%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU553^1 : 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 : n013.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 09:35:14 AM UTC 2026
% Result : Theorem 2.06s 1.51s
% Output : Refutation 2.06s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 2
% Syntax : Number of formulae : 12 ( 4 unt; 0 typ; 0 def)
% Number of atoms : 1071 ( 297 equ; 0 cnn)
% Maximal formula atoms : 71 ( 89 avg)
% Number of connectives : 525 ( 10 ~; 0 |; 212 &; 22 @)
% ( 0 <=>; 281 =>; 0 <=; 0 <~>)
% Maximal formula depth : 74 ( 44 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 166 ( 163 usr; 73 con; 0-3 aty)
% Number of variables : 14 ( 0 ^; 8 !; 6 ?; 14 :)
% 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_95,type,
sP1: $i > $i > $o ).
thf(func_def_96,type,
sP2: $i > $o ).
thf(func_def_97,type,
sP3: $i > $i > $o ).
thf(func_def_98,type,
sP4: $i > $i > $o ).
thf(func_def_100,type,
sK6: $i > $i ).
thf(func_def_101,type,
sK7: $i > $i > $i ).
thf(func_def_110,type,
sK16: $i > $o ).
thf(func_def_112,type,
sK18: ( $i > $o ) > $i > $i ).
thf(func_def_113,type,
sK19: $i > $i ).
thf(func_def_116,type,
sK22: $i > $i ).
thf(func_def_117,type,
sK23: $i > $o ).
thf(func_def_118,type,
sK24: $i > $i ).
thf(func_def_119,type,
sK25: ( $i > $o ) > $i ).
thf(func_def_120,type,
sK26: $i > $o ).
thf(func_def_122,type,
sK28: $i > ( $i > $o ) > $i ).
thf(func_def_126,type,
sK32: ( $i > $o ) > $i > $i ).
thf(func_def_128,type,
sK34: $i > $o ).
thf(func_def_132,type,
sK38: $i > $i > $i ).
thf(func_def_133,type,
sK39: $i > ( $i > $o ) > $i ).
thf(func_def_134,type,
sK40: $i > $o ).
thf(func_def_148,type,
sK54: $o > $i > $i > $i ).
thf(func_def_152,type,
sK58: $i > $o ).
thf(func_def_155,type,
sK61: $i > ( $i > $o ) > $i ).
thf(func_def_157,type,
sK63: $i > $o ).
thf(func_def_160,type,
sK66: $i > ( $i > $o ) > $i ).
thf(func_def_161,type,
sK67: $i > $i > $o ).
thf(func_def_163,type,
sK69: $i > $i ).
thf(func_def_164,type,
sK70: $i > $i ).
thf(func_def_165,type,
sK71: $i > ( $i > $i > $o ) > $i ).
thf(func_def_166,type,
sK72: $i > ( $i > $i > $o ) > $i ).
thf(func_def_167,type,
sK73: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_171,type,
sK77: $i > $o ).
thf(func_def_178,type,
sK84: $i > $o ).
thf(func_def_179,type,
sK85: $i > $i ).
thf(func_def_180,type,
sK86: ( $i > $o ) > $i ).
thf(func_def_183,type,
sK89: $i > $o ).
thf(func_def_193,type,
sK99: ( $i > $o ) > $i > $i ).
thf(func_def_195,type,
sK101: $i > $o ).
thf(func_def_197,type,
sK103: $i > ( $i > $o ) > $i ).
thf(func_def_198,type,
sK104: $i > $o ).
thf(func_def_201,type,
sK107: ( $i > $o ) > $i ).
thf(func_def_202,type,
sK108: ( $i > $o ) > $i ).
thf(func_def_203,type,
sK109: $i > $o ).
thf(func_def_206,type,
sK112: $i > $o ).
thf(func_def_208,type,
sK114: $i > $o ).
thf(func_def_209,type,
sK115: $i > $o ).
thf(func_def_210,type,
sK116: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_211,type,
sK117: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_214,type,
sK120: $i > $o ).
thf(func_def_216,type,
sK122: $i > $o ).
thf(func_def_225,type,
sK131: $i > $o ).
thf(func_def_232,type,
sK138: $i > $i ).
thf(func_def_234,type,
sK140: ( $i > $o ) > $i ).
thf(func_def_235,type,
sK141: $i > $o ).
thf(func_def_236,type,
sK142: $i > $i > $i ).
thf(func_def_237,type,
sK143: $i > $i > $i ).
thf(func_def_238,type,
sK144: $i > $i > $i ).
thf(func_def_239,type,
sK145: $i > $i > $i ).
thf(func_def_240,type,
sK146: $i > $i > $i > $i ).
thf(func_def_241,type,
sK147: $i > $i ).
thf(func_def_242,type,
sK148: $i > $i ).
thf(func_def_243,type,
sK149: $i > $i ).
thf(func_def_244,type,
sK150: $i > $i ).
thf(func_def_245,type,
sK151: $i > $i > $i ).
thf(func_def_246,type,
sK152: $i > $i > $i > $i ).
thf(func_def_247,type,
sK153: $i > $i > $i > $i ).
thf(func_def_248,type,
sK154: $i > $i > $i ).
thf(func_def_249,type,
sK155: $i > $i > $i ).
thf(func_def_250,type,
sK156: $i > $i ).
thf(func_def_251,type,
sK157: $i > $i > $i ).
thf(func_def_253,type,
sK159: $i > $i ).
thf(func_def_254,type,
sK160: $i > $i ).
thf(func_def_258,type,
sK164: $i > $o ).
thf(func_def_261,type,
sK167: $i > $o ).
thf(func_def_264,type,
sK170: $i > $i ).
thf(func_def_272,type,
sK178: $i > $i > $i ).
thf(func_def_275,type,
sK181: $i > $i ).
thf(func_def_279,type,
sK185: $i > $i > $i ).
thf(func_def_280,type,
sK186: $i > $i > $i ).
thf(func_def_291,type,
sK197: $i > $i > $i ).
thf(func_def_294,type,
sK200: $i > $i ).
thf(func_def_298,type,
ph204:
!>[X0: $tType] : X0 ).
thf(f1405,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f1241]) ).
thf(f1241,plain,
( ( powerset @ sK180 )
!= ( powerset @ sK180 ) ),
inference(definition_unfolding,[status(thm)],[f915,f916]) ).
thf(f916,plain,
sK179 = sK180,
inference(cnf_transformation,[status(thm)],[f615]) ).
thf(f615,plain,
( ( subPowSU = $true )
& ( omega0Ax = $true )
& ( replAx = $true )
& ( inCongP = $true )
& ( exuI1 = $true )
& ( setadjoinIR = $true )
& ( emptysetimpfalse = $true )
& ( setunionAx = $true )
& ( setunionI = $true )
& ( prop2set2propI = $true )
& ( dsetconstrI = $true )
& ( dsetconstrER = $true )
& ( quantDeMorgan2 = $true )
& ( emptyinunitempty = $true )
& ( exu__Cong = $true )
& ( setbeta = $true )
& ( exuE3u = $true )
& ( powersetI = $true )
& ( notinemptyset = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( emptyInPowerset = $true )
& ( setadjoinIL = $true )
& ( emptysetE = $true )
& ( nonemptyI1 = $true )
& ( omegaIndAx = $true )
& ( prop2setI = $true )
& ( prop2setE = $true )
& ( quantDeMorgan3 = $true )
& ( emptyI = $true )
& ( notdallE = $true )
& ( omegaSAx = $true )
& ( upairsetIR = $true )
& ( exuE1 = $true )
& ( notinsingleton = $true )
& ( setadjoinE = $true )
& ( setextAx = $true )
& ( emptyset__Cong = $true )
& ( descrp = $true )
& ( emptysetAx = $true )
& ( exuE2 = $true )
& ( in__Cong = $true )
& ( quantDeMorgan4 = $true )
& ( nonemptyE1 = $true )
& ( upairsetIL = $true )
& ( emptyE1 = $true )
& ( quantDeMorgan1 = $true )
& ( singletonsswitch = $true )
& ( setadjoin__Cong = $true )
& ( eqinunit = $true )
& ( foundationAx = $true )
& ( setoftrueEq = $true )
& ( exuE3e = $true )
& ( setunionE = $true )
& ( wellorderingAx = $true )
& ( emptyinPowerset = $true )
& ( powersetAx = $true )
& ( setext = $true )
& ( upairsetE = $true )
& ( setadjoinOr = $true )
& ( exuI2 = $true )
& ( dsetconstrEL = $true )
& ( exuI3 = $true )
& ( nonemptyI = $true )
& ( notdexE = $true )
& ( setadjoinAx = $true )
& ( noeltsimpempty = $true )
& ( vacuousDall = $true )
& ( sK179 = sK180 )
& ( ( powerset @ sK179 )
!= ( powerset @ sK180 ) )
& ( powersetE = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK179,sK180])],[f295,f614]) ).
thf(f614,plain,
( ? [X0: $i,X1: $i] :
( ( X0 = X1 )
& ( ( powerset @ X1 )
!= ( powerset @ X0 ) ) )
=> ( ( sK179 = sK180 )
& ( ( powerset @ sK179 )
!= ( powerset @ sK180 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f295,plain,
( ( subPowSU = $true )
& ( omega0Ax = $true )
& ( replAx = $true )
& ( inCongP = $true )
& ( exuI1 = $true )
& ( setadjoinIR = $true )
& ( emptysetimpfalse = $true )
& ( setunionAx = $true )
& ( setunionI = $true )
& ( prop2set2propI = $true )
& ( dsetconstrI = $true )
& ( dsetconstrER = $true )
& ( quantDeMorgan2 = $true )
& ( emptyinunitempty = $true )
& ( exu__Cong = $true )
& ( setbeta = $true )
& ( exuE3u = $true )
& ( powersetI = $true )
& ( notinemptyset = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( emptyInPowerset = $true )
& ( setadjoinIL = $true )
& ( emptysetE = $true )
& ( nonemptyI1 = $true )
& ( omegaIndAx = $true )
& ( prop2setI = $true )
& ( prop2setE = $true )
& ( quantDeMorgan3 = $true )
& ( emptyI = $true )
& ( notdallE = $true )
& ( omegaSAx = $true )
& ( upairsetIR = $true )
& ( exuE1 = $true )
& ( notinsingleton = $true )
& ( setadjoinE = $true )
& ( setextAx = $true )
& ( emptyset__Cong = $true )
& ( descrp = $true )
& ( emptysetAx = $true )
& ( exuE2 = $true )
& ( in__Cong = $true )
& ( quantDeMorgan4 = $true )
& ( nonemptyE1 = $true )
& ( upairsetIL = $true )
& ( emptyE1 = $true )
& ( quantDeMorgan1 = $true )
& ( singletonsswitch = $true )
& ( setadjoin__Cong = $true )
& ( eqinunit = $true )
& ( foundationAx = $true )
& ( setoftrueEq = $true )
& ( exuE3e = $true )
& ( setunionE = $true )
& ( wellorderingAx = $true )
& ( emptyinPowerset = $true )
& ( powersetAx = $true )
& ( setext = $true )
& ( upairsetE = $true )
& ( setadjoinOr = $true )
& ( exuI2 = $true )
& ( dsetconstrEL = $true )
& ( exuI3 = $true )
& ( nonemptyI = $true )
& ( notdexE = $true )
& ( setadjoinAx = $true )
& ( noeltsimpempty = $true )
& ( vacuousDall = $true )
& ? [X0: $i,X1: $i] :
( ( X0 = X1 )
& ( ( powerset @ X1 )
!= ( powerset @ X0 ) ) )
& ( powersetE = $true ) ),
inference(flattening,[status(thm)],[f294]) ).
thf(f294,plain,
( ? [X0: $i,X1: $i] :
( ( X0 = X1 )
& ( ( powerset @ X1 )
!= ( powerset @ X0 ) ) )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f108]) ).
thf(f108,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ( ( powerset @ X1 )
= ( powerset @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f107]) ).
thf(f107,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ! [X0: $i,X1: $i] :
( ( X0 = X1 )
=> ( ( powerset @ X1 )
= ( powerset @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f73]) ).
thf(f73,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( powerset @ X3 )
= ( powerset @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f72]) ).
thf(f72,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ! [X3: $i,X4: $i] :
( ( X3 = X4 )
=> ( ( powerset @ X3 )
= ( powerset @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.5INz166dx8/DTF2THF_11109.p',powerset__Cong) ).
thf(f915,plain,
( ( powerset @ sK179 )
!= ( powerset @ sK180 ) ),
inference(cnf_transformation,[status(thm)],[f615]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU553^1 : TPTP v9.2.1. Bugfixed v4.0.0.
% 0.00/0.09 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.11/0.29 % Computer : n013.cluster.edu
% 0.11/0.29 % Model : x86_64 x86_64
% 0.11/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.29 % Memory : 8042.1875MB
% 0.11/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.29 % CPULimit : 300
% 0.11/0.29 % WCLimit : 300
% 0.11/0.29 % DateTime : Tue Feb 24 12:26:08 EST 2026
% 0.11/0.29 % CPUTime :
% 0.11/0.29 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.38 ---- Original DTF file ---
% 0.17/0.38 thf(spec,logic,$$dhol).
% 0.17/0.38 %------------------------------------------------------------------------------
% 0.17/0.38 % File : SEU553^1 : TPTP v9.2.1. Bugfixed v4.0.0.
% 0.17/0.38 % Domain : Set Theory
% 0.17/0.38 % Problem : A simple congruence property of powerset
% 0.17/0.38 % Version : Especial.
% 0.17/0.38 % English : (forall A:i.forall B:i.A = B -> powerset A = powerset B)
% 0.17/0.38
% 0.17/0.38 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.17/0.38 % Source : [Bro08]
% 0.17/0.38 % Names : ZFC055g [Bro08]
% 0.17/0.38
% 0.17/0.38 % Status : Theorem
% 0.17/0.38 % 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, 0.17 v5.4.0, 0.20 v5.3.0, 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
% 0.17/0.38 % Syntax : Number of formulae : 153 ( 71 unt; 81 typ; 71 def)
% 0.17/0.38 % Number of atoms : 389 ( 105 equ; 0 cnn)
% 0.17/0.38 % Maximal formula atoms : 71 ( 5 avg)
% 0.17/0.38 % Number of connectives : 632 ( 21 ~; 5 |; 27 &; 392 @)
% 0.17/0.38 % ( 12 <=>; 175 =>; 0 <=; 0 <~>)
% 0.17/0.38 % Maximal formula depth : 73 ( 2 avg)
% 0.17/0.38 % Number of types : 2 ( 0 usr)
% 0.17/0.38 % Number of type conns : 42 ( 42 >; 0 *; 0 +; 0 <<)
% 0.17/0.38 % Number of symbols : 84 ( 81 usr; 73 con; 0-2 aty)
% 0.17/0.38 % Number of variables : 220 ( 21 ^; 171 !; 28 ?; 220 :)
% 0.17/0.38 % SPC : TH0_THM_EQU_NAR
% 0.17/0.38
% 0.17/0.38 % Comments : http://mathgate.info/detsetitem.php?id=32
% 0.17/0.38 % Bugfixes : v4.0.0 - Adding missing type delaration for in
% 0.17/0.38 %------------------------------------------------------------------------------
% 0.17/0.38 thf(in_type,type,
% 0.17/0.38 in: $i > $i > $o ).
% 0.17/0.38
% 0.17/0.38 thf(exu_type,type,
% 0.17/0.38 exu: ( $i > $o ) > $o ).
% 0.17/0.38
% 0.17/0.38 thf(exu,definition,
% 0.17/0.38 ( exu
% 0.17/0.38 = ( ^ [Xphi: $i > $o] :
% 0.17/0.38 ? [Xx: $i] :
% 0.17/0.38 ( ( Xphi @ Xx )
% 0.17/0.38 & ! [Xy: $i] :
% 0.17/0.38 ( ( Xphi @ Xy )
% 0.17/0.38 => ( Xx = Xy ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(setextAx_type,type,
% 0.17/0.38 setextAx: $o ).
% 0.17/0.38
% 0.17/0.38 thf(setextAx,definition,
% 0.17/0.38 ( setextAx
% 0.17/0.38 = ( ! [A: $i,B: $i] :
% 0.17/0.38 ( ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 <=> ( in @ Xx @ B ) )
% 0.17/0.38 => ( A = B ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(emptyset_type,type,
% 0.17/0.38 emptyset: $i ).
% 0.17/0.38
% 0.17/0.38 thf(emptysetAx_type,type,
% 0.17/0.38 emptysetAx: $o ).
% 0.17/0.38
% 0.17/0.38 thf(emptysetAx,definition,
% 0.17/0.38 ( emptysetAx
% 0.17/0.38 = ( ! [Xx: $i] :
% 0.17/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(setadjoin_type,type,
% 0.17/0.38 setadjoin: $i > $i > $i ).
% 0.17/0.38
% 0.17/0.38 thf(setadjoinAx_type,type,
% 0.17/0.38 setadjoinAx: $o ).
% 0.17/0.38
% 0.17/0.38 thf(setadjoinAx,definition,
% 0.17/0.38 ( setadjoinAx
% 0.17/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.38 <=> ( ( Xy = Xx )
% 0.17/0.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(powerset_type,type,
% 0.17/0.38 powerset: $i > $i ).
% 0.17/0.38
% 0.17/0.38 thf(powersetAx_type,type,
% 0.17/0.38 powersetAx: $o ).
% 0.17/0.38
% 0.17/0.38 thf(powersetAx,definition,
% 0.17/0.38 ( powersetAx
% 0.17/0.38 = ( ! [A: $i,B: $i] :
% 0.17/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.38 <=> ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ B )
% 0.17/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(setunion_type,type,
% 0.17/0.38 setunion: $i > $i ).
% 0.17/0.38
% 0.17/0.38 thf(setunionAx_type,type,
% 0.17/0.38 setunionAx: $o ).
% 0.17/0.38
% 0.17/0.38 thf(setunionAx,definition,
% 0.17/0.38 ( setunionAx
% 0.17/0.38 = ( ! [A: $i,Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.38 <=> ? [B: $i] :
% 0.17/0.38 ( ( in @ Xx @ B )
% 0.17/0.38 & ( in @ B @ A ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(omega_type,type,
% 0.17/0.38 omega: $i ).
% 0.17/0.38
% 0.17/0.38 thf(omega0Ax_type,type,
% 0.17/0.38 omega0Ax: $o ).
% 0.17/0.38
% 0.17/0.38 thf(omega0Ax,definition,
% 0.17/0.38 ( omega0Ax
% 0.17/0.38 = ( in @ emptyset @ omega ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(omegaSAx_type,type,
% 0.17/0.38 omegaSAx: $o ).
% 0.17/0.38
% 0.17/0.38 thf(omegaSAx,definition,
% 0.17/0.38 ( omegaSAx
% 0.17/0.38 = ( ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ omega )
% 0.17/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(omegaIndAx_type,type,
% 0.17/0.38 omegaIndAx: $o ).
% 0.17/0.38
% 0.17/0.38 thf(omegaIndAx,definition,
% 0.17/0.38 ( omegaIndAx
% 0.17/0.38 = ( ! [A: $i] :
% 0.17/0.38 ( ( ( in @ emptyset @ A )
% 0.17/0.38 & ! [Xx: $i] :
% 0.17/0.38 ( ( ( in @ Xx @ omega )
% 0.17/0.38 & ( in @ Xx @ A ) )
% 0.17/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.17/0.38 => ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ omega )
% 0.17/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(replAx_type,type,
% 0.17/0.38 replAx: $o ).
% 0.17/0.38
% 0.17/0.38 thf(replAx,definition,
% 0.17/0.38 ( replAx
% 0.17/0.38 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.17/0.38 ( ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 => ( exu
% 0.17/0.38 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.17/0.38 => ? [B: $i] :
% 0.17/0.38 ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ B )
% 0.17/0.38 <=> ? [Xy: $i] :
% 0.17/0.38 ( ( in @ Xy @ A )
% 0.17/0.38 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(foundationAx_type,type,
% 0.17/0.38 foundationAx: $o ).
% 0.17/0.38
% 0.17/0.38 thf(foundationAx,definition,
% 0.17/0.38 ( foundationAx
% 0.17/0.38 = ( ! [A: $i] :
% 0.17/0.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.17/0.38 => ? [B: $i] :
% 0.17/0.38 ( ( in @ B @ A )
% 0.17/0.38 & ~ ? [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ B )
% 0.17/0.38 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(wellorderingAx_type,type,
% 0.17/0.38 wellorderingAx: $o ).
% 0.17/0.38
% 0.17/0.38 thf(wellorderingAx,definition,
% 0.17/0.38 ( wellorderingAx
% 0.17/0.38 = ( ! [A: $i] :
% 0.17/0.38 ? [B: $i] :
% 0.17/0.38 ( ! [C: $i] :
% 0.17/0.38 ( ( in @ C @ B )
% 0.17/0.38 => ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ C )
% 0.17/0.38 => ( in @ Xx @ A ) ) )
% 0.17/0.38 & ! [Xx: $i,Xy: $i] :
% 0.17/0.38 ( ( ( in @ Xx @ A )
% 0.17/0.38 & ( in @ Xy @ A ) )
% 0.17/0.38 => ( ! [C: $i] :
% 0.17/0.38 ( ( in @ C @ B )
% 0.17/0.38 => ( ( in @ Xx @ C )
% 0.17/0.38 <=> ( in @ Xy @ C ) ) )
% 0.17/0.38 => ( Xx = Xy ) ) )
% 0.17/0.38 & ! [C: $i,D: $i] :
% 0.17/0.38 ( ( ( in @ C @ B )
% 0.17/0.38 & ( in @ D @ B ) )
% 0.17/0.38 => ( ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ C )
% 0.17/0.38 => ( in @ Xx @ D ) )
% 0.17/0.38 | ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ D )
% 0.17/0.38 => ( in @ Xx @ C ) ) ) )
% 0.17/0.38 & ! [C: $i] :
% 0.17/0.38 ( ( ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ C )
% 0.17/0.38 => ( in @ Xx @ A ) )
% 0.17/0.38 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.17/0.38 => ? [D: $i,Xx: $i] :
% 0.17/0.38 ( ( in @ D @ B )
% 0.17/0.38 & ( in @ Xx @ C )
% 0.17/0.38 & ~ ? [Xy: $i] :
% 0.17/0.38 ( ( in @ Xy @ D )
% 0.17/0.38 & ( in @ Xy @ C ) )
% 0.17/0.38 & ! [E: $i] :
% 0.17/0.38 ( ( in @ E @ B )
% 0.17/0.38 => ( ! [Xy: $i] :
% 0.17/0.38 ( ( in @ Xy @ E )
% 0.17/0.38 => ( in @ Xy @ D ) )
% 0.17/0.38 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(descr_type,type,
% 0.17/0.38 descr: ( $i > $o ) > $i ).
% 0.17/0.38
% 0.17/0.38 thf(descrp_type,type,
% 0.17/0.38 descrp: $o ).
% 0.17/0.38
% 0.17/0.38 thf(descrp,definition,
% 0.17/0.38 ( descrp
% 0.17/0.38 = ( ! [Xphi: $i > $o] :
% 0.17/0.38 ( ( exu
% 0.17/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.38 => ( Xphi
% 0.17/0.38 @ ( descr
% 0.17/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(dsetconstr_type,type,
% 0.17/0.38 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.17/0.38
% 0.17/0.38 thf(dsetconstrI_type,type,
% 0.17/0.38 dsetconstrI: $o ).
% 0.17/0.38
% 0.17/0.38 thf(dsetconstrI,definition,
% 0.17/0.38 ( dsetconstrI
% 0.17/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 => ( ( Xphi @ Xx )
% 0.17/0.38 => ( in @ Xx
% 0.17/0.38 @ ( dsetconstr @ A
% 0.17/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(dsetconstrEL_type,type,
% 0.17/0.38 dsetconstrEL: $o ).
% 0.17/0.38
% 0.17/0.38 thf(dsetconstrEL,definition,
% 0.17/0.38 ( dsetconstrEL
% 0.17/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.38 ( ( in @ Xx
% 0.17/0.38 @ ( dsetconstr @ A
% 0.17/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.38 => ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(dsetconstrER_type,type,
% 0.17/0.38 dsetconstrER: $o ).
% 0.17/0.38
% 0.17/0.38 thf(dsetconstrER,definition,
% 0.17/0.38 ( dsetconstrER
% 0.17/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.38 ( ( in @ Xx
% 0.17/0.38 @ ( dsetconstr @ A
% 0.17/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.38 => ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(exuE1_type,type,
% 0.17/0.38 exuE1: $o ).
% 0.17/0.38
% 0.17/0.38 thf(exuE1,definition,
% 0.17/0.38 ( exuE1
% 0.17/0.38 = ( ! [Xphi: $i > $o] :
% 0.17/0.38 ( ( exu
% 0.17/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.38 => ? [Xx: $i] :
% 0.17/0.38 ( ( Xphi @ Xx )
% 0.17/0.38 & ! [Xy: $i] :
% 0.17/0.38 ( ( Xphi @ Xy )
% 0.17/0.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(prop2set_type,type,
% 0.17/0.38 prop2set: $o > $i ).
% 0.17/0.38
% 0.17/0.38 thf(prop2setE_type,type,
% 0.17/0.38 prop2setE: $o ).
% 0.17/0.38
% 0.17/0.38 thf(prop2setE,definition,
% 0.17/0.38 ( prop2setE
% 0.17/0.38 = ( ! [Xphi: $o,Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.17/0.38 => Xphi ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(emptysetE_type,type,
% 0.17/0.38 emptysetE: $o ).
% 0.17/0.38
% 0.17/0.38 thf(emptysetE,definition,
% 0.17/0.38 ( emptysetE
% 0.17/0.38 = ( ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ emptyset )
% 0.17/0.38 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(emptysetimpfalse_type,type,
% 0.17/0.38 emptysetimpfalse: $o ).
% 0.17/0.38
% 0.17/0.38 thf(emptysetimpfalse,definition,
% 0.17/0.38 ( emptysetimpfalse
% 0.17/0.38 = ( ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ emptyset )
% 0.17/0.38 => $false ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(notinemptyset_type,type,
% 0.17/0.38 notinemptyset: $o ).
% 0.17/0.38
% 0.17/0.38 thf(notinemptyset,definition,
% 0.17/0.38 ( notinemptyset
% 0.17/0.38 = ( ! [Xx: $i] :
% 0.17/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(exuE3e_type,type,
% 0.17/0.38 exuE3e: $o ).
% 0.17/0.38
% 0.17/0.38 thf(exuE3e,definition,
% 0.17/0.38 ( exuE3e
% 0.17/0.38 = ( ! [Xphi: $i > $o] :
% 0.17/0.38 ( ( exu
% 0.17/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.38 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(setext_type,type,
% 0.17/0.38 setext: $o ).
% 0.17/0.38
% 0.17/0.38 thf(setext,definition,
% 0.17/0.38 ( setext
% 0.17/0.38 = ( ! [A: $i,B: $i] :
% 0.17/0.38 ( ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 => ( in @ Xx @ B ) )
% 0.17/0.38 => ( ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ B )
% 0.17/0.38 => ( in @ Xx @ A ) )
% 0.17/0.38 => ( A = B ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(emptyI_type,type,
% 0.17/0.38 emptyI: $o ).
% 0.17/0.38
% 0.17/0.38 thf(emptyI,definition,
% 0.17/0.38 ( emptyI
% 0.17/0.38 = ( ! [A: $i] :
% 0.17/0.38 ( ! [Xx: $i] :
% 0.17/0.38 ~ ( in @ Xx @ A )
% 0.17/0.38 => ( A = emptyset ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(noeltsimpempty_type,type,
% 0.17/0.38 noeltsimpempty: $o ).
% 0.17/0.38
% 0.17/0.38 thf(noeltsimpempty,definition,
% 0.17/0.38 ( noeltsimpempty
% 0.17/0.38 = ( ! [A: $i] :
% 0.17/0.38 ( ! [Xx: $i] :
% 0.17/0.38 ~ ( in @ Xx @ A )
% 0.17/0.38 => ( A = emptyset ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(setbeta_type,type,
% 0.17/0.38 setbeta: $o ).
% 0.17/0.38
% 0.17/0.38 thf(setbeta,definition,
% 0.17/0.38 ( setbeta
% 0.17/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 => ( ( in @ Xx
% 0.17/0.38 @ ( dsetconstr @ A
% 0.17/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.17/0.38 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(nonempty_type,type,
% 0.17/0.38 nonempty: $i > $o ).
% 0.17/0.38
% 0.17/0.38 thf(nonempty,definition,
% 0.17/0.38 ( nonempty
% 0.17/0.38 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(nonemptyE1_type,type,
% 0.17/0.38 nonemptyE1: $o ).
% 0.17/0.38
% 0.17/0.38 thf(nonemptyE1,definition,
% 0.17/0.38 ( nonemptyE1
% 0.17/0.38 = ( ! [A: $i] :
% 0.17/0.38 ( ( nonempty @ A )
% 0.17/0.38 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(nonemptyI_type,type,
% 0.17/0.38 nonemptyI: $o ).
% 0.17/0.38
% 0.17/0.38 thf(nonemptyI,definition,
% 0.17/0.38 ( nonemptyI
% 0.17/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 => ( ( Xphi @ Xx )
% 0.17/0.38 => ( nonempty
% 0.17/0.38 @ ( dsetconstr @ A
% 0.17/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(nonemptyI1_type,type,
% 0.17/0.38 nonemptyI1: $o ).
% 0.17/0.38
% 0.17/0.38 thf(nonemptyI1,definition,
% 0.17/0.38 ( nonemptyI1
% 0.17/0.38 = ( ! [A: $i] :
% 0.17/0.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.17/0.38 => ( nonempty @ A ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(setadjoinIL_type,type,
% 0.17/0.38 setadjoinIL: $o ).
% 0.17/0.38
% 0.17/0.38 thf(setadjoinIL,definition,
% 0.17/0.38 ( setadjoinIL
% 0.17/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(emptyinunitempty_type,type,
% 0.17/0.38 emptyinunitempty: $o ).
% 0.17/0.38
% 0.17/0.38 thf(emptyinunitempty,definition,
% 0.17/0.38 ( emptyinunitempty
% 0.17/0.38 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(setadjoinIR_type,type,
% 0.17/0.38 setadjoinIR: $o ).
% 0.17/0.38
% 0.17/0.38 thf(setadjoinIR,definition,
% 0.17/0.38 ( setadjoinIR
% 0.17/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.38 ( ( in @ Xy @ A )
% 0.17/0.38 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(setadjoinE_type,type,
% 0.17/0.38 setadjoinE: $o ).
% 0.17/0.38
% 0.17/0.38 thf(setadjoinE,definition,
% 0.17/0.38 ( setadjoinE
% 0.17/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.38 => ! [Xphi: $o] :
% 0.17/0.38 ( ( ( Xy = Xx )
% 0.17/0.38 => Xphi )
% 0.17/0.38 => ( ( ( in @ Xy @ A )
% 0.17/0.38 => Xphi )
% 0.17/0.38 => Xphi ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(setadjoinOr_type,type,
% 0.17/0.38 setadjoinOr: $o ).
% 0.17/0.38
% 0.17/0.38 thf(setadjoinOr,definition,
% 0.17/0.38 ( setadjoinOr
% 0.17/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.17/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.17/0.38 => ( ( Xy = Xx )
% 0.17/0.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(setoftrueEq_type,type,
% 0.17/0.38 setoftrueEq: $o ).
% 0.17/0.38
% 0.17/0.38 thf(setoftrueEq,definition,
% 0.17/0.38 ( setoftrueEq
% 0.17/0.38 = ( ! [A: $i] :
% 0.17/0.38 ( ( dsetconstr @ A
% 0.17/0.38 @ ^ [Xx: $i] : $true )
% 0.17/0.38 = A ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(powersetI_type,type,
% 0.17/0.38 powersetI: $o ).
% 0.17/0.38
% 0.17/0.38 thf(powersetI,definition,
% 0.17/0.38 ( powersetI
% 0.17/0.38 = ( ! [A: $i,B: $i] :
% 0.17/0.38 ( ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ B )
% 0.17/0.38 => ( in @ Xx @ A ) )
% 0.17/0.38 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(emptyinPowerset_type,type,
% 0.17/0.38 emptyinPowerset: $o ).
% 0.17/0.38
% 0.17/0.38 thf(emptyinPowerset,definition,
% 0.17/0.38 ( emptyinPowerset
% 0.17/0.38 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(emptyInPowerset_type,type,
% 0.17/0.38 emptyInPowerset: $o ).
% 0.17/0.38
% 0.17/0.38 thf(emptyInPowerset,definition,
% 0.17/0.38 ( emptyInPowerset
% 0.17/0.38 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(powersetE_type,type,
% 0.17/0.38 powersetE: $o ).
% 0.17/0.38
% 0.17/0.38 thf(powersetE,definition,
% 0.17/0.38 ( powersetE
% 0.17/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.17/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.17/0.38 => ( ( in @ Xx @ B )
% 0.17/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(setunionI_type,type,
% 0.17/0.38 setunionI: $o ).
% 0.17/0.38
% 0.17/0.38 thf(setunionI,definition,
% 0.17/0.38 ( setunionI
% 0.17/0.38 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.17/0.38 ( ( in @ Xx @ B )
% 0.17/0.38 => ( ( in @ B @ A )
% 0.17/0.38 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(setunionE_type,type,
% 0.17/0.38 setunionE: $o ).
% 0.17/0.38
% 0.17/0.38 thf(setunionE,definition,
% 0.17/0.38 ( setunionE
% 0.17/0.38 = ( ! [A: $i,Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ ( setunion @ A ) )
% 0.17/0.38 => ! [Xphi: $o] :
% 0.17/0.38 ( ! [B: $i] :
% 0.17/0.38 ( ( in @ Xx @ B )
% 0.17/0.38 => ( ( in @ B @ A )
% 0.17/0.38 => Xphi ) )
% 0.17/0.38 => Xphi ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(subPowSU_type,type,
% 0.17/0.38 subPowSU: $o ).
% 0.17/0.38
% 0.17/0.38 thf(subPowSU,definition,
% 0.17/0.38 ( subPowSU
% 0.17/0.38 = ( ! [A: $i,Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(exuE2_type,type,
% 0.17/0.38 exuE2: $o ).
% 0.17/0.38
% 0.17/0.38 thf(exuE2,definition,
% 0.17/0.38 ( exuE2
% 0.17/0.38 = ( ! [Xphi: $i > $o] :
% 0.17/0.38 ( ( exu
% 0.17/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.38 => ? [Xx: $i] :
% 0.17/0.38 ! [Xy: $i] :
% 0.17/0.38 ( ( Xphi @ Xy )
% 0.17/0.38 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(nonemptyImpWitness_type,type,
% 0.17/0.38 nonemptyImpWitness: $o ).
% 0.17/0.38
% 0.17/0.38 thf(nonemptyImpWitness,definition,
% 0.17/0.38 ( nonemptyImpWitness
% 0.17/0.38 = ( ! [A: $i] :
% 0.17/0.38 ( ( nonempty @ A )
% 0.17/0.38 => ? [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 & $true ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(uniqinunit_type,type,
% 0.17/0.38 uniqinunit: $o ).
% 0.17/0.38
% 0.17/0.38 thf(uniqinunit,definition,
% 0.17/0.38 ( uniqinunit
% 0.17/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.38 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.38 => ( Xx = Xy ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(notinsingleton_type,type,
% 0.17/0.38 notinsingleton: $o ).
% 0.17/0.38
% 0.17/0.38 thf(notinsingleton,definition,
% 0.17/0.38 ( notinsingleton
% 0.17/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.38 ( ( Xx != Xy )
% 0.17/0.38 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(eqinunit_type,type,
% 0.17/0.38 eqinunit: $o ).
% 0.17/0.38
% 0.17/0.38 thf(eqinunit,definition,
% 0.17/0.38 ( eqinunit
% 0.17/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.38 ( ( Xx = Xy )
% 0.17/0.38 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(singletonsswitch_type,type,
% 0.17/0.38 singletonsswitch: $o ).
% 0.17/0.38
% 0.17/0.38 thf(singletonsswitch,definition,
% 0.17/0.38 ( singletonsswitch
% 0.17/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.38 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.17/0.38 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(upairsetE_type,type,
% 0.17/0.38 upairsetE: $o ).
% 0.17/0.38
% 0.17/0.38 thf(upairsetE,definition,
% 0.17/0.38 ( upairsetE
% 0.17/0.38 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.17/0.38 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.17/0.38 => ( ( Xz = Xx )
% 0.17/0.38 | ( Xz = Xy ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(upairsetIL_type,type,
% 0.17/0.38 upairsetIL: $o ).
% 0.17/0.38
% 0.17/0.38 thf(upairsetIL,definition,
% 0.17/0.38 ( upairsetIL
% 0.17/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(upairsetIR_type,type,
% 0.17/0.38 upairsetIR: $o ).
% 0.17/0.38
% 0.17/0.38 thf(upairsetIR,definition,
% 0.17/0.38 ( upairsetIR
% 0.17/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(emptyE1_type,type,
% 0.17/0.38 emptyE1: $o ).
% 0.17/0.38
% 0.17/0.38 thf(emptyE1,definition,
% 0.17/0.38 ( emptyE1
% 0.17/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.38 ( ? [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 & ( Xphi @ Xx ) )
% 0.17/0.38 => ( ( ( dsetconstr @ A
% 0.17/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.38 = emptyset )
% 0.17/0.38 => $false ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(vacuousDall_type,type,
% 0.17/0.38 vacuousDall: $o ).
% 0.17/0.38
% 0.17/0.38 thf(vacuousDall,definition,
% 0.17/0.38 ( vacuousDall
% 0.17/0.38 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ emptyset )
% 0.17/0.38 => ( Xphi @ Xx ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(quantDeMorgan1_type,type,
% 0.17/0.38 quantDeMorgan1: $o ).
% 0.17/0.38
% 0.17/0.38 thf(quantDeMorgan1,definition,
% 0.17/0.38 ( quantDeMorgan1
% 0.17/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.38 ( ~ ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 => ( Xphi @ Xx ) )
% 0.17/0.38 => ? [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(quantDeMorgan2_type,type,
% 0.17/0.38 quantDeMorgan2: $o ).
% 0.17/0.38
% 0.17/0.38 thf(quantDeMorgan2,definition,
% 0.17/0.38 ( quantDeMorgan2
% 0.17/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.38 ( ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 => ~ ( Xphi @ Xx ) )
% 0.17/0.38 => ~ ? [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(quantDeMorgan3_type,type,
% 0.17/0.38 quantDeMorgan3: $o ).
% 0.17/0.38
% 0.17/0.38 thf(quantDeMorgan3,definition,
% 0.17/0.38 ( quantDeMorgan3
% 0.17/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.38 ( ~ ? [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 & ( Xphi @ Xx ) )
% 0.17/0.38 => ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(quantDeMorgan4_type,type,
% 0.17/0.38 quantDeMorgan4: $o ).
% 0.17/0.38
% 0.17/0.38 thf(quantDeMorgan4,definition,
% 0.17/0.38 ( quantDeMorgan4
% 0.17/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.38 ( ? [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 & ~ ( Xphi @ Xx ) )
% 0.17/0.38 => ~ ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(prop2setI_type,type,
% 0.17/0.38 prop2setI: $o ).
% 0.17/0.38
% 0.17/0.38 thf(prop2setI,definition,
% 0.17/0.38 ( prop2setI
% 0.17/0.38 = ( ! [Xphi: $o] :
% 0.17/0.38 ( Xphi
% 0.17/0.38 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(set2prop_type,type,
% 0.17/0.38 set2prop: $i > $o ).
% 0.17/0.38
% 0.17/0.38 thf(prop2set2propI_type,type,
% 0.17/0.38 prop2set2propI: $o ).
% 0.17/0.38
% 0.17/0.38 thf(prop2set2propI,definition,
% 0.17/0.38 ( prop2set2propI
% 0.17/0.38 = ( ! [Xphi: $o] :
% 0.17/0.38 ( Xphi
% 0.17/0.38 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(notdexE_type,type,
% 0.17/0.38 notdexE: $o ).
% 0.17/0.38
% 0.17/0.38 thf(notdexE,definition,
% 0.17/0.38 ( notdexE
% 0.17/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.38 ( ~ ? [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 & ( Xphi @ Xx ) )
% 0.17/0.38 => ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(notdallE_type,type,
% 0.17/0.38 notdallE: $o ).
% 0.17/0.38
% 0.17/0.38 thf(notdallE,definition,
% 0.17/0.38 ( notdallE
% 0.17/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.17/0.38 ( ~ ! [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 => ( Xphi @ Xx ) )
% 0.17/0.38 => ? [Xx: $i] :
% 0.17/0.38 ( ( in @ Xx @ A )
% 0.17/0.38 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(exuI1_type,type,
% 0.17/0.38 exuI1: $o ).
% 0.17/0.38
% 0.17/0.38 thf(exuI1,definition,
% 0.17/0.38 ( exuI1
% 0.17/0.38 = ( ! [Xphi: $i > $o] :
% 0.17/0.38 ( ? [Xx: $i] :
% 0.17/0.38 ( ( Xphi @ Xx )
% 0.17/0.38 & ! [Xy: $i] :
% 0.17/0.38 ( ( Xphi @ Xy )
% 0.17/0.38 => ( Xx = Xy ) ) )
% 0.17/0.38 => ( exu
% 0.17/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(exuI3_type,type,
% 0.17/0.38 exuI3: $o ).
% 0.17/0.38
% 0.17/0.38 thf(exuI3,definition,
% 0.17/0.38 ( exuI3
% 0.17/0.38 = ( ! [Xphi: $i > $o] :
% 0.17/0.38 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.17/0.38 => ( ! [Xx: $i,Xy: $i] :
% 0.17/0.38 ( ( Xphi @ Xx )
% 0.17/0.38 => ( ( Xphi @ Xy )
% 0.17/0.38 => ( Xx = Xy ) ) )
% 0.17/0.38 => ( exu
% 0.17/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(exuI2_type,type,
% 0.17/0.38 exuI2: $o ).
% 0.17/0.38
% 0.17/0.38 thf(exuI2,definition,
% 0.17/0.38 ( exuI2
% 0.17/0.38 = ( ! [Xphi: $i > $o] :
% 0.17/0.38 ( ? [Xx: $i] :
% 0.17/0.38 ! [Xy: $i] :
% 0.17/0.38 ( ( Xphi @ Xy )
% 0.17/0.38 <=> ( Xy = Xx ) )
% 0.17/0.38 => ( exu
% 0.17/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(inCongP_type,type,
% 0.17/0.38 inCongP: $o ).
% 0.17/0.38
% 0.17/0.38 thf(inCongP,definition,
% 0.17/0.38 ( inCongP
% 0.17/0.38 = ( ! [A: $i,B: $i] :
% 0.17/0.38 ( ( A = B )
% 0.17/0.38 => ! [Xx: $i,Xy: $i] :
% 0.17/0.38 ( ( Xx = Xy )
% 0.17/0.38 => ( ( in @ Xx @ A )
% 0.17/0.38 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(in__Cong_type,type,
% 0.17/0.38 in__Cong: $o ).
% 0.17/0.38
% 0.17/0.38 thf(in__Cong,definition,
% 0.17/0.38 ( in__Cong
% 0.17/0.38 = ( ! [A: $i,B: $i] :
% 0.17/0.38 ( ( A = B )
% 0.17/0.38 => ! [Xx: $i,Xy: $i] :
% 0.17/0.38 ( ( Xx = Xy )
% 0.17/0.38 => ( ( in @ Xx @ A )
% 0.17/0.38 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(exuE3u_type,type,
% 0.17/0.38 exuE3u: $o ).
% 0.17/0.38
% 0.17/0.38 thf(exuE3u,definition,
% 0.17/0.38 ( exuE3u
% 0.17/0.38 = ( ! [Xphi: $i > $o] :
% 0.17/0.38 ( ( exu
% 0.17/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.38 => ! [Xx: $i,Xy: $i] :
% 0.17/0.38 ( ( Xphi @ Xx )
% 0.17/0.38 => ( ( Xphi @ Xy )
% 0.17/0.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(exu__Cong_type,type,
% 0.17/0.38 exu__Cong: $o ).
% 0.17/0.38
% 0.17/0.38 thf(exu__Cong,definition,
% 0.17/0.38 ( exu__Cong
% 0.17/0.38 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.17/0.38 ( ! [Xx: $i,Xy: $i] :
% 0.17/0.38 ( ( Xx = Xy )
% 0.17/0.38 => ( ( Xphi @ Xx )
% 0.17/0.38 <=> ( Xpsi @ Xy ) ) )
% 0.17/0.38 => ( ( exu
% 0.17/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.17/0.38 <=> ( exu
% 0.17/0.38 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(emptyset__Cong_type,type,
% 0.17/0.38 emptyset__Cong: $o ).
% 0.17/0.38
% 0.17/0.38 thf(emptyset__Cong,definition,
% 0.17/0.38 ( emptyset__Cong
% 0.17/0.38 = ( emptyset = emptyset ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(setadjoin__Cong_type,type,
% 0.17/0.38 setadjoin__Cong: $o ).
% 0.17/0.38
% 0.17/0.38 thf(setadjoin__Cong,definition,
% 0.17/0.38 ( setadjoin__Cong
% 0.17/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.17/0.38 ( ( Xx = Xy )
% 0.17/0.38 => ! [Xz: $i,Xu: $i] :
% 0.17/0.38 ( ( Xz = Xu )
% 0.17/0.38 => ( ( setadjoin @ Xx @ Xz )
% 0.17/0.38 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 thf(powerset__Cong,conjecture,
% 0.17/0.38 ( setextAx
% 0.17/0.38 => ( emptysetAx
% 0.17/0.38 => ( setadjoinAx
% 0.17/0.38 => ( powersetAx
% 0.17/0.38 => ( setunionAx
% 0.17/0.38 => ( omega0Ax
% 0.17/0.38 => ( omegaSAx
% 0.17/0.38 => ( omegaIndAx
% 0.17/0.38 => ( replAx
% 0.17/0.38 => ( foundationAx
% 0.17/0.38 => ( wellorderingAx
% 0.17/0.38 => ( descrp
% 0.17/0.38 => ( dsetconstrI
% 0.17/0.38 => ( dsetconstrEL
% 0.17/0.38 => ( dsetconstrER
% 0.17/0.38 => ( exuE1
% 0.17/0.38 => ( prop2setE
% 0.17/0.38 => ( emptysetE
% 0.17/0.38 => ( emptysetimpfalse
% 0.17/0.38 => ( notinemptyset
% 0.17/0.38 => ( exuE3e
% 0.17/0.38 => ( setext
% 0.17/0.38 => ( emptyI
% 0.17/0.38 => ( noeltsimpempty
% 0.17/0.38 => ( setbeta
% 0.17/0.38 => ( nonemptyE1
% 0.17/0.38 => ( nonemptyI
% 0.17/0.38 => ( nonemptyI1
% 0.17/0.38 => ( setadjoinIL
% 0.17/0.38 => ( emptyinunitempty
% 0.17/0.38 => ( setadjoinIR
% 0.17/0.38 => ( setadjoinE
% 0.17/0.38 => ( setadjoinOr
% 0.17/0.38 => ( setoftrueEq
% 0.17/0.38 => ( powersetI
% 0.17/0.38 => ( emptyinPowerset
% 0.17/0.38 => ( emptyInPowerset
% 0.17/0.38 => ( powersetE
% 0.17/0.38 => ( setunionI
% 0.17/0.38 => ( setunionE
% 0.17/0.38 => ( subPowSU
% 0.17/0.38 => ( exuE2
% 0.17/0.38 => ( nonemptyImpWitness
% 0.17/0.38 => ( uniqinunit
% 0.17/0.38 => ( notinsingleton
% 0.17/0.38 => ( eqinunit
% 0.17/0.38 => ( singletonsswitch
% 0.17/0.38 => ( upairsetE
% 0.17/0.38 => ( upairsetIL
% 0.17/0.38 => ( upairsetIR
% 0.17/0.38 => ( emptyE1
% 0.17/0.38 => ( vacuousDall
% 0.17/0.38 => ( quantDeMorgan1
% 0.17/0.38 => ( quantDeMorgan2
% 0.17/0.38 => ( quantDeMorgan3
% 0.17/0.38 => ( quantDeMorgan4
% 0.17/0.38 => ( prop2setI
% 0.17/0.38 => ( prop2set2propI
% 0.17/0.38 => ( notdexE
% 0.17/0.38 => ( notdallE
% 0.17/0.38 => ( exuI1
% 0.17/0.38 => ( exuI3
% 0.17/0.38 => ( exuI2
% 0.17/0.38 => ( inCongP
% 0.17/0.38 => ( in__Cong
% 0.17/0.38 => ( exuE3u
% 0.17/0.38 => ( exu__Cong
% 0.17/0.38 => ( emptyset__Cong
% 0.17/0.38 => ( setadjoin__Cong
% 0.17/0.38 => ! [A: $i,B: $i] :
% 0.17/0.38 ( ( A = B )
% 0.17/0.38 => ( ( powerset @ A )
% 0.17/0.38 = ( powerset @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.17/0.38
% 0.17/0.38 %------------------------------------------------------------------------------
% 0.17/0.38 ------------------------
% 1.95/1.44 ---- Embedded in THF ---
% 1.95/1.44 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.95/1.44 %%% Generated on Tue Feb 24 12:26:09 EST 2026
% 1.95/1.44 %%% using '$$dhol' embedding, version 1.3.0.
% 1.95/1.44 %%% Logic specification used:
% 1.95/1.44 %%% thf(spec, logic, $$dhol).
% 1.95/1.44
% 1.95/1.44 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.5INz166dx8/DTF2DTF11109.p
% See solution above
% 1.95/1.44 ------------------------
% 1.95/1.46 ---- Cleaned THF ---
% 1.95/1.46 thf(in_type,type,
% 1.95/1.46 in: $i > $i > $o ).
% 1.95/1.46
% 1.95/1.46 thf(exu_type,type,
% 1.95/1.46 exu: ( $i > $o ) > $o ).
% 1.95/1.46
% 1.95/1.46 thf(exu,definition,
% 1.95/1.46 ( exu
% 1.95/1.46 = ( ^ [Xphi: $i > $o] :
% 1.95/1.46 ? [Xx: $i] :
% 1.95/1.46 ( ( Xphi @ Xx )
% 1.95/1.46 & ! [Xy: $i] :
% 1.95/1.46 ( ( Xphi @ Xy )
% 1.95/1.46 => ( Xx = Xy ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(setextAx_type,type,
% 1.95/1.46 setextAx: $o ).
% 1.95/1.46
% 1.95/1.46 thf(setextAx,definition,
% 1.95/1.46 ( setextAx
% 1.95/1.46 = ( ! [A: $i,B: $i] :
% 1.95/1.46 ( ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 <=> ( in @ Xx @ B ) )
% 1.95/1.46 => ( A = B ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(emptyset_type,type,
% 1.95/1.46 emptyset: $i ).
% 1.95/1.46
% 1.95/1.46 thf(emptysetAx_type,type,
% 1.95/1.46 emptysetAx: $o ).
% 1.95/1.46
% 1.95/1.46 thf(emptysetAx,definition,
% 1.95/1.46 ( emptysetAx
% 1.95/1.46 = ( ! [Xx: $i] :
% 1.95/1.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(setadjoin_type,type,
% 1.95/1.46 setadjoin: $i > $i > $i ).
% 1.95/1.46
% 1.95/1.46 thf(setadjoinAx_type,type,
% 1.95/1.46 setadjoinAx: $o ).
% 1.95/1.46
% 1.95/1.46 thf(setadjoinAx,definition,
% 1.95/1.46 ( setadjoinAx
% 1.95/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.95/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.95/1.46 <=> ( ( Xy = Xx )
% 1.95/1.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(powerset_type,type,
% 1.95/1.46 powerset: $i > $i ).
% 1.95/1.46
% 1.95/1.46 thf(powersetAx_type,type,
% 1.95/1.46 powersetAx: $o ).
% 1.95/1.46
% 1.95/1.46 thf(powersetAx,definition,
% 1.95/1.46 ( powersetAx
% 1.95/1.46 = ( ! [A: $i,B: $i] :
% 1.95/1.46 ( ( in @ B @ ( powerset @ A ) )
% 1.95/1.46 <=> ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ B )
% 1.95/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(setunion_type,type,
% 1.95/1.46 setunion: $i > $i ).
% 1.95/1.46
% 1.95/1.46 thf(setunionAx_type,type,
% 1.95/1.46 setunionAx: $o ).
% 1.95/1.46
% 1.95/1.46 thf(setunionAx,definition,
% 1.95/1.46 ( setunionAx
% 1.95/1.46 = ( ! [A: $i,Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ ( setunion @ A ) )
% 1.95/1.46 <=> ? [B: $i] :
% 1.95/1.46 ( ( in @ Xx @ B )
% 1.95/1.46 & ( in @ B @ A ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(omega_type,type,
% 1.95/1.46 omega: $i ).
% 1.95/1.46
% 1.95/1.46 thf(omega0Ax_type,type,
% 1.95/1.46 omega0Ax: $o ).
% 1.95/1.46
% 1.95/1.46 thf(omega0Ax,definition,
% 1.95/1.46 ( omega0Ax
% 1.95/1.46 = ( in @ emptyset @ omega ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(omegaSAx_type,type,
% 1.95/1.46 omegaSAx: $o ).
% 1.95/1.46
% 1.95/1.46 thf(omegaSAx,definition,
% 1.95/1.46 ( omegaSAx
% 1.95/1.46 = ( ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ omega )
% 1.95/1.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(omegaIndAx_type,type,
% 1.95/1.46 omegaIndAx: $o ).
% 1.95/1.46
% 1.95/1.46 thf(omegaIndAx,definition,
% 1.95/1.46 ( omegaIndAx
% 1.95/1.46 = ( ! [A: $i] :
% 1.95/1.46 ( ( ( in @ emptyset @ A )
% 1.95/1.46 & ! [Xx: $i] :
% 1.95/1.46 ( ( ( in @ Xx @ omega )
% 1.95/1.46 & ( in @ Xx @ A ) )
% 1.95/1.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 1.95/1.46 => ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ omega )
% 1.95/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(replAx_type,type,
% 1.95/1.46 replAx: $o ).
% 1.95/1.46
% 1.95/1.46 thf(replAx,definition,
% 1.95/1.46 ( replAx
% 1.95/1.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 1.95/1.46 ( ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 => ( exu
% 1.95/1.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 1.95/1.46 => ? [B: $i] :
% 1.95/1.46 ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ B )
% 1.95/1.46 <=> ? [Xy: $i] :
% 1.95/1.46 ( ( in @ Xy @ A )
% 1.95/1.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(foundationAx_type,type,
% 1.95/1.46 foundationAx: $o ).
% 1.95/1.46
% 1.95/1.46 thf(foundationAx,definition,
% 1.95/1.46 ( foundationAx
% 1.95/1.46 = ( ! [A: $i] :
% 1.95/1.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.95/1.46 => ? [B: $i] :
% 1.95/1.46 ( ( in @ B @ A )
% 1.95/1.46 & ~ ? [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ B )
% 1.95/1.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(wellorderingAx_type,type,
% 1.95/1.46 wellorderingAx: $o ).
% 1.95/1.46
% 1.95/1.46 thf(wellorderingAx,definition,
% 1.95/1.46 ( wellorderingAx
% 1.95/1.46 = ( ! [A: $i] :
% 1.95/1.46 ? [B: $i] :
% 1.95/1.46 ( ! [C: $i] :
% 1.95/1.46 ( ( in @ C @ B )
% 1.95/1.46 => ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ C )
% 1.95/1.46 => ( in @ Xx @ A ) ) )
% 1.95/1.46 & ! [Xx: $i,Xy: $i] :
% 1.95/1.46 ( ( ( in @ Xx @ A )
% 1.95/1.46 & ( in @ Xy @ A ) )
% 1.95/1.46 => ( ! [C: $i] :
% 1.95/1.46 ( ( in @ C @ B )
% 1.95/1.46 => ( ( in @ Xx @ C )
% 1.95/1.46 <=> ( in @ Xy @ C ) ) )
% 1.95/1.46 => ( Xx = Xy ) ) )
% 1.95/1.46 & ! [C: $i,D: $i] :
% 1.95/1.46 ( ( ( in @ C @ B )
% 1.95/1.46 & ( in @ D @ B ) )
% 1.95/1.46 => ( ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ C )
% 1.95/1.46 => ( in @ Xx @ D ) )
% 1.95/1.46 | ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ D )
% 1.95/1.46 => ( in @ Xx @ C ) ) ) )
% 1.95/1.46 & ! [C: $i] :
% 1.95/1.46 ( ( ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ C )
% 1.95/1.46 => ( in @ Xx @ A ) )
% 1.95/1.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 1.95/1.46 => ? [D: $i,Xx: $i] :
% 1.95/1.46 ( ( in @ D @ B )
% 1.95/1.46 & ( in @ Xx @ C )
% 1.95/1.46 & ~ ? [Xy: $i] :
% 1.95/1.46 ( ( in @ Xy @ D )
% 1.95/1.46 & ( in @ Xy @ C ) )
% 1.95/1.46 & ! [E: $i] :
% 1.95/1.46 ( ( in @ E @ B )
% 1.95/1.46 => ( ! [Xy: $i] :
% 1.95/1.46 ( ( in @ Xy @ E )
% 1.95/1.46 => ( in @ Xy @ D ) )
% 1.95/1.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(descr_type,type,
% 1.95/1.46 descr: ( $i > $o ) > $i ).
% 1.95/1.46
% 1.95/1.46 thf(descrp_type,type,
% 1.95/1.46 descrp: $o ).
% 1.95/1.46
% 1.95/1.46 thf(descrp,definition,
% 1.95/1.46 ( descrp
% 1.95/1.46 = ( ! [Xphi: $i > $o] :
% 1.95/1.46 ( ( exu
% 1.95/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.95/1.46 => ( Xphi
% 1.95/1.46 @ ( descr
% 1.95/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(dsetconstr_type,type,
% 1.95/1.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 1.95/1.46
% 1.95/1.46 thf(dsetconstrI_type,type,
% 1.95/1.46 dsetconstrI: $o ).
% 1.95/1.46
% 1.95/1.46 thf(dsetconstrI,definition,
% 1.95/1.46 ( dsetconstrI
% 1.95/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 => ( ( Xphi @ Xx )
% 1.95/1.46 => ( in @ Xx
% 1.95/1.46 @ ( dsetconstr @ A
% 1.95/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(dsetconstrEL_type,type,
% 1.95/1.46 dsetconstrEL: $o ).
% 1.95/1.46
% 1.95/1.46 thf(dsetconstrEL,definition,
% 1.95/1.46 ( dsetconstrEL
% 1.95/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.95/1.46 ( ( in @ Xx
% 1.95/1.46 @ ( dsetconstr @ A
% 1.95/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.95/1.46 => ( in @ Xx @ A ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(dsetconstrER_type,type,
% 1.95/1.46 dsetconstrER: $o ).
% 1.95/1.46
% 1.95/1.46 thf(dsetconstrER,definition,
% 1.95/1.46 ( dsetconstrER
% 1.95/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.95/1.46 ( ( in @ Xx
% 1.95/1.46 @ ( dsetconstr @ A
% 1.95/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.95/1.46 => ( Xphi @ Xx ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(exuE1_type,type,
% 1.95/1.46 exuE1: $o ).
% 1.95/1.46
% 1.95/1.46 thf(exuE1,definition,
% 1.95/1.46 ( exuE1
% 1.95/1.46 = ( ! [Xphi: $i > $o] :
% 1.95/1.46 ( ( exu
% 1.95/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.95/1.46 => ? [Xx: $i] :
% 1.95/1.46 ( ( Xphi @ Xx )
% 1.95/1.46 & ! [Xy: $i] :
% 1.95/1.46 ( ( Xphi @ Xy )
% 1.95/1.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(prop2set_type,type,
% 1.95/1.46 prop2set: $o > $i ).
% 1.95/1.46
% 1.95/1.46 thf(prop2setE_type,type,
% 1.95/1.46 prop2setE: $o ).
% 1.95/1.46
% 1.95/1.46 thf(prop2setE,definition,
% 1.95/1.46 ( prop2setE
% 1.95/1.46 = ( ! [Xphi: $o,Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 1.95/1.46 => Xphi ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(emptysetE_type,type,
% 1.95/1.46 emptysetE: $o ).
% 1.95/1.46
% 1.95/1.46 thf(emptysetE,definition,
% 1.95/1.46 ( emptysetE
% 1.95/1.46 = ( ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ emptyset )
% 1.95/1.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(emptysetimpfalse_type,type,
% 1.95/1.46 emptysetimpfalse: $o ).
% 1.95/1.46
% 1.95/1.46 thf(emptysetimpfalse,definition,
% 1.95/1.46 ( emptysetimpfalse
% 1.95/1.46 = ( ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ emptyset )
% 1.95/1.46 => $false ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(notinemptyset_type,type,
% 1.95/1.46 notinemptyset: $o ).
% 1.95/1.46
% 1.95/1.46 thf(notinemptyset,definition,
% 1.95/1.46 ( notinemptyset
% 1.95/1.46 = ( ! [Xx: $i] :
% 1.95/1.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(exuE3e_type,type,
% 1.95/1.46 exuE3e: $o ).
% 1.95/1.46
% 1.95/1.46 thf(exuE3e,definition,
% 1.95/1.46 ( exuE3e
% 1.95/1.46 = ( ! [Xphi: $i > $o] :
% 1.95/1.46 ( ( exu
% 1.95/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.95/1.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(setext_type,type,
% 1.95/1.46 setext: $o ).
% 1.95/1.46
% 1.95/1.46 thf(setext,definition,
% 1.95/1.46 ( setext
% 1.95/1.46 = ( ! [A: $i,B: $i] :
% 1.95/1.46 ( ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 => ( in @ Xx @ B ) )
% 1.95/1.46 => ( ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ B )
% 1.95/1.46 => ( in @ Xx @ A ) )
% 1.95/1.46 => ( A = B ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(emptyI_type,type,
% 1.95/1.46 emptyI: $o ).
% 1.95/1.46
% 1.95/1.46 thf(emptyI,definition,
% 1.95/1.46 ( emptyI
% 1.95/1.46 = ( ! [A: $i] :
% 1.95/1.46 ( ! [Xx: $i] :
% 1.95/1.46 ~ ( in @ Xx @ A )
% 1.95/1.46 => ( A = emptyset ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(noeltsimpempty_type,type,
% 1.95/1.46 noeltsimpempty: $o ).
% 1.95/1.46
% 1.95/1.46 thf(noeltsimpempty,definition,
% 1.95/1.46 ( noeltsimpempty
% 1.95/1.46 = ( ! [A: $i] :
% 1.95/1.46 ( ! [Xx: $i] :
% 1.95/1.46 ~ ( in @ Xx @ A )
% 1.95/1.46 => ( A = emptyset ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(setbeta_type,type,
% 1.95/1.46 setbeta: $o ).
% 1.95/1.46
% 1.95/1.46 thf(setbeta,definition,
% 1.95/1.46 ( setbeta
% 1.95/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 => ( ( in @ Xx
% 1.95/1.46 @ ( dsetconstr @ A
% 1.95/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 1.95/1.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(nonempty_type,type,
% 1.95/1.46 nonempty: $i > $o ).
% 1.95/1.46
% 1.95/1.46 thf(nonempty,definition,
% 1.95/1.46 ( nonempty
% 1.95/1.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(nonemptyE1_type,type,
% 1.95/1.46 nonemptyE1: $o ).
% 1.95/1.46
% 1.95/1.46 thf(nonemptyE1,definition,
% 1.95/1.46 ( nonemptyE1
% 1.95/1.46 = ( ! [A: $i] :
% 1.95/1.46 ( ( nonempty @ A )
% 1.95/1.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(nonemptyI_type,type,
% 1.95/1.46 nonemptyI: $o ).
% 1.95/1.46
% 1.95/1.46 thf(nonemptyI,definition,
% 1.95/1.46 ( nonemptyI
% 1.95/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 => ( ( Xphi @ Xx )
% 1.95/1.46 => ( nonempty
% 1.95/1.46 @ ( dsetconstr @ A
% 1.95/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(nonemptyI1_type,type,
% 1.95/1.46 nonemptyI1: $o ).
% 1.95/1.46
% 1.95/1.46 thf(nonemptyI1,definition,
% 1.95/1.46 ( nonemptyI1
% 1.95/1.46 = ( ! [A: $i] :
% 1.95/1.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 1.95/1.46 => ( nonempty @ A ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(setadjoinIL_type,type,
% 1.95/1.46 setadjoinIL: $o ).
% 1.95/1.46
% 1.95/1.46 thf(setadjoinIL,definition,
% 1.95/1.46 ( setadjoinIL
% 1.95/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(emptyinunitempty_type,type,
% 1.95/1.46 emptyinunitempty: $o ).
% 1.95/1.46
% 1.95/1.46 thf(emptyinunitempty,definition,
% 1.95/1.46 ( emptyinunitempty
% 1.95/1.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(setadjoinIR_type,type,
% 1.95/1.46 setadjoinIR: $o ).
% 1.95/1.46
% 1.95/1.46 thf(setadjoinIR,definition,
% 1.95/1.46 ( setadjoinIR
% 1.95/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.95/1.46 ( ( in @ Xy @ A )
% 1.95/1.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(setadjoinE_type,type,
% 1.95/1.46 setadjoinE: $o ).
% 1.95/1.46
% 1.95/1.46 thf(setadjoinE,definition,
% 1.95/1.46 ( setadjoinE
% 1.95/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.95/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.95/1.46 => ! [Xphi: $o] :
% 1.95/1.46 ( ( ( Xy = Xx )
% 1.95/1.46 => Xphi )
% 1.95/1.46 => ( ( ( in @ Xy @ A )
% 1.95/1.46 => Xphi )
% 1.95/1.46 => Xphi ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(setadjoinOr_type,type,
% 1.95/1.46 setadjoinOr: $o ).
% 1.95/1.46
% 1.95/1.46 thf(setadjoinOr,definition,
% 1.95/1.46 ( setadjoinOr
% 1.95/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 1.95/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 1.95/1.46 => ( ( Xy = Xx )
% 1.95/1.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(setoftrueEq_type,type,
% 1.95/1.46 setoftrueEq: $o ).
% 1.95/1.46
% 1.95/1.46 thf(setoftrueEq,definition,
% 1.95/1.46 ( setoftrueEq
% 1.95/1.46 = ( ! [A: $i] :
% 1.95/1.46 ( ( dsetconstr @ A
% 1.95/1.46 @ ^ [Xx: $i] : $true )
% 1.95/1.46 = A ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(powersetI_type,type,
% 1.95/1.46 powersetI: $o ).
% 1.95/1.46
% 1.95/1.46 thf(powersetI,definition,
% 1.95/1.46 ( powersetI
% 1.95/1.46 = ( ! [A: $i,B: $i] :
% 1.95/1.46 ( ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ B )
% 1.95/1.46 => ( in @ Xx @ A ) )
% 1.95/1.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(emptyinPowerset_type,type,
% 1.95/1.46 emptyinPowerset: $o ).
% 1.95/1.46
% 1.95/1.46 thf(emptyinPowerset,definition,
% 1.95/1.46 ( emptyinPowerset
% 1.95/1.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(emptyInPowerset_type,type,
% 1.95/1.46 emptyInPowerset: $o ).
% 1.95/1.46
% 1.95/1.46 thf(emptyInPowerset,definition,
% 1.95/1.46 ( emptyInPowerset
% 1.95/1.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(powersetE_type,type,
% 1.95/1.46 powersetE: $o ).
% 1.95/1.46
% 1.95/1.46 thf(powersetE,definition,
% 1.95/1.46 ( powersetE
% 1.95/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 1.95/1.46 ( ( in @ B @ ( powerset @ A ) )
% 1.95/1.46 => ( ( in @ Xx @ B )
% 1.95/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(setunionI_type,type,
% 1.95/1.46 setunionI: $o ).
% 1.95/1.46
% 1.95/1.46 thf(setunionI,definition,
% 1.95/1.46 ( setunionI
% 1.95/1.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 1.95/1.46 ( ( in @ Xx @ B )
% 1.95/1.46 => ( ( in @ B @ A )
% 1.95/1.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(setunionE_type,type,
% 1.95/1.46 setunionE: $o ).
% 1.95/1.46
% 1.95/1.46 thf(setunionE,definition,
% 1.95/1.46 ( setunionE
% 1.95/1.46 = ( ! [A: $i,Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ ( setunion @ A ) )
% 1.95/1.46 => ! [Xphi: $o] :
% 1.95/1.46 ( ! [B: $i] :
% 1.95/1.46 ( ( in @ Xx @ B )
% 1.95/1.46 => ( ( in @ B @ A )
% 1.95/1.46 => Xphi ) )
% 1.95/1.46 => Xphi ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(subPowSU_type,type,
% 1.95/1.46 subPowSU: $o ).
% 1.95/1.46
% 1.95/1.46 thf(subPowSU,definition,
% 1.95/1.46 ( subPowSU
% 1.95/1.46 = ( ! [A: $i,Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(exuE2_type,type,
% 1.95/1.46 exuE2: $o ).
% 1.95/1.46
% 1.95/1.46 thf(exuE2,definition,
% 1.95/1.46 ( exuE2
% 1.95/1.46 = ( ! [Xphi: $i > $o] :
% 1.95/1.46 ( ( exu
% 1.95/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.95/1.46 => ? [Xx: $i] :
% 1.95/1.46 ! [Xy: $i] :
% 1.95/1.46 ( ( Xphi @ Xy )
% 1.95/1.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(nonemptyImpWitness_type,type,
% 1.95/1.46 nonemptyImpWitness: $o ).
% 1.95/1.46
% 1.95/1.46 thf(nonemptyImpWitness,definition,
% 1.95/1.46 ( nonemptyImpWitness
% 1.95/1.46 = ( ! [A: $i] :
% 1.95/1.46 ( ( nonempty @ A )
% 1.95/1.46 => ? [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 & $true ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(uniqinunit_type,type,
% 1.95/1.46 uniqinunit: $o ).
% 1.95/1.46
% 1.95/1.46 thf(uniqinunit,definition,
% 1.95/1.46 ( uniqinunit
% 1.95/1.46 = ( ! [Xx: $i,Xy: $i] :
% 1.95/1.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 1.95/1.46 => ( Xx = Xy ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(notinsingleton_type,type,
% 1.95/1.46 notinsingleton: $o ).
% 1.95/1.46
% 1.95/1.46 thf(notinsingleton,definition,
% 1.95/1.46 ( notinsingleton
% 1.95/1.46 = ( ! [Xx: $i,Xy: $i] :
% 1.95/1.46 ( ( Xx != Xy )
% 1.95/1.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(eqinunit_type,type,
% 1.95/1.46 eqinunit: $o ).
% 1.95/1.46
% 1.95/1.46 thf(eqinunit,definition,
% 1.95/1.46 ( eqinunit
% 1.95/1.46 = ( ! [Xx: $i,Xy: $i] :
% 1.95/1.46 ( ( Xx = Xy )
% 1.95/1.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(singletonsswitch_type,type,
% 1.95/1.46 singletonsswitch: $o ).
% 1.95/1.46
% 1.95/1.46 thf(singletonsswitch,definition,
% 1.95/1.46 ( singletonsswitch
% 1.95/1.46 = ( ! [Xx: $i,Xy: $i] :
% 1.95/1.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 1.95/1.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(upairsetE_type,type,
% 1.95/1.46 upairsetE: $o ).
% 1.95/1.46
% 1.95/1.46 thf(upairsetE,definition,
% 1.95/1.46 ( upairsetE
% 1.95/1.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 1.95/1.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 1.95/1.46 => ( ( Xz = Xx )
% 1.95/1.46 | ( Xz = Xy ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(upairsetIL_type,type,
% 1.95/1.46 upairsetIL: $o ).
% 1.95/1.46
% 1.95/1.46 thf(upairsetIL,definition,
% 1.95/1.46 ( upairsetIL
% 1.95/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(upairsetIR_type,type,
% 1.95/1.46 upairsetIR: $o ).
% 1.95/1.46
% 1.95/1.46 thf(upairsetIR,definition,
% 1.95/1.46 ( upairsetIR
% 1.95/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(emptyE1_type,type,
% 1.95/1.46 emptyE1: $o ).
% 1.95/1.46
% 1.95/1.46 thf(emptyE1,definition,
% 1.95/1.46 ( emptyE1
% 1.95/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 1.95/1.46 ( ? [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 & ( Xphi @ Xx ) )
% 1.95/1.46 => ( ( ( dsetconstr @ A
% 1.95/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.95/1.46 = emptyset )
% 1.95/1.46 => $false ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(vacuousDall_type,type,
% 1.95/1.46 vacuousDall: $o ).
% 1.95/1.46
% 1.95/1.46 thf(vacuousDall,definition,
% 1.95/1.46 ( vacuousDall
% 1.95/1.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ emptyset )
% 1.95/1.46 => ( Xphi @ Xx ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(quantDeMorgan1_type,type,
% 1.95/1.46 quantDeMorgan1: $o ).
% 1.95/1.46
% 1.95/1.46 thf(quantDeMorgan1,definition,
% 1.95/1.46 ( quantDeMorgan1
% 1.95/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 1.95/1.46 ( ~ ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 => ( Xphi @ Xx ) )
% 1.95/1.46 => ? [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(quantDeMorgan2_type,type,
% 1.95/1.46 quantDeMorgan2: $o ).
% 1.95/1.46
% 1.95/1.46 thf(quantDeMorgan2,definition,
% 1.95/1.46 ( quantDeMorgan2
% 1.95/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 1.95/1.46 ( ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 => ~ ( Xphi @ Xx ) )
% 1.95/1.46 => ~ ? [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(quantDeMorgan3_type,type,
% 1.95/1.46 quantDeMorgan3: $o ).
% 1.95/1.46
% 1.95/1.46 thf(quantDeMorgan3,definition,
% 1.95/1.46 ( quantDeMorgan3
% 1.95/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 1.95/1.46 ( ~ ? [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 & ( Xphi @ Xx ) )
% 1.95/1.46 => ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(quantDeMorgan4_type,type,
% 1.95/1.46 quantDeMorgan4: $o ).
% 1.95/1.46
% 1.95/1.46 thf(quantDeMorgan4,definition,
% 1.95/1.46 ( quantDeMorgan4
% 1.95/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 1.95/1.46 ( ? [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 & ~ ( Xphi @ Xx ) )
% 1.95/1.46 => ~ ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(prop2setI_type,type,
% 1.95/1.46 prop2setI: $o ).
% 1.95/1.46
% 1.95/1.46 thf(prop2setI,definition,
% 1.95/1.46 ( prop2setI
% 1.95/1.46 = ( ! [Xphi: $o] :
% 1.95/1.46 ( Xphi
% 1.95/1.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(set2prop_type,type,
% 1.95/1.46 set2prop: $i > $o ).
% 1.95/1.46
% 1.95/1.46 thf(prop2set2propI_type,type,
% 1.95/1.46 prop2set2propI: $o ).
% 1.95/1.46
% 1.95/1.46 thf(prop2set2propI,definition,
% 1.95/1.46 ( prop2set2propI
% 1.95/1.46 = ( ! [Xphi: $o] :
% 1.95/1.46 ( Xphi
% 1.95/1.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(notdexE_type,type,
% 1.95/1.46 notdexE: $o ).
% 1.95/1.46
% 1.95/1.46 thf(notdexE,definition,
% 1.95/1.46 ( notdexE
% 1.95/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 1.95/1.46 ( ~ ? [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 & ( Xphi @ Xx ) )
% 1.95/1.46 => ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(notdallE_type,type,
% 1.95/1.46 notdallE: $o ).
% 1.95/1.46
% 1.95/1.46 thf(notdallE,definition,
% 1.95/1.46 ( notdallE
% 1.95/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 1.95/1.46 ( ~ ! [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 => ( Xphi @ Xx ) )
% 1.95/1.46 => ? [Xx: $i] :
% 1.95/1.46 ( ( in @ Xx @ A )
% 1.95/1.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(exuI1_type,type,
% 1.95/1.46 exuI1: $o ).
% 1.95/1.46
% 1.95/1.46 thf(exuI1,definition,
% 1.95/1.46 ( exuI1
% 1.95/1.46 = ( ! [Xphi: $i > $o] :
% 1.95/1.46 ( ? [Xx: $i] :
% 1.95/1.46 ( ( Xphi @ Xx )
% 1.95/1.46 & ! [Xy: $i] :
% 1.95/1.46 ( ( Xphi @ Xy )
% 1.95/1.46 => ( Xx = Xy ) ) )
% 1.95/1.46 => ( exu
% 1.95/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(exuI3_type,type,
% 1.95/1.46 exuI3: $o ).
% 1.95/1.46
% 1.95/1.46 thf(exuI3,definition,
% 1.95/1.46 ( exuI3
% 1.95/1.46 = ( ! [Xphi: $i > $o] :
% 1.95/1.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 1.95/1.46 => ( ! [Xx: $i,Xy: $i] :
% 1.95/1.46 ( ( Xphi @ Xx )
% 1.95/1.46 => ( ( Xphi @ Xy )
% 1.95/1.46 => ( Xx = Xy ) ) )
% 1.95/1.46 => ( exu
% 1.95/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(exuI2_type,type,
% 1.95/1.46 exuI2: $o ).
% 1.95/1.46
% 1.95/1.46 thf(exuI2,definition,
% 1.95/1.46 ( exuI2
% 1.95/1.46 = ( ! [Xphi: $i > $o] :
% 1.95/1.46 ( ? [Xx: $i] :
% 1.95/1.46 ! [Xy: $i] :
% 1.95/1.46 ( ( Xphi @ Xy )
% 1.95/1.46 <=> ( Xy = Xx ) )
% 1.95/1.46 => ( exu
% 1.95/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(inCongP_type,type,
% 1.95/1.46 inCongP: $o ).
% 1.95/1.46
% 1.95/1.46 thf(inCongP,definition,
% 1.95/1.46 ( inCongP
% 1.95/1.46 = ( ! [A: $i,B: $i] :
% 1.95/1.46 ( ( A = B )
% 1.95/1.46 => ! [Xx: $i,Xy: $i] :
% 1.95/1.46 ( ( Xx = Xy )
% 1.95/1.46 => ( ( in @ Xx @ A )
% 1.95/1.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(in__Cong_type,type,
% 1.95/1.46 in__Cong: $o ).
% 1.95/1.46
% 1.95/1.46 thf(in__Cong,definition,
% 1.95/1.46 ( in__Cong
% 1.95/1.46 = ( ! [A: $i,B: $i] :
% 1.95/1.46 ( ( A = B )
% 1.95/1.46 => ! [Xx: $i,Xy: $i] :
% 1.95/1.46 ( ( Xx = Xy )
% 1.95/1.46 => ( ( in @ Xx @ A )
% 1.95/1.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(exuE3u_type,type,
% 1.95/1.46 exuE3u: $o ).
% 1.95/1.46
% 1.95/1.46 thf(exuE3u,definition,
% 1.95/1.46 ( exuE3u
% 1.95/1.46 = ( ! [Xphi: $i > $o] :
% 1.95/1.46 ( ( exu
% 1.95/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.95/1.46 => ! [Xx: $i,Xy: $i] :
% 1.95/1.46 ( ( Xphi @ Xx )
% 1.95/1.46 => ( ( Xphi @ Xy )
% 1.95/1.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(exu__Cong_type,type,
% 1.95/1.46 exu__Cong: $o ).
% 1.95/1.46
% 1.95/1.46 thf(exu__Cong,definition,
% 1.95/1.46 ( exu__Cong
% 1.95/1.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 1.95/1.46 ( ! [Xx: $i,Xy: $i] :
% 1.95/1.46 ( ( Xx = Xy )
% 1.95/1.46 => ( ( Xphi @ Xx )
% 1.95/1.46 <=> ( Xpsi @ Xy ) ) )
% 1.95/1.46 => ( ( exu
% 1.95/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 1.95/1.46 <=> ( exu
% 1.95/1.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(emptyset__Cong_type,type,
% 1.95/1.46 emptyset__Cong: $o ).
% 1.95/1.46
% 1.95/1.46 thf(emptyset__Cong,definition,
% 1.95/1.46 ( emptyset__Cong
% 1.95/1.46 = ( emptyset = emptyset ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(setadjoin__Cong_type,type,
% 1.95/1.46 setadjoin__Cong: $o ).
% 1.95/1.46
% 1.95/1.46 thf(setadjoin__Cong,definition,
% 1.95/1.46 ( setadjoin__Cong
% 1.95/1.46 = ( ! [Xx: $i,Xy: $i] :
% 1.95/1.46 ( ( Xx = Xy )
% 1.95/1.46 => ! [Xz: $i,Xu: $i] :
% 1.95/1.46 ( ( Xz = Xu )
% 1.95/1.46 => ( ( setadjoin @ Xx @ Xz )
% 1.95/1.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 1.95/1.46
% 1.95/1.46 thf(powerset__Cong,conjecture,
% 1.95/1.46 ( setextAx
% 1.95/1.46 => ( emptysetAx
% 1.95/1.46 => ( setadjoinAx
% 1.95/1.46 => ( powersetAx
% 1.95/1.46 => ( setunionAx
% 1.95/1.46 => ( omega0Ax
% 1.95/1.46 => ( omegaSAx
% 1.95/1.46 => ( omegaIndAx
% 1.95/1.46 => ( replAx
% 1.95/1.46 => ( foundationAx
% 1.95/1.46 => ( wellorderingAx
% 1.95/1.46 => ( descrp
% 1.95/1.46 => ( dsetconstrI
% 1.95/1.46 => ( dsetconstrEL
% 1.95/1.46 => ( dsetconstrER
% 1.95/1.46 => ( exuE1
% 1.95/1.46 => ( prop2setE
% 1.95/1.46 => ( emptysetE
% 1.95/1.46 => ( emptysetimpfalse
% 1.95/1.46 => ( notinemptyset
% 1.95/1.46 => ( exuE3e
% 1.95/1.46 => ( setext
% 1.95/1.46 => ( emptyI
% 1.95/1.46 => ( noeltsimpempty
% 1.95/1.46 => ( setbeta
% 1.95/1.46 => ( nonemptyE1
% 1.95/1.46 => ( nonemptyI
% 1.95/1.46 => ( nonemptyI1
% 1.95/1.46 => ( setadjoinIL
% 1.95/1.46 => ( emptyinunitempty
% 1.95/1.46 => ( setadjoinIR
% 1.95/1.46 => ( setadjoinE
% 1.95/1.46 => ( setadjoinOr
% 1.95/1.46 => ( setoftrueEq
% 1.95/1.46 => ( powersetI
% 1.95/1.46 => ( emptyinPowerset
% 1.95/1.46 => ( emptyInPowerset
% 1.95/1.46 => ( powersetE
% 1.95/1.46 => ( setunionI
% 1.95/1.46 => ( setunionE
% 1.95/1.46 => ( subPowSU
% 1.95/1.46 => ( exuE2
% 1.95/1.46 => ( nonemptyImpWitness
% 1.95/1.46 => ( uniqinunit
% 1.95/1.46 => ( notinsingleton
% 1.95/1.46 => ( eqinunit
% 1.95/1.46 => ( singletonsswitch
% 1.95/1.46 => ( upairsetE
% 1.95/1.46 => ( upairsetIL
% 1.95/1.46 => ( upairsetIR
% 1.95/1.46 => ( emptyE1
% 1.95/1.46 => ( vacuousDall
% 1.95/1.46 => ( quantDeMorgan1
% 1.95/1.46 => ( quantDeMorgan2
% 1.95/1.46 => ( quantDeMorgan3
% 1.95/1.46 => ( quantDeMorgan4
% 1.95/1.46 => ( prop2setI
% 1.95/1.46 => ( prop2set2propI
% 1.95/1.46 => ( notdexE
% 1.95/1.46 => ( notdallE
% 1.95/1.46 => ( exuI1
% 1.95/1.46 => ( exuI3
% 1.95/1.46 => ( exuI2
% 1.95/1.46 => ( inCongP
% 1.95/1.46 => ( in__Cong
% 1.95/1.46 => ( exuE3u
% 1.95/1.46 => ( exu__Cong
% 1.95/1.46 => ( emptyset__Cong
% 1.95/1.46 => ( setadjoin__Cong
% 1.95/1.46 => ! [A: $i,B: $i] :
% 1.95/1.46 ( ( A = B )
% 1.95/1.46 => ( ( powerset @ A )
% 1.95/1.46 = ( powerset @ B ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 1.95/1.46 ------------------------
% 1.95/1.48 % (11209)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_11109 for (2999ds/4Mi)
% 1.95/1.48 % (11211)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_11109 for (2999ds/2Mi)
% 1.95/1.48 % (11212)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_11109 for (2999ds/2Mi)
% 1.95/1.48 % (11210)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_11109 for (2999ds/27Mi)
% 1.95/1.48 % (11208)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_11109 for (2999ds/183Mi)
% 1.95/1.48 % (11214)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_11109 for (2999ds/18Mi)
% 1.95/1.48 % (11213)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_11109 for (2999ds/275Mi)
% 1.95/1.48 % (11211)Instruction limit reached!
% 1.95/1.48 % (11211)------------------------------
% 1.95/1.48 % (11211)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.95/1.48 % (11211)Termination reason: Unknown
% 1.95/1.48 % (11211)Termination phase: shuffling
% 1.95/1.48
% 1.95/1.48 % (11211)Memory used [KB]: 1151
% 1.95/1.48 % (11211)Time elapsed: 0.003 s
% 1.95/1.48 % (11211)Instructions burned: 2 (million)
% 1.95/1.48 % (11211)------------------------------
% 1.95/1.48 % (11211)------------------------------
% 1.95/1.48 % (11212)Instruction limit reached!
% 1.95/1.48 % (11212)------------------------------
% 1.95/1.48 % (11212)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.95/1.48 % (11212)Termination reason: Unknown
% 1.95/1.48 % (11212)Termination phase: shuffling
% 1.95/1.48
% 1.95/1.48 % (11212)Memory used [KB]: 1151
% 1.95/1.48 % (11212)Time elapsed: 0.003 s
% 1.95/1.48 % (11212)Instructions burned: 3 (million)
% 1.95/1.48 % (11212)------------------------------
% 1.95/1.48 % (11212)------------------------------
% 1.95/1.48 % (11209)Instruction limit reached!
% 1.95/1.48 % (11209)------------------------------
% 1.95/1.48 % (11209)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.95/1.48 % (11209)Termination reason: Unknown
% 1.95/1.48 % (11209)Termination phase: shuffling
% 1.95/1.48
% 1.95/1.48 % (11209)Memory used [KB]: 1151
% 1.95/1.48 % (11209)Time elapsed: 0.004 s
% 1.95/1.48 % (11209)Instructions burned: 4 (million)
% 1.95/1.48 % (11209)------------------------------
% 1.95/1.48 % (11209)------------------------------
% 1.95/1.49 % (11214)Instruction limit reached!
% 1.95/1.49 % (11214)------------------------------
% 1.95/1.49 % (11214)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.95/1.49 % (11214)Termination reason: Unknown
% 1.95/1.49 % (11214)Termination phase: Preprocessing 3
% 1.95/1.49
% 1.95/1.49 % (11214)Memory used [KB]: 1663
% 1.95/1.49 % (11214)Time elapsed: 0.012 s
% 1.95/1.49 % (11214)Instructions burned: 19 (million)
% 1.95/1.49 % (11214)------------------------------
% 1.95/1.49 % (11214)------------------------------
% 1.95/1.49 % (11210)Instruction limit reached!
% 1.95/1.49 % (11210)------------------------------
% 1.95/1.49 % (11210)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 1.95/1.49 % (11210)Termination reason: Unknown
% 1.95/1.49 % (11210)Termination phase: Property scanning
% 1.95/1.49
% 1.95/1.49 % (11210)Memory used [KB]: 1407
% 1.95/1.49 % (11210)Time elapsed: 0.014 s
% 1.95/1.49 % (11210)Instructions burned: 27 (million)
% 1.95/1.49 % (11210)------------------------------
% 1.95/1.49 % (11210)------------------------------
% 2.06/1.50 % (11215)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11109 for (2999ds/3Mi)
% 2.06/1.50 % (11216)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_11109 for (2999ds/37Mi)
% 2.06/1.50 % (11217)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_11109 for (2999ds/15Mi)
% 2.06/1.50 % (11215)Instruction limit reached!
% 2.06/1.50 % (11215)------------------------------
% 2.06/1.50 % (11215)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.06/1.50 % (11215)Termination reason: Unknown
% 2.06/1.50 % (11215)Termination phase: shuffling
% 2.06/1.50
% 2.06/1.50 % (11215)Memory used [KB]: 1151
% 2.06/1.50 % (11215)Time elapsed: 0.003 s
% 2.06/1.50 % (11215)Instructions burned: 4 (million)
% 2.06/1.50 % (11215)------------------------------
% 2.06/1.50 % (11215)------------------------------
% 2.06/1.50 % (11213)First to succeed.
% 2.06/1.51 % (11217)Instruction limit reached!
% 2.06/1.51 % (11217)------------------------------
% 2.06/1.51 % (11217)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.06/1.51 % (11217)Termination reason: Unknown
% 2.06/1.51 % (11217)Termination phase: Preprocessing 3
% 2.06/1.51
% 2.06/1.51 % (11217)Memory used [KB]: 1535
% 2.06/1.51 % (11217)Time elapsed: 0.009 s
% 2.06/1.51 % (11217)Instructions burned: 16 (million)
% 2.06/1.51 % (11217)------------------------------
% 2.06/1.51 % (11217)------------------------------
% 2.06/1.51 % (11218)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_11109 for (2999ds/3Mi)
% 2.06/1.51 % (11213)Refutation found. Thanks to Tanya!
% 2.06/1.51 % SZS status Theorem for DTF2THF_11109
% 2.06/1.51 % SZS output start Proof for DTF2THF_11109
% 2.06/1.51 thf(func_def_0, type, in: $i > $i > $o).
% 2.06/1.51 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.06/1.51 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.06/1.51 thf(func_def_8, type, powerset: $i > $i).
% 2.06/1.51 thf(func_def_10, type, setunion: $i > $i).
% 2.06/1.51 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.06/1.51 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.06/1.51 thf(func_def_26, type, prop2set: $o > $i).
% 2.06/1.51 thf(func_def_36, type, nonempty: $i > $o).
% 2.06/1.51 thf(func_def_69, type, set2prop: $i > $o).
% 2.06/1.51 thf(func_def_95, type, sP1: $i > $i > $o).
% 2.06/1.51 thf(func_def_96, type, sP2: $i > $o).
% 2.06/1.51 thf(func_def_97, type, sP3: $i > $i > $o).
% 2.06/1.51 thf(func_def_98, type, sP4: $i > $i > $o).
% 2.06/1.51 thf(func_def_100, type, sK6: $i > $i).
% 2.06/1.51 thf(func_def_101, type, sK7: $i > $i > $i).
% 2.06/1.51 thf(func_def_110, type, sK16: $i > $o).
% 2.06/1.51 thf(func_def_112, type, sK18: ($i > $o) > $i > $i).
% 2.06/1.51 thf(func_def_113, type, sK19: $i > $i).
% 2.06/1.51 thf(func_def_116, type, sK22: $i > $i).
% 2.06/1.51 thf(func_def_117, type, sK23: $i > $o).
% 2.06/1.51 thf(func_def_118, type, sK24: $i > $i).
% 2.06/1.51 thf(func_def_119, type, sK25: ($i > $o) > $i).
% 2.06/1.51 thf(func_def_120, type, sK26: $i > $o).
% 2.06/1.51 thf(func_def_122, type, sK28: $i > ($i > $o) > $i).
% 2.06/1.51 thf(func_def_126, type, sK32: ($i > $o) > $i > $i).
% 2.06/1.51 thf(func_def_128, type, sK34: $i > $o).
% 2.06/1.51 thf(func_def_132, type, sK38: $i > $i > $i).
% 2.06/1.51 thf(func_def_133, type, sK39: $i > ($i > $o) > $i).
% 2.06/1.51 thf(func_def_134, type, sK40: $i > $o).
% 2.06/1.51 thf(func_def_148, type, sK54: $o > $i > $i > $i).
% 2.06/1.51 thf(func_def_152, type, sK58: $i > $o).
% 2.06/1.51 thf(func_def_155, type, sK61: $i > ($i > $o) > $i).
% 2.06/1.51 thf(func_def_157, type, sK63: $i > $o).
% 2.06/1.51 thf(func_def_160, type, sK66: $i > ($i > $o) > $i).
% 2.06/1.51 thf(func_def_161, type, sK67: $i > $i > $o).
% 2.06/1.51 thf(func_def_163, type, sK69: $i > $i).
% 2.06/1.51 thf(func_def_164, type, sK70: $i > $i).
% 2.06/1.51 thf(func_def_165, type, sK71: $i > ($i > $i > $o) > $i).
% 2.06/1.51 thf(func_def_166, type, sK72: $i > ($i > $i > $o) > $i).
% 2.06/1.51 thf(func_def_167, type, sK73: $i > $i > ($i > $i > $o) > $i).
% 2.06/1.51 thf(func_def_171, type, sK77: $i > $o).
% 2.06/1.51 thf(func_def_178, type, sK84: $i > $o).
% 2.06/1.51 thf(func_def_179, type, sK85: $i > $i).
% 2.06/1.51 thf(func_def_180, type, sK86: ($i > $o) > $i).
% 2.06/1.51 thf(func_def_183, type, sK89: $i > $o).
% 2.06/1.51 thf(func_def_193, type, sK99: ($i > $o) > $i > $i).
% 2.06/1.51 thf(func_def_195, type, sK101: $i > $o).
% 2.06/1.51 thf(func_def_197, type, sK103: $i > ($i > $o) > $i).
% 2.06/1.51 thf(func_def_198, type, sK104: $i > $o).
% 2.06/1.51 thf(func_def_201, type, sK107: ($i > $o) > $i).
% 2.06/1.51 thf(func_def_202, type, sK108: ($i > $o) > $i).
% 2.06/1.51 thf(func_def_203, type, sK109: $i > $o).
% 2.06/1.51 thf(func_def_206, type, sK112: $i > $o).
% 2.06/1.51 thf(func_def_208, type, sK114: $i > $o).
% 2.06/1.51 thf(func_def_209, type, sK115: $i > $o).
% 2.06/1.51 thf(func_def_210, type, sK116: ($i > $o) > ($i > $o) > $i).
% 2.06/1.51 thf(func_def_211, type, sK117: ($i > $o) > ($i > $o) > $i).
% 2.06/1.51 thf(func_def_214, type, sK120: $i > $o).
% 2.06/1.51 thf(func_def_216, type, sK122: $i > $o).
% 2.06/1.51 thf(func_def_225, type, sK131: $i > $o).
% 2.06/1.51 thf(func_def_232, type, sK138: $i > $i).
% 2.06/1.51 thf(func_def_234, type, sK140: ($i > $o) > $i).
% 2.06/1.51 thf(func_def_235, type, sK141: $i > $o).
% 2.06/1.51 thf(func_def_236, type, sK142: $i > $i > $i).
% 2.06/1.51 thf(func_def_237, type, sK143: $i > $i > $i).
% 2.06/1.51 thf(func_def_238, type, sK144: $i > $i > $i).
% 2.06/1.51 thf(func_def_239, type, sK145: $i > $i > $i).
% 2.06/1.51 thf(func_def_240, type, sK146: $i > $i > $i > $i).
% 2.06/1.51 thf(func_def_241, type, sK147: $i > $i).
% 2.06/1.51 thf(func_def_242, type, sK148: $i > $i).
% 2.06/1.51 thf(func_def_243, type, sK149: $i > $i).
% 2.06/1.51 thf(func_def_244, type, sK150: $i > $i).
% 2.06/1.51 thf(func_def_245, type, sK151: $i > $i > $i).
% 2.06/1.51 thf(func_def_246, type, sK152: $i > $i > $i > $i).
% 2.06/1.51 thf(func_def_247, type, sK153: $i > $i > $i > $i).
% 2.06/1.51 thf(func_def_248, type, sK154: $i > $i > $i).
% 2.06/1.51 thf(func_def_249, type, sK155: $i > $i > $i).
% 2.06/1.51 thf(func_def_250, type, sK156: $i > $i).
% 2.06/1.51 thf(func_def_251, type, sK157: $i > $i > $i).
% 2.06/1.51 thf(func_def_253, type, sK159: $i > $i).
% 2.06/1.51 thf(func_def_254, type, sK160: $i > $i).
% 2.06/1.51 thf(func_def_258, type, sK164: $i > $o).
% 2.06/1.51 thf(func_def_261, type, sK167: $i > $o).
% 2.06/1.51 thf(func_def_264, type, sK170: $i > $i).
% 2.06/1.51 thf(func_def_272, type, sK178: $i > $i > $i).
% 2.06/1.51 thf(func_def_275, type, sK181: $i > $i).
% 2.06/1.51 thf(func_def_279, type, sK185: $i > $i > $i).
% 2.06/1.51 thf(func_def_280, type, sK186: $i > $i > $i).
% 2.06/1.51 thf(func_def_291, type, sK197: $i > $i > $i).
% 2.06/1.51 thf(func_def_294, type, sK200: $i > $i).
% 2.06/1.51 thf(func_def_298, type, ph204: !>[X0: $tType]:(X0)).
% 2.06/1.51 thf(f1405,plain,(
% 2.06/1.51 $false),
% 2.06/1.51 inference(trivial_inequality_removal,[status(thm)],[f1241])).
% 2.06/1.51 thf(f1241,plain,(
% 2.06/1.51 ((powerset @ sK180) != (powerset @ sK180))),
% 2.06/1.51 inference(definition_unfolding,[status(thm)],[f915,f916])).
% 2.06/1.51 thf(f916,plain,(
% 2.06/1.51 (sK179 = sK180)),
% 2.06/1.51 inference(cnf_transformation,[status(thm)],[f615])).
% 2.06/1.51 thf(f615,plain,(
% 2.06/1.51 (subPowSU = $true) & (omega0Ax = $true) & (replAx = $true) & (inCongP = $true) & (exuI1 = $true) & (setadjoinIR = $true) & (emptysetimpfalse = $true) & (setunionAx = $true) & (setunionI = $true) & (prop2set2propI = $true) & (dsetconstrI = $true) & (dsetconstrER = $true) & (quantDeMorgan2 = $true) & (emptyinunitempty = $true) & (exu__Cong = $true) & (setbeta = $true) & (exuE3u = $true) & (powersetI = $true) & (notinemptyset = $true) & (uniqinunit = $true) & (nonemptyImpWitness = $true) & (emptyInPowerset = $true) & (setadjoinIL = $true) & (emptysetE = $true) & (nonemptyI1 = $true) & (omegaIndAx = $true) & (prop2setI = $true) & (prop2setE = $true) & (quantDeMorgan3 = $true) & (emptyI = $true) & (notdallE = $true) & (omegaSAx = $true) & (upairsetIR = $true) & (exuE1 = $true) & (notinsingleton = $true) & (setadjoinE = $true) & (setextAx = $true) & (emptyset__Cong = $true) & (descrp = $true) & (emptysetAx = $true) & (exuE2 = $true) & (in__Cong = $true) & (quantDeMorgan4 = $true) & (nonemptyE1 = $true) & (upairsetIL = $true) & (emptyE1 = $true) & (quantDeMorgan1 = $true) & (singletonsswitch = $true) & (setadjoin__Cong = $true) & (eqinunit = $true) & (foundationAx = $true) & (setoftrueEq = $true) & (exuE3e = $true) & (setunionE = $true) & (wellorderingAx = $true) & (emptyinPowerset = $true) & (powersetAx = $true) & (setext = $true) & (upairsetE = $true) & (setadjoinOr = $true) & (exuI2 = $true) & (dsetconstrEL = $true) & (exuI3 = $true) & (nonemptyI = $true) & (notdexE = $true) & (setadjoinAx = $true) & (noeltsimpempty = $true) & (vacuousDall = $true) & ((sK179 = sK180) & ((powerset @ sK179) != (powerset @ sK180))) & (powersetE = $true)),
% 2.06/1.51 inference(skolemisation,[status(esa),new_symbols(skolem,[sK179,sK180])],[f295,f614])).
% 2.06/1.51 thf(f614,plain,(
% 2.06/1.51 ? [X0 : $i,X1 : $i] : ((X0 = X1) & ((powerset @ X1) != (powerset @ X0))) => ((sK179 = sK180) & ((powerset @ sK179) != (powerset @ sK180)))),
% 2.06/1.51 introduced(definition,[],[choice_axiom])).
% 2.06/1.51 thf(f295,plain,(
% 2.06/1.51 (subPowSU = $true) & (omega0Ax = $true) & (replAx = $true) & (inCongP = $true) & (exuI1 = $true) & (setadjoinIR = $true) & (emptysetimpfalse = $true) & (setunionAx = $true) & (setunionI = $true) & (prop2set2propI = $true) & (dsetconstrI = $true) & (dsetconstrER = $true) & (quantDeMorgan2 = $true) & (emptyinunitempty = $true) & (exu__Cong = $true) & (setbeta = $true) & (exuE3u = $true) & (powersetI = $true) & (notinemptyset = $true) & (uniqinunit = $true) & (nonemptyImpWitness = $true) & (emptyInPowerset = $true) & (setadjoinIL = $true) & (emptysetE = $true) & (nonemptyI1 = $true) & (omegaIndAx = $true) & (prop2setI = $true) & (prop2setE = $true) & (quantDeMorgan3 = $true) & (emptyI = $true) & (notdallE = $true) & (omegaSAx = $true) & (upairsetIR = $true) & (exuE1 = $true) & (notinsingleton = $true) & (setadjoinE = $true) & (setextAx = $true) & (emptyset__Cong = $true) & (descrp = $true) & (emptysetAx = $true) & (exuE2 = $true) & (in__Cong = $true) & (quantDeMorgan4 = $true) & (nonemptyE1 = $true) & (upairsetIL = $true) & (emptyE1 = $true) & (quantDeMorgan1 = $true) & (singletonsswitch = $true) & (setadjoin__Cong = $true) & (eqinunit = $true) & (foundationAx = $true) & (setoftrueEq = $true) & (exuE3e = $true) & (setunionE = $true) & (wellorderingAx = $true) & (emptyinPowerset = $true) & (powersetAx = $true) & (setext = $true) & (upairsetE = $true) & (setadjoinOr = $true) & (exuI2 = $true) & (dsetconstrEL = $true) & (exuI3 = $true) & (nonemptyI = $true) & (notdexE = $true) & (setadjoinAx = $true) & (noeltsimpempty = $true) & (vacuousDall = $true) & ? [X0 : $i,X1 : $i] : ((X0 = X1) & ((powerset @ X1) != (powerset @ X0))) & (powersetE = $true)),
% 2.06/1.51 inference(flattening,[status(thm)],[f294])).
% 2.06/1.51 thf(f294,plain,(
% 2.06/1.51 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : ((X0 = X1) & ((powerset @ X1) != (powerset @ X0))) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.06/1.51 inference(ennf_transformation,[status(thm)],[f108])).
% 2.06/1.51 thf(f108,plain,(
% 2.06/1.51 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ! [X0 : $i,X1 : $i] : ((X0 = X1) => ((powerset @ X1) = (powerset @ X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.06/1.51 inference(fool_elimination,[status(thm)],[f107])).
% 2.06/1.51 thf(f107,plain,(
% 2.06/1.51 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => ! [X0 : $i,X1 : $i] : ((X0 = X1) => ((powerset @ X1) = (powerset @ X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.06/1.51 inference(rectify,[status(thm)],[f73])).
% 2.06/1.51 thf(f73,negated_conjecture,(
% 2.06/1.51 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => ! [X3 : $i,X4 : $i] : ((X3 = X4) => ((powerset @ X3) = (powerset @ X4))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.06/1.51 inference(negated_conjecture,[status(cth)],[f72])).
% 2.06/1.51 thf(f72,conjecture,(
% 2.06/1.51 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => ! [X3 : $i,X4 : $i] : ((X3 = X4) => ((powerset @ X3) = (powerset @ X4)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.06/1.51 file('/export/starexec/sandbox/tmp/tmp.5INz166dx8/DTF2THF_11109.p',powerset__Cong)).
% 2.06/1.51 thf(f915,plain,(
% 2.06/1.51 ((powerset @ sK179) != (powerset @ sK180))),
% 2.06/1.51 inference(cnf_transformation,[status(thm)],[f615])).
% 2.06/1.51 % SZS output end Proof for DTF2THF_11109
% 2.06/1.51 % (11213)------------------------------
% 2.06/1.51 % (11213)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.06/1.51 % (11213)Termination reason: Refutation
% 2.06/1.51
% 2.06/1.51 % (11213)Memory used [KB]: 6524
% 2.06/1.51 % (11213)Time elapsed: 0.027 s
% 2.06/1.51 % (11213)Instructions burned: 49 (million)
% 2.06/1.51 % (11213)------------------------------
% 2.06/1.51 % (11213)------------------------------
% 2.06/1.51 % (11207)Success in time 0.043 s
%------------------------------------------------------------------------------