%------------------------------------------------------------------------------
% File : E---3.5.1
% Problem : SEU698^1 : TPTP v9.3.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : run_E /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% Computer : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Sep 30 08:25:15 AM UTC 2026
% Result : Theorem 4.23s 1.01s
% Output : CNFRefutation 4.23s
% Verified :
% SZS Type : Refutation
% Derivation depth : 7
% Number of leaves : 47
% Syntax : Number of formulae : 87 ( 18 unt; 0 typ; 0 def)
% Number of atoms : 1530 ( 126 equ; 0 cnn)
% Maximal formula atoms : 436 ( 17 avg)
% Number of connectives : 4367 ( 160 ~; 204 |; 280 &;2761 @)
% ( 61 <=>; 901 =>; 0 <=; 0 <~>)
% Maximal formula depth : 412 ( 21 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 96 ( 96 >; 0 *; 0 +; 0 <<)
% Number of symbols : 263 ( 260 usr; 220 con; 0-4 aty)
% Number of variables : 723 ( 87 ^; 616 !; 20 ?; 723 :)
% Comments :
%------------------------------------------------------------------------------
thf(decl_23,type,
in: $i > $i > $o ).
thf(decl_25,type,
setextAx: $o ).
thf(decl_26,type,
emptyset: $i ).
thf(decl_27,type,
emptysetAx: $o ).
thf(decl_28,type,
setadjoin: $i > $i > $i ).
thf(decl_29,type,
setadjoinAx: $o ).
thf(decl_31,type,
powersetAx: $o ).
thf(decl_32,type,
setunion: $i > $i ).
thf(decl_33,type,
setunionAx: $o ).
thf(decl_35,type,
omega0Ax: $o ).
thf(decl_36,type,
omegaSAx: $o ).
thf(decl_37,type,
omegaIndAx: $o ).
thf(decl_38,type,
replAx: $o ).
thf(decl_39,type,
foundationAx: $o ).
thf(decl_40,type,
wellorderingAx: $o ).
thf(decl_42,type,
descrp: $o ).
thf(decl_43,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(decl_44,type,
dsetconstrI: $o ).
thf(decl_45,type,
dsetconstrEL: $o ).
thf(decl_46,type,
dsetconstrER: $o ).
thf(decl_47,type,
exuE1: $o ).
thf(decl_49,type,
prop2setE: $o ).
thf(decl_50,type,
emptysetE: $o ).
thf(decl_51,type,
emptysetimpfalse: $o ).
thf(decl_52,type,
notinemptyset: $o ).
thf(decl_53,type,
exuE3e: $o ).
thf(decl_54,type,
setext: $o ).
thf(decl_55,type,
emptyI: $o ).
thf(decl_56,type,
noeltsimpempty: $o ).
thf(decl_57,type,
setbeta: $o ).
thf(decl_59,type,
nonemptyE1: $o ).
thf(decl_60,type,
nonemptyI: $o ).
thf(decl_61,type,
nonemptyI1: $o ).
thf(decl_62,type,
setadjoinIL: $o ).
thf(decl_63,type,
emptyinunitempty: $o ).
thf(decl_64,type,
setadjoinIR: $o ).
thf(decl_65,type,
setadjoinE: $o ).
thf(decl_66,type,
setadjoinOr: $o ).
thf(decl_67,type,
setoftrueEq: $o ).
thf(decl_68,type,
powersetI: $o ).
thf(decl_69,type,
emptyinPowerset: $o ).
thf(decl_70,type,
emptyInPowerset: $o ).
thf(decl_71,type,
powersetE: $o ).
thf(decl_72,type,
setunionI: $o ).
thf(decl_73,type,
setunionE: $o ).
thf(decl_74,type,
subPowSU: $o ).
thf(decl_75,type,
exuE2: $o ).
thf(decl_76,type,
nonemptyImpWitness: $o ).
thf(decl_77,type,
uniqinunit: $o ).
thf(decl_78,type,
notinsingleton: $o ).
thf(decl_79,type,
eqinunit: $o ).
thf(decl_80,type,
singletonsswitch: $o ).
thf(decl_81,type,
upairsetE: $o ).
thf(decl_82,type,
upairsetIL: $o ).
thf(decl_83,type,
upairsetIR: $o ).
thf(decl_84,type,
emptyE1: $o ).
thf(decl_85,type,
vacuousDall: $o ).
thf(decl_86,type,
quantDeMorgan1: $o ).
thf(decl_87,type,
quantDeMorgan2: $o ).
thf(decl_88,type,
quantDeMorgan3: $o ).
thf(decl_89,type,
quantDeMorgan4: $o ).
thf(decl_90,type,
prop2setI: $o ).
thf(decl_92,type,
prop2set2propI: $o ).
thf(decl_93,type,
notdexE: $o ).
thf(decl_94,type,
notdallE: $o ).
thf(decl_95,type,
exuI1: $o ).
thf(decl_96,type,
exuI3: $o ).
thf(decl_97,type,
exuI2: $o ).
thf(decl_98,type,
inCongP: $o ).
thf(decl_99,type,
in__Cong: $o ).
thf(decl_100,type,
exuE3u: $o ).
thf(decl_101,type,
exu__Cong: $o ).
thf(decl_102,type,
emptyset__Cong: $o ).
thf(decl_103,type,
setadjoin__Cong: $o ).
thf(decl_104,type,
powerset__Cong: $o ).
thf(decl_105,type,
setunion__Cong: $o ).
thf(decl_106,type,
omega__Cong: $o ).
thf(decl_107,type,
exuEu: $o ).
thf(decl_108,type,
descr__Cong: $o ).
thf(decl_109,type,
dsetconstr__Cong: $o ).
thf(decl_110,type,
subset: $i > $i > $o ).
thf(decl_113,type,
subsetI1: $o ).
thf(decl_114,type,
eqimpsubset2: $o ).
thf(decl_115,type,
eqimpsubset1: $o ).
thf(decl_116,type,
subsetI2: $o ).
thf(decl_117,type,
emptysetsubset: $o ).
thf(decl_118,type,
subsetE: $o ).
thf(decl_119,type,
subsetE2: $o ).
thf(decl_120,type,
notsubsetI: $o ).
thf(decl_121,type,
notequalI1: $o ).
thf(decl_122,type,
notequalI2: $o ).
thf(decl_123,type,
subsetRefl: $o ).
thf(decl_124,type,
subsetTrans: $o ).
thf(decl_125,type,
setadjoinSub: $o ).
thf(decl_126,type,
setadjoinSub2: $o ).
thf(decl_127,type,
subset2powerset: $o ).
thf(decl_128,type,
setextsub: $o ).
thf(decl_129,type,
subsetemptysetimpeq: $o ).
thf(decl_130,type,
powersetI1: $o ).
thf(decl_131,type,
powersetE1: $o ).
thf(decl_132,type,
inPowerset: $o ).
thf(decl_133,type,
powersetsubset: $o ).
thf(decl_134,type,
sepInPowerset: $o ).
thf(decl_135,type,
sepSubset: $o ).
thf(decl_137,type,
binunionIL: $o ).
thf(decl_138,type,
upairset2IR: $o ).
thf(decl_139,type,
binunionIR: $o ).
thf(decl_140,type,
binunionEcases: $o ).
thf(decl_141,type,
binunionE: $o ).
thf(decl_142,type,
binunionLsub: $o ).
thf(decl_143,type,
binunionRsub: $o ).
thf(decl_145,type,
binintersectI: $o ).
thf(decl_146,type,
binintersectSubset5: $o ).
thf(decl_147,type,
binintersectEL: $o ).
thf(decl_148,type,
binintersectLsub: $o ).
thf(decl_149,type,
binintersectSubset2: $o ).
thf(decl_150,type,
binintersectSubset3: $o ).
thf(decl_151,type,
binintersectER: $o ).
thf(decl_152,type,
disjointsetsI1: $o ).
thf(decl_153,type,
binintersectRsub: $o ).
thf(decl_154,type,
binintersectSubset4: $o ).
thf(decl_155,type,
binintersectSubset1: $o ).
thf(decl_156,type,
bs114d: $o ).
thf(decl_159,type,
setminusI: $o ).
thf(decl_160,type,
setminusEL: $o ).
thf(decl_161,type,
setminusER: $o ).
thf(decl_162,type,
setminusSubset2: $o ).
thf(decl_163,type,
setminusERneg: $o ).
thf(decl_164,type,
setminusELneg: $o ).
thf(decl_165,type,
setminusILneg: $o ).
thf(decl_166,type,
setminusIRneg: $o ).
thf(decl_167,type,
setminusLsub: $o ).
thf(decl_168,type,
setminusSubset1: $o ).
thf(decl_170,type,
symdiffE: $o ).
thf(decl_171,type,
symdiffI1: $o ).
thf(decl_172,type,
symdiffI2: $o ).
thf(decl_173,type,
symdiffIneg1: $o ).
thf(decl_174,type,
symdiffIneg2: $o ).
thf(decl_175,type,
iskpair: $i > $o ).
thf(decl_176,type,
secondinupair: $o ).
thf(decl_177,type,
setukpairIL: $o ).
thf(decl_178,type,
setukpairIR: $o ).
thf(decl_179,type,
kpairiskpair: $o ).
thf(decl_180,type,
kpair: $i > $i > $i ).
thf(decl_181,type,
kpairp: $o ).
thf(decl_182,type,
cartprod: $i > $i > $i ).
thf(decl_183,type,
singletonsubset: $o ).
thf(decl_184,type,
singletoninpowerset: $o ).
thf(decl_185,type,
singletoninpowunion: $o ).
thf(decl_186,type,
upairset2E: $o ).
thf(decl_187,type,
upairsubunion: $o ).
thf(decl_188,type,
upairinpowunion: $o ).
thf(decl_189,type,
ubforcartprodlem1: $o ).
thf(decl_190,type,
ubforcartprodlem2: $o ).
thf(decl_191,type,
ubforcartprodlem3: $o ).
thf(decl_192,type,
cartprodpairin: $o ).
thf(decl_193,type,
cartprodmempair1: $o ).
thf(decl_194,type,
cartprodmempair: $o ).
thf(decl_195,type,
setunionE2: $o ).
thf(decl_196,type,
setunionsingleton1: $o ).
thf(decl_197,type,
setunionsingleton2: $o ).
thf(decl_198,type,
setunionsingleton: $o ).
thf(decl_199,type,
singleton: $i > $o ).
thf(decl_200,type,
singletonprop: $o ).
thf(decl_202,type,
ex1E1: $o ).
thf(decl_203,type,
ex1I: $o ).
thf(decl_204,type,
ex1I2: $o ).
thf(decl_205,type,
singletonsuniq: $o ).
thf(decl_210,type,
setukpairinjL1: $o ).
thf(decl_211,type,
kfstsingleton: $o ).
thf(decl_212,type,
theprop: $o ).
thf(decl_213,type,
kfst: $i > $i ).
thf(decl_214,type,
kfstpairEq: $o ).
thf(decl_215,type,
cartprodfstin: $o ).
thf(decl_216,type,
setukpairinjL2: $o ).
thf(decl_217,type,
setukpairinjL: $o ).
thf(decl_218,type,
setukpairinjR11: $o ).
thf(decl_219,type,
setukpairinjR12: $o ).
thf(decl_220,type,
setukpairinjR1: $o ).
thf(decl_221,type,
upairequniteq: $o ).
thf(decl_222,type,
setukpairinjR2: $o ).
thf(decl_223,type,
setukpairinjR: $o ).
thf(decl_224,type,
ksndsingleton: $o ).
thf(decl_225,type,
ksnd: $i > $i ).
thf(decl_226,type,
ksndpairEq: $o ).
thf(decl_227,type,
kpairsurjEq: $o ).
thf(decl_228,type,
cartprodsndin: $o ).
thf(decl_229,type,
cartprodpairmemEL: $o ).
thf(decl_230,type,
cartprodpairmemER: $o ).
thf(decl_231,type,
cartprodmempaircEq: $o ).
thf(decl_232,type,
cartprodfstpairEq: $o ).
thf(decl_233,type,
cartprodsndpairEq: $o ).
thf(decl_234,type,
cartprodpairsurjEq: $o ).
thf(decl_235,type,
breln: $i > $i > $i > $o ).
thf(decl_236,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(decl_237,type,
dpsetconstrI: $o ).
thf(decl_238,type,
dpsetconstrSub: $o ).
thf(decl_239,type,
setOfPairsIsBReln: $o ).
thf(decl_240,type,
dpsetconstrERa: $o ).
thf(decl_241,type,
dpsetconstrEL1: $o ).
thf(decl_242,type,
dpsetconstrEL2: $o ).
thf(decl_243,type,
dpsetconstrER: $o ).
thf(decl_244,type,
func: $i > $i > $i > $o ).
thf(decl_245,type,
funcSet: $i > $i > $i ).
thf(decl_246,type,
funcImageSingleton: $o ).
thf(decl_247,type,
apProp: $o ).
thf(decl_248,type,
ap: $i > $i > $i > $i > $i ).
thf(decl_249,type,
app: $o ).
thf(decl_250,type,
infuncsetfunc: $o ).
thf(decl_251,type,
ap2p: $o ).
thf(decl_252,type,
funcinfuncset: $o ).
thf(decl_253,type,
lamProp: $o ).
thf(decl_254,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(decl_255,type,
lamp: $o ).
thf(decl_256,type,
lam2p: $o ).
thf(decl_257,type,
brelnall1: $o ).
thf(decl_258,type,
brelnall2: $o ).
thf(decl_259,type,
ex1E2: $o ).
thf(decl_260,type,
funcGraphProp1: $o ).
thf(decl_261,type,
funcGraphProp3: $o ).
thf(decl_262,type,
funcGraphProp2: $o ).
thf(decl_263,type,
funcextLem: $o ).
thf(decl_264,type,
funcGraphProp4: $o ).
thf(decl_265,type,
subbreln: $o ).
thf(decl_266,type,
eqbreln: $o ).
thf(decl_267,type,
funcext: $o ).
thf(decl_268,type,
funcext2: $o ).
thf(decl_269,type,
ap2apEq1: $o ).
thf(decl_270,type,
ap2apEq2: $o ).
thf(decl_271,type,
beta1: $o ).
thf(decl_272,type,
eta1: $o ).
thf(decl_273,type,
lam2lamEq: $o ).
thf(decl_274,type,
esk1_3: $i > $i > $i > $i ).
thf(decl_275,type,
esk2_3: $i > $i > $i > $i ).
thf(decl_276,type,
esk3_2: $i > ( $i > $o ) > $i ).
thf(decl_277,type,
esk4_2: $i > ( $i > $o ) > $i ).
thf(decl_278,type,
esk5_2: $i > ( $i > $o ) > $i ).
thf(decl_279,type,
esk6_1: $i > $i ).
thf(decl_280,type,
esk7_1: $i > $i ).
thf(decl_281,type,
esk8_4: $i > $i > $i > $i > $i ).
thf(decl_282,type,
esk9_3: $i > $i > ( $i > $i ) > $i ).
thf(decl_283,type,
esk10_3: $i > $i > ( $i > $i ) > $i ).
thf(decl_284,type,
esk11_3: $i > $i > ( $i > $i ) > $i ).
thf(decl_285,type,
esk12_4: $i > $i > $i > ( $i > $o ) > $i ).
thf(decl_286,type,
esk13_4: $i > $i > $i > ( $i > $o ) > $i ).
thf(decl_287,type,
esk14_4: $i > $i > $i > ( $i > $o ) > $i ).
thf(decl_288,type,
esk15_4: $i > $i > $i > ( $i > $o ) > $i ).
thf(decl_289,type,
esk16_4: $i > $i > $i > $i > $i ).
thf(decl_290,type,
esk17_4: $i > $i > $i > $i > $i ).
thf(decl_291,type,
esk18_4: $i > $i > $i > $i > $i ).
thf(decl_292,type,
esk19_4: $i > $i > $i > $i > $i ).
thf(decl_293,type,
esk20_4: $i > $i > $i > $i > $i ).
thf(decl_294,type,
esk21_4: $i > $i > $i > $i > $i ).
thf(decl_295,type,
esk22_4: $i > $i > $i > $i > $i ).
thf(decl_296,type,
esk23_3: $i > $i > ( $i > $i ) > $i ).
thf(decl_297,type,
esk24_3: $i > $i > ( $i > $i ) > $i ).
thf(decl_298,type,
esk25_0: $i ).
thf(decl_299,type,
esk26_0: $i ).
thf(decl_300,type,
esk27_0: $i > $i ).
thf(decl_301,type,
esk28_0: $i ).
thf(funcImageSingleton,axiom,
( funcImageSingleton
<=> ! [X4: $i,X5: $i,X29: $i] :
( ( func @ X4 @ X5 @ X29 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( singleton
@ ( dsetconstr @ X5
@ ^ [X3: $i] : ( in @ ( kpair @ X2 @ X3 ) @ X29 ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',funcImageSingleton) ).
thf(singleton,axiom,
( singleton
= ( ^ [X4: $i] :
? [X2: $i] :
( ( in @ X2 @ X4 )
& ( X4
= ( setadjoin @ X2 @ emptyset ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',singleton) ).
thf(singletonprop,axiom,
( singletonprop
<=> ! [X4: $i,X1: $i > $o] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X4 )
=> ( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ? [X2: $i] :
( ( in @ X2 @ X4 )
& ( X1 @ X2 ) )
=> ( singleton
@ ( dsetconstr @ X4
@ ^ [X2: $i] : ( X1 @ X2 ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',singletonprop) ).
thf(ksndsingleton,axiom,
( ksndsingleton
<=> ! [X18: $i] :
( ( iskpair @ X18 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X18 )
@ ^ [X2: $i] :
( X18
= ( kpair @ ( kfst @ X18 ) @ X2 ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',ksndsingleton) ).
thf(kfstsingleton,axiom,
( kfstsingleton
<=> ! [X18: $i] :
( ( iskpair @ X18 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X18 )
@ ^ [X2: $i] : ( in @ ( setadjoin @ X2 @ emptyset ) @ X18 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',kfstsingleton) ).
thf(breln,axiom,
( breln
= ( ^ [X4: $i,X5: $i,X7: $i] : ( subset @ X7 @ ( cartprod @ X4 @ X5 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',breln) ).
thf(setOfPairsIsBReln,axiom,
( setOfPairsIsBReln
<=> ! [X4: $i,X5: $i,X24: $i > $i > $o] :
( breln @ X4 @ X5
@ ( dpsetconstr @ X4 @ X5
@ ^ [X2: $i,X3: $i] : ( X24 @ X2 @ X3 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',setOfPairsIsBReln) ).
thf(beta2,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ! [X4: $i,X5: $i,X39: $i > $i] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( X39 @ X2 ) @ X5 ) )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( ( ap @ X4 @ X5
@ ( lam @ X4 @ X5
@ ^ [X3: $i] : ( X39 @ X3 ) )
@ X2 )
= ( X39 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',beta2) ).
thf(theprop,axiom,
( theprop
<=> ! [X21: $i] :
( ( singleton @ X21 )
=> ( in @ ( setunion @ X21 ) @ X21 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',theprop) ).
thf(eqbreln,axiom,
( eqbreln
<=> ! [X4: $i,X5: $i,X34: $i] :
( ( breln @ X4 @ X5 @ X34 )
=> ! [X36: $i] :
( ( breln @ X4 @ X5 @ X36 )
=> ( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( in @ ( kpair @ X2 @ X3 ) @ X34 )
=> ( in @ ( kpair @ X2 @ X3 ) @ X36 ) ) ) )
=> ( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( in @ ( kpair @ X2 @ X3 ) @ X36 )
=> ( in @ ( kpair @ X2 @ X3 ) @ X34 ) ) ) )
=> ( X34 = X36 ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',eqbreln) ).
thf(subbreln,axiom,
( subbreln
<=> ! [X4: $i,X5: $i,X34: $i] :
( ( breln @ X4 @ X5 @ X34 )
=> ! [X36: $i] :
( ( breln @ X4 @ X5 @ X36 )
=> ( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( in @ ( kpair @ X2 @ X3 ) @ X34 )
=> ( in @ ( kpair @ X2 @ X3 ) @ X36 ) ) ) )
=> ( subset @ X34 @ X36 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',subbreln) ).
thf(brelnall2,axiom,
( brelnall2
<=> ! [X4: $i,X5: $i,X34: $i] :
( ( breln @ X4 @ X5 @ X34 )
=> ! [X1: $i > $o] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( in @ ( kpair @ X2 @ X3 ) @ X34 )
=> ( X1 @ ( kpair @ X2 @ X3 ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ X34 )
=> ( X1 @ X2 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',brelnall2) ).
thf(brelnall1,axiom,
( brelnall1
<=> ! [X4: $i,X5: $i,X34: $i] :
( ( breln @ X4 @ X5 @ X34 )
=> ! [X1: $i > $o] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( in @ ( kpair @ X2 @ X3 ) @ X34 )
=> ( X1 @ ( kpair @ X2 @ X3 ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ X34 )
=> ( X1 @ X2 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',brelnall1) ).
thf(lamProp,axiom,
( lamProp
<=> ! [X4: $i,X5: $i,X30: $i > $i] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( X30 @ X2 ) @ X5 ) )
=> ( func @ X4 @ X5
@ ( dpsetconstr @ X4 @ X5
@ ^ [X2: $i,X3: $i] :
( ( X30 @ X2 )
= X3 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',lamProp) ).
thf(dpsetconstrER,axiom,
( dpsetconstrER
<=> ! [X4: $i,X5: $i,X28: $i > $i > $o,X2: $i,X3: $i] :
( ( in @ ( kpair @ X2 @ X3 )
@ ( dpsetconstr @ X4 @ X5
@ ^ [X14: $i,X18: $i] : ( X28 @ X14 @ X18 ) ) )
=> ( X28 @ X2 @ X3 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',dpsetconstrER) ).
thf(dpsetconstrEL2,axiom,
( dpsetconstrEL2
<=> ! [X4: $i,X5: $i,X27: $i > $i > $o,X2: $i,X3: $i] :
( ( in @ ( kpair @ X2 @ X3 )
@ ( dpsetconstr @ X4 @ X5
@ ^ [X14: $i,X18: $i] : ( X27 @ X14 @ X18 ) ) )
=> ( in @ X3 @ X5 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',dpsetconstrEL2) ).
thf(dpsetconstrEL1,axiom,
( dpsetconstrEL1
<=> ! [X4: $i,X5: $i,X26: $i > $i > $o,X2: $i,X3: $i] :
( ( in @ ( kpair @ X2 @ X3 )
@ ( dpsetconstr @ X4 @ X5
@ ^ [X14: $i,X18: $i] : ( X26 @ X14 @ X18 ) ) )
=> ( in @ X2 @ X4 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',dpsetconstrEL1) ).
thf(dpsetconstrERa,axiom,
( dpsetconstrERa
<=> ! [X4: $i,X5: $i,X25: $i > $i > $o,X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( in @ ( kpair @ X2 @ X3 )
@ ( dpsetconstr @ X4 @ X5
@ ^ [X14: $i,X18: $i] : ( X25 @ X14 @ X18 ) ) )
=> ( X25 @ X2 @ X3 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',dpsetconstrERa) ).
thf(dpsetconstrSub,axiom,
( dpsetconstrSub
<=> ! [X4: $i,X5: $i,X23: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X4 @ X5
@ ^ [X2: $i,X3: $i] : ( X23 @ X2 @ X3 ) )
@ ( cartprod @ X4 @ X5 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',dpsetconstrSub) ).
thf(dpsetconstrI,axiom,
( dpsetconstrI
<=> ! [X4: $i,X5: $i,X22: $i > $i > $o,X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( X22 @ X2 @ X3 )
=> ( in @ ( kpair @ X2 @ X3 )
@ ( dpsetconstr @ X4 @ X5
@ ^ [X14: $i,X18: $i] : ( X22 @ X14 @ X18 ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',dpsetconstrI) ).
thf(lam2lamEq,axiom,
( lam2lamEq
<=> ! [X4: $i,X5: $i,X38: $i > $i] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( X38 @ X2 ) @ X5 ) )
=> ( ( lam @ X4 @ X5
@ ^ [X2: $i] : ( X38 @ X2 ) )
= ( lam @ X4 @ X5
@ ^ [X2: $i] : ( X38 @ X2 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',lam2lamEq) ).
thf(eta1,axiom,
( eta1
<=> ! [X4: $i,X5: $i,X29: $i] :
( ( func @ X4 @ X5 @ X29 )
=> ( ( lam @ X4 @ X5
@ ^ [X2: $i] : ( ap @ X4 @ X5 @ X29 @ X2 ) )
= X29 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',eta1) ).
thf(beta1,axiom,
( beta1
<=> ! [X4: $i,X5: $i,X37: $i > $i] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( X37 @ X2 ) @ X5 ) )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( ( ap @ X4 @ X5
@ ( lam @ X4 @ X5
@ ^ [X3: $i] : ( X37 @ X3 ) )
@ X2 )
= ( X37 @ X2 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',beta1) ).
thf(lam2p,axiom,
( lam2p
<=> ! [X4: $i,X5: $i,X33: $i > $i] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( X33 @ X2 ) @ X5 ) )
=> ( in
@ ( lam @ X4 @ X5
@ ^ [X2: $i] : ( X33 @ X2 ) )
@ ( funcSet @ X4 @ X5 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',lam2p) ).
thf(lamp,axiom,
( lamp
<=> ! [X4: $i,X5: $i,X32: $i > $i] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( X32 @ X2 ) @ X5 ) )
=> ( func @ X4 @ X5
@ ( lam @ X4 @ X5
@ ^ [X2: $i] : ( X32 @ X2 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',lamp) ).
thf(cartprodsndpairEq,axiom,
( cartprodsndpairEq
<=> ! [X4: $i,X5: $i,X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( ksnd @ ( kpair @ X2 @ X3 ) )
= X3 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',cartprodsndpairEq) ).
thf(ksndpairEq,axiom,
( ksndpairEq
<=> ! [X2: $i,X3: $i] :
( ( ksnd @ ( kpair @ X2 @ X3 ) )
= X3 ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',ksndpairEq) ).
thf(cartprodfstpairEq,axiom,
( cartprodfstpairEq
<=> ! [X4: $i,X5: $i,X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( kfst @ ( kpair @ X2 @ X3 ) )
= X2 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',cartprodfstpairEq) ).
thf(kfstpairEq,axiom,
( kfstpairEq
<=> ! [X2: $i,X3: $i] :
( ( kfst @ ( kpair @ X2 @ X3 ) )
= X2 ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',kfstpairEq) ).
thf(kpairsurjEq,axiom,
( kpairsurjEq
<=> ! [X18: $i] :
( ( iskpair @ X18 )
=> ( ( kpair @ ( kfst @ X18 ) @ ( ksnd @ X18 ) )
= X18 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',kpairsurjEq) ).
thf(kpairp,axiom,
( kpairp
<=> ! [X2: $i,X3: $i] : ( iskpair @ ( kpair @ X2 @ X3 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',kpairp) ).
thf(kpairiskpair,axiom,
( kpairiskpair
<=> ! [X2: $i,X3: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X2 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X2 @ ( setadjoin @ X3 @ emptyset ) ) @ emptyset ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',kpairiskpair) ).
thf(cartprodpairsurjEq,axiom,
( cartprodpairsurjEq
<=> ! [X4: $i,X5: $i,X18: $i] :
( ( in @ X18 @ ( cartprod @ X4 @ X5 ) )
=> ( ( kpair @ ( kfst @ X18 ) @ ( ksnd @ X18 ) )
= X18 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',cartprodpairsurjEq) ).
thf(cartprodpairmemER,axiom,
( cartprodpairmemER
<=> ! [X4: $i,X5: $i,X2: $i,X3: $i] :
( ( in @ ( kpair @ X2 @ X3 ) @ ( cartprod @ X4 @ X5 ) )
=> ( in @ X3 @ X5 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',cartprodpairmemER) ).
thf(cartprodpairmemEL,axiom,
( cartprodpairmemEL
<=> ! [X4: $i,X5: $i,X2: $i,X3: $i] :
( ( in @ ( kpair @ X2 @ X3 ) @ ( cartprod @ X4 @ X5 ) )
=> ( in @ X2 @ X4 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',cartprodpairmemEL) ).
thf(cartprodsndin,axiom,
( cartprodsndin
<=> ! [X4: $i,X5: $i,X18: $i] :
( ( in @ X18 @ ( cartprod @ X4 @ X5 ) )
=> ( in @ ( ksnd @ X18 ) @ X5 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',cartprodsndin) ).
thf(cartprodfstin,axiom,
( cartprodfstin
<=> ! [X4: $i,X5: $i,X18: $i] :
( ( in @ X18 @ ( cartprod @ X4 @ X5 ) )
=> ( in @ ( kfst @ X18 ) @ X4 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',cartprodfstin) ).
thf(cartprodmempair,axiom,
( cartprodmempair
<=> ! [X4: $i,X5: $i,X18: $i] :
( ( in @ X18 @ ( cartprod @ X4 @ X5 ) )
=> ( iskpair @ X18 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',cartprodmempair) ).
thf(cartprodmempair1,axiom,
( cartprodmempair1
<=> ! [X4: $i,X5: $i,X18: $i] :
( ( in @ X18 @ ( cartprod @ X4 @ X5 ) )
=> ? [X2: $i] :
( ( in @ X2 @ X4 )
& ? [X3: $i] :
( ( in @ X3 @ X5 )
& ( X18
= ( kpair @ X2 @ X3 ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',cartprodmempair1) ).
thf(cartprodpairin,axiom,
( cartprodpairin
<=> ! [X4: $i,X5: $i,X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( in @ ( kpair @ X2 @ X3 ) @ ( cartprod @ X4 @ X5 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',cartprodpairin) ).
thf(ap2apEq1,axiom,
( ap2apEq1
<=> ! [X4: $i,X5: $i,X29: $i] :
( ( in @ X29 @ ( funcSet @ X4 @ X5 ) )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( ( ap @ X4 @ X5 @ X29 @ X2 )
= ( ap @ X4 @ X5 @ X29 @ X2 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',ap2apEq1) ).
thf(funcext2,axiom,
( funcext2
<=> ! [X4: $i,X5: $i,X29: $i] :
( ( in @ X29 @ ( funcSet @ X4 @ X5 ) )
=> ! [X35: $i] :
( ( in @ X35 @ ( funcSet @ X4 @ X5 ) )
=> ( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( ( ap @ X4 @ X5 @ X29 @ X2 )
= ( ap @ X4 @ X5 @ X35 @ X2 ) ) )
=> ( X29 = X35 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',funcext2) ).
thf(funcGraphProp4,axiom,
( funcGraphProp4
<=> ! [X4: $i,X5: $i,X29: $i] :
( ( in @ X29 @ ( funcSet @ X4 @ X5 ) )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( in @ ( kpair @ X2 @ X3 ) @ X29 )
=> ( ( ap @ X4 @ X5 @ X29 @ X2 )
= X3 ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',funcGraphProp4) ).
thf(funcGraphProp3,axiom,
( funcGraphProp3
<=> ! [X4: $i,X5: $i,X29: $i] :
( ( in @ X29 @ ( funcSet @ X4 @ X5 ) )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( kpair @ X2 @ ( ap @ X4 @ X5 @ X29 @ X2 ) ) @ X29 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',funcGraphProp3) ).
thf(funcinfuncset,axiom,
( funcinfuncset
<=> ! [X4: $i,X5: $i,X29: $i] :
( ( func @ X4 @ X5 @ X29 )
=> ( in @ X29 @ ( funcSet @ X4 @ X5 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',funcinfuncset) ).
thf(ap2p,axiom,
( ap2p
<=> ! [X4: $i,X5: $i,X29: $i] :
( ( in @ X29 @ ( funcSet @ X4 @ X5 ) )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( ap @ X4 @ X5 @ X29 @ X2 ) @ X5 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',ap2p) ).
thf(infuncsetfunc,axiom,
( infuncsetfunc
<=> ! [X4: $i,X5: $i,X29: $i] :
( ( in @ X29 @ ( funcSet @ X4 @ X5 ) )
=> ( func @ X4 @ X5 @ X29 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p',infuncsetfunc) ).
thf(c_0_47,plain,
( funcImageSingleton
<=> ! [X4: $i,X5: $i,X29: $i] :
( ( func @ X4 @ X5 @ X29 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( singleton
@ ( dsetconstr @ X5
@ ^ [Z0: $i] : ( in @ ( kpair @ X2 @ Z0 ) @ X29 ) ) ) ) ) ),
inference(fof_simplification,[status(thm)],[funcImageSingleton]) ).
thf(c_0_48,plain,
( singleton
= ( ^ [Z0: $i] :
? [X2: $i] :
( ( in @ X2 @ Z0 )
& ( Z0
= ( setadjoin @ X2 @ emptyset ) ) ) ) ),
inference(fof_simplification,[status(thm)],[singleton]) ).
thf(c_0_49,plain,
( singletonprop
<=> ! [X4: $i,X1: $i > $o] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X4 )
=> ( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ? [X2: $i] :
( ( in @ X2 @ X4 )
& ( X1 @ X2 ) )
=> ( singleton
@ ( dsetconstr @ X4
@ ^ [Z0: $i] : ( X1 @ Z0 ) ) ) ) ) ),
inference(fof_simplification,[status(thm)],[singletonprop]) ).
thf(c_0_50,plain,
( ksndsingleton
<=> ! [X18: $i] :
( ( iskpair @ X18 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X18 )
@ ^ [Z0: $i] :
( X18
= ( kpair @ ( kfst @ X18 ) @ Z0 ) ) ) ) ) ),
inference(fof_simplification,[status(thm)],[ksndsingleton]) ).
thf(c_0_51,plain,
( kfstsingleton
<=> ! [X18: $i] :
( ( iskpair @ X18 )
=> ( singleton
@ ( dsetconstr @ ( setunion @ X18 )
@ ^ [Z0: $i] : ( in @ ( setadjoin @ Z0 @ emptyset ) @ X18 ) ) ) ) ),
inference(fof_simplification,[status(thm)],[kfstsingleton]) ).
thf(c_0_52,plain,
( breln
= ( ^ [Z0: $i,Z1: $i,Z2: $i] : ( subset @ Z2 @ ( cartprod @ Z0 @ Z1 ) ) ) ),
inference(fof_simplification,[status(thm)],[breln]) ).
thf(c_0_53,plain,
( setOfPairsIsBReln
<=> ! [X4: $i,X5: $i,X24: $i > $i > $o] :
( breln @ X4 @ X5
@ ( dpsetconstr @ X4 @ X5
@ ^ [Z0: $i,Z1: $i] : ( X24 @ Z0 @ Z1 ) ) ) ),
inference(fof_simplification,[status(thm)],[setOfPairsIsBReln]) ).
thf(c_0_54,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ! [X4: $i,X5: $i,X39: $i > $i] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( X39 @ X2 ) @ X5 ) )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( ( ap @ X4 @ X5
@ ( lam @ X4 @ X5
@ ^ [X3: $i] : ( X39 @ X3 ) )
@ X2 )
= ( X39 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(assume_negation,[status(cth)],[beta2]) ).
thf(c_0_55,plain,
( funcImageSingleton
= ( ! [X4: $i,X5: $i,X29: $i] :
( ( func @ X4 @ X5 @ X29 )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ? [X219: $i] :
( ( in @ X219
@ ( dsetconstr @ X5
@ ^ [Z0: $i] : ( in @ ( kpair @ X2 @ Z0 ) @ X29 ) ) )
& ( ( dsetconstr @ X5
@ ^ [Z0: $i] : ( in @ ( kpair @ X2 @ Z0 ) @ X29 ) )
= ( setadjoin @ X219 @ emptyset ) ) ) ) ) ) ),
inference(apply_def,[status(thm)],[c_0_47,c_0_48]) ).
thf(c_0_56,axiom,
( theprop
= ( ! [X21: $i] :
( ? [X217: $i] :
( ( in @ X217 @ X21 )
& ( X21
= ( setadjoin @ X217 @ emptyset ) ) )
=> ( in @ ( setunion @ X21 ) @ X21 ) ) ) ),
inference(apply_def,[status(thm)],[theprop,c_0_48]) ).
thf(c_0_57,plain,
( singletonprop
= ( ! [X4: $i,X1: $i > $o] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X4 )
=> ( ( X1 @ X2 )
=> ( ( X1 @ X3 )
=> ( X2 = X3 ) ) ) ) )
=> ( ? [X2: $i] :
( ( in @ X2 @ X4 )
& ( X1 @ X2 ) )
=> ? [X214: $i] :
( ( in @ X214
@ ( dsetconstr @ X4
@ ^ [Z0: $i] : ( X1 @ Z0 ) ) )
& ( ( dsetconstr @ X4
@ ^ [Z0: $i] : ( X1 @ Z0 ) )
= ( setadjoin @ X214 @ emptyset ) ) ) ) ) ) ),
inference(apply_def,[status(thm)],[c_0_49,c_0_48]) ).
thf(c_0_58,plain,
( ksndsingleton
= ( ! [X18: $i] :
( ( iskpair @ X18 )
=> ? [X218: $i] :
( ( in @ X218
@ ( dsetconstr @ ( setunion @ X18 )
@ ^ [Z0: $i] :
( X18
= ( kpair @ ( kfst @ X18 ) @ Z0 ) ) ) )
& ( ( dsetconstr @ ( setunion @ X18 )
@ ^ [Z0: $i] :
( X18
= ( kpair @ ( kfst @ X18 ) @ Z0 ) ) )
= ( setadjoin @ X218 @ emptyset ) ) ) ) ) ),
inference(apply_def,[status(thm)],[c_0_50,c_0_48]) ).
thf(c_0_59,plain,
( kfstsingleton
= ( ! [X18: $i] :
( ( iskpair @ X18 )
=> ? [X216: $i] :
( ( in @ X216
@ ( dsetconstr @ ( setunion @ X18 )
@ ^ [Z0: $i] : ( in @ ( setadjoin @ Z0 @ emptyset ) @ X18 ) ) )
& ( ( dsetconstr @ ( setunion @ X18 )
@ ^ [Z0: $i] : ( in @ ( setadjoin @ Z0 @ emptyset ) @ X18 ) )
= ( setadjoin @ X216 @ emptyset ) ) ) ) ) ),
inference(apply_def,[status(thm)],[c_0_51,c_0_48]) ).
thf(c_0_60,axiom,
( eqbreln
= ( ! [X4: $i,X5: $i,X34: $i] :
( ( subset @ X34 @ ( cartprod @ X4 @ X5 ) )
=> ! [X36: $i] :
( ( subset @ X36 @ ( cartprod @ X4 @ X5 ) )
=> ( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( in @ ( kpair @ X2 @ X3 ) @ X34 )
=> ( in @ ( kpair @ X2 @ X3 ) @ X36 ) ) ) )
=> ( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( in @ ( kpair @ X2 @ X3 ) @ X36 )
=> ( in @ ( kpair @ X2 @ X3 ) @ X34 ) ) ) )
=> ( X34 = X36 ) ) ) ) ) ) ),
inference(apply_def,[status(thm)],[eqbreln,c_0_52]) ).
thf(c_0_61,axiom,
( subbreln
= ( ! [X4: $i,X5: $i,X34: $i] :
( ( subset @ X34 @ ( cartprod @ X4 @ X5 ) )
=> ! [X36: $i] :
( ( subset @ X36 @ ( cartprod @ X4 @ X5 ) )
=> ( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( in @ ( kpair @ X2 @ X3 ) @ X34 )
=> ( in @ ( kpair @ X2 @ X3 ) @ X36 ) ) ) )
=> ( subset @ X34 @ X36 ) ) ) ) ) ),
inference(apply_def,[status(thm)],[subbreln,c_0_52]) ).
thf(c_0_62,axiom,
( brelnall2
= ( ! [X4: $i,X5: $i,X34: $i] :
( ( subset @ X34 @ ( cartprod @ X4 @ X5 ) )
=> ! [X1: $i > $o] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( in @ ( kpair @ X2 @ X3 ) @ X34 )
=> ( X1 @ ( kpair @ X2 @ X3 ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ X34 )
=> ( X1 @ X2 ) ) ) ) ) ),
inference(apply_def,[status(thm)],[brelnall2,c_0_52]) ).
thf(c_0_63,axiom,
( brelnall1
= ( ! [X4: $i,X5: $i,X34: $i] :
( ( subset @ X34 @ ( cartprod @ X4 @ X5 ) )
=> ! [X1: $i > $o] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( in @ ( kpair @ X2 @ X3 ) @ X34 )
=> ( X1 @ ( kpair @ X2 @ X3 ) ) ) ) )
=> ! [X2: $i] :
( ( in @ X2 @ X34 )
=> ( X1 @ X2 ) ) ) ) ) ),
inference(apply_def,[status(thm)],[brelnall1,c_0_52]) ).
thf(c_0_64,plain,
( lamProp
<=> ! [X4: $i,X5: $i,X30: $i > $i] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( X30 @ X2 ) @ X5 ) )
=> ( func @ X4 @ X5
@ ( dpsetconstr @ X4 @ X5
@ ^ [Z0: $i,Z1: $i] :
( ( X30 @ Z0 )
= Z1 ) ) ) ) ),
inference(fof_simplification,[status(thm)],[lamProp]) ).
thf(c_0_65,plain,
( dpsetconstrER
<=> ! [X4: $i,X5: $i,X28: $i > $i > $o,X2: $i,X3: $i] :
( ( in @ ( kpair @ X2 @ X3 )
@ ( dpsetconstr @ X4 @ X5
@ ^ [Z0: $i,Z1: $i] : ( X28 @ Z0 @ Z1 ) ) )
=> ( X28 @ X2 @ X3 ) ) ),
inference(fof_simplification,[status(thm)],[dpsetconstrER]) ).
thf(c_0_66,plain,
( dpsetconstrEL2
<=> ! [X4: $i,X5: $i,X27: $i > $i > $o,X2: $i,X3: $i] :
( ( in @ ( kpair @ X2 @ X3 )
@ ( dpsetconstr @ X4 @ X5
@ ^ [Z0: $i,Z1: $i] : ( X27 @ Z0 @ Z1 ) ) )
=> ( in @ X3 @ X5 ) ) ),
inference(fof_simplification,[status(thm)],[dpsetconstrEL2]) ).
thf(c_0_67,plain,
( dpsetconstrEL1
<=> ! [X4: $i,X5: $i,X26: $i > $i > $o,X2: $i,X3: $i] :
( ( in @ ( kpair @ X2 @ X3 )
@ ( dpsetconstr @ X4 @ X5
@ ^ [Z0: $i,Z1: $i] : ( X26 @ Z0 @ Z1 ) ) )
=> ( in @ X2 @ X4 ) ) ),
inference(fof_simplification,[status(thm)],[dpsetconstrEL1]) ).
thf(c_0_68,plain,
( dpsetconstrERa
<=> ! [X4: $i,X5: $i,X25: $i > $i > $o,X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( in @ ( kpair @ X2 @ X3 )
@ ( dpsetconstr @ X4 @ X5
@ ^ [Z0: $i,Z1: $i] : ( X25 @ Z0 @ Z1 ) ) )
=> ( X25 @ X2 @ X3 ) ) ) ) ),
inference(fof_simplification,[status(thm)],[dpsetconstrERa]) ).
thf(c_0_69,plain,
( setOfPairsIsBReln
= ( ! [X4: $i,X5: $i,X24: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X4 @ X5
@ ^ [Z0: $i,Z1: $i] : ( X24 @ Z0 @ Z1 ) )
@ ( cartprod @ X4 @ X5 ) ) ) ),
inference(apply_def,[status(thm)],[c_0_53,c_0_52]) ).
thf(c_0_70,plain,
( dpsetconstrSub
<=> ! [X4: $i,X5: $i,X23: $i > $i > $o] :
( subset
@ ( dpsetconstr @ X4 @ X5
@ ^ [Z0: $i,Z1: $i] : ( X23 @ Z0 @ Z1 ) )
@ ( cartprod @ X4 @ X5 ) ) ),
inference(fof_simplification,[status(thm)],[dpsetconstrSub]) ).
thf(c_0_71,plain,
( dpsetconstrI
<=> ! [X4: $i,X5: $i,X22: $i > $i > $o,X2: $i] :
( ( in @ X2 @ X4 )
=> ! [X3: $i] :
( ( in @ X3 @ X5 )
=> ( ( X22 @ X2 @ X3 )
=> ( in @ ( kpair @ X2 @ X3 )
@ ( dpsetconstr @ X4 @ X5
@ ^ [Z0: $i,Z1: $i] : ( X22 @ Z0 @ Z1 ) ) ) ) ) ) ),
inference(fof_simplification,[status(thm)],[dpsetconstrI]) ).
thf(c_0_72,plain,
( lam2lamEq
<=> ! [X4: $i,X5: $i,X38: $i > $i] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( X38 @ X2 ) @ X5 ) )
=> ( ( lam @ X4 @ X5
@ ^ [Z0: $i] : ( X38 @ Z0 ) )
= ( lam @ X4 @ X5
@ ^ [Z0: $i] : ( X38 @ Z0 ) ) ) ) ),
inference(fof_simplification,[status(thm)],[lam2lamEq]) ).
thf(c_0_73,plain,
( eta1
<=> ! [X4: $i,X5: $i,X29: $i] :
( ( func @ X4 @ X5 @ X29 )
=> ( ( lam @ X4 @ X5
@ ^ [Z0: $i] : ( ap @ X4 @ X5 @ X29 @ Z0 ) )
= X29 ) ) ),
inference(fof_simplification,[status(thm)],[eta1]) ).
thf(c_0_74,plain,
( beta1
<=> ! [X4: $i,X5: $i,X37: $i > $i] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( X37 @ X2 ) @ X5 ) )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( ( ap @ X4 @ X5
@ ( lam @ X4 @ X5
@ ^ [Z0: $i] : ( X37 @ Z0 ) )
@ X2 )
= ( X37 @ X2 ) ) ) ) ),
inference(fof_simplification,[status(thm)],[beta1]) ).
thf(c_0_75,plain,
( lam2p
<=> ! [X4: $i,X5: $i,X33: $i > $i] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( X33 @ X2 ) @ X5 ) )
=> ( in
@ ( lam @ X4 @ X5
@ ^ [Z0: $i] : ( X33 @ Z0 ) )
@ ( funcSet @ X4 @ X5 ) ) ) ),
inference(fof_simplification,[status(thm)],[lam2p]) ).
thf(c_0_76,plain,
( lamp
<=> ! [X4: $i,X5: $i,X32: $i > $i] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( X32 @ X2 ) @ X5 ) )
=> ( func @ X4 @ X5
@ ( lam @ X4 @ X5
@ ^ [Z0: $i] : ( X32 @ Z0 ) ) ) ) ),
inference(fof_simplification,[status(thm)],[lamp]) ).
thf(c_0_77,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( ! [X220: $i,X221: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ X220 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X220 @ ( setadjoin @ X221 @ emptyset ) ) @ emptyset ) ) )
=> ( ! [X222: $i,X223: $i] : ( iskpair @ ( kpair @ X222 @ X223 ) )
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( ! [X224: $i,X225: $i,X226: $i] :
( ( in @ X226 @ X224 )
=> ! [X227: $i] :
( ( in @ X227 @ X225 )
=> ( in @ ( kpair @ X226 @ X227 ) @ ( cartprod @ X224 @ X225 ) ) ) )
=> ( ! [X228: $i,X229: $i,X230: $i] :
( ( in @ X230 @ ( cartprod @ X228 @ X229 ) )
=> ? [X231: $i] :
( ( in @ X231 @ X228 )
& ? [X232: $i] :
( ( in @ X232 @ X229 )
& ( X230
= ( kpair @ X231 @ X232 ) ) ) ) )
=> ( ! [X233: $i,X234: $i,X235: $i] :
( ( in @ X235 @ ( cartprod @ X233 @ X234 ) )
=> ( iskpair @ X235 ) )
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( ! [X236: $i,X237: $i > $o] :
( ! [X238: $i] :
( ( in @ X238 @ X236 )
=> ! [X239: $i] :
( ( in @ X239 @ X236 )
=> ( ( X237 @ X238 )
=> ( ( X237 @ X239 )
=> ( X238 = X239 ) ) ) ) )
=> ( ? [X240: $i] :
( ( in @ X240 @ X236 )
& ( X237 @ X240 ) )
=> ? [X241: $i] :
( ( in @ X241 @ ( dsetconstr @ X236 @ X237 ) )
& ( ( dsetconstr @ X236 @ X237 )
= ( setadjoin @ X241 @ emptyset ) ) ) ) )
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( ! [X242: $i] :
( ( iskpair @ X242 )
=> ? [X243: $i] :
( ( in @ X243
@ ( dsetconstr @ ( setunion @ X242 )
@ ^ [Z0: $i] : ( in @ ( setadjoin @ Z0 @ emptyset ) @ X242 ) ) )
& ( ( dsetconstr @ ( setunion @ X242 )
@ ^ [Z0: $i] : ( in @ ( setadjoin @ Z0 @ emptyset ) @ X242 ) )
= ( setadjoin @ X243 @ emptyset ) ) ) )
=> ( ! [X244: $i] :
( ? [X245: $i] :
( ( in @ X245 @ X244 )
& ( X244
= ( setadjoin @ X245 @ emptyset ) ) )
=> ( in @ ( setunion @ X244 ) @ X244 ) )
=> ( ! [X246: $i,X247: $i] :
( ( kfst @ ( kpair @ X246 @ X247 ) )
= X246 )
=> ( ! [X248: $i,X249: $i,X250: $i] :
( ( in @ X250 @ ( cartprod @ X248 @ X249 ) )
=> ( in @ ( kfst @ X250 ) @ X248 ) )
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ! [X251: $i] :
( ( iskpair @ X251 )
=> ? [X252: $i] :
( ( in @ X252
@ ( dsetconstr @ ( setunion @ X251 )
@ ^ [Z0: $i] :
( X251
= ( kpair @ ( kfst @ X251 ) @ Z0 ) ) ) )
& ( ( dsetconstr @ ( setunion @ X251 )
@ ^ [Z0: $i] :
( X251
= ( kpair @ ( kfst @ X251 ) @ Z0 ) ) )
= ( setadjoin @ X252 @ emptyset ) ) ) )
=> ( ! [X253: $i,X254: $i] :
( ( ksnd @ ( kpair @ X253 @ X254 ) )
= X254 )
=> ( ! [X255: $i] :
( ( iskpair @ X255 )
=> ( ( kpair @ ( kfst @ X255 ) @ ( ksnd @ X255 ) )
= X255 ) )
=> ( ! [X256: $i,X257: $i,X258: $i] :
( ( in @ X258 @ ( cartprod @ X256 @ X257 ) )
=> ( in @ ( ksnd @ X258 ) @ X257 ) )
=> ( ! [X259: $i,X260: $i,X261: $i,X262: $i] :
( ( in @ ( kpair @ X261 @ X262 ) @ ( cartprod @ X259 @ X260 ) )
=> ( in @ X261 @ X259 ) )
=> ( ! [X263: $i,X264: $i,X265: $i,X266: $i] :
( ( in @ ( kpair @ X265 @ X266 ) @ ( cartprod @ X263 @ X264 ) )
=> ( in @ X266 @ X264 ) )
=> ( cartprodmempaircEq
=> ( ! [X267: $i,X268: $i,X269: $i] :
( ( in @ X269 @ X267 )
=> ! [X270: $i] :
( ( in @ X270 @ X268 )
=> ( ( kfst @ ( kpair @ X269 @ X270 ) )
= X269 ) ) )
=> ( ! [X271: $i,X272: $i,X273: $i] :
( ( in @ X273 @ X271 )
=> ! [X274: $i] :
( ( in @ X274 @ X272 )
=> ( ( ksnd @ ( kpair @ X273 @ X274 ) )
= X274 ) ) )
=> ( ! [X275: $i,X276: $i,X277: $i] :
( ( in @ X277 @ ( cartprod @ X275 @ X276 ) )
=> ( ( kpair @ ( kfst @ X277 ) @ ( ksnd @ X277 ) )
= X277 ) )
=> ( ! [X278: $i,X279: $i,X280: $i > $i > $o,X281: $i] :
( ( in @ X281 @ X278 )
=> ! [X282: $i] :
( ( in @ X282 @ X279 )
=> ( ( X280 @ X281 @ X282 )
=> ( in @ ( kpair @ X281 @ X282 ) @ ( dpsetconstr @ X278 @ X279 @ X280 ) ) ) ) )
=> ( ! [X283: $i,X284: $i,X285: $i > $i > $o] : ( subset @ ( dpsetconstr @ X283 @ X284 @ X285 ) @ ( cartprod @ X283 @ X284 ) )
=> ( ! [X286: $i,X287: $i,X288: $i > $i > $o] : ( subset @ ( dpsetconstr @ X286 @ X287 @ X288 ) @ ( cartprod @ X286 @ X287 ) )
=> ( ! [X289: $i,X290: $i,X291: $i > $i > $o,X292: $i] :
( ( in @ X292 @ X289 )
=> ! [X293: $i] :
( ( in @ X293 @ X290 )
=> ( ( in @ ( kpair @ X292 @ X293 ) @ ( dpsetconstr @ X289 @ X290 @ X291 ) )
=> ( X291 @ X292 @ X293 ) ) ) )
=> ( ! [X294: $i,X295: $i,X296: $i > $i > $o,X297: $i,X298: $i] :
( ( in @ ( kpair @ X297 @ X298 ) @ ( dpsetconstr @ X294 @ X295 @ X296 ) )
=> ( in @ X297 @ X294 ) )
=> ( ! [X299: $i,X300: $i,X301: $i > $i > $o,X302: $i,X303: $i] :
( ( in @ ( kpair @ X302 @ X303 ) @ ( dpsetconstr @ X299 @ X300 @ X301 ) )
=> ( in @ X303 @ X300 ) )
=> ( ! [X304: $i,X305: $i,X306: $i > $i > $o,X307: $i,X308: $i] :
( ( in @ ( kpair @ X307 @ X308 ) @ ( dpsetconstr @ X304 @ X305 @ X306 ) )
=> ( X306 @ X307 @ X308 ) )
=> ( ! [X309: $i,X310: $i,X311: $i] :
( ( func @ X309 @ X310 @ X311 )
=> ! [X312: $i] :
( ( in @ X312 @ X309 )
=> ? [X313: $i] :
( ( in @ X313
@ ( dsetconstr @ X310
@ ^ [Z0: $i] : ( in @ ( kpair @ X312 @ Z0 ) @ X311 ) ) )
& ( ( dsetconstr @ X310
@ ^ [Z0: $i] : ( in @ ( kpair @ X312 @ Z0 ) @ X311 ) )
= ( setadjoin @ X313 @ emptyset ) ) ) ) )
=> ( apProp
=> ( app
=> ( ! [X314: $i,X315: $i,X316: $i] :
( ( in @ X316 @ ( funcSet @ X314 @ X315 ) )
=> ( func @ X314 @ X315 @ X316 ) )
=> ( ! [X317: $i,X318: $i,X319: $i] :
( ( in @ X319 @ ( funcSet @ X317 @ X318 ) )
=> ! [X320: $i] :
( ( in @ X320 @ X317 )
=> ( in @ ( ap @ X317 @ X318 @ X319 @ X320 ) @ X318 ) ) )
=> ( ! [X321: $i,X322: $i,X323: $i] :
( ( func @ X321 @ X322 @ X323 )
=> ( in @ X323 @ ( funcSet @ X321 @ X322 ) ) )
=> ( ! [X324: $i,X325: $i,X326: $i > $i] :
( ! [X327: $i] :
( ( in @ X327 @ X324 )
=> ( in @ ( X326 @ X327 ) @ X325 ) )
=> ( func @ X324 @ X325
@ ( dpsetconstr @ X324 @ X325
@ ^ [Z0: $i] : ( $eq @ ( X326 @ Z0 ) ) ) ) )
=> ( ! [X328: $i,X329: $i,X330: $i > $i] :
( ! [X331: $i] :
( ( in @ X331 @ X328 )
=> ( in @ ( X330 @ X331 ) @ X329 ) )
=> ( func @ X328 @ X329 @ ( lam @ X328 @ X329 @ X330 ) ) )
=> ( ! [X332: $i,X333: $i,X334: $i > $i] :
( ! [X335: $i] :
( ( in @ X335 @ X332 )
=> ( in @ ( X334 @ X335 ) @ X333 ) )
=> ( in @ ( lam @ X332 @ X333 @ X334 ) @ ( funcSet @ X332 @ X333 ) ) )
=> ( ! [X336: $i,X337: $i,X338: $i] :
( ( subset @ X338 @ ( cartprod @ X336 @ X337 ) )
=> ! [X339: $i > $o] :
( ! [X340: $i] :
( ( in @ X340 @ X336 )
=> ! [X341: $i] :
( ( in @ X341 @ X337 )
=> ( ( in @ ( kpair @ X340 @ X341 ) @ X338 )
=> ( X339 @ ( kpair @ X340 @ X341 ) ) ) ) )
=> ! [X342: $i] :
( ( in @ X342 @ X338 )
=> ( X339 @ X342 ) ) ) )
=> ( ! [X343: $i,X344: $i,X345: $i] :
( ( subset @ X345 @ ( cartprod @ X343 @ X344 ) )
=> ! [X346: $i > $o] :
( ! [X347: $i] :
( ( in @ X347 @ X343 )
=> ! [X348: $i] :
( ( in @ X348 @ X344 )
=> ( ( in @ ( kpair @ X347 @ X348 ) @ X345 )
=> ( X346 @ ( kpair @ X347 @ X348 ) ) ) ) )
=> ! [X349: $i] :
( ( in @ X349 @ X345 )
=> ( X346 @ X349 ) ) ) )
=> ( ex1E2
=> ( funcGraphProp1
=> ( ! [X350: $i,X351: $i,X352: $i] :
( ( in @ X352 @ ( funcSet @ X350 @ X351 ) )
=> ! [X353: $i] :
( ( in @ X353 @ X350 )
=> ( in @ ( kpair @ X353 @ ( ap @ X350 @ X351 @ X352 @ X353 ) ) @ X352 ) ) )
=> ( funcGraphProp2
=> ( funcextLem
=> ( ! [X354: $i,X355: $i,X356: $i] :
( ( in @ X356 @ ( funcSet @ X354 @ X355 ) )
=> ! [X357: $i] :
( ( in @ X357 @ X354 )
=> ! [X358: $i] :
( ( in @ X358 @ X355 )
=> ( ( in @ ( kpair @ X357 @ X358 ) @ X356 )
=> ( ( ap @ X354 @ X355 @ X356 @ X357 )
= X358 ) ) ) ) )
=> ( ! [X359: $i,X360: $i,X361: $i] :
( ( subset @ X361 @ ( cartprod @ X359 @ X360 ) )
=> ! [X362: $i] :
( ( subset @ X362 @ ( cartprod @ X359 @ X360 ) )
=> ( ! [X363: $i] :
( ( in @ X363 @ X359 )
=> ! [X364: $i] :
( ( in @ X364 @ X360 )
=> ( ( in @ ( kpair @ X363 @ X364 ) @ X361 )
=> ( in @ ( kpair @ X363 @ X364 ) @ X362 ) ) ) )
=> ( subset @ X361 @ X362 ) ) ) )
=> ( ! [X365: $i,X366: $i,X367: $i] :
( ( subset @ X367 @ ( cartprod @ X365 @ X366 ) )
=> ! [X368: $i] :
( ( subset @ X368 @ ( cartprod @ X365 @ X366 ) )
=> ( ! [X369: $i] :
( ( in @ X369 @ X365 )
=> ! [X370: $i] :
( ( in @ X370 @ X366 )
=> ( ( in @ ( kpair @ X369 @ X370 ) @ X367 )
=> ( in @ ( kpair @ X369 @ X370 ) @ X368 ) ) ) )
=> ( ! [X371: $i] :
( ( in @ X371 @ X365 )
=> ! [X372: $i] :
( ( in @ X372 @ X366 )
=> ( ( in @ ( kpair @ X371 @ X372 ) @ X368 )
=> ( in @ ( kpair @ X371 @ X372 ) @ X367 ) ) ) )
=> ( X367 = X368 ) ) ) ) )
=> ( funcext
=> ( ! [X373: $i,X374: $i,X375: $i] :
( ( in @ X375 @ ( funcSet @ X373 @ X374 ) )
=> ! [X376: $i] :
( ( in @ X376 @ ( funcSet @ X373 @ X374 ) )
=> ( ! [X377: $i] :
( ( in @ X377 @ X373 )
=> ( ( ap @ X373 @ X374 @ X375 @ X377 )
= ( ap @ X373 @ X374 @ X376 @ X377 ) ) )
=> ( X375 = X376 ) ) ) )
=> ( ! [X378: $i,X379: $i,X380: $i] :
( ( in @ X380 @ ( funcSet @ X378 @ X379 ) )
=> ! [X381: $i] :
( ( in @ X381 @ X378 )
=> ( ( ap @ X378 @ X379 @ X380 @ X381 )
= ( ap @ X378 @ X379 @ X380 @ X381 ) ) ) )
=> ( ap2apEq2
=> ( ! [X382: $i,X383: $i,X384: $i > $i] :
( ! [X385: $i] :
( ( in @ X385 @ X382 )
=> ( in @ ( X384 @ X385 ) @ X383 ) )
=> ! [X386: $i] :
( ( in @ X386 @ X382 )
=> ( ( ap @ X382 @ X383 @ ( lam @ X382 @ X383 @ X384 ) @ X386 )
= ( X384 @ X386 ) ) ) )
=> ( ! [X387: $i,X388: $i,X389: $i] :
( ( func @ X387 @ X388 @ X389 )
=> ( ( lam @ X387 @ X388 @ ( ap @ X387 @ X388 @ X389 ) )
= X389 ) )
=> ( ! [X390: $i,X391: $i,X392: $i > $i] :
( ! [X393: $i] :
( ( in @ X393 @ X390 )
=> ( in @ ( X392 @ X393 ) @ X391 ) )
=> ( ( lam @ X390 @ X391 @ X392 )
= ( lam @ X390 @ X391 @ X392 ) ) )
=> ! [X4: $i,X5: $i,X39: $i > $i] :
( ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( in @ ( X39 @ X2 ) @ X5 ) )
=> ! [X2: $i] :
( ( in @ X2 @ X4 )
=> ( ( ap @ X4 @ X5 @ ( lam @ X4 @ X5 @ X39 ) @ X2 )
= ( X39 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_unroll,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(apply_def,[status(thm)],[inference(fof_simplification,[status(thm)],[c_0_54]),c_0_55]),c_0_56]),c_0_57]),cartprodsndpairEq]),ksndpairEq]),cartprodfstpairEq]),kfstpairEq]),kpairsurjEq]),c_0_58]),c_0_59]),kpairp]),kpairiskpair]),c_0_60]),c_0_61]),c_0_62]),c_0_63]),cartprodpairsurjEq]),cartprodpairmemER]),cartprodpairmemEL]),cartprodsndin]),cartprodfstin]),cartprodmempair]),cartprodmempair1]),cartprodpairin]),ap2apEq1]),funcext2]),funcGraphProp4]),funcGraphProp3]),funcinfuncset]),ap2p]),infuncsetfunc]),c_0_64]),c_0_65]),c_0_66]),c_0_67]),c_0_68]),c_0_69]),c_0_70]),c_0_71]),c_0_72]),c_0_73]),c_0_74]),c_0_75]),c_0_76])]) ).
thf(c_0_78,negated_conjecture,
! [X397: $i,X398: $i,X399: $i,X400: $i,X401: $i,X402: $i,X403: $i,X404: $i,X405: $i,X406: $i,X407: $i,X410: $i,X411: $i,X412: $i,X413: $i,X414: $i > $o,X417: $i,X419: $i,X421: $i,X422: $i,X423: $i,X424: $i,X425: $i,X426: $i,X427: $i,X428: $i,X430: $i,X431: $i,X432: $i,X433: $i,X434: $i,X435: $i,X436: $i,X437: $i,X438: $i,X439: $i,X440: $i,X441: $i,X442: $i,X443: $i,X444: $i,X445: $i,X446: $i,X447: $i,X448: $i,X449: $i,X450: $i,X451: $i,X452: $i,X453: $i,X454: $i,X455: $i,X456: $i,X457: $i > $i > $o,X458: $i,X459: $i,X460: $i,X461: $i,X462: $i > $i > $o,X463: $i,X464: $i,X465: $i > $i > $o,X466: $i,X467: $i,X468: $i > $i > $o,X469: $i,X470: $i,X471: $i,X472: $i,X473: $i > $i > $o,X474: $i,X475: $i,X476: $i,X477: $i,X478: $i > $i > $o,X479: $i,X480: $i,X481: $i,X482: $i,X483: $i > $i > $o,X484: $i,X485: $i,X486: $i,X487: $i,X488: $i,X489: $i,X491: $i,X492: $i,X493: $i,X494: $i,X495: $i,X496: $i,X497: $i,X498: $i,X499: $i,X500: $i,X501: $i,X502: $i,X503: $i > $i,X505: $i,X506: $i,X507: $i > $i,X509: $i,X510: $i,X511: $i > $i,X513: $i,X514: $i,X515: $i,X516: $i > $o,X519: $i,X520: $i,X521: $i,X522: $i,X523: $i > $o,X526: $i,X527: $i,X528: $i,X529: $i,X530: $i,X531: $i,X532: $i,X533: $i,X534: $i,X535: $i,X536: $i,X537: $i,X538: $i,X539: $i,X542: $i,X543: $i,X544: $i,X545: $i,X550: $i,X551: $i,X552: $i,X553: $i,X555: $i,X556: $i,X557: $i,X558: $i,X559: $i,X560: $i,X561: $i > $i,X563: $i,X564: $i,X565: $i,X566: $i,X567: $i,X568: $i,X569: $i > $i,X574: $i] :
( setextAx
& emptysetAx
& setadjoinAx
& powersetAx
& setunionAx
& omega0Ax
& omegaSAx
& omegaIndAx
& replAx
& foundationAx
& wellorderingAx
& descrp
& dsetconstrI
& dsetconstrEL
& dsetconstrER
& exuE1
& prop2setE
& emptysetE
& emptysetimpfalse
& notinemptyset
& exuE3e
& setext
& emptyI
& noeltsimpempty
& setbeta
& nonemptyE1
& nonemptyI
& nonemptyI1
& setadjoinIL
& emptyinunitempty
& setadjoinIR
& setadjoinE
& setadjoinOr
& setoftrueEq
& powersetI
& emptyinPowerset
& emptyInPowerset
& powersetE
& setunionI
& setunionE
& subPowSU
& exuE2
& nonemptyImpWitness
& uniqinunit
& notinsingleton
& eqinunit
& singletonsswitch
& upairsetE
& upairsetIL
& upairsetIR
& emptyE1
& vacuousDall
& quantDeMorgan1
& quantDeMorgan2
& quantDeMorgan3
& quantDeMorgan4
& prop2setI
& prop2set2propI
& notdexE
& notdallE
& exuI1
& exuI3
& exuI2
& inCongP
& in__Cong
& exuE3u
& exu__Cong
& emptyset__Cong
& setadjoin__Cong
& powerset__Cong
& setunion__Cong
& omega__Cong
& exuEu
& descr__Cong
& dsetconstr__Cong
& subsetI1
& eqimpsubset2
& eqimpsubset1
& subsetI2
& emptysetsubset
& subsetE
& subsetE2
& notsubsetI
& notequalI1
& notequalI2
& subsetRefl
& subsetTrans
& setadjoinSub
& setadjoinSub2
& subset2powerset
& setextsub
& subsetemptysetimpeq
& powersetI1
& powersetE1
& inPowerset
& powersetsubset
& sepInPowerset
& sepSubset
& binunionIL
& upairset2IR
& binunionIR
& binunionEcases
& binunionE
& binunionLsub
& binunionRsub
& binintersectI
& binintersectSubset5
& binintersectEL
& binintersectLsub
& binintersectSubset2
& binintersectSubset3
& binintersectER
& disjointsetsI1
& binintersectRsub
& binintersectSubset4
& binintersectSubset1
& bs114d
& setminusI
& setminusEL
& setminusER
& setminusSubset2
& setminusERneg
& setminusELneg
& setminusILneg
& setminusIRneg
& setminusLsub
& setminusSubset1
& symdiffE
& symdiffI1
& symdiffI2
& symdiffIneg1
& symdiffIneg2
& secondinupair
& setukpairIL
& setukpairIR
& ( iskpair @ ( setadjoin @ ( setadjoin @ X397 @ emptyset ) @ ( setadjoin @ ( setadjoin @ X397 @ ( setadjoin @ X398 @ emptyset ) ) @ emptyset ) ) )
& ( iskpair @ ( kpair @ X399 @ X400 ) )
& singletonsubset
& singletoninpowerset
& singletoninpowunion
& upairset2E
& upairsubunion
& upairinpowunion
& ubforcartprodlem1
& ubforcartprodlem2
& ubforcartprodlem3
& ( ~ ( in @ X403 @ X401 )
| ~ ( in @ X404 @ X402 )
| ( in @ ( kpair @ X403 @ X404 ) @ ( cartprod @ X401 @ X402 ) ) )
& ( ( in @ ( esk1_3 @ X405 @ X406 @ X407 ) @ X405 )
| ~ ( in @ X407 @ ( cartprod @ X405 @ X406 ) ) )
& ( ( in @ ( esk2_3 @ X405 @ X406 @ X407 ) @ X406 )
| ~ ( in @ X407 @ ( cartprod @ X405 @ X406 ) ) )
& ( ( X407
= ( kpair @ ( esk1_3 @ X405 @ X406 @ X407 ) @ ( esk2_3 @ X405 @ X406 @ X407 ) ) )
| ~ ( in @ X407 @ ( cartprod @ X405 @ X406 ) ) )
& ( ~ ( in @ X412 @ ( cartprod @ X410 @ X411 ) )
| ( iskpair @ X412 ) )
& setunionE2
& setunionsingleton1
& setunionsingleton2
& setunionsingleton
& ( ( in @ ( esk5_2 @ X413 @ X414 ) @ ( dsetconstr @ X413 @ X414 ) )
| ~ ( in @ X417 @ X413 )
| ~ ( X414 @ X417 )
| ( in @ ( esk3_2 @ X413 @ X414 ) @ X413 ) )
& ( ( ( dsetconstr @ X413 @ X414 )
= ( setadjoin @ ( esk5_2 @ X413 @ X414 ) @ emptyset ) )
| ~ ( in @ X417 @ X413 )
| ~ ( X414 @ X417 )
| ( in @ ( esk3_2 @ X413 @ X414 ) @ X413 ) )
& ( ( in @ ( esk5_2 @ X413 @ X414 ) @ ( dsetconstr @ X413 @ X414 ) )
| ~ ( in @ X417 @ X413 )
| ~ ( X414 @ X417 )
| ( in @ ( esk4_2 @ X413 @ X414 ) @ X413 ) )
& ( ( ( dsetconstr @ X413 @ X414 )
= ( setadjoin @ ( esk5_2 @ X413 @ X414 ) @ emptyset ) )
| ~ ( in @ X417 @ X413 )
| ~ ( X414 @ X417 )
| ( in @ ( esk4_2 @ X413 @ X414 ) @ X413 ) )
& ( ( in @ ( esk5_2 @ X413 @ X414 ) @ ( dsetconstr @ X413 @ X414 ) )
| ~ ( in @ X417 @ X413 )
| ~ ( X414 @ X417 )
| ( X414 @ ( esk3_2 @ X413 @ X414 ) ) )
& ( ( ( dsetconstr @ X413 @ X414 )
= ( setadjoin @ ( esk5_2 @ X413 @ X414 ) @ emptyset ) )
| ~ ( in @ X417 @ X413 )
| ~ ( X414 @ X417 )
| ( X414 @ ( esk3_2 @ X413 @ X414 ) ) )
& ( ( in @ ( esk5_2 @ X413 @ X414 ) @ ( dsetconstr @ X413 @ X414 ) )
| ~ ( in @ X417 @ X413 )
| ~ ( X414 @ X417 )
| ( X414 @ ( esk4_2 @ X413 @ X414 ) ) )
& ( ( ( dsetconstr @ X413 @ X414 )
= ( setadjoin @ ( esk5_2 @ X413 @ X414 ) @ emptyset ) )
| ~ ( in @ X417 @ X413 )
| ~ ( X414 @ X417 )
| ( X414 @ ( esk4_2 @ X413 @ X414 ) ) )
& ( ( in @ ( esk5_2 @ X413 @ X414 ) @ ( dsetconstr @ X413 @ X414 ) )
| ~ ( in @ X417 @ X413 )
| ~ ( X414 @ X417 )
| ( ( esk3_2 @ X413 @ X414 )
!= ( esk4_2 @ X413 @ X414 ) ) )
& ( ( ( dsetconstr @ X413 @ X414 )
= ( setadjoin @ ( esk5_2 @ X413 @ X414 ) @ emptyset ) )
| ~ ( in @ X417 @ X413 )
| ~ ( X414 @ X417 )
| ( ( esk3_2 @ X413 @ X414 )
!= ( esk4_2 @ X413 @ X414 ) ) )
& ex1E1
& ex1I
& ex1I2
& singletonsuniq
& setukpairinjL1
& ( ( in @ ( esk6_1 @ X419 )
@ ( dsetconstr @ ( setunion @ X419 )
@ ^ [Z0: $i] : ( in @ ( setadjoin @ Z0 @ emptyset ) @ X419 ) ) )
| ~ ( iskpair @ X419 ) )
& ( ( ( dsetconstr @ ( setunion @ X419 )
@ ^ [Z0: $i] : ( in @ ( setadjoin @ Z0 @ emptyset ) @ X419 ) )
= ( setadjoin @ ( esk6_1 @ X419 ) @ emptyset ) )
| ~ ( iskpair @ X419 ) )
& ( ~ ( in @ X422 @ X421 )
| ( X421
!= ( setadjoin @ X422 @ emptyset ) )
| ( in @ ( setunion @ X421 ) @ X421 ) )
& ( ( kfst @ ( kpair @ X423 @ X424 ) )
= X423 )
& ( ~ ( in @ X427 @ ( cartprod @ X425 @ X426 ) )
| ( in @ ( kfst @ X427 ) @ X425 ) )
& setukpairinjL2
& setukpairinjL
& setukpairinjR11
& setukpairinjR12
& setukpairinjR1
& upairequniteq
& setukpairinjR2
& setukpairinjR
& ( ( in @ ( esk7_1 @ X428 )
@ ( dsetconstr @ ( setunion @ X428 )
@ ^ [Z0: $i] :
( X428
= ( kpair @ ( kfst @ X428 ) @ Z0 ) ) ) )
| ~ ( iskpair @ X428 ) )
& ( ( ( dsetconstr @ ( setunion @ X428 )
@ ^ [Z0: $i] :
( X428
= ( kpair @ ( kfst @ X428 ) @ Z0 ) ) )
= ( setadjoin @ ( esk7_1 @ X428 ) @ emptyset ) )
| ~ ( iskpair @ X428 ) )
& ( ( ksnd @ ( kpair @ X430 @ X431 ) )
= X431 )
& ( ~ ( iskpair @ X432 )
| ( ( kpair @ ( kfst @ X432 ) @ ( ksnd @ X432 ) )
= X432 ) )
& ( ~ ( in @ X435 @ ( cartprod @ X433 @ X434 ) )
| ( in @ ( ksnd @ X435 ) @ X434 ) )
& ( ~ ( in @ ( kpair @ X438 @ X439 ) @ ( cartprod @ X436 @ X437 ) )
| ( in @ X438 @ X436 ) )
& ( ~ ( in @ ( kpair @ X442 @ X443 ) @ ( cartprod @ X440 @ X441 ) )
| ( in @ X443 @ X441 ) )
& cartprodmempaircEq
& ( ~ ( in @ X446 @ X444 )
| ~ ( in @ X447 @ X445 )
| ( ( kfst @ ( kpair @ X446 @ X447 ) )
= X446 ) )
& ( ~ ( in @ X450 @ X448 )
| ~ ( in @ X451 @ X449 )
| ( ( ksnd @ ( kpair @ X450 @ X451 ) )
= X451 ) )
& ( ~ ( in @ X454 @ ( cartprod @ X452 @ X453 ) )
| ( ( kpair @ ( kfst @ X454 ) @ ( ksnd @ X454 ) )
= X454 ) )
& ( ~ ( in @ X458 @ X455 )
| ~ ( in @ X459 @ X456 )
| ~ ( X457 @ X458 @ X459 )
| ( in @ ( kpair @ X458 @ X459 ) @ ( dpsetconstr @ X455 @ X456 @ X457 ) ) )
& ( subset @ ( dpsetconstr @ X460 @ X461 @ X462 ) @ ( cartprod @ X460 @ X461 ) )
& ( subset @ ( dpsetconstr @ X463 @ X464 @ X465 ) @ ( cartprod @ X463 @ X464 ) )
& ( ~ ( in @ X469 @ X466 )
| ~ ( in @ X470 @ X467 )
| ~ ( in @ ( kpair @ X469 @ X470 ) @ ( dpsetconstr @ X466 @ X467 @ X468 ) )
| ( X468 @ X469 @ X470 ) )
& ( ~ ( in @ ( kpair @ X474 @ X475 ) @ ( dpsetconstr @ X471 @ X472 @ X473 ) )
| ( in @ X474 @ X471 ) )
& ( ~ ( in @ ( kpair @ X479 @ X480 ) @ ( dpsetconstr @ X476 @ X477 @ X478 ) )
| ( in @ X480 @ X477 ) )
& ( ~ ( in @ ( kpair @ X484 @ X485 ) @ ( dpsetconstr @ X481 @ X482 @ X483 ) )
| ( X483 @ X484 @ X485 ) )
& ( ( in @ ( esk8_4 @ X486 @ X487 @ X488 @ X489 )
@ ( dsetconstr @ X487
@ ^ [Z0: $i] : ( in @ ( kpair @ X489 @ Z0 ) @ X488 ) ) )
| ~ ( in @ X489 @ X486 )
| ~ ( func @ X486 @ X487 @ X488 ) )
& ( ( ( dsetconstr @ X487
@ ^ [Z0: $i] : ( in @ ( kpair @ X489 @ Z0 ) @ X488 ) )
= ( setadjoin @ ( esk8_4 @ X486 @ X487 @ X488 @ X489 ) @ emptyset ) )
| ~ ( in @ X489 @ X486 )
| ~ ( func @ X486 @ X487 @ X488 ) )
& apProp
& app
& ( ~ ( in @ X493 @ ( funcSet @ X491 @ X492 ) )
| ( func @ X491 @ X492 @ X493 ) )
& ( ~ ( in @ X496 @ ( funcSet @ X494 @ X495 ) )
| ~ ( in @ X497 @ X494 )
| ( in @ ( ap @ X494 @ X495 @ X496 @ X497 ) @ X495 ) )
& ( ~ ( func @ X498 @ X499 @ X500 )
| ( in @ X500 @ ( funcSet @ X498 @ X499 ) ) )
& ( ( in @ ( esk9_3 @ X501 @ X502 @ X503 ) @ X501 )
| ( func @ X501 @ X502
@ ( dpsetconstr @ X501 @ X502
@ ^ [Z0: $i] : ( $eq @ ( X503 @ Z0 ) ) ) ) )
& ( ~ ( in @ ( X503 @ ( esk9_3 @ X501 @ X502 @ X503 ) ) @ X502 )
| ( func @ X501 @ X502
@ ( dpsetconstr @ X501 @ X502
@ ^ [Z0: $i] : ( $eq @ ( X503 @ Z0 ) ) ) ) )
& ( ( in @ ( esk10_3 @ X505 @ X506 @ X507 ) @ X505 )
| ( func @ X505 @ X506 @ ( lam @ X505 @ X506 @ X507 ) ) )
& ( ~ ( in @ ( X507 @ ( esk10_3 @ X505 @ X506 @ X507 ) ) @ X506 )
| ( func @ X505 @ X506 @ ( lam @ X505 @ X506 @ X507 ) ) )
& ( ( in @ ( esk11_3 @ X509 @ X510 @ X511 ) @ X509 )
| ( in @ ( lam @ X509 @ X510 @ X511 ) @ ( funcSet @ X509 @ X510 ) ) )
& ( ~ ( in @ ( X511 @ ( esk11_3 @ X509 @ X510 @ X511 ) ) @ X510 )
| ( in @ ( lam @ X509 @ X510 @ X511 ) @ ( funcSet @ X509 @ X510 ) ) )
& ( ( in @ ( esk12_4 @ X513 @ X514 @ X515 @ X516 ) @ X513 )
| ~ ( in @ X519 @ X515 )
| ( X516 @ X519 )
| ~ ( subset @ X515 @ ( cartprod @ X513 @ X514 ) ) )
& ( ( in @ ( esk13_4 @ X513 @ X514 @ X515 @ X516 ) @ X514 )
| ~ ( in @ X519 @ X515 )
| ( X516 @ X519 )
| ~ ( subset @ X515 @ ( cartprod @ X513 @ X514 ) ) )
& ( ( in @ ( kpair @ ( esk12_4 @ X513 @ X514 @ X515 @ X516 ) @ ( esk13_4 @ X513 @ X514 @ X515 @ X516 ) ) @ X515 )
| ~ ( in @ X519 @ X515 )
| ( X516 @ X519 )
| ~ ( subset @ X515 @ ( cartprod @ X513 @ X514 ) ) )
& ( ~ ( X516 @ ( kpair @ ( esk12_4 @ X513 @ X514 @ X515 @ X516 ) @ ( esk13_4 @ X513 @ X514 @ X515 @ X516 ) ) )
| ~ ( in @ X519 @ X515 )
| ( X516 @ X519 )
| ~ ( subset @ X515 @ ( cartprod @ X513 @ X514 ) ) )
& ( ( in @ ( esk14_4 @ X520 @ X521 @ X522 @ X523 ) @ X520 )
| ~ ( in @ X526 @ X522 )
| ( X523 @ X526 )
| ~ ( subset @ X522 @ ( cartprod @ X520 @ X521 ) ) )
& ( ( in @ ( esk15_4 @ X520 @ X521 @ X522 @ X523 ) @ X521 )
| ~ ( in @ X526 @ X522 )
| ( X523 @ X526 )
| ~ ( subset @ X522 @ ( cartprod @ X520 @ X521 ) ) )
& ( ( in @ ( kpair @ ( esk14_4 @ X520 @ X521 @ X522 @ X523 ) @ ( esk15_4 @ X520 @ X521 @ X522 @ X523 ) ) @ X522 )
| ~ ( in @ X526 @ X522 )
| ( X523 @ X526 )
| ~ ( subset @ X522 @ ( cartprod @ X520 @ X521 ) ) )
& ( ~ ( X523 @ ( kpair @ ( esk14_4 @ X520 @ X521 @ X522 @ X523 ) @ ( esk15_4 @ X520 @ X521 @ X522 @ X523 ) ) )
| ~ ( in @ X526 @ X522 )
| ( X523 @ X526 )
| ~ ( subset @ X522 @ ( cartprod @ X520 @ X521 ) ) )
& ex1E2
& funcGraphProp1
& ( ~ ( in @ X529 @ ( funcSet @ X527 @ X528 ) )
| ~ ( in @ X530 @ X527 )
| ( in @ ( kpair @ X530 @ ( ap @ X527 @ X528 @ X529 @ X530 ) ) @ X529 ) )
& funcGraphProp2
& funcextLem
& ( ~ ( in @ X533 @ ( funcSet @ X531 @ X532 ) )
| ~ ( in @ X534 @ X531 )
| ~ ( in @ X535 @ X532 )
| ~ ( in @ ( kpair @ X534 @ X535 ) @ X533 )
| ( ( ap @ X531 @ X532 @ X533 @ X534 )
= X535 ) )
& ( ( in @ ( esk16_4 @ X536 @ X537 @ X538 @ X539 ) @ X536 )
| ( subset @ X538 @ X539 )
| ~ ( subset @ X539 @ ( cartprod @ X536 @ X537 ) )
| ~ ( subset @ X538 @ ( cartprod @ X536 @ X537 ) ) )
& ( ( in @ ( esk17_4 @ X536 @ X537 @ X538 @ X539 ) @ X537 )
| ( subset @ X538 @ X539 )
| ~ ( subset @ X539 @ ( cartprod @ X536 @ X537 ) )
| ~ ( subset @ X538 @ ( cartprod @ X536 @ X537 ) ) )
& ( ( in @ ( kpair @ ( esk16_4 @ X536 @ X537 @ X538 @ X539 ) @ ( esk17_4 @ X536 @ X537 @ X538 @ X539 ) ) @ X538 )
| ( subset @ X538 @ X539 )
| ~ ( subset @ X539 @ ( cartprod @ X536 @ X537 ) )
| ~ ( subset @ X538 @ ( cartprod @ X536 @ X537 ) ) )
& ( ~ ( in @ ( kpair @ ( esk16_4 @ X536 @ X537 @ X538 @ X539 ) @ ( esk17_4 @ X536 @ X537 @ X538 @ X539 ) ) @ X539 )
| ( subset @ X538 @ X539 )
| ~ ( subset @ X539 @ ( cartprod @ X536 @ X537 ) )
| ~ ( subset @ X538 @ ( cartprod @ X536 @ X537 ) ) )
& ( ( in @ ( esk20_4 @ X542 @ X543 @ X544 @ X545 ) @ X542 )
| ( X544 = X545 )
| ( in @ ( esk18_4 @ X542 @ X543 @ X544 @ X545 ) @ X542 )
| ~ ( subset @ X545 @ ( cartprod @ X542 @ X543 ) )
| ~ ( subset @ X544 @ ( cartprod @ X542 @ X543 ) ) )
& ( ( in @ ( esk21_4 @ X542 @ X543 @ X544 @ X545 ) @ X543 )
| ( X544 = X545 )
| ( in @ ( esk18_4 @ X542 @ X543 @ X544 @ X545 ) @ X542 )
| ~ ( subset @ X545 @ ( cartprod @ X542 @ X543 ) )
| ~ ( subset @ X544 @ ( cartprod @ X542 @ X543 ) ) )
& ( ( in @ ( kpair @ ( esk20_4 @ X542 @ X543 @ X544 @ X545 ) @ ( esk21_4 @ X542 @ X543 @ X544 @ X545 ) ) @ X545 )
| ( X544 = X545 )
| ( in @ ( esk18_4 @ X542 @ X543 @ X544 @ X545 ) @ X542 )
| ~ ( subset @ X545 @ ( cartprod @ X542 @ X543 ) )
| ~ ( subset @ X544 @ ( cartprod @ X542 @ X543 ) ) )
& ( ~ ( in @ ( kpair @ ( esk20_4 @ X542 @ X543 @ X544 @ X545 ) @ ( esk21_4 @ X542 @ X543 @ X544 @ X545 ) ) @ X544 )
| ( X544 = X545 )
| ( in @ ( esk18_4 @ X542 @ X543 @ X544 @ X545 ) @ X542 )
| ~ ( subset @ X545 @ ( cartprod @ X542 @ X543 ) )
| ~ ( subset @ X544 @ ( cartprod @ X542 @ X543 ) ) )
& ( ( in @ ( esk20_4 @ X542 @ X543 @ X544 @ X545 ) @ X542 )
| ( X544 = X545 )
| ( in @ ( esk19_4 @ X542 @ X543 @ X544 @ X545 ) @ X543 )
| ~ ( subset @ X545 @ ( cartprod @ X542 @ X543 ) )
| ~ ( subset @ X544 @ ( cartprod @ X542 @ X543 ) ) )
& ( ( in @ ( esk21_4 @ X542 @ X543 @ X544 @ X545 ) @ X543 )
| ( X544 = X545 )
| ( in @ ( esk19_4 @ X542 @ X543 @ X544 @ X545 ) @ X543 )
| ~ ( subset @ X545 @ ( cartprod @ X542 @ X543 ) )
| ~ ( subset @ X544 @ ( cartprod @ X542 @ X543 ) ) )
& ( ( in @ ( kpair @ ( esk20_4 @ X542 @ X543 @ X544 @ X545 ) @ ( esk21_4 @ X542 @ X543 @ X544 @ X545 ) ) @ X545 )
| ( X544 = X545 )
| ( in @ ( esk19_4 @ X542 @ X543 @ X544 @ X545 ) @ X543 )
| ~ ( subset @ X545 @ ( cartprod @ X542 @ X543 ) )
| ~ ( subset @ X544 @ ( cartprod @ X542 @ X543 ) ) )
& ( ~ ( in @ ( kpair @ ( esk20_4 @ X542 @ X543 @ X544 @ X545 ) @ ( esk21_4 @ X542 @ X543 @ X544 @ X545 ) ) @ X544 )
| ( X544 = X545 )
| ( in @ ( esk19_4 @ X542 @ X543 @ X544 @ X545 ) @ X543 )
| ~ ( subset @ X545 @ ( cartprod @ X542 @ X543 ) )
| ~ ( subset @ X544 @ ( cartprod @ X542 @ X543 ) ) )
& ( ( in @ ( esk20_4 @ X542 @ X543 @ X544 @ X545 ) @ X542 )
| ( X544 = X545 )
| ( in @ ( kpair @ ( esk18_4 @ X542 @ X543 @ X544 @ X545 ) @ ( esk19_4 @ X542 @ X543 @ X544 @ X545 ) ) @ X544 )
| ~ ( subset @ X545 @ ( cartprod @ X542 @ X543 ) )
| ~ ( subset @ X544 @ ( cartprod @ X542 @ X543 ) ) )
& ( ( in @ ( esk21_4 @ X542 @ X543 @ X544 @ X545 ) @ X543 )
| ( X544 = X545 )
| ( in @ ( kpair @ ( esk18_4 @ X542 @ X543 @ X544 @ X545 ) @ ( esk19_4 @ X542 @ X543 @ X544 @ X545 ) ) @ X544 )
| ~ ( subset @ X545 @ ( cartprod @ X542 @ X543 ) )
| ~ ( subset @ X544 @ ( cartprod @ X542 @ X543 ) ) )
& ( ( in @ ( kpair @ ( esk20_4 @ X542 @ X543 @ X544 @ X545 ) @ ( esk21_4 @ X542 @ X543 @ X544 @ X545 ) ) @ X545 )
| ( X544 = X545 )
| ( in @ ( kpair @ ( esk18_4 @ X542 @ X543 @ X544 @ X545 ) @ ( esk19_4 @ X542 @ X543 @ X544 @ X545 ) ) @ X544 )
| ~ ( subset @ X545 @ ( cartprod @ X542 @ X543 ) )
| ~ ( subset @ X544 @ ( cartprod @ X542 @ X543 ) ) )
& ( ~ ( in @ ( kpair @ ( esk20_4 @ X542 @ X543 @ X544 @ X545 ) @ ( esk21_4 @ X542 @ X543 @ X544 @ X545 ) ) @ X544 )
| ( X544 = X545 )
| ( in @ ( kpair @ ( esk18_4 @ X542 @ X543 @ X544 @ X545 ) @ ( esk19_4 @ X542 @ X543 @ X544 @ X545 ) ) @ X544 )
| ~ ( subset @ X545 @ ( cartprod @ X542 @ X543 ) )
| ~ ( subset @ X544 @ ( cartprod @ X542 @ X543 ) ) )
& ( ( in @ ( esk20_4 @ X542 @ X543 @ X544 @ X545 ) @ X542 )
| ( X544 = X545 )
| ~ ( in @ ( kpair @ ( esk18_4 @ X542 @ X543 @ X544 @ X545 ) @ ( esk19_4 @ X542 @ X543 @ X544 @ X545 ) ) @ X545 )
| ~ ( subset @ X545 @ ( cartprod @ X542 @ X543 ) )
| ~ ( subset @ X544 @ ( cartprod @ X542 @ X543 ) ) )
& ( ( in @ ( esk21_4 @ X542 @ X543 @ X544 @ X545 ) @ X543 )
| ( X544 = X545 )
| ~ ( in @ ( kpair @ ( esk18_4 @ X542 @ X543 @ X544 @ X545 ) @ ( esk19_4 @ X542 @ X543 @ X544 @ X545 ) ) @ X545 )
| ~ ( subset @ X545 @ ( cartprod @ X542 @ X543 ) )
| ~ ( subset @ X544 @ ( cartprod @ X542 @ X543 ) ) )
& ( ( in @ ( kpair @ ( esk20_4 @ X542 @ X543 @ X544 @ X545 ) @ ( esk21_4 @ X542 @ X543 @ X544 @ X545 ) ) @ X545 )
| ( X544 = X545 )
| ~ ( in @ ( kpair @ ( esk18_4 @ X542 @ X543 @ X544 @ X545 ) @ ( esk19_4 @ X542 @ X543 @ X544 @ X545 ) ) @ X545 )
| ~ ( subset @ X545 @ ( cartprod @ X542 @ X543 ) )
| ~ ( subset @ X544 @ ( cartprod @ X542 @ X543 ) ) )
& ( ~ ( in @ ( kpair @ ( esk20_4 @ X542 @ X543 @ X544 @ X545 ) @ ( esk21_4 @ X542 @ X543 @ X544 @ X545 ) ) @ X544 )
| ( X544 = X545 )
| ~ ( in @ ( kpair @ ( esk18_4 @ X542 @ X543 @ X544 @ X545 ) @ ( esk19_4 @ X542 @ X543 @ X544 @ X545 ) ) @ X545 )
| ~ ( subset @ X545 @ ( cartprod @ X542 @ X543 ) )
| ~ ( subset @ X544 @ ( cartprod @ X542 @ X543 ) ) )
& funcext
& ( ( in @ ( esk22_4 @ X550 @ X551 @ X552 @ X553 ) @ X550 )
| ( X552 = X553 )
| ~ ( in @ X553 @ ( funcSet @ X550 @ X551 ) )
| ~ ( in @ X552 @ ( funcSet @ X550 @ X551 ) ) )
& ( ( ( ap @ X550 @ X551 @ X552 @ ( esk22_4 @ X550 @ X551 @ X552 @ X553 ) )
!= ( ap @ X550 @ X551 @ X553 @ ( esk22_4 @ X550 @ X551 @ X552 @ X553 ) ) )
| ( X552 = X553 )
| ~ ( in @ X553 @ ( funcSet @ X550 @ X551 ) )
| ~ ( in @ X552 @ ( funcSet @ X550 @ X551 ) ) )
& ( ~ ( in @ X557 @ ( funcSet @ X555 @ X556 ) )
| ~ ( in @ X558 @ X555 )
| ( ( ap @ X555 @ X556 @ X557 @ X558 )
= ( ap @ X555 @ X556 @ X557 @ X558 ) ) )
& ap2apEq2
& ( ( in @ ( esk23_3 @ X559 @ X560 @ X561 ) @ X559 )
| ~ ( in @ X563 @ X559 )
| ( ( ap @ X559 @ X560 @ ( lam @ X559 @ X560 @ X561 ) @ X563 )
= ( X561 @ X563 ) ) )
& ( ~ ( in @ ( X561 @ ( esk23_3 @ X559 @ X560 @ X561 ) ) @ X560 )
| ~ ( in @ X563 @ X559 )
| ( ( ap @ X559 @ X560 @ ( lam @ X559 @ X560 @ X561 ) @ X563 )
= ( X561 @ X563 ) ) )
& ( ~ ( func @ X564 @ X565 @ X566 )
| ( ( lam @ X564 @ X565 @ ( ap @ X564 @ X565 @ X566 ) )
= X566 ) )
& ( ( in @ ( esk24_3 @ X567 @ X568 @ X569 ) @ X567 )
| ( ( lam @ X567 @ X568 @ X569 )
= ( lam @ X567 @ X568 @ X569 ) ) )
& ( ~ ( in @ ( X569 @ ( esk24_3 @ X567 @ X568 @ X569 ) ) @ X568 )
| ( ( lam @ X567 @ X568 @ X569 )
= ( lam @ X567 @ X568 @ X569 ) ) )
& ( ~ ( in @ X574 @ esk25_0 )
| ( in @ ( esk27_0 @ X574 ) @ esk26_0 ) )
& ( in @ esk28_0 @ esk25_0 )
& ( ( ap @ esk25_0 @ esk26_0 @ ( lam @ esk25_0 @ esk26_0 @ esk27_0 ) @ esk28_0 )
!= ( esk27_0 @ esk28_0 ) ) ),
inference(distribute,[status(thm)],[inference(fof_nnf,[status(thm)],[inference(shift_quantors,[status(thm)],[inference(skolemize,[status(esa)],[inference(variable_rename,[status(thm)],[inference(fof_nnf,[status(thm)],[c_0_77])])])])])]) ).
thf(c_0_79,negated_conjecture,
! [X30: $i > $i,X4: $i,X3: $i,X2: $i] :
( ( ( ap @ X2 @ X3 @ ( lam @ X2 @ X3 @ X30 ) @ X4 )
= ( X30 @ X4 ) )
| ~ ( in @ ( X30 @ ( esk23_3 @ X2 @ X3 @ X30 ) ) @ X3 )
| ~ ( in @ X4 @ X2 ) ),
inference(split_conjunct,[status(thm)],[c_0_78]) ).
thf(c_0_80,negated_conjecture,
! [X2: $i] :
( ( in @ ( esk27_0 @ X2 ) @ esk26_0 )
| ~ ( in @ X2 @ esk25_0 ) ),
inference(split_conjunct,[status(thm)],[c_0_78]) ).
thf(c_0_81,negated_conjecture,
( ( ap @ esk25_0 @ esk26_0 @ ( lam @ esk25_0 @ esk26_0 @ esk27_0 ) @ esk28_0 )
!= ( esk27_0 @ esk28_0 ) ),
inference(split_conjunct,[status(thm)],[c_0_78]) ).
thf(c_0_82,negated_conjecture,
! [X30: $i > $i,X4: $i,X3: $i,X2: $i] :
( ( in @ ( esk23_3 @ X2 @ X3 @ X30 ) @ X2 )
| ( ( ap @ X2 @ X3 @ ( lam @ X2 @ X3 @ X30 ) @ X4 )
= ( X30 @ X4 ) )
| ~ ( in @ X4 @ X2 ) ),
inference(split_conjunct,[status(thm)],[c_0_78]) ).
thf(c_0_83,negated_conjecture,
in @ esk28_0 @ esk25_0,
inference(split_conjunct,[status(thm)],[c_0_78]) ).
thf(c_0_84,negated_conjecture,
! [X3: $i,X2: $i] :
( ( ( ap @ X2 @ esk26_0 @ ( lam @ X2 @ esk26_0 @ esk27_0 ) @ X3 )
= ( esk27_0 @ X3 ) )
| ~ ( in @ ( esk23_3 @ X2 @ esk26_0 @ esk27_0 ) @ esk25_0 )
| ~ ( in @ X3 @ X2 ) ),
inference(spm,[status(thm)],[c_0_79,c_0_80]) ).
thf(c_0_85,negated_conjecture,
in @ ( esk23_3 @ esk25_0 @ esk26_0 @ esk27_0 ) @ esk25_0,
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_81,c_0_82]),c_0_83])]) ).
thf(c_0_86,negated_conjecture,
$false,
inference(cn,[status(thm)],[inference(rw,[status(thm)],[inference(rw,[status(thm)],[inference(spm,[status(thm)],[c_0_81,c_0_84]),c_0_85]),c_0_83])]),
[proof] ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : SEU698^1 : TPTP v9.3.1. Released v3.7.0.
% 0.00/0.05 % Command : run_E /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% 0.11/0.18 % Computer : n011.cluster.edu
% 0.11/0.18 % Model : x86_64 x86_64
% 0.11/0.18 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.18 % Memory : 8046.5625MB
% 0.11/0.18 % OS : Linux 6.8.0-71-generic
% 0.11/0.18 % CPULimit : 300
% 0.11/0.18 % WCLimit : 300
% 0.11/0.18 % DateTime : Tue Sep 29 14:11:46 UTC 2026
% 0.11/0.19 % CPUTime :
% 0.11/0.19 Running run_E /export/starexec/sandbox2/benchmark/theBenchmark.p 300 THM
% 0.11/0.23 Running higher-order theorem proving
% 0.21/0.28 Running: /export/starexec/sandbox2/solver/bin/eprover-ho --delete-bad-limit=2000000000 --definitional-cnf=24 -s --print-statistics -R --print-version --proof-object --auto-schedule=8 --cpu-limit=300 /export/starexec/sandbox2/tmp/tmp.bEM0E4m0Z4/E---3.1_150737.p
% 4.23/1.01 % Version: 3.5.1-ho
% 4.23/1.01 % Preprocessing class: HSLMSLSSLLLCHSA.
% 4.23/1.01 % Scheduled 5 strats onto 8 cores with 300 seconds (2400 total)
% 4.23/1.01 % Starting ho_unfolding_5 with 1200s (4) cores
% 4.23/1.01 % Starting pre_casc_3 with 300s (1) cores
% 4.23/1.01 % Starting new_ho_10_cnf2 with 300s (1) cores
% 4.23/1.01 % Starting full_lambda_10 with 300s (1) cores
% 4.23/1.01 % Starting ehoh_best_nonlift_rwall with 300s (1) cores
% 4.23/1.01 % pre_casc_3 with pid 150751 completed with status 0
% 4.23/1.01 % Result found by pre_casc_3
% 4.23/1.01 % Preprocessing class: HSLMSLSSLLLCHSA.
% 4.23/1.01 % Scheduled 5 strats onto 8 cores with 300 seconds (2400 total)
% 4.23/1.01 % Starting ho_unfolding_5 with 1200s (4) cores
% 4.23/1.01 % Starting pre_casc_3 with 300s (1) cores
% 4.23/1.01 % (lift_lambdas = 1, lambda_to_forall = 1,unroll_only_formulas = 1, sine = GSinE(CountFormulas,hypos,10,,5,20000,3.0,true))
% 4.23/1.01 % SinE strategy is GSinE(CountFormulas,hypos,10,,5,20000,3.0,true)
% 4.23/1.01 % Search class: HGUSM-FSLM32-MHSMMFBN
% 4.23/1.01 % partial match(2): HGHSM-FSLM32-MHSMMSBN
% 4.23/1.01 % Scheduled 11 strats onto 1 cores with 300 seconds (300 total)
% 4.23/1.01 % Starting SubtermCWHack with 31s (1) cores
% 4.23/1.01 % SubtermCWHack with pid 150757 completed with status 0
% 4.23/1.01 % Result found by SubtermCWHack
% 4.23/1.01 % Preprocessing class: HSLMSLSSLLLCHSA.
% 4.23/1.01 % Scheduled 5 strats onto 8 cores with 300 seconds (2400 total)
% 4.23/1.01 % Starting ho_unfolding_5 with 1200s (4) cores
% 4.23/1.01 % Starting pre_casc_3 with 300s (1) cores
% 4.23/1.01 % (lift_lambdas = 1, lambda_to_forall = 1,unroll_only_formulas = 1, sine = GSinE(CountFormulas,hypos,10,,5,20000,3.0,true))
% 4.23/1.01 % SinE strategy is GSinE(CountFormulas,hypos,10,,5,20000,3.0,true)
% 4.23/1.01 % Search class: HGUSM-FSLM32-MHSMMFBN
% 4.23/1.01 % partial match(2): HGHSM-FSLM32-MHSMMSBN
% 4.23/1.01 % Scheduled 11 strats onto 1 cores with 300 seconds (300 total)
% 4.23/1.01 % Starting SubtermCWHack with 31s (1) cores
% 4.23/1.01 % Preprocessing time : 0.004 s
% 4.23/1.01
% 4.23/1.01 % Proof found!
% 4.23/1.01 % SZS status Theorem
% 4.23/1.01 % SZS output start CNFRefutation
% See solution above
% 4.23/1.01 % Parsed axioms : 472
% 4.23/1.01 % Removed by relevancy pruning/SinE : 423
% 4.23/1.01 % Initial clauses : 269
% 4.23/1.01 % Removed in clause preprocessing : 3
% 4.23/1.01 % Initial clauses in saturation : 266
% 4.23/1.01 % Processed clauses : 1729
% 4.23/1.01 % ...of these trivial : 4
% 4.23/1.01 % ...subsumed : 785
% 4.23/1.01 % ...remaining for further processing : 940
% 4.23/1.01 % Other redundant clauses eliminated : 2
% 4.23/1.01 % Clauses deleted for lack of memory : 0
% 4.23/1.01 % Backward-subsumed : 6
% 4.23/1.01 % Backward-rewritten : 0
% 4.23/1.01 % Generated clauses : 19870
% 4.23/1.01 % ...of the previous two non-redundant : 19634
% 4.23/1.01 % ...aggressively subsumed : 0
% 4.23/1.01 % Contextual simplify-reflections : 18
% 4.23/1.01 % Paramodulations : 19844
% 4.23/1.01 % Factorizations : 0
% 4.23/1.01 % NegExts : 0
% 4.23/1.01 % Equation resolutions : 23
% 4.23/1.01 % Disequality decompositions : 0
% 4.23/1.01 % Total rewrite steps : 4622
% 4.23/1.01 % ...of those cached : 4240
% 4.23/1.01 % Propositional unsat checks : 0
% 4.23/1.01 % Propositional check models : 0
% 4.23/1.01 % Propositional check unsatisfiable : 0
% 4.23/1.01 % Propositional clauses : 0
% 4.23/1.01 % Propositional clauses after purity: 0
% 4.23/1.01 % Propositional unsat core size : 0
% 4.23/1.01 % Propositional preprocessing time : 0.000
% 4.23/1.01 % Propositional encoding time : 0.000
% 4.23/1.01 % Propositional solver time : 0.000
% 4.23/1.01 % Success case prop preproc time : 0.000
% 4.23/1.01 % Success case prop encoding time : 0.000
% 4.23/1.01 % Success case prop solver time : 0.000
% 4.23/1.01 % Current number of processed clauses : 932
% 4.23/1.01 % Positive orientable unit clauses : 186
% 4.23/1.01 % Positive unorientable unit clauses: 0
% 4.23/1.01 % Negative unit clauses : 1
% 4.23/1.01 % Non-unit-clauses : 745
% 4.23/1.01 % Current number of unprocessed clauses: 18165
% 4.23/1.01 % ...number of literals in the above : 80377
% 4.23/1.01 % Current number of archived formulas : 0
% 4.23/1.01 % Current number of archived clauses : 7
% 4.23/1.01 % Clause-clause subsumption calls (NU) : 68577
% 4.23/1.01 % Rec. Clause-clause subsumption calls : 42328
% 4.23/1.01 % Non-unit clause-clause subsumptions : 809
% 4.23/1.01 % Unit Clause-clause subsumption calls : 282
% 4.23/1.01 % Rewrite failures with RHS unbound : 0
% 4.23/1.01 % BW rewrite match attempts : 2
% 4.23/1.01 % BW rewrite match successes : 0
% 4.23/1.01 % Condensation attempts : 0
% 4.23/1.01 % Condensation successes : 0
% 4.23/1.01 % Termbank termtop insertions : 509091
% 4.23/1.01 % Search garbage collected termcells : 9396
% 4.23/1.01
% 4.23/1.01 % -------------------------------------------------
% 4.23/1.01 % User time : 0.596 s
% 4.23/1.01 % System time : 0.029 s
% 4.23/1.01 % Total time : 0.625 s
% 4.23/1.01 % Maximum resident set size: 6952 pages
% 4.23/1.01
% 4.23/1.01 % -------------------------------------------------
% 4.23/1.01 % User time : 0.610 s
% 4.23/1.01 % System time : 0.034 s
% 4.23/1.01 % Total time : 0.644 s
% 4.23/1.01 % Maximum resident set size: 5376 pages
% 4.23/1.01 % E exiting
%------------------------------------------------------------------------------