%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU556^1 : 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 : n017.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 09:35:15 AM UTC 2026
% Result : Theorem 2.04s 1.47s
% Output : Refutation 2.04s
% Verified :
% SZS Type : Refutation
% Derivation depth : 21
% Number of leaves : 4
% Syntax : Number of formulae : 33 ( 19 unt; 0 typ; 0 def)
% Number of atoms : 1495 ( 435 equ; 0 cnn)
% Maximal formula atoms : 76 ( 45 avg)
% Number of connectives : 753 ( 12 ~; 4 |; 243 &; 95 @)
% ( 0 <=>; 388 =>; 0 <=; 0 <~>)
% Maximal formula depth : 81 ( 20 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 13 ( 13 >; 0 *; 0 +; 0 <<)
% Number of symbols : 158 ( 153 usr; 79 con; 0-3 aty)
% ( 7 !!; 4 ??; 0 @@+; 0 @@-)
% Number of variables : 66 ( 30 ^; 21 !; 15 ?; 66 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_101,type,
sK0: $i > $o ).
thf(func_def_105,type,
sK4: $i > $i > $i ).
thf(func_def_106,type,
sK5: $i > ( $i > $i > $o ) > $i ).
thf(func_def_107,type,
sK6: $i > ( $i > $i > $o ) > $i ).
thf(func_def_108,type,
sK7: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_109,type,
sK8: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_110,type,
sK9: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_111,type,
sK10: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_112,type,
sK11: ( $i > $o ) > $i > $i ).
thf(func_def_113,type,
sK12: ( $i > $o ) > $i > $i ).
thf(func_def_114,type,
sK13: $i > $i ).
thf(func_def_115,type,
sK14: $i > $i ).
thf(func_def_116,type,
sK15: $i > $i ).
thf(func_def_117,type,
sK16: $i > $i > $i ).
thf(func_def_118,type,
sK17: $i > $i > $i ).
thf(func_def_119,type,
sK18: $i > $i > $i ).
thf(func_def_120,type,
sK19: $i > $i > $i ).
thf(func_def_121,type,
sK20: $i > $i > $i > $i ).
thf(func_def_122,type,
sK21: $o > $i > $i > $i ).
thf(func_def_123,type,
sK22: ( $i > $o ) > $i ).
thf(func_def_124,type,
sK23: ( $i > $o ) > $i > $i ).
thf(func_def_125,type,
sK24: ( $i > $o ) > $i > $i ).
thf(func_def_126,type,
sK25: $i > $i > $i ).
thf(func_def_127,type,
sK26: ( $i > $o ) > $i ).
thf(func_def_128,type,
sK27: ( $i > $o ) > $i > $i ).
thf(func_def_129,type,
sK28: $i > $i ).
thf(func_def_130,type,
sK29: ( $i > $o ) > $i > $i ).
thf(func_def_131,type,
sK30: ( $i > $o ) > $i > $i ).
thf(func_def_132,type,
sK31: $i > ( $i > $o ) > $i ).
thf(func_def_133,type,
sK32: ( $i > $o ) > $i ).
thf(func_def_134,type,
sK33: ( $i > $o ) > $i > $i ).
thf(func_def_135,type,
sK34: ( $i > $o ) > $i > $i ).
thf(func_def_136,type,
sK35: $i > $i ).
thf(func_def_137,type,
sK36: $i > $i ).
thf(func_def_139,type,
sK38: $i > $i > $i ).
thf(func_def_140,type,
sK39: ( $i > $o ) > $i > $i ).
thf(func_def_141,type,
sK40: ( $i > $o ) > $i > $i ).
thf(func_def_142,type,
sK41: $i > $i > $i ).
thf(func_def_143,type,
sK42: ( $i > $o ) > $i ).
thf(func_def_144,type,
sK43: ( $i > $o ) > $i > $i ).
thf(func_def_145,type,
sK44: ( $i > $o ) > $i > $i ).
thf(func_def_146,type,
sK45: $i > $i ).
thf(func_def_147,type,
sK46: ( $i > $o ) > $i > $i ).
thf(func_def_148,type,
sK47: ( $i > $o ) > $i ).
thf(func_def_149,type,
sK48: ( $i > $o ) > $i ).
thf(func_def_150,type,
sK49: ( $i > $o ) > $i ).
thf(func_def_151,type,
sK50: $i > $i > $i ).
thf(func_def_152,type,
sK51: $i > $i > $i ).
thf(func_def_153,type,
sK52: $i > $i > $i ).
thf(func_def_154,type,
sK53: ( $i > $o ) > $i ).
thf(func_def_155,type,
sK54: $i > ( $i > $o ) > $i ).
thf(func_def_156,type,
sK55: $i > ( $i > $o ) > $i ).
thf(func_def_157,type,
sK56: ( $i > $o ) > $i ).
thf(func_def_158,type,
sK57: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_159,type,
sK58: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_160,type,
sK59: $i > ( $i > $o ) > $i ).
thf(func_def_161,type,
sK60: $i > ( $i > $o ) > $i ).
thf(func_def_162,type,
sK61: $i > ( $i > $o ) > $i ).
thf(func_def_163,type,
sK62: $i > ( $i > $o ) > $i ).
thf(func_def_164,type,
sK63: $i > ( $i > $o ) > $i ).
thf(func_def_165,type,
sK64: $i > ( $i > $o ) > $i ).
thf(func_def_166,type,
sK65: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_167,type,
sK66: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_168,type,
sK67: $i > ( $i > $o ) > $i ).
thf(func_def_169,type,
sK68: ( $i > $o ) > $i ).
thf(func_def_170,type,
sK69: ( $i > $o ) > $i ).
thf(func_def_171,type,
sK70: ( $i > $o ) > $i ).
thf(func_def_172,type,
sK71: ( $i > $o ) > $i ).
thf(f1528,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1526,f288]) ).
thf(f288,plain,
sK2 != sK1,
inference(cnf_transformation,[status(thm)],[f227]) ).
thf(f227,plain,
( ( emptysetE = $true )
& ( setadjoinIR = $true )
& ( emptyinPowerset = $true )
& ( emptyE1 = $true )
& ( setext = $true )
& ( subPowSU = $true )
& ( exu__Cong = $true )
& ( notdallE = $true )
& ( uniqinunit = $true )
& ( powersetAx = $true )
& ( exuE3u = $true )
& ( replAx = $true )
& ( vacuousDall = $true )
& ( noeltsimpempty = $true )
& ( setbeta = $true )
& ( emptysetimpfalse = $true )
& ( setadjoinIL = $true )
& ( omegaSAx = $true )
& ( upairsetIR = $true )
& ( setadjoinOr = $true )
& ( setadjoin__Cong = $true )
& ( in__Cong = $true )
& ( exuI2 = $true )
& ( omega__Cong = $true )
& ( $true
= ( exu
@ ^ [Y0: $i] : ( sK0 @ Y0 ) ) )
& ( sK2 != sK1 )
& ( $true
= ( sK0 @ sK1 ) )
& ( $true
= ( sK0 @ sK2 ) )
& ( quantDeMorgan3 = $true )
& ( upairsetE = $true )
& ( quantDeMorgan2 = $true )
& ( emptyset__Cong = $true )
& ( powerset__Cong = $true )
& ( nonemptyI = $true )
& ( setunionI = $true )
& ( powersetI = $true )
& ( setunionE = $true )
& ( omegaIndAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( emptyInPowerset = $true )
& ( exuI1 = $true )
& ( wellorderingAx = $true )
& ( singletonsswitch = $true )
& ( nonemptyE1 = $true )
& ( quantDeMorgan4 = $true )
& ( setadjoinAx = $true )
& ( powersetE = $true )
& ( setunion__Cong = $true )
& ( notdexE = $true )
& ( exuI3 = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( upairsetIL = $true )
& ( notinsingleton = $true )
& ( emptysetAx = $true )
& ( dsetconstrI = $true )
& ( dsetconstrER = $true )
& ( nonemptyImpWitness = $true )
& ( prop2setE = $true )
& ( dsetconstrEL = $true )
& ( emptyI = $true )
& ( foundationAx = $true )
& ( inCongP = $true )
& ( setoftrueEq = $true )
& ( prop2setI = $true )
& ( prop2set2propI = $true )
& ( setextAx = $true )
& ( descrp = $true )
& ( quantDeMorgan1 = $true )
& ( emptyinunitempty = $true )
& ( exuE1 = $true )
& ( nonemptyI1 = $true )
& ( setadjoinE = $true )
& ( eqinunit = $true )
& ( exuE2 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f224,f226,f225]) ).
thf(f225,plain,
( ? [X0: $i > $o] :
( ( ( exu
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
= $true )
& ? [X1: $i,X2: $i] :
( ( X1 != X2 )
& ( ( X0 @ X1 )
= $true )
& ( ( X0 @ X2 )
= $true ) ) )
=> ( ( $true
= ( exu
@ ^ [Y0: $i] : ( sK0 @ Y0 ) ) )
& ? [X2: $i,X1: $i] :
( ( X1 != X2 )
& ( $true
= ( sK0 @ X1 ) )
& ( $true
= ( sK0 @ X2 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f226,plain,
( ? [X2: $i,X1: $i] :
( ( X1 != X2 )
& ( $true
= ( sK0 @ X1 ) )
& ( $true
= ( sK0 @ X2 ) ) )
=> ( ( sK2 != sK1 )
& ( $true
= ( sK0 @ sK1 ) )
& ( $true
= ( sK0 @ sK2 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f224,plain,
( ( emptysetE = $true )
& ( setadjoinIR = $true )
& ( emptyinPowerset = $true )
& ( emptyE1 = $true )
& ( setext = $true )
& ( subPowSU = $true )
& ( exu__Cong = $true )
& ( notdallE = $true )
& ( uniqinunit = $true )
& ( powersetAx = $true )
& ( exuE3u = $true )
& ( replAx = $true )
& ( vacuousDall = $true )
& ( noeltsimpempty = $true )
& ( setbeta = $true )
& ( emptysetimpfalse = $true )
& ( setadjoinIL = $true )
& ( omegaSAx = $true )
& ( upairsetIR = $true )
& ( setadjoinOr = $true )
& ( setadjoin__Cong = $true )
& ( in__Cong = $true )
& ( exuI2 = $true )
& ( omega__Cong = $true )
& ? [X0: $i > $o] :
( ( ( exu
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
= $true )
& ? [X1: $i,X2: $i] :
( ( X1 != X2 )
& ( ( X0 @ X1 )
= $true )
& ( ( X0 @ X2 )
= $true ) ) )
& ( quantDeMorgan3 = $true )
& ( upairsetE = $true )
& ( quantDeMorgan2 = $true )
& ( emptyset__Cong = $true )
& ( powerset__Cong = $true )
& ( nonemptyI = $true )
& ( setunionI = $true )
& ( powersetI = $true )
& ( setunionE = $true )
& ( omegaIndAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( emptyInPowerset = $true )
& ( exuI1 = $true )
& ( wellorderingAx = $true )
& ( singletonsswitch = $true )
& ( nonemptyE1 = $true )
& ( quantDeMorgan4 = $true )
& ( setadjoinAx = $true )
& ( powersetE = $true )
& ( setunion__Cong = $true )
& ( notdexE = $true )
& ( exuI3 = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( upairsetIL = $true )
& ( notinsingleton = $true )
& ( emptysetAx = $true )
& ( dsetconstrI = $true )
& ( dsetconstrER = $true )
& ( nonemptyImpWitness = $true )
& ( prop2setE = $true )
& ( dsetconstrEL = $true )
& ( emptyI = $true )
& ( foundationAx = $true )
& ( inCongP = $true )
& ( setoftrueEq = $true )
& ( prop2setI = $true )
& ( prop2set2propI = $true )
& ( setextAx = $true )
& ( descrp = $true )
& ( quantDeMorgan1 = $true )
& ( emptyinunitempty = $true )
& ( exuE1 = $true )
& ( nonemptyI1 = $true )
& ( setadjoinE = $true )
& ( eqinunit = $true )
& ( exuE2 = $true ) ),
inference(flattening,[status(thm)],[f223]) ).
thf(f223,plain,
( ? [X0: $i > $o] :
( ? [X1: $i,X2: $i] :
( ( X1 != X2 )
& ( ( X0 @ X1 )
= $true )
& ( ( X0 @ X2 )
= $true ) )
& ( ( exu
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
= $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)],[f222]) ).
thf(f222,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 )
=> ! [X0: $i > $o] :
( ( ( exu
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
= $true )
=> ! [X1: $i,X2: $i] :
( ( ( X0 @ X2 )
= $true )
=> ( ( ( X0 @ X1 )
= $true )
=> ( X1 = X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f110]) ).
thf(f110,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 )
=> ! [X0: $i > $o] :
( ( ( exu
@ ^ [Y0: $i] : ( X0 @ Y0 ) )
= $true )
=> ! [X2: $i,X3: $i] :
( ( $true
= ( X0 @ X3 ) )
=> ( ( ( X0 @ X2 )
= $true )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f109]) ).
thf(f109,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
=> ! [X0: $i > $o] :
( ( exu
@ ^ [X1: $i] : ( X0 @ X1 ) )
=> ! [X2: $i,X3: $i] :
( ( X0 @ X3 )
=> ( ( X0 @ X2 )
=> ( X2 = X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f76]) ).
thf(f76,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
=> ! [X0: $i > $o] :
( ( exu
@ ^ [X1: $i] : ( X0 @ X1 ) )
=> ! [X2: $i,X1: $i] :
( ( X0 @ X1 )
=> ( ( X0 @ X2 )
=> ( X1 = X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f75]) ).
thf(f75,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
=> ! [X0: $i > $o] :
( ( exu
@ ^ [X1: $i] : ( X0 @ X1 ) )
=> ! [X2: $i,X1: $i] :
( ( X0 @ X1 )
=> ( ( X0 @ X2 )
=> ( X1 = X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.5KNgBg82Lh/DTF2THF_8219.p',exuEu) ).
thf(f1526,plain,
sK2 = sK1,
inference(superposition,[status(thm)],[f1522,f1523]) ).
thf(f1523,plain,
sK37 = sK1,
inference(trivial_inequality_removal,[status(thm)],[f1512]) ).
thf(f1512,plain,
( ( $true = $false )
| ( sK37 = sK1 ) ),
inference(superposition,[status(thm)],[f994,f287]) ).
thf(f287,plain,
( $true
= ( sK0 @ sK1 ) ),
inference(cnf_transformation,[status(thm)],[f227]) ).
thf(f994,plain,
! [X1: $i] :
( ( ( sK0 @ X1 )
= $false )
| ( sK37 = X1 ) ),
inference(equality_proxy_clausification,[status(thm)],[f993]) ).
thf(f993,plain,
! [X1: $i] :
( ( $true
= ( sK37 = X1 ) )
| ( ( sK0 @ X1 )
= $false ) ),
inference(binary_proxy_clausification,[status(thm)],[f992]) ).
thf(f992,plain,
! [X1: $i] :
( ( ( sK0 @ X1 )
=> ( sK37 = X1 ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f991]) ).
thf(f991,plain,
! [X1: $i] :
( ( ^ [Y0: $i] :
( ( sK0 @ Y0 )
=> ( sK37 = Y0 ) )
@ X1 )
= $true ),
inference(pi_clausification,[status(thm)],[f989]) ).
thf(f989,plain,
( $true
= ( !! @ $i
@ ^ [Y0: $i] :
( ( sK0 @ Y0 )
=> ( sK37 = Y0 ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f988]) ).
thf(f988,plain,
( $true
= ( ( sK0 @ sK37 )
& ( !! @ $i
@ ^ [Y0: $i] :
( ( sK0 @ Y0 )
=> ( sK37 = Y0 ) ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f987]) ).
thf(f987,plain,
( $true
= ( ^ [Y0: $i] :
( ( sK0 @ Y0 )
& ( !! @ $i
@ ^ [Y1: $i] :
( ( sK0 @ Y1 )
=> ( Y0 = Y1 ) ) ) )
@ sK37 ) ),
inference(sigma_clausification,[status(thm)],[f986]) ).
thf(f986,plain,
( ( ?? @ $i
@ ^ [Y0: $i] :
( ( sK0 @ Y0 )
& ( !! @ $i
@ ^ [Y1: $i] :
( ( sK0 @ Y1 )
=> ( Y0 = Y1 ) ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f381]) ).
thf(f381,plain,
( $true
= ( ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) )
@ ^ [Y0: $i] : ( sK0 @ Y0 ) ) ),
inference(definition_unfolding,[status(thm)],[f289,f339]) ).
thf(f339,plain,
( exu
= ( ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f100]) ).
thf(f100,plain,
( exu
= ( ^ [Y0: $i > $o] :
( ?? @ $i
@ ^ [Y1: $i] :
( ( Y0 @ Y1 )
& ( !! @ $i
@ ^ [Y2: $i] :
( ( Y0 @ Y2 )
=> ( Y1 = Y2 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f99]) ).
thf(f99,plain,
( exu
= ( ^ [X0: $i > $o] :
? [X1: $i] :
( ! [X2: $i] :
( ( X0 @ X2 )
=> ( X1 = X2 ) )
& ( X0 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f1]) ).
thf(f1,axiom,
( exu
= ( ^ [X0: $i > $o] :
? [X1: $i] :
( ! [X2: $i] :
( ( X0 @ X2 )
=> ( X1 = X2 ) )
& ( X0 @ X1 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.5KNgBg82Lh/DTF2THF_8219.p',exu) ).
thf(f289,plain,
( $true
= ( exu
@ ^ [Y0: $i] : ( sK0 @ Y0 ) ) ),
inference(cnf_transformation,[status(thm)],[f227]) ).
thf(f1522,plain,
sK37 = sK2,
inference(trivial_inequality_removal,[status(thm)],[f1517]) ).
thf(f1517,plain,
( ( sK37 = sK2 )
| ( $true = $false ) ),
inference(superposition,[status(thm)],[f286,f994]) ).
thf(f286,plain,
( $true
= ( sK0 @ sK2 ) ),
inference(cnf_transformation,[status(thm)],[f227]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU556^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n017.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 12:24:56 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : SEU556^1 : TPTP v9.2.1. Released v3.7.0.
% 0.30/0.46 % Domain : Set Theory
% 0.30/0.46 % Problem : Preliminary Notions - Equality Laws
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English : (! phi:i>o.exu (^ x:i.phi x) -> (! x:i.! y:i.phi x -> phi y ->
% 0.30/0.46 % x = y))
% 0.30/0.46
% 0.30/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.30/0.46 % Source : [Bro08]
% 0.30/0.46 % Names : ZFC058g [Bro08]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.33 v4.0.1, 0.67 v4.0.0, 0.33 v3.7.0
% 0.30/0.46 % Syntax : Number of formulae : 159 ( 74 unt; 84 typ; 74 def)
% 0.30/0.46 % Number of atoms : 403 ( 112 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 74 ( 5 avg)
% 0.30/0.46 % Number of connectives : 645 ( 21 ~; 5 |; 27 &; 398 @)
% 0.30/0.46 % ( 12 <=>; 182 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 80 ( 2 avg)
% 0.30/0.46 % Number of types : 2 ( 0 usr)
% 0.30/0.46 % Number of type conns : 43 ( 43 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 87 ( 84 usr; 76 con; 0-2 aty)
% 0.30/0.46 % Number of variables : 226 ( 22 ^; 176 !; 28 ?; 226 :)
% 0.30/0.46 % SPC : TH0_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : http://mathgate.info/detsetitem.php?id=146
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 thf(in_type,type,
% 0.30/0.46 in: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu_type,type,
% 0.30/0.46 exu: ( $i > $o ) > $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu,definition,
% 0.30/0.46 ( exu
% 0.30/0.46 = ( ^ [Xphi: $i > $o] :
% 0.30/0.46 ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextAx_type,type,
% 0.30/0.46 setextAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextAx,definition,
% 0.30/0.46 ( setextAx
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 <=> ( in @ Xx @ B ) )
% 0.30/0.46 => ( A = B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset_type,type,
% 0.30/0.46 emptyset: $i ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetAx_type,type,
% 0.30/0.46 emptysetAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetAx,definition,
% 0.30/0.46 ( emptysetAx
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin_type,type,
% 0.30/0.46 setadjoin: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinAx_type,type,
% 0.30/0.46 setadjoinAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinAx,definition,
% 0.30/0.46 ( setadjoinAx
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 <=> ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powerset_type,type,
% 0.30/0.46 powerset: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(powersetAx_type,type,
% 0.30/0.46 powersetAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetAx,definition,
% 0.30/0.46 ( powersetAx
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 <=> ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunion_type,type,
% 0.30/0.46 setunion: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setunionAx_type,type,
% 0.30/0.46 setunionAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionAx,definition,
% 0.30/0.46 ( setunionAx
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 <=> ? [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omega_type,type,
% 0.30/0.46 omega: $i ).
% 0.30/0.46
% 0.30/0.46 thf(omega0Ax_type,type,
% 0.30/0.46 omega0Ax: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omega0Ax,definition,
% 0.30/0.46 ( omega0Ax
% 0.30/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omegaSAx_type,type,
% 0.30/0.46 omegaSAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omegaSAx,definition,
% 0.30/0.46 ( omegaSAx
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ omega )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omegaIndAx_type,type,
% 0.30/0.46 omegaIndAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omegaIndAx,definition,
% 0.30/0.46 ( omegaIndAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( ( in @ emptyset @ A )
% 0.30/0.46 & ! [Xx: $i] :
% 0.30/0.46 ( ( ( in @ Xx @ omega )
% 0.30/0.46 & ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ omega )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(replAx_type,type,
% 0.30/0.46 replAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(replAx,definition,
% 0.30/0.46 ( replAx
% 0.30/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.30/0.46 => ? [B: $i] :
% 0.30/0.46 ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 <=> ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(foundationAx_type,type,
% 0.30/0.46 foundationAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(foundationAx,definition,
% 0.30/0.46 ( foundationAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ? [B: $i] :
% 0.30/0.46 ( ( in @ B @ A )
% 0.30/0.46 & ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(wellorderingAx_type,type,
% 0.30/0.46 wellorderingAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(wellorderingAx,definition,
% 0.30/0.46 ( wellorderingAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ? [B: $i] :
% 0.30/0.46 ( ! [C: $i] :
% 0.30/0.46 ( ( in @ C @ B )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ A ) ) )
% 0.30/0.46 & ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( ( in @ Xx @ A )
% 0.30/0.46 & ( in @ Xy @ A ) )
% 0.30/0.46 => ( ! [C: $i] :
% 0.30/0.46 ( ( in @ C @ B )
% 0.30/0.46 => ( ( in @ Xx @ C )
% 0.30/0.46 <=> ( in @ Xy @ C ) ) )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 & ! [C: $i,D: $i] :
% 0.30/0.46 ( ( ( in @ C @ B )
% 0.30/0.46 & ( in @ D @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ D ) )
% 0.30/0.46 | ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ D )
% 0.30/0.46 => ( in @ Xx @ C ) ) ) )
% 0.30/0.46 & ! [C: $i] :
% 0.30/0.46 ( ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.30/0.46 => ? [D: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ D @ B )
% 0.30/0.46 & ( in @ Xx @ C )
% 0.30/0.46 & ~ ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ D )
% 0.30/0.46 & ( in @ Xy @ C ) )
% 0.30/0.46 & ! [E: $i] :
% 0.30/0.46 ( ( in @ E @ B )
% 0.30/0.46 => ( ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ E )
% 0.30/0.46 => ( in @ Xy @ D ) )
% 0.30/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(descr_type,type,
% 0.30/0.46 descr: ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(descrp_type,type,
% 0.30/0.46 descrp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(descrp,definition,
% 0.30/0.46 ( descrp
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 @ ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr_type,type,
% 0.30/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrI_type,type,
% 0.30/0.46 dsetconstrI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrI,definition,
% 0.30/0.46 ( dsetconstrI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrEL_type,type,
% 0.30/0.46 dsetconstrEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrEL,definition,
% 0.30/0.46 ( dsetconstrEL
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrER_type,type,
% 0.30/0.46 dsetconstrER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrER,definition,
% 0.30/0.46 ( dsetconstrER
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE1_type,type,
% 0.30/0.46 exuE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE1,definition,
% 0.30/0.46 ( exuE1
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set_type,type,
% 0.30/0.46 prop2set: $o > $i ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setE_type,type,
% 0.30/0.46 prop2setE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setE,definition,
% 0.30/0.46 ( prop2setE
% 0.30/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetE_type,type,
% 0.30/0.46 emptysetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetE,definition,
% 0.30/0.46 ( emptysetE
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetimpfalse_type,type,
% 0.30/0.46 emptysetimpfalse: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetimpfalse,definition,
% 0.30/0.46 ( emptysetimpfalse
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => $false ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notinemptyset_type,type,
% 0.30/0.46 notinemptyset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notinemptyset,definition,
% 0.30/0.46 ( notinemptyset
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3e_type,type,
% 0.30/0.46 exuE3e: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3e,definition,
% 0.30/0.46 ( exuE3e
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setext_type,type,
% 0.30/0.46 setext: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setext,definition,
% 0.30/0.46 ( setext
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( A = B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyI_type,type,
% 0.30/0.46 emptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyI,definition,
% 0.30/0.46 ( emptyI
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ A )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(noeltsimpempty_type,type,
% 0.30/0.46 noeltsimpempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(noeltsimpempty,definition,
% 0.30/0.46 ( noeltsimpempty
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ A )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setbeta_type,type,
% 0.30/0.46 setbeta: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setbeta,definition,
% 0.30/0.46 ( setbeta
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonempty_type,type,
% 0.30/0.46 nonempty: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonempty,definition,
% 0.30/0.46 ( nonempty
% 0.30/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1_type,type,
% 0.30/0.46 nonemptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1,definition,
% 0.30/0.46 ( nonemptyE1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI_type,type,
% 0.30/0.46 nonemptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI,definition,
% 0.30/0.46 ( nonemptyI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( nonempty
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1_type,type,
% 0.30/0.46 nonemptyI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1,definition,
% 0.30/0.46 ( nonemptyI1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL_type,type,
% 0.30/0.46 setadjoinIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL,definition,
% 0.30/0.46 ( setadjoinIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty_type,type,
% 0.30/0.46 emptyinunitempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty,definition,
% 0.30/0.46 ( emptyinunitempty
% 0.30/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR_type,type,
% 0.30/0.46 setadjoinIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR,definition,
% 0.30/0.46 ( setadjoinIR
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE_type,type,
% 0.30/0.46 setadjoinE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE,definition,
% 0.30/0.46 ( setadjoinE
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ( ( Xy = Xx )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xy @ A )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr_type,type,
% 0.30/0.46 setadjoinOr: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr,definition,
% 0.30/0.46 ( setadjoinOr
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq_type,type,
% 0.30/0.46 setoftrueEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq,definition,
% 0.30/0.46 ( setoftrueEq
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : $true )
% 0.30/0.46 = A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI_type,type,
% 0.30/0.46 powersetI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI,definition,
% 0.30/0.46 ( powersetI
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset_type,type,
% 0.30/0.46 emptyinPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset,definition,
% 0.30/0.46 ( emptyinPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset_type,type,
% 0.30/0.46 emptyInPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset,definition,
% 0.30/0.46 ( emptyInPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE_type,type,
% 0.30/0.46 powersetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE,definition,
% 0.30/0.46 ( powersetE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI_type,type,
% 0.30/0.46 setunionI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI,definition,
% 0.30/0.46 ( setunionI
% 0.30/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE_type,type,
% 0.30/0.46 setunionE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE,definition,
% 0.30/0.46 ( setunionE
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ! [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU_type,type,
% 0.30/0.46 subPowSU: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU,definition,
% 0.30/0.46 ( subPowSU
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2_type,type,
% 0.30/0.46 exuE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2,definition,
% 0.30/0.46 ( exuE2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness_type,type,
% 0.30/0.46 nonemptyImpWitness: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness,definition,
% 0.30/0.46 ( nonemptyImpWitness
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & $true ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(uniqinunit_type,type,
% 0.30/0.46 uniqinunit: $o ).
% 0.30/0.46
% 0.30/0.46 thf(uniqinunit,definition,
% 0.30/0.46 ( uniqinunit
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notinsingleton_type,type,
% 0.30/0.46 notinsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notinsingleton,definition,
% 0.30/0.46 ( notinsingleton
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx != Xy )
% 0.30/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqinunit_type,type,
% 0.30/0.46 eqinunit: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqinunit,definition,
% 0.30/0.46 ( eqinunit
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsswitch_type,type,
% 0.30/0.46 singletonsswitch: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsswitch,definition,
% 0.30/0.46 ( singletonsswitch
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetE_type,type,
% 0.30/0.46 upairsetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetE,definition,
% 0.30/0.46 ( upairsetE
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.46 => ( ( Xz = Xx )
% 0.30/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIL_type,type,
% 0.30/0.46 upairsetIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIL,definition,
% 0.30/0.46 ( upairsetIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIR_type,type,
% 0.30/0.46 upairsetIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIR,definition,
% 0.30/0.46 ( upairsetIR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyE1_type,type,
% 0.30/0.46 emptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyE1,definition,
% 0.30/0.46 ( emptyE1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ( ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = emptyset )
% 0.30/0.46 => $false ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(vacuousDall_type,type,
% 0.30/0.46 vacuousDall: $o ).
% 0.30/0.46
% 0.30/0.46 thf(vacuousDall,definition,
% 0.30/0.46 ( vacuousDall
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan1_type,type,
% 0.30/0.46 quantDeMorgan1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan1,definition,
% 0.30/0.46 ( quantDeMorgan1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan2_type,type,
% 0.30/0.46 quantDeMorgan2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan2,definition,
% 0.30/0.46 ( quantDeMorgan2
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) )
% 0.30/0.46 => ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan3_type,type,
% 0.30/0.46 quantDeMorgan3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan3,definition,
% 0.30/0.46 ( quantDeMorgan3
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan4_type,type,
% 0.30/0.46 quantDeMorgan4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan4,definition,
% 0.30/0.46 ( quantDeMorgan4
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) )
% 0.30/0.46 => ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setI_type,type,
% 0.30/0.46 prop2setI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setI,definition,
% 0.30/0.46 ( prop2setI
% 0.30/0.46 = ( ! [Xphi: $o] :
% 0.30/0.46 ( Xphi
% 0.30/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(set2prop_type,type,
% 0.30/0.46 set2prop: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set2propI_type,type,
% 0.30/0.46 prop2set2propI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set2propI,definition,
% 0.30/0.46 ( prop2set2propI
% 0.30/0.46 = ( ! [Xphi: $o] :
% 0.30/0.46 ( Xphi
% 0.30/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notdexE_type,type,
% 0.30/0.46 notdexE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notdexE,definition,
% 0.30/0.46 ( notdexE
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notdallE_type,type,
% 0.30/0.46 notdallE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notdallE,definition,
% 0.30/0.46 ( notdallE
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI1_type,type,
% 0.30/0.46 exuI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI1,definition,
% 0.30/0.46 ( exuI1
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI3_type,type,
% 0.30/0.46 exuI3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI3,definition,
% 0.30/0.46 ( exuI3
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.30/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI2_type,type,
% 0.30/0.46 exuI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI2,definition,
% 0.30/0.46 ( exuI2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inCongP_type,type,
% 0.30/0.46 inCongP: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inCongP,definition,
% 0.30/0.46 ( inCongP
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(in__Cong_type,type,
% 0.30/0.46 in__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(in__Cong,definition,
% 0.30/0.46 ( in__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3u_type,type,
% 0.30/0.46 exuE3u: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3u,definition,
% 0.30/0.46 ( exuE3u
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exu__Cong_type,type,
% 0.30/0.46 exu__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu__Cong,definition,
% 0.30/0.46 ( exu__Cong
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 <=> ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset__Cong_type,type,
% 0.30/0.46 emptyset__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset__Cong,definition,
% 0.30/0.46 ( emptyset__Cong
% 0.30/0.46 = ( emptyset = emptyset ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin__Cong_type,type,
% 0.30/0.46 setadjoin__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin__Cong,definition,
% 0.30/0.46 ( setadjoin__Cong
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ! [Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( Xz = Xu )
% 0.30/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.30/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powerset__Cong_type,type,
% 0.30/0.46 powerset__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powerset__Cong,definition,
% 0.30/0.46 ( powerset__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( powerset @ A )
% 0.30/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunion__Cong_type,type,
% 0.30/0.46 setunion__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunion__Cong,definition,
% 0.30/0.46 ( setunion__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( setunion @ A )
% 0.30/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omega__Cong_type,type,
% 0.30/0.46 omega__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omega__Cong,definition,
% 0.30/0.46 ( omega__Cong
% 0.30/0.46 = ( omega = omega ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuEu,conjecture,
% 0.30/0.46 ( setextAx
% 0.30/0.46 => ( emptysetAx
% 0.30/0.46 => ( setadjoinAx
% 0.30/0.46 => ( powersetAx
% 0.30/0.46 => ( setunionAx
% 0.30/0.46 => ( omega0Ax
% 0.30/0.46 => ( omegaSAx
% 0.30/0.46 => ( omegaIndAx
% 0.30/0.46 => ( replAx
% 0.30/0.46 => ( foundationAx
% 0.30/0.46 => ( wellorderingAx
% 0.30/0.46 => ( descrp
% 0.30/0.46 => ( dsetconstrI
% 0.30/0.46 => ( dsetconstrEL
% 0.30/0.46 => ( dsetconstrER
% 0.30/0.46 => ( exuE1
% 0.30/0.46 => ( prop2setE
% 0.30/0.46 => ( emptysetE
% 0.30/0.46 => ( emptysetimpfalse
% 0.30/0.46 => ( notinemptyset
% 0.30/0.46 => ( exuE3e
% 0.30/0.46 => ( setext
% 0.30/0.46 => ( emptyI
% 0.30/0.46 => ( noeltsimpempty
% 0.30/0.46 => ( setbeta
% 0.30/0.46 => ( nonemptyE1
% 0.30/0.46 => ( nonemptyI
% 0.30/0.46 => ( nonemptyI1
% 0.30/0.46 => ( setadjoinIL
% 0.30/0.46 => ( emptyinunitempty
% 0.30/0.46 => ( setadjoinIR
% 0.30/0.46 => ( setadjoinE
% 0.30/0.46 => ( setadjoinOr
% 0.30/0.46 => ( setoftrueEq
% 0.30/0.46 => ( powersetI
% 0.30/0.46 => ( emptyinPowerset
% 0.30/0.46 => ( emptyInPowerset
% 0.30/0.46 => ( powersetE
% 0.30/0.46 => ( setunionI
% 0.30/0.46 => ( setunionE
% 0.30/0.46 => ( subPowSU
% 0.30/0.46 => ( exuE2
% 0.30/0.46 => ( nonemptyImpWitness
% 0.30/0.46 => ( uniqinunit
% 0.30/0.46 => ( notinsingleton
% 0.30/0.46 => ( eqinunit
% 0.30/0.46 => ( singletonsswitch
% 0.30/0.46 => ( upairsetE
% 0.30/0.46 => ( upairsetIL
% 0.30/0.46 => ( upairsetIR
% 0.30/0.46 => ( emptyE1
% 0.30/0.46 => ( vacuousDall
% 0.30/0.46 => ( quantDeMorgan1
% 0.30/0.46 => ( quantDeMorgan2
% 0.30/0.46 => ( quantDeMorgan3
% 0.30/0.46 => ( quantDeMorgan4
% 0.30/0.46 => ( prop2setI
% 0.30/0.46 => ( prop2set2propI
% 0.30/0.46 => ( notdexE
% 0.30/0.46 => ( notdallE
% 0.30/0.46 => ( exuI1
% 0.30/0.46 => ( exuI3
% 0.30/0.46 => ( exuI2
% 0.30/0.46 => ( inCongP
% 0.30/0.46 => ( in__Cong
% 0.30/0.46 => ( exuE3u
% 0.30/0.46 => ( exu__Cong
% 0.30/0.46 => ( emptyset__Cong
% 0.30/0.46 => ( setadjoin__Cong
% 0.30/0.46 => ( powerset__Cong
% 0.30/0.46 => ( setunion__Cong
% 0.30/0.46 => ( omega__Cong
% 0.30/0.46 => ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 1.81/1.39 ---- Embedded in THF ---
% 1.81/1.39 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.81/1.39 %%% Generated on Tue Feb 24 12:24:57 EST 2026
% 1.81/1.39 %%% using '$$dhol' embedding, version 1.3.0.
% 1.81/1.39 %%% Logic specification used:
% 1.81/1.39 %%% thf(spec, logic, $$dhol).
% 1.81/1.39
% 1.81/1.39 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.5KNgBg82Lh/DTF2DTF8219.p
% See solution above
% 1.81/1.39 ------------------------
% 2.04/1.41 ---- Cleaned THF ---
% 2.04/1.41 thf(in_type,type,
% 2.04/1.41 in: $i > $i > $o ).
% 2.04/1.41
% 2.04/1.41 thf(exu_type,type,
% 2.04/1.41 exu: ( $i > $o ) > $o ).
% 2.04/1.41
% 2.04/1.41 thf(exu,definition,
% 2.04/1.41 ( exu
% 2.04/1.41 = ( ^ [Xphi: $i > $o] :
% 2.04/1.41 ? [Xx: $i] :
% 2.04/1.41 ( ( Xphi @ Xx )
% 2.04/1.41 & ! [Xy: $i] :
% 2.04/1.41 ( ( Xphi @ Xy )
% 2.04/1.41 => ( Xx = Xy ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setextAx_type,type,
% 2.04/1.41 setextAx: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setextAx,definition,
% 2.04/1.41 ( setextAx
% 2.04/1.41 = ( ! [A: $i,B: $i] :
% 2.04/1.41 ( ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 <=> ( in @ Xx @ B ) )
% 2.04/1.41 => ( A = B ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(emptyset_type,type,
% 2.04/1.41 emptyset: $i ).
% 2.04/1.41
% 2.04/1.41 thf(emptysetAx_type,type,
% 2.04/1.41 emptysetAx: $o ).
% 2.04/1.41
% 2.04/1.41 thf(emptysetAx,definition,
% 2.04/1.41 ( emptysetAx
% 2.04/1.41 = ( ! [Xx: $i] :
% 2.04/1.41 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoin_type,type,
% 2.04/1.41 setadjoin: $i > $i > $i ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinAx_type,type,
% 2.04/1.41 setadjoinAx: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinAx,definition,
% 2.04/1.41 ( setadjoinAx
% 2.04/1.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.04/1.41 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.04/1.41 <=> ( ( Xy = Xx )
% 2.04/1.41 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(powerset_type,type,
% 2.04/1.41 powerset: $i > $i ).
% 2.04/1.41
% 2.04/1.41 thf(powersetAx_type,type,
% 2.04/1.41 powersetAx: $o ).
% 2.04/1.41
% 2.04/1.41 thf(powersetAx,definition,
% 2.04/1.41 ( powersetAx
% 2.04/1.41 = ( ! [A: $i,B: $i] :
% 2.04/1.41 ( ( in @ B @ ( powerset @ A ) )
% 2.04/1.41 <=> ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ B )
% 2.04/1.41 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setunion_type,type,
% 2.04/1.41 setunion: $i > $i ).
% 2.04/1.41
% 2.04/1.41 thf(setunionAx_type,type,
% 2.04/1.41 setunionAx: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setunionAx,definition,
% 2.04/1.41 ( setunionAx
% 2.04/1.41 = ( ! [A: $i,Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ ( setunion @ A ) )
% 2.04/1.41 <=> ? [B: $i] :
% 2.04/1.41 ( ( in @ Xx @ B )
% 2.04/1.41 & ( in @ B @ A ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(omega_type,type,
% 2.04/1.41 omega: $i ).
% 2.04/1.41
% 2.04/1.41 thf(omega0Ax_type,type,
% 2.04/1.41 omega0Ax: $o ).
% 2.04/1.41
% 2.04/1.41 thf(omega0Ax,definition,
% 2.04/1.41 ( omega0Ax
% 2.04/1.41 = ( in @ emptyset @ omega ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(omegaSAx_type,type,
% 2.04/1.41 omegaSAx: $o ).
% 2.04/1.41
% 2.04/1.41 thf(omegaSAx,definition,
% 2.04/1.41 ( omegaSAx
% 2.04/1.41 = ( ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ omega )
% 2.04/1.41 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(omegaIndAx_type,type,
% 2.04/1.41 omegaIndAx: $o ).
% 2.04/1.41
% 2.04/1.41 thf(omegaIndAx,definition,
% 2.04/1.41 ( omegaIndAx
% 2.04/1.41 = ( ! [A: $i] :
% 2.04/1.41 ( ( ( in @ emptyset @ A )
% 2.04/1.41 & ! [Xx: $i] :
% 2.04/1.41 ( ( ( in @ Xx @ omega )
% 2.04/1.41 & ( in @ Xx @ A ) )
% 2.04/1.41 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.04/1.41 => ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ omega )
% 2.04/1.41 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(replAx_type,type,
% 2.04/1.41 replAx: $o ).
% 2.04/1.41
% 2.04/1.41 thf(replAx,definition,
% 2.04/1.41 ( replAx
% 2.04/1.41 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.04/1.41 ( ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 => ( exu
% 2.04/1.41 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.04/1.41 => ? [B: $i] :
% 2.04/1.41 ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ B )
% 2.04/1.41 <=> ? [Xy: $i] :
% 2.04/1.41 ( ( in @ Xy @ A )
% 2.04/1.41 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(foundationAx_type,type,
% 2.04/1.41 foundationAx: $o ).
% 2.04/1.41
% 2.04/1.41 thf(foundationAx,definition,
% 2.04/1.41 ( foundationAx
% 2.04/1.41 = ( ! [A: $i] :
% 2.04/1.41 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.04/1.41 => ? [B: $i] :
% 2.04/1.41 ( ( in @ B @ A )
% 2.04/1.41 & ~ ? [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ B )
% 2.04/1.41 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(wellorderingAx_type,type,
% 2.04/1.41 wellorderingAx: $o ).
% 2.04/1.41
% 2.04/1.41 thf(wellorderingAx,definition,
% 2.04/1.41 ( wellorderingAx
% 2.04/1.41 = ( ! [A: $i] :
% 2.04/1.41 ? [B: $i] :
% 2.04/1.41 ( ! [C: $i] :
% 2.04/1.41 ( ( in @ C @ B )
% 2.04/1.41 => ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ C )
% 2.04/1.41 => ( in @ Xx @ A ) ) )
% 2.04/1.41 & ! [Xx: $i,Xy: $i] :
% 2.04/1.41 ( ( ( in @ Xx @ A )
% 2.04/1.41 & ( in @ Xy @ A ) )
% 2.04/1.41 => ( ! [C: $i] :
% 2.04/1.41 ( ( in @ C @ B )
% 2.04/1.41 => ( ( in @ Xx @ C )
% 2.04/1.41 <=> ( in @ Xy @ C ) ) )
% 2.04/1.41 => ( Xx = Xy ) ) )
% 2.04/1.41 & ! [C: $i,D: $i] :
% 2.04/1.41 ( ( ( in @ C @ B )
% 2.04/1.41 & ( in @ D @ B ) )
% 2.04/1.41 => ( ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ C )
% 2.04/1.41 => ( in @ Xx @ D ) )
% 2.04/1.41 | ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ D )
% 2.04/1.41 => ( in @ Xx @ C ) ) ) )
% 2.04/1.41 & ! [C: $i] :
% 2.04/1.41 ( ( ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ C )
% 2.04/1.41 => ( in @ Xx @ A ) )
% 2.04/1.41 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.04/1.41 => ? [D: $i,Xx: $i] :
% 2.04/1.41 ( ( in @ D @ B )
% 2.04/1.41 & ( in @ Xx @ C )
% 2.04/1.41 & ~ ? [Xy: $i] :
% 2.04/1.41 ( ( in @ Xy @ D )
% 2.04/1.41 & ( in @ Xy @ C ) )
% 2.04/1.41 & ! [E: $i] :
% 2.04/1.41 ( ( in @ E @ B )
% 2.04/1.41 => ( ! [Xy: $i] :
% 2.04/1.41 ( ( in @ Xy @ E )
% 2.04/1.41 => ( in @ Xy @ D ) )
% 2.04/1.41 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(descr_type,type,
% 2.04/1.41 descr: ( $i > $o ) > $i ).
% 2.04/1.41
% 2.04/1.41 thf(descrp_type,type,
% 2.04/1.41 descrp: $o ).
% 2.04/1.41
% 2.04/1.41 thf(descrp,definition,
% 2.04/1.41 ( descrp
% 2.04/1.41 = ( ! [Xphi: $i > $o] :
% 2.04/1.41 ( ( exu
% 2.04/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.04/1.41 => ( Xphi
% 2.04/1.41 @ ( descr
% 2.04/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(dsetconstr_type,type,
% 2.04/1.41 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.04/1.41
% 2.04/1.41 thf(dsetconstrI_type,type,
% 2.04/1.41 dsetconstrI: $o ).
% 2.04/1.41
% 2.04/1.41 thf(dsetconstrI,definition,
% 2.04/1.41 ( dsetconstrI
% 2.04/1.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 => ( ( Xphi @ Xx )
% 2.04/1.41 => ( in @ Xx
% 2.04/1.41 @ ( dsetconstr @ A
% 2.04/1.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(dsetconstrEL_type,type,
% 2.04/1.41 dsetconstrEL: $o ).
% 2.04/1.41
% 2.04/1.41 thf(dsetconstrEL,definition,
% 2.04/1.41 ( dsetconstrEL
% 2.04/1.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.04/1.41 ( ( in @ Xx
% 2.04/1.41 @ ( dsetconstr @ A
% 2.04/1.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.04/1.41 => ( in @ Xx @ A ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(dsetconstrER_type,type,
% 2.04/1.41 dsetconstrER: $o ).
% 2.04/1.41
% 2.04/1.41 thf(dsetconstrER,definition,
% 2.04/1.41 ( dsetconstrER
% 2.04/1.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.04/1.41 ( ( in @ Xx
% 2.04/1.41 @ ( dsetconstr @ A
% 2.04/1.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.04/1.41 => ( Xphi @ Xx ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(exuE1_type,type,
% 2.04/1.41 exuE1: $o ).
% 2.04/1.41
% 2.04/1.41 thf(exuE1,definition,
% 2.04/1.41 ( exuE1
% 2.04/1.41 = ( ! [Xphi: $i > $o] :
% 2.04/1.41 ( ( exu
% 2.04/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.04/1.41 => ? [Xx: $i] :
% 2.04/1.41 ( ( Xphi @ Xx )
% 2.04/1.41 & ! [Xy: $i] :
% 2.04/1.41 ( ( Xphi @ Xy )
% 2.04/1.41 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(prop2set_type,type,
% 2.04/1.41 prop2set: $o > $i ).
% 2.04/1.41
% 2.04/1.41 thf(prop2setE_type,type,
% 2.04/1.41 prop2setE: $o ).
% 2.04/1.41
% 2.04/1.41 thf(prop2setE,definition,
% 2.04/1.41 ( prop2setE
% 2.04/1.41 = ( ! [Xphi: $o,Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.04/1.41 => Xphi ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(emptysetE_type,type,
% 2.04/1.41 emptysetE: $o ).
% 2.04/1.41
% 2.04/1.41 thf(emptysetE,definition,
% 2.04/1.41 ( emptysetE
% 2.04/1.41 = ( ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ emptyset )
% 2.04/1.41 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(emptysetimpfalse_type,type,
% 2.04/1.41 emptysetimpfalse: $o ).
% 2.04/1.41
% 2.04/1.41 thf(emptysetimpfalse,definition,
% 2.04/1.41 ( emptysetimpfalse
% 2.04/1.41 = ( ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ emptyset )
% 2.04/1.41 => $false ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(notinemptyset_type,type,
% 2.04/1.41 notinemptyset: $o ).
% 2.04/1.41
% 2.04/1.41 thf(notinemptyset,definition,
% 2.04/1.41 ( notinemptyset
% 2.04/1.41 = ( ! [Xx: $i] :
% 2.04/1.41 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(exuE3e_type,type,
% 2.04/1.41 exuE3e: $o ).
% 2.04/1.41
% 2.04/1.41 thf(exuE3e,definition,
% 2.04/1.41 ( exuE3e
% 2.04/1.41 = ( ! [Xphi: $i > $o] :
% 2.04/1.41 ( ( exu
% 2.04/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.04/1.41 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setext_type,type,
% 2.04/1.41 setext: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setext,definition,
% 2.04/1.41 ( setext
% 2.04/1.41 = ( ! [A: $i,B: $i] :
% 2.04/1.41 ( ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 => ( in @ Xx @ B ) )
% 2.04/1.41 => ( ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ B )
% 2.04/1.41 => ( in @ Xx @ A ) )
% 2.04/1.41 => ( A = B ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(emptyI_type,type,
% 2.04/1.41 emptyI: $o ).
% 2.04/1.41
% 2.04/1.41 thf(emptyI,definition,
% 2.04/1.41 ( emptyI
% 2.04/1.41 = ( ! [A: $i] :
% 2.04/1.41 ( ! [Xx: $i] :
% 2.04/1.41 ~ ( in @ Xx @ A )
% 2.04/1.41 => ( A = emptyset ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(noeltsimpempty_type,type,
% 2.04/1.41 noeltsimpempty: $o ).
% 2.04/1.41
% 2.04/1.41 thf(noeltsimpempty,definition,
% 2.04/1.41 ( noeltsimpempty
% 2.04/1.41 = ( ! [A: $i] :
% 2.04/1.41 ( ! [Xx: $i] :
% 2.04/1.41 ~ ( in @ Xx @ A )
% 2.04/1.41 => ( A = emptyset ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setbeta_type,type,
% 2.04/1.41 setbeta: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setbeta,definition,
% 2.04/1.41 ( setbeta
% 2.04/1.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 => ( ( in @ Xx
% 2.04/1.41 @ ( dsetconstr @ A
% 2.04/1.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.04/1.41 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(nonempty_type,type,
% 2.04/1.41 nonempty: $i > $o ).
% 2.04/1.41
% 2.04/1.41 thf(nonempty,definition,
% 2.04/1.41 ( nonempty
% 2.04/1.41 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(nonemptyE1_type,type,
% 2.04/1.41 nonemptyE1: $o ).
% 2.04/1.41
% 2.04/1.41 thf(nonemptyE1,definition,
% 2.04/1.41 ( nonemptyE1
% 2.04/1.41 = ( ! [A: $i] :
% 2.04/1.41 ( ( nonempty @ A )
% 2.04/1.41 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(nonemptyI_type,type,
% 2.04/1.41 nonemptyI: $o ).
% 2.04/1.41
% 2.04/1.41 thf(nonemptyI,definition,
% 2.04/1.41 ( nonemptyI
% 2.04/1.41 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 => ( ( Xphi @ Xx )
% 2.04/1.41 => ( nonempty
% 2.04/1.41 @ ( dsetconstr @ A
% 2.04/1.41 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(nonemptyI1_type,type,
% 2.04/1.41 nonemptyI1: $o ).
% 2.04/1.41
% 2.04/1.41 thf(nonemptyI1,definition,
% 2.04/1.41 ( nonemptyI1
% 2.04/1.41 = ( ! [A: $i] :
% 2.04/1.41 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.04/1.41 => ( nonempty @ A ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinIL_type,type,
% 2.04/1.41 setadjoinIL: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinIL,definition,
% 2.04/1.41 ( setadjoinIL
% 2.04/1.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(emptyinunitempty_type,type,
% 2.04/1.41 emptyinunitempty: $o ).
% 2.04/1.41
% 2.04/1.41 thf(emptyinunitempty,definition,
% 2.04/1.41 ( emptyinunitempty
% 2.04/1.41 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinIR_type,type,
% 2.04/1.41 setadjoinIR: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinIR,definition,
% 2.04/1.41 ( setadjoinIR
% 2.04/1.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.04/1.41 ( ( in @ Xy @ A )
% 2.04/1.41 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinE_type,type,
% 2.04/1.41 setadjoinE: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinE,definition,
% 2.04/1.41 ( setadjoinE
% 2.04/1.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.04/1.41 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.04/1.41 => ! [Xphi: $o] :
% 2.04/1.41 ( ( ( Xy = Xx )
% 2.04/1.41 => Xphi )
% 2.04/1.41 => ( ( ( in @ Xy @ A )
% 2.04/1.41 => Xphi )
% 2.04/1.41 => Xphi ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinOr_type,type,
% 2.04/1.41 setadjoinOr: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoinOr,definition,
% 2.04/1.41 ( setadjoinOr
% 2.04/1.41 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.04/1.41 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.04/1.41 => ( ( Xy = Xx )
% 2.04/1.41 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setoftrueEq_type,type,
% 2.04/1.41 setoftrueEq: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setoftrueEq,definition,
% 2.04/1.41 ( setoftrueEq
% 2.04/1.41 = ( ! [A: $i] :
% 2.04/1.41 ( ( dsetconstr @ A
% 2.04/1.41 @ ^ [Xx: $i] : $true )
% 2.04/1.41 = A ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(powersetI_type,type,
% 2.04/1.41 powersetI: $o ).
% 2.04/1.41
% 2.04/1.41 thf(powersetI,definition,
% 2.04/1.41 ( powersetI
% 2.04/1.41 = ( ! [A: $i,B: $i] :
% 2.04/1.41 ( ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ B )
% 2.04/1.41 => ( in @ Xx @ A ) )
% 2.04/1.41 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(emptyinPowerset_type,type,
% 2.04/1.41 emptyinPowerset: $o ).
% 2.04/1.41
% 2.04/1.41 thf(emptyinPowerset,definition,
% 2.04/1.41 ( emptyinPowerset
% 2.04/1.41 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(emptyInPowerset_type,type,
% 2.04/1.41 emptyInPowerset: $o ).
% 2.04/1.41
% 2.04/1.41 thf(emptyInPowerset,definition,
% 2.04/1.41 ( emptyInPowerset
% 2.04/1.41 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(powersetE_type,type,
% 2.04/1.41 powersetE: $o ).
% 2.04/1.41
% 2.04/1.41 thf(powersetE,definition,
% 2.04/1.41 ( powersetE
% 2.04/1.41 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.04/1.41 ( ( in @ B @ ( powerset @ A ) )
% 2.04/1.41 => ( ( in @ Xx @ B )
% 2.04/1.41 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setunionI_type,type,
% 2.04/1.41 setunionI: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setunionI,definition,
% 2.04/1.41 ( setunionI
% 2.04/1.41 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.04/1.41 ( ( in @ Xx @ B )
% 2.04/1.41 => ( ( in @ B @ A )
% 2.04/1.41 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setunionE_type,type,
% 2.04/1.41 setunionE: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setunionE,definition,
% 2.04/1.41 ( setunionE
% 2.04/1.41 = ( ! [A: $i,Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ ( setunion @ A ) )
% 2.04/1.41 => ! [Xphi: $o] :
% 2.04/1.41 ( ! [B: $i] :
% 2.04/1.41 ( ( in @ Xx @ B )
% 2.04/1.41 => ( ( in @ B @ A )
% 2.04/1.41 => Xphi ) )
% 2.04/1.41 => Xphi ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(subPowSU_type,type,
% 2.04/1.41 subPowSU: $o ).
% 2.04/1.41
% 2.04/1.41 thf(subPowSU,definition,
% 2.04/1.41 ( subPowSU
% 2.04/1.41 = ( ! [A: $i,Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(exuE2_type,type,
% 2.04/1.41 exuE2: $o ).
% 2.04/1.41
% 2.04/1.41 thf(exuE2,definition,
% 2.04/1.41 ( exuE2
% 2.04/1.41 = ( ! [Xphi: $i > $o] :
% 2.04/1.41 ( ( exu
% 2.04/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.04/1.41 => ? [Xx: $i] :
% 2.04/1.41 ! [Xy: $i] :
% 2.04/1.41 ( ( Xphi @ Xy )
% 2.04/1.41 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(nonemptyImpWitness_type,type,
% 2.04/1.41 nonemptyImpWitness: $o ).
% 2.04/1.41
% 2.04/1.41 thf(nonemptyImpWitness,definition,
% 2.04/1.41 ( nonemptyImpWitness
% 2.04/1.41 = ( ! [A: $i] :
% 2.04/1.41 ( ( nonempty @ A )
% 2.04/1.41 => ? [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 & $true ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(uniqinunit_type,type,
% 2.04/1.41 uniqinunit: $o ).
% 2.04/1.41
% 2.04/1.41 thf(uniqinunit,definition,
% 2.04/1.41 ( uniqinunit
% 2.04/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.04/1.41 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.04/1.41 => ( Xx = Xy ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(notinsingleton_type,type,
% 2.04/1.41 notinsingleton: $o ).
% 2.04/1.41
% 2.04/1.41 thf(notinsingleton,definition,
% 2.04/1.41 ( notinsingleton
% 2.04/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.04/1.41 ( ( Xx != Xy )
% 2.04/1.41 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(eqinunit_type,type,
% 2.04/1.41 eqinunit: $o ).
% 2.04/1.41
% 2.04/1.41 thf(eqinunit,definition,
% 2.04/1.41 ( eqinunit
% 2.04/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.04/1.41 ( ( Xx = Xy )
% 2.04/1.41 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(singletonsswitch_type,type,
% 2.04/1.41 singletonsswitch: $o ).
% 2.04/1.41
% 2.04/1.41 thf(singletonsswitch,definition,
% 2.04/1.41 ( singletonsswitch
% 2.04/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.04/1.41 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.04/1.41 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(upairsetE_type,type,
% 2.04/1.41 upairsetE: $o ).
% 2.04/1.41
% 2.04/1.41 thf(upairsetE,definition,
% 2.04/1.41 ( upairsetE
% 2.04/1.41 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.04/1.41 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.04/1.41 => ( ( Xz = Xx )
% 2.04/1.41 | ( Xz = Xy ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(upairsetIL_type,type,
% 2.04/1.41 upairsetIL: $o ).
% 2.04/1.41
% 2.04/1.41 thf(upairsetIL,definition,
% 2.04/1.41 ( upairsetIL
% 2.04/1.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(upairsetIR_type,type,
% 2.04/1.41 upairsetIR: $o ).
% 2.04/1.41
% 2.04/1.41 thf(upairsetIR,definition,
% 2.04/1.41 ( upairsetIR
% 2.04/1.41 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(emptyE1_type,type,
% 2.04/1.41 emptyE1: $o ).
% 2.04/1.41
% 2.04/1.41 thf(emptyE1,definition,
% 2.04/1.41 ( emptyE1
% 2.04/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.04/1.41 ( ? [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 & ( Xphi @ Xx ) )
% 2.04/1.41 => ( ( ( dsetconstr @ A
% 2.04/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.04/1.41 = emptyset )
% 2.04/1.41 => $false ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(vacuousDall_type,type,
% 2.04/1.41 vacuousDall: $o ).
% 2.04/1.41
% 2.04/1.41 thf(vacuousDall,definition,
% 2.04/1.41 ( vacuousDall
% 2.04/1.41 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ emptyset )
% 2.04/1.41 => ( Xphi @ Xx ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(quantDeMorgan1_type,type,
% 2.04/1.41 quantDeMorgan1: $o ).
% 2.04/1.41
% 2.04/1.41 thf(quantDeMorgan1,definition,
% 2.04/1.41 ( quantDeMorgan1
% 2.04/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.04/1.41 ( ~ ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 => ( Xphi @ Xx ) )
% 2.04/1.41 => ? [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(quantDeMorgan2_type,type,
% 2.04/1.41 quantDeMorgan2: $o ).
% 2.04/1.41
% 2.04/1.41 thf(quantDeMorgan2,definition,
% 2.04/1.41 ( quantDeMorgan2
% 2.04/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.04/1.41 ( ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 => ~ ( Xphi @ Xx ) )
% 2.04/1.41 => ~ ? [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(quantDeMorgan3_type,type,
% 2.04/1.41 quantDeMorgan3: $o ).
% 2.04/1.41
% 2.04/1.41 thf(quantDeMorgan3,definition,
% 2.04/1.41 ( quantDeMorgan3
% 2.04/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.04/1.41 ( ~ ? [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 & ( Xphi @ Xx ) )
% 2.04/1.41 => ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(quantDeMorgan4_type,type,
% 2.04/1.41 quantDeMorgan4: $o ).
% 2.04/1.41
% 2.04/1.41 thf(quantDeMorgan4,definition,
% 2.04/1.41 ( quantDeMorgan4
% 2.04/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.04/1.41 ( ? [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 & ~ ( Xphi @ Xx ) )
% 2.04/1.41 => ~ ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(prop2setI_type,type,
% 2.04/1.41 prop2setI: $o ).
% 2.04/1.41
% 2.04/1.41 thf(prop2setI,definition,
% 2.04/1.41 ( prop2setI
% 2.04/1.41 = ( ! [Xphi: $o] :
% 2.04/1.41 ( Xphi
% 2.04/1.41 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(set2prop_type,type,
% 2.04/1.41 set2prop: $i > $o ).
% 2.04/1.41
% 2.04/1.41 thf(prop2set2propI_type,type,
% 2.04/1.41 prop2set2propI: $o ).
% 2.04/1.41
% 2.04/1.41 thf(prop2set2propI,definition,
% 2.04/1.41 ( prop2set2propI
% 2.04/1.41 = ( ! [Xphi: $o] :
% 2.04/1.41 ( Xphi
% 2.04/1.41 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(notdexE_type,type,
% 2.04/1.41 notdexE: $o ).
% 2.04/1.41
% 2.04/1.41 thf(notdexE,definition,
% 2.04/1.41 ( notdexE
% 2.04/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.04/1.41 ( ~ ? [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 & ( Xphi @ Xx ) )
% 2.04/1.41 => ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(notdallE_type,type,
% 2.04/1.41 notdallE: $o ).
% 2.04/1.41
% 2.04/1.41 thf(notdallE,definition,
% 2.04/1.41 ( notdallE
% 2.04/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.04/1.41 ( ~ ! [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 => ( Xphi @ Xx ) )
% 2.04/1.41 => ? [Xx: $i] :
% 2.04/1.41 ( ( in @ Xx @ A )
% 2.04/1.41 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(exuI1_type,type,
% 2.04/1.41 exuI1: $o ).
% 2.04/1.41
% 2.04/1.41 thf(exuI1,definition,
% 2.04/1.41 ( exuI1
% 2.04/1.41 = ( ! [Xphi: $i > $o] :
% 2.04/1.41 ( ? [Xx: $i] :
% 2.04/1.41 ( ( Xphi @ Xx )
% 2.04/1.41 & ! [Xy: $i] :
% 2.04/1.41 ( ( Xphi @ Xy )
% 2.04/1.41 => ( Xx = Xy ) ) )
% 2.04/1.41 => ( exu
% 2.04/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(exuI3_type,type,
% 2.04/1.41 exuI3: $o ).
% 2.04/1.41
% 2.04/1.41 thf(exuI3,definition,
% 2.04/1.41 ( exuI3
% 2.04/1.41 = ( ! [Xphi: $i > $o] :
% 2.04/1.41 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.04/1.41 => ( ! [Xx: $i,Xy: $i] :
% 2.04/1.41 ( ( Xphi @ Xx )
% 2.04/1.41 => ( ( Xphi @ Xy )
% 2.04/1.41 => ( Xx = Xy ) ) )
% 2.04/1.41 => ( exu
% 2.04/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(exuI2_type,type,
% 2.04/1.41 exuI2: $o ).
% 2.04/1.41
% 2.04/1.41 thf(exuI2,definition,
% 2.04/1.41 ( exuI2
% 2.04/1.41 = ( ! [Xphi: $i > $o] :
% 2.04/1.41 ( ? [Xx: $i] :
% 2.04/1.41 ! [Xy: $i] :
% 2.04/1.41 ( ( Xphi @ Xy )
% 2.04/1.41 <=> ( Xy = Xx ) )
% 2.04/1.41 => ( exu
% 2.04/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(inCongP_type,type,
% 2.04/1.41 inCongP: $o ).
% 2.04/1.41
% 2.04/1.41 thf(inCongP,definition,
% 2.04/1.41 ( inCongP
% 2.04/1.41 = ( ! [A: $i,B: $i] :
% 2.04/1.41 ( ( A = B )
% 2.04/1.41 => ! [Xx: $i,Xy: $i] :
% 2.04/1.41 ( ( Xx = Xy )
% 2.04/1.41 => ( ( in @ Xx @ A )
% 2.04/1.41 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(in__Cong_type,type,
% 2.04/1.41 in__Cong: $o ).
% 2.04/1.41
% 2.04/1.41 thf(in__Cong,definition,
% 2.04/1.41 ( in__Cong
% 2.04/1.41 = ( ! [A: $i,B: $i] :
% 2.04/1.41 ( ( A = B )
% 2.04/1.41 => ! [Xx: $i,Xy: $i] :
% 2.04/1.41 ( ( Xx = Xy )
% 2.04/1.41 => ( ( in @ Xx @ A )
% 2.04/1.41 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(exuE3u_type,type,
% 2.04/1.41 exuE3u: $o ).
% 2.04/1.41
% 2.04/1.41 thf(exuE3u,definition,
% 2.04/1.41 ( exuE3u
% 2.04/1.41 = ( ! [Xphi: $i > $o] :
% 2.04/1.41 ( ( exu
% 2.04/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.04/1.41 => ! [Xx: $i,Xy: $i] :
% 2.04/1.41 ( ( Xphi @ Xx )
% 2.04/1.41 => ( ( Xphi @ Xy )
% 2.04/1.41 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(exu__Cong_type,type,
% 2.04/1.41 exu__Cong: $o ).
% 2.04/1.41
% 2.04/1.41 thf(exu__Cong,definition,
% 2.04/1.41 ( exu__Cong
% 2.04/1.41 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.04/1.41 ( ! [Xx: $i,Xy: $i] :
% 2.04/1.41 ( ( Xx = Xy )
% 2.04/1.41 => ( ( Xphi @ Xx )
% 2.04/1.41 <=> ( Xpsi @ Xy ) ) )
% 2.04/1.41 => ( ( exu
% 2.04/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.04/1.41 <=> ( exu
% 2.04/1.41 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(emptyset__Cong_type,type,
% 2.04/1.41 emptyset__Cong: $o ).
% 2.04/1.41
% 2.04/1.41 thf(emptyset__Cong,definition,
% 2.04/1.41 ( emptyset__Cong
% 2.04/1.41 = ( emptyset = emptyset ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoin__Cong_type,type,
% 2.04/1.41 setadjoin__Cong: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setadjoin__Cong,definition,
% 2.04/1.41 ( setadjoin__Cong
% 2.04/1.41 = ( ! [Xx: $i,Xy: $i] :
% 2.04/1.41 ( ( Xx = Xy )
% 2.04/1.41 => ! [Xz: $i,Xu: $i] :
% 2.04/1.41 ( ( Xz = Xu )
% 2.04/1.41 => ( ( setadjoin @ Xx @ Xz )
% 2.04/1.41 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(powerset__Cong_type,type,
% 2.04/1.41 powerset__Cong: $o ).
% 2.04/1.41
% 2.04/1.41 thf(powerset__Cong,definition,
% 2.04/1.41 ( powerset__Cong
% 2.04/1.41 = ( ! [A: $i,B: $i] :
% 2.04/1.41 ( ( A = B )
% 2.04/1.41 => ( ( powerset @ A )
% 2.04/1.41 = ( powerset @ B ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(setunion__Cong_type,type,
% 2.04/1.41 setunion__Cong: $o ).
% 2.04/1.41
% 2.04/1.41 thf(setunion__Cong,definition,
% 2.04/1.41 ( setunion__Cong
% 2.04/1.41 = ( ! [A: $i,B: $i] :
% 2.04/1.41 ( ( A = B )
% 2.04/1.41 => ( ( setunion @ A )
% 2.04/1.41 = ( setunion @ B ) ) ) ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(omega__Cong_type,type,
% 2.04/1.41 omega__Cong: $o ).
% 2.04/1.41
% 2.04/1.41 thf(omega__Cong,definition,
% 2.04/1.41 ( omega__Cong
% 2.04/1.41 = ( omega = omega ) ) ).
% 2.04/1.41
% 2.04/1.41 thf(exuEu,conjecture,
% 2.04/1.41 ( setextAx
% 2.04/1.41 => ( emptysetAx
% 2.04/1.41 => ( setadjoinAx
% 2.04/1.41 => ( powersetAx
% 2.04/1.41 => ( setunionAx
% 2.04/1.41 => ( omega0Ax
% 2.04/1.41 => ( omegaSAx
% 2.04/1.41 => ( omegaIndAx
% 2.04/1.41 => ( replAx
% 2.04/1.41 => ( foundationAx
% 2.04/1.41 => ( wellorderingAx
% 2.04/1.41 => ( descrp
% 2.04/1.41 => ( dsetconstrI
% 2.04/1.41 => ( dsetconstrEL
% 2.04/1.41 => ( dsetconstrER
% 2.04/1.41 => ( exuE1
% 2.04/1.41 => ( prop2setE
% 2.04/1.41 => ( emptysetE
% 2.04/1.41 => ( emptysetimpfalse
% 2.04/1.41 => ( notinemptyset
% 2.04/1.41 => ( exuE3e
% 2.04/1.41 => ( setext
% 2.04/1.41 => ( emptyI
% 2.04/1.41 => ( noeltsimpempty
% 2.04/1.41 => ( setbeta
% 2.04/1.41 => ( nonemptyE1
% 2.04/1.41 => ( nonemptyI
% 2.04/1.41 => ( nonemptyI1
% 2.04/1.41 => ( setadjoinIL
% 2.04/1.41 => ( emptyinunitempty
% 2.04/1.41 => ( setadjoinIR
% 2.04/1.41 => ( setadjoinE
% 2.04/1.41 => ( setadjoinOr
% 2.04/1.41 => ( setoftrueEq
% 2.04/1.41 => ( powersetI
% 2.04/1.41 => ( emptyinPowerset
% 2.04/1.41 => ( emptyInPowerset
% 2.04/1.41 => ( powersetE
% 2.04/1.41 => ( setunionI
% 2.04/1.41 => ( setunionE
% 2.04/1.41 => ( subPowSU
% 2.04/1.41 => ( exuE2
% 2.04/1.41 => ( nonemptyImpWitness
% 2.04/1.41 => ( uniqinunit
% 2.04/1.41 => ( notinsingleton
% 2.04/1.41 => ( eqinunit
% 2.04/1.41 => ( singletonsswitch
% 2.04/1.41 => ( upairsetE
% 2.04/1.41 => ( upairsetIL
% 2.04/1.41 => ( upairsetIR
% 2.04/1.41 => ( emptyE1
% 2.04/1.41 => ( vacuousDall
% 2.04/1.41 => ( quantDeMorgan1
% 2.04/1.41 => ( quantDeMorgan2
% 2.04/1.41 => ( quantDeMorgan3
% 2.04/1.41 => ( quantDeMorgan4
% 2.04/1.41 => ( prop2setI
% 2.04/1.41 => ( prop2set2propI
% 2.04/1.41 => ( notdexE
% 2.04/1.41 => ( notdallE
% 2.04/1.41 => ( exuI1
% 2.04/1.41 => ( exuI3
% 2.04/1.41 => ( exuI2
% 2.04/1.41 => ( inCongP
% 2.04/1.41 => ( in__Cong
% 2.04/1.41 => ( exuE3u
% 2.04/1.41 => ( exu__Cong
% 2.04/1.41 => ( emptyset__Cong
% 2.04/1.41 => ( setadjoin__Cong
% 2.04/1.41 => ( powerset__Cong
% 2.04/1.41 => ( setunion__Cong
% 2.04/1.41 => ( omega__Cong
% 2.04/1.41 => ! [Xphi: $i > $o] :
% 2.04/1.41 ( ( exu
% 2.04/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.04/1.41 => ! [Xx: $i,Xy: $i] :
% 2.04/1.41 ( ( Xphi @ Xx )
% 2.04/1.41 => ( ( Xphi @ Xy )
% 2.04/1.41 => ( Xx = Xy ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.04/1.41 ------------------------
% 2.04/1.43 % (8455)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_8219 for (2999ds/183Mi)
% 2.04/1.43 % (8461)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_8219 for (2999ds/18Mi)
% 2.04/1.43 % (8461)Instruction limit reached!
% 2.04/1.43 % (8461)------------------------------
% 2.04/1.43 % (8461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.43 % (8461)Termination reason: Unknown
% 2.04/1.43 % (8461)Termination phase: Preprocessing 3
% 2.04/1.43
% 2.04/1.43 % (8461)Memory used [KB]: 1535
% 2.04/1.43 % (8461)Time elapsed: 0.007 s
% 2.04/1.43 % (8461)Instructions burned: 18 (million)
% 2.04/1.43 % (8461)------------------------------
% 2.04/1.43 % (8461)------------------------------
% 2.04/1.43 % (8456)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_8219 for (2999ds/4Mi)
% 2.04/1.43 % (8457)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_8219 for (2999ds/27Mi)
% 2.04/1.43 % (8458)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_8219 for (2999ds/2Mi)
% 2.04/1.43 % (8459)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_8219 for (2999ds/2Mi)
% 2.04/1.43 % (8460)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_8219 for (2999ds/275Mi)
% 2.04/1.43 % (8456)Instruction limit reached!
% 2.04/1.43 % (8456)------------------------------
% 2.04/1.43 % (8456)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.43 % (8456)Termination reason: Unknown
% 2.04/1.43 % (8456)Termination phase: shuffling
% 2.04/1.43
% 2.04/1.43 % (8456)Memory used [KB]: 1151
% 2.04/1.43 % (8456)Time elapsed: 0.004 s
% 2.04/1.43 % (8456)Instructions burned: 4 (million)
% 2.04/1.43 % (8456)------------------------------
% 2.04/1.43 % (8456)------------------------------
% 2.04/1.43 % (8458)Instruction limit reached!
% 2.04/1.43 % (8458)------------------------------
% 2.04/1.43 % (8458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.43 % (8458)Termination reason: Unknown
% 2.04/1.43 % (8458)Termination phase: shuffling
% 2.04/1.43
% 2.04/1.43 % (8458)Memory used [KB]: 1023
% 2.04/1.43 % (8458)Time elapsed: 0.003 s
% 2.04/1.43 % (8458)Instructions burned: 2 (million)
% 2.04/1.43 % (8458)------------------------------
% 2.04/1.43 % (8458)------------------------------
% 2.04/1.43 % (8459)Instruction limit reached!
% 2.04/1.43 % (8459)------------------------------
% 2.04/1.43 % (8459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.43 % (8459)Termination reason: Unknown
% 2.04/1.43 % (8459)Termination phase: shuffling
% 2.04/1.43
% 2.04/1.43 % (8459)Memory used [KB]: 1023
% 2.04/1.43 % (8459)Time elapsed: 0.003 s
% 2.04/1.43 % (8459)Instructions burned: 2 (million)
% 2.04/1.43 % (8459)------------------------------
% 2.04/1.43 % (8459)------------------------------
% 2.04/1.44 % (8466)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_8219 for (2999ds/3Mi)
% 2.04/1.44 % (8466)Instruction limit reached!
% 2.04/1.44 % (8466)------------------------------
% 2.04/1.44 % (8466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.44 % (8466)Termination reason: Unknown
% 2.04/1.44 % (8466)Termination phase: shuffling
% 2.04/1.44
% 2.04/1.44 % (8466)Memory used [KB]: 1151
% 2.04/1.44 % (8466)Time elapsed: 0.002 s
% 2.04/1.44 % (8466)Instructions burned: 4 (million)
% 2.04/1.44 % (8466)------------------------------
% 2.04/1.44 % (8466)------------------------------
% 2.04/1.45 % (8457)Instruction limit reached!
% 2.04/1.45 % (8457)------------------------------
% 2.04/1.45 % (8457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.45 % (8457)Termination reason: Unknown
% 2.04/1.45 % (8457)Termination phase: Function definition elimination
% 2.04/1.45
% 2.04/1.45 % (8457)Memory used [KB]: 1407
% 2.04/1.45 % (8457)Time elapsed: 0.016 s
% 2.04/1.45 % (8457)Instructions burned: 28 (million)
% 2.04/1.45 % (8457)------------------------------
% 2.04/1.45 % (8457)------------------------------
% 2.04/1.45 % (8475)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_8219 for (2999ds/1041Mi)
% 2.04/1.45 % (8470)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_8219 for (2999ds/37Mi)
% 2.04/1.45 % (8471)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_8219 for (2999ds/15Mi)
% 2.04/1.45 % (8472)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_8219 for (2999ds/3Mi)
% 2.04/1.45 % (8472)Instruction limit reached!
% 2.04/1.45 % (8472)------------------------------
% 2.04/1.45 % (8472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.45 % (8472)Termination reason: Unknown
% 2.04/1.45 % (8472)Termination phase: shuffling
% 2.04/1.45
% 2.04/1.45 % (8472)Memory used [KB]: 1151
% 2.04/1.45 % (8472)Time elapsed: 0.004 s
% 2.04/1.45 % (8472)Instructions burned: 4 (million)
% 2.04/1.45 % (8472)------------------------------
% 2.04/1.45 % (8472)------------------------------
% 2.04/1.46 % (8471)Instruction limit reached!
% 2.04/1.46 % (8471)------------------------------
% 2.04/1.46 % (8471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.46 % (8471)Termination reason: Unknown
% 2.04/1.46 % (8471)Termination phase: Preprocessing 3
% 2.04/1.46
% 2.04/1.46 % (8471)Memory used [KB]: 1535
% 2.04/1.46 % (8471)Time elapsed: 0.011 s
% 2.04/1.46 % (8471)Instructions burned: 16 (million)
% 2.04/1.46 % (8471)------------------------------
% 2.04/1.46 % (8471)------------------------------
% 2.04/1.46 % (8455)First to succeed.
% 2.04/1.46 % (8476)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_8219 for (2999ds/7Mi)
% 2.04/1.47 % (8455)Refutation found. Thanks to Tanya!
% 2.04/1.47 % SZS status Theorem for DTF2THF_8219
% 2.04/1.47 % SZS output start Proof for DTF2THF_8219
% 2.04/1.47 thf(func_def_0, type, in: $i > $i > $o).
% 2.04/1.47 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.04/1.47 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.04/1.47 thf(func_def_8, type, powerset: $i > $i).
% 2.04/1.47 thf(func_def_10, type, setunion: $i > $i).
% 2.04/1.47 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.04/1.47 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.04/1.47 thf(func_def_26, type, prop2set: $o > $i).
% 2.04/1.47 thf(func_def_36, type, nonempty: $i > $o).
% 2.04/1.47 thf(func_def_69, type, set2prop: $i > $o).
% 2.04/1.47 thf(func_def_101, type, sK0: $i > $o).
% 2.04/1.47 thf(func_def_105, type, sK4: $i > $i > $i).
% 2.04/1.47 thf(func_def_106, type, sK5: $i > ($i > $i > $o) > $i).
% 2.04/1.47 thf(func_def_107, type, sK6: $i > ($i > $i > $o) > $i).
% 2.04/1.47 thf(func_def_108, type, sK7: $i > $i > ($i > $i > $o) > $i).
% 2.04/1.47 thf(func_def_109, type, sK8: $i > $i > ($i > $i > $o) > $i).
% 2.04/1.47 thf(func_def_110, type, sK9: $i > $i > ($i > $i > $o) > $i).
% 2.04/1.47 thf(func_def_111, type, sK10: $i > $i > ($i > $i > $o) > $i).
% 2.04/1.47 thf(func_def_112, type, sK11: ($i > $o) > $i > $i).
% 2.04/1.47 thf(func_def_113, type, sK12: ($i > $o) > $i > $i).
% 2.04/1.47 thf(func_def_114, type, sK13: $i > $i).
% 2.04/1.47 thf(func_def_115, type, sK14: $i > $i).
% 2.04/1.47 thf(func_def_116, type, sK15: $i > $i).
% 2.04/1.47 thf(func_def_117, type, sK16: $i > $i > $i).
% 2.04/1.47 thf(func_def_118, type, sK17: $i > $i > $i).
% 2.04/1.47 thf(func_def_119, type, sK18: $i > $i > $i).
% 2.04/1.47 thf(func_def_120, type, sK19: $i > $i > $i).
% 2.04/1.47 thf(func_def_121, type, sK20: $i > $i > $i > $i).
% 2.04/1.47 thf(func_def_122, type, sK21: $o > $i > $i > $i).
% 2.04/1.47 thf(func_def_123, type, sK22: ($i > $o) > $i).
% 2.04/1.47 thf(func_def_124, type, sK23: ($i > $o) > $i > $i).
% 2.04/1.47 thf(func_def_125, type, sK24: ($i > $o) > $i > $i).
% 2.04/1.47 thf(func_def_126, type, sK25: $i > $i > $i).
% 2.04/1.47 thf(func_def_127, type, sK26: ($i > $o) > $i).
% 2.04/1.47 thf(func_def_128, type, sK27: ($i > $o) > $i > $i).
% 2.04/1.47 thf(func_def_129, type, sK28: $i > $i).
% 2.04/1.47 thf(func_def_130, type, sK29: ($i > $o) > $i > $i).
% 2.04/1.47 thf(func_def_131, type, sK30: ($i > $o) > $i > $i).
% 2.04/1.47 thf(func_def_132, type, sK31: $i > ($i > $o) > $i).
% 2.04/1.47 thf(func_def_133, type, sK32: ($i > $o) > $i).
% 2.04/1.47 thf(func_def_134, type, sK33: ($i > $o) > $i > $i).
% 2.04/1.47 thf(func_def_135, type, sK34: ($i > $o) > $i > $i).
% 2.04/1.47 thf(func_def_136, type, sK35: $i > $i).
% 2.04/1.47 thf(func_def_137, type, sK36: $i > $i).
% 2.04/1.47 thf(func_def_139, type, sK38: $i > $i > $i).
% 2.04/1.47 thf(func_def_140, type, sK39: ($i > $o) > $i > $i).
% 2.04/1.47 thf(func_def_141, type, sK40: ($i > $o) > $i > $i).
% 2.04/1.47 thf(func_def_142, type, sK41: $i > $i > $i).
% 2.04/1.47 thf(func_def_143, type, sK42: ($i > $o) > $i).
% 2.04/1.47 thf(func_def_144, type, sK43: ($i > $o) > $i > $i).
% 2.04/1.47 thf(func_def_145, type, sK44: ($i > $o) > $i > $i).
% 2.04/1.47 thf(func_def_146, type, sK45: $i > $i).
% 2.04/1.47 thf(func_def_147, type, sK46: ($i > $o) > $i > $i).
% 2.04/1.47 thf(func_def_148, type, sK47: ($i > $o) > $i).
% 2.04/1.47 thf(func_def_149, type, sK48: ($i > $o) > $i).
% 2.04/1.47 thf(func_def_150, type, sK49: ($i > $o) > $i).
% 2.04/1.47 thf(func_def_151, type, sK50: $i > $i > $i).
% 2.04/1.47 thf(func_def_152, type, sK51: $i > $i > $i).
% 2.04/1.47 thf(func_def_153, type, sK52: $i > $i > $i).
% 2.04/1.47 thf(func_def_154, type, sK53: ($i > $o) > $i).
% 2.04/1.47 thf(func_def_155, type, sK54: $i > ($i > $o) > $i).
% 2.04/1.47 thf(func_def_156, type, sK55: $i > ($i > $o) > $i).
% 2.04/1.47 thf(func_def_157, type, sK56: ($i > $o) > $i).
% 2.04/1.47 thf(func_def_158, type, sK57: ($i > $o) > ($i > $o) > $i).
% 2.04/1.47 thf(func_def_159, type, sK58: ($i > $o) > ($i > $o) > $i).
% 2.04/1.47 thf(func_def_160, type, sK59: $i > ($i > $o) > $i).
% 2.04/1.47 thf(func_def_161, type, sK60: $i > ($i > $o) > $i).
% 2.04/1.47 thf(func_def_162, type, sK61: $i > ($i > $o) > $i).
% 2.04/1.47 thf(func_def_163, type, sK62: $i > ($i > $o) > $i).
% 2.04/1.47 thf(func_def_164, type, sK63: $i > ($i > $o) > $i).
% 2.04/1.47 thf(func_def_165, type, sK64: $i > ($i > $o) > $i).
% 2.04/1.47 thf(func_def_166, type, sK65: ($i > $o) > ($i > $o) > $i).
% 2.04/1.47 thf(func_def_167, type, sK66: ($i > $o) > ($i > $o) > $i).
% 2.04/1.47 thf(func_def_168, type, sK67: $i > ($i > $o) > $i).
% 2.04/1.47 thf(func_def_169, type, sK68: ($i > $o) > $i).
% 2.04/1.47 thf(func_def_170, type, sK69: ($i > $o) > $i).
% 2.04/1.47 thf(func_def_171, type, sK70: ($i > $o) > $i).
% 2.04/1.47 thf(func_def_172, type, sK71: ($i > $o) > $i).
% 2.04/1.47 thf(f1528,plain,(
% 2.04/1.47 $false),
% 2.04/1.47 inference(subsumption_resolution,[status(thm)],[f1526,f288])).
% 2.04/1.47 thf(f288,plain,(
% 2.04/1.47 (sK2 != sK1)),
% 2.04/1.47 inference(cnf_transformation,[status(thm)],[f227])).
% 2.04/1.47 thf(f227,plain,(
% 2.04/1.47 (emptysetE = $true) & (setadjoinIR = $true) & (emptyinPowerset = $true) & (emptyE1 = $true) & (setext = $true) & (subPowSU = $true) & (exu__Cong = $true) & (notdallE = $true) & (uniqinunit = $true) & (powersetAx = $true) & (exuE3u = $true) & (replAx = $true) & (vacuousDall = $true) & (noeltsimpempty = $true) & (setbeta = $true) & (emptysetimpfalse = $true) & (setadjoinIL = $true) & (omegaSAx = $true) & (upairsetIR = $true) & (setadjoinOr = $true) & (setadjoin__Cong = $true) & (in__Cong = $true) & (exuI2 = $true) & (omega__Cong = $true) & (($true = (exu @ (^[Y0 : $i]: (sK0 @ Y0)))) & ((sK2 != sK1) & ($true = (sK0 @ sK1)) & ($true = (sK0 @ sK2)))) & (quantDeMorgan3 = $true) & (upairsetE = $true) & (quantDeMorgan2 = $true) & (emptyset__Cong = $true) & (powerset__Cong = $true) & (nonemptyI = $true) & (setunionI = $true) & (powersetI = $true) & (setunionE = $true) & (omegaIndAx = $true) & (omega0Ax = $true) & (setunionAx = $true) & (emptyInPowerset = $true) & (exuI1 = $true) & (wellorderingAx = $true) & (singletonsswitch = $true) & (nonemptyE1 = $true) & (quantDeMorgan4 = $true) & (setadjoinAx = $true) & (powersetE = $true) & (setunion__Cong = $true) & (notdexE = $true) & (exuI3 = $true) & (exuE3e = $true) & (notinemptyset = $true) & (upairsetIL = $true) & (notinsingleton = $true) & (emptysetAx = $true) & (dsetconstrI = $true) & (dsetconstrER = $true) & (nonemptyImpWitness = $true) & (prop2setE = $true) & (dsetconstrEL = $true) & (emptyI = $true) & (foundationAx = $true) & (inCongP = $true) & (setoftrueEq = $true) & (prop2setI = $true) & (prop2set2propI = $true) & (setextAx = $true) & (descrp = $true) & (quantDeMorgan1 = $true) & (emptyinunitempty = $true) & (exuE1 = $true) & (nonemptyI1 = $true) & (setadjoinE = $true) & (eqinunit = $true) & (exuE2 = $true)),
% 2.04/1.47 inference(skolemisation,[status(esa),new_symbols(skolem,[sK0,sK1,sK2])],[f224,f226,f225])).
% 2.04/1.47 thf(f225,plain,(
% 2.04/1.47 ? [X0 : $i > $o] : (((exu @ (^[Y0 : $i]: (X0 @ Y0))) = $true) & ? [X1 : $i,X2 : $i] : ((X1 != X2) & ((X0 @ X1) = $true) & ((X0 @ X2) = $true))) => (($true = (exu @ (^[Y0 : $i]: (sK0 @ Y0)))) & ? [X2 : $i,X1 : $i] : ((X1 != X2) & ($true = (sK0 @ X1)) & ($true = (sK0 @ X2))))),
% 2.04/1.47 introduced(definition,[],[choice_axiom])).
% 2.04/1.47 thf(f226,plain,(
% 2.04/1.47 ? [X2 : $i,X1 : $i] : ((X1 != X2) & ($true = (sK0 @ X1)) & ($true = (sK0 @ X2))) => ((sK2 != sK1) & ($true = (sK0 @ sK1)) & ($true = (sK0 @ sK2)))),
% 2.04/1.47 introduced(definition,[],[choice_axiom])).
% 2.04/1.47 thf(f224,plain,(
% 2.04/1.47 (emptysetE = $true) & (setadjoinIR = $true) & (emptyinPowerset = $true) & (emptyE1 = $true) & (setext = $true) & (subPowSU = $true) & (exu__Cong = $true) & (notdallE = $true) & (uniqinunit = $true) & (powersetAx = $true) & (exuE3u = $true) & (replAx = $true) & (vacuousDall = $true) & (noeltsimpempty = $true) & (setbeta = $true) & (emptysetimpfalse = $true) & (setadjoinIL = $true) & (omegaSAx = $true) & (upairsetIR = $true) & (setadjoinOr = $true) & (setadjoin__Cong = $true) & (in__Cong = $true) & (exuI2 = $true) & (omega__Cong = $true) & ? [X0 : $i > $o] : (((exu @ (^[Y0 : $i]: (X0 @ Y0))) = $true) & ? [X1 : $i,X2 : $i] : ((X1 != X2) & ((X0 @ X1) = $true) & ((X0 @ X2) = $true))) & (quantDeMorgan3 = $true) & (upairsetE = $true) & (quantDeMorgan2 = $true) & (emptyset__Cong = $true) & (powerset__Cong = $true) & (nonemptyI = $true) & (setunionI = $true) & (powersetI = $true) & (setunionE = $true) & (omegaIndAx = $true) & (omega0Ax = $true) & (setunionAx = $true) & (emptyInPowerset = $true) & (exuI1 = $true) & (wellorderingAx = $true) & (singletonsswitch = $true) & (nonemptyE1 = $true) & (quantDeMorgan4 = $true) & (setadjoinAx = $true) & (powersetE = $true) & (setunion__Cong = $true) & (notdexE = $true) & (exuI3 = $true) & (exuE3e = $true) & (notinemptyset = $true) & (upairsetIL = $true) & (notinsingleton = $true) & (emptysetAx = $true) & (dsetconstrI = $true) & (dsetconstrER = $true) & (nonemptyImpWitness = $true) & (prop2setE = $true) & (dsetconstrEL = $true) & (emptyI = $true) & (foundationAx = $true) & (inCongP = $true) & (setoftrueEq = $true) & (prop2setI = $true) & (prop2set2propI = $true) & (setextAx = $true) & (descrp = $true) & (quantDeMorgan1 = $true) & (emptyinunitempty = $true) & (exuE1 = $true) & (nonemptyI1 = $true) & (setadjoinE = $true) & (eqinunit = $true) & (exuE2 = $true)),
% 2.04/1.47 inference(flattening,[status(thm)],[f223])).
% 2.04/1.47 thf(f223,plain,(
% 2.04/1.47 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i > $o] : (? [X1 : $i,X2 : $i] : (((X1 != X2) & ((X0 @ X1) = $true)) & ((X0 @ X2) = $true)) & ((exu @ (^[Y0 : $i]: (X0 @ Y0))) = $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)),
% 2.04/1.47 inference(ennf_transformation,[status(thm)],[f222])).
% 2.04/1.47 thf(f222,plain,(
% 2.04/1.47 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((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) => ! [X0 : $i > $o] : (((exu @ (^[Y0 : $i]: (X0 @ Y0))) = $true) => ! [X1 : $i,X2 : $i] : (((X0 @ X2) = $true) => (((X0 @ X1) = $true) => (X1 = X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.04/1.47 inference(rectify,[status(thm)],[f110])).
% 2.04/1.47 thf(f110,plain,(
% 2.04/1.47 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((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) => ! [X0 : $i > $o] : (((exu @ (^[Y0 : $i]: (X0 @ Y0))) = $true) => ! [X2 : $i,X3 : $i] : (($true = (X0 @ X3)) => (((X0 @ X2) = $true) => (X2 = X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.04/1.47 inference(fool_elimination,[status(thm)],[f109])).
% 2.04/1.47 thf(f109,plain,(
% 2.04/1.47 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (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 => ! [X0 : $i > $o] : ((exu @ (^[X1 : $i] : (X0 @ X1))) => ! [X2 : $i,X3 : $i] : ((X0 @ X3) => ((X0 @ X2) => (X2 = X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.04/1.47 inference(rectify,[status(thm)],[f76])).
% 2.04/1.47 thf(f76,negated_conjecture,(
% 2.04/1.47 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (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 => ! [X0 : $i > $o] : ((exu @ (^[X1 : $i] : (X0 @ X1))) => ! [X2 : $i,X1 : $i] : ((X0 @ X1) => ((X0 @ X2) => (X1 = X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.04/1.47 inference(negated_conjecture,[status(cth)],[f75])).
% 2.04/1.47 thf(f75,conjecture,(
% 2.04/1.47 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (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 => ! [X0 : $i > $o] : ((exu @ (^[X1 : $i] : (X0 @ X1))) => ! [X2 : $i,X1 : $i] : ((X0 @ X1) => ((X0 @ X2) => (X1 = X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.04/1.47 file('/export/starexec/sandbox2/tmp/tmp.5KNgBg82Lh/DTF2THF_8219.p',exuEu)).
% 2.04/1.47 thf(f1526,plain,(
% 2.04/1.47 (sK2 = sK1)),
% 2.04/1.47 inference(superposition,[status(thm)],[f1522,f1523])).
% 2.04/1.47 thf(f1523,plain,(
% 2.04/1.47 (sK37 = sK1)),
% 2.04/1.47 inference(trivial_inequality_removal,[status(thm)],[f1512])).
% 2.04/1.47 thf(f1512,plain,(
% 2.04/1.47 ($true = $false) | (sK37 = sK1)),
% 2.04/1.47 inference(superposition,[status(thm)],[f994,f287])).
% 2.04/1.47 thf(f287,plain,(
% 2.04/1.47 ($true = (sK0 @ sK1))),
% 2.04/1.47 inference(cnf_transformation,[status(thm)],[f227])).
% 2.04/1.47 thf(f994,plain,(
% 2.04/1.47 ( ! [X1 : $i] : (((sK0 @ X1) = $false) | (sK37 = X1)) )),
% 2.04/1.47 inference(equality_proxy_clausification,[status(thm)],[f993])).
% 2.04/1.47 thf(f993,plain,(
% 2.04/1.47 ( ! [X1 : $i] : (($true = (sK37 = X1)) | ((sK0 @ X1) = $false)) )),
% 2.04/1.47 inference(binary_proxy_clausification,[status(thm)],[f992])).
% 2.04/1.47 thf(f992,plain,(
% 2.04/1.47 ( ! [X1 : $i] : ((((sK0 @ X1) => (sK37 = X1)) = $true)) )),
% 2.04/1.47 inference(beta_eta_normalization,[status(thm)],[f991])).
% 2.04/1.47 thf(f991,plain,(
% 2.04/1.47 ( ! [X1 : $i] : ((((^[Y0 : $i]: ((sK0 @ Y0) => (sK37 = Y0))) @ X1) = $true)) )),
% 2.04/1.47 inference(pi_clausification,[status(thm)],[f989])).
% 2.04/1.47 thf(f989,plain,(
% 2.04/1.47 ($true = (!! @ $i @ (^[Y0 : $i]: ((sK0 @ Y0) => (sK37 = Y0)))))),
% 2.04/1.47 inference(binary_proxy_clausification,[status(thm)],[f988])).
% 2.04/1.47 thf(f988,plain,(
% 2.04/1.47 ($true = ((sK0 @ sK37) & (!! @ $i @ (^[Y0 : $i]: ((sK0 @ Y0) => (sK37 = Y0))))))),
% 2.04/1.47 inference(beta_eta_normalization,[status(thm)],[f987])).
% 2.04/1.47 thf(f987,plain,(
% 2.04/1.47 ($true = ((^[Y0 : $i]: ((sK0 @ Y0) & (!! @ $i @ (^[Y1 : $i]: ((sK0 @ Y1) => (Y0 = Y1)))))) @ sK37))),
% 2.04/1.47 inference(sigma_clausification,[status(thm)],[f986])).
% 2.04/1.47 thf(f986,plain,(
% 2.04/1.47 ((?? @ $i @ (^[Y0 : $i]: ((sK0 @ Y0) & (!! @ $i @ (^[Y1 : $i]: ((sK0 @ Y1) => (Y0 = Y1))))))) = $true)),
% 2.04/1.47 inference(beta_eta_normalization,[status(thm)],[f381])).
% 2.04/1.47 thf(f381,plain,(
% 2.04/1.47 ($true = ((^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (!! @ $i @ (^[Y2 : $i]: ((Y0 @ Y2) => (Y1 = Y2)))))))) @ (^[Y0 : $i]: (sK0 @ Y0))))),
% 2.04/1.47 inference(definition_unfolding,[status(thm)],[f289,f339])).
% 2.04/1.47 thf(f339,plain,(
% 2.04/1.47 (exu = (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (!! @ $i @ (^[Y2 : $i]: ((Y0 @ Y2) => (Y1 = Y2)))))))))),
% 2.04/1.47 inference(cnf_transformation,[status(thm)],[f100])).
% 2.04/1.47 thf(f100,plain,(
% 2.04/1.47 (exu = (^[Y0 : $i > $o]: (?? @ $i @ (^[Y1 : $i]: ((Y0 @ Y1) & (!! @ $i @ (^[Y2 : $i]: ((Y0 @ Y2) => (Y1 = Y2)))))))))),
% 2.04/1.47 inference(fool_elimination,[status(thm)],[f99])).
% 2.04/1.47 thf(f99,plain,(
% 2.04/1.47 (exu = (^[X0 : $i > $o] : (? [X1 : $i] : (! [X2 : $i] : ((X0 @ X2) => (X1 = X2)) & (X0 @ X1)))))),
% 2.04/1.47 inference(rectify,[status(thm)],[f1])).
% 2.04/1.47 thf(f1,axiom,(
% 2.04/1.47 (exu = (^[X0 : $i > $o] : (? [X1 : $i] : (! [X2 : $i] : ((X0 @ X2) => (X1 = X2)) & (X0 @ X1)))))),
% 2.04/1.47 file('/export/starexec/sandbox2/tmp/tmp.5KNgBg82Lh/DTF2THF_8219.p',exu)).
% 2.04/1.47 thf(f289,plain,(
% 2.04/1.47 ($true = (exu @ (^[Y0 : $i]: (sK0 @ Y0))))),
% 2.04/1.47 inference(cnf_transformation,[status(thm)],[f227])).
% 2.04/1.47 thf(f1522,plain,(
% 2.04/1.47 (sK37 = sK2)),
% 2.04/1.47 inference(trivial_inequality_removal,[status(thm)],[f1517])).
% 2.04/1.47 thf(f1517,plain,(
% 2.04/1.47 (sK37 = sK2) | ($true = $false)),
% 2.04/1.47 inference(superposition,[status(thm)],[f286,f994])).
% 2.04/1.47 thf(f286,plain,(
% 2.04/1.47 ($true = (sK0 @ sK2))),
% 2.04/1.47 inference(cnf_transformation,[status(thm)],[f227])).
% 2.04/1.47 % SZS output end Proof for DTF2THF_8219
% 2.04/1.47 % (8455)------------------------------
% 2.04/1.47 % (8455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.04/1.47 % (8455)Termination reason: Refutation
% 2.04/1.47
% 2.04/1.47 % (8455)Memory used [KB]: 6652
% 2.04/1.47 % (8455)Time elapsed: 0.040 s
% 2.04/1.47 % (8455)Instructions burned: 127 (million)
% 2.04/1.47 % (8455)------------------------------
% 2.04/1.47 % (8455)------------------------------
% 2.04/1.47 % (8451)Success in time 0.05 s
%------------------------------------------------------------------------------