%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU548^1 : TPTP v9.2.1. Bugfixed v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n027.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 09:35:12 AM UTC 2026
% Result : Theorem 2.11s 1.58s
% Output : Refutation 2.11s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 3
% Syntax : Number of formulae : 14 ( 5 unt; 0 typ; 0 def)
% Number of atoms : 1051 ( 288 equ; 0 cnn)
% Maximal formula atoms : 68 ( 75 avg)
% Number of connectives : 537 ( 12 ~; 0 |; 204 &; 52 @)
% ( 3 <=>; 266 =>; 0 <=; 0 <~>)
% Maximal formula depth : 75 ( 37 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 158 ( 155 usr; 70 con; 0-3 aty)
% Number of variables : 32 ( 0 ^; 16 !; 16 ?; 32 :)
% 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_90,type,
sP1: $i > $i > $o ).
thf(func_def_91,type,
sP2: $i > $o ).
thf(func_def_92,type,
sP3: $i > $i > $o ).
thf(func_def_93,type,
sP4: $i > $i > $o ).
thf(func_def_95,type,
sK6: $i > $o ).
thf(func_def_105,type,
sK16: $i > $o ).
thf(func_def_107,type,
sK18: ( $i > $o ) > $i > $i ).
thf(func_def_112,type,
sK23: $i > $i > $i ).
thf(func_def_119,type,
sK30: $i > $o ).
thf(func_def_123,type,
sK34: $i > $i ).
thf(func_def_131,type,
sK42: $i > $i > $i ).
thf(func_def_134,type,
sK45: $i > $i > $i ).
thf(func_def_138,type,
sK49: $i > $o ).
thf(func_def_141,type,
sK52: ( $i > $o ) > $i > $i ).
thf(func_def_143,type,
sK54: $i > $o ).
thf(func_def_146,type,
sK57: $i > $o ).
thf(func_def_149,type,
sK60: $i > ( $i > $o ) > $i ).
thf(func_def_156,type,
sK67: $i > $o ).
thf(func_def_157,type,
sK68: $i > $o ).
thf(func_def_158,type,
sK69: $i > $i ).
thf(func_def_159,type,
sK70: ( $i > $o ) > $i ).
thf(func_def_165,type,
sK76: $i > $o ).
thf(func_def_166,type,
sK77: ( $i > $o ) > $i ).
thf(func_def_167,type,
sK78: $i > $i > $i ).
thf(func_def_176,type,
sK87: $i > $o ).
thf(func_def_177,type,
sK88: $i > $o ).
thf(func_def_178,type,
sK89: $i > $i ).
thf(func_def_179,type,
sK90: ( $i > $o ) > $i ).
thf(func_def_183,type,
sK94: $i > $i > $i ).
thf(func_def_184,type,
sK95: $i > $i > $i ).
thf(func_def_187,type,
sK98: $i > $o ).
thf(func_def_189,type,
sK100: ( $i > $o ) > $i ).
thf(func_def_190,type,
sK101: ( $i > $o ) > $i ).
thf(func_def_191,type,
sK102: $i > ( $i > $o ) > $i ).
thf(func_def_192,type,
sK103: $i > $o ).
thf(func_def_196,type,
sK107: $i > $i ).
thf(func_def_197,type,
sK108: $i > $i ).
thf(func_def_198,type,
sK109: $i > $o ).
thf(func_def_200,type,
sK111: $i > $o ).
thf(func_def_205,type,
sK116: $i > ( $i > $o ) > $i ).
thf(func_def_206,type,
sK117: $i > $o ).
thf(func_def_209,type,
sK120: $i > $i > $o ).
thf(func_def_210,type,
sK121: $i > $i ).
thf(func_def_211,type,
sK122: $i > $i ).
thf(func_def_212,type,
sK123: ( $i > $i > $o ) > $i > $i ).
thf(func_def_213,type,
sK124: ( $i > $i > $o ) > $i > $i ).
thf(func_def_214,type,
sK125: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_215,type,
sK126: $i > $i ).
thf(func_def_219,type,
sK130: $i > $i ).
thf(func_def_227,type,
sK138: $i > $i > $i ).
thf(func_def_228,type,
sK139: $i > $i > $i ).
thf(func_def_229,type,
sK140: $i > $i > $i ).
thf(func_def_230,type,
sK141: $i > $i > $i ).
thf(func_def_231,type,
sK142: $i > $i > $i > $i ).
thf(func_def_232,type,
sK143: $i > $i ).
thf(func_def_233,type,
sK144: $i > $i ).
thf(func_def_234,type,
sK145: $i > $i ).
thf(func_def_235,type,
sK146: $i > $i ).
thf(func_def_236,type,
sK147: $i > $i > $i ).
thf(func_def_237,type,
sK148: $i > $i > $i > $i ).
thf(func_def_238,type,
sK149: $i > $i > $i > $i ).
thf(func_def_239,type,
sK150: $i > $i > $i ).
thf(func_def_240,type,
sK151: $i > $i > $i ).
thf(func_def_242,type,
sK153: $i > $i ).
thf(func_def_243,type,
sK154: $i > $i ).
thf(func_def_244,type,
sK155: $i > $i ).
thf(func_def_245,type,
sK156: $i > $i > $i ).
thf(func_def_249,type,
sK160: $o > $i > $i > $i ).
thf(func_def_260,type,
sK171: $i > $i ).
thf(func_def_262,type,
sK173: $i > $i ).
thf(func_def_263,type,
sK174: $i > ( $i > $o ) > $i ).
thf(func_def_264,type,
sK175: $i > $o ).
thf(func_def_268,type,
sK179: ( $i > $o ) > $i > $i ).
thf(func_def_270,type,
sK181: $i > $o ).
thf(func_def_275,type,
sK186: ( $i > $o ) > $i > $i ).
thf(func_def_277,type,
sK188: $i > $o ).
thf(func_def_280,type,
ph191:
!>[X0: $tType] : X0 ).
thf(f1269,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f1071]) ).
thf(f1071,plain,
( ( in @ sK84 @ sK82 )
!= ( in @ sK84 @ sK82 ) ),
inference(definition_unfolding,[status(thm)],[f762,f763,f764]) ).
thf(f764,plain,
sK81 = sK82,
inference(cnf_transformation,[status(thm)],[f430]) ).
thf(f430,plain,
( ( quantDeMorgan3 = $true )
& ( notinemptyset = $true )
& ( noeltsimpempty = $true )
& ( emptysetE = $true )
& ( exuE1 = $true )
& ( nonemptyI1 = $true )
& ( uniqinunit = $true )
& ( prop2set2propI = $true )
& ( singletonsswitch = $true )
& ( omega0Ax = $true )
& ( notdallE = $true )
& ( nonemptyE1 = $true )
& ( upairsetE = $true )
& ( emptyinPowerset = $true )
& ( emptyInPowerset = $true )
& ( inCongP = $true )
& ( upairsetIR = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( notdexE = $true )
& ( exuE3e = $true )
& ( omegaIndAx = $true )
& ( quantDeMorgan1 = $true )
& ( prop2setI = $true )
& ( eqinunit = $true )
& ( dsetconstrI = $true )
& ( dsetconstrER = $true )
& ( exuI2 = $true )
& ( prop2setE = $true )
& ( setextAx = $true )
& ( replAx = $true )
& ( sK81 = sK82 )
& ( sK83 = sK84 )
& ( ( in @ sK84 @ sK82 )
!= ( in @ sK83 @ sK81 ) )
& ( quantDeMorgan4 = $true )
& ( emptyE1 = $true )
& ( setadjoinIR = $true )
& ( notinsingleton = $true )
& ( setadjoinIL = $true )
& ( omegaSAx = $true )
& ( setadjoinE = $true )
& ( descrp = $true )
& ( vacuousDall = $true )
& ( subPowSU = $true )
& ( setadjoinOr = $true )
& ( upairsetIL = $true )
& ( powersetE = $true )
& ( setadjoinAx = $true )
& ( powersetAx = $true )
& ( emptysetAx = $true )
& ( setext = $true )
& ( quantDeMorgan2 = $true )
& ( emptysetimpfalse = $true )
& ( nonemptyI = $true )
& ( setunionI = $true )
& ( dsetconstrEL = $true )
& ( exuI1 = $true )
& ( emptyinunitempty = $true )
& ( setunionAx = $true )
& ( setunionE = $true )
& ( setoftrueEq = $true )
& ( powersetI = $true )
& ( emptyI = $true )
& ( exuE2 = $true )
& ( nonemptyImpWitness = $true )
& ( exuI3 = $true )
& ( setbeta = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK81,sK82,sK83,sK84])],[f273,f429,f428]) ).
thf(f428,plain,
( ? [X0: $i,X1: $i] :
( ( X0 = X1 )
& ? [X2: $i,X3: $i] :
( ( X2 = X3 )
& ( ( in @ X3 @ X1 )
!= ( in @ X2 @ X0 ) ) ) )
=> ( ( sK81 = sK82 )
& ? [X3: $i,X2: $i] :
( ( X2 = X3 )
& ( ( in @ X3 @ sK82 )
!= ( in @ X2 @ sK81 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f429,plain,
( ? [X3: $i,X2: $i] :
( ( X2 = X3 )
& ( ( in @ X3 @ sK82 )
!= ( in @ X2 @ sK81 ) ) )
=> ( ( sK83 = sK84 )
& ( ( in @ sK84 @ sK82 )
!= ( in @ sK83 @ sK81 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f273,plain,
( ( quantDeMorgan3 = $true )
& ( notinemptyset = $true )
& ( noeltsimpempty = $true )
& ( emptysetE = $true )
& ( exuE1 = $true )
& ( nonemptyI1 = $true )
& ( uniqinunit = $true )
& ( prop2set2propI = $true )
& ( singletonsswitch = $true )
& ( omega0Ax = $true )
& ( notdallE = $true )
& ( nonemptyE1 = $true )
& ( upairsetE = $true )
& ( emptyinPowerset = $true )
& ( emptyInPowerset = $true )
& ( inCongP = $true )
& ( upairsetIR = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( notdexE = $true )
& ( exuE3e = $true )
& ( omegaIndAx = $true )
& ( quantDeMorgan1 = $true )
& ( prop2setI = $true )
& ( eqinunit = $true )
& ( dsetconstrI = $true )
& ( dsetconstrER = $true )
& ( exuI2 = $true )
& ( prop2setE = $true )
& ( setextAx = $true )
& ( replAx = $true )
& ? [X0: $i,X1: $i] :
( ( X0 = X1 )
& ? [X2: $i,X3: $i] :
( ( X2 = X3 )
& ( ( in @ X3 @ X1 )
!= ( in @ X2 @ X0 ) ) ) )
& ( quantDeMorgan4 = $true )
& ( emptyE1 = $true )
& ( setadjoinIR = $true )
& ( notinsingleton = $true )
& ( setadjoinIL = $true )
& ( omegaSAx = $true )
& ( setadjoinE = $true )
& ( descrp = $true )
& ( vacuousDall = $true )
& ( subPowSU = $true )
& ( setadjoinOr = $true )
& ( upairsetIL = $true )
& ( powersetE = $true )
& ( setadjoinAx = $true )
& ( powersetAx = $true )
& ( emptysetAx = $true )
& ( setext = $true )
& ( quantDeMorgan2 = $true )
& ( emptysetimpfalse = $true )
& ( nonemptyI = $true )
& ( setunionI = $true )
& ( dsetconstrEL = $true )
& ( exuI1 = $true )
& ( emptyinunitempty = $true )
& ( setunionAx = $true )
& ( setunionE = $true )
& ( setoftrueEq = $true )
& ( powersetI = $true )
& ( emptyI = $true )
& ( exuE2 = $true )
& ( nonemptyImpWitness = $true )
& ( exuI3 = $true )
& ( setbeta = $true ) ),
inference(flattening,[status(thm)],[f272]) ).
thf(f272,plain,
( ? [X0: $i,X1: $i] :
( ( X0 = X1 )
& ? [X2: $i,X3: $i] :
( ( X2 = X3 )
& ( ( in @ X3 @ X1 )
!= ( in @ X2 @ X0 ) ) ) )
& ( 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)],[f151]) ).
thf(f151,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 )
=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ! [X2: $i,X3: $i] :
( ( X2 = X3 )
=> ( ( in @ X3 @ X1 )
= ( in @ X2 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f150]) ).
thf(f150,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
=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ! [X2: $i,X3: $i] :
( ( X2 = X3 )
=> ( ( in @ X2 @ X0 )
<=> ( in @ X3 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f68]) ).
thf(f68,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
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( in @ X1 @ X3 )
<=> ( in @ X2 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f67]) ).
thf(f67,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
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ! [X1: $i,X2: $i] :
( ( X1 = X2 )
=> ( ( in @ X1 @ X3 )
<=> ( in @ X2 @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.Ifh9i1xfml/DTF2THF_30943.p',in__Cong) ).
thf(f763,plain,
sK83 = sK84,
inference(cnf_transformation,[status(thm)],[f430]) ).
thf(f762,plain,
( ( in @ sK84 @ sK82 )
!= ( in @ sK83 @ sK81 ) ),
inference(cnf_transformation,[status(thm)],[f430]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU548^1 : TPTP v9.2.1. Bugfixed v4.0.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n027.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:26:52 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.21/0.46 ---- Original DTF file ---
% 0.21/0.46 thf(spec,logic,$$dhol).
% 0.21/0.46 %------------------------------------------------------------------------------
% 0.21/0.46 % File : SEU548^1 : TPTP v9.2.1. Bugfixed v4.0.0.
% 0.21/0.46 % Domain : Set Theory
% 0.21/0.46 % Problem : A simple congruence property of in
% 0.21/0.46 % Version : Especial.
% 0.21/0.46 % English : (forall A:i.forall B:i.A = B -> (forall x:i.forall y:i.x = y ->
% 0.21/0.46 % (in x A <-> in y B)))
% 0.21/0.46
% 0.21/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.21/0.46 % Source : [Bro08]
% 0.21/0.46 % Names : ZFC050g [Bro08]
% 0.21/0.46
% 0.21/0.46 % Status : Theorem
% 0.21/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.33 v4.0.1, 0.67 v4.0.0
% 0.21/0.46 % Syntax : Number of formulae : 143 ( 66 unt; 76 typ; 66 def)
% 0.21/0.46 % Number of atoms : 363 ( 92 equ; 0 cnn)
% 0.21/0.46 % Maximal formula atoms : 68 ( 5 avg)
% 0.21/0.46 % Number of connectives : 601 ( 21 ~; 5 |; 27 &; 376 @)
% 0.21/0.46 % ( 10 <=>; 162 =>; 0 <=; 0 <~>)
% 0.21/0.46 % Maximal formula depth : 74 ( 2 avg)
% 0.21/0.46 % Number of types : 2 ( 0 usr)
% 0.21/0.46 % Number of type conns : 39 ( 39 >; 0 *; 0 +; 0 <<)
% 0.21/0.46 % Number of symbols : 79 ( 76 usr; 68 con; 0-2 aty)
% 0.21/0.46 % Number of variables : 204 ( 18 ^; 158 !; 28 ?; 204 :)
% 0.21/0.46 % SPC : TH0_THM_EQU_NAR
% 0.21/0.46
% 0.21/0.46 % Comments : http://mathgate.info/detsetitem.php?id=1
% 0.21/0.46 % Bugfixes : v4.0.0 - Adding missing type delaration for in
% 0.21/0.46 %------------------------------------------------------------------------------
% 0.21/0.46 thf(in_type,type,
% 0.21/0.46 in: $i > $i > $o ).
% 0.21/0.46
% 0.21/0.46 thf(exu_type,type,
% 0.21/0.46 exu: ( $i > $o ) > $o ).
% 0.21/0.46
% 0.21/0.46 thf(exu,definition,
% 0.21/0.46 ( exu
% 0.21/0.46 = ( ^ [Xphi: $i > $o] :
% 0.21/0.46 ? [Xx: $i] :
% 0.21/0.46 ( ( Xphi @ Xx )
% 0.21/0.46 & ! [Xy: $i] :
% 0.21/0.46 ( ( Xphi @ Xy )
% 0.21/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setextAx_type,type,
% 0.21/0.46 setextAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setextAx,definition,
% 0.21/0.46 ( setextAx
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 <=> ( in @ Xx @ B ) )
% 0.21/0.46 => ( A = B ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptyset_type,type,
% 0.21/0.46 emptyset: $i ).
% 0.21/0.46
% 0.21/0.46 thf(emptysetAx_type,type,
% 0.21/0.46 emptysetAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptysetAx,definition,
% 0.21/0.46 ( emptysetAx
% 0.21/0.46 = ( ! [Xx: $i] :
% 0.21/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoin_type,type,
% 0.21/0.46 setadjoin: $i > $i > $i ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinAx_type,type,
% 0.21/0.46 setadjoinAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinAx,definition,
% 0.21/0.46 ( setadjoinAx
% 0.21/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.21/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.21/0.46 <=> ( ( Xy = Xx )
% 0.21/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(powerset_type,type,
% 0.21/0.46 powerset: $i > $i ).
% 0.21/0.46
% 0.21/0.46 thf(powersetAx_type,type,
% 0.21/0.46 powersetAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(powersetAx,definition,
% 0.21/0.46 ( powersetAx
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.21/0.46 <=> ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setunion_type,type,
% 0.21/0.46 setunion: $i > $i ).
% 0.21/0.46
% 0.21/0.46 thf(setunionAx_type,type,
% 0.21/0.46 setunionAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setunionAx,definition,
% 0.21/0.46 ( setunionAx
% 0.21/0.46 = ( ! [A: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.21/0.46 <=> ? [B: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(omega_type,type,
% 0.21/0.46 omega: $i ).
% 0.21/0.46
% 0.21/0.46 thf(omega0Ax_type,type,
% 0.21/0.46 omega0Ax: $o ).
% 0.21/0.46
% 0.21/0.46 thf(omega0Ax,definition,
% 0.21/0.46 ( omega0Ax
% 0.21/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(omegaSAx_type,type,
% 0.21/0.46 omegaSAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(omegaSAx,definition,
% 0.21/0.46 ( omegaSAx
% 0.21/0.46 = ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ omega )
% 0.21/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(omegaIndAx_type,type,
% 0.21/0.46 omegaIndAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(omegaIndAx,definition,
% 0.21/0.46 ( omegaIndAx
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ( ( in @ emptyset @ A )
% 0.21/0.46 & ! [Xx: $i] :
% 0.21/0.46 ( ( ( in @ Xx @ omega )
% 0.21/0.46 & ( in @ Xx @ A ) )
% 0.21/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.21/0.46 => ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ omega )
% 0.21/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(replAx_type,type,
% 0.21/0.46 replAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(replAx,definition,
% 0.21/0.46 ( replAx
% 0.21/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( exu
% 0.21/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.21/0.46 => ? [B: $i] :
% 0.21/0.46 ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 <=> ? [Xy: $i] :
% 0.21/0.46 ( ( in @ Xy @ A )
% 0.21/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(foundationAx_type,type,
% 0.21/0.46 foundationAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(foundationAx,definition,
% 0.21/0.46 ( foundationAx
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.21/0.46 => ? [B: $i] :
% 0.21/0.46 ( ( in @ B @ A )
% 0.21/0.46 & ~ ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(wellorderingAx_type,type,
% 0.21/0.46 wellorderingAx: $o ).
% 0.21/0.46
% 0.21/0.46 thf(wellorderingAx,definition,
% 0.21/0.46 ( wellorderingAx
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ? [B: $i] :
% 0.21/0.46 ( ! [C: $i] :
% 0.21/0.46 ( ( in @ C @ B )
% 0.21/0.46 => ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ C )
% 0.21/0.46 => ( in @ Xx @ A ) ) )
% 0.21/0.46 & ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( ( in @ Xx @ A )
% 0.21/0.46 & ( in @ Xy @ A ) )
% 0.21/0.46 => ( ! [C: $i] :
% 0.21/0.46 ( ( in @ C @ B )
% 0.21/0.46 => ( ( in @ Xx @ C )
% 0.21/0.46 <=> ( in @ Xy @ C ) ) )
% 0.21/0.46 => ( Xx = Xy ) ) )
% 0.21/0.46 & ! [C: $i,D: $i] :
% 0.21/0.46 ( ( ( in @ C @ B )
% 0.21/0.46 & ( in @ D @ B ) )
% 0.21/0.46 => ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ C )
% 0.21/0.46 => ( in @ Xx @ D ) )
% 0.21/0.46 | ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ D )
% 0.21/0.46 => ( in @ Xx @ C ) ) ) )
% 0.21/0.46 & ! [C: $i] :
% 0.21/0.46 ( ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ C )
% 0.21/0.46 => ( in @ Xx @ A ) )
% 0.21/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.21/0.46 => ? [D: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ D @ B )
% 0.21/0.46 & ( in @ Xx @ C )
% 0.21/0.46 & ~ ? [Xy: $i] :
% 0.21/0.46 ( ( in @ Xy @ D )
% 0.21/0.46 & ( in @ Xy @ C ) )
% 0.21/0.46 & ! [E: $i] :
% 0.21/0.46 ( ( in @ E @ B )
% 0.21/0.46 => ( ! [Xy: $i] :
% 0.21/0.46 ( ( in @ Xy @ E )
% 0.21/0.46 => ( in @ Xy @ D ) )
% 0.21/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(descr_type,type,
% 0.21/0.46 descr: ( $i > $o ) > $i ).
% 0.21/0.46
% 0.21/0.46 thf(descrp_type,type,
% 0.21/0.46 descrp: $o ).
% 0.21/0.46
% 0.21/0.46 thf(descrp,definition,
% 0.21/0.46 ( descrp
% 0.21/0.46 = ( ! [Xphi: $i > $o] :
% 0.21/0.46 ( ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 => ( Xphi
% 0.21/0.46 @ ( descr
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(dsetconstr_type,type,
% 0.21/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.21/0.46
% 0.21/0.46 thf(dsetconstrI_type,type,
% 0.21/0.46 dsetconstrI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(dsetconstrI,definition,
% 0.21/0.46 ( dsetconstrI
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( ( Xphi @ Xx )
% 0.21/0.46 => ( in @ Xx
% 0.21/0.46 @ ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(dsetconstrEL_type,type,
% 0.21/0.46 dsetconstrEL: $o ).
% 0.21/0.46
% 0.21/0.46 thf(dsetconstrEL,definition,
% 0.21/0.46 ( dsetconstrEL
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx
% 0.21/0.46 @ ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.21/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(dsetconstrER_type,type,
% 0.21/0.46 dsetconstrER: $o ).
% 0.21/0.46
% 0.21/0.46 thf(dsetconstrER,definition,
% 0.21/0.46 ( dsetconstrER
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx
% 0.21/0.46 @ ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.21/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(exuE1_type,type,
% 0.21/0.46 exuE1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(exuE1,definition,
% 0.21/0.46 ( exuE1
% 0.21/0.46 = ( ! [Xphi: $i > $o] :
% 0.21/0.46 ( ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 => ? [Xx: $i] :
% 0.21/0.46 ( ( Xphi @ Xx )
% 0.21/0.46 & ! [Xy: $i] :
% 0.21/0.46 ( ( Xphi @ Xy )
% 0.21/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(prop2set_type,type,
% 0.21/0.46 prop2set: $o > $i ).
% 0.21/0.46
% 0.21/0.46 thf(prop2setE_type,type,
% 0.21/0.46 prop2setE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(prop2setE,definition,
% 0.21/0.46 ( prop2setE
% 0.21/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.21/0.46 => Xphi ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptysetE_type,type,
% 0.21/0.46 emptysetE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptysetE,definition,
% 0.21/0.46 ( emptysetE
% 0.21/0.46 = ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ emptyset )
% 0.21/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptysetimpfalse_type,type,
% 0.21/0.46 emptysetimpfalse: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptysetimpfalse,definition,
% 0.21/0.46 ( emptysetimpfalse
% 0.21/0.46 = ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ emptyset )
% 0.21/0.46 => $false ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(notinemptyset_type,type,
% 0.21/0.46 notinemptyset: $o ).
% 0.21/0.46
% 0.21/0.46 thf(notinemptyset,definition,
% 0.21/0.46 ( notinemptyset
% 0.21/0.46 = ( ! [Xx: $i] :
% 0.21/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(exuE3e_type,type,
% 0.21/0.46 exuE3e: $o ).
% 0.21/0.46
% 0.21/0.46 thf(exuE3e,definition,
% 0.21/0.46 ( exuE3e
% 0.21/0.46 = ( ! [Xphi: $i > $o] :
% 0.21/0.46 ( ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setext_type,type,
% 0.21/0.46 setext: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setext,definition,
% 0.21/0.46 ( setext
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( in @ Xx @ B ) )
% 0.21/0.46 => ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 => ( in @ Xx @ A ) )
% 0.21/0.46 => ( A = B ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptyI_type,type,
% 0.21/0.46 emptyI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptyI,definition,
% 0.21/0.46 ( emptyI
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ~ ( in @ Xx @ A )
% 0.21/0.46 => ( A = emptyset ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(noeltsimpempty_type,type,
% 0.21/0.46 noeltsimpempty: $o ).
% 0.21/0.46
% 0.21/0.46 thf(noeltsimpempty,definition,
% 0.21/0.46 ( noeltsimpempty
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ~ ( in @ Xx @ A )
% 0.21/0.46 => ( A = emptyset ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setbeta_type,type,
% 0.21/0.46 setbeta: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setbeta,definition,
% 0.21/0.46 ( setbeta
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( ( in @ Xx
% 0.21/0.46 @ ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.21/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(nonempty_type,type,
% 0.21/0.46 nonempty: $i > $o ).
% 0.21/0.46
% 0.21/0.46 thf(nonempty,definition,
% 0.21/0.46 ( nonempty
% 0.21/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(nonemptyE1_type,type,
% 0.21/0.46 nonemptyE1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(nonemptyE1,definition,
% 0.21/0.46 ( nonemptyE1
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ( nonempty @ A )
% 0.21/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(nonemptyI_type,type,
% 0.21/0.46 nonemptyI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(nonemptyI,definition,
% 0.21/0.46 ( nonemptyI
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( ( Xphi @ Xx )
% 0.21/0.46 => ( nonempty
% 0.21/0.46 @ ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(nonemptyI1_type,type,
% 0.21/0.46 nonemptyI1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(nonemptyI1,definition,
% 0.21/0.46 ( nonemptyI1
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.21/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinIL_type,type,
% 0.21/0.46 setadjoinIL: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinIL,definition,
% 0.21/0.46 ( setadjoinIL
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptyinunitempty_type,type,
% 0.21/0.46 emptyinunitempty: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptyinunitempty,definition,
% 0.21/0.46 ( emptyinunitempty
% 0.21/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinIR_type,type,
% 0.21/0.46 setadjoinIR: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinIR,definition,
% 0.21/0.46 ( setadjoinIR
% 0.21/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.21/0.46 ( ( in @ Xy @ A )
% 0.21/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinE_type,type,
% 0.21/0.46 setadjoinE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinE,definition,
% 0.21/0.46 ( setadjoinE
% 0.21/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.21/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.21/0.46 => ! [Xphi: $o] :
% 0.21/0.46 ( ( ( Xy = Xx )
% 0.21/0.46 => Xphi )
% 0.21/0.46 => ( ( ( in @ Xy @ A )
% 0.21/0.46 => Xphi )
% 0.21/0.46 => Xphi ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinOr_type,type,
% 0.21/0.46 setadjoinOr: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setadjoinOr,definition,
% 0.21/0.46 ( setadjoinOr
% 0.21/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.21/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.21/0.46 => ( ( Xy = Xx )
% 0.21/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setoftrueEq_type,type,
% 0.21/0.46 setoftrueEq: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setoftrueEq,definition,
% 0.21/0.46 ( setoftrueEq
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xx: $i] : $true )
% 0.21/0.46 = A ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(powersetI_type,type,
% 0.21/0.46 powersetI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(powersetI,definition,
% 0.21/0.46 ( powersetI
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 => ( in @ Xx @ A ) )
% 0.21/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptyinPowerset_type,type,
% 0.21/0.46 emptyinPowerset: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptyinPowerset,definition,
% 0.21/0.46 ( emptyinPowerset
% 0.21/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptyInPowerset_type,type,
% 0.21/0.46 emptyInPowerset: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptyInPowerset,definition,
% 0.21/0.46 ( emptyInPowerset
% 0.21/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(powersetE_type,type,
% 0.21/0.46 powersetE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(powersetE,definition,
% 0.21/0.46 ( powersetE
% 0.21/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.21/0.46 => ( ( in @ Xx @ B )
% 0.21/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setunionI_type,type,
% 0.21/0.46 setunionI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setunionI,definition,
% 0.21/0.46 ( setunionI
% 0.21/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 => ( ( in @ B @ A )
% 0.21/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(setunionE_type,type,
% 0.21/0.46 setunionE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(setunionE,definition,
% 0.21/0.46 ( setunionE
% 0.21/0.46 = ( ! [A: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.21/0.46 => ! [Xphi: $o] :
% 0.21/0.46 ( ! [B: $i] :
% 0.21/0.46 ( ( in @ Xx @ B )
% 0.21/0.46 => ( ( in @ B @ A )
% 0.21/0.46 => Xphi ) )
% 0.21/0.46 => Xphi ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(subPowSU_type,type,
% 0.21/0.46 subPowSU: $o ).
% 0.21/0.46
% 0.21/0.46 thf(subPowSU,definition,
% 0.21/0.46 ( subPowSU
% 0.21/0.46 = ( ! [A: $i,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(exuE2_type,type,
% 0.21/0.46 exuE2: $o ).
% 0.21/0.46
% 0.21/0.46 thf(exuE2,definition,
% 0.21/0.46 ( exuE2
% 0.21/0.46 = ( ! [Xphi: $i > $o] :
% 0.21/0.46 ( ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 => ? [Xx: $i] :
% 0.21/0.46 ! [Xy: $i] :
% 0.21/0.46 ( ( Xphi @ Xy )
% 0.21/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(nonemptyImpWitness_type,type,
% 0.21/0.46 nonemptyImpWitness: $o ).
% 0.21/0.46
% 0.21/0.46 thf(nonemptyImpWitness,definition,
% 0.21/0.46 ( nonemptyImpWitness
% 0.21/0.46 = ( ! [A: $i] :
% 0.21/0.46 ( ( nonempty @ A )
% 0.21/0.46 => ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & $true ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(uniqinunit_type,type,
% 0.21/0.46 uniqinunit: $o ).
% 0.21/0.46
% 0.21/0.46 thf(uniqinunit,definition,
% 0.21/0.46 ( uniqinunit
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.21/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(notinsingleton_type,type,
% 0.21/0.46 notinsingleton: $o ).
% 0.21/0.46
% 0.21/0.46 thf(notinsingleton,definition,
% 0.21/0.46 ( notinsingleton
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( Xx != Xy )
% 0.21/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(eqinunit_type,type,
% 0.21/0.46 eqinunit: $o ).
% 0.21/0.46
% 0.21/0.46 thf(eqinunit,definition,
% 0.21/0.46 ( eqinunit
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( Xx = Xy )
% 0.21/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(singletonsswitch_type,type,
% 0.21/0.46 singletonsswitch: $o ).
% 0.21/0.46
% 0.21/0.46 thf(singletonsswitch,definition,
% 0.21/0.46 ( singletonsswitch
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.21/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(upairsetE_type,type,
% 0.21/0.46 upairsetE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(upairsetE,definition,
% 0.21/0.46 ( upairsetE
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.21/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.21/0.46 => ( ( Xz = Xx )
% 0.21/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(upairsetIL_type,type,
% 0.21/0.46 upairsetIL: $o ).
% 0.21/0.46
% 0.21/0.46 thf(upairsetIL,definition,
% 0.21/0.46 ( upairsetIL
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(upairsetIR_type,type,
% 0.21/0.46 upairsetIR: $o ).
% 0.21/0.46
% 0.21/0.46 thf(upairsetIR,definition,
% 0.21/0.46 ( upairsetIR
% 0.21/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(emptyE1_type,type,
% 0.21/0.46 emptyE1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(emptyE1,definition,
% 0.21/0.46 ( emptyE1
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.21/0.46 ( ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & ( Xphi @ Xx ) )
% 0.21/0.46 => ( ( ( dsetconstr @ A
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.21/0.46 = emptyset )
% 0.21/0.46 => $false ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(vacuousDall_type,type,
% 0.21/0.46 vacuousDall: $o ).
% 0.21/0.46
% 0.21/0.46 thf(vacuousDall,definition,
% 0.21/0.46 ( vacuousDall
% 0.21/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ emptyset )
% 0.21/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(quantDeMorgan1_type,type,
% 0.21/0.46 quantDeMorgan1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(quantDeMorgan1,definition,
% 0.21/0.46 ( quantDeMorgan1
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.21/0.46 ( ~ ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( Xphi @ Xx ) )
% 0.21/0.46 => ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(quantDeMorgan2_type,type,
% 0.21/0.46 quantDeMorgan2: $o ).
% 0.21/0.46
% 0.21/0.46 thf(quantDeMorgan2,definition,
% 0.21/0.46 ( quantDeMorgan2
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.21/0.46 ( ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ~ ( Xphi @ Xx ) )
% 0.21/0.46 => ~ ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(quantDeMorgan3_type,type,
% 0.21/0.46 quantDeMorgan3: $o ).
% 0.21/0.46
% 0.21/0.46 thf(quantDeMorgan3,definition,
% 0.21/0.46 ( quantDeMorgan3
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.21/0.46 ( ~ ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & ( Xphi @ Xx ) )
% 0.21/0.46 => ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(quantDeMorgan4_type,type,
% 0.21/0.46 quantDeMorgan4: $o ).
% 0.21/0.46
% 0.21/0.46 thf(quantDeMorgan4,definition,
% 0.21/0.46 ( quantDeMorgan4
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.21/0.46 ( ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & ~ ( Xphi @ Xx ) )
% 0.21/0.46 => ~ ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(prop2setI_type,type,
% 0.21/0.46 prop2setI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(prop2setI,definition,
% 0.21/0.46 ( prop2setI
% 0.21/0.46 = ( ! [Xphi: $o] :
% 0.21/0.46 ( Xphi
% 0.21/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(set2prop_type,type,
% 0.21/0.46 set2prop: $i > $o ).
% 0.21/0.46
% 0.21/0.46 thf(prop2set2propI_type,type,
% 0.21/0.46 prop2set2propI: $o ).
% 0.21/0.46
% 0.21/0.46 thf(prop2set2propI,definition,
% 0.21/0.46 ( prop2set2propI
% 0.21/0.46 = ( ! [Xphi: $o] :
% 0.21/0.46 ( Xphi
% 0.21/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(notdexE_type,type,
% 0.21/0.46 notdexE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(notdexE,definition,
% 0.21/0.46 ( notdexE
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.21/0.46 ( ~ ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & ( Xphi @ Xx ) )
% 0.21/0.46 => ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(notdallE_type,type,
% 0.21/0.46 notdallE: $o ).
% 0.21/0.46
% 0.21/0.46 thf(notdallE,definition,
% 0.21/0.46 ( notdallE
% 0.21/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.21/0.46 ( ~ ! [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 => ( Xphi @ Xx ) )
% 0.21/0.46 => ? [Xx: $i] :
% 0.21/0.46 ( ( in @ Xx @ A )
% 0.21/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(exuI1_type,type,
% 0.21/0.46 exuI1: $o ).
% 0.21/0.46
% 0.21/0.46 thf(exuI1,definition,
% 0.21/0.46 ( exuI1
% 0.21/0.46 = ( ! [Xphi: $i > $o] :
% 0.21/0.46 ( ? [Xx: $i] :
% 0.21/0.46 ( ( Xphi @ Xx )
% 0.21/0.46 & ! [Xy: $i] :
% 0.21/0.46 ( ( Xphi @ Xy )
% 0.21/0.46 => ( Xx = Xy ) ) )
% 0.21/0.46 => ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(exuI3_type,type,
% 0.21/0.46 exuI3: $o ).
% 0.21/0.46
% 0.21/0.46 thf(exuI3,definition,
% 0.21/0.46 ( exuI3
% 0.21/0.46 = ( ! [Xphi: $i > $o] :
% 0.21/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.21/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( Xphi @ Xx )
% 0.21/0.46 => ( ( Xphi @ Xy )
% 0.21/0.46 => ( Xx = Xy ) ) )
% 0.21/0.46 => ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(exuI2_type,type,
% 0.21/0.46 exuI2: $o ).
% 0.21/0.46
% 0.21/0.46 thf(exuI2,definition,
% 0.21/0.46 ( exuI2
% 0.21/0.46 = ( ! [Xphi: $i > $o] :
% 0.21/0.46 ( ? [Xx: $i] :
% 0.21/0.46 ! [Xy: $i] :
% 0.21/0.46 ( ( Xphi @ Xy )
% 0.21/0.46 <=> ( Xy = Xx ) )
% 0.21/0.46 => ( exu
% 0.21/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(inCongP_type,type,
% 0.21/0.46 inCongP: $o ).
% 0.21/0.46
% 0.21/0.46 thf(inCongP,definition,
% 0.21/0.46 ( inCongP
% 0.21/0.46 = ( ! [A: $i,B: $i] :
% 0.21/0.46 ( ( A = B )
% 0.21/0.46 => ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( Xx = Xy )
% 0.21/0.46 => ( ( in @ Xx @ A )
% 0.21/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 thf(in__Cong,conjecture,
% 0.21/0.46 ( setextAx
% 0.21/0.46 => ( emptysetAx
% 0.21/0.46 => ( setadjoinAx
% 0.21/0.46 => ( powersetAx
% 0.21/0.46 => ( setunionAx
% 0.21/0.46 => ( omega0Ax
% 0.21/0.46 => ( omegaSAx
% 0.21/0.46 => ( omegaIndAx
% 0.21/0.46 => ( replAx
% 0.21/0.46 => ( foundationAx
% 0.21/0.46 => ( wellorderingAx
% 0.21/0.46 => ( descrp
% 0.21/0.46 => ( dsetconstrI
% 0.21/0.46 => ( dsetconstrEL
% 0.21/0.46 => ( dsetconstrER
% 0.21/0.46 => ( exuE1
% 0.21/0.46 => ( prop2setE
% 0.21/0.46 => ( emptysetE
% 0.21/0.46 => ( emptysetimpfalse
% 0.21/0.46 => ( notinemptyset
% 0.21/0.46 => ( exuE3e
% 0.21/0.46 => ( setext
% 0.21/0.46 => ( emptyI
% 0.21/0.46 => ( noeltsimpempty
% 0.21/0.46 => ( setbeta
% 0.21/0.46 => ( nonemptyE1
% 0.21/0.46 => ( nonemptyI
% 0.21/0.46 => ( nonemptyI1
% 0.21/0.46 => ( setadjoinIL
% 0.21/0.46 => ( emptyinunitempty
% 0.21/0.46 => ( setadjoinIR
% 0.21/0.46 => ( setadjoinE
% 0.21/0.46 => ( setadjoinOr
% 0.21/0.46 => ( setoftrueEq
% 0.21/0.46 => ( powersetI
% 0.21/0.46 => ( emptyinPowerset
% 0.21/0.46 => ( emptyInPowerset
% 0.21/0.46 => ( powersetE
% 0.21/0.46 => ( setunionI
% 0.21/0.46 => ( setunionE
% 0.21/0.46 => ( subPowSU
% 0.21/0.46 => ( exuE2
% 0.21/0.46 => ( nonemptyImpWitness
% 0.21/0.46 => ( uniqinunit
% 0.21/0.46 => ( notinsingleton
% 0.21/0.46 => ( eqinunit
% 0.21/0.46 => ( singletonsswitch
% 0.21/0.46 => ( upairsetE
% 0.21/0.46 => ( upairsetIL
% 0.21/0.46 => ( upairsetIR
% 0.21/0.46 => ( emptyE1
% 0.21/0.46 => ( vacuousDall
% 0.21/0.46 => ( quantDeMorgan1
% 0.21/0.46 => ( quantDeMorgan2
% 0.21/0.46 => ( quantDeMorgan3
% 0.21/0.46 => ( quantDeMorgan4
% 0.21/0.46 => ( prop2setI
% 0.21/0.46 => ( prop2set2propI
% 0.21/0.46 => ( notdexE
% 0.21/0.46 => ( notdallE
% 0.21/0.46 => ( exuI1
% 0.21/0.46 => ( exuI3
% 0.21/0.46 => ( exuI2
% 0.21/0.46 => ( inCongP
% 0.21/0.46 => ! [A: $i,B: $i] :
% 0.21/0.46 ( ( A = B )
% 0.21/0.46 => ! [Xx: $i,Xy: $i] :
% 0.21/0.46 ( ( Xx = Xy )
% 0.21/0.46 => ( ( in @ Xx @ A )
% 0.21/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.21/0.46
% 0.21/0.46 %------------------------------------------------------------------------------
% 0.21/0.46 ------------------------
% 2.11/1.51 ---- Embedded in THF ---
% 2.11/1.51 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.11/1.51 %%% Generated on Tue Feb 24 12:26:53 EST 2026
% 2.11/1.51 %%% using '$$dhol' embedding, version 1.3.0.
% 2.11/1.51 %%% Logic specification used:
% 2.11/1.51 %%% thf(spec, logic, $$dhol).
% 2.11/1.51
% 2.11/1.51 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.Ifh9i1xfml/DTF2DTF30943.p
% See solution above
% 2.11/1.51 ------------------------
% 2.11/1.53 ---- Cleaned THF ---
% 2.11/1.53 thf(in_type,type,
% 2.11/1.53 in: $i > $i > $o ).
% 2.11/1.53
% 2.11/1.53 thf(exu_type,type,
% 2.11/1.53 exu: ( $i > $o ) > $o ).
% 2.11/1.53
% 2.11/1.53 thf(exu,definition,
% 2.11/1.53 ( exu
% 2.11/1.53 = ( ^ [Xphi: $i > $o] :
% 2.11/1.53 ? [Xx: $i] :
% 2.11/1.53 ( ( Xphi @ Xx )
% 2.11/1.53 & ! [Xy: $i] :
% 2.11/1.53 ( ( Xphi @ Xy )
% 2.11/1.53 => ( Xx = Xy ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(setextAx_type,type,
% 2.11/1.53 setextAx: $o ).
% 2.11/1.53
% 2.11/1.53 thf(setextAx,definition,
% 2.11/1.53 ( setextAx
% 2.11/1.53 = ( ! [A: $i,B: $i] :
% 2.11/1.53 ( ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 <=> ( in @ Xx @ B ) )
% 2.11/1.53 => ( A = B ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(emptyset_type,type,
% 2.11/1.53 emptyset: $i ).
% 2.11/1.53
% 2.11/1.53 thf(emptysetAx_type,type,
% 2.11/1.53 emptysetAx: $o ).
% 2.11/1.53
% 2.11/1.53 thf(emptysetAx,definition,
% 2.11/1.53 ( emptysetAx
% 2.11/1.53 = ( ! [Xx: $i] :
% 2.11/1.53 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(setadjoin_type,type,
% 2.11/1.53 setadjoin: $i > $i > $i ).
% 2.11/1.53
% 2.11/1.53 thf(setadjoinAx_type,type,
% 2.11/1.53 setadjoinAx: $o ).
% 2.11/1.53
% 2.11/1.53 thf(setadjoinAx,definition,
% 2.11/1.53 ( setadjoinAx
% 2.11/1.53 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.11/1.53 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.11/1.53 <=> ( ( Xy = Xx )
% 2.11/1.53 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(powerset_type,type,
% 2.11/1.53 powerset: $i > $i ).
% 2.11/1.53
% 2.11/1.53 thf(powersetAx_type,type,
% 2.11/1.53 powersetAx: $o ).
% 2.11/1.53
% 2.11/1.53 thf(powersetAx,definition,
% 2.11/1.53 ( powersetAx
% 2.11/1.53 = ( ! [A: $i,B: $i] :
% 2.11/1.53 ( ( in @ B @ ( powerset @ A ) )
% 2.11/1.53 <=> ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ B )
% 2.11/1.53 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(setunion_type,type,
% 2.11/1.53 setunion: $i > $i ).
% 2.11/1.53
% 2.11/1.53 thf(setunionAx_type,type,
% 2.11/1.53 setunionAx: $o ).
% 2.11/1.53
% 2.11/1.53 thf(setunionAx,definition,
% 2.11/1.53 ( setunionAx
% 2.11/1.53 = ( ! [A: $i,Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ ( setunion @ A ) )
% 2.11/1.53 <=> ? [B: $i] :
% 2.11/1.53 ( ( in @ Xx @ B )
% 2.11/1.53 & ( in @ B @ A ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(omega_type,type,
% 2.11/1.53 omega: $i ).
% 2.11/1.53
% 2.11/1.53 thf(omega0Ax_type,type,
% 2.11/1.53 omega0Ax: $o ).
% 2.11/1.53
% 2.11/1.53 thf(omega0Ax,definition,
% 2.11/1.53 ( omega0Ax
% 2.11/1.53 = ( in @ emptyset @ omega ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(omegaSAx_type,type,
% 2.11/1.53 omegaSAx: $o ).
% 2.11/1.53
% 2.11/1.53 thf(omegaSAx,definition,
% 2.11/1.53 ( omegaSAx
% 2.11/1.53 = ( ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ omega )
% 2.11/1.53 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(omegaIndAx_type,type,
% 2.11/1.53 omegaIndAx: $o ).
% 2.11/1.53
% 2.11/1.53 thf(omegaIndAx,definition,
% 2.11/1.53 ( omegaIndAx
% 2.11/1.53 = ( ! [A: $i] :
% 2.11/1.53 ( ( ( in @ emptyset @ A )
% 2.11/1.53 & ! [Xx: $i] :
% 2.11/1.53 ( ( ( in @ Xx @ omega )
% 2.11/1.53 & ( in @ Xx @ A ) )
% 2.11/1.53 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.11/1.53 => ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ omega )
% 2.11/1.53 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(replAx_type,type,
% 2.11/1.53 replAx: $o ).
% 2.11/1.53
% 2.11/1.53 thf(replAx,definition,
% 2.11/1.53 ( replAx
% 2.11/1.53 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.11/1.53 ( ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 => ( exu
% 2.11/1.53 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.11/1.53 => ? [B: $i] :
% 2.11/1.53 ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ B )
% 2.11/1.53 <=> ? [Xy: $i] :
% 2.11/1.53 ( ( in @ Xy @ A )
% 2.11/1.53 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(foundationAx_type,type,
% 2.11/1.53 foundationAx: $o ).
% 2.11/1.53
% 2.11/1.53 thf(foundationAx,definition,
% 2.11/1.53 ( foundationAx
% 2.11/1.53 = ( ! [A: $i] :
% 2.11/1.53 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.11/1.53 => ? [B: $i] :
% 2.11/1.53 ( ( in @ B @ A )
% 2.11/1.53 & ~ ? [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ B )
% 2.11/1.53 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(wellorderingAx_type,type,
% 2.11/1.53 wellorderingAx: $o ).
% 2.11/1.53
% 2.11/1.53 thf(wellorderingAx,definition,
% 2.11/1.53 ( wellorderingAx
% 2.11/1.53 = ( ! [A: $i] :
% 2.11/1.53 ? [B: $i] :
% 2.11/1.53 ( ! [C: $i] :
% 2.11/1.53 ( ( in @ C @ B )
% 2.11/1.53 => ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ C )
% 2.11/1.53 => ( in @ Xx @ A ) ) )
% 2.11/1.53 & ! [Xx: $i,Xy: $i] :
% 2.11/1.53 ( ( ( in @ Xx @ A )
% 2.11/1.53 & ( in @ Xy @ A ) )
% 2.11/1.53 => ( ! [C: $i] :
% 2.11/1.53 ( ( in @ C @ B )
% 2.11/1.53 => ( ( in @ Xx @ C )
% 2.11/1.53 <=> ( in @ Xy @ C ) ) )
% 2.11/1.53 => ( Xx = Xy ) ) )
% 2.11/1.53 & ! [C: $i,D: $i] :
% 2.11/1.53 ( ( ( in @ C @ B )
% 2.11/1.53 & ( in @ D @ B ) )
% 2.11/1.53 => ( ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ C )
% 2.11/1.53 => ( in @ Xx @ D ) )
% 2.11/1.53 | ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ D )
% 2.11/1.53 => ( in @ Xx @ C ) ) ) )
% 2.11/1.53 & ! [C: $i] :
% 2.11/1.53 ( ( ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ C )
% 2.11/1.53 => ( in @ Xx @ A ) )
% 2.11/1.53 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.11/1.53 => ? [D: $i,Xx: $i] :
% 2.11/1.53 ( ( in @ D @ B )
% 2.11/1.53 & ( in @ Xx @ C )
% 2.11/1.53 & ~ ? [Xy: $i] :
% 2.11/1.53 ( ( in @ Xy @ D )
% 2.11/1.53 & ( in @ Xy @ C ) )
% 2.11/1.53 & ! [E: $i] :
% 2.11/1.53 ( ( in @ E @ B )
% 2.11/1.53 => ( ! [Xy: $i] :
% 2.11/1.53 ( ( in @ Xy @ E )
% 2.11/1.53 => ( in @ Xy @ D ) )
% 2.11/1.53 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(descr_type,type,
% 2.11/1.53 descr: ( $i > $o ) > $i ).
% 2.11/1.53
% 2.11/1.53 thf(descrp_type,type,
% 2.11/1.53 descrp: $o ).
% 2.11/1.53
% 2.11/1.53 thf(descrp,definition,
% 2.11/1.53 ( descrp
% 2.11/1.53 = ( ! [Xphi: $i > $o] :
% 2.11/1.53 ( ( exu
% 2.11/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.11/1.53 => ( Xphi
% 2.11/1.53 @ ( descr
% 2.11/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(dsetconstr_type,type,
% 2.11/1.53 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.11/1.53
% 2.11/1.53 thf(dsetconstrI_type,type,
% 2.11/1.53 dsetconstrI: $o ).
% 2.11/1.53
% 2.11/1.53 thf(dsetconstrI,definition,
% 2.11/1.53 ( dsetconstrI
% 2.11/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 => ( ( Xphi @ Xx )
% 2.11/1.53 => ( in @ Xx
% 2.11/1.53 @ ( dsetconstr @ A
% 2.11/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(dsetconstrEL_type,type,
% 2.11/1.53 dsetconstrEL: $o ).
% 2.11/1.53
% 2.11/1.53 thf(dsetconstrEL,definition,
% 2.11/1.53 ( dsetconstrEL
% 2.11/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.11/1.53 ( ( in @ Xx
% 2.11/1.53 @ ( dsetconstr @ A
% 2.11/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.11/1.53 => ( in @ Xx @ A ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(dsetconstrER_type,type,
% 2.11/1.53 dsetconstrER: $o ).
% 2.11/1.53
% 2.11/1.53 thf(dsetconstrER,definition,
% 2.11/1.53 ( dsetconstrER
% 2.11/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.11/1.53 ( ( in @ Xx
% 2.11/1.53 @ ( dsetconstr @ A
% 2.11/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.11/1.53 => ( Xphi @ Xx ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(exuE1_type,type,
% 2.11/1.53 exuE1: $o ).
% 2.11/1.53
% 2.11/1.53 thf(exuE1,definition,
% 2.11/1.53 ( exuE1
% 2.11/1.53 = ( ! [Xphi: $i > $o] :
% 2.11/1.53 ( ( exu
% 2.11/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.11/1.53 => ? [Xx: $i] :
% 2.11/1.53 ( ( Xphi @ Xx )
% 2.11/1.53 & ! [Xy: $i] :
% 2.11/1.53 ( ( Xphi @ Xy )
% 2.11/1.53 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(prop2set_type,type,
% 2.11/1.53 prop2set: $o > $i ).
% 2.11/1.53
% 2.11/1.53 thf(prop2setE_type,type,
% 2.11/1.53 prop2setE: $o ).
% 2.11/1.53
% 2.11/1.53 thf(prop2setE,definition,
% 2.11/1.53 ( prop2setE
% 2.11/1.53 = ( ! [Xphi: $o,Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.11/1.53 => Xphi ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(emptysetE_type,type,
% 2.11/1.53 emptysetE: $o ).
% 2.11/1.53
% 2.11/1.53 thf(emptysetE,definition,
% 2.11/1.53 ( emptysetE
% 2.11/1.53 = ( ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ emptyset )
% 2.11/1.53 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(emptysetimpfalse_type,type,
% 2.11/1.53 emptysetimpfalse: $o ).
% 2.11/1.53
% 2.11/1.53 thf(emptysetimpfalse,definition,
% 2.11/1.53 ( emptysetimpfalse
% 2.11/1.53 = ( ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ emptyset )
% 2.11/1.53 => $false ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(notinemptyset_type,type,
% 2.11/1.53 notinemptyset: $o ).
% 2.11/1.53
% 2.11/1.53 thf(notinemptyset,definition,
% 2.11/1.53 ( notinemptyset
% 2.11/1.53 = ( ! [Xx: $i] :
% 2.11/1.53 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(exuE3e_type,type,
% 2.11/1.53 exuE3e: $o ).
% 2.11/1.53
% 2.11/1.53 thf(exuE3e,definition,
% 2.11/1.53 ( exuE3e
% 2.11/1.53 = ( ! [Xphi: $i > $o] :
% 2.11/1.53 ( ( exu
% 2.11/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.11/1.53 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(setext_type,type,
% 2.11/1.53 setext: $o ).
% 2.11/1.53
% 2.11/1.53 thf(setext,definition,
% 2.11/1.53 ( setext
% 2.11/1.53 = ( ! [A: $i,B: $i] :
% 2.11/1.53 ( ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 => ( in @ Xx @ B ) )
% 2.11/1.53 => ( ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ B )
% 2.11/1.53 => ( in @ Xx @ A ) )
% 2.11/1.53 => ( A = B ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(emptyI_type,type,
% 2.11/1.53 emptyI: $o ).
% 2.11/1.53
% 2.11/1.53 thf(emptyI,definition,
% 2.11/1.53 ( emptyI
% 2.11/1.53 = ( ! [A: $i] :
% 2.11/1.53 ( ! [Xx: $i] :
% 2.11/1.53 ~ ( in @ Xx @ A )
% 2.11/1.53 => ( A = emptyset ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(noeltsimpempty_type,type,
% 2.11/1.53 noeltsimpempty: $o ).
% 2.11/1.53
% 2.11/1.53 thf(noeltsimpempty,definition,
% 2.11/1.53 ( noeltsimpempty
% 2.11/1.53 = ( ! [A: $i] :
% 2.11/1.53 ( ! [Xx: $i] :
% 2.11/1.53 ~ ( in @ Xx @ A )
% 2.11/1.53 => ( A = emptyset ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(setbeta_type,type,
% 2.11/1.53 setbeta: $o ).
% 2.11/1.53
% 2.11/1.53 thf(setbeta,definition,
% 2.11/1.53 ( setbeta
% 2.11/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 => ( ( in @ Xx
% 2.11/1.53 @ ( dsetconstr @ A
% 2.11/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.11/1.53 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(nonempty_type,type,
% 2.11/1.53 nonempty: $i > $o ).
% 2.11/1.53
% 2.11/1.53 thf(nonempty,definition,
% 2.11/1.53 ( nonempty
% 2.11/1.53 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(nonemptyE1_type,type,
% 2.11/1.53 nonemptyE1: $o ).
% 2.11/1.53
% 2.11/1.53 thf(nonemptyE1,definition,
% 2.11/1.53 ( nonemptyE1
% 2.11/1.53 = ( ! [A: $i] :
% 2.11/1.53 ( ( nonempty @ A )
% 2.11/1.53 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(nonemptyI_type,type,
% 2.11/1.53 nonemptyI: $o ).
% 2.11/1.53
% 2.11/1.53 thf(nonemptyI,definition,
% 2.11/1.53 ( nonemptyI
% 2.11/1.53 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 => ( ( Xphi @ Xx )
% 2.11/1.53 => ( nonempty
% 2.11/1.53 @ ( dsetconstr @ A
% 2.11/1.53 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(nonemptyI1_type,type,
% 2.11/1.53 nonemptyI1: $o ).
% 2.11/1.53
% 2.11/1.53 thf(nonemptyI1,definition,
% 2.11/1.53 ( nonemptyI1
% 2.11/1.53 = ( ! [A: $i] :
% 2.11/1.53 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.11/1.53 => ( nonempty @ A ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(setadjoinIL_type,type,
% 2.11/1.53 setadjoinIL: $o ).
% 2.11/1.53
% 2.11/1.53 thf(setadjoinIL,definition,
% 2.11/1.53 ( setadjoinIL
% 2.11/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(emptyinunitempty_type,type,
% 2.11/1.53 emptyinunitempty: $o ).
% 2.11/1.53
% 2.11/1.53 thf(emptyinunitempty,definition,
% 2.11/1.53 ( emptyinunitempty
% 2.11/1.53 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(setadjoinIR_type,type,
% 2.11/1.53 setadjoinIR: $o ).
% 2.11/1.53
% 2.11/1.53 thf(setadjoinIR,definition,
% 2.11/1.53 ( setadjoinIR
% 2.11/1.53 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.11/1.53 ( ( in @ Xy @ A )
% 2.11/1.53 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(setadjoinE_type,type,
% 2.11/1.53 setadjoinE: $o ).
% 2.11/1.53
% 2.11/1.53 thf(setadjoinE,definition,
% 2.11/1.53 ( setadjoinE
% 2.11/1.53 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.11/1.53 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.11/1.53 => ! [Xphi: $o] :
% 2.11/1.53 ( ( ( Xy = Xx )
% 2.11/1.53 => Xphi )
% 2.11/1.53 => ( ( ( in @ Xy @ A )
% 2.11/1.53 => Xphi )
% 2.11/1.53 => Xphi ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(setadjoinOr_type,type,
% 2.11/1.53 setadjoinOr: $o ).
% 2.11/1.53
% 2.11/1.53 thf(setadjoinOr,definition,
% 2.11/1.53 ( setadjoinOr
% 2.11/1.53 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.11/1.53 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.11/1.53 => ( ( Xy = Xx )
% 2.11/1.53 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(setoftrueEq_type,type,
% 2.11/1.53 setoftrueEq: $o ).
% 2.11/1.53
% 2.11/1.53 thf(setoftrueEq,definition,
% 2.11/1.53 ( setoftrueEq
% 2.11/1.53 = ( ! [A: $i] :
% 2.11/1.53 ( ( dsetconstr @ A
% 2.11/1.53 @ ^ [Xx: $i] : $true )
% 2.11/1.53 = A ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(powersetI_type,type,
% 2.11/1.53 powersetI: $o ).
% 2.11/1.53
% 2.11/1.53 thf(powersetI,definition,
% 2.11/1.53 ( powersetI
% 2.11/1.53 = ( ! [A: $i,B: $i] :
% 2.11/1.53 ( ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ B )
% 2.11/1.53 => ( in @ Xx @ A ) )
% 2.11/1.53 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(emptyinPowerset_type,type,
% 2.11/1.53 emptyinPowerset: $o ).
% 2.11/1.53
% 2.11/1.53 thf(emptyinPowerset,definition,
% 2.11/1.53 ( emptyinPowerset
% 2.11/1.53 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(emptyInPowerset_type,type,
% 2.11/1.53 emptyInPowerset: $o ).
% 2.11/1.53
% 2.11/1.53 thf(emptyInPowerset,definition,
% 2.11/1.53 ( emptyInPowerset
% 2.11/1.53 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(powersetE_type,type,
% 2.11/1.53 powersetE: $o ).
% 2.11/1.53
% 2.11/1.53 thf(powersetE,definition,
% 2.11/1.53 ( powersetE
% 2.11/1.53 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.11/1.53 ( ( in @ B @ ( powerset @ A ) )
% 2.11/1.53 => ( ( in @ Xx @ B )
% 2.11/1.53 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(setunionI_type,type,
% 2.11/1.53 setunionI: $o ).
% 2.11/1.53
% 2.11/1.53 thf(setunionI,definition,
% 2.11/1.53 ( setunionI
% 2.11/1.53 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.11/1.53 ( ( in @ Xx @ B )
% 2.11/1.53 => ( ( in @ B @ A )
% 2.11/1.53 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(setunionE_type,type,
% 2.11/1.53 setunionE: $o ).
% 2.11/1.53
% 2.11/1.53 thf(setunionE,definition,
% 2.11/1.53 ( setunionE
% 2.11/1.53 = ( ! [A: $i,Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ ( setunion @ A ) )
% 2.11/1.53 => ! [Xphi: $o] :
% 2.11/1.53 ( ! [B: $i] :
% 2.11/1.53 ( ( in @ Xx @ B )
% 2.11/1.53 => ( ( in @ B @ A )
% 2.11/1.53 => Xphi ) )
% 2.11/1.53 => Xphi ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(subPowSU_type,type,
% 2.11/1.53 subPowSU: $o ).
% 2.11/1.53
% 2.11/1.53 thf(subPowSU,definition,
% 2.11/1.53 ( subPowSU
% 2.11/1.53 = ( ! [A: $i,Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(exuE2_type,type,
% 2.11/1.53 exuE2: $o ).
% 2.11/1.53
% 2.11/1.53 thf(exuE2,definition,
% 2.11/1.53 ( exuE2
% 2.11/1.53 = ( ! [Xphi: $i > $o] :
% 2.11/1.53 ( ( exu
% 2.11/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.11/1.53 => ? [Xx: $i] :
% 2.11/1.53 ! [Xy: $i] :
% 2.11/1.53 ( ( Xphi @ Xy )
% 2.11/1.53 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(nonemptyImpWitness_type,type,
% 2.11/1.53 nonemptyImpWitness: $o ).
% 2.11/1.53
% 2.11/1.53 thf(nonemptyImpWitness,definition,
% 2.11/1.53 ( nonemptyImpWitness
% 2.11/1.53 = ( ! [A: $i] :
% 2.11/1.53 ( ( nonempty @ A )
% 2.11/1.53 => ? [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 & $true ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(uniqinunit_type,type,
% 2.11/1.53 uniqinunit: $o ).
% 2.11/1.53
% 2.11/1.53 thf(uniqinunit,definition,
% 2.11/1.53 ( uniqinunit
% 2.11/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.11/1.53 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.11/1.53 => ( Xx = Xy ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(notinsingleton_type,type,
% 2.11/1.53 notinsingleton: $o ).
% 2.11/1.53
% 2.11/1.53 thf(notinsingleton,definition,
% 2.11/1.53 ( notinsingleton
% 2.11/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.11/1.53 ( ( Xx != Xy )
% 2.11/1.53 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(eqinunit_type,type,
% 2.11/1.53 eqinunit: $o ).
% 2.11/1.53
% 2.11/1.53 thf(eqinunit,definition,
% 2.11/1.53 ( eqinunit
% 2.11/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.11/1.53 ( ( Xx = Xy )
% 2.11/1.53 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(singletonsswitch_type,type,
% 2.11/1.53 singletonsswitch: $o ).
% 2.11/1.53
% 2.11/1.53 thf(singletonsswitch,definition,
% 2.11/1.53 ( singletonsswitch
% 2.11/1.53 = ( ! [Xx: $i,Xy: $i] :
% 2.11/1.53 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.11/1.53 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(upairsetE_type,type,
% 2.11/1.53 upairsetE: $o ).
% 2.11/1.53
% 2.11/1.53 thf(upairsetE,definition,
% 2.11/1.53 ( upairsetE
% 2.11/1.53 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.11/1.53 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.11/1.53 => ( ( Xz = Xx )
% 2.11/1.53 | ( Xz = Xy ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(upairsetIL_type,type,
% 2.11/1.53 upairsetIL: $o ).
% 2.11/1.53
% 2.11/1.53 thf(upairsetIL,definition,
% 2.11/1.53 ( upairsetIL
% 2.11/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(upairsetIR_type,type,
% 2.11/1.53 upairsetIR: $o ).
% 2.11/1.53
% 2.11/1.53 thf(upairsetIR,definition,
% 2.11/1.53 ( upairsetIR
% 2.11/1.53 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(emptyE1_type,type,
% 2.11/1.53 emptyE1: $o ).
% 2.11/1.53
% 2.11/1.53 thf(emptyE1,definition,
% 2.11/1.53 ( emptyE1
% 2.11/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.11/1.53 ( ? [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 & ( Xphi @ Xx ) )
% 2.11/1.53 => ( ( ( dsetconstr @ A
% 2.11/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.11/1.53 = emptyset )
% 2.11/1.53 => $false ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(vacuousDall_type,type,
% 2.11/1.53 vacuousDall: $o ).
% 2.11/1.53
% 2.11/1.53 thf(vacuousDall,definition,
% 2.11/1.53 ( vacuousDall
% 2.11/1.53 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ emptyset )
% 2.11/1.53 => ( Xphi @ Xx ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(quantDeMorgan1_type,type,
% 2.11/1.53 quantDeMorgan1: $o ).
% 2.11/1.53
% 2.11/1.53 thf(quantDeMorgan1,definition,
% 2.11/1.53 ( quantDeMorgan1
% 2.11/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.11/1.53 ( ~ ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 => ( Xphi @ Xx ) )
% 2.11/1.53 => ? [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(quantDeMorgan2_type,type,
% 2.11/1.53 quantDeMorgan2: $o ).
% 2.11/1.53
% 2.11/1.53 thf(quantDeMorgan2,definition,
% 2.11/1.53 ( quantDeMorgan2
% 2.11/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.11/1.53 ( ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 => ~ ( Xphi @ Xx ) )
% 2.11/1.53 => ~ ? [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(quantDeMorgan3_type,type,
% 2.11/1.53 quantDeMorgan3: $o ).
% 2.11/1.53
% 2.11/1.53 thf(quantDeMorgan3,definition,
% 2.11/1.53 ( quantDeMorgan3
% 2.11/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.11/1.53 ( ~ ? [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 & ( Xphi @ Xx ) )
% 2.11/1.53 => ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(quantDeMorgan4_type,type,
% 2.11/1.53 quantDeMorgan4: $o ).
% 2.11/1.53
% 2.11/1.53 thf(quantDeMorgan4,definition,
% 2.11/1.53 ( quantDeMorgan4
% 2.11/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.11/1.53 ( ? [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 & ~ ( Xphi @ Xx ) )
% 2.11/1.53 => ~ ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(prop2setI_type,type,
% 2.11/1.53 prop2setI: $o ).
% 2.11/1.53
% 2.11/1.53 thf(prop2setI,definition,
% 2.11/1.53 ( prop2setI
% 2.11/1.53 = ( ! [Xphi: $o] :
% 2.11/1.53 ( Xphi
% 2.11/1.53 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(set2prop_type,type,
% 2.11/1.53 set2prop: $i > $o ).
% 2.11/1.53
% 2.11/1.53 thf(prop2set2propI_type,type,
% 2.11/1.53 prop2set2propI: $o ).
% 2.11/1.53
% 2.11/1.53 thf(prop2set2propI,definition,
% 2.11/1.53 ( prop2set2propI
% 2.11/1.53 = ( ! [Xphi: $o] :
% 2.11/1.53 ( Xphi
% 2.11/1.53 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(notdexE_type,type,
% 2.11/1.53 notdexE: $o ).
% 2.11/1.53
% 2.11/1.53 thf(notdexE,definition,
% 2.11/1.53 ( notdexE
% 2.11/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.11/1.53 ( ~ ? [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 & ( Xphi @ Xx ) )
% 2.11/1.53 => ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(notdallE_type,type,
% 2.11/1.53 notdallE: $o ).
% 2.11/1.53
% 2.11/1.53 thf(notdallE,definition,
% 2.11/1.53 ( notdallE
% 2.11/1.53 = ( ! [A: $i,Xphi: $i > $o] :
% 2.11/1.53 ( ~ ! [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 => ( Xphi @ Xx ) )
% 2.11/1.53 => ? [Xx: $i] :
% 2.11/1.53 ( ( in @ Xx @ A )
% 2.11/1.53 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(exuI1_type,type,
% 2.11/1.53 exuI1: $o ).
% 2.11/1.53
% 2.11/1.53 thf(exuI1,definition,
% 2.11/1.53 ( exuI1
% 2.11/1.53 = ( ! [Xphi: $i > $o] :
% 2.11/1.53 ( ? [Xx: $i] :
% 2.11/1.53 ( ( Xphi @ Xx )
% 2.11/1.53 & ! [Xy: $i] :
% 2.11/1.53 ( ( Xphi @ Xy )
% 2.11/1.53 => ( Xx = Xy ) ) )
% 2.11/1.53 => ( exu
% 2.11/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(exuI3_type,type,
% 2.11/1.53 exuI3: $o ).
% 2.11/1.53
% 2.11/1.53 thf(exuI3,definition,
% 2.11/1.53 ( exuI3
% 2.11/1.53 = ( ! [Xphi: $i > $o] :
% 2.11/1.53 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.11/1.53 => ( ! [Xx: $i,Xy: $i] :
% 2.11/1.53 ( ( Xphi @ Xx )
% 2.11/1.53 => ( ( Xphi @ Xy )
% 2.11/1.53 => ( Xx = Xy ) ) )
% 2.11/1.53 => ( exu
% 2.11/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(exuI2_type,type,
% 2.11/1.53 exuI2: $o ).
% 2.11/1.53
% 2.11/1.53 thf(exuI2,definition,
% 2.11/1.53 ( exuI2
% 2.11/1.53 = ( ! [Xphi: $i > $o] :
% 2.11/1.53 ( ? [Xx: $i] :
% 2.11/1.53 ! [Xy: $i] :
% 2.11/1.53 ( ( Xphi @ Xy )
% 2.11/1.53 <=> ( Xy = Xx ) )
% 2.11/1.53 => ( exu
% 2.11/1.53 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(inCongP_type,type,
% 2.11/1.53 inCongP: $o ).
% 2.11/1.53
% 2.11/1.53 thf(inCongP,definition,
% 2.11/1.53 ( inCongP
% 2.11/1.53 = ( ! [A: $i,B: $i] :
% 2.11/1.53 ( ( A = B )
% 2.11/1.53 => ! [Xx: $i,Xy: $i] :
% 2.11/1.53 ( ( Xx = Xy )
% 2.11/1.53 => ( ( in @ Xx @ A )
% 2.11/1.53 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.11/1.53
% 2.11/1.53 thf(in__Cong,conjecture,
% 2.11/1.53 ( setextAx
% 2.11/1.53 => ( emptysetAx
% 2.11/1.53 => ( setadjoinAx
% 2.11/1.53 => ( powersetAx
% 2.11/1.53 => ( setunionAx
% 2.11/1.53 => ( omega0Ax
% 2.11/1.53 => ( omegaSAx
% 2.11/1.53 => ( omegaIndAx
% 2.11/1.53 => ( replAx
% 2.11/1.53 => ( foundationAx
% 2.11/1.53 => ( wellorderingAx
% 2.11/1.53 => ( descrp
% 2.11/1.53 => ( dsetconstrI
% 2.11/1.53 => ( dsetconstrEL
% 2.11/1.53 => ( dsetconstrER
% 2.11/1.53 => ( exuE1
% 2.11/1.53 => ( prop2setE
% 2.11/1.53 => ( emptysetE
% 2.11/1.53 => ( emptysetimpfalse
% 2.11/1.53 => ( notinemptyset
% 2.11/1.53 => ( exuE3e
% 2.11/1.53 => ( setext
% 2.11/1.53 => ( emptyI
% 2.11/1.53 => ( noeltsimpempty
% 2.11/1.53 => ( setbeta
% 2.11/1.53 => ( nonemptyE1
% 2.11/1.53 => ( nonemptyI
% 2.11/1.53 => ( nonemptyI1
% 2.11/1.53 => ( setadjoinIL
% 2.11/1.53 => ( emptyinunitempty
% 2.11/1.53 => ( setadjoinIR
% 2.11/1.53 => ( setadjoinE
% 2.11/1.53 => ( setadjoinOr
% 2.11/1.53 => ( setoftrueEq
% 2.11/1.53 => ( powersetI
% 2.11/1.53 => ( emptyinPowerset
% 2.11/1.53 => ( emptyInPowerset
% 2.11/1.53 => ( powersetE
% 2.11/1.53 => ( setunionI
% 2.11/1.53 => ( setunionE
% 2.11/1.53 => ( subPowSU
% 2.11/1.53 => ( exuE2
% 2.11/1.53 => ( nonemptyImpWitness
% 2.11/1.53 => ( uniqinunit
% 2.11/1.53 => ( notinsingleton
% 2.11/1.53 => ( eqinunit
% 2.11/1.53 => ( singletonsswitch
% 2.11/1.53 => ( upairsetE
% 2.11/1.53 => ( upairsetIL
% 2.11/1.53 => ( upairsetIR
% 2.11/1.53 => ( emptyE1
% 2.11/1.53 => ( vacuousDall
% 2.11/1.53 => ( quantDeMorgan1
% 2.11/1.53 => ( quantDeMorgan2
% 2.11/1.53 => ( quantDeMorgan3
% 2.11/1.53 => ( quantDeMorgan4
% 2.11/1.53 => ( prop2setI
% 2.11/1.53 => ( prop2set2propI
% 2.11/1.53 => ( notdexE
% 2.11/1.53 => ( notdallE
% 2.11/1.53 => ( exuI1
% 2.11/1.53 => ( exuI3
% 2.11/1.53 => ( exuI2
% 2.11/1.53 => ( inCongP
% 2.11/1.53 => ! [A: $i,B: $i] :
% 2.11/1.53 ( ( A = B )
% 2.11/1.53 => ! [Xx: $i,Xy: $i] :
% 2.11/1.53 ( ( Xx = Xy )
% 2.11/1.53 => ( ( in @ Xx @ A )
% 2.11/1.53 <=> ( in @ Xy @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.11/1.53 ------------------------
% 2.11/1.56 % (31046)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_30943 for (2999ds/2Mi)
% 2.11/1.56 % (31047)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_30943 for (2999ds/275Mi)
% 2.11/1.56 % (31043)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_30943 for (2999ds/4Mi)
% 2.11/1.56 % (31042)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_30943 for (2999ds/183Mi)
% 2.11/1.56 % (31048)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_30943 for (2999ds/18Mi)
% 2.11/1.56 % (31045)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_30943 for (2999ds/2Mi)
% 2.11/1.56 % (31044)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_30943 for (2999ds/27Mi)
% 2.11/1.56 % (31045)Instruction limit reached!
% 2.11/1.56 % (31045)------------------------------
% 2.11/1.56 % (31045)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.11/1.56 % (31045)Termination reason: Unknown
% 2.11/1.56 % (31045)Termination phase: shuffling
% 2.11/1.56
% 2.11/1.56 % (31045)Memory used [KB]: 1151
% 2.11/1.56 % (31045)Time elapsed: 0.003 s
% 2.11/1.56 % (31045)Instructions burned: 3 (million)
% 2.11/1.56 % (31045)------------------------------
% 2.11/1.56 % (31045)------------------------------
% 2.11/1.56 % (31046)Instruction limit reached!
% 2.11/1.56 % (31046)------------------------------
% 2.11/1.56 % (31046)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.11/1.56 % (31046)Termination reason: Unknown
% 2.11/1.56 % (31046)Termination phase: shuffling
% 2.11/1.56
% 2.11/1.56 % (31046)Memory used [KB]: 1151
% 2.11/1.56 % (31046)Time elapsed: 0.003 s
% 2.11/1.56 % (31046)Instructions burned: 3 (million)
% 2.11/1.56 % (31046)------------------------------
% 2.11/1.56 % (31046)------------------------------
% 2.11/1.56 % (31043)Instruction limit reached!
% 2.11/1.56 % (31043)------------------------------
% 2.11/1.56 % (31043)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.11/1.56 % (31043)Termination reason: Unknown
% 2.11/1.56 % (31043)Termination phase: shuffling
% 2.11/1.56
% 2.11/1.56 % (31043)Memory used [KB]: 1151
% 2.11/1.56 % (31043)Time elapsed: 0.004 s
% 2.11/1.56 % (31043)Instructions burned: 4 (million)
% 2.11/1.56 % (31043)------------------------------
% 2.11/1.56 % (31043)------------------------------
% 2.11/1.56 % (31048)Instruction limit reached!
% 2.11/1.56 % (31048)------------------------------
% 2.11/1.56 % (31048)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.11/1.56 % (31048)Termination reason: Unknown
% 2.11/1.56 % (31048)Termination phase: Preprocessing 3
% 2.11/1.56
% 2.11/1.56 % (31048)Memory used [KB]: 1663
% 2.11/1.56 % (31048)Time elapsed: 0.011 s
% 2.11/1.56 % (31048)Instructions burned: 19 (million)
% 2.11/1.56 % (31048)------------------------------
% 2.11/1.56 % (31048)------------------------------
% 2.11/1.57 % (31044)Instruction limit reached!
% 2.11/1.57 % (31044)------------------------------
% 2.11/1.57 % (31044)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.11/1.57 % (31044)Termination reason: Unknown
% 2.11/1.57 % (31044)Termination phase: Property scanning
% 2.11/1.57
% 2.11/1.57 % (31044)Memory used [KB]: 1407
% 2.11/1.57 % (31044)Time elapsed: 0.014 s
% 2.11/1.57 % (31044)Instructions burned: 27 (million)
% 2.11/1.57 % (31044)------------------------------
% 2.11/1.57 % (31044)------------------------------
% 2.11/1.57 % (31049)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_30943 for (2999ds/3Mi)
% 2.11/1.57 % (31050)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_30943 for (2999ds/37Mi)
% 2.11/1.57 % (31051)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_30943 for (2999ds/15Mi)
% 2.11/1.57 % (31049)Instruction limit reached!
% 2.11/1.57 % (31049)------------------------------
% 2.11/1.57 % (31049)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.11/1.57 % (31049)Termination reason: Unknown
% 2.11/1.57 % (31049)Termination phase: shuffling
% 2.11/1.57
% 2.11/1.57 % (31049)Memory used [KB]: 1151
% 2.11/1.57 % (31049)Time elapsed: 0.003 s
% 2.11/1.57 % (31049)Instructions burned: 4 (million)
% 2.11/1.57 % (31049)------------------------------
% 2.11/1.57 % (31049)------------------------------
% 2.11/1.58 % (31047)First to succeed.
% 2.11/1.58 % (31047)Refutation found. Thanks to Tanya!
% 2.11/1.58 % SZS status Theorem for DTF2THF_30943
% 2.11/1.58 % SZS output start Proof for DTF2THF_30943
% 2.11/1.58 thf(func_def_0, type, in: $i > $i > $o).
% 2.11/1.58 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.11/1.58 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.11/1.58 thf(func_def_8, type, powerset: $i > $i).
% 2.11/1.58 thf(func_def_10, type, setunion: $i > $i).
% 2.11/1.58 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.11/1.58 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.11/1.58 thf(func_def_26, type, prop2set: $o > $i).
% 2.11/1.58 thf(func_def_36, type, nonempty: $i > $o).
% 2.11/1.58 thf(func_def_69, type, set2prop: $i > $o).
% 2.11/1.58 thf(func_def_90, type, sP1: $i > $i > $o).
% 2.11/1.58 thf(func_def_91, type, sP2: $i > $o).
% 2.11/1.58 thf(func_def_92, type, sP3: $i > $i > $o).
% 2.11/1.58 thf(func_def_93, type, sP4: $i > $i > $o).
% 2.11/1.58 thf(func_def_95, type, sK6: $i > $o).
% 2.11/1.58 thf(func_def_105, type, sK16: $i > $o).
% 2.11/1.58 thf(func_def_107, type, sK18: ($i > $o) > $i > $i).
% 2.11/1.58 thf(func_def_112, type, sK23: $i > $i > $i).
% 2.11/1.58 thf(func_def_119, type, sK30: $i > $o).
% 2.11/1.58 thf(func_def_123, type, sK34: $i > $i).
% 2.11/1.58 thf(func_def_131, type, sK42: $i > $i > $i).
% 2.11/1.58 thf(func_def_134, type, sK45: $i > $i > $i).
% 2.11/1.58 thf(func_def_138, type, sK49: $i > $o).
% 2.11/1.58 thf(func_def_141, type, sK52: ($i > $o) > $i > $i).
% 2.11/1.58 thf(func_def_143, type, sK54: $i > $o).
% 2.11/1.58 thf(func_def_146, type, sK57: $i > $o).
% 2.11/1.58 thf(func_def_149, type, sK60: $i > ($i > $o) > $i).
% 2.11/1.58 thf(func_def_156, type, sK67: $i > $o).
% 2.11/1.58 thf(func_def_157, type, sK68: $i > $o).
% 2.11/1.58 thf(func_def_158, type, sK69: $i > $i).
% 2.11/1.58 thf(func_def_159, type, sK70: ($i > $o) > $i).
% 2.11/1.58 thf(func_def_165, type, sK76: $i > $o).
% 2.11/1.58 thf(func_def_166, type, sK77: ($i > $o) > $i).
% 2.11/1.58 thf(func_def_167, type, sK78: $i > $i > $i).
% 2.11/1.58 thf(func_def_176, type, sK87: $i > $o).
% 2.11/1.58 thf(func_def_177, type, sK88: $i > $o).
% 2.11/1.58 thf(func_def_178, type, sK89: $i > $i).
% 2.11/1.58 thf(func_def_179, type, sK90: ($i > $o) > $i).
% 2.11/1.58 thf(func_def_183, type, sK94: $i > $i > $i).
% 2.11/1.58 thf(func_def_184, type, sK95: $i > $i > $i).
% 2.11/1.58 thf(func_def_187, type, sK98: $i > $o).
% 2.11/1.58 thf(func_def_189, type, sK100: ($i > $o) > $i).
% 2.11/1.58 thf(func_def_190, type, sK101: ($i > $o) > $i).
% 2.11/1.58 thf(func_def_191, type, sK102: $i > ($i > $o) > $i).
% 2.11/1.58 thf(func_def_192, type, sK103: $i > $o).
% 2.11/1.58 thf(func_def_196, type, sK107: $i > $i).
% 2.11/1.58 thf(func_def_197, type, sK108: $i > $i).
% 2.11/1.58 thf(func_def_198, type, sK109: $i > $o).
% 2.11/1.58 thf(func_def_200, type, sK111: $i > $o).
% 2.11/1.58 thf(func_def_205, type, sK116: $i > ($i > $o) > $i).
% 2.11/1.58 thf(func_def_206, type, sK117: $i > $o).
% 2.11/1.58 thf(func_def_209, type, sK120: $i > $i > $o).
% 2.11/1.58 thf(func_def_210, type, sK121: $i > $i).
% 2.11/1.58 thf(func_def_211, type, sK122: $i > $i).
% 2.11/1.58 thf(func_def_212, type, sK123: ($i > $i > $o) > $i > $i).
% 2.11/1.58 thf(func_def_213, type, sK124: ($i > $i > $o) > $i > $i).
% 2.11/1.58 thf(func_def_214, type, sK125: $i > ($i > $i > $o) > $i > $i).
% 2.11/1.58 thf(func_def_215, type, sK126: $i > $i).
% 2.11/1.58 thf(func_def_219, type, sK130: $i > $i).
% 2.11/1.58 thf(func_def_227, type, sK138: $i > $i > $i).
% 2.11/1.58 thf(func_def_228, type, sK139: $i > $i > $i).
% 2.11/1.58 thf(func_def_229, type, sK140: $i > $i > $i).
% 2.11/1.58 thf(func_def_230, type, sK141: $i > $i > $i).
% 2.11/1.58 thf(func_def_231, type, sK142: $i > $i > $i > $i).
% 2.11/1.58 thf(func_def_232, type, sK143: $i > $i).
% 2.11/1.58 thf(func_def_233, type, sK144: $i > $i).
% 2.11/1.58 thf(func_def_234, type, sK145: $i > $i).
% 2.11/1.58 thf(func_def_235, type, sK146: $i > $i).
% 2.11/1.58 thf(func_def_236, type, sK147: $i > $i > $i).
% 2.11/1.58 thf(func_def_237, type, sK148: $i > $i > $i > $i).
% 2.11/1.58 thf(func_def_238, type, sK149: $i > $i > $i > $i).
% 2.11/1.58 thf(func_def_239, type, sK150: $i > $i > $i).
% 2.11/1.58 thf(func_def_240, type, sK151: $i > $i > $i).
% 2.11/1.58 thf(func_def_242, type, sK153: $i > $i).
% 2.11/1.58 thf(func_def_243, type, sK154: $i > $i).
% 2.11/1.58 thf(func_def_244, type, sK155: $i > $i).
% 2.11/1.58 thf(func_def_245, type, sK156: $i > $i > $i).
% 2.11/1.58 thf(func_def_249, type, sK160: $o > $i > $i > $i).
% 2.11/1.58 thf(func_def_260, type, sK171: $i > $i).
% 2.11/1.58 thf(func_def_262, type, sK173: $i > $i).
% 2.11/1.58 thf(func_def_263, type, sK174: $i > ($i > $o) > $i).
% 2.11/1.58 thf(func_def_264, type, sK175: $i > $o).
% 2.11/1.58 thf(func_def_268, type, sK179: ($i > $o) > $i > $i).
% 2.11/1.58 thf(func_def_270, type, sK181: $i > $o).
% 2.11/1.58 thf(func_def_275, type, sK186: ($i > $o) > $i > $i).
% 2.11/1.58 thf(func_def_277, type, sK188: $i > $o).
% 2.11/1.58 thf(func_def_280, type, ph191: !>[X0: $tType]:(X0)).
% 2.11/1.58 thf(f1269,plain,(
% 2.11/1.58 $false),
% 2.11/1.58 inference(trivial_inequality_removal,[status(thm)],[f1071])).
% 2.11/1.58 thf(f1071,plain,(
% 2.11/1.58 ((in @ sK84 @ sK82) != (in @ sK84 @ sK82))),
% 2.11/1.58 inference(definition_unfolding,[status(thm)],[f762,f763,f764])).
% 2.11/1.58 thf(f764,plain,(
% 2.11/1.58 (sK81 = sK82)),
% 2.11/1.58 inference(cnf_transformation,[status(thm)],[f430])).
% 2.11/1.58 thf(f430,plain,(
% 2.11/1.58 (quantDeMorgan3 = $true) & (notinemptyset = $true) & (noeltsimpempty = $true) & (emptysetE = $true) & (exuE1 = $true) & (nonemptyI1 = $true) & (uniqinunit = $true) & (prop2set2propI = $true) & (singletonsswitch = $true) & (omega0Ax = $true) & (notdallE = $true) & (nonemptyE1 = $true) & (upairsetE = $true) & (emptyinPowerset = $true) & (emptyInPowerset = $true) & (inCongP = $true) & (upairsetIR = $true) & (wellorderingAx = $true) & (foundationAx = $true) & (notdexE = $true) & (exuE3e = $true) & (omegaIndAx = $true) & (quantDeMorgan1 = $true) & (prop2setI = $true) & (eqinunit = $true) & (dsetconstrI = $true) & (dsetconstrER = $true) & (exuI2 = $true) & (prop2setE = $true) & (setextAx = $true) & (replAx = $true) & ((sK81 = sK82) & ((sK83 = sK84) & ((in @ sK84 @ sK82) != (in @ sK83 @ sK81)))) & (quantDeMorgan4 = $true) & (emptyE1 = $true) & (setadjoinIR = $true) & (notinsingleton = $true) & (setadjoinIL = $true) & (omegaSAx = $true) & (setadjoinE = $true) & (descrp = $true) & (vacuousDall = $true) & (subPowSU = $true) & (setadjoinOr = $true) & (upairsetIL = $true) & (powersetE = $true) & (setadjoinAx = $true) & (powersetAx = $true) & (emptysetAx = $true) & (setext = $true) & (quantDeMorgan2 = $true) & (emptysetimpfalse = $true) & (nonemptyI = $true) & (setunionI = $true) & (dsetconstrEL = $true) & (exuI1 = $true) & (emptyinunitempty = $true) & (setunionAx = $true) & (setunionE = $true) & (setoftrueEq = $true) & (powersetI = $true) & (emptyI = $true) & (exuE2 = $true) & (nonemptyImpWitness = $true) & (exuI3 = $true) & (setbeta = $true)),
% 2.11/1.58 inference(skolemisation,[status(esa),new_symbols(skolem,[sK81,sK82,sK83,sK84])],[f273,f429,f428])).
% 2.11/1.58 thf(f428,plain,(
% 2.11/1.58 ? [X0 : $i,X1 : $i] : ((X0 = X1) & ? [X2 : $i,X3 : $i] : ((X2 = X3) & ((in @ X3 @ X1) != (in @ X2 @ X0)))) => ((sK81 = sK82) & ? [X3 : $i,X2 : $i] : ((X2 = X3) & ((in @ X3 @ sK82) != (in @ X2 @ sK81))))),
% 2.11/1.58 introduced(definition,[],[choice_axiom])).
% 2.11/1.58 thf(f429,plain,(
% 2.11/1.58 ? [X3 : $i,X2 : $i] : ((X2 = X3) & ((in @ X3 @ sK82) != (in @ X2 @ sK81))) => ((sK83 = sK84) & ((in @ sK84 @ sK82) != (in @ sK83 @ sK81)))),
% 2.11/1.58 introduced(definition,[],[choice_axiom])).
% 2.11/1.58 thf(f273,plain,(
% 2.11/1.58 (quantDeMorgan3 = $true) & (notinemptyset = $true) & (noeltsimpempty = $true) & (emptysetE = $true) & (exuE1 = $true) & (nonemptyI1 = $true) & (uniqinunit = $true) & (prop2set2propI = $true) & (singletonsswitch = $true) & (omega0Ax = $true) & (notdallE = $true) & (nonemptyE1 = $true) & (upairsetE = $true) & (emptyinPowerset = $true) & (emptyInPowerset = $true) & (inCongP = $true) & (upairsetIR = $true) & (wellorderingAx = $true) & (foundationAx = $true) & (notdexE = $true) & (exuE3e = $true) & (omegaIndAx = $true) & (quantDeMorgan1 = $true) & (prop2setI = $true) & (eqinunit = $true) & (dsetconstrI = $true) & (dsetconstrER = $true) & (exuI2 = $true) & (prop2setE = $true) & (setextAx = $true) & (replAx = $true) & ? [X0 : $i,X1 : $i] : ((X0 = X1) & ? [X2 : $i,X3 : $i] : ((X2 = X3) & ((in @ X3 @ X1) != (in @ X2 @ X0)))) & (quantDeMorgan4 = $true) & (emptyE1 = $true) & (setadjoinIR = $true) & (notinsingleton = $true) & (setadjoinIL = $true) & (omegaSAx = $true) & (setadjoinE = $true) & (descrp = $true) & (vacuousDall = $true) & (subPowSU = $true) & (setadjoinOr = $true) & (upairsetIL = $true) & (powersetE = $true) & (setadjoinAx = $true) & (powersetAx = $true) & (emptysetAx = $true) & (setext = $true) & (quantDeMorgan2 = $true) & (emptysetimpfalse = $true) & (nonemptyI = $true) & (setunionI = $true) & (dsetconstrEL = $true) & (exuI1 = $true) & (emptyinunitempty = $true) & (setunionAx = $true) & (setunionE = $true) & (setoftrueEq = $true) & (powersetI = $true) & (emptyI = $true) & (exuE2 = $true) & (nonemptyImpWitness = $true) & (exuI3 = $true) & (setbeta = $true)),
% 2.11/1.58 inference(flattening,[status(thm)],[f272])).
% 2.11/1.58 thf(f272,plain,(
% 2.11/1.58 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : ((X0 = X1) & ? [X2 : $i,X3 : $i] : ((X2 = X3) & ((in @ X3 @ X1) != (in @ X2 @ X0)))) & (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)),
% 2.11/1.58 inference(ennf_transformation,[status(thm)],[f151])).
% 2.11/1.58 thf(f151,plain,(
% 2.11/1.58 ~((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) => ! [X0 : $i,X1 : $i] : ((X0 = X1) => ! [X2 : $i,X3 : $i] : ((X2 = X3) => ((in @ X3 @ X1) = (in @ X2 @ X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.11/1.58 inference(fool_elimination,[status(thm)],[f150])).
% 2.11/1.58 thf(f150,plain,(
% 2.11/1.58 ~(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 => ! [X0 : $i,X1 : $i] : ((X0 = X1) => ! [X2 : $i,X3 : $i] : ((X2 = X3) => ((in @ X2 @ X0) <=> (in @ X3 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.11/1.58 inference(rectify,[status(thm)],[f68])).
% 2.11/1.58 thf(f68,negated_conjecture,(
% 2.11/1.58 ~(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 => ! [X3 : $i,X4 : $i] : ((X3 = X4) => ! [X1 : $i,X2 : $i] : ((X1 = X2) => ((in @ X1 @ X3) <=> (in @ X2 @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.11/1.58 inference(negated_conjecture,[status(cth)],[f67])).
% 2.11/1.58 thf(f67,conjecture,(
% 2.11/1.58 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 => ! [X3 : $i,X4 : $i] : ((X3 = X4) => ! [X1 : $i,X2 : $i] : ((X1 = X2) => ((in @ X1 @ X3) <=> (in @ X2 @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.11/1.58 file('/export/starexec/sandbox/tmp/tmp.Ifh9i1xfml/DTF2THF_30943.p',in__Cong)).
% 2.11/1.58 thf(f763,plain,(
% 2.11/1.58 (sK83 = sK84)),
% 2.11/1.58 inference(cnf_transformation,[status(thm)],[f430])).
% 2.11/1.58 thf(f762,plain,(
% 2.11/1.58 ((in @ sK84 @ sK82) != (in @ sK83 @ sK81))),
% 2.11/1.58 inference(cnf_transformation,[status(thm)],[f430])).
% 2.11/1.58 % SZS output end Proof for DTF2THF_30943
% 2.11/1.58 % (31047)------------------------------
% 2.11/1.58 % (31047)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.11/1.58 % (31047)Termination reason: Refutation
% 2.11/1.58
% 2.11/1.58 % (31047)Memory used [KB]: 6524
% 2.11/1.58 % (31047)Time elapsed: 0.025 s
% 2.11/1.58 % (31047)Instructions burned: 45 (million)
% 2.11/1.58 % (31047)------------------------------
% 2.11/1.58 % (31047)------------------------------
% 2.11/1.58 % (31041)Success in time 0.041 s
%------------------------------------------------------------------------------