,plain,(
% 1.56/1.06 ((eqimpsubset2 = $true) | ? [X0 : $i,X1 : $i] : ((X0 = X1) & ((subset @ X0 @ X1) != $true))) & (! [X0 : $i,X1 : $i] : ((X0 != X1) | ((subset @ X0 @ X1) = $true)) | (eqimpsubset2 != $true))),
% 1.56/1.06 inference(nnf_transformation,[status(thm)],[f306])).
% 1.56/1.06 thf(f306,plain,(
% 1.56/1.06 (eqimpsubset2 = $true) <=> ! [X0 : $i,X1 : $i] : ((X0 != X1) | ((subset @ X0 @ X1) = $true))),
% 1.56/1.06 inference(ennf_transformation,[status(thm)],[f129])).
% 1.56/1.06 thf(f129,plain,(
% 1.56/1.06 ! [X0 : $i,X1 : $i] : ((X0 = X1) => ((subset @ X0 @ X1) = $true)) <=> (eqimpsubset2 = $true)),
% 1.56/1.06 inference(fool_elimination,[status(thm)],[f128])).
% 1.56/1.06 thf(f128,plain,(
% 1.56/1.06 (! [X0 : $i,X1 : $i] : ((X0 = X1) => (subset @ X0 @ X1)) = eqimpsubset2)),
% 1.56/1.06 inference(rectify,[status(thm)],[f79])).
% 1.56/1.06 thf(f79,axiom,(
% 1.56/1.06 (! [X4 : $i,X3 : $i] : ((X3 = X4) => (subset @ X4 @ X3)) = eqimpsubset2)),
% 1.56/1.06 file('/export/starexec/sandbox2/tmp/tmp.HAHRJ9zahH/DTF2THF_17640.p',eqimpsubset2)).
% 1.56/1.06 thf(f1354,plain,(
% 1.56/1.06 ((subset @ sK103 @ sK103) != $true)),
% 1.56/1.06 inference(definition_unfolding,[status(thm)],[f1009,f1010])).
% 1.56/1.06 thf(f1010,plain,(
% 1.56/1.06 (sK102 = sK103)),
% 1.56/1.06 inference(cnf_transformation,[status(thm)],[f543])).
% 1.56/1.06 thf(f1009,plain,(
% 1.56/1.06 ((subset @ sK102 @ sK103) != $true)),
% 1.56/1.06 inference(cnf_transformation,[status(thm)],[f543])).
% 1.56/1.06 % SZS output end Proof for DTF2THF_17640
% 1.56/1.06 % (17831)------------------------------
% 1.56/1.06 % (17831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.56/1.06 % (17831)Termination reason: Refutation
% 1.56/1.06
% 1.56/1.06 % (17831)Memory used [KB]: 6908
% 1.56/1.06 % (17831)Time elapsed: 0.029 s
% 1.56/1.06 % (17831)Instructions burned: 89 (million)
% 1.56/1.06 % (17831)------------------------------
% 1.56/1.06 % (17831)------------------------------
% 1.56/1.06 % (17825)Success in time 0.036 s
%------------------------------------------------------------------------------