%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU685^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n031.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 09:36:23 AM UTC 2026
% Result : Theorem 2.87s 1.88s
% Output : Refutation 2.87s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 8
% Syntax : Number of formulae : 45 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 3552 ( 936 equ; 0 cnn)
% Maximal formula atoms : 204 ( 78 avg)
% Number of connectives : 2054 ( 66 ~; 42 |; 638 &; 478 @)
% ( 4 <=>; 826 =>; 0 <=; 0 <~>)
% Maximal formula depth : 216 ( 37 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 381 ( 378 usr; 214 con; 0-4 aty)
% Number of variables : 109 ( 0 ^; 72 !; 37 ?; 109 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_254,type,
sP2: $i > $i > $o ).
thf(func_def_255,type,
sP3: $i > $o ).
thf(func_def_256,type,
sP4: $i > $i > $o ).
thf(func_def_257,type,
sP5: $i > $i > $o ).
thf(func_def_259,type,
sP7: $i > $i > $i > $o > $o ).
thf(func_def_268,type,
sK16: $i > $i > $o ).
thf(func_def_273,type,
sK21: $i > $i > $o ).
thf(func_def_277,type,
sK25: $i > $i ).
thf(func_def_281,type,
sK29: $o > $i > $i > $i ).
thf(func_def_282,type,
sK30: ( $i > $o ) > $i > $i ).
thf(func_def_284,type,
sK32: $i > $o ).
thf(func_def_296,type,
sK44: ( $i > $o ) > $i ).
thf(func_def_297,type,
sK45: ( $i > $o ) > $i ).
thf(func_def_298,type,
sK46: $i > $o ).
thf(func_def_300,type,
sK48: $i > $o ).
thf(func_def_304,type,
sK52: $i > $o ).
thf(func_def_307,type,
sK55: $i > ( $i > $o ) > $i ).
thf(func_def_312,type,
sK60: $i > $i > $i ).
thf(func_def_333,type,
sK81: $i > $o ).
thf(func_def_334,type,
sK82: ( $i > $o ) > $i > $i ).
thf(func_def_337,type,
sK85: $i > ( $i > $o ) > $i ).
thf(func_def_338,type,
sK86: $i > $o ).
thf(func_def_345,type,
sK93: $i > $o ).
thf(func_def_364,type,
sK112: ( $i > $o ) > $i ).
thf(func_def_365,type,
sK113: $i > $o ).
thf(func_def_370,type,
sK118: $i > $i > $o ).
thf(func_def_374,type,
sK122: $i > $o ).
thf(func_def_376,type,
sK124: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_377,type,
sK125: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_390,type,
sK138: $i > $i > $i ).
thf(func_def_410,type,
sK158: $i > $i ).
thf(func_def_411,type,
sK159: ( $i > $i ) > $i > $i > $i ).
thf(func_def_417,type,
sK165: $i > $i > $i ).
thf(func_def_420,type,
sK168: $i > $i ).
thf(func_def_428,type,
sK176: $i > $o ).
thf(func_def_429,type,
sK177: $i > $o ).
thf(func_def_430,type,
sK178: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_431,type,
sK179: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_437,type,
sK185: $i > $i ).
thf(func_def_445,type,
sK193: $i > ( $i > $o ) > $i ).
thf(func_def_446,type,
sK194: $i > $o ).
thf(func_def_448,type,
sK196: $i > $o ).
thf(func_def_454,type,
sK202: $i > $i > $o ).
thf(func_def_455,type,
sK203: $i > $i ).
thf(func_def_456,type,
sK204: $i > $i ).
thf(func_def_457,type,
sK205: ( $i > $i > $o ) > $i > $i ).
thf(func_def_458,type,
sK206: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_459,type,
sK207: ( $i > $i > $o ) > $i > $i ).
thf(func_def_461,type,
sK209: $i > $i > $o ).
thf(func_def_468,type,
sK216: $i > $o ).
thf(func_def_469,type,
sK217: $i > $o ).
thf(func_def_470,type,
sK218: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_471,type,
sK219: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_480,type,
sK228: $i > $o ).
thf(func_def_482,type,
sK230: $i > $o ).
thf(func_def_484,type,
sK232: $i > $o ).
thf(func_def_503,type,
sK251: $i > $o ).
thf(func_def_505,type,
sK253: $i > $i ).
thf(func_def_508,type,
sK256: $i > $i ).
thf(func_def_510,type,
sK258: $i > $i ).
thf(func_def_512,type,
sK260: $i > $o ).
thf(func_def_514,type,
sK262: ( $i > $o ) > $i > $i ).
thf(func_def_515,type,
sK263: ( $i > $o ) > $i > $i ).
thf(func_def_524,type,
sK272: $i > $i > $i ).
thf(func_def_541,type,
sK289: $i > $o ).
thf(func_def_542,type,
sK290: $i > $i ).
thf(func_def_543,type,
sK291: ( $i > $o ) > $i ).
thf(func_def_547,type,
sK295: $i > $i > $o ).
thf(func_def_555,type,
sK303: $i > $i > $o ).
thf(func_def_560,type,
sK308: $i > ( $i > $o ) > $i ).
thf(func_def_561,type,
sK309: $i > $o ).
thf(func_def_564,type,
sK312: $i > $i > $i > $i ).
thf(func_def_565,type,
sK313: $i > $i > $i > $i ).
thf(func_def_571,type,
sK319: $i > $o ).
thf(func_def_577,type,
sK325: $i > $i ).
thf(func_def_578,type,
sK326: ( $i > $i ) > $i > $i > $i ).
thf(func_def_591,type,
sK339: $i > $i > $i ).
thf(func_def_624,type,
sK372: $i > $i > $i ).
thf(func_def_625,type,
sK373: $i > $i > $i ).
thf(func_def_626,type,
sK374: $i > $i > $i ).
thf(func_def_627,type,
sK375: $i > $i > $i ).
thf(func_def_628,type,
sK376: $i > $i > $i > $i ).
thf(func_def_629,type,
sK377: $i > $i ).
thf(func_def_630,type,
sK378: $i > $i ).
thf(func_def_631,type,
sK379: $i > $i ).
thf(func_def_632,type,
sK380: $i > $i ).
thf(func_def_633,type,
sK381: $i > $i > $i ).
thf(func_def_634,type,
sK382: $i > $i > $i > $i ).
thf(func_def_635,type,
sK383: $i > $i > $i > $i ).
thf(func_def_636,type,
sK384: $i > $i > $i ).
thf(func_def_637,type,
sK385: $i > $i > $i ).
thf(func_def_638,type,
sK386: $i > $i ).
thf(func_def_639,type,
sK387: $i > $i > $i ).
thf(func_def_641,type,
sK389: $i > $i ).
thf(func_def_642,type,
sK390: $i > $i ).
thf(func_def_650,type,
sK398: $i > $o ).
thf(func_def_664,type,
sK412: $i > $o ).
thf(func_def_666,type,
sK414: ( $i > $o ) > $i > $i ).
thf(func_def_667,type,
sK415: ( $i > $o ) > $i > $i ).
thf(func_def_673,type,
sK421: $i > $o ).
thf(func_def_674,type,
sK422: $i > $i ).
thf(func_def_675,type,
sK423: ( $i > $o ) > $i ).
thf(func_def_687,type,
sK435: $i > $o ).
thf(func_def_691,type,
sK439: $i > $o ).
thf(func_def_692,type,
sK440: $i > $o ).
thf(func_def_693,type,
sK441: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_694,type,
sK442: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_699,type,
sK447: $i > $o ).
thf(func_def_702,type,
sK450: $i > ( $i > $o ) > $i ).
thf(func_def_703,type,
sK451: $i > $o ).
thf(func_def_710,type,
sK458: $i > $i ).
thf(func_def_712,type,
sK460: $i > ( $i > $i ) > $i > $i ).
thf(func_def_714,type,
sK462: $i > $i ).
thf(func_def_717,type,
sK465: $i > $o ).
thf(func_def_723,type,
sK471: $i > $i > $i ).
thf(func_def_726,type,
sK474: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_727,type,
sK475: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_731,type,
sK479: $i > $o ).
thf(func_def_746,type,
sK494: $i > $i > $i ).
thf(func_def_748,type,
sK496: $i > $o ).
thf(func_def_750,type,
sK498: $i > ( $i > $o ) > $i > $i ).
thf(func_def_761,type,
sK509: $i > $i > $i ).
thf(func_def_762,type,
sK510: $i > $i > $i ).
thf(func_def_766,type,
sK514: $i > $i > $i ).
thf(func_def_786,type,
sK534: $i > $o ).
thf(func_def_788,type,
sK536: ( $i > $o ) > $i > $i ).
thf(func_def_794,type,
sK542: $i > ( $i > $o ) > $i ).
thf(func_def_795,type,
sK543: $i > $o ).
thf(func_def_830,type,
sK578: $i > $i > $o ).
thf(func_def_836,type,
sK584: $i > $o ).
thf(func_def_853,type,
ph601:
!>[X0: $tType] : X0 ).
thf(f2979,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f2978]) ).
thf(f2978,plain,
$true != $true,
inference(superposition,[status(thm)],[f2977,f2955]) ).
thf(f2955,plain,
( $true
= ( func @ sK129 @ sK130 @ sK131 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2954]) ).
thf(f2954,plain,
( ( $true != $true )
| ( $true
= ( func @ sK129 @ sK130 @ sK131 ) ) ),
inference(forward_demodulation,[status(thm)],[f2950,f2083]) ).
thf(f2083,plain,
infuncsetfunc = $true,
inference(cnf_transformation,[status(thm)],[f1070]) ).
thf(f1070,plain,
( ( setext = $true )
& ( kpairiskpair = $true )
& ( binunionIR = $true )
& ( setminusEL = $true )
& ( cartprodsndpairEq = $true )
& ( subsetI1 = $true )
& ( powersetsubset = $true )
& ( emptyinPowerset = $true )
& ( cartprodmempair = $true )
& ( exu__Cong = $true )
& ( disjointsetsI1 = $true )
& ( powersetAx = $true )
& ( prop2setE = $true )
& ( dsetconstrER = $true )
& ( binintersectSubset1 = $true )
& ( quantDeMorgan2 = $true )
& ( exuI2 = $true )
& ( omegaSAx = $true )
& ( quantDeMorgan1 = $true )
& ( ex1I2 = $true )
& ( eqimpsubset1 = $true )
& ( powerset__Cong = $true )
& ( setextsub = $true )
& ( setminusLsub = $true )
& ( setadjoinSub2 = $true )
& ( dpsetconstrSub = $true )
& ( setukpairinjL1 = $true )
& ( notdallE = $true )
& ( secondinupair = $true )
& ( cartprodfstin = $true )
& ( notinemptyset = $true )
& ( subsetE2 = $true )
& ( cartprodpairin = $true )
& ( brelnall1 = $true )
& ( omega__Cong = $true )
& ( cartprodpairmemEL = $true )
& ( setunionI = $true )
& ( $true
= ( in @ sK131 @ ( funcSet @ sK129 @ sK130 ) ) )
& ( $true
= ( in @ sK132 @ sK129 ) )
& ( ( in @ ( kpair @ sK132 @ ( ap @ sK129 @ sK130 @ sK131 @ sK132 ) ) @ sK131 )
!= $true )
& ( sepSubset = $true )
& ( singletonsuniq = $true )
& ( setextAx = $true )
& ( cartprodpairmemER = $true )
& ( upairsetE = $true )
& ( cartprodmempaircEq = $true )
& ( singletoninpowunion = $true )
& ( upairset2IR = $true )
& ( cartprodsndin = $true )
& ( prop2setI = $true )
& ( kpairsurjEq = $true )
& ( eqimpsubset2 = $true )
& ( quantDeMorgan4 = $true )
& ( emptyInPowerset = $true )
& ( binunionIL = $true )
& ( dpsetconstrEL2 = $true )
& ( cartprodmempair1 = $true )
& ( notequalI1 = $true )
& ( binintersectI = $true )
& ( inPowerset = $true )
& ( theprop = $true )
& ( ubforcartprodlem1 = $true )
& ( setukpairinjR2 = $true )
& ( subsetemptysetimpeq = $true )
& ( ksndpairEq = $true )
& ( descr__Cong = $true )
& ( emptyinunitempty = $true )
& ( subset2powerset = $true )
& ( brelnall2 = $true )
& ( notdexE = $true )
& ( dsetconstrI = $true )
& ( notinsingleton = $true )
& ( dpsetconstrER = $true )
& ( symdiffIneg2 = $true )
& ( upairset2E = $true )
& ( emptysetsubset = $true )
& ( eqinunit = $true )
& ( ex1E2 = $true )
& ( setminusER = $true )
& ( vacuousDall = $true )
& ( binunionLsub = $true )
& ( cartprodfstpairEq = $true )
& ( binintersectSubset5 = $true )
& ( ex1I = $true )
& ( setunionAx = $true )
& ( singletonsubset = $true )
& ( setadjoinIL = $true )
& ( exuE1 = $true )
& ( subsetRefl = $true )
& ( binintersectSubset3 = $true )
& ( infuncsetfunc = $true )
& ( lam2p = $true )
& ( setadjoinIR = $true )
& ( powersetE1 = $true )
& ( setunionE = $true )
& ( foundationAx = $true )
& ( dpsetconstrERa = $true )
& ( emptysetAx = $true )
& ( lamProp = $true )
& ( singletonsswitch = $true )
& ( exuEu = $true )
& ( emptysetimpfalse = $true )
& ( setminusELneg = $true )
& ( setminusSubset2 = $true )
& ( powersetI = $true )
& ( symdiffIneg1 = $true )
& ( kfstpairEq = $true )
& ( kpairp = $true )
& ( funcGraphProp1 = $true )
& ( exuI3 = $true )
& ( binintersectER = $true )
& ( setOfPairsIsBReln = $true )
& ( binintersectEL = $true )
& ( nonemptyI1 = $true )
& ( singletonprop = $true )
& ( binunionEcases = $true )
& ( powersetE = $true )
& ( prop2set2propI = $true )
& ( noeltsimpempty = $true )
& ( setukpairinjR1 = $true )
& ( setunion__Cong = $true )
& ( replAx = $true )
& ( inCongP = $true )
& ( ksndsingleton = $true )
& ( ex1E1 = $true )
& ( setunionsingleton2 = $true )
& ( wellorderingAx = $true )
& ( upairinpowunion = $true )
& ( setminusI = $true )
& ( dpsetconstrEL1 = $true )
& ( ap2p = $true )
& ( setminusSubset1 = $true )
& ( setukpairIR = $true )
& ( subPowSU = $true )
& ( setukpairinjR12 = $true )
& ( upairequniteq = $true )
& ( funcImageSingleton = $true )
& ( setminusERneg = $true )
& ( subsetTrans = $true )
& ( setoftrueEq = $true )
& ( binunionE = $true )
& ( exuE3u = $true )
& ( nonemptyE1 = $true )
& ( exuE3e = $true )
& ( setadjoin__Cong = $true )
& ( setunionE2 = $true )
& ( binintersectSubset4 = $true )
& ( kfstsingleton = $true )
& ( omega0Ax = $true )
& ( sepInPowerset = $true )
& ( uniqinunit = $true )
& ( notequalI2 = $true )
& ( setadjoinE = $true )
& ( funcinfuncset = $true )
& ( omegaIndAx = $true )
& ( binunionRsub = $true )
& ( symdiffE = $true )
& ( setukpairinjL2 = $true )
& ( subsetI2 = $true )
& ( binintersectSubset2 = $true )
& ( ubforcartprodlem3 = $true )
& ( setunionsingleton = $true )
& ( cartprodpairsurjEq = $true )
& ( notsubsetI = $true )
& ( upairsubunion = $true )
& ( dsetconstr__Cong = $true )
& ( setunionsingleton1 = $true )
& ( apProp = $true )
& ( setukpairinjR11 = $true )
& ( setminusILneg = $true )
& ( binintersectRsub = $true )
& ( setadjoinOr = $true )
& ( upairsetIR = $true )
& ( exuE2 = $true )
& ( setukpairinjR = $true )
& ( singletoninpowerset = $true )
& ( dpsetconstrI = $true )
& ( emptyset__Cong = $true )
& ( emptyE1 = $true )
& ( ubforcartprodlem2 = $true )
& ( upairsetIL = $true )
& ( nonemptyI = $true )
& ( setadjoinSub = $true )
& ( setadjoinAx = $true )
& ( powersetI1 = $true )
& ( setminusIRneg = $true )
& ( emptysetE = $true )
& ( nonemptyImpWitness = $true )
& ( app = $true )
& ( quantDeMorgan3 = $true )
& ( setbeta = $true )
& ( in__Cong = $true )
& ( setukpairIL = $true )
& ( binintersectLsub = $true )
& ( symdiffI1 = $true )
& ( dsetconstrEL = $true )
& ( setukpairinjL = $true )
& ( subsetE = $true )
& ( descrp = $true )
& ( emptyI = $true )
& ( lamp = $true )
& ( exuI1 = $true )
& ( bs114d = $true )
& ( symdiffI2 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK129,sK130,sK131,sK132])],[f850,f1069,f1068]) ).
thf(f1068,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( funcSet @ X0 @ X1 ) ) )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ( $true
!= ( in @ ( kpair @ X3 @ ( ap @ X0 @ X1 @ X2 @ X3 ) ) @ X2 ) ) ) )
=> ( ( $true
= ( in @ sK131 @ ( funcSet @ sK129 @ sK130 ) ) )
& ? [X3: $i] :
( ( ( in @ X3 @ sK129 )
= $true )
& ( ( in @ ( kpair @ X3 @ ( ap @ sK129 @ sK130 @ sK131 @ X3 ) ) @ sK131 )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1069,plain,
( ? [X3: $i] :
( ( ( in @ X3 @ sK129 )
= $true )
& ( ( in @ ( kpair @ X3 @ ( ap @ sK129 @ sK130 @ sK131 @ X3 ) ) @ sK131 )
!= $true ) )
=> ( ( $true
= ( in @ sK132 @ sK129 ) )
& ( ( in @ ( kpair @ sK132 @ ( ap @ sK129 @ sK130 @ sK131 @ sK132 ) ) @ sK131 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f850,plain,
( ( setext = $true )
& ( kpairiskpair = $true )
& ( binunionIR = $true )
& ( setminusEL = $true )
& ( cartprodsndpairEq = $true )
& ( subsetI1 = $true )
& ( powersetsubset = $true )
& ( emptyinPowerset = $true )
& ( cartprodmempair = $true )
& ( exu__Cong = $true )
& ( disjointsetsI1 = $true )
& ( powersetAx = $true )
& ( prop2setE = $true )
& ( dsetconstrER = $true )
& ( binintersectSubset1 = $true )
& ( quantDeMorgan2 = $true )
& ( exuI2 = $true )
& ( omegaSAx = $true )
& ( quantDeMorgan1 = $true )
& ( ex1I2 = $true )
& ( eqimpsubset1 = $true )
& ( powerset__Cong = $true )
& ( setextsub = $true )
& ( setminusLsub = $true )
& ( setadjoinSub2 = $true )
& ( dpsetconstrSub = $true )
& ( setukpairinjL1 = $true )
& ( notdallE = $true )
& ( secondinupair = $true )
& ( cartprodfstin = $true )
& ( notinemptyset = $true )
& ( subsetE2 = $true )
& ( cartprodpairin = $true )
& ( brelnall1 = $true )
& ( omega__Cong = $true )
& ( cartprodpairmemEL = $true )
& ( setunionI = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( funcSet @ X0 @ X1 ) ) )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ( $true
!= ( in @ ( kpair @ X3 @ ( ap @ X0 @ X1 @ X2 @ X3 ) ) @ X2 ) ) ) )
& ( sepSubset = $true )
& ( singletonsuniq = $true )
& ( setextAx = $true )
& ( cartprodpairmemER = $true )
& ( upairsetE = $true )
& ( cartprodmempaircEq = $true )
& ( singletoninpowunion = $true )
& ( upairset2IR = $true )
& ( cartprodsndin = $true )
& ( prop2setI = $true )
& ( kpairsurjEq = $true )
& ( eqimpsubset2 = $true )
& ( quantDeMorgan4 = $true )
& ( emptyInPowerset = $true )
& ( binunionIL = $true )
& ( dpsetconstrEL2 = $true )
& ( cartprodmempair1 = $true )
& ( notequalI1 = $true )
& ( binintersectI = $true )
& ( inPowerset = $true )
& ( theprop = $true )
& ( ubforcartprodlem1 = $true )
& ( setukpairinjR2 = $true )
& ( subsetemptysetimpeq = $true )
& ( ksndpairEq = $true )
& ( descr__Cong = $true )
& ( emptyinunitempty = $true )
& ( subset2powerset = $true )
& ( brelnall2 = $true )
& ( notdexE = $true )
& ( dsetconstrI = $true )
& ( notinsingleton = $true )
& ( dpsetconstrER = $true )
& ( symdiffIneg2 = $true )
& ( upairset2E = $true )
& ( emptysetsubset = $true )
& ( eqinunit = $true )
& ( ex1E2 = $true )
& ( setminusER = $true )
& ( vacuousDall = $true )
& ( binunionLsub = $true )
& ( cartprodfstpairEq = $true )
& ( binintersectSubset5 = $true )
& ( ex1I = $true )
& ( setunionAx = $true )
& ( singletonsubset = $true )
& ( setadjoinIL = $true )
& ( exuE1 = $true )
& ( subsetRefl = $true )
& ( binintersectSubset3 = $true )
& ( infuncsetfunc = $true )
& ( lam2p = $true )
& ( setadjoinIR = $true )
& ( powersetE1 = $true )
& ( setunionE = $true )
& ( foundationAx = $true )
& ( dpsetconstrERa = $true )
& ( emptysetAx = $true )
& ( lamProp = $true )
& ( singletonsswitch = $true )
& ( exuEu = $true )
& ( emptysetimpfalse = $true )
& ( setminusELneg = $true )
& ( setminusSubset2 = $true )
& ( powersetI = $true )
& ( symdiffIneg1 = $true )
& ( kfstpairEq = $true )
& ( kpairp = $true )
& ( funcGraphProp1 = $true )
& ( exuI3 = $true )
& ( binintersectER = $true )
& ( setOfPairsIsBReln = $true )
& ( binintersectEL = $true )
& ( nonemptyI1 = $true )
& ( singletonprop = $true )
& ( binunionEcases = $true )
& ( powersetE = $true )
& ( prop2set2propI = $true )
& ( noeltsimpempty = $true )
& ( setukpairinjR1 = $true )
& ( setunion__Cong = $true )
& ( replAx = $true )
& ( inCongP = $true )
& ( ksndsingleton = $true )
& ( ex1E1 = $true )
& ( setunionsingleton2 = $true )
& ( wellorderingAx = $true )
& ( upairinpowunion = $true )
& ( setminusI = $true )
& ( dpsetconstrEL1 = $true )
& ( ap2p = $true )
& ( setminusSubset1 = $true )
& ( setukpairIR = $true )
& ( subPowSU = $true )
& ( setukpairinjR12 = $true )
& ( upairequniteq = $true )
& ( funcImageSingleton = $true )
& ( setminusERneg = $true )
& ( subsetTrans = $true )
& ( setoftrueEq = $true )
& ( binunionE = $true )
& ( exuE3u = $true )
& ( nonemptyE1 = $true )
& ( exuE3e = $true )
& ( setadjoin__Cong = $true )
& ( setunionE2 = $true )
& ( binintersectSubset4 = $true )
& ( kfstsingleton = $true )
& ( omega0Ax = $true )
& ( sepInPowerset = $true )
& ( uniqinunit = $true )
& ( notequalI2 = $true )
& ( setadjoinE = $true )
& ( funcinfuncset = $true )
& ( omegaIndAx = $true )
& ( binunionRsub = $true )
& ( symdiffE = $true )
& ( setukpairinjL2 = $true )
& ( subsetI2 = $true )
& ( binintersectSubset2 = $true )
& ( ubforcartprodlem3 = $true )
& ( setunionsingleton = $true )
& ( cartprodpairsurjEq = $true )
& ( notsubsetI = $true )
& ( upairsubunion = $true )
& ( dsetconstr__Cong = $true )
& ( setunionsingleton1 = $true )
& ( apProp = $true )
& ( setukpairinjR11 = $true )
& ( setminusILneg = $true )
& ( binintersectRsub = $true )
& ( setadjoinOr = $true )
& ( upairsetIR = $true )
& ( exuE2 = $true )
& ( setukpairinjR = $true )
& ( singletoninpowerset = $true )
& ( dpsetconstrI = $true )
& ( emptyset__Cong = $true )
& ( emptyE1 = $true )
& ( ubforcartprodlem2 = $true )
& ( upairsetIL = $true )
& ( nonemptyI = $true )
& ( setadjoinSub = $true )
& ( setadjoinAx = $true )
& ( powersetI1 = $true )
& ( setminusIRneg = $true )
& ( emptysetE = $true )
& ( nonemptyImpWitness = $true )
& ( app = $true )
& ( quantDeMorgan3 = $true )
& ( setbeta = $true )
& ( in__Cong = $true )
& ( setukpairIL = $true )
& ( binintersectLsub = $true )
& ( symdiffI1 = $true )
& ( dsetconstrEL = $true )
& ( setukpairinjL = $true )
& ( subsetE = $true )
& ( descrp = $true )
& ( emptyI = $true )
& ( lamp = $true )
& ( exuI1 = $true )
& ( bs114d = $true )
& ( symdiffI2 = $true ) ),
inference(flattening,[status(thm)],[f849]) ).
thf(f849,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( funcSet @ X0 @ X1 ) ) )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ( $true
!= ( in @ ( kpair @ X3 @ ( ap @ X0 @ X1 @ X2 @ X3 ) ) @ X2 ) ) ) )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f434]) ).
thf(f434,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( funcSet @ X0 @ X1 ) ) )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
=> ( $true
= ( in @ ( kpair @ X3 @ ( ap @ X0 @ X1 @ X2 @ X3 ) ) @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f433]) ).
thf(f433,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( 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
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( funcSet @ X0 @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( in @ ( kpair @ X3 @ ( ap @ X0 @ X1 @ X2 @ X3 ) ) @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f210]) ).
thf(f210,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
=> ! [X3: $i,X4: $i,X13: $i] :
( ( in @ X13 @ ( funcSet @ X3 @ X4 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ ( kpair @ X1 @ ( ap @ X3 @ X4 @ X13 @ X1 ) ) @ X13 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f209]) ).
thf(f209,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
=> ! [X3: $i,X4: $i,X13: $i] :
( ( in @ X13 @ ( funcSet @ X3 @ X4 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ ( kpair @ X1 @ ( ap @ X3 @ X4 @ X13 @ X1 ) ) @ X13 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.ClpnksWlLJ/DTF2THF_22405.p',funcGraphProp3) ).
thf(f2950,plain,
( ( $true
= ( func @ sK129 @ sK130 @ sK131 ) )
| ( infuncsetfunc != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f2948]) ).
thf(f2948,plain,
( ( infuncsetfunc != $true )
| ( $true != $true )
| ( $true
= ( func @ sK129 @ sK130 @ sK131 ) ) ),
inference(superposition,[status(thm)],[f2289,f2136]) ).
thf(f2136,plain,
( $true
= ( in @ sK131 @ ( funcSet @ sK129 @ sK130 ) ) ),
inference(cnf_transformation,[status(thm)],[f1070]) ).
thf(f2289,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) )
| ( $true
= ( func @ X2 @ X1 @ X0 ) )
| ( infuncsetfunc != $true ) ),
inference(cnf_transformation,[status(thm)],[f1222]) ).
thf(f1222,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) )
| ( $true
= ( func @ X2 @ X1 @ X0 ) ) )
| ( infuncsetfunc != $true ) )
& ( ( infuncsetfunc = $true )
| ( ( $true
= ( in @ sK213 @ ( funcSet @ sK215 @ sK214 ) ) )
& ( ( func @ sK215 @ sK214 @ sK213 )
!= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK213,sK214,sK215])],[f1220,f1221]) ).
thf(f1221,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X3 @ ( funcSet @ X5 @ X4 ) ) )
& ( $true
!= ( func @ X5 @ X4 @ X3 ) ) )
=> ( ( $true
= ( in @ sK213 @ ( funcSet @ sK215 @ sK214 ) ) )
& ( ( func @ sK215 @ sK214 @ sK213 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1220,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) )
| ( $true
= ( func @ X2 @ X1 @ X0 ) ) )
| ( infuncsetfunc != $true ) )
& ( ( infuncsetfunc = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X3 @ ( funcSet @ X5 @ X4 ) ) )
& ( $true
!= ( func @ X5 @ X4 @ X3 ) ) ) ) ),
inference(rectify,[status(thm)],[f1219]) ).
thf(f1219,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) )
| ( $true
= ( func @ X2 @ X1 @ X0 ) ) )
| ( infuncsetfunc != $true ) )
& ( ( infuncsetfunc = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) )
& ( $true
!= ( func @ X2 @ X1 @ X0 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f683]) ).
thf(f683,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) )
| ( $true
= ( func @ X2 @ X1 @ X0 ) ) )
<=> ( infuncsetfunc = $true ) ),
inference(ennf_transformation,[status(thm)],[f377]) ).
thf(f377,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) )
=> ( $true
= ( func @ X2 @ X1 @ X0 ) ) )
<=> ( infuncsetfunc = $true ) ),
inference(fool_elimination,[status(thm)],[f376]) ).
thf(f376,plain,
( infuncsetfunc
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ ( funcSet @ X2 @ X1 ) )
=> ( func @ X2 @ X1 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f198]) ).
thf(f198,axiom,
( infuncsetfunc
= ( ! [X13: $i,X4: $i,X3: $i] :
( ( in @ X13 @ ( funcSet @ X3 @ X4 ) )
=> ( func @ X3 @ X4 @ X13 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.ClpnksWlLJ/DTF2THF_22405.p',infuncsetfunc) ).
thf(f2977,plain,
( $true
!= ( func @ sK129 @ sK130 @ sK131 ) ),
inference(trivial_inequality_removal,[status(thm)],[f2976]) ).
thf(f2976,plain,
( ( $true
!= ( func @ sK129 @ sK130 @ sK131 ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f2975,f2135]) ).
thf(f2135,plain,
( $true
= ( in @ sK132 @ sK129 ) ),
inference(cnf_transformation,[status(thm)],[f1070]) ).
thf(f2975,plain,
( ( $true
!= ( in @ sK132 @ sK129 ) )
| ( $true
!= ( func @ sK129 @ sK130 @ sK131 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2974]) ).
thf(f2974,plain,
( ( $true
!= ( in @ sK132 @ sK129 ) )
| ( $true
!= ( func @ sK129 @ sK130 @ sK131 ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f2962,f2065]) ).
thf(f2065,plain,
funcGraphProp1 = $true,
inference(cnf_transformation,[status(thm)],[f1070]) ).
thf(f2962,plain,
( ( funcGraphProp1 != $true )
| ( $true
!= ( func @ sK129 @ sK130 @ sK131 ) )
| ( $true
!= ( in @ sK132 @ sK129 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2956]) ).
thf(f2956,plain,
( ( $true
!= ( in @ sK132 @ sK129 ) )
| ( $true != $true )
| ( funcGraphProp1 != $true )
| ( $true
!= ( func @ sK129 @ sK130 @ sK131 ) ) ),
inference(superposition,[status(thm)],[f2134,f2733]) ).
thf(f2733,plain,
! [X6: $i,X7: $i,X4: $i,X5: $i] :
( ( ( in @ ( kpair @ X7 @ ( ap @ X6 @ X4 @ X5 @ X7 ) ) @ X5 )
= $true )
| ( ( func @ X6 @ X4 @ X5 )
!= $true )
| ( funcGraphProp1 != $true )
| ( ( in @ X7 @ X6 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1745]) ).
thf(f1745,plain,
( ( ( funcGraphProp1 = $true )
| ( ( ( func @ sK548 @ sK546 @ sK547 )
= $true )
& ( ( in @ ( kpair @ sK549 @ ( ap @ sK548 @ sK546 @ sK547 @ sK549 ) ) @ sK547 )
!= $true )
& ( ( in @ sK549 @ sK548 )
= $true ) ) )
& ( ! [X4: $i,X5: $i,X6: $i] :
( ( ( func @ X6 @ X4 @ X5 )
!= $true )
| ! [X7: $i] :
( ( ( in @ ( kpair @ X7 @ ( ap @ X6 @ X4 @ X5 @ X7 ) ) @ X5 )
= $true )
| ( ( in @ X7 @ X6 )
!= $true ) ) )
| ( funcGraphProp1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK546,sK547,sK548,sK549])],[f1742,f1744,f1743]) ).
thf(f1743,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X2 @ X0 @ X1 ) )
& ? [X3: $i] :
( ( ( in @ ( kpair @ X3 @ ( ap @ X2 @ X0 @ X1 @ X3 ) ) @ X1 )
!= $true )
& ( $true
= ( in @ X3 @ X2 ) ) ) )
=> ( ( ( func @ sK548 @ sK546 @ sK547 )
= $true )
& ? [X3: $i] :
( ( $true
!= ( in @ ( kpair @ X3 @ ( ap @ sK548 @ sK546 @ sK547 @ X3 ) ) @ sK547 ) )
& ( $true
= ( in @ X3 @ sK548 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1744,plain,
( ? [X3: $i] :
( ( $true
!= ( in @ ( kpair @ X3 @ ( ap @ sK548 @ sK546 @ sK547 @ X3 ) ) @ sK547 ) )
& ( $true
= ( in @ X3 @ sK548 ) ) )
=> ( ( ( in @ ( kpair @ sK549 @ ( ap @ sK548 @ sK546 @ sK547 @ sK549 ) ) @ sK547 )
!= $true )
& ( ( in @ sK549 @ sK548 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1742,plain,
( ( ( funcGraphProp1 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X2 @ X0 @ X1 ) )
& ? [X3: $i] :
( ( ( in @ ( kpair @ X3 @ ( ap @ X2 @ X0 @ X1 @ X3 ) ) @ X1 )
!= $true )
& ( $true
= ( in @ X3 @ X2 ) ) ) ) )
& ( ! [X4: $i,X5: $i,X6: $i] :
( ( ( func @ X6 @ X4 @ X5 )
!= $true )
| ! [X7: $i] :
( ( ( in @ ( kpair @ X7 @ ( ap @ X6 @ X4 @ X5 @ X7 ) ) @ X5 )
= $true )
| ( ( in @ X7 @ X6 )
!= $true ) ) )
| ( funcGraphProp1 != $true ) ) ),
inference(rectify,[status(thm)],[f1741]) ).
thf(f1741,plain,
( ( ( funcGraphProp1 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X2 @ X0 @ X1 ) )
& ? [X3: $i] :
( ( ( in @ ( kpair @ X3 @ ( ap @ X2 @ X0 @ X1 @ X3 ) ) @ X1 )
!= $true )
& ( $true
= ( in @ X3 @ X2 ) ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( func @ X2 @ X0 @ X1 ) )
| ! [X3: $i] :
( ( ( in @ ( kpair @ X3 @ ( ap @ X2 @ X0 @ X1 @ X3 ) ) @ X1 )
= $true )
| ( $true
!= ( in @ X3 @ X2 ) ) ) )
| ( funcGraphProp1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f782]) ).
thf(f782,plain,
( ( funcGraphProp1 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( func @ X2 @ X0 @ X1 ) )
| ! [X3: $i] :
( ( ( in @ ( kpair @ X3 @ ( ap @ X2 @ X0 @ X1 @ X3 ) ) @ X1 )
= $true )
| ( $true
!= ( in @ X3 @ X2 ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f317]) ).
thf(f317,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X2 @ X0 @ X1 ) )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X2 ) )
=> ( ( in @ ( kpair @ X3 @ ( ap @ X2 @ X0 @ X1 @ X3 ) ) @ X1 )
= $true ) ) )
<=> ( funcGraphProp1 = $true ) ),
inference(fool_elimination,[status(thm)],[f316]) ).
thf(f316,plain,
( funcGraphProp1
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( func @ X2 @ X0 @ X1 )
=> ! [X3: $i] :
( ( in @ X3 @ X2 )
=> ( in @ ( kpair @ X3 @ ( ap @ X2 @ X0 @ X1 @ X3 ) ) @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f208]) ).
thf(f208,axiom,
( funcGraphProp1
= ( ! [X4: $i,X13: $i,X3: $i] :
( ( func @ X3 @ X4 @ X13 )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ ( kpair @ X1 @ ( ap @ X3 @ X4 @ X13 @ X1 ) ) @ X13 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.ClpnksWlLJ/DTF2THF_22405.p',funcGraphProp1) ).
thf(f2134,plain,
( ( in @ ( kpair @ sK132 @ ( ap @ sK129 @ sK130 @ sK131 @ sK132 ) ) @ sK131 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1070]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU685^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.09 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.11/0.29 % Computer : n031.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 12:31:48 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
% 0.18/0.38 ---- Original DTF file ---
% 0.18/0.38 thf(spec,logic,$$dhol).
% 0.18/0.38 %------------------------------------------------------------------------------
% 0.18/0.38 % File : SEU685^1 : TPTP v9.2.1. Released v3.7.0.
% 0.18/0.38 % Domain : Set Theory
% 0.18/0.38 % Problem : Functions - Extensionality and Beta Reduction
% 0.18/0.38 % Version : Especial.
% 0.18/0.38 % English : (! A:i.! B:i.! f:i.in f (funcSet A B) -> (! x:i.in x A ->
% 0.18/0.38 % in (kpair x (ap A B f x)) f))
% 0.18/0.38
% 0.18/0.38 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.18/0.38 % Source : [Bro08]
% 0.18/0.38 % Names : ZFC187g [Bro08]
% 0.18/0.38
% 0.18/0.38 % Status : Theorem
% 0.18/0.38 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.40 v8.2.0, 0.62 v8.1.0, 0.55 v7.5.0, 0.43 v7.4.0, 0.44 v7.2.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.18/0.38 % Syntax : Number of formulae : 447 ( 208 unt; 238 typ; 208 def)
% 0.18/0.38 % Number of atoms : 1114 ( 304 equ; 0 cnn)
% 0.18/0.38 % Maximal formula atoms : 204 ( 5 avg)
% 0.18/0.38 % Number of connectives : 2056 ( 47 ~; 7 |; 36 &;1451 @)
% 0.18/0.38 % ( 14 <=>; 501 =>; 0 <=; 0 <~>)
% 0.18/0.38 % Maximal formula depth : 215 ( 2 avg)
% 0.18/0.38 % Number of types : 2 ( 0 usr)
% 0.18/0.38 % Number of type conns : 126 ( 126 >; 0 *; 0 +; 0 <<)
% 0.18/0.38 % Number of symbols : 241 ( 238 usr; 205 con; 0-4 aty)
% 0.18/0.38 % Number of variables : 666 ( 73 ^; 557 !; 36 ?; 666 :)
% 0.18/0.38 % SPC : TH0_THM_EQU_NAR
% 0.18/0.38
% 0.18/0.38 % Comments : http://mathgate.info/detsetitem.php?id=412
% 0.18/0.38 %------------------------------------------------------------------------------
% 0.18/0.38 thf(in_type,type,
% 0.18/0.38 in: $i > $i > $o ).
% 0.18/0.38
% 0.18/0.38 thf(exu_type,type,
% 0.18/0.38 exu: ( $i > $o ) > $o ).
% 0.18/0.38
% 0.18/0.38 thf(exu,definition,
% 0.18/0.38 ( exu
% 0.18/0.38 = ( ^ [Xphi: $i > $o] :
% 0.18/0.38 ? [Xx: $i] :
% 0.18/0.38 ( ( Xphi @ Xx )
% 0.18/0.38 & ! [Xy: $i] :
% 0.18/0.38 ( ( Xphi @ Xy )
% 0.18/0.38 => ( Xx = Xy ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setextAx_type,type,
% 0.18/0.38 setextAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setextAx,definition,
% 0.18/0.38 ( setextAx
% 0.18/0.38 = ( ! [A: $i,B: $i] :
% 0.18/0.38 ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 <=> ( in @ Xx @ B ) )
% 0.18/0.38 => ( A = B ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(emptyset_type,type,
% 0.18/0.38 emptyset: $i ).
% 0.18/0.38
% 0.18/0.38 thf(emptysetAx_type,type,
% 0.18/0.38 emptysetAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(emptysetAx,definition,
% 0.18/0.38 ( emptysetAx
% 0.18/0.38 = ( ! [Xx: $i] :
% 0.18/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoin_type,type,
% 0.18/0.38 setadjoin: $i > $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoinAx_type,type,
% 0.18/0.38 setadjoinAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoinAx,definition,
% 0.18/0.38 ( setadjoinAx
% 0.18/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.38 <=> ( ( Xy = Xx )
% 0.18/0.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(powerset_type,type,
% 0.18/0.38 powerset: $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(powersetAx_type,type,
% 0.18/0.38 powersetAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(powersetAx,definition,
% 0.18/0.38 ( powersetAx
% 0.18/0.38 = ( ! [A: $i,B: $i] :
% 0.18/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.18/0.38 <=> ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setunion_type,type,
% 0.18/0.38 setunion: $i > $i ).
% 0.18/0.38
% 0.18/0.38 thf(setunionAx_type,type,
% 0.18/0.38 setunionAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setunionAx,definition,
% 0.18/0.38 ( setunionAx
% 0.18/0.38 = ( ! [A: $i,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.38 <=> ? [B: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 & ( in @ B @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(omega_type,type,
% 0.18/0.38 omega: $i ).
% 0.18/0.38
% 0.18/0.38 thf(omega0Ax_type,type,
% 0.18/0.38 omega0Ax: $o ).
% 0.18/0.38
% 0.18/0.38 thf(omega0Ax,definition,
% 0.18/0.38 ( omega0Ax
% 0.18/0.38 = ( in @ emptyset @ omega ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(omegaSAx_type,type,
% 0.18/0.38 omegaSAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(omegaSAx,definition,
% 0.18/0.38 ( omegaSAx
% 0.18/0.38 = ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ omega )
% 0.18/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(omegaIndAx_type,type,
% 0.18/0.38 omegaIndAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(omegaIndAx,definition,
% 0.18/0.38 ( omegaIndAx
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ( ( ( in @ emptyset @ A )
% 0.18/0.38 & ! [Xx: $i] :
% 0.18/0.38 ( ( ( in @ Xx @ omega )
% 0.18/0.38 & ( in @ Xx @ A ) )
% 0.18/0.38 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.18/0.38 => ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ omega )
% 0.18/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(replAx_type,type,
% 0.18/0.38 replAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(replAx,definition,
% 0.18/0.38 ( replAx
% 0.18/0.38 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.18/0.38 ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( exu
% 0.18/0.38 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.18/0.38 => ? [B: $i] :
% 0.18/0.38 ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 <=> ? [Xy: $i] :
% 0.18/0.38 ( ( in @ Xy @ A )
% 0.18/0.38 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(foundationAx_type,type,
% 0.18/0.38 foundationAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(foundationAx,definition,
% 0.18/0.38 ( foundationAx
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.18/0.38 => ? [B: $i] :
% 0.18/0.38 ( ( in @ B @ A )
% 0.18/0.38 & ~ ? [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(wellorderingAx_type,type,
% 0.18/0.38 wellorderingAx: $o ).
% 0.18/0.38
% 0.18/0.38 thf(wellorderingAx,definition,
% 0.18/0.38 ( wellorderingAx
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ? [B: $i] :
% 0.18/0.38 ( ! [C: $i] :
% 0.18/0.38 ( ( in @ C @ B )
% 0.18/0.38 => ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ C )
% 0.18/0.38 => ( in @ Xx @ A ) ) )
% 0.18/0.38 & ! [Xx: $i,Xy: $i] :
% 0.18/0.38 ( ( ( in @ Xx @ A )
% 0.18/0.38 & ( in @ Xy @ A ) )
% 0.18/0.38 => ( ! [C: $i] :
% 0.18/0.38 ( ( in @ C @ B )
% 0.18/0.38 => ( ( in @ Xx @ C )
% 0.18/0.38 <=> ( in @ Xy @ C ) ) )
% 0.18/0.38 => ( Xx = Xy ) ) )
% 0.18/0.38 & ! [C: $i,D: $i] :
% 0.18/0.38 ( ( ( in @ C @ B )
% 0.18/0.38 & ( in @ D @ B ) )
% 0.18/0.38 => ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ C )
% 0.18/0.38 => ( in @ Xx @ D ) )
% 0.18/0.38 | ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ D )
% 0.18/0.38 => ( in @ Xx @ C ) ) ) )
% 0.18/0.38 & ! [C: $i] :
% 0.18/0.38 ( ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ C )
% 0.18/0.38 => ( in @ Xx @ A ) )
% 0.18/0.38 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.18/0.38 => ? [D: $i,Xx: $i] :
% 0.18/0.38 ( ( in @ D @ B )
% 0.18/0.38 & ( in @ Xx @ C )
% 0.18/0.38 & ~ ? [Xy: $i] :
% 0.18/0.38 ( ( in @ Xy @ D )
% 0.18/0.38 & ( in @ Xy @ C ) )
% 0.18/0.38 & ! [E: $i] :
% 0.18/0.38 ( ( in @ E @ B )
% 0.18/0.38 => ( ! [Xy: $i] :
% 0.18/0.38 ( ( in @ Xy @ E )
% 0.18/0.38 => ( in @ Xy @ D ) )
% 0.18/0.38 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(descr_type,type,
% 0.18/0.38 descr: ( $i > $o ) > $i ).
% 0.18/0.38
% 0.18/0.38 thf(descrp_type,type,
% 0.18/0.38 descrp: $o ).
% 0.18/0.38
% 0.18/0.38 thf(descrp,definition,
% 0.18/0.38 ( descrp
% 0.18/0.38 = ( ! [Xphi: $i > $o] :
% 0.18/0.38 ( ( exu
% 0.18/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.38 => ( Xphi
% 0.18/0.38 @ ( descr
% 0.18/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstr_type,type,
% 0.18/0.38 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstrI_type,type,
% 0.18/0.38 dsetconstrI: $o ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstrI,definition,
% 0.18/0.38 ( dsetconstrI
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( ( Xphi @ Xx )
% 0.18/0.38 => ( in @ Xx
% 0.18/0.38 @ ( dsetconstr @ A
% 0.18/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstrEL_type,type,
% 0.18/0.38 dsetconstrEL: $o ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstrEL,definition,
% 0.18/0.38 ( dsetconstrEL
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx
% 0.18/0.38 @ ( dsetconstr @ A
% 0.18/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.38 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstrER_type,type,
% 0.18/0.38 dsetconstrER: $o ).
% 0.18/0.38
% 0.18/0.38 thf(dsetconstrER,definition,
% 0.18/0.38 ( dsetconstrER
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx
% 0.18/0.38 @ ( dsetconstr @ A
% 0.18/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.38 => ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(exuE1_type,type,
% 0.18/0.38 exuE1: $o ).
% 0.18/0.38
% 0.18/0.38 thf(exuE1,definition,
% 0.18/0.38 ( exuE1
% 0.18/0.38 = ( ! [Xphi: $i > $o] :
% 0.18/0.38 ( ( exu
% 0.18/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.38 => ? [Xx: $i] :
% 0.18/0.38 ( ( Xphi @ Xx )
% 0.18/0.38 & ! [Xy: $i] :
% 0.18/0.38 ( ( Xphi @ Xy )
% 0.18/0.38 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(prop2set_type,type,
% 0.18/0.38 prop2set: $o > $i ).
% 0.18/0.38
% 0.18/0.38 thf(prop2setE_type,type,
% 0.18/0.38 prop2setE: $o ).
% 0.18/0.38
% 0.18/0.38 thf(prop2setE,definition,
% 0.18/0.38 ( prop2setE
% 0.18/0.38 = ( ! [Xphi: $o,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.18/0.38 => Xphi ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(emptysetE_type,type,
% 0.18/0.38 emptysetE: $o ).
% 0.18/0.38
% 0.18/0.38 thf(emptysetE,definition,
% 0.18/0.38 ( emptysetE
% 0.18/0.38 = ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ emptyset )
% 0.18/0.38 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(emptysetimpfalse_type,type,
% 0.18/0.38 emptysetimpfalse: $o ).
% 0.18/0.38
% 0.18/0.38 thf(emptysetimpfalse,definition,
% 0.18/0.38 ( emptysetimpfalse
% 0.18/0.38 = ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ emptyset )
% 0.18/0.38 => $false ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(notinemptyset_type,type,
% 0.18/0.38 notinemptyset: $o ).
% 0.18/0.38
% 0.18/0.38 thf(notinemptyset,definition,
% 0.18/0.38 ( notinemptyset
% 0.18/0.38 = ( ! [Xx: $i] :
% 0.18/0.38 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(exuE3e_type,type,
% 0.18/0.38 exuE3e: $o ).
% 0.18/0.38
% 0.18/0.38 thf(exuE3e,definition,
% 0.18/0.38 ( exuE3e
% 0.18/0.38 = ( ! [Xphi: $i > $o] :
% 0.18/0.38 ( ( exu
% 0.18/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.38 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setext_type,type,
% 0.18/0.38 setext: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setext,definition,
% 0.18/0.38 ( setext
% 0.18/0.38 = ( ! [A: $i,B: $i] :
% 0.18/0.38 ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( in @ Xx @ B ) )
% 0.18/0.38 => ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 => ( in @ Xx @ A ) )
% 0.18/0.38 => ( A = B ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(emptyI_type,type,
% 0.18/0.38 emptyI: $o ).
% 0.18/0.38
% 0.18/0.38 thf(emptyI,definition,
% 0.18/0.38 ( emptyI
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ( ! [Xx: $i] :
% 0.18/0.38 ~ ( in @ Xx @ A )
% 0.18/0.38 => ( A = emptyset ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(noeltsimpempty_type,type,
% 0.18/0.38 noeltsimpempty: $o ).
% 0.18/0.38
% 0.18/0.38 thf(noeltsimpempty,definition,
% 0.18/0.38 ( noeltsimpempty
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ( ! [Xx: $i] :
% 0.18/0.38 ~ ( in @ Xx @ A )
% 0.18/0.38 => ( A = emptyset ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setbeta_type,type,
% 0.18/0.38 setbeta: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setbeta,definition,
% 0.18/0.38 ( setbeta
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( ( in @ Xx
% 0.18/0.38 @ ( dsetconstr @ A
% 0.18/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.18/0.38 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(nonempty_type,type,
% 0.18/0.38 nonempty: $i > $o ).
% 0.18/0.38
% 0.18/0.38 thf(nonempty,definition,
% 0.18/0.38 ( nonempty
% 0.18/0.38 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(nonemptyE1_type,type,
% 0.18/0.38 nonemptyE1: $o ).
% 0.18/0.38
% 0.18/0.38 thf(nonemptyE1,definition,
% 0.18/0.38 ( nonemptyE1
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ( ( nonempty @ A )
% 0.18/0.38 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(nonemptyI_type,type,
% 0.18/0.38 nonemptyI: $o ).
% 0.18/0.38
% 0.18/0.38 thf(nonemptyI,definition,
% 0.18/0.38 ( nonemptyI
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( ( Xphi @ Xx )
% 0.18/0.38 => ( nonempty
% 0.18/0.38 @ ( dsetconstr @ A
% 0.18/0.38 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(nonemptyI1_type,type,
% 0.18/0.38 nonemptyI1: $o ).
% 0.18/0.38
% 0.18/0.38 thf(nonemptyI1,definition,
% 0.18/0.38 ( nonemptyI1
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.18/0.38 => ( nonempty @ A ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoinIL_type,type,
% 0.18/0.38 setadjoinIL: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoinIL,definition,
% 0.18/0.38 ( setadjoinIL
% 0.18/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(emptyinunitempty_type,type,
% 0.18/0.38 emptyinunitempty: $o ).
% 0.18/0.38
% 0.18/0.38 thf(emptyinunitempty,definition,
% 0.18/0.38 ( emptyinunitempty
% 0.18/0.38 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoinIR_type,type,
% 0.18/0.38 setadjoinIR: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoinIR,definition,
% 0.18/0.38 ( setadjoinIR
% 0.18/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.38 ( ( in @ Xy @ A )
% 0.18/0.38 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoinE_type,type,
% 0.18/0.38 setadjoinE: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoinE,definition,
% 0.18/0.38 ( setadjoinE
% 0.18/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.38 => ! [Xphi: $o] :
% 0.18/0.38 ( ( ( Xy = Xx )
% 0.18/0.38 => Xphi )
% 0.18/0.38 => ( ( ( in @ Xy @ A )
% 0.18/0.38 => Xphi )
% 0.18/0.38 => Xphi ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoinOr_type,type,
% 0.18/0.38 setadjoinOr: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setadjoinOr,definition,
% 0.18/0.38 ( setadjoinOr
% 0.18/0.38 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.18/0.38 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.18/0.38 => ( ( Xy = Xx )
% 0.18/0.38 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setoftrueEq_type,type,
% 0.18/0.38 setoftrueEq: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setoftrueEq,definition,
% 0.18/0.38 ( setoftrueEq
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ( ( dsetconstr @ A
% 0.18/0.38 @ ^ [Xx: $i] : $true )
% 0.18/0.38 = A ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(powersetI_type,type,
% 0.18/0.38 powersetI: $o ).
% 0.18/0.38
% 0.18/0.38 thf(powersetI,definition,
% 0.18/0.38 ( powersetI
% 0.18/0.38 = ( ! [A: $i,B: $i] :
% 0.18/0.38 ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 => ( in @ Xx @ A ) )
% 0.18/0.38 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(emptyinPowerset_type,type,
% 0.18/0.38 emptyinPowerset: $o ).
% 0.18/0.38
% 0.18/0.38 thf(emptyinPowerset,definition,
% 0.18/0.38 ( emptyinPowerset
% 0.18/0.38 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(emptyInPowerset_type,type,
% 0.18/0.38 emptyInPowerset: $o ).
% 0.18/0.38
% 0.18/0.38 thf(emptyInPowerset,definition,
% 0.18/0.38 ( emptyInPowerset
% 0.18/0.38 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(powersetE_type,type,
% 0.18/0.38 powersetE: $o ).
% 0.18/0.38
% 0.18/0.38 thf(powersetE,definition,
% 0.18/0.38 ( powersetE
% 0.18/0.38 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.38 ( ( in @ B @ ( powerset @ A ) )
% 0.18/0.38 => ( ( in @ Xx @ B )
% 0.18/0.38 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setunionI_type,type,
% 0.18/0.38 setunionI: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setunionI,definition,
% 0.18/0.38 ( setunionI
% 0.18/0.38 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 => ( ( in @ B @ A )
% 0.18/0.38 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(setunionE_type,type,
% 0.18/0.38 setunionE: $o ).
% 0.18/0.38
% 0.18/0.38 thf(setunionE,definition,
% 0.18/0.38 ( setunionE
% 0.18/0.38 = ( ! [A: $i,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.38 => ! [Xphi: $o] :
% 0.18/0.38 ( ! [B: $i] :
% 0.18/0.38 ( ( in @ Xx @ B )
% 0.18/0.38 => ( ( in @ B @ A )
% 0.18/0.38 => Xphi ) )
% 0.18/0.38 => Xphi ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(subPowSU_type,type,
% 0.18/0.38 subPowSU: $o ).
% 0.18/0.38
% 0.18/0.38 thf(subPowSU,definition,
% 0.18/0.38 ( subPowSU
% 0.18/0.38 = ( ! [A: $i,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(exuE2_type,type,
% 0.18/0.38 exuE2: $o ).
% 0.18/0.38
% 0.18/0.38 thf(exuE2,definition,
% 0.18/0.38 ( exuE2
% 0.18/0.38 = ( ! [Xphi: $i > $o] :
% 0.18/0.38 ( ( exu
% 0.18/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.38 => ? [Xx: $i] :
% 0.18/0.38 ! [Xy: $i] :
% 0.18/0.38 ( ( Xphi @ Xy )
% 0.18/0.38 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(nonemptyImpWitness_type,type,
% 0.18/0.38 nonemptyImpWitness: $o ).
% 0.18/0.38
% 0.18/0.38 thf(nonemptyImpWitness,definition,
% 0.18/0.38 ( nonemptyImpWitness
% 0.18/0.38 = ( ! [A: $i] :
% 0.18/0.38 ( ( nonempty @ A )
% 0.18/0.38 => ? [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 & $true ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(uniqinunit_type,type,
% 0.18/0.38 uniqinunit: $o ).
% 0.18/0.38
% 0.18/0.38 thf(uniqinunit,definition,
% 0.18/0.38 ( uniqinunit
% 0.18/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.38 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.38 => ( Xx = Xy ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(notinsingleton_type,type,
% 0.18/0.38 notinsingleton: $o ).
% 0.18/0.38
% 0.18/0.38 thf(notinsingleton,definition,
% 0.18/0.38 ( notinsingleton
% 0.18/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.38 ( ( Xx != Xy )
% 0.18/0.38 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(eqinunit_type,type,
% 0.18/0.38 eqinunit: $o ).
% 0.18/0.38
% 0.18/0.38 thf(eqinunit,definition,
% 0.18/0.38 ( eqinunit
% 0.18/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.38 ( ( Xx = Xy )
% 0.18/0.38 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(singletonsswitch_type,type,
% 0.18/0.38 singletonsswitch: $o ).
% 0.18/0.38
% 0.18/0.38 thf(singletonsswitch,definition,
% 0.18/0.38 ( singletonsswitch
% 0.18/0.38 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.38 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.38 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(upairsetE_type,type,
% 0.18/0.38 upairsetE: $o ).
% 0.18/0.38
% 0.18/0.38 thf(upairsetE,definition,
% 0.18/0.38 ( upairsetE
% 0.18/0.38 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.18/0.38 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.18/0.38 => ( ( Xz = Xx )
% 0.18/0.38 | ( Xz = Xy ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(upairsetIL_type,type,
% 0.18/0.38 upairsetIL: $o ).
% 0.18/0.38
% 0.18/0.38 thf(upairsetIL,definition,
% 0.18/0.38 ( upairsetIL
% 0.18/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(upairsetIR_type,type,
% 0.18/0.38 upairsetIR: $o ).
% 0.18/0.38
% 0.18/0.38 thf(upairsetIR,definition,
% 0.18/0.38 ( upairsetIR
% 0.18/0.38 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(emptyE1_type,type,
% 0.18/0.38 emptyE1: $o ).
% 0.18/0.38
% 0.18/0.38 thf(emptyE1,definition,
% 0.18/0.38 ( emptyE1
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.38 ( ? [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 & ( Xphi @ Xx ) )
% 0.18/0.38 => ( ( ( dsetconstr @ A
% 0.18/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.38 = emptyset )
% 0.18/0.38 => $false ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(vacuousDall_type,type,
% 0.18/0.38 vacuousDall: $o ).
% 0.18/0.38
% 0.18/0.38 thf(vacuousDall,definition,
% 0.18/0.38 ( vacuousDall
% 0.18/0.38 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ emptyset )
% 0.18/0.38 => ( Xphi @ Xx ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(quantDeMorgan1_type,type,
% 0.18/0.38 quantDeMorgan1: $o ).
% 0.18/0.38
% 0.18/0.38 thf(quantDeMorgan1,definition,
% 0.18/0.38 ( quantDeMorgan1
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.38 ( ~ ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( Xphi @ Xx ) )
% 0.18/0.38 => ? [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(quantDeMorgan2_type,type,
% 0.18/0.38 quantDeMorgan2: $o ).
% 0.18/0.38
% 0.18/0.38 thf(quantDeMorgan2,definition,
% 0.18/0.38 ( quantDeMorgan2
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.38 ( ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ~ ( Xphi @ Xx ) )
% 0.18/0.38 => ~ ? [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(quantDeMorgan3_type,type,
% 0.18/0.38 quantDeMorgan3: $o ).
% 0.18/0.38
% 0.18/0.38 thf(quantDeMorgan3,definition,
% 0.18/0.38 ( quantDeMorgan3
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.38 ( ~ ? [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 & ( Xphi @ Xx ) )
% 0.18/0.38 => ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(quantDeMorgan4_type,type,
% 0.18/0.38 quantDeMorgan4: $o ).
% 0.18/0.38
% 0.18/0.38 thf(quantDeMorgan4,definition,
% 0.18/0.38 ( quantDeMorgan4
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.38 ( ? [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 & ~ ( Xphi @ Xx ) )
% 0.18/0.38 => ~ ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(prop2setI_type,type,
% 0.18/0.38 prop2setI: $o ).
% 0.18/0.38
% 0.18/0.38 thf(prop2setI,definition,
% 0.18/0.38 ( prop2setI
% 0.18/0.38 = ( ! [Xphi: $o] :
% 0.18/0.38 ( Xphi
% 0.18/0.38 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(set2prop_type,type,
% 0.18/0.38 set2prop: $i > $o ).
% 0.18/0.38
% 0.18/0.38 thf(prop2set2propI_type,type,
% 0.18/0.38 prop2set2propI: $o ).
% 0.18/0.38
% 0.18/0.38 thf(prop2set2propI,definition,
% 0.18/0.38 ( prop2set2propI
% 0.18/0.38 = ( ! [Xphi: $o] :
% 0.18/0.38 ( Xphi
% 0.18/0.38 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(notdexE_type,type,
% 0.18/0.38 notdexE: $o ).
% 0.18/0.38
% 0.18/0.38 thf(notdexE,definition,
% 0.18/0.38 ( notdexE
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.38 ( ~ ? [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 & ( Xphi @ Xx ) )
% 0.18/0.38 => ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(notdallE_type,type,
% 0.18/0.38 notdallE: $o ).
% 0.18/0.38
% 0.18/0.38 thf(notdallE,definition,
% 0.18/0.38 ( notdallE
% 0.18/0.38 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.38 ( ~ ! [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 => ( Xphi @ Xx ) )
% 0.18/0.38 => ? [Xx: $i] :
% 0.18/0.38 ( ( in @ Xx @ A )
% 0.18/0.38 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(exuI1_type,type,
% 0.18/0.38 exuI1: $o ).
% 0.18/0.38
% 0.18/0.38 thf(exuI1,definition,
% 0.18/0.38 ( exuI1
% 0.18/0.38 = ( ! [Xphi: $i > $o] :
% 0.18/0.38 ( ? [Xx: $i] :
% 0.18/0.38 ( ( Xphi @ Xx )
% 0.18/0.38 & ! [Xy: $i] :
% 0.18/0.38 ( ( Xphi @ Xy )
% 0.18/0.38 => ( Xx = Xy ) ) )
% 0.18/0.38 => ( exu
% 0.18/0.38 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.38
% 0.18/0.38 thf(exuI3_type,type,
% 0.18/0.38 exuI3: $o ).
% 0.18/0.38
% 0.18/0.38 thf(exuI3,definition,
% 0.18/0.38 ( exuI3
% 0.18/0.38 = ( ! [Xphi: $i > $o] :
% 0.18/0.38 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.18/0.38 => ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xx )
% 0.18/0.39 => ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xx = Xy ) ) )
% 0.18/0.39 => ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuI2_type,type,
% 0.18/0.39 exuI2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuI2,definition,
% 0.18/0.39 ( exuI2
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ? [Xx: $i] :
% 0.18/0.39 ! [Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xy )
% 0.18/0.39 <=> ( Xy = Xx ) )
% 0.18/0.39 => ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(inCongP_type,type,
% 0.18/0.39 inCongP: $o ).
% 0.18/0.39
% 0.18/0.39 thf(inCongP,definition,
% 0.18/0.39 ( inCongP
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(in__Cong_type,type,
% 0.18/0.39 in__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(in__Cong,definition,
% 0.18/0.39 ( in__Cong
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ( ( in @ Xx @ A )
% 0.18/0.39 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuE3u_type,type,
% 0.18/0.39 exuE3u: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuE3u,definition,
% 0.18/0.39 ( exuE3u
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xx )
% 0.18/0.39 => ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exu__Cong_type,type,
% 0.18/0.39 exu__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exu__Cong,definition,
% 0.18/0.39 ( exu__Cong
% 0.18/0.39 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.18/0.39 ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ( ( Xphi @ Xx )
% 0.18/0.39 <=> ( Xpsi @ Xy ) ) )
% 0.18/0.39 => ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 <=> ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptyset__Cong_type,type,
% 0.18/0.39 emptyset__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptyset__Cong,definition,
% 0.18/0.39 ( emptyset__Cong
% 0.18/0.39 = ( emptyset = emptyset ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoin__Cong_type,type,
% 0.18/0.39 setadjoin__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoin__Cong,definition,
% 0.18/0.39 ( setadjoin__Cong
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ! [Xz: $i,Xu: $i] :
% 0.18/0.39 ( ( Xz = Xu )
% 0.18/0.39 => ( ( setadjoin @ Xx @ Xz )
% 0.18/0.39 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(powerset__Cong_type,type,
% 0.18/0.39 powerset__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(powerset__Cong,definition,
% 0.18/0.39 ( powerset__Cong
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ( ( powerset @ A )
% 0.18/0.39 = ( powerset @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setunion__Cong_type,type,
% 0.18/0.39 setunion__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setunion__Cong,definition,
% 0.18/0.39 ( setunion__Cong
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ( ( setunion @ A )
% 0.18/0.39 = ( setunion @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(omega__Cong_type,type,
% 0.18/0.39 omega__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(omega__Cong,definition,
% 0.18/0.39 ( omega__Cong
% 0.18/0.39 = ( omega = omega ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(exuEu_type,type,
% 0.18/0.39 exuEu: $o ).
% 0.18/0.39
% 0.18/0.39 thf(exuEu,definition,
% 0.18/0.39 ( exuEu
% 0.18/0.39 = ( ! [Xphi: $i > $o] :
% 0.18/0.39 ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xphi @ Xx )
% 0.18/0.39 => ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(descr__Cong_type,type,
% 0.18/0.39 descr__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(descr__Cong,definition,
% 0.18/0.39 ( descr__Cong
% 0.18/0.39 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.18/0.39 ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ( ( Xphi @ Xx )
% 0.18/0.39 <=> ( Xpsi @ Xy ) ) )
% 0.18/0.39 => ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ( ( exu
% 0.18/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.18/0.39 => ( ( descr
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 = ( descr
% 0.18/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(dsetconstr__Cong_type,type,
% 0.18/0.39 dsetconstr__Cong: $o ).
% 0.18/0.39
% 0.18/0.39 thf(dsetconstr__Cong,definition,
% 0.18/0.39 ( dsetconstr__Cong
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( ( Xx = Xy )
% 0.18/0.39 => ( ( Xphi @ Xx )
% 0.18/0.39 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.18/0.39 => ( ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 = ( dsetconstr @ B
% 0.18/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subset_type,type,
% 0.18/0.39 subset: $i > $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(disjoint_type,type,
% 0.18/0.39 disjoint: $i > $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(setsmeet_type,type,
% 0.18/0.39 setsmeet: $i > $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetI1_type,type,
% 0.18/0.39 subsetI1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetI1,definition,
% 0.18/0.39 ( subsetI1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ B ) )
% 0.18/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(eqimpsubset2_type,type,
% 0.18/0.39 eqimpsubset2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(eqimpsubset2,definition,
% 0.18/0.39 ( eqimpsubset2
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ( subset @ B @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(eqimpsubset1_type,type,
% 0.18/0.39 eqimpsubset1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(eqimpsubset1,definition,
% 0.18/0.39 ( eqimpsubset1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( A = B )
% 0.18/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetI2_type,type,
% 0.18/0.39 subsetI2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetI2,definition,
% 0.18/0.39 ( subsetI2
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ B ) )
% 0.18/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(emptysetsubset_type,type,
% 0.18/0.39 emptysetsubset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(emptysetsubset,definition,
% 0.18/0.39 ( emptysetsubset
% 0.18/0.39 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetE_type,type,
% 0.18/0.39 subsetE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetE,definition,
% 0.18/0.39 ( subsetE
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetE2_type,type,
% 0.18/0.39 subsetE2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetE2,definition,
% 0.18/0.39 ( subsetE2
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notsubsetI_type,type,
% 0.18/0.39 notsubsetI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notsubsetI,definition,
% 0.18/0.39 ( notsubsetI
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notequalI1_type,type,
% 0.18/0.39 notequalI1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notequalI1,definition,
% 0.18/0.39 ( notequalI1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ~ ( subset @ A @ B )
% 0.18/0.39 => ( A != B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(notequalI2_type,type,
% 0.18/0.39 notequalI2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(notequalI2,definition,
% 0.18/0.39 ( notequalI2
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ( A != B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetRefl_type,type,
% 0.18/0.39 subsetRefl: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetRefl,definition,
% 0.18/0.39 ( subsetRefl
% 0.18/0.39 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetTrans_type,type,
% 0.18/0.39 subsetTrans: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetTrans,definition,
% 0.18/0.39 ( subsetTrans
% 0.18/0.39 = ( ! [A: $i,B: $i,C: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( ( subset @ B @ C )
% 0.18/0.39 => ( subset @ A @ C ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinSub_type,type,
% 0.18/0.39 setadjoinSub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinSub,definition,
% 0.18/0.39 ( setadjoinSub
% 0.18/0.39 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinSub2_type,type,
% 0.18/0.39 setadjoinSub2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setadjoinSub2,definition,
% 0.18/0.39 ( setadjoinSub2
% 0.18/0.39 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subset2powerset_type,type,
% 0.18/0.39 subset2powerset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subset2powerset,definition,
% 0.18/0.39 ( subset2powerset
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setextsub_type,type,
% 0.18/0.39 setextsub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setextsub,definition,
% 0.18/0.39 ( setextsub
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( ( subset @ B @ A )
% 0.18/0.39 => ( A = B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(subsetemptysetimpeq_type,type,
% 0.18/0.39 subsetemptysetimpeq: $o ).
% 0.18/0.39
% 0.18/0.39 thf(subsetemptysetimpeq,definition,
% 0.18/0.39 ( subsetemptysetimpeq
% 0.18/0.39 = ( ! [A: $i] :
% 0.18/0.39 ( ( subset @ A @ emptyset )
% 0.18/0.39 => ( A = emptyset ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(powersetI1_type,type,
% 0.18/0.39 powersetI1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(powersetI1,definition,
% 0.18/0.39 ( powersetI1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( subset @ B @ A )
% 0.18/0.39 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(powersetE1_type,type,
% 0.18/0.39 powersetE1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(powersetE1,definition,
% 0.18/0.39 ( powersetE1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.18/0.39 => ( subset @ B @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(inPowerset_type,type,
% 0.18/0.39 inPowerset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(inPowerset,definition,
% 0.18/0.39 ( inPowerset
% 0.18/0.39 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(powersetsubset_type,type,
% 0.18/0.39 powersetsubset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(powersetsubset,definition,
% 0.18/0.39 ( powersetsubset
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(sepInPowerset_type,type,
% 0.18/0.39 sepInPowerset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(sepInPowerset,definition,
% 0.18/0.39 ( sepInPowerset
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( in
% 0.18/0.39 @ ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 @ ( powerset @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(sepSubset_type,type,
% 0.18/0.39 sepSubset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(sepSubset,definition,
% 0.18/0.39 ( sepSubset
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( subset
% 0.18/0.39 @ ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 @ A ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binunion_type,type,
% 0.18/0.39 binunion: $i > $i > $i ).
% 0.18/0.39
% 0.18/0.39 thf(binunionIL_type,type,
% 0.18/0.39 binunionIL: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binunionIL,definition,
% 0.18/0.39 ( binunionIL
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(upairset2IR_type,type,
% 0.18/0.39 upairset2IR: $o ).
% 0.18/0.39
% 0.18/0.39 thf(upairset2IR,definition,
% 0.18/0.39 ( upairset2IR
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binunionIR_type,type,
% 0.18/0.39 binunionIR: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binunionIR,definition,
% 0.18/0.39 ( binunionIR
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ B )
% 0.18/0.39 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binunionEcases_type,type,
% 0.18/0.39 binunionEcases: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binunionEcases,definition,
% 0.18/0.39 ( binunionEcases
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.18/0.39 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.18/0.39 => ( ( ( in @ Xx @ A )
% 0.18/0.39 => Xphi )
% 0.18/0.39 => ( ( ( in @ Xx @ B )
% 0.18/0.39 => Xphi )
% 0.18/0.39 => Xphi ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binunionE_type,type,
% 0.18/0.39 binunionE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binunionE,definition,
% 0.18/0.39 ( binunionE
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.18/0.39 => ( ( in @ Xx @ A )
% 0.18/0.39 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binunionLsub_type,type,
% 0.18/0.39 binunionLsub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binunionLsub,definition,
% 0.18/0.39 ( binunionLsub
% 0.18/0.39 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binunionRsub_type,type,
% 0.18/0.39 binunionRsub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binunionRsub,definition,
% 0.18/0.39 ( binunionRsub
% 0.18/0.39 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersect_type,type,
% 0.18/0.39 binintersect: $i > $i > $i ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectI_type,type,
% 0.18/0.39 binintersectI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectI,definition,
% 0.18/0.39 ( binintersectI
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ( in @ Xx @ B )
% 0.18/0.39 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset5_type,type,
% 0.18/0.39 binintersectSubset5: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset5,definition,
% 0.18/0.39 ( binintersectSubset5
% 0.18/0.39 = ( ! [A: $i,B: $i,C: $i] :
% 0.18/0.39 ( ( subset @ C @ A )
% 0.18/0.39 => ( ( subset @ C @ B )
% 0.18/0.39 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectEL_type,type,
% 0.18/0.39 binintersectEL: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectEL,definition,
% 0.18/0.39 ( binintersectEL
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.18/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectLsub_type,type,
% 0.18/0.39 binintersectLsub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectLsub,definition,
% 0.18/0.39 ( binintersectLsub
% 0.18/0.39 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset2_type,type,
% 0.18/0.39 binintersectSubset2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset2,definition,
% 0.18/0.39 ( binintersectSubset2
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( ( binintersect @ A @ B )
% 0.18/0.39 = A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset3_type,type,
% 0.18/0.39 binintersectSubset3: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset3,definition,
% 0.18/0.39 ( binintersectSubset3
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( ( binintersect @ A @ B )
% 0.18/0.39 = B )
% 0.18/0.39 => ( subset @ B @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectER_type,type,
% 0.18/0.39 binintersectER: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectER,definition,
% 0.18/0.39 ( binintersectER
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.18/0.39 => ( in @ Xx @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(disjointsetsI1_type,type,
% 0.18/0.39 disjointsetsI1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(disjointsetsI1,definition,
% 0.18/0.39 ( disjointsetsI1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ~ ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ( in @ Xx @ B ) )
% 0.18/0.39 => ( ( binintersect @ A @ B )
% 0.18/0.39 = emptyset ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectRsub_type,type,
% 0.18/0.39 binintersectRsub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectRsub,definition,
% 0.18/0.39 ( binintersectRsub
% 0.18/0.39 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset4_type,type,
% 0.18/0.39 binintersectSubset4: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset4,definition,
% 0.18/0.39 ( binintersectSubset4
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( subset @ B @ A )
% 0.18/0.39 => ( ( binintersect @ A @ B )
% 0.18/0.39 = B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset1_type,type,
% 0.18/0.39 binintersectSubset1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(binintersectSubset1,definition,
% 0.18/0.39 ( binintersectSubset1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( ( binintersect @ A @ B )
% 0.18/0.39 = A )
% 0.18/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(bs114d_type,type,
% 0.18/0.39 bs114d: $o ).
% 0.18/0.39
% 0.18/0.39 thf(bs114d,definition,
% 0.18/0.39 ( bs114d
% 0.18/0.39 = ( ! [A: $i,B: $i,C: $i] :
% 0.18/0.39 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.18/0.39 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(regular_type,type,
% 0.18/0.39 regular: $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminus_type,type,
% 0.18/0.39 setminus: $i > $i > $i ).
% 0.18/0.39
% 0.18/0.39 thf(setminusI_type,type,
% 0.18/0.39 setminusI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusI,definition,
% 0.18/0.39 ( setminusI
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusEL_type,type,
% 0.18/0.39 setminusEL: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusEL,definition,
% 0.18/0.39 ( setminusEL
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.18/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusER_type,type,
% 0.18/0.39 setminusER: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusER,definition,
% 0.18/0.39 ( setminusER
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.18/0.39 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusSubset2_type,type,
% 0.18/0.39 setminusSubset2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusSubset2,definition,
% 0.18/0.39 ( setminusSubset2
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( subset @ A @ B )
% 0.18/0.39 => ( ( setminus @ A @ B )
% 0.18/0.39 = emptyset ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusERneg_type,type,
% 0.18/0.39 setminusERneg: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusERneg,definition,
% 0.18/0.39 ( setminusERneg
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.18/0.39 => ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusELneg_type,type,
% 0.18/0.39 setminusELneg: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusELneg,definition,
% 0.18/0.39 ( setminusELneg
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusILneg_type,type,
% 0.18/0.39 setminusILneg: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusILneg,definition,
% 0.18/0.39 ( setminusILneg
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ~ ( in @ Xx @ A )
% 0.18/0.39 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusIRneg_type,type,
% 0.18/0.39 setminusIRneg: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusIRneg,definition,
% 0.18/0.39 ( setminusIRneg
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ B )
% 0.18/0.39 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusLsub_type,type,
% 0.18/0.39 setminusLsub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusLsub,definition,
% 0.18/0.39 ( setminusLsub
% 0.18/0.39 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setminusSubset1_type,type,
% 0.18/0.39 setminusSubset1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setminusSubset1,definition,
% 0.18/0.39 ( setminusSubset1
% 0.18/0.39 = ( ! [A: $i,B: $i] :
% 0.18/0.39 ( ( ( setminus @ A @ B )
% 0.18/0.39 = emptyset )
% 0.18/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(symdiff_type,type,
% 0.18/0.39 symdiff: $i > $i > $i ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffE_type,type,
% 0.18/0.39 symdiffE: $o ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffE,definition,
% 0.18/0.39 ( symdiffE
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.18/0.39 => ! [Xphi: $o] :
% 0.18/0.39 ( ( ( in @ Xx @ A )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => Xphi ) )
% 0.18/0.39 => ( ( ~ ( in @ Xx @ A )
% 0.18/0.39 => ( ( in @ Xx @ B )
% 0.18/0.39 => Xphi ) )
% 0.18/0.39 => Xphi ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffI1_type,type,
% 0.18/0.39 symdiffI1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffI1,definition,
% 0.18/0.39 ( symdiffI1
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffI2_type,type,
% 0.18/0.39 symdiffI2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffI2,definition,
% 0.18/0.39 ( symdiffI2
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ~ ( in @ Xx @ A )
% 0.18/0.39 => ( ( in @ Xx @ B )
% 0.18/0.39 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffIneg1_type,type,
% 0.18/0.39 symdiffIneg1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffIneg1,definition,
% 0.18/0.39 ( symdiffIneg1
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ( in @ Xx @ B )
% 0.18/0.39 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffIneg2_type,type,
% 0.18/0.39 symdiffIneg2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(symdiffIneg2,definition,
% 0.18/0.39 ( symdiffIneg2
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ~ ( in @ Xx @ A )
% 0.18/0.39 => ( ~ ( in @ Xx @ B )
% 0.18/0.39 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(iskpair_type,type,
% 0.18/0.39 iskpair: $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(secondinupair_type,type,
% 0.18/0.39 secondinupair: $o ).
% 0.18/0.39
% 0.18/0.39 thf(secondinupair,definition,
% 0.18/0.39 ( secondinupair
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairIL_type,type,
% 0.18/0.39 setukpairIL: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairIL,definition,
% 0.18/0.39 ( setukpairIL
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairIR_type,type,
% 0.18/0.39 setukpairIR: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairIR,definition,
% 0.18/0.39 ( setukpairIR
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(kpairiskpair_type,type,
% 0.18/0.39 kpairiskpair: $o ).
% 0.18/0.39
% 0.18/0.39 thf(kpairiskpair,definition,
% 0.18/0.39 ( kpairiskpair
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(kpair_type,type,
% 0.18/0.39 kpair: $i > $i > $i ).
% 0.18/0.39
% 0.18/0.39 thf(kpairp_type,type,
% 0.18/0.39 kpairp: $o ).
% 0.18/0.39
% 0.18/0.39 thf(kpairp,definition,
% 0.18/0.39 ( kpairp
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(cartprod_type,type,
% 0.18/0.39 cartprod: $i > $i > $i ).
% 0.18/0.39
% 0.18/0.39 thf(singletonsubset_type,type,
% 0.18/0.39 singletonsubset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(singletonsubset,definition,
% 0.18/0.39 ( singletonsubset
% 0.18/0.39 = ( ! [A: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(singletoninpowerset_type,type,
% 0.18/0.39 singletoninpowerset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(singletoninpowerset,definition,
% 0.18/0.39 ( singletoninpowerset
% 0.18/0.39 = ( ! [A: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(singletoninpowunion_type,type,
% 0.18/0.39 singletoninpowunion: $o ).
% 0.18/0.39
% 0.18/0.39 thf(singletoninpowunion,definition,
% 0.18/0.39 ( singletoninpowunion
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(upairset2E_type,type,
% 0.18/0.39 upairset2E: $o ).
% 0.18/0.39
% 0.18/0.39 thf(upairset2E,definition,
% 0.18/0.39 ( upairset2E
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.18/0.39 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.18/0.39 => ( ( Xz = Xx )
% 0.18/0.39 | ( Xz = Xy ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(upairsubunion_type,type,
% 0.18/0.39 upairsubunion: $o ).
% 0.18/0.39
% 0.18/0.39 thf(upairsubunion,definition,
% 0.18/0.39 ( upairsubunion
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(upairinpowunion_type,type,
% 0.18/0.39 upairinpowunion: $o ).
% 0.18/0.39
% 0.18/0.39 thf(upairinpowunion,definition,
% 0.18/0.39 ( upairinpowunion
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(ubforcartprodlem1_type,type,
% 0.18/0.39 ubforcartprodlem1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(ubforcartprodlem1,definition,
% 0.18/0.39 ( ubforcartprodlem1
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(ubforcartprodlem2_type,type,
% 0.18/0.39 ubforcartprodlem2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(ubforcartprodlem2,definition,
% 0.18/0.39 ( ubforcartprodlem2
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(ubforcartprodlem3_type,type,
% 0.18/0.39 ubforcartprodlem3: $o ).
% 0.18/0.39
% 0.18/0.39 thf(ubforcartprodlem3,definition,
% 0.18/0.39 ( ubforcartprodlem3
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodpairin_type,type,
% 0.18/0.39 cartprodpairin: $o ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodpairin,definition,
% 0.18/0.39 ( cartprodpairin
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodmempair1_type,type,
% 0.18/0.39 cartprodmempair1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodmempair1,definition,
% 0.18/0.39 ( cartprodmempair1
% 0.18/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.18/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.18/0.39 => ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ? [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 & ( Xu
% 0.18/0.39 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodmempair_type,type,
% 0.18/0.39 cartprodmempair: $o ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodmempair,definition,
% 0.18/0.39 ( cartprodmempair
% 0.18/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.18/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.18/0.39 => ( iskpair @ Xu ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setunionE2_type,type,
% 0.18/0.39 setunionE2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setunionE2,definition,
% 0.18/0.39 ( setunionE2
% 0.18/0.39 = ( ! [A: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.18/0.39 => ? [X: $i] :
% 0.18/0.39 ( ( in @ X @ A )
% 0.18/0.39 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setunionsingleton1_type,type,
% 0.18/0.39 setunionsingleton1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setunionsingleton1,definition,
% 0.18/0.39 ( setunionsingleton1
% 0.18/0.39 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setunionsingleton2_type,type,
% 0.18/0.39 setunionsingleton2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setunionsingleton2,definition,
% 0.18/0.39 ( setunionsingleton2
% 0.18/0.39 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setunionsingleton_type,type,
% 0.18/0.39 setunionsingleton: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setunionsingleton,definition,
% 0.18/0.39 ( setunionsingleton
% 0.18/0.39 = ( ! [Xx: $i] :
% 0.18/0.39 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.18/0.39 = Xx ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(singleton_type,type,
% 0.18/0.39 singleton: $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(singleton,definition,
% 0.18/0.39 ( singleton
% 0.18/0.39 = ( ^ [A: $i] :
% 0.18/0.39 ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ( A
% 0.18/0.39 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(singletonprop_type,type,
% 0.18/0.39 singletonprop: $o ).
% 0.18/0.39
% 0.18/0.39 thf(singletonprop,definition,
% 0.18/0.39 ( singletonprop
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ A )
% 0.18/0.39 => ( ( Xphi @ Xx )
% 0.18/0.39 => ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xx = Xy ) ) ) ) )
% 0.18/0.39 => ( ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ( Xphi @ Xx ) )
% 0.18/0.39 => ( singleton
% 0.18/0.39 @ ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(ex1_type,type,
% 0.18/0.39 ex1: $i > ( $i > $o ) > $o ).
% 0.18/0.39
% 0.18/0.39 thf(ex1,definition,
% 0.18/0.39 ( ex1
% 0.18/0.39 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( singleton
% 0.18/0.39 @ ( dsetconstr @ A
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(ex1E1_type,type,
% 0.18/0.39 ex1E1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(ex1E1,definition,
% 0.18/0.39 ( ex1E1
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ( ex1 @ A
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(ex1I_type,type,
% 0.18/0.39 ex1I: $o ).
% 0.18/0.39
% 0.18/0.39 thf(ex1I,definition,
% 0.18/0.39 ( ex1I
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ( Xphi @ Xx )
% 0.18/0.39 => ( ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ A )
% 0.18/0.39 => ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xy = Xx ) ) )
% 0.18/0.39 => ( ex1 @ A
% 0.18/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(ex1I2_type,type,
% 0.18/0.39 ex1I2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(ex1I2,definition,
% 0.18/0.39 ( ex1I2
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ A )
% 0.18/0.39 => ( ( Xphi @ Xx )
% 0.18/0.39 => ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xx = Xy ) ) ) ) )
% 0.18/0.39 => ( ? [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 & ( Xphi @ Xx ) )
% 0.18/0.39 => ( ex1 @ A
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(singletonsuniq_type,type,
% 0.18/0.39 singletonsuniq: $o ).
% 0.18/0.39
% 0.18/0.39 thf(singletonsuniq,definition,
% 0.18/0.39 ( singletonsuniq
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( ( setadjoin @ Xx @ emptyset )
% 0.18/0.39 = ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(atmost1p_type,type,
% 0.18/0.39 atmost1p: $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(atleast2p_type,type,
% 0.18/0.39 atleast2p: $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(atmost2p_type,type,
% 0.18/0.39 atmost2p: $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(upairsetp_type,type,
% 0.18/0.39 upairsetp: $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairinjL1_type,type,
% 0.18/0.39 setukpairinjL1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairinjL1,definition,
% 0.18/0.39 ( setukpairinjL1
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.18/0.39 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.18/0.39 => ( Xx = Xz ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(kfstsingleton_type,type,
% 0.18/0.39 kfstsingleton: $o ).
% 0.18/0.39
% 0.18/0.39 thf(kfstsingleton,definition,
% 0.18/0.39 ( kfstsingleton
% 0.18/0.39 = ( ! [Xu: $i] :
% 0.18/0.39 ( ( iskpair @ Xu )
% 0.18/0.39 => ( singleton
% 0.18/0.39 @ ( dsetconstr @ ( setunion @ Xu )
% 0.18/0.39 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(theprop_type,type,
% 0.18/0.39 theprop: $o ).
% 0.18/0.39
% 0.18/0.39 thf(theprop,definition,
% 0.18/0.39 ( theprop
% 0.18/0.39 = ( ! [X: $i] :
% 0.18/0.39 ( ( singleton @ X )
% 0.18/0.39 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(kfst_type,type,
% 0.18/0.39 kfst: $i > $i ).
% 0.18/0.39
% 0.18/0.39 thf(kfstpairEq_type,type,
% 0.18/0.39 kfstpairEq: $o ).
% 0.18/0.39
% 0.18/0.39 thf(kfstpairEq,definition,
% 0.18/0.39 ( kfstpairEq
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.18/0.39 = Xx ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodfstin_type,type,
% 0.18/0.39 cartprodfstin: $o ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodfstin,definition,
% 0.18/0.39 ( cartprodfstin
% 0.18/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.18/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.18/0.39 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairinjL2_type,type,
% 0.18/0.39 setukpairinjL2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairinjL2,definition,
% 0.18/0.39 ( setukpairinjL2
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.18/0.39 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.18/0.39 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.18/0.39 => ( Xx = Xz ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairinjL_type,type,
% 0.18/0.39 setukpairinjL: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairinjL,definition,
% 0.18/0.39 ( setukpairinjL
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.18/0.39 ( ( ( kpair @ Xx @ Xy )
% 0.18/0.39 = ( kpair @ Xz @ Xu ) )
% 0.18/0.39 => ( Xx = Xz ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairinjR11_type,type,
% 0.18/0.39 setukpairinjR11: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairinjR11,definition,
% 0.18/0.39 ( setukpairinjR11
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.39 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairinjR12_type,type,
% 0.18/0.39 setukpairinjR12: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairinjR12,definition,
% 0.18/0.39 ( setukpairinjR12
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xx = Xy )
% 0.18/0.39 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.18/0.39 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairinjR1_type,type,
% 0.18/0.39 setukpairinjR1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairinjR1,definition,
% 0.18/0.39 ( setukpairinjR1
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.18/0.39 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.18/0.39 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.18/0.39 => ( ( Xz = Xu )
% 0.18/0.39 => ( Xy = Xu ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(upairequniteq_type,type,
% 0.18/0.39 upairequniteq: $o ).
% 0.18/0.39
% 0.18/0.39 thf(upairequniteq,definition,
% 0.18/0.39 ( upairequniteq
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.18/0.39 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.18/0.39 = ( setadjoin @ Xz @ emptyset ) )
% 0.18/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairinjR2_type,type,
% 0.18/0.39 setukpairinjR2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairinjR2,definition,
% 0.18/0.39 ( setukpairinjR2
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.18/0.39 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.18/0.39 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.18/0.39 => ( Xy = Xu ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairinjR_type,type,
% 0.18/0.39 setukpairinjR: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setukpairinjR,definition,
% 0.18/0.39 ( setukpairinjR
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.18/0.39 ( ( ( kpair @ Xx @ Xy )
% 0.18/0.39 = ( kpair @ Xz @ Xu ) )
% 0.18/0.39 => ( Xy = Xu ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(ksndsingleton_type,type,
% 0.18/0.39 ksndsingleton: $o ).
% 0.18/0.39
% 0.18/0.39 thf(ksndsingleton,definition,
% 0.18/0.39 ( ksndsingleton
% 0.18/0.39 = ( ! [Xu: $i] :
% 0.18/0.39 ( ( iskpair @ Xu )
% 0.18/0.39 => ( singleton
% 0.18/0.39 @ ( dsetconstr @ ( setunion @ Xu )
% 0.18/0.39 @ ^ [Xx: $i] :
% 0.18/0.39 ( Xu
% 0.18/0.39 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(ksnd_type,type,
% 0.18/0.39 ksnd: $i > $i ).
% 0.18/0.39
% 0.18/0.39 thf(ksndpairEq_type,type,
% 0.18/0.39 ksndpairEq: $o ).
% 0.18/0.39
% 0.18/0.39 thf(ksndpairEq,definition,
% 0.18/0.39 ( ksndpairEq
% 0.18/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.18/0.39 = Xy ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(kpairsurjEq_type,type,
% 0.18/0.39 kpairsurjEq: $o ).
% 0.18/0.39
% 0.18/0.39 thf(kpairsurjEq,definition,
% 0.18/0.39 ( kpairsurjEq
% 0.18/0.39 = ( ! [Xu: $i] :
% 0.18/0.39 ( ( iskpair @ Xu )
% 0.18/0.39 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.18/0.39 = Xu ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodsndin_type,type,
% 0.18/0.39 cartprodsndin: $o ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodsndin,definition,
% 0.18/0.39 ( cartprodsndin
% 0.18/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.18/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.18/0.39 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodpairmemEL_type,type,
% 0.18/0.39 cartprodpairmemEL: $o ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodpairmemEL,definition,
% 0.18/0.39 ( cartprodpairmemEL
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.18/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodpairmemER_type,type,
% 0.18/0.39 cartprodpairmemER: $o ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodpairmemER,definition,
% 0.18/0.39 ( cartprodpairmemER
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.18/0.39 => ( in @ Xy @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodmempaircEq_type,type,
% 0.18/0.39 cartprodmempaircEq: $o ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodmempaircEq,definition,
% 0.18/0.39 ( cartprodmempaircEq
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( ( kpair @ Xx @ Xy )
% 0.18/0.39 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodfstpairEq_type,type,
% 0.18/0.39 cartprodfstpairEq: $o ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodfstpairEq,definition,
% 0.18/0.39 ( cartprodfstpairEq
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.18/0.39 = Xx ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodsndpairEq_type,type,
% 0.18/0.39 cartprodsndpairEq: $o ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodsndpairEq,definition,
% 0.18/0.39 ( cartprodsndpairEq
% 0.18/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.18/0.39 = Xy ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodpairsurjEq_type,type,
% 0.18/0.39 cartprodpairsurjEq: $o ).
% 0.18/0.39
% 0.18/0.39 thf(cartprodpairsurjEq,definition,
% 0.18/0.39 ( cartprodpairsurjEq
% 0.18/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.18/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.18/0.39 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.18/0.39 = Xu ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(breln_type,type,
% 0.18/0.39 breln: $i > $i > $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(breln,definition,
% 0.18/0.39 ( breln
% 0.18/0.39 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(dpsetconstr_type,type,
% 0.18/0.39 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.18/0.39
% 0.18/0.39 thf(dpsetconstrI_type,type,
% 0.18/0.39 dpsetconstrI: $o ).
% 0.18/0.39
% 0.18/0.39 thf(dpsetconstrI,definition,
% 0.18/0.39 ( dpsetconstrI
% 0.18/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( ( Xphi @ Xx @ Xy )
% 0.18/0.39 => ( in @ ( kpair @ Xx @ Xy )
% 0.18/0.39 @ ( dpsetconstr @ A @ B
% 0.18/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(dpsetconstrSub_type,type,
% 0.18/0.39 dpsetconstrSub: $o ).
% 0.18/0.39
% 0.18/0.39 thf(dpsetconstrSub,definition,
% 0.18/0.39 ( dpsetconstrSub
% 0.18/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.18/0.39 ( subset
% 0.18/0.39 @ ( dpsetconstr @ A @ B
% 0.18/0.39 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.18/0.39 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(setOfPairsIsBReln_type,type,
% 0.18/0.39 setOfPairsIsBReln: $o ).
% 0.18/0.39
% 0.18/0.39 thf(setOfPairsIsBReln,definition,
% 0.18/0.39 ( setOfPairsIsBReln
% 0.18/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.18/0.39 ( breln @ A @ B
% 0.18/0.39 @ ( dpsetconstr @ A @ B
% 0.18/0.39 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(dpsetconstrERa_type,type,
% 0.18/0.39 dpsetconstrERa: $o ).
% 0.18/0.39
% 0.18/0.39 thf(dpsetconstrERa,definition,
% 0.18/0.39 ( dpsetconstrERa
% 0.18/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.18/0.39 @ ( dpsetconstr @ A @ B
% 0.18/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.18/0.39 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(dpsetconstrEL1_type,type,
% 0.18/0.39 dpsetconstrEL1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(dpsetconstrEL1,definition,
% 0.18/0.39 ( dpsetconstrEL1
% 0.18/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( in @ ( kpair @ Xx @ Xy )
% 0.18/0.39 @ ( dpsetconstr @ A @ B
% 0.18/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.18/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(dpsetconstrEL2_type,type,
% 0.18/0.39 dpsetconstrEL2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(dpsetconstrEL2,definition,
% 0.18/0.39 ( dpsetconstrEL2
% 0.18/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( in @ ( kpair @ Xx @ Xy )
% 0.18/0.39 @ ( dpsetconstr @ A @ B
% 0.18/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.18/0.39 => ( in @ Xy @ B ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(dpsetconstrER_type,type,
% 0.18/0.39 dpsetconstrER: $o ).
% 0.18/0.39
% 0.18/0.39 thf(dpsetconstrER,definition,
% 0.18/0.39 ( dpsetconstrER
% 0.18/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( in @ ( kpair @ Xx @ Xy )
% 0.18/0.39 @ ( dpsetconstr @ A @ B
% 0.18/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.18/0.39 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(func_type,type,
% 0.18/0.39 func: $i > $i > $i > $o ).
% 0.18/0.39
% 0.18/0.39 thf(func,definition,
% 0.18/0.39 ( func
% 0.18/0.39 = ( ^ [A: $i,B: $i,R: $i] :
% 0.18/0.39 ( ( breln @ A @ B @ R )
% 0.18/0.39 & ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( ex1 @ B
% 0.18/0.39 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(funcSet_type,type,
% 0.18/0.39 funcSet: $i > $i > $i ).
% 0.18/0.39
% 0.18/0.39 thf(funcImageSingleton_type,type,
% 0.18/0.39 funcImageSingleton: $o ).
% 0.18/0.39
% 0.18/0.39 thf(funcImageSingleton,definition,
% 0.18/0.39 ( funcImageSingleton
% 0.18/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.18/0.39 ( ( func @ A @ B @ Xf )
% 0.18/0.39 => ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( singleton
% 0.18/0.39 @ ( dsetconstr @ B
% 0.18/0.39 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(apProp_type,type,
% 0.18/0.39 apProp: $o ).
% 0.18/0.39
% 0.18/0.39 thf(apProp,definition,
% 0.18/0.39 ( apProp
% 0.18/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.18/0.39 ( ( func @ A @ B @ Xf )
% 0.18/0.39 => ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in
% 0.18/0.39 @ ( setunion
% 0.18/0.39 @ ( dsetconstr @ B
% 0.18/0.39 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.18/0.39 @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(ap_type,type,
% 0.18/0.39 ap: $i > $i > $i > $i > $i ).
% 0.18/0.39
% 0.18/0.39 thf(app_type,type,
% 0.18/0.39 app: $o ).
% 0.18/0.39
% 0.18/0.39 thf(app,definition,
% 0.18/0.39 ( app
% 0.18/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.18/0.39 ( ( func @ A @ B @ Xf )
% 0.18/0.39 => ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(infuncsetfunc_type,type,
% 0.18/0.39 infuncsetfunc: $o ).
% 0.18/0.39
% 0.18/0.39 thf(infuncsetfunc,definition,
% 0.18/0.39 ( infuncsetfunc
% 0.18/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.18/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.18/0.39 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(ap2p_type,type,
% 0.18/0.39 ap2p: $o ).
% 0.18/0.39
% 0.18/0.39 thf(ap2p,definition,
% 0.18/0.39 ( ap2p
% 0.18/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.18/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.18/0.39 => ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(funcinfuncset_type,type,
% 0.18/0.39 funcinfuncset: $o ).
% 0.18/0.39
% 0.18/0.39 thf(funcinfuncset,definition,
% 0.18/0.39 ( funcinfuncset
% 0.18/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.18/0.39 ( ( func @ A @ B @ Xf )
% 0.18/0.39 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(lamProp_type,type,
% 0.18/0.39 lamProp: $o ).
% 0.18/0.39
% 0.18/0.39 thf(lamProp,definition,
% 0.18/0.39 ( lamProp
% 0.18/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.18/0.39 => ( func @ A @ B
% 0.18/0.39 @ ( dpsetconstr @ A @ B
% 0.18/0.39 @ ^ [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xf @ Xx )
% 0.18/0.39 = Xy ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(lam_type,type,
% 0.18/0.39 lam: $i > $i > ( $i > $i ) > $i ).
% 0.18/0.39
% 0.18/0.39 thf(lam,definition,
% 0.18/0.39 ( lam
% 0.18/0.39 = ( ^ [A: $i,B: $i,Xf: $i > $i] :
% 0.18/0.39 ( dpsetconstr @ A @ B
% 0.18/0.39 @ ^ [Xx: $i,Xy: $i] :
% 0.18/0.39 ( ( Xf @ Xx )
% 0.18/0.39 = Xy ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(lamp_type,type,
% 0.18/0.39 lamp: $o ).
% 0.18/0.39
% 0.18/0.39 thf(lamp,definition,
% 0.18/0.39 ( lamp
% 0.18/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.18/0.39 => ( func @ A @ B
% 0.18/0.39 @ ( lam @ A @ B
% 0.18/0.39 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(lam2p_type,type,
% 0.18/0.39 lam2p: $o ).
% 0.18/0.39
% 0.18/0.39 thf(lam2p,definition,
% 0.18/0.39 ( lam2p
% 0.18/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.18/0.39 => ( in
% 0.18/0.39 @ ( lam @ A @ B
% 0.18/0.39 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.18/0.39 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(brelnall1_type,type,
% 0.18/0.39 brelnall1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(brelnall1,definition,
% 0.18/0.39 ( brelnall1
% 0.18/0.39 = ( ! [A: $i,B: $i,R: $i] :
% 0.18/0.39 ( ( breln @ A @ B @ R )
% 0.18/0.39 => ! [Xphi: $i > $o] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.18/0.39 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.18/0.39 => ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ R )
% 0.18/0.39 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(brelnall2_type,type,
% 0.18/0.39 brelnall2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(brelnall2,definition,
% 0.18/0.39 ( brelnall2
% 0.18/0.39 = ( ! [A: $i,B: $i,R: $i] :
% 0.18/0.39 ( ( breln @ A @ B @ R )
% 0.18/0.39 => ! [Xphi: $i > $o] :
% 0.18/0.39 ( ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ B )
% 0.18/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.18/0.39 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.18/0.39 => ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ R )
% 0.18/0.39 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(ex1E2_type,type,
% 0.18/0.39 ex1E2: $o ).
% 0.18/0.39
% 0.18/0.39 thf(ex1E2,definition,
% 0.18/0.39 ( ex1E2
% 0.18/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.18/0.39 ( ( ex1 @ A
% 0.18/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.18/0.39 => ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ! [Xy: $i] :
% 0.18/0.39 ( ( in @ Xy @ A )
% 0.18/0.39 => ( ( Xphi @ Xx )
% 0.18/0.39 => ( ( Xphi @ Xy )
% 0.18/0.39 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(funcGraphProp1_type,type,
% 0.18/0.39 funcGraphProp1: $o ).
% 0.18/0.39
% 0.18/0.39 thf(funcGraphProp1,definition,
% 0.18/0.39 ( funcGraphProp1
% 0.18/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.18/0.39 ( ( func @ A @ B @ Xf )
% 0.18/0.39 => ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 thf(funcGraphProp3,conjecture,
% 0.18/0.39 ( setextAx
% 0.18/0.39 => ( emptysetAx
% 0.18/0.39 => ( setadjoinAx
% 0.18/0.39 => ( powersetAx
% 0.18/0.39 => ( setunionAx
% 0.18/0.39 => ( omega0Ax
% 0.18/0.39 => ( omegaSAx
% 0.18/0.39 => ( omegaIndAx
% 0.18/0.39 => ( replAx
% 0.18/0.39 => ( foundationAx
% 0.18/0.39 => ( wellorderingAx
% 0.18/0.39 => ( descrp
% 0.18/0.39 => ( dsetconstrI
% 0.18/0.39 => ( dsetconstrEL
% 0.18/0.39 => ( dsetconstrER
% 0.18/0.39 => ( exuE1
% 0.18/0.39 => ( prop2setE
% 0.18/0.39 => ( emptysetE
% 0.18/0.39 => ( emptysetimpfalse
% 0.18/0.39 => ( notinemptyset
% 0.18/0.39 => ( exuE3e
% 0.18/0.39 => ( setext
% 0.18/0.39 => ( emptyI
% 0.18/0.39 => ( noeltsimpempty
% 0.18/0.39 => ( setbeta
% 0.18/0.39 => ( nonemptyE1
% 0.18/0.39 => ( nonemptyI
% 0.18/0.39 => ( nonemptyI1
% 0.18/0.39 => ( setadjoinIL
% 0.18/0.39 => ( emptyinunitempty
% 0.18/0.39 => ( setadjoinIR
% 0.18/0.39 => ( setadjoinE
% 0.18/0.39 => ( setadjoinOr
% 0.18/0.39 => ( setoftrueEq
% 0.18/0.39 => ( powersetI
% 0.18/0.39 => ( emptyinPowerset
% 0.18/0.39 => ( emptyInPowerset
% 0.18/0.39 => ( powersetE
% 0.18/0.39 => ( setunionI
% 0.18/0.39 => ( setunionE
% 0.18/0.39 => ( subPowSU
% 0.18/0.39 => ( exuE2
% 0.18/0.39 => ( nonemptyImpWitness
% 0.18/0.39 => ( uniqinunit
% 0.18/0.39 => ( notinsingleton
% 0.18/0.39 => ( eqinunit
% 0.18/0.39 => ( singletonsswitch
% 0.18/0.39 => ( upairsetE
% 0.18/0.39 => ( upairsetIL
% 0.18/0.39 => ( upairsetIR
% 0.18/0.39 => ( emptyE1
% 0.18/0.39 => ( vacuousDall
% 0.18/0.39 => ( quantDeMorgan1
% 0.18/0.39 => ( quantDeMorgan2
% 0.18/0.39 => ( quantDeMorgan3
% 0.18/0.39 => ( quantDeMorgan4
% 0.18/0.39 => ( prop2setI
% 0.18/0.39 => ( prop2set2propI
% 0.18/0.39 => ( notdexE
% 0.18/0.39 => ( notdallE
% 0.18/0.39 => ( exuI1
% 0.18/0.39 => ( exuI3
% 0.18/0.39 => ( exuI2
% 0.18/0.39 => ( inCongP
% 0.18/0.39 => ( in__Cong
% 0.18/0.39 => ( exuE3u
% 0.18/0.39 => ( exu__Cong
% 0.18/0.39 => ( emptyset__Cong
% 0.18/0.39 => ( setadjoin__Cong
% 0.18/0.39 => ( powerset__Cong
% 0.18/0.39 => ( setunion__Cong
% 0.18/0.39 => ( omega__Cong
% 0.18/0.39 => ( exuEu
% 0.18/0.39 => ( descr__Cong
% 0.18/0.39 => ( dsetconstr__Cong
% 0.18/0.39 => ( subsetI1
% 0.18/0.39 => ( eqimpsubset2
% 0.18/0.39 => ( eqimpsubset1
% 0.18/0.39 => ( subsetI2
% 0.18/0.39 => ( emptysetsubset
% 0.18/0.39 => ( subsetE
% 0.18/0.39 => ( subsetE2
% 0.18/0.39 => ( notsubsetI
% 0.18/0.39 => ( notequalI1
% 0.18/0.39 => ( notequalI2
% 0.18/0.39 => ( subsetRefl
% 0.18/0.39 => ( subsetTrans
% 0.18/0.39 => ( setadjoinSub
% 0.18/0.39 => ( setadjoinSub2
% 0.18/0.39 => ( subset2powerset
% 0.18/0.39 => ( setextsub
% 0.18/0.39 => ( subsetemptysetimpeq
% 0.18/0.39 => ( powersetI1
% 0.18/0.39 => ( powersetE1
% 0.18/0.39 => ( inPowerset
% 0.18/0.39 => ( powersetsubset
% 0.18/0.39 => ( sepInPowerset
% 0.18/0.39 => ( sepSubset
% 0.18/0.39 => ( binunionIL
% 0.18/0.39 => ( upairset2IR
% 0.18/0.39 => ( binunionIR
% 0.18/0.39 => ( binunionEcases
% 0.18/0.39 => ( binunionE
% 0.18/0.39 => ( binunionLsub
% 0.18/0.39 => ( binunionRsub
% 0.18/0.39 => ( binintersectI
% 0.18/0.39 => ( binintersectSubset5
% 0.18/0.39 => ( binintersectEL
% 0.18/0.39 => ( binintersectLsub
% 0.18/0.39 => ( binintersectSubset2
% 0.18/0.39 => ( binintersectSubset3
% 0.18/0.39 => ( binintersectER
% 0.18/0.39 => ( disjointsetsI1
% 0.18/0.39 => ( binintersectRsub
% 0.18/0.39 => ( binintersectSubset4
% 0.18/0.39 => ( binintersectSubset1
% 0.18/0.39 => ( bs114d
% 0.18/0.39 => ( setminusI
% 0.18/0.39 => ( setminusEL
% 0.18/0.39 => ( setminusER
% 0.18/0.39 => ( setminusSubset2
% 0.18/0.39 => ( setminusERneg
% 0.18/0.39 => ( setminusELneg
% 0.18/0.39 => ( setminusILneg
% 0.18/0.39 => ( setminusIRneg
% 0.18/0.39 => ( setminusLsub
% 0.18/0.39 => ( setminusSubset1
% 0.18/0.39 => ( symdiffE
% 0.18/0.39 => ( symdiffI1
% 0.18/0.39 => ( symdiffI2
% 0.18/0.39 => ( symdiffIneg1
% 0.18/0.39 => ( symdiffIneg2
% 0.18/0.39 => ( secondinupair
% 0.18/0.39 => ( setukpairIL
% 0.18/0.39 => ( setukpairIR
% 0.18/0.39 => ( kpairiskpair
% 0.18/0.39 => ( kpairp
% 0.18/0.39 => ( singletonsubset
% 0.18/0.39 => ( singletoninpowerset
% 0.18/0.39 => ( singletoninpowunion
% 0.18/0.39 => ( upairset2E
% 0.18/0.39 => ( upairsubunion
% 0.18/0.39 => ( upairinpowunion
% 0.18/0.39 => ( ubforcartprodlem1
% 0.18/0.39 => ( ubforcartprodlem2
% 0.18/0.39 => ( ubforcartprodlem3
% 0.18/0.39 => ( cartprodpairin
% 0.18/0.39 => ( cartprodmempair1
% 0.18/0.39 => ( cartprodmempair
% 0.18/0.39 => ( setunionE2
% 0.18/0.39 => ( setunionsingleton1
% 0.18/0.39 => ( setunionsingleton2
% 0.18/0.39 => ( setunionsingleton
% 0.18/0.39 => ( singletonprop
% 0.18/0.39 => ( ex1E1
% 0.18/0.39 => ( ex1I
% 0.18/0.39 => ( ex1I2
% 0.18/0.39 => ( singletonsuniq
% 0.18/0.39 => ( setukpairinjL1
% 0.18/0.39 => ( kfstsingleton
% 0.18/0.39 => ( theprop
% 0.18/0.39 => ( kfstpairEq
% 0.18/0.39 => ( cartprodfstin
% 0.18/0.39 => ( setukpairinjL2
% 0.18/0.39 => ( setukpairinjL
% 0.18/0.39 => ( setukpairinjR11
% 0.18/0.39 => ( setukpairinjR12
% 0.18/0.39 => ( setukpairinjR1
% 0.18/0.39 => ( upairequniteq
% 0.18/0.39 => ( setukpairinjR2
% 0.18/0.39 => ( setukpairinjR
% 0.18/0.39 => ( ksndsingleton
% 0.18/0.39 => ( ksndpairEq
% 0.18/0.39 => ( kpairsurjEq
% 0.18/0.39 => ( cartprodsndin
% 0.18/0.39 => ( cartprodpairmemEL
% 0.18/0.39 => ( cartprodpairmemER
% 0.18/0.39 => ( cartprodmempaircEq
% 0.18/0.39 => ( cartprodfstpairEq
% 0.18/0.39 => ( cartprodsndpairEq
% 0.18/0.39 => ( cartprodpairsurjEq
% 0.18/0.39 => ( dpsetconstrI
% 0.18/0.39 => ( dpsetconstrSub
% 0.18/0.39 => ( setOfPairsIsBReln
% 0.18/0.39 => ( dpsetconstrERa
% 0.18/0.39 => ( dpsetconstrEL1
% 0.18/0.39 => ( dpsetconstrEL2
% 0.18/0.39 => ( dpsetconstrER
% 0.18/0.39 => ( funcImageSingleton
% 0.18/0.39 => ( apProp
% 0.18/0.39 => ( app
% 0.18/0.39 => ( infuncsetfunc
% 0.18/0.39 => ( ap2p
% 0.18/0.39 => ( funcinfuncset
% 0.18/0.39 => ( lamProp
% 0.18/0.39 => ( lamp
% 0.18/0.39 => ( lam2p
% 0.18/0.39 => ( brelnall1
% 0.18/0.39 => ( brelnall2
% 0.18/0.39 => ( ex1E2
% 0.18/0.39 => ( funcGraphProp1
% 0.18/0.39 => ! [A: $i,B: $i,Xf: $i] :
% 0.18/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.18/0.39 => ! [Xx: $i] :
% 0.18/0.39 ( ( in @ Xx @ A )
% 0.18/0.39 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.18/0.39
% 0.18/0.39 %------------------------------------------------------------------------------
% 0.18/0.39 ------------------------
% 2.58/1.59 ---- Embedded in THF ---
% 2.58/1.59 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.58/1.59 %%% Generated on Tue Feb 24 12:31:49 EST 2026
% 2.58/1.59 %%% using '$$dhol' embedding, version 1.3.0.
% 2.58/1.59 %%% Logic specification used:
% 2.58/1.59 %%% thf(spec, logic, $$dhol).
% 2.58/1.59
% 2.58/1.59 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.ClpnksWlLJ/DTF2DTF22405.p
% See solution above
% 2.58/1.59 ------------------------
% 2.83/1.63 ---- Cleaned THF ---
% 2.83/1.63 thf(in_type,type,
% 2.83/1.63 in: $i > $i > $o ).
% 2.83/1.63
% 2.83/1.63 thf(exu_type,type,
% 2.83/1.63 exu: ( $i > $o ) > $o ).
% 2.83/1.63
% 2.83/1.63 thf(exu,definition,
% 2.83/1.63 ( exu
% 2.83/1.63 = ( ^ [Xphi: $i > $o] :
% 2.83/1.63 ? [Xx: $i] :
% 2.83/1.63 ( ( Xphi @ Xx )
% 2.83/1.63 & ! [Xy: $i] :
% 2.83/1.63 ( ( Xphi @ Xy )
% 2.83/1.63 => ( Xx = Xy ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setextAx_type,type,
% 2.83/1.63 setextAx: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setextAx,definition,
% 2.83/1.63 ( setextAx
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 <=> ( in @ Xx @ B ) )
% 2.83/1.63 => ( A = B ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(emptyset_type,type,
% 2.83/1.63 emptyset: $i ).
% 2.83/1.63
% 2.83/1.63 thf(emptysetAx_type,type,
% 2.83/1.63 emptysetAx: $o ).
% 2.83/1.63
% 2.83/1.63 thf(emptysetAx,definition,
% 2.83/1.63 ( emptysetAx
% 2.83/1.63 = ( ! [Xx: $i] :
% 2.83/1.63 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoin_type,type,
% 2.83/1.63 setadjoin: $i > $i > $i ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoinAx_type,type,
% 2.83/1.63 setadjoinAx: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoinAx,definition,
% 2.83/1.63 ( setadjoinAx
% 2.83/1.63 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.83/1.63 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.83/1.63 <=> ( ( Xy = Xx )
% 2.83/1.63 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(powerset_type,type,
% 2.83/1.63 powerset: $i > $i ).
% 2.83/1.63
% 2.83/1.63 thf(powersetAx_type,type,
% 2.83/1.63 powersetAx: $o ).
% 2.83/1.63
% 2.83/1.63 thf(powersetAx,definition,
% 2.83/1.63 ( powersetAx
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( in @ B @ ( powerset @ A ) )
% 2.83/1.63 <=> ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ B )
% 2.83/1.63 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setunion_type,type,
% 2.83/1.63 setunion: $i > $i ).
% 2.83/1.63
% 2.83/1.63 thf(setunionAx_type,type,
% 2.83/1.63 setunionAx: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setunionAx,definition,
% 2.83/1.63 ( setunionAx
% 2.83/1.63 = ( ! [A: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ ( setunion @ A ) )
% 2.83/1.63 <=> ? [B: $i] :
% 2.83/1.63 ( ( in @ Xx @ B )
% 2.83/1.63 & ( in @ B @ A ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(omega_type,type,
% 2.83/1.63 omega: $i ).
% 2.83/1.63
% 2.83/1.63 thf(omega0Ax_type,type,
% 2.83/1.63 omega0Ax: $o ).
% 2.83/1.63
% 2.83/1.63 thf(omega0Ax,definition,
% 2.83/1.63 ( omega0Ax
% 2.83/1.63 = ( in @ emptyset @ omega ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(omegaSAx_type,type,
% 2.83/1.63 omegaSAx: $o ).
% 2.83/1.63
% 2.83/1.63 thf(omegaSAx,definition,
% 2.83/1.63 ( omegaSAx
% 2.83/1.63 = ( ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ omega )
% 2.83/1.63 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(omegaIndAx_type,type,
% 2.83/1.63 omegaIndAx: $o ).
% 2.83/1.63
% 2.83/1.63 thf(omegaIndAx,definition,
% 2.83/1.63 ( omegaIndAx
% 2.83/1.63 = ( ! [A: $i] :
% 2.83/1.63 ( ( ( in @ emptyset @ A )
% 2.83/1.63 & ! [Xx: $i] :
% 2.83/1.63 ( ( ( in @ Xx @ omega )
% 2.83/1.63 & ( in @ Xx @ A ) )
% 2.83/1.63 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.83/1.63 => ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ omega )
% 2.83/1.63 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(replAx_type,type,
% 2.83/1.63 replAx: $o ).
% 2.83/1.63
% 2.83/1.63 thf(replAx,definition,
% 2.83/1.63 ( replAx
% 2.83/1.63 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.83/1.63 ( ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( exu
% 2.83/1.63 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.83/1.63 => ? [B: $i] :
% 2.83/1.63 ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ B )
% 2.83/1.63 <=> ? [Xy: $i] :
% 2.83/1.63 ( ( in @ Xy @ A )
% 2.83/1.63 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(foundationAx_type,type,
% 2.83/1.63 foundationAx: $o ).
% 2.83/1.63
% 2.83/1.63 thf(foundationAx,definition,
% 2.83/1.63 ( foundationAx
% 2.83/1.63 = ( ! [A: $i] :
% 2.83/1.63 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.83/1.63 => ? [B: $i] :
% 2.83/1.63 ( ( in @ B @ A )
% 2.83/1.63 & ~ ? [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ B )
% 2.83/1.63 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(wellorderingAx_type,type,
% 2.83/1.63 wellorderingAx: $o ).
% 2.83/1.63
% 2.83/1.63 thf(wellorderingAx,definition,
% 2.83/1.63 ( wellorderingAx
% 2.83/1.63 = ( ! [A: $i] :
% 2.83/1.63 ? [B: $i] :
% 2.83/1.63 ( ! [C: $i] :
% 2.83/1.63 ( ( in @ C @ B )
% 2.83/1.63 => ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ C )
% 2.83/1.63 => ( in @ Xx @ A ) ) )
% 2.83/1.63 & ! [Xx: $i,Xy: $i] :
% 2.83/1.63 ( ( ( in @ Xx @ A )
% 2.83/1.63 & ( in @ Xy @ A ) )
% 2.83/1.63 => ( ! [C: $i] :
% 2.83/1.63 ( ( in @ C @ B )
% 2.83/1.63 => ( ( in @ Xx @ C )
% 2.83/1.63 <=> ( in @ Xy @ C ) ) )
% 2.83/1.63 => ( Xx = Xy ) ) )
% 2.83/1.63 & ! [C: $i,D: $i] :
% 2.83/1.63 ( ( ( in @ C @ B )
% 2.83/1.63 & ( in @ D @ B ) )
% 2.83/1.63 => ( ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ C )
% 2.83/1.63 => ( in @ Xx @ D ) )
% 2.83/1.63 | ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ D )
% 2.83/1.63 => ( in @ Xx @ C ) ) ) )
% 2.83/1.63 & ! [C: $i] :
% 2.83/1.63 ( ( ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ C )
% 2.83/1.63 => ( in @ Xx @ A ) )
% 2.83/1.63 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.83/1.63 => ? [D: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ D @ B )
% 2.83/1.63 & ( in @ Xx @ C )
% 2.83/1.63 & ~ ? [Xy: $i] :
% 2.83/1.63 ( ( in @ Xy @ D )
% 2.83/1.63 & ( in @ Xy @ C ) )
% 2.83/1.63 & ! [E: $i] :
% 2.83/1.63 ( ( in @ E @ B )
% 2.83/1.63 => ( ! [Xy: $i] :
% 2.83/1.63 ( ( in @ Xy @ E )
% 2.83/1.63 => ( in @ Xy @ D ) )
% 2.83/1.63 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(descr_type,type,
% 2.83/1.63 descr: ( $i > $o ) > $i ).
% 2.83/1.63
% 2.83/1.63 thf(descrp_type,type,
% 2.83/1.63 descrp: $o ).
% 2.83/1.63
% 2.83/1.63 thf(descrp,definition,
% 2.83/1.63 ( descrp
% 2.83/1.63 = ( ! [Xphi: $i > $o] :
% 2.83/1.63 ( ( exu
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.83/1.63 => ( Xphi
% 2.83/1.63 @ ( descr
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(dsetconstr_type,type,
% 2.83/1.63 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.83/1.63
% 2.83/1.63 thf(dsetconstrI_type,type,
% 2.83/1.63 dsetconstrI: $o ).
% 2.83/1.63
% 2.83/1.63 thf(dsetconstrI,definition,
% 2.83/1.63 ( dsetconstrI
% 2.83/1.63 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( ( Xphi @ Xx )
% 2.83/1.63 => ( in @ Xx
% 2.83/1.63 @ ( dsetconstr @ A
% 2.83/1.63 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(dsetconstrEL_type,type,
% 2.83/1.63 dsetconstrEL: $o ).
% 2.83/1.63
% 2.83/1.63 thf(dsetconstrEL,definition,
% 2.83/1.63 ( dsetconstrEL
% 2.83/1.63 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx
% 2.83/1.63 @ ( dsetconstr @ A
% 2.83/1.63 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.83/1.63 => ( in @ Xx @ A ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(dsetconstrER_type,type,
% 2.83/1.63 dsetconstrER: $o ).
% 2.83/1.63
% 2.83/1.63 thf(dsetconstrER,definition,
% 2.83/1.63 ( dsetconstrER
% 2.83/1.63 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx
% 2.83/1.63 @ ( dsetconstr @ A
% 2.83/1.63 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.83/1.63 => ( Xphi @ Xx ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(exuE1_type,type,
% 2.83/1.63 exuE1: $o ).
% 2.83/1.63
% 2.83/1.63 thf(exuE1,definition,
% 2.83/1.63 ( exuE1
% 2.83/1.63 = ( ! [Xphi: $i > $o] :
% 2.83/1.63 ( ( exu
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.83/1.63 => ? [Xx: $i] :
% 2.83/1.63 ( ( Xphi @ Xx )
% 2.83/1.63 & ! [Xy: $i] :
% 2.83/1.63 ( ( Xphi @ Xy )
% 2.83/1.63 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(prop2set_type,type,
% 2.83/1.63 prop2set: $o > $i ).
% 2.83/1.63
% 2.83/1.63 thf(prop2setE_type,type,
% 2.83/1.63 prop2setE: $o ).
% 2.83/1.63
% 2.83/1.63 thf(prop2setE,definition,
% 2.83/1.63 ( prop2setE
% 2.83/1.63 = ( ! [Xphi: $o,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.83/1.63 => Xphi ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(emptysetE_type,type,
% 2.83/1.63 emptysetE: $o ).
% 2.83/1.63
% 2.83/1.63 thf(emptysetE,definition,
% 2.83/1.63 ( emptysetE
% 2.83/1.63 = ( ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ emptyset )
% 2.83/1.63 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(emptysetimpfalse_type,type,
% 2.83/1.63 emptysetimpfalse: $o ).
% 2.83/1.63
% 2.83/1.63 thf(emptysetimpfalse,definition,
% 2.83/1.63 ( emptysetimpfalse
% 2.83/1.63 = ( ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ emptyset )
% 2.83/1.63 => $false ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(notinemptyset_type,type,
% 2.83/1.63 notinemptyset: $o ).
% 2.83/1.63
% 2.83/1.63 thf(notinemptyset,definition,
% 2.83/1.63 ( notinemptyset
% 2.83/1.63 = ( ! [Xx: $i] :
% 2.83/1.63 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(exuE3e_type,type,
% 2.83/1.63 exuE3e: $o ).
% 2.83/1.63
% 2.83/1.63 thf(exuE3e,definition,
% 2.83/1.63 ( exuE3e
% 2.83/1.63 = ( ! [Xphi: $i > $o] :
% 2.83/1.63 ( ( exu
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.83/1.63 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setext_type,type,
% 2.83/1.63 setext: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setext,definition,
% 2.83/1.63 ( setext
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( in @ Xx @ B ) )
% 2.83/1.63 => ( ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ B )
% 2.83/1.63 => ( in @ Xx @ A ) )
% 2.83/1.63 => ( A = B ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(emptyI_type,type,
% 2.83/1.63 emptyI: $o ).
% 2.83/1.63
% 2.83/1.63 thf(emptyI,definition,
% 2.83/1.63 ( emptyI
% 2.83/1.63 = ( ! [A: $i] :
% 2.83/1.63 ( ! [Xx: $i] :
% 2.83/1.63 ~ ( in @ Xx @ A )
% 2.83/1.63 => ( A = emptyset ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(noeltsimpempty_type,type,
% 2.83/1.63 noeltsimpempty: $o ).
% 2.83/1.63
% 2.83/1.63 thf(noeltsimpempty,definition,
% 2.83/1.63 ( noeltsimpempty
% 2.83/1.63 = ( ! [A: $i] :
% 2.83/1.63 ( ! [Xx: $i] :
% 2.83/1.63 ~ ( in @ Xx @ A )
% 2.83/1.63 => ( A = emptyset ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setbeta_type,type,
% 2.83/1.63 setbeta: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setbeta,definition,
% 2.83/1.63 ( setbeta
% 2.83/1.63 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( ( in @ Xx
% 2.83/1.63 @ ( dsetconstr @ A
% 2.83/1.63 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.83/1.63 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(nonempty_type,type,
% 2.83/1.63 nonempty: $i > $o ).
% 2.83/1.63
% 2.83/1.63 thf(nonempty,definition,
% 2.83/1.63 ( nonempty
% 2.83/1.63 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(nonemptyE1_type,type,
% 2.83/1.63 nonemptyE1: $o ).
% 2.83/1.63
% 2.83/1.63 thf(nonemptyE1,definition,
% 2.83/1.63 ( nonemptyE1
% 2.83/1.63 = ( ! [A: $i] :
% 2.83/1.63 ( ( nonempty @ A )
% 2.83/1.63 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(nonemptyI_type,type,
% 2.83/1.63 nonemptyI: $o ).
% 2.83/1.63
% 2.83/1.63 thf(nonemptyI,definition,
% 2.83/1.63 ( nonemptyI
% 2.83/1.63 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( ( Xphi @ Xx )
% 2.83/1.63 => ( nonempty
% 2.83/1.63 @ ( dsetconstr @ A
% 2.83/1.63 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(nonemptyI1_type,type,
% 2.83/1.63 nonemptyI1: $o ).
% 2.83/1.63
% 2.83/1.63 thf(nonemptyI1,definition,
% 2.83/1.63 ( nonemptyI1
% 2.83/1.63 = ( ! [A: $i] :
% 2.83/1.63 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.83/1.63 => ( nonempty @ A ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoinIL_type,type,
% 2.83/1.63 setadjoinIL: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoinIL,definition,
% 2.83/1.63 ( setadjoinIL
% 2.83/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(emptyinunitempty_type,type,
% 2.83/1.63 emptyinunitempty: $o ).
% 2.83/1.63
% 2.83/1.63 thf(emptyinunitempty,definition,
% 2.83/1.63 ( emptyinunitempty
% 2.83/1.63 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoinIR_type,type,
% 2.83/1.63 setadjoinIR: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoinIR,definition,
% 2.83/1.63 ( setadjoinIR
% 2.83/1.63 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.83/1.63 ( ( in @ Xy @ A )
% 2.83/1.63 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoinE_type,type,
% 2.83/1.63 setadjoinE: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoinE,definition,
% 2.83/1.63 ( setadjoinE
% 2.83/1.63 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.83/1.63 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.83/1.63 => ! [Xphi: $o] :
% 2.83/1.63 ( ( ( Xy = Xx )
% 2.83/1.63 => Xphi )
% 2.83/1.63 => ( ( ( in @ Xy @ A )
% 2.83/1.63 => Xphi )
% 2.83/1.63 => Xphi ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoinOr_type,type,
% 2.83/1.63 setadjoinOr: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoinOr,definition,
% 2.83/1.63 ( setadjoinOr
% 2.83/1.63 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.83/1.63 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.83/1.63 => ( ( Xy = Xx )
% 2.83/1.63 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setoftrueEq_type,type,
% 2.83/1.63 setoftrueEq: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setoftrueEq,definition,
% 2.83/1.63 ( setoftrueEq
% 2.83/1.63 = ( ! [A: $i] :
% 2.83/1.63 ( ( dsetconstr @ A
% 2.83/1.63 @ ^ [Xx: $i] : $true )
% 2.83/1.63 = A ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(powersetI_type,type,
% 2.83/1.63 powersetI: $o ).
% 2.83/1.63
% 2.83/1.63 thf(powersetI,definition,
% 2.83/1.63 ( powersetI
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ B )
% 2.83/1.63 => ( in @ Xx @ A ) )
% 2.83/1.63 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(emptyinPowerset_type,type,
% 2.83/1.63 emptyinPowerset: $o ).
% 2.83/1.63
% 2.83/1.63 thf(emptyinPowerset,definition,
% 2.83/1.63 ( emptyinPowerset
% 2.83/1.63 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(emptyInPowerset_type,type,
% 2.83/1.63 emptyInPowerset: $o ).
% 2.83/1.63
% 2.83/1.63 thf(emptyInPowerset,definition,
% 2.83/1.63 ( emptyInPowerset
% 2.83/1.63 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(powersetE_type,type,
% 2.83/1.63 powersetE: $o ).
% 2.83/1.63
% 2.83/1.63 thf(powersetE,definition,
% 2.83/1.63 ( powersetE
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ B @ ( powerset @ A ) )
% 2.83/1.63 => ( ( in @ Xx @ B )
% 2.83/1.63 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setunionI_type,type,
% 2.83/1.63 setunionI: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setunionI,definition,
% 2.83/1.63 ( setunionI
% 2.83/1.63 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.83/1.63 ( ( in @ Xx @ B )
% 2.83/1.63 => ( ( in @ B @ A )
% 2.83/1.63 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setunionE_type,type,
% 2.83/1.63 setunionE: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setunionE,definition,
% 2.83/1.63 ( setunionE
% 2.83/1.63 = ( ! [A: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ ( setunion @ A ) )
% 2.83/1.63 => ! [Xphi: $o] :
% 2.83/1.63 ( ! [B: $i] :
% 2.83/1.63 ( ( in @ Xx @ B )
% 2.83/1.63 => ( ( in @ B @ A )
% 2.83/1.63 => Xphi ) )
% 2.83/1.63 => Xphi ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(subPowSU_type,type,
% 2.83/1.63 subPowSU: $o ).
% 2.83/1.63
% 2.83/1.63 thf(subPowSU,definition,
% 2.83/1.63 ( subPowSU
% 2.83/1.63 = ( ! [A: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(exuE2_type,type,
% 2.83/1.63 exuE2: $o ).
% 2.83/1.63
% 2.83/1.63 thf(exuE2,definition,
% 2.83/1.63 ( exuE2
% 2.83/1.63 = ( ! [Xphi: $i > $o] :
% 2.83/1.63 ( ( exu
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.83/1.63 => ? [Xx: $i] :
% 2.83/1.63 ! [Xy: $i] :
% 2.83/1.63 ( ( Xphi @ Xy )
% 2.83/1.63 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(nonemptyImpWitness_type,type,
% 2.83/1.63 nonemptyImpWitness: $o ).
% 2.83/1.63
% 2.83/1.63 thf(nonemptyImpWitness,definition,
% 2.83/1.63 ( nonemptyImpWitness
% 2.83/1.63 = ( ! [A: $i] :
% 2.83/1.63 ( ( nonempty @ A )
% 2.83/1.63 => ? [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 & $true ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(uniqinunit_type,type,
% 2.83/1.63 uniqinunit: $o ).
% 2.83/1.63
% 2.83/1.63 thf(uniqinunit,definition,
% 2.83/1.63 ( uniqinunit
% 2.83/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.83/1.63 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.83/1.63 => ( Xx = Xy ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(notinsingleton_type,type,
% 2.83/1.63 notinsingleton: $o ).
% 2.83/1.63
% 2.83/1.63 thf(notinsingleton,definition,
% 2.83/1.63 ( notinsingleton
% 2.83/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.83/1.63 ( ( Xx != Xy )
% 2.83/1.63 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(eqinunit_type,type,
% 2.83/1.63 eqinunit: $o ).
% 2.83/1.63
% 2.83/1.63 thf(eqinunit,definition,
% 2.83/1.63 ( eqinunit
% 2.83/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.83/1.63 ( ( Xx = Xy )
% 2.83/1.63 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(singletonsswitch_type,type,
% 2.83/1.63 singletonsswitch: $o ).
% 2.83/1.63
% 2.83/1.63 thf(singletonsswitch,definition,
% 2.83/1.63 ( singletonsswitch
% 2.83/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.83/1.63 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.83/1.63 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(upairsetE_type,type,
% 2.83/1.63 upairsetE: $o ).
% 2.83/1.63
% 2.83/1.63 thf(upairsetE,definition,
% 2.83/1.63 ( upairsetE
% 2.83/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.83/1.63 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.83/1.63 => ( ( Xz = Xx )
% 2.83/1.63 | ( Xz = Xy ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(upairsetIL_type,type,
% 2.83/1.63 upairsetIL: $o ).
% 2.83/1.63
% 2.83/1.63 thf(upairsetIL,definition,
% 2.83/1.63 ( upairsetIL
% 2.83/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(upairsetIR_type,type,
% 2.83/1.63 upairsetIR: $o ).
% 2.83/1.63
% 2.83/1.63 thf(upairsetIR,definition,
% 2.83/1.63 ( upairsetIR
% 2.83/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(emptyE1_type,type,
% 2.83/1.63 emptyE1: $o ).
% 2.83/1.63
% 2.83/1.63 thf(emptyE1,definition,
% 2.83/1.63 ( emptyE1
% 2.83/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.83/1.63 ( ? [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 & ( Xphi @ Xx ) )
% 2.83/1.63 => ( ( ( dsetconstr @ A
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.83/1.63 = emptyset )
% 2.83/1.63 => $false ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(vacuousDall_type,type,
% 2.83/1.63 vacuousDall: $o ).
% 2.83/1.63
% 2.83/1.63 thf(vacuousDall,definition,
% 2.83/1.63 ( vacuousDall
% 2.83/1.63 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ emptyset )
% 2.83/1.63 => ( Xphi @ Xx ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(quantDeMorgan1_type,type,
% 2.83/1.63 quantDeMorgan1: $o ).
% 2.83/1.63
% 2.83/1.63 thf(quantDeMorgan1,definition,
% 2.83/1.63 ( quantDeMorgan1
% 2.83/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.83/1.63 ( ~ ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( Xphi @ Xx ) )
% 2.83/1.63 => ? [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(quantDeMorgan2_type,type,
% 2.83/1.63 quantDeMorgan2: $o ).
% 2.83/1.63
% 2.83/1.63 thf(quantDeMorgan2,definition,
% 2.83/1.63 ( quantDeMorgan2
% 2.83/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.83/1.63 ( ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ~ ( Xphi @ Xx ) )
% 2.83/1.63 => ~ ? [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(quantDeMorgan3_type,type,
% 2.83/1.63 quantDeMorgan3: $o ).
% 2.83/1.63
% 2.83/1.63 thf(quantDeMorgan3,definition,
% 2.83/1.63 ( quantDeMorgan3
% 2.83/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.83/1.63 ( ~ ? [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 & ( Xphi @ Xx ) )
% 2.83/1.63 => ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(quantDeMorgan4_type,type,
% 2.83/1.63 quantDeMorgan4: $o ).
% 2.83/1.63
% 2.83/1.63 thf(quantDeMorgan4,definition,
% 2.83/1.63 ( quantDeMorgan4
% 2.83/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.83/1.63 ( ? [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 & ~ ( Xphi @ Xx ) )
% 2.83/1.63 => ~ ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(prop2setI_type,type,
% 2.83/1.63 prop2setI: $o ).
% 2.83/1.63
% 2.83/1.63 thf(prop2setI,definition,
% 2.83/1.63 ( prop2setI
% 2.83/1.63 = ( ! [Xphi: $o] :
% 2.83/1.63 ( Xphi
% 2.83/1.63 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(set2prop_type,type,
% 2.83/1.63 set2prop: $i > $o ).
% 2.83/1.63
% 2.83/1.63 thf(prop2set2propI_type,type,
% 2.83/1.63 prop2set2propI: $o ).
% 2.83/1.63
% 2.83/1.63 thf(prop2set2propI,definition,
% 2.83/1.63 ( prop2set2propI
% 2.83/1.63 = ( ! [Xphi: $o] :
% 2.83/1.63 ( Xphi
% 2.83/1.63 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(notdexE_type,type,
% 2.83/1.63 notdexE: $o ).
% 2.83/1.63
% 2.83/1.63 thf(notdexE,definition,
% 2.83/1.63 ( notdexE
% 2.83/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.83/1.63 ( ~ ? [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 & ( Xphi @ Xx ) )
% 2.83/1.63 => ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(notdallE_type,type,
% 2.83/1.63 notdallE: $o ).
% 2.83/1.63
% 2.83/1.63 thf(notdallE,definition,
% 2.83/1.63 ( notdallE
% 2.83/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.83/1.63 ( ~ ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( Xphi @ Xx ) )
% 2.83/1.63 => ? [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(exuI1_type,type,
% 2.83/1.63 exuI1: $o ).
% 2.83/1.63
% 2.83/1.63 thf(exuI1,definition,
% 2.83/1.63 ( exuI1
% 2.83/1.63 = ( ! [Xphi: $i > $o] :
% 2.83/1.63 ( ? [Xx: $i] :
% 2.83/1.63 ( ( Xphi @ Xx )
% 2.83/1.63 & ! [Xy: $i] :
% 2.83/1.63 ( ( Xphi @ Xy )
% 2.83/1.63 => ( Xx = Xy ) ) )
% 2.83/1.63 => ( exu
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(exuI3_type,type,
% 2.83/1.63 exuI3: $o ).
% 2.83/1.63
% 2.83/1.63 thf(exuI3,definition,
% 2.83/1.63 ( exuI3
% 2.83/1.63 = ( ! [Xphi: $i > $o] :
% 2.83/1.63 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.83/1.63 => ( ! [Xx: $i,Xy: $i] :
% 2.83/1.63 ( ( Xphi @ Xx )
% 2.83/1.63 => ( ( Xphi @ Xy )
% 2.83/1.63 => ( Xx = Xy ) ) )
% 2.83/1.63 => ( exu
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(exuI2_type,type,
% 2.83/1.63 exuI2: $o ).
% 2.83/1.63
% 2.83/1.63 thf(exuI2,definition,
% 2.83/1.63 ( exuI2
% 2.83/1.63 = ( ! [Xphi: $i > $o] :
% 2.83/1.63 ( ? [Xx: $i] :
% 2.83/1.63 ! [Xy: $i] :
% 2.83/1.63 ( ( Xphi @ Xy )
% 2.83/1.63 <=> ( Xy = Xx ) )
% 2.83/1.63 => ( exu
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(inCongP_type,type,
% 2.83/1.63 inCongP: $o ).
% 2.83/1.63
% 2.83/1.63 thf(inCongP,definition,
% 2.83/1.63 ( inCongP
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( A = B )
% 2.83/1.63 => ! [Xx: $i,Xy: $i] :
% 2.83/1.63 ( ( Xx = Xy )
% 2.83/1.63 => ( ( in @ Xx @ A )
% 2.83/1.63 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(in__Cong_type,type,
% 2.83/1.63 in__Cong: $o ).
% 2.83/1.63
% 2.83/1.63 thf(in__Cong,definition,
% 2.83/1.63 ( in__Cong
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( A = B )
% 2.83/1.63 => ! [Xx: $i,Xy: $i] :
% 2.83/1.63 ( ( Xx = Xy )
% 2.83/1.63 => ( ( in @ Xx @ A )
% 2.83/1.63 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(exuE3u_type,type,
% 2.83/1.63 exuE3u: $o ).
% 2.83/1.63
% 2.83/1.63 thf(exuE3u,definition,
% 2.83/1.63 ( exuE3u
% 2.83/1.63 = ( ! [Xphi: $i > $o] :
% 2.83/1.63 ( ( exu
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.83/1.63 => ! [Xx: $i,Xy: $i] :
% 2.83/1.63 ( ( Xphi @ Xx )
% 2.83/1.63 => ( ( Xphi @ Xy )
% 2.83/1.63 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(exu__Cong_type,type,
% 2.83/1.63 exu__Cong: $o ).
% 2.83/1.63
% 2.83/1.63 thf(exu__Cong,definition,
% 2.83/1.63 ( exu__Cong
% 2.83/1.63 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.83/1.63 ( ! [Xx: $i,Xy: $i] :
% 2.83/1.63 ( ( Xx = Xy )
% 2.83/1.63 => ( ( Xphi @ Xx )
% 2.83/1.63 <=> ( Xpsi @ Xy ) ) )
% 2.83/1.63 => ( ( exu
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.83/1.63 <=> ( exu
% 2.83/1.63 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(emptyset__Cong_type,type,
% 2.83/1.63 emptyset__Cong: $o ).
% 2.83/1.63
% 2.83/1.63 thf(emptyset__Cong,definition,
% 2.83/1.63 ( emptyset__Cong
% 2.83/1.63 = ( emptyset = emptyset ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoin__Cong_type,type,
% 2.83/1.63 setadjoin__Cong: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoin__Cong,definition,
% 2.83/1.63 ( setadjoin__Cong
% 2.83/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.83/1.63 ( ( Xx = Xy )
% 2.83/1.63 => ! [Xz: $i,Xu: $i] :
% 2.83/1.63 ( ( Xz = Xu )
% 2.83/1.63 => ( ( setadjoin @ Xx @ Xz )
% 2.83/1.63 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(powerset__Cong_type,type,
% 2.83/1.63 powerset__Cong: $o ).
% 2.83/1.63
% 2.83/1.63 thf(powerset__Cong,definition,
% 2.83/1.63 ( powerset__Cong
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( A = B )
% 2.83/1.63 => ( ( powerset @ A )
% 2.83/1.63 = ( powerset @ B ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setunion__Cong_type,type,
% 2.83/1.63 setunion__Cong: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setunion__Cong,definition,
% 2.83/1.63 ( setunion__Cong
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( A = B )
% 2.83/1.63 => ( ( setunion @ A )
% 2.83/1.63 = ( setunion @ B ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(omega__Cong_type,type,
% 2.83/1.63 omega__Cong: $o ).
% 2.83/1.63
% 2.83/1.63 thf(omega__Cong,definition,
% 2.83/1.63 ( omega__Cong
% 2.83/1.63 = ( omega = omega ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(exuEu_type,type,
% 2.83/1.63 exuEu: $o ).
% 2.83/1.63
% 2.83/1.63 thf(exuEu,definition,
% 2.83/1.63 ( exuEu
% 2.83/1.63 = ( ! [Xphi: $i > $o] :
% 2.83/1.63 ( ( exu
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.83/1.63 => ! [Xx: $i,Xy: $i] :
% 2.83/1.63 ( ( Xphi @ Xx )
% 2.83/1.63 => ( ( Xphi @ Xy )
% 2.83/1.63 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(descr__Cong_type,type,
% 2.83/1.63 descr__Cong: $o ).
% 2.83/1.63
% 2.83/1.63 thf(descr__Cong,definition,
% 2.83/1.63 ( descr__Cong
% 2.83/1.63 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.83/1.63 ( ! [Xx: $i,Xy: $i] :
% 2.83/1.63 ( ( Xx = Xy )
% 2.83/1.63 => ( ( Xphi @ Xx )
% 2.83/1.63 <=> ( Xpsi @ Xy ) ) )
% 2.83/1.63 => ( ( exu
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.83/1.63 => ( ( exu
% 2.83/1.63 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.83/1.63 => ( ( descr
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.83/1.63 = ( descr
% 2.83/1.63 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(dsetconstr__Cong_type,type,
% 2.83/1.63 dsetconstr__Cong: $o ).
% 2.83/1.63
% 2.83/1.63 thf(dsetconstr__Cong,definition,
% 2.83/1.63 ( dsetconstr__Cong
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( A = B )
% 2.83/1.63 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.83/1.63 ( ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ! [Xy: $i] :
% 2.83/1.63 ( ( in @ Xy @ B )
% 2.83/1.63 => ( ( Xx = Xy )
% 2.83/1.63 => ( ( Xphi @ Xx )
% 2.83/1.63 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.83/1.63 => ( ( dsetconstr @ A
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.83/1.63 = ( dsetconstr @ B
% 2.83/1.63 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(subset_type,type,
% 2.83/1.63 subset: $i > $i > $o ).
% 2.83/1.63
% 2.83/1.63 thf(disjoint_type,type,
% 2.83/1.63 disjoint: $i > $i > $o ).
% 2.83/1.63
% 2.83/1.63 thf(setsmeet_type,type,
% 2.83/1.63 setsmeet: $i > $i > $o ).
% 2.83/1.63
% 2.83/1.63 thf(subsetI1_type,type,
% 2.83/1.63 subsetI1: $o ).
% 2.83/1.63
% 2.83/1.63 thf(subsetI1,definition,
% 2.83/1.63 ( subsetI1
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( in @ Xx @ B ) )
% 2.83/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(eqimpsubset2_type,type,
% 2.83/1.63 eqimpsubset2: $o ).
% 2.83/1.63
% 2.83/1.63 thf(eqimpsubset2,definition,
% 2.83/1.63 ( eqimpsubset2
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( A = B )
% 2.83/1.63 => ( subset @ B @ A ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(eqimpsubset1_type,type,
% 2.83/1.63 eqimpsubset1: $o ).
% 2.83/1.63
% 2.83/1.63 thf(eqimpsubset1,definition,
% 2.83/1.63 ( eqimpsubset1
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( A = B )
% 2.83/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(subsetI2_type,type,
% 2.83/1.63 subsetI2: $o ).
% 2.83/1.63
% 2.83/1.63 thf(subsetI2,definition,
% 2.83/1.63 ( subsetI2
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ! [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( in @ Xx @ B ) )
% 2.83/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(emptysetsubset_type,type,
% 2.83/1.63 emptysetsubset: $o ).
% 2.83/1.63
% 2.83/1.63 thf(emptysetsubset,definition,
% 2.83/1.63 ( emptysetsubset
% 2.83/1.63 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(subsetE_type,type,
% 2.83/1.63 subsetE: $o ).
% 2.83/1.63
% 2.83/1.63 thf(subsetE,definition,
% 2.83/1.63 ( subsetE
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( subset @ A @ B )
% 2.83/1.63 => ( ( in @ Xx @ A )
% 2.83/1.63 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(subsetE2_type,type,
% 2.83/1.63 subsetE2: $o ).
% 2.83/1.63
% 2.83/1.63 thf(subsetE2,definition,
% 2.83/1.63 ( subsetE2
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( subset @ A @ B )
% 2.83/1.63 => ( ~ ( in @ Xx @ B )
% 2.83/1.63 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(notsubsetI_type,type,
% 2.83/1.63 notsubsetI: $o ).
% 2.83/1.63
% 2.83/1.63 thf(notsubsetI,definition,
% 2.83/1.63 ( notsubsetI
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( ~ ( in @ Xx @ B )
% 2.83/1.63 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(notequalI1_type,type,
% 2.83/1.63 notequalI1: $o ).
% 2.83/1.63
% 2.83/1.63 thf(notequalI1,definition,
% 2.83/1.63 ( notequalI1
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ~ ( subset @ A @ B )
% 2.83/1.63 => ( A != B ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(notequalI2_type,type,
% 2.83/1.63 notequalI2: $o ).
% 2.83/1.63
% 2.83/1.63 thf(notequalI2,definition,
% 2.83/1.63 ( notequalI2
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( ~ ( in @ Xx @ B )
% 2.83/1.63 => ( A != B ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(subsetRefl_type,type,
% 2.83/1.63 subsetRefl: $o ).
% 2.83/1.63
% 2.83/1.63 thf(subsetRefl,definition,
% 2.83/1.63 ( subsetRefl
% 2.83/1.63 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(subsetTrans_type,type,
% 2.83/1.63 subsetTrans: $o ).
% 2.83/1.63
% 2.83/1.63 thf(subsetTrans,definition,
% 2.83/1.63 ( subsetTrans
% 2.83/1.63 = ( ! [A: $i,B: $i,C: $i] :
% 2.83/1.63 ( ( subset @ A @ B )
% 2.83/1.63 => ( ( subset @ B @ C )
% 2.83/1.63 => ( subset @ A @ C ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoinSub_type,type,
% 2.83/1.63 setadjoinSub: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoinSub,definition,
% 2.83/1.63 ( setadjoinSub
% 2.83/1.63 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoinSub2_type,type,
% 2.83/1.63 setadjoinSub2: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setadjoinSub2,definition,
% 2.83/1.63 ( setadjoinSub2
% 2.83/1.63 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.83/1.63 ( ( subset @ A @ B )
% 2.83/1.63 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(subset2powerset_type,type,
% 2.83/1.63 subset2powerset: $o ).
% 2.83/1.63
% 2.83/1.63 thf(subset2powerset,definition,
% 2.83/1.63 ( subset2powerset
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( subset @ A @ B )
% 2.83/1.63 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setextsub_type,type,
% 2.83/1.63 setextsub: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setextsub,definition,
% 2.83/1.63 ( setextsub
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( subset @ A @ B )
% 2.83/1.63 => ( ( subset @ B @ A )
% 2.83/1.63 => ( A = B ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(subsetemptysetimpeq_type,type,
% 2.83/1.63 subsetemptysetimpeq: $o ).
% 2.83/1.63
% 2.83/1.63 thf(subsetemptysetimpeq,definition,
% 2.83/1.63 ( subsetemptysetimpeq
% 2.83/1.63 = ( ! [A: $i] :
% 2.83/1.63 ( ( subset @ A @ emptyset )
% 2.83/1.63 => ( A = emptyset ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(powersetI1_type,type,
% 2.83/1.63 powersetI1: $o ).
% 2.83/1.63
% 2.83/1.63 thf(powersetI1,definition,
% 2.83/1.63 ( powersetI1
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( subset @ B @ A )
% 2.83/1.63 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(powersetE1_type,type,
% 2.83/1.63 powersetE1: $o ).
% 2.83/1.63
% 2.83/1.63 thf(powersetE1,definition,
% 2.83/1.63 ( powersetE1
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( in @ B @ ( powerset @ A ) )
% 2.83/1.63 => ( subset @ B @ A ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(inPowerset_type,type,
% 2.83/1.63 inPowerset: $o ).
% 2.83/1.63
% 2.83/1.63 thf(inPowerset,definition,
% 2.83/1.63 ( inPowerset
% 2.83/1.63 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(powersetsubset_type,type,
% 2.83/1.63 powersetsubset: $o ).
% 2.83/1.63
% 2.83/1.63 thf(powersetsubset,definition,
% 2.83/1.63 ( powersetsubset
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( subset @ A @ B )
% 2.83/1.63 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(sepInPowerset_type,type,
% 2.83/1.63 sepInPowerset: $o ).
% 2.83/1.63
% 2.83/1.63 thf(sepInPowerset,definition,
% 2.83/1.63 ( sepInPowerset
% 2.83/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.83/1.63 ( in
% 2.83/1.63 @ ( dsetconstr @ A
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.83/1.63 @ ( powerset @ A ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(sepSubset_type,type,
% 2.83/1.63 sepSubset: $o ).
% 2.83/1.63
% 2.83/1.63 thf(sepSubset,definition,
% 2.83/1.63 ( sepSubset
% 2.83/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.83/1.63 ( subset
% 2.83/1.63 @ ( dsetconstr @ A
% 2.83/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.83/1.63 @ A ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(binunion_type,type,
% 2.83/1.63 binunion: $i > $i > $i ).
% 2.83/1.63
% 2.83/1.63 thf(binunionIL_type,type,
% 2.83/1.63 binunionIL: $o ).
% 2.83/1.63
% 2.83/1.63 thf(binunionIL,definition,
% 2.83/1.63 ( binunionIL
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(upairset2IR_type,type,
% 2.83/1.63 upairset2IR: $o ).
% 2.83/1.63
% 2.83/1.63 thf(upairset2IR,definition,
% 2.83/1.63 ( upairset2IR
% 2.83/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(binunionIR_type,type,
% 2.83/1.63 binunionIR: $o ).
% 2.83/1.63
% 2.83/1.63 thf(binunionIR,definition,
% 2.83/1.63 ( binunionIR
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ B )
% 2.83/1.63 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(binunionEcases_type,type,
% 2.83/1.63 binunionEcases: $o ).
% 2.83/1.63
% 2.83/1.63 thf(binunionEcases,definition,
% 2.83/1.63 ( binunionEcases
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.83/1.63 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.83/1.63 => ( ( ( in @ Xx @ A )
% 2.83/1.63 => Xphi )
% 2.83/1.63 => ( ( ( in @ Xx @ B )
% 2.83/1.63 => Xphi )
% 2.83/1.63 => Xphi ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(binunionE_type,type,
% 2.83/1.63 binunionE: $o ).
% 2.83/1.63
% 2.83/1.63 thf(binunionE,definition,
% 2.83/1.63 ( binunionE
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.83/1.63 => ( ( in @ Xx @ A )
% 2.83/1.63 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(binunionLsub_type,type,
% 2.83/1.63 binunionLsub: $o ).
% 2.83/1.63
% 2.83/1.63 thf(binunionLsub,definition,
% 2.83/1.63 ( binunionLsub
% 2.83/1.63 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(binunionRsub_type,type,
% 2.83/1.63 binunionRsub: $o ).
% 2.83/1.63
% 2.83/1.63 thf(binunionRsub,definition,
% 2.83/1.63 ( binunionRsub
% 2.83/1.63 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(binintersect_type,type,
% 2.83/1.63 binintersect: $i > $i > $i ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectI_type,type,
% 2.83/1.63 binintersectI: $o ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectI,definition,
% 2.83/1.63 ( binintersectI
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( ( in @ Xx @ B )
% 2.83/1.63 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectSubset5_type,type,
% 2.83/1.63 binintersectSubset5: $o ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectSubset5,definition,
% 2.83/1.63 ( binintersectSubset5
% 2.83/1.63 = ( ! [A: $i,B: $i,C: $i] :
% 2.83/1.63 ( ( subset @ C @ A )
% 2.83/1.63 => ( ( subset @ C @ B )
% 2.83/1.63 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectEL_type,type,
% 2.83/1.63 binintersectEL: $o ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectEL,definition,
% 2.83/1.63 ( binintersectEL
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.83/1.63 => ( in @ Xx @ A ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectLsub_type,type,
% 2.83/1.63 binintersectLsub: $o ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectLsub,definition,
% 2.83/1.63 ( binintersectLsub
% 2.83/1.63 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectSubset2_type,type,
% 2.83/1.63 binintersectSubset2: $o ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectSubset2,definition,
% 2.83/1.63 ( binintersectSubset2
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( subset @ A @ B )
% 2.83/1.63 => ( ( binintersect @ A @ B )
% 2.83/1.63 = A ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectSubset3_type,type,
% 2.83/1.63 binintersectSubset3: $o ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectSubset3,definition,
% 2.83/1.63 ( binintersectSubset3
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( ( binintersect @ A @ B )
% 2.83/1.63 = B )
% 2.83/1.63 => ( subset @ B @ A ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectER_type,type,
% 2.83/1.63 binintersectER: $o ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectER,definition,
% 2.83/1.63 ( binintersectER
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.83/1.63 => ( in @ Xx @ B ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(disjointsetsI1_type,type,
% 2.83/1.63 disjointsetsI1: $o ).
% 2.83/1.63
% 2.83/1.63 thf(disjointsetsI1,definition,
% 2.83/1.63 ( disjointsetsI1
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ~ ? [Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 & ( in @ Xx @ B ) )
% 2.83/1.63 => ( ( binintersect @ A @ B )
% 2.83/1.63 = emptyset ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectRsub_type,type,
% 2.83/1.63 binintersectRsub: $o ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectRsub,definition,
% 2.83/1.63 ( binintersectRsub
% 2.83/1.63 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectSubset4_type,type,
% 2.83/1.63 binintersectSubset4: $o ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectSubset4,definition,
% 2.83/1.63 ( binintersectSubset4
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( subset @ B @ A )
% 2.83/1.63 => ( ( binintersect @ A @ B )
% 2.83/1.63 = B ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectSubset1_type,type,
% 2.83/1.63 binintersectSubset1: $o ).
% 2.83/1.63
% 2.83/1.63 thf(binintersectSubset1,definition,
% 2.83/1.63 ( binintersectSubset1
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( ( binintersect @ A @ B )
% 2.83/1.63 = A )
% 2.83/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(bs114d_type,type,
% 2.83/1.63 bs114d: $o ).
% 2.83/1.63
% 2.83/1.63 thf(bs114d,definition,
% 2.83/1.63 ( bs114d
% 2.83/1.63 = ( ! [A: $i,B: $i,C: $i] :
% 2.83/1.63 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.83/1.63 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(regular_type,type,
% 2.83/1.63 regular: $i > $o ).
% 2.83/1.63
% 2.83/1.63 thf(setminus_type,type,
% 2.83/1.63 setminus: $i > $i > $i ).
% 2.83/1.63
% 2.83/1.63 thf(setminusI_type,type,
% 2.83/1.63 setminusI: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setminusI,definition,
% 2.83/1.63 ( setminusI
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( ~ ( in @ Xx @ B )
% 2.83/1.63 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setminusEL_type,type,
% 2.83/1.63 setminusEL: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setminusEL,definition,
% 2.83/1.63 ( setminusEL
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.83/1.63 => ( in @ Xx @ A ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setminusER_type,type,
% 2.83/1.63 setminusER: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setminusER,definition,
% 2.83/1.63 ( setminusER
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.83/1.63 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setminusSubset2_type,type,
% 2.83/1.63 setminusSubset2: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setminusSubset2,definition,
% 2.83/1.63 ( setminusSubset2
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( subset @ A @ B )
% 2.83/1.63 => ( ( setminus @ A @ B )
% 2.83/1.63 = emptyset ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setminusERneg_type,type,
% 2.83/1.63 setminusERneg: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setminusERneg,definition,
% 2.83/1.63 ( setminusERneg
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.83/1.63 => ( ( in @ Xx @ A )
% 2.83/1.63 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setminusELneg_type,type,
% 2.83/1.63 setminusELneg: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setminusELneg,definition,
% 2.83/1.63 ( setminusELneg
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.83/1.63 => ( ~ ( in @ Xx @ B )
% 2.83/1.63 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setminusILneg_type,type,
% 2.83/1.63 setminusILneg: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setminusILneg,definition,
% 2.83/1.63 ( setminusILneg
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ~ ( in @ Xx @ A )
% 2.83/1.63 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setminusIRneg_type,type,
% 2.83/1.63 setminusIRneg: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setminusIRneg,definition,
% 2.83/1.63 ( setminusIRneg
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ B )
% 2.83/1.63 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setminusLsub_type,type,
% 2.83/1.63 setminusLsub: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setminusLsub,definition,
% 2.83/1.63 ( setminusLsub
% 2.83/1.63 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(setminusSubset1_type,type,
% 2.83/1.63 setminusSubset1: $o ).
% 2.83/1.63
% 2.83/1.63 thf(setminusSubset1,definition,
% 2.83/1.63 ( setminusSubset1
% 2.83/1.63 = ( ! [A: $i,B: $i] :
% 2.83/1.63 ( ( ( setminus @ A @ B )
% 2.83/1.63 = emptyset )
% 2.83/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(symdiff_type,type,
% 2.83/1.63 symdiff: $i > $i > $i ).
% 2.83/1.63
% 2.83/1.63 thf(symdiffE_type,type,
% 2.83/1.63 symdiffE: $o ).
% 2.83/1.63
% 2.83/1.63 thf(symdiffE,definition,
% 2.83/1.63 ( symdiffE
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.83/1.63 => ! [Xphi: $o] :
% 2.83/1.63 ( ( ( in @ Xx @ A )
% 2.83/1.63 => ( ~ ( in @ Xx @ B )
% 2.83/1.63 => Xphi ) )
% 2.83/1.63 => ( ( ~ ( in @ Xx @ A )
% 2.83/1.63 => ( ( in @ Xx @ B )
% 2.83/1.63 => Xphi ) )
% 2.83/1.63 => Xphi ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(symdiffI1_type,type,
% 2.83/1.63 symdiffI1: $o ).
% 2.83/1.63
% 2.83/1.63 thf(symdiffI1,definition,
% 2.83/1.63 ( symdiffI1
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( ~ ( in @ Xx @ B )
% 2.83/1.63 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(symdiffI2_type,type,
% 2.83/1.63 symdiffI2: $o ).
% 2.83/1.63
% 2.83/1.63 thf(symdiffI2,definition,
% 2.83/1.63 ( symdiffI2
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ~ ( in @ Xx @ A )
% 2.83/1.63 => ( ( in @ Xx @ B )
% 2.83/1.63 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(symdiffIneg1_type,type,
% 2.83/1.63 symdiffIneg1: $o ).
% 2.83/1.63
% 2.83/1.63 thf(symdiffIneg1,definition,
% 2.83/1.63 ( symdiffIneg1
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ( in @ Xx @ A )
% 2.83/1.63 => ( ( in @ Xx @ B )
% 2.83/1.63 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(symdiffIneg2_type,type,
% 2.83/1.63 symdiffIneg2: $o ).
% 2.83/1.63
% 2.83/1.63 thf(symdiffIneg2,definition,
% 2.83/1.63 ( symdiffIneg2
% 2.83/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.63 ( ~ ( in @ Xx @ A )
% 2.83/1.63 => ( ~ ( in @ Xx @ B )
% 2.83/1.63 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.83/1.63
% 2.83/1.63 thf(iskpair_type,type,
% 2.83/1.63 iskpair: $i > $o ).
% 2.83/1.63
% 2.83/1.63 thf(secondinupair_type,type,
% 2.83/1.63 secondinupair: $o ).
% 2.83/1.63
% 2.83/1.64 thf(secondinupair,definition,
% 2.83/1.64 ( secondinupair
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairIL_type,type,
% 2.83/1.64 setukpairIL: $o ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairIL,definition,
% 2.83/1.64 ( setukpairIL
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairIR_type,type,
% 2.83/1.64 setukpairIR: $o ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairIR,definition,
% 2.83/1.64 ( setukpairIR
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(kpairiskpair_type,type,
% 2.83/1.64 kpairiskpair: $o ).
% 2.83/1.64
% 2.83/1.64 thf(kpairiskpair,definition,
% 2.83/1.64 ( kpairiskpair
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(kpair_type,type,
% 2.83/1.64 kpair: $i > $i > $i ).
% 2.83/1.64
% 2.83/1.64 thf(kpairp_type,type,
% 2.83/1.64 kpairp: $o ).
% 2.83/1.64
% 2.83/1.64 thf(kpairp,definition,
% 2.83/1.64 ( kpairp
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(cartprod_type,type,
% 2.83/1.64 cartprod: $i > $i > $i ).
% 2.83/1.64
% 2.83/1.64 thf(singletonsubset_type,type,
% 2.83/1.64 singletonsubset: $o ).
% 2.83/1.64
% 2.83/1.64 thf(singletonsubset,definition,
% 2.83/1.64 ( singletonsubset
% 2.83/1.64 = ( ! [A: $i,Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(singletoninpowerset_type,type,
% 2.83/1.64 singletoninpowerset: $o ).
% 2.83/1.64
% 2.83/1.64 thf(singletoninpowerset,definition,
% 2.83/1.64 ( singletoninpowerset
% 2.83/1.64 = ( ! [A: $i,Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(singletoninpowunion_type,type,
% 2.83/1.64 singletoninpowunion: $o ).
% 2.83/1.64
% 2.83/1.64 thf(singletoninpowunion,definition,
% 2.83/1.64 ( singletoninpowunion
% 2.83/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(upairset2E_type,type,
% 2.83/1.64 upairset2E: $o ).
% 2.83/1.64
% 2.83/1.64 thf(upairset2E,definition,
% 2.83/1.64 ( upairset2E
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.83/1.64 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.83/1.64 => ( ( Xz = Xx )
% 2.83/1.64 | ( Xz = Xy ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(upairsubunion_type,type,
% 2.83/1.64 upairsubunion: $o ).
% 2.83/1.64
% 2.83/1.64 thf(upairsubunion,definition,
% 2.83/1.64 ( upairsubunion
% 2.83/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ B )
% 2.83/1.64 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(upairinpowunion_type,type,
% 2.83/1.64 upairinpowunion: $o ).
% 2.83/1.64
% 2.83/1.64 thf(upairinpowunion,definition,
% 2.83/1.64 ( upairinpowunion
% 2.83/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ B )
% 2.83/1.64 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(ubforcartprodlem1_type,type,
% 2.83/1.64 ubforcartprodlem1: $o ).
% 2.83/1.64
% 2.83/1.64 thf(ubforcartprodlem1,definition,
% 2.83/1.64 ( ubforcartprodlem1
% 2.83/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ B )
% 2.83/1.64 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(ubforcartprodlem2_type,type,
% 2.83/1.64 ubforcartprodlem2: $o ).
% 2.83/1.64
% 2.83/1.64 thf(ubforcartprodlem2,definition,
% 2.83/1.64 ( ubforcartprodlem2
% 2.83/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ B )
% 2.83/1.64 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(ubforcartprodlem3_type,type,
% 2.83/1.64 ubforcartprodlem3: $o ).
% 2.83/1.64
% 2.83/1.64 thf(ubforcartprodlem3,definition,
% 2.83/1.64 ( ubforcartprodlem3
% 2.83/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ B )
% 2.83/1.64 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodpairin_type,type,
% 2.83/1.64 cartprodpairin: $o ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodpairin,definition,
% 2.83/1.64 ( cartprodpairin
% 2.83/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ B )
% 2.83/1.64 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodmempair1_type,type,
% 2.83/1.64 cartprodmempair1: $o ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodmempair1,definition,
% 2.83/1.64 ( cartprodmempair1
% 2.83/1.64 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.83/1.64 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.83/1.64 => ? [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 & ? [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ B )
% 2.83/1.64 & ( Xu
% 2.83/1.64 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodmempair_type,type,
% 2.83/1.64 cartprodmempair: $o ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodmempair,definition,
% 2.83/1.64 ( cartprodmempair
% 2.83/1.64 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.83/1.64 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.83/1.64 => ( iskpair @ Xu ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(setunionE2_type,type,
% 2.83/1.64 setunionE2: $o ).
% 2.83/1.64
% 2.83/1.64 thf(setunionE2,definition,
% 2.83/1.64 ( setunionE2
% 2.83/1.64 = ( ! [A: $i,Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ ( setunion @ A ) )
% 2.83/1.64 => ? [X: $i] :
% 2.83/1.64 ( ( in @ X @ A )
% 2.83/1.64 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(setunionsingleton1_type,type,
% 2.83/1.64 setunionsingleton1: $o ).
% 2.83/1.64
% 2.83/1.64 thf(setunionsingleton1,definition,
% 2.83/1.64 ( setunionsingleton1
% 2.83/1.64 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(setunionsingleton2_type,type,
% 2.83/1.64 setunionsingleton2: $o ).
% 2.83/1.64
% 2.83/1.64 thf(setunionsingleton2,definition,
% 2.83/1.64 ( setunionsingleton2
% 2.83/1.64 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(setunionsingleton_type,type,
% 2.83/1.64 setunionsingleton: $o ).
% 2.83/1.64
% 2.83/1.64 thf(setunionsingleton,definition,
% 2.83/1.64 ( setunionsingleton
% 2.83/1.64 = ( ! [Xx: $i] :
% 2.83/1.64 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.83/1.64 = Xx ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(singleton_type,type,
% 2.83/1.64 singleton: $i > $o ).
% 2.83/1.64
% 2.83/1.64 thf(singleton,definition,
% 2.83/1.64 ( singleton
% 2.83/1.64 = ( ^ [A: $i] :
% 2.83/1.64 ? [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 & ( A
% 2.83/1.64 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(singletonprop_type,type,
% 2.83/1.64 singletonprop: $o ).
% 2.83/1.64
% 2.83/1.64 thf(singletonprop,definition,
% 2.83/1.64 ( singletonprop
% 2.83/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 2.83/1.64 ( ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ A )
% 2.83/1.64 => ( ( Xphi @ Xx )
% 2.83/1.64 => ( ( Xphi @ Xy )
% 2.83/1.64 => ( Xx = Xy ) ) ) ) )
% 2.83/1.64 => ( ? [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 & ( Xphi @ Xx ) )
% 2.83/1.64 => ( singleton
% 2.83/1.64 @ ( dsetconstr @ A
% 2.83/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(ex1_type,type,
% 2.83/1.64 ex1: $i > ( $i > $o ) > $o ).
% 2.83/1.64
% 2.83/1.64 thf(ex1,definition,
% 2.83/1.64 ( ex1
% 2.83/1.64 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.83/1.64 ( singleton
% 2.83/1.64 @ ( dsetconstr @ A
% 2.83/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(ex1E1_type,type,
% 2.83/1.64 ex1E1: $o ).
% 2.83/1.64
% 2.83/1.64 thf(ex1E1,definition,
% 2.83/1.64 ( ex1E1
% 2.83/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 2.83/1.64 ( ( ex1 @ A
% 2.83/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.83/1.64 => ? [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(ex1I_type,type,
% 2.83/1.64 ex1I: $o ).
% 2.83/1.64
% 2.83/1.64 thf(ex1I,definition,
% 2.83/1.64 ( ex1I
% 2.83/1.64 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ( ( Xphi @ Xx )
% 2.83/1.64 => ( ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ A )
% 2.83/1.64 => ( ( Xphi @ Xy )
% 2.83/1.64 => ( Xy = Xx ) ) )
% 2.83/1.64 => ( ex1 @ A
% 2.83/1.64 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(ex1I2_type,type,
% 2.83/1.64 ex1I2: $o ).
% 2.83/1.64
% 2.83/1.64 thf(ex1I2,definition,
% 2.83/1.64 ( ex1I2
% 2.83/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 2.83/1.64 ( ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ A )
% 2.83/1.64 => ( ( Xphi @ Xx )
% 2.83/1.64 => ( ( Xphi @ Xy )
% 2.83/1.64 => ( Xx = Xy ) ) ) ) )
% 2.83/1.64 => ( ? [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 & ( Xphi @ Xx ) )
% 2.83/1.64 => ( ex1 @ A
% 2.83/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(singletonsuniq_type,type,
% 2.83/1.64 singletonsuniq: $o ).
% 2.83/1.64
% 2.83/1.64 thf(singletonsuniq,definition,
% 2.83/1.64 ( singletonsuniq
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i] :
% 2.83/1.64 ( ( ( setadjoin @ Xx @ emptyset )
% 2.83/1.64 = ( setadjoin @ Xy @ emptyset ) )
% 2.83/1.64 => ( Xx = Xy ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(atmost1p_type,type,
% 2.83/1.64 atmost1p: $i > $o ).
% 2.83/1.64
% 2.83/1.64 thf(atleast2p_type,type,
% 2.83/1.64 atleast2p: $i > $o ).
% 2.83/1.64
% 2.83/1.64 thf(atmost2p_type,type,
% 2.83/1.64 atmost2p: $i > $o ).
% 2.83/1.64
% 2.83/1.64 thf(upairsetp_type,type,
% 2.83/1.64 upairsetp: $i > $o ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairinjL1_type,type,
% 2.83/1.64 setukpairinjL1: $o ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairinjL1,definition,
% 2.83/1.64 ( setukpairinjL1
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.83/1.64 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.83/1.64 => ( Xx = Xz ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(kfstsingleton_type,type,
% 2.83/1.64 kfstsingleton: $o ).
% 2.83/1.64
% 2.83/1.64 thf(kfstsingleton,definition,
% 2.83/1.64 ( kfstsingleton
% 2.83/1.64 = ( ! [Xu: $i] :
% 2.83/1.64 ( ( iskpair @ Xu )
% 2.83/1.64 => ( singleton
% 2.83/1.64 @ ( dsetconstr @ ( setunion @ Xu )
% 2.83/1.64 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(theprop_type,type,
% 2.83/1.64 theprop: $o ).
% 2.83/1.64
% 2.83/1.64 thf(theprop,definition,
% 2.83/1.64 ( theprop
% 2.83/1.64 = ( ! [X: $i] :
% 2.83/1.64 ( ( singleton @ X )
% 2.83/1.64 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(kfst_type,type,
% 2.83/1.64 kfst: $i > $i ).
% 2.83/1.64
% 2.83/1.64 thf(kfstpairEq_type,type,
% 2.83/1.64 kfstpairEq: $o ).
% 2.83/1.64
% 2.83/1.64 thf(kfstpairEq,definition,
% 2.83/1.64 ( kfstpairEq
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i] :
% 2.83/1.64 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.83/1.64 = Xx ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodfstin_type,type,
% 2.83/1.64 cartprodfstin: $o ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodfstin,definition,
% 2.83/1.64 ( cartprodfstin
% 2.83/1.64 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.83/1.64 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.83/1.64 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairinjL2_type,type,
% 2.83/1.64 setukpairinjL2: $o ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairinjL2,definition,
% 2.83/1.64 ( setukpairinjL2
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.83/1.64 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.83/1.64 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.83/1.64 => ( Xx = Xz ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairinjL_type,type,
% 2.83/1.64 setukpairinjL: $o ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairinjL,definition,
% 2.83/1.64 ( setukpairinjL
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.83/1.64 ( ( ( kpair @ Xx @ Xy )
% 2.83/1.64 = ( kpair @ Xz @ Xu ) )
% 2.83/1.64 => ( Xx = Xz ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairinjR11_type,type,
% 2.83/1.64 setukpairinjR11: $o ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairinjR11,definition,
% 2.83/1.64 ( setukpairinjR11
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i] :
% 2.83/1.64 ( ( Xx = Xy )
% 2.83/1.64 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.83/1.64 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairinjR12_type,type,
% 2.83/1.64 setukpairinjR12: $o ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairinjR12,definition,
% 2.83/1.64 ( setukpairinjR12
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i] :
% 2.83/1.64 ( ( Xx = Xy )
% 2.83/1.64 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.83/1.64 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairinjR1_type,type,
% 2.83/1.64 setukpairinjR1: $o ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairinjR1,definition,
% 2.83/1.64 ( setukpairinjR1
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.83/1.64 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.83/1.64 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.83/1.64 => ( ( Xz = Xu )
% 2.83/1.64 => ( Xy = Xu ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(upairequniteq_type,type,
% 2.83/1.64 upairequniteq: $o ).
% 2.83/1.64
% 2.83/1.64 thf(upairequniteq,definition,
% 2.83/1.64 ( upairequniteq
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.83/1.64 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.83/1.64 = ( setadjoin @ Xz @ emptyset ) )
% 2.83/1.64 => ( Xx = Xy ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairinjR2_type,type,
% 2.83/1.64 setukpairinjR2: $o ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairinjR2,definition,
% 2.83/1.64 ( setukpairinjR2
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.83/1.64 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.83/1.64 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.83/1.64 => ( Xy = Xu ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairinjR_type,type,
% 2.83/1.64 setukpairinjR: $o ).
% 2.83/1.64
% 2.83/1.64 thf(setukpairinjR,definition,
% 2.83/1.64 ( setukpairinjR
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.83/1.64 ( ( ( kpair @ Xx @ Xy )
% 2.83/1.64 = ( kpair @ Xz @ Xu ) )
% 2.83/1.64 => ( Xy = Xu ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(ksndsingleton_type,type,
% 2.83/1.64 ksndsingleton: $o ).
% 2.83/1.64
% 2.83/1.64 thf(ksndsingleton,definition,
% 2.83/1.64 ( ksndsingleton
% 2.83/1.64 = ( ! [Xu: $i] :
% 2.83/1.64 ( ( iskpair @ Xu )
% 2.83/1.64 => ( singleton
% 2.83/1.64 @ ( dsetconstr @ ( setunion @ Xu )
% 2.83/1.64 @ ^ [Xx: $i] :
% 2.83/1.64 ( Xu
% 2.83/1.64 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(ksnd_type,type,
% 2.83/1.64 ksnd: $i > $i ).
% 2.83/1.64
% 2.83/1.64 thf(ksndpairEq_type,type,
% 2.83/1.64 ksndpairEq: $o ).
% 2.83/1.64
% 2.83/1.64 thf(ksndpairEq,definition,
% 2.83/1.64 ( ksndpairEq
% 2.83/1.64 = ( ! [Xx: $i,Xy: $i] :
% 2.83/1.64 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.83/1.64 = Xy ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(kpairsurjEq_type,type,
% 2.83/1.64 kpairsurjEq: $o ).
% 2.83/1.64
% 2.83/1.64 thf(kpairsurjEq,definition,
% 2.83/1.64 ( kpairsurjEq
% 2.83/1.64 = ( ! [Xu: $i] :
% 2.83/1.64 ( ( iskpair @ Xu )
% 2.83/1.64 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.83/1.64 = Xu ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodsndin_type,type,
% 2.83/1.64 cartprodsndin: $o ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodsndin,definition,
% 2.83/1.64 ( cartprodsndin
% 2.83/1.64 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.83/1.64 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.83/1.64 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodpairmemEL_type,type,
% 2.83/1.64 cartprodpairmemEL: $o ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodpairmemEL,definition,
% 2.83/1.64 ( cartprodpairmemEL
% 2.83/1.64 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.83/1.64 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.83/1.64 => ( in @ Xx @ A ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodpairmemER_type,type,
% 2.83/1.64 cartprodpairmemER: $o ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodpairmemER,definition,
% 2.83/1.64 ( cartprodpairmemER
% 2.83/1.64 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.83/1.64 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.83/1.64 => ( in @ Xy @ B ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodmempaircEq_type,type,
% 2.83/1.64 cartprodmempaircEq: $o ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodmempaircEq,definition,
% 2.83/1.64 ( cartprodmempaircEq
% 2.83/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ B )
% 2.83/1.64 => ( ( kpair @ Xx @ Xy )
% 2.83/1.64 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodfstpairEq_type,type,
% 2.83/1.64 cartprodfstpairEq: $o ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodfstpairEq,definition,
% 2.83/1.64 ( cartprodfstpairEq
% 2.83/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ B )
% 2.83/1.64 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.83/1.64 = Xx ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodsndpairEq_type,type,
% 2.83/1.64 cartprodsndpairEq: $o ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodsndpairEq,definition,
% 2.83/1.64 ( cartprodsndpairEq
% 2.83/1.64 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ B )
% 2.83/1.64 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.83/1.64 = Xy ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodpairsurjEq_type,type,
% 2.83/1.64 cartprodpairsurjEq: $o ).
% 2.83/1.64
% 2.83/1.64 thf(cartprodpairsurjEq,definition,
% 2.83/1.64 ( cartprodpairsurjEq
% 2.83/1.64 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.83/1.64 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.83/1.64 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.83/1.64 = Xu ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(breln_type,type,
% 2.83/1.64 breln: $i > $i > $i > $o ).
% 2.83/1.64
% 2.83/1.64 thf(breln,definition,
% 2.83/1.64 ( breln
% 2.83/1.64 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(dpsetconstr_type,type,
% 2.83/1.64 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.83/1.64
% 2.83/1.64 thf(dpsetconstrI_type,type,
% 2.83/1.64 dpsetconstrI: $o ).
% 2.83/1.64
% 2.83/1.64 thf(dpsetconstrI,definition,
% 2.83/1.64 ( dpsetconstrI
% 2.83/1.64 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ B )
% 2.83/1.64 => ( ( Xphi @ Xx @ Xy )
% 2.83/1.64 => ( in @ ( kpair @ Xx @ Xy )
% 2.83/1.64 @ ( dpsetconstr @ A @ B
% 2.83/1.64 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(dpsetconstrSub_type,type,
% 2.83/1.64 dpsetconstrSub: $o ).
% 2.83/1.64
% 2.83/1.64 thf(dpsetconstrSub,definition,
% 2.83/1.64 ( dpsetconstrSub
% 2.83/1.64 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.83/1.64 ( subset
% 2.83/1.64 @ ( dpsetconstr @ A @ B
% 2.83/1.64 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.83/1.64 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(setOfPairsIsBReln_type,type,
% 2.83/1.64 setOfPairsIsBReln: $o ).
% 2.83/1.64
% 2.83/1.64 thf(setOfPairsIsBReln,definition,
% 2.83/1.64 ( setOfPairsIsBReln
% 2.83/1.64 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.83/1.64 ( breln @ A @ B
% 2.83/1.64 @ ( dpsetconstr @ A @ B
% 2.83/1.64 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(dpsetconstrERa_type,type,
% 2.83/1.64 dpsetconstrERa: $o ).
% 2.83/1.64
% 2.83/1.64 thf(dpsetconstrERa,definition,
% 2.83/1.64 ( dpsetconstrERa
% 2.83/1.64 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ B )
% 2.83/1.64 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.83/1.64 @ ( dpsetconstr @ A @ B
% 2.83/1.64 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.83/1.64 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(dpsetconstrEL1_type,type,
% 2.83/1.64 dpsetconstrEL1: $o ).
% 2.83/1.64
% 2.83/1.64 thf(dpsetconstrEL1,definition,
% 2.83/1.64 ( dpsetconstrEL1
% 2.83/1.64 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.83/1.64 ( ( in @ ( kpair @ Xx @ Xy )
% 2.83/1.64 @ ( dpsetconstr @ A @ B
% 2.83/1.64 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.83/1.64 => ( in @ Xx @ A ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(dpsetconstrEL2_type,type,
% 2.83/1.64 dpsetconstrEL2: $o ).
% 2.83/1.64
% 2.83/1.64 thf(dpsetconstrEL2,definition,
% 2.83/1.64 ( dpsetconstrEL2
% 2.83/1.64 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.83/1.64 ( ( in @ ( kpair @ Xx @ Xy )
% 2.83/1.64 @ ( dpsetconstr @ A @ B
% 2.83/1.64 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.83/1.64 => ( in @ Xy @ B ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(dpsetconstrER_type,type,
% 2.83/1.64 dpsetconstrER: $o ).
% 2.83/1.64
% 2.83/1.64 thf(dpsetconstrER,definition,
% 2.83/1.64 ( dpsetconstrER
% 2.83/1.64 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.83/1.64 ( ( in @ ( kpair @ Xx @ Xy )
% 2.83/1.64 @ ( dpsetconstr @ A @ B
% 2.83/1.64 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.83/1.64 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(func_type,type,
% 2.83/1.64 func: $i > $i > $i > $o ).
% 2.83/1.64
% 2.83/1.64 thf(func,definition,
% 2.83/1.64 ( func
% 2.83/1.64 = ( ^ [A: $i,B: $i,R: $i] :
% 2.83/1.64 ( ( breln @ A @ B @ R )
% 2.83/1.64 & ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ( ex1 @ B
% 2.83/1.64 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(funcSet_type,type,
% 2.83/1.64 funcSet: $i > $i > $i ).
% 2.83/1.64
% 2.83/1.64 thf(funcImageSingleton_type,type,
% 2.83/1.64 funcImageSingleton: $o ).
% 2.83/1.64
% 2.83/1.64 thf(funcImageSingleton,definition,
% 2.83/1.64 ( funcImageSingleton
% 2.83/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.83/1.64 ( ( func @ A @ B @ Xf )
% 2.83/1.64 => ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ( singleton
% 2.83/1.64 @ ( dsetconstr @ B
% 2.83/1.64 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(apProp_type,type,
% 2.83/1.64 apProp: $o ).
% 2.83/1.64
% 2.83/1.64 thf(apProp,definition,
% 2.83/1.64 ( apProp
% 2.83/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.83/1.64 ( ( func @ A @ B @ Xf )
% 2.83/1.64 => ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ( in
% 2.83/1.64 @ ( setunion
% 2.83/1.64 @ ( dsetconstr @ B
% 2.83/1.64 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.83/1.64 @ B ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(ap_type,type,
% 2.83/1.64 ap: $i > $i > $i > $i > $i ).
% 2.83/1.64
% 2.83/1.64 thf(app_type,type,
% 2.83/1.64 app: $o ).
% 2.83/1.64
% 2.83/1.64 thf(app,definition,
% 2.83/1.64 ( app
% 2.83/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.83/1.64 ( ( func @ A @ B @ Xf )
% 2.83/1.64 => ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(infuncsetfunc_type,type,
% 2.83/1.64 infuncsetfunc: $o ).
% 2.83/1.64
% 2.83/1.64 thf(infuncsetfunc,definition,
% 2.83/1.64 ( infuncsetfunc
% 2.83/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.83/1.64 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.83/1.64 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(ap2p_type,type,
% 2.83/1.64 ap2p: $o ).
% 2.83/1.64
% 2.83/1.64 thf(ap2p,definition,
% 2.83/1.64 ( ap2p
% 2.83/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.83/1.64 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.83/1.64 => ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(funcinfuncset_type,type,
% 2.83/1.64 funcinfuncset: $o ).
% 2.83/1.64
% 2.83/1.64 thf(funcinfuncset,definition,
% 2.83/1.64 ( funcinfuncset
% 2.83/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.83/1.64 ( ( func @ A @ B @ Xf )
% 2.83/1.64 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(lamProp_type,type,
% 2.83/1.64 lamProp: $o ).
% 2.83/1.64
% 2.83/1.64 thf(lamProp,definition,
% 2.83/1.64 ( lamProp
% 2.83/1.64 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.83/1.64 ( ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.83/1.64 => ( func @ A @ B
% 2.83/1.64 @ ( dpsetconstr @ A @ B
% 2.83/1.64 @ ^ [Xx: $i,Xy: $i] :
% 2.83/1.64 ( ( Xf @ Xx )
% 2.83/1.64 = Xy ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(lam_type,type,
% 2.83/1.64 lam: $i > $i > ( $i > $i ) > $i ).
% 2.83/1.64
% 2.83/1.64 thf(lam,definition,
% 2.83/1.64 ( lam
% 2.83/1.64 = ( ^ [A: $i,B: $i,Xf: $i > $i] :
% 2.83/1.64 ( dpsetconstr @ A @ B
% 2.83/1.64 @ ^ [Xx: $i,Xy: $i] :
% 2.83/1.64 ( ( Xf @ Xx )
% 2.83/1.64 = Xy ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(lamp_type,type,
% 2.83/1.64 lamp: $o ).
% 2.83/1.64
% 2.83/1.64 thf(lamp,definition,
% 2.83/1.64 ( lamp
% 2.83/1.64 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.83/1.64 ( ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.83/1.64 => ( func @ A @ B
% 2.83/1.64 @ ( lam @ A @ B
% 2.83/1.64 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(lam2p_type,type,
% 2.83/1.64 lam2p: $o ).
% 2.83/1.64
% 2.83/1.64 thf(lam2p,definition,
% 2.83/1.64 ( lam2p
% 2.83/1.64 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.83/1.64 ( ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.83/1.64 => ( in
% 2.83/1.64 @ ( lam @ A @ B
% 2.83/1.64 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.83/1.64 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(brelnall1_type,type,
% 2.83/1.64 brelnall1: $o ).
% 2.83/1.64
% 2.83/1.64 thf(brelnall1,definition,
% 2.83/1.64 ( brelnall1
% 2.83/1.64 = ( ! [A: $i,B: $i,R: $i] :
% 2.83/1.64 ( ( breln @ A @ B @ R )
% 2.83/1.64 => ! [Xphi: $i > $o] :
% 2.83/1.64 ( ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ B )
% 2.83/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.83/1.64 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.83/1.64 => ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ R )
% 2.83/1.64 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(brelnall2_type,type,
% 2.83/1.64 brelnall2: $o ).
% 2.83/1.64
% 2.83/1.64 thf(brelnall2,definition,
% 2.83/1.64 ( brelnall2
% 2.83/1.64 = ( ! [A: $i,B: $i,R: $i] :
% 2.83/1.64 ( ( breln @ A @ B @ R )
% 2.83/1.64 => ! [Xphi: $i > $o] :
% 2.83/1.64 ( ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ B )
% 2.83/1.64 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.83/1.64 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.83/1.64 => ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ R )
% 2.83/1.64 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(ex1E2_type,type,
% 2.83/1.64 ex1E2: $o ).
% 2.83/1.64
% 2.83/1.64 thf(ex1E2,definition,
% 2.83/1.64 ( ex1E2
% 2.83/1.64 = ( ! [A: $i,Xphi: $i > $o] :
% 2.83/1.64 ( ( ex1 @ A
% 2.83/1.64 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.83/1.64 => ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ! [Xy: $i] :
% 2.83/1.64 ( ( in @ Xy @ A )
% 2.83/1.64 => ( ( Xphi @ Xx )
% 2.83/1.64 => ( ( Xphi @ Xy )
% 2.83/1.64 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(funcGraphProp1_type,type,
% 2.83/1.64 funcGraphProp1: $o ).
% 2.83/1.64
% 2.83/1.64 thf(funcGraphProp1,definition,
% 2.83/1.64 ( funcGraphProp1
% 2.83/1.64 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.83/1.64 ( ( func @ A @ B @ Xf )
% 2.83/1.64 => ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.83/1.64
% 2.83/1.64 thf(funcGraphProp3,conjecture,
% 2.83/1.64 ( setextAx
% 2.83/1.64 => ( emptysetAx
% 2.83/1.64 => ( setadjoinAx
% 2.83/1.64 => ( powersetAx
% 2.83/1.64 => ( setunionAx
% 2.83/1.64 => ( omega0Ax
% 2.83/1.64 => ( omegaSAx
% 2.83/1.64 => ( omegaIndAx
% 2.83/1.64 => ( replAx
% 2.83/1.64 => ( foundationAx
% 2.83/1.64 => ( wellorderingAx
% 2.83/1.64 => ( descrp
% 2.83/1.64 => ( dsetconstrI
% 2.83/1.64 => ( dsetconstrEL
% 2.83/1.64 => ( dsetconstrER
% 2.83/1.64 => ( exuE1
% 2.83/1.64 => ( prop2setE
% 2.83/1.64 => ( emptysetE
% 2.83/1.64 => ( emptysetimpfalse
% 2.83/1.64 => ( notinemptyset
% 2.83/1.64 => ( exuE3e
% 2.83/1.64 => ( setext
% 2.83/1.64 => ( emptyI
% 2.83/1.64 => ( noeltsimpempty
% 2.83/1.64 => ( setbeta
% 2.83/1.64 => ( nonemptyE1
% 2.83/1.64 => ( nonemptyI
% 2.83/1.64 => ( nonemptyI1
% 2.83/1.64 => ( setadjoinIL
% 2.83/1.64 => ( emptyinunitempty
% 2.83/1.64 => ( setadjoinIR
% 2.83/1.64 => ( setadjoinE
% 2.83/1.64 => ( setadjoinOr
% 2.83/1.64 => ( setoftrueEq
% 2.83/1.64 => ( powersetI
% 2.83/1.64 => ( emptyinPowerset
% 2.83/1.64 => ( emptyInPowerset
% 2.83/1.64 => ( powersetE
% 2.83/1.64 => ( setunionI
% 2.83/1.64 => ( setunionE
% 2.83/1.64 => ( subPowSU
% 2.83/1.64 => ( exuE2
% 2.83/1.64 => ( nonemptyImpWitness
% 2.83/1.64 => ( uniqinunit
% 2.83/1.64 => ( notinsingleton
% 2.83/1.64 => ( eqinunit
% 2.83/1.64 => ( singletonsswitch
% 2.83/1.64 => ( upairsetE
% 2.83/1.64 => ( upairsetIL
% 2.83/1.64 => ( upairsetIR
% 2.83/1.64 => ( emptyE1
% 2.83/1.64 => ( vacuousDall
% 2.83/1.64 => ( quantDeMorgan1
% 2.83/1.64 => ( quantDeMorgan2
% 2.83/1.64 => ( quantDeMorgan3
% 2.83/1.64 => ( quantDeMorgan4
% 2.83/1.64 => ( prop2setI
% 2.83/1.64 => ( prop2set2propI
% 2.83/1.64 => ( notdexE
% 2.83/1.64 => ( notdallE
% 2.83/1.64 => ( exuI1
% 2.83/1.64 => ( exuI3
% 2.83/1.64 => ( exuI2
% 2.83/1.64 => ( inCongP
% 2.83/1.64 => ( in__Cong
% 2.83/1.64 => ( exuE3u
% 2.83/1.64 => ( exu__Cong
% 2.83/1.64 => ( emptyset__Cong
% 2.83/1.64 => ( setadjoin__Cong
% 2.83/1.64 => ( powerset__Cong
% 2.83/1.64 => ( setunion__Cong
% 2.83/1.64 => ( omega__Cong
% 2.83/1.64 => ( exuEu
% 2.83/1.64 => ( descr__Cong
% 2.83/1.64 => ( dsetconstr__Cong
% 2.83/1.64 => ( subsetI1
% 2.83/1.64 => ( eqimpsubset2
% 2.83/1.64 => ( eqimpsubset1
% 2.83/1.64 => ( subsetI2
% 2.83/1.64 => ( emptysetsubset
% 2.83/1.64 => ( subsetE
% 2.83/1.64 => ( subsetE2
% 2.83/1.64 => ( notsubsetI
% 2.83/1.64 => ( notequalI1
% 2.83/1.64 => ( notequalI2
% 2.83/1.64 => ( subsetRefl
% 2.83/1.64 => ( subsetTrans
% 2.83/1.64 => ( setadjoinSub
% 2.83/1.64 => ( setadjoinSub2
% 2.83/1.64 => ( subset2powerset
% 2.83/1.64 => ( setextsub
% 2.83/1.64 => ( subsetemptysetimpeq
% 2.83/1.64 => ( powersetI1
% 2.83/1.64 => ( powersetE1
% 2.83/1.64 => ( inPowerset
% 2.83/1.64 => ( powersetsubset
% 2.83/1.64 => ( sepInPowerset
% 2.83/1.64 => ( sepSubset
% 2.83/1.64 => ( binunionIL
% 2.83/1.64 => ( upairset2IR
% 2.83/1.64 => ( binunionIR
% 2.83/1.64 => ( binunionEcases
% 2.83/1.64 => ( binunionE
% 2.83/1.64 => ( binunionLsub
% 2.83/1.64 => ( binunionRsub
% 2.83/1.64 => ( binintersectI
% 2.83/1.64 => ( binintersectSubset5
% 2.83/1.64 => ( binintersectEL
% 2.83/1.64 => ( binintersectLsub
% 2.83/1.64 => ( binintersectSubset2
% 2.83/1.64 => ( binintersectSubset3
% 2.83/1.64 => ( binintersectER
% 2.83/1.64 => ( disjointsetsI1
% 2.83/1.64 => ( binintersectRsub
% 2.83/1.64 => ( binintersectSubset4
% 2.83/1.64 => ( binintersectSubset1
% 2.83/1.64 => ( bs114d
% 2.83/1.64 => ( setminusI
% 2.83/1.64 => ( setminusEL
% 2.83/1.64 => ( setminusER
% 2.83/1.64 => ( setminusSubset2
% 2.83/1.64 => ( setminusERneg
% 2.83/1.64 => ( setminusELneg
% 2.83/1.64 => ( setminusILneg
% 2.83/1.64 => ( setminusIRneg
% 2.83/1.64 => ( setminusLsub
% 2.83/1.64 => ( setminusSubset1
% 2.83/1.64 => ( symdiffE
% 2.83/1.64 => ( symdiffI1
% 2.83/1.64 => ( symdiffI2
% 2.83/1.64 => ( symdiffIneg1
% 2.83/1.64 => ( symdiffIneg2
% 2.83/1.64 => ( secondinupair
% 2.83/1.64 => ( setukpairIL
% 2.83/1.64 => ( setukpairIR
% 2.83/1.64 => ( kpairiskpair
% 2.83/1.64 => ( kpairp
% 2.83/1.64 => ( singletonsubset
% 2.83/1.64 => ( singletoninpowerset
% 2.83/1.64 => ( singletoninpowunion
% 2.83/1.64 => ( upairset2E
% 2.83/1.64 => ( upairsubunion
% 2.83/1.64 => ( upairinpowunion
% 2.83/1.64 => ( ubforcartprodlem1
% 2.83/1.64 => ( ubforcartprodlem2
% 2.83/1.64 => ( ubforcartprodlem3
% 2.83/1.64 => ( cartprodpairin
% 2.83/1.64 => ( cartprodmempair1
% 2.83/1.64 => ( cartprodmempair
% 2.83/1.64 => ( setunionE2
% 2.83/1.64 => ( setunionsingleton1
% 2.83/1.64 => ( setunionsingleton2
% 2.83/1.64 => ( setunionsingleton
% 2.83/1.64 => ( singletonprop
% 2.83/1.64 => ( ex1E1
% 2.83/1.64 => ( ex1I
% 2.83/1.64 => ( ex1I2
% 2.83/1.64 => ( singletonsuniq
% 2.83/1.64 => ( setukpairinjL1
% 2.83/1.64 => ( kfstsingleton
% 2.83/1.64 => ( theprop
% 2.83/1.64 => ( kfstpairEq
% 2.83/1.64 => ( cartprodfstin
% 2.83/1.64 => ( setukpairinjL2
% 2.83/1.64 => ( setukpairinjL
% 2.83/1.64 => ( setukpairinjR11
% 2.83/1.64 => ( setukpairinjR12
% 2.83/1.64 => ( setukpairinjR1
% 2.83/1.64 => ( upairequniteq
% 2.83/1.64 => ( setukpairinjR2
% 2.83/1.64 => ( setukpairinjR
% 2.83/1.64 => ( ksndsingleton
% 2.83/1.64 => ( ksndpairEq
% 2.83/1.64 => ( kpairsurjEq
% 2.83/1.64 => ( cartprodsndin
% 2.83/1.64 => ( cartprodpairmemEL
% 2.83/1.64 => ( cartprodpairmemER
% 2.83/1.64 => ( cartprodmempaircEq
% 2.83/1.64 => ( cartprodfstpairEq
% 2.83/1.64 => ( cartprodsndpairEq
% 2.83/1.64 => ( cartprodpairsurjEq
% 2.83/1.64 => ( dpsetconstrI
% 2.83/1.64 => ( dpsetconstrSub
% 2.83/1.64 => ( setOfPairsIsBReln
% 2.83/1.64 => ( dpsetconstrERa
% 2.83/1.64 => ( dpsetconstrEL1
% 2.83/1.64 => ( dpsetconstrEL2
% 2.83/1.64 => ( dpsetconstrER
% 2.83/1.64 => ( funcImageSingleton
% 2.83/1.64 => ( apProp
% 2.83/1.64 => ( app
% 2.83/1.64 => ( infuncsetfunc
% 2.83/1.64 => ( ap2p
% 2.83/1.64 => ( funcinfuncset
% 2.83/1.64 => ( lamProp
% 2.83/1.64 => ( lamp
% 2.83/1.64 => ( lam2p
% 2.83/1.64 => ( brelnall1
% 2.83/1.64 => ( brelnall2
% 2.83/1.64 => ( ex1E2
% 2.83/1.64 => ( funcGraphProp1
% 2.83/1.64 => ! [A: $i,B: $i,Xf: $i] :
% 2.83/1.64 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.83/1.64 => ! [Xx: $i] :
% 2.83/1.64 ( ( in @ Xx @ A )
% 2.83/1.64 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.83/1.64 ------------------------
% 2.83/1.67 % (22525)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/18Mi)
% 2.83/1.67 % (22520)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_22405 for (2999ds/4Mi)
% 2.83/1.67 % (22524)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_22405 for (2999ds/275Mi)
% 2.83/1.67 % (22521)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_22405 for (2999ds/27Mi)
% 2.83/1.67 % (22523)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_22405 for (2999ds/2Mi)
% 2.83/1.67 % (22519)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/183Mi)
% 2.83/1.67 % (22522)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/2Mi)
% 2.83/1.67 % (22522)Instruction limit reached!
% 2.83/1.67 % (22522)------------------------------
% 2.83/1.67 % (22522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.67 % (22522)Termination reason: Unknown
% 2.83/1.67 % (22523)Instruction limit reached!
% 2.83/1.67 % (22523)------------------------------
% 2.83/1.67 % (22523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.67 % (22523)Termination reason: Unknown
% 2.83/1.67 % (22523)Termination phase: shuffling
% 2.83/1.67
% 2.83/1.67 % (22523)Memory used [KB]: 1407
% 2.83/1.67 % (22523)Time elapsed: 0.003 s
% 2.83/1.67 % (22523)Instructions burned: 2 (million)
% 2.83/1.67 % (22523)------------------------------
% 2.83/1.67 % (22523)------------------------------
% 2.83/1.67 % (22522)Termination phase: shuffling
% 2.83/1.67
% 2.83/1.67 % (22522)Memory used [KB]: 1407
% 2.83/1.67 % (22522)Time elapsed: 0.003 s
% 2.83/1.67 % (22522)Instructions burned: 2 (million)
% 2.83/1.67 % (22522)------------------------------
% 2.83/1.67 % (22522)------------------------------
% 2.83/1.67 % (22520)Instruction limit reached!
% 2.83/1.67 % (22520)------------------------------
% 2.83/1.67 % (22520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.67 % (22520)Termination reason: Unknown
% 2.83/1.67 % (22520)Termination phase: shuffling
% 2.83/1.67
% 2.83/1.67 % (22520)Memory used [KB]: 1407
% 2.83/1.67 % (22520)Time elapsed: 0.004 s
% 2.83/1.67 % (22520)Instructions burned: 4 (million)
% 2.83/1.67 % (22520)------------------------------
% 2.83/1.67 % (22520)------------------------------
% 2.83/1.68 % (22525)Instruction limit reached!
% 2.83/1.68 % (22525)------------------------------
% 2.83/1.68 % (22525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.68 % (22525)Termination reason: Unknown
% 2.83/1.68 % (22525)Termination phase: shuffling
% 2.83/1.68
% 2.83/1.68 % (22525)Memory used [KB]: 1663
% 2.83/1.68 % (22525)Time elapsed: 0.011 s
% 2.83/1.68 % (22525)Instructions burned: 18 (million)
% 2.83/1.68 % (22525)------------------------------
% 2.83/1.68 % (22525)------------------------------
% 2.83/1.68 % (22526)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/3Mi)
% 2.83/1.68 % (22521)Instruction limit reached!
% 2.83/1.68 % (22521)------------------------------
% 2.83/1.68 % (22521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.68 % (22521)Termination reason: Unknown
% 2.83/1.68 % (22521)Termination phase: shuffling
% 2.83/1.68
% 2.83/1.68 % (22521)Memory used [KB]: 1918
% 2.83/1.68 % (22521)Time elapsed: 0.015 s
% 2.83/1.68 % (22521)Instructions burned: 28 (million)
% 2.83/1.68 % (22521)------------------------------
% 2.83/1.68 % (22521)------------------------------
% 2.83/1.68 % (22526)Instruction limit reached!
% 2.83/1.68 % (22526)------------------------------
% 2.83/1.68 % (22526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.68 % (22526)Termination reason: Unknown
% 2.83/1.68 % (22526)Termination phase: shuffling
% 2.83/1.68
% 2.83/1.68 % (22526)Memory used [KB]: 1407
% 2.83/1.68 % (22526)Time elapsed: 0.002 s
% 2.83/1.68 % (22526)Instructions burned: 3 (million)
% 2.83/1.68 % (22526)------------------------------
% 2.83/1.68 % (22526)------------------------------
% 2.83/1.69 % (22527)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_22405 for (2999ds/37Mi)
% 2.83/1.69 % (22528)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_22405 for (2999ds/15Mi)
% 2.83/1.69 % (22529)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_22405 for (2999ds/3Mi)
% 2.83/1.69 % (22529)Instruction limit reached!
% 2.83/1.69 % (22529)------------------------------
% 2.83/1.69 % (22529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.69 % (22529)Termination reason: Unknown
% 2.83/1.69 % (22529)Termination phase: shuffling
% 2.83/1.69
% 2.83/1.69 % (22529)Memory used [KB]: 1407
% 2.83/1.69 % (22529)Time elapsed: 0.002 s
% 2.83/1.69 % (22529)Instructions burned: 4 (million)
% 2.83/1.69 % (22529)------------------------------
% 2.83/1.69 % (22529)------------------------------
% 2.83/1.70 % (22528)Instruction limit reached!
% 2.83/1.70 % (22528)------------------------------
% 2.83/1.70 % (22528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.83/1.70 % (22528)Termination reason: Unknown
% 2.83/1.70 % (22528)Termination phase: shuffling
% 2.83/1.70
% 2.83/1.70 % (22528)Memory used [KB]: 1663
% 2.83/1.70 % (22528)Time elapsed: 0.009 s
% 2.83/1.70 % (22528)Instructions burned: 15 (million)
% 2.83/1.70 % (22528)------------------------------
% 2.83/1.70 % (22528)------------------------------
% 2.87/1.70 % (22531)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/7Mi)
% 2.87/1.70 % (22530)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/1041Mi)
% 2.87/1.70 % (22531)Instruction limit reached!
% 2.87/1.70 % (22531)------------------------------
% 2.87/1.70 % (22531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.70 % (22531)Termination reason: Unknown
% 2.87/1.70 % (22531)Termination phase: shuffling
% 2.87/1.70
% 2.87/1.70 % (22531)Memory used [KB]: 1535
% 2.87/1.70 % (22531)Time elapsed: 0.004 s
% 2.87/1.70 % (22531)Instructions burned: 9 (million)
% 2.87/1.70 % (22531)------------------------------
% 2.87/1.70 % (22531)------------------------------
% 2.87/1.71 % (22527)Instruction limit reached!
% 2.87/1.71 % (22527)------------------------------
% 2.87/1.71 % (22527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.71 % (22527)Termination reason: Unknown
% 2.87/1.71 % (22527)Termination phase: Property scanning
% 2.87/1.71
% 2.87/1.71 % (22527)Memory used [KB]: 2174
% 2.87/1.71 % (22527)Time elapsed: 0.019 s
% 2.87/1.71 % (22527)Instructions burned: 38 (million)
% 2.87/1.71 % (22527)------------------------------
% 2.87/1.71 % (22527)------------------------------
% 2.87/1.71 % (22533)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/3Mi)
% 2.87/1.71 % (22533)Instruction limit reached!
% 2.87/1.71 % (22533)------------------------------
% 2.87/1.71 % (22533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.71 % (22533)Termination reason: Unknown
% 2.87/1.71 % (22533)Termination phase: shuffling
% 2.87/1.71
% 2.87/1.71 % (22533)Memory used [KB]: 1407
% 2.87/1.71 % (22533)Time elapsed: 0.003 s
% 2.87/1.71 % (22533)Instructions burned: 3 (million)
% 2.87/1.71 % (22533)------------------------------
% 2.87/1.71 % (22533)------------------------------
% 2.87/1.71 % (22532)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/16Mi)
% 2.87/1.72 % (22535)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/3Mi)
% 2.87/1.72 % (22535)Instruction limit reached!
% 2.87/1.72 % (22535)------------------------------
% 2.87/1.72 % (22535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.72 % (22535)Termination reason: Unknown
% 2.87/1.72 % (22535)Termination phase: shuffling
% 2.87/1.72
% 2.87/1.72 % (22535)Memory used [KB]: 1407
% 2.87/1.72 % (22535)Time elapsed: 0.003 s
% 2.87/1.72 % (22535)Instructions burned: 3 (million)
% 2.87/1.72 % (22535)------------------------------
% 2.87/1.72 % (22535)------------------------------
% 2.87/1.72 % (22532)Instruction limit reached!
% 2.87/1.72 % (22532)------------------------------
% 2.87/1.72 % (22532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.72 % (22532)Termination reason: Unknown
% 2.87/1.72 % (22532)Termination phase: shuffling
% 2.87/1.72
% 2.87/1.72 % (22532)Memory used [KB]: 1663
% 2.87/1.72 % (22532)Time elapsed: 0.009 s
% 2.87/1.72 % (22532)Instructions burned: 16 (million)
% 2.87/1.72 % (22532)------------------------------
% 2.87/1.72 % (22532)------------------------------
% 2.87/1.72 % (22536)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/7Mi)
% 2.87/1.72 % (22536)Instruction limit reached!
% 2.87/1.72 % (22536)------------------------------
% 2.87/1.72 % (22536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.72 % (22536)Termination reason: Unknown
% 2.87/1.72 % (22536)Termination phase: shuffling
% 2.87/1.72
% 2.87/1.72 % (22536)Memory used [KB]: 1535
% 2.87/1.72 % (22536)Time elapsed: 0.005 s
% 2.87/1.72 % (22536)Instructions burned: 8 (million)
% 2.87/1.72 % (22536)------------------------------
% 2.87/1.72 % (22536)------------------------------
% 2.87/1.73 % (22537)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/3Mi)
% 2.87/1.73 % (22537)Instruction limit reached!
% 2.87/1.73 % (22537)------------------------------
% 2.87/1.73 % (22537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.73 % (22537)Termination reason: Unknown
% 2.87/1.73 % (22537)Termination phase: shuffling
% 2.87/1.73
% 2.87/1.73 % (22537)Memory used [KB]: 1407
% 2.87/1.73 % (22537)Time elapsed: 0.003 s
% 2.87/1.73 % (22537)Instructions burned: 3 (million)
% 2.87/1.73 % (22537)------------------------------
% 2.87/1.73 % (22537)------------------------------
% 2.87/1.73 % (22538)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/4Mi)
% 2.87/1.74 % (22538)Instruction limit reached!
% 2.87/1.74 % (22538)------------------------------
% 2.87/1.74 % (22538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.74 % (22538)Termination reason: Unknown
% 2.87/1.74 % (22538)Termination phase: shuffling
% 2.87/1.74
% 2.87/1.74 % (22538)Memory used [KB]: 1407
% 2.87/1.74 % (22538)Time elapsed: 0.004 s
% 2.87/1.74 % (22538)Instructions burned: 5 (million)
% 2.87/1.74 % (22538)------------------------------
% 2.87/1.74 % (22538)------------------------------
% 2.87/1.74 % (22539)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/18Mi)
% 2.87/1.74 % (22540)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/710Mi)
% 2.87/1.75 % (22539)Instruction limit reached!
% 2.87/1.75 % (22539)------------------------------
% 2.87/1.75 % (22539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.75 % (22539)Termination reason: Unknown
% 2.87/1.75 % (22539)Termination phase: shuffling
% 2.87/1.75
% 2.87/1.75 % (22539)Memory used [KB]: 1791
% 2.87/1.75 % (22539)Time elapsed: 0.010 s
% 2.87/1.75 % (22539)Instructions burned: 18 (million)
% 2.87/1.75 % (22539)------------------------------
% 2.87/1.75 % (22539)------------------------------
% 2.87/1.75 % (22541)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/6Mi)
% 2.87/1.75 % (22541)Instruction limit reached!
% 2.87/1.75 % (22541)------------------------------
% 2.87/1.75 % (22541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.75 % (22541)Termination reason: Unknown
% 2.87/1.75 % (22541)Termination phase: shuffling
% 2.87/1.75
% 2.87/1.75 % (22541)Memory used [KB]: 1535
% 2.87/1.75 % (22541)Time elapsed: 0.005 s
% 2.87/1.75 % (22541)Instructions burned: 8 (million)
% 2.87/1.75 % (22541)------------------------------
% 2.87/1.75 % (22541)------------------------------
% 2.87/1.75 % (22542)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/902Mi)
% 2.87/1.75 % (22519)Instruction limit reached!
% 2.87/1.75 % (22519)------------------------------
% 2.87/1.75 % (22519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.75 % (22519)Termination reason: Unknown
% 2.87/1.75 % (22519)Termination phase: Saturation
% 2.87/1.75
% 2.87/1.75 % (22519)Memory used [KB]: 7419
% 2.87/1.75 % (22519)Time elapsed: 0.084 s
% 2.87/1.75 % (22519)Instructions burned: 184 (million)
% 2.87/1.75 % (22519)------------------------------
% 2.87/1.75 % (22519)------------------------------
% 2.87/1.76 % (22543)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_22405 for (2999ds/21Mi)
% 2.87/1.77 % (22545)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_22405 for (2998ds/5Mi)
% 2.87/1.77 % (22545)Instruction limit reached!
% 2.87/1.77 % (22545)------------------------------
% 2.87/1.77 % (22545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.77 % (22545)Termination reason: Unknown
% 2.87/1.77 % (22545)Termination phase: shuffling
% 2.87/1.77
% 2.87/1.77 % (22545)Memory used [KB]: 1407
% 2.87/1.77 % (22545)Time elapsed: 0.004 s
% 2.87/1.77 % (22545)Instructions burned: 6 (million)
% 2.87/1.77 % (22545)------------------------------
% 2.87/1.77 % (22545)------------------------------
% 2.87/1.77 % (22546)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_22405 for (2998ds/6Mi)
% 2.87/1.77 % (22543)Instruction limit reached!
% 2.87/1.77 % (22543)------------------------------
% 2.87/1.77 % (22543)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.77 % (22543)Termination reason: Unknown
% 2.87/1.77 % (22543)Termination phase: shuffling
% 2.87/1.77
% 2.87/1.77 % (22543)Memory used [KB]: 1791
% 2.87/1.77 % (22543)Time elapsed: 0.012 s
% 2.87/1.77 % (22543)Instructions burned: 23 (million)
% 2.87/1.77 % (22543)------------------------------
% 2.87/1.77 % (22543)------------------------------
% 2.87/1.77 % (22546)Instruction limit reached!
% 2.87/1.77 % (22546)------------------------------
% 2.87/1.77 % (22546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.77 % (22546)Termination reason: Unknown
% 2.87/1.77 % (22546)Termination phase: shuffling
% 2.87/1.77
% 2.87/1.77 % (22546)Memory used [KB]: 1535
% 2.87/1.77 % (22546)Time elapsed: 0.005 s
% 2.87/1.77 % (22546)Instructions burned: 7 (million)
% 2.87/1.77 % (22546)------------------------------
% 2.87/1.77 % (22546)------------------------------
% 2.87/1.79 % (22547)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_22405 for (2998ds/377Mi)
% 2.87/1.79 % (22548)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_22405 for (2998ds/779Mi)
% 2.87/1.79 % (22549)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_22405 for (2998ds/19Mi)
% 2.87/1.80 % (22549)Instruction limit reached!
% 2.87/1.80 % (22549)------------------------------
% 2.87/1.80 % (22549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.80 % (22549)Termination reason: Unknown
% 2.87/1.80 % (22549)Termination phase: shuffling
% 2.87/1.80
% 2.87/1.80 % (22549)Memory used [KB]: 1791
% 2.87/1.80 % (22549)Time elapsed: 0.010 s
% 2.87/1.80 % (22549)Instructions burned: 19 (million)
% 2.87/1.80 % (22549)------------------------------
% 2.87/1.80 % (22549)------------------------------
% 2.87/1.80 % (22524)Instruction limit reached!
% 2.87/1.80 % (22524)------------------------------
% 2.87/1.80 % (22524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.80 % (22524)Termination reason: Unknown
% 2.87/1.80 % (22524)Termination phase: Saturation
% 2.87/1.80
% 2.87/1.80 % (22524)Memory used [KB]: 8699
% 2.87/1.80 % (22524)Time elapsed: 0.136 s
% 2.87/1.80 % (22524)Instructions burned: 275 (million)
% 2.87/1.80 % (22524)------------------------------
% 2.87/1.80 % (22524)------------------------------
% 2.87/1.81 % (22550)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_22405 for (2998ds/879Mi)
% 2.87/1.82 % (22551)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_22405 for (2998ds/17Mi)
% 2.87/1.83 % (22551)Instruction limit reached!
% 2.87/1.83 % (22551)------------------------------
% 2.87/1.83 % (22551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.83 % (22551)Termination reason: Unknown
% 2.87/1.83 % (22551)Termination phase: shuffling
% 2.87/1.83
% 2.87/1.83 % (22551)Memory used [KB]: 1791
% 2.87/1.83 % (22551)Time elapsed: 0.010 s
% 2.87/1.83 % (22551)Instructions burned: 18 (million)
% 2.87/1.83 % (22551)------------------------------
% 2.87/1.83 % (22551)------------------------------
% 2.87/1.85 % (22552)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_22405 for (2998ds/3Mi)
% 2.87/1.85 % (22552)Instruction limit reached!
% 2.87/1.85 % (22552)------------------------------
% 2.87/1.85 % (22552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.85 % (22552)Termination reason: Unknown
% 2.87/1.85 % (22552)Termination phase: shuffling
% 2.87/1.85
% 2.87/1.85 % (22552)Memory used [KB]: 1407
% 2.87/1.85 % (22552)Time elapsed: 0.003 s
% 2.87/1.85 % (22552)Instructions burned: 3 (million)
% 2.87/1.85 % (22552)------------------------------
% 2.87/1.85 % (22552)------------------------------
% 2.87/1.86 % (22553)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_22405 for (2998ds/30Mi)
% 2.87/1.87 % (22542)First to succeed.
% 2.87/1.88 % (22553)Instruction limit reached!
% 2.87/1.88 % (22553)------------------------------
% 2.87/1.88 % (22553)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.88 % (22553)Termination reason: Unknown
% 2.87/1.88 % (22553)Termination phase: shuffling
% 2.87/1.88
% 2.87/1.88 % (22553)Memory used [KB]: 1918
% 2.87/1.88 % (22553)Time elapsed: 0.016 s
% 2.87/1.88 % (22553)Instructions burned: 31 (million)
% 2.87/1.88 % (22553)------------------------------
% 2.87/1.88 % (22553)------------------------------
% 2.87/1.88 % (22542)Refutation found. Thanks to Tanya!
% 2.87/1.88 % SZS status Theorem for DTF2THF_22405
% 2.87/1.88 % SZS output start Proof for DTF2THF_22405
% 2.87/1.88 thf(func_def_0, type, in: $i > $i > $o).
% 2.87/1.88 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.87/1.88 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.87/1.88 thf(func_def_8, type, powerset: $i > $i).
% 2.87/1.88 thf(func_def_10, type, setunion: $i > $i).
% 2.87/1.88 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.87/1.88 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.87/1.88 thf(func_def_26, type, prop2set: $o > $i).
% 2.87/1.88 thf(func_def_36, type, nonempty: $i > $o).
% 2.87/1.88 thf(func_def_69, type, set2prop: $i > $o).
% 2.87/1.88 thf(func_def_88, type, subset: $i > $i > $o).
% 2.87/1.88 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.87/1.88 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.87/1.88 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.87/1.88 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.87/1.88 thf(func_def_135, type, regular: $i > $o).
% 2.87/1.88 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.87/1.88 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.87/1.88 thf(func_def_153, type, iskpair: $i > $o).
% 2.87/1.88 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.87/1.88 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.87/1.88 thf(func_def_177, type, singleton: $i > $o).
% 2.87/1.88 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.87/1.88 thf(func_def_184, type, atmost1p: $i > $o).
% 2.87/1.88 thf(func_def_185, type, atleast2p: $i > $o).
% 2.87/1.88 thf(func_def_186, type, atmost2p: $i > $o).
% 2.87/1.88 thf(func_def_187, type, upairsetp: $i > $o).
% 2.87/1.88 thf(func_def_191, type, kfst: $i > $i).
% 2.87/1.88 thf(func_def_203, type, ksnd: $i > $i).
% 2.87/1.88 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.87/1.88 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.87/1.88 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.87/1.88 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.87/1.88 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.87/1.88 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.87/1.88 thf(func_def_254, type, sP2: $i > $i > $o).
% 2.87/1.88 thf(func_def_255, type, sP3: $i > $o).
% 2.87/1.88 thf(func_def_256, type, sP4: $i > $i > $o).
% 2.87/1.88 thf(func_def_257, type, sP5: $i > $i > $o).
% 2.87/1.88 thf(func_def_259, type, sP7: $i > $i > $i > $o > $o).
% 2.87/1.88 thf(func_def_268, type, sK16: $i > $i > $o).
% 2.87/1.88 thf(func_def_273, type, sK21: $i > $i > $o).
% 2.87/1.88 thf(func_def_277, type, sK25: $i > $i).
% 2.87/1.88 thf(func_def_281, type, sK29: $o > $i > $i > $i).
% 2.87/1.88 thf(func_def_282, type, sK30: ($i > $o) > $i > $i).
% 2.87/1.88 thf(func_def_284, type, sK32: $i > $o).
% 2.87/1.88 thf(func_def_296, type, sK44: ($i > $o) > $i).
% 2.87/1.88 thf(func_def_297, type, sK45: ($i > $o) > $i).
% 2.87/1.88 thf(func_def_298, type, sK46: $i > $o).
% 2.87/1.88 thf(func_def_300, type, sK48: $i > $o).
% 2.87/1.88 thf(func_def_304, type, sK52: $i > $o).
% 2.87/1.88 thf(func_def_307, type, sK55: $i > ($i > $o) > $i).
% 2.87/1.88 thf(func_def_312, type, sK60: $i > $i > $i).
% 2.87/1.88 thf(func_def_333, type, sK81: $i > $o).
% 2.87/1.88 thf(func_def_334, type, sK82: ($i > $o) > $i > $i).
% 2.87/1.88 thf(func_def_337, type, sK85: $i > ($i > $o) > $i).
% 2.87/1.88 thf(func_def_338, type, sK86: $i > $o).
% 2.87/1.88 thf(func_def_345, type, sK93: $i > $o).
% 2.87/1.88 thf(func_def_364, type, sK112: ($i > $o) > $i).
% 2.87/1.88 thf(func_def_365, type, sK113: $i > $o).
% 2.87/1.88 thf(func_def_370, type, sK118: $i > $i > $o).
% 2.87/1.88 thf(func_def_374, type, sK122: $i > $o).
% 2.87/1.88 thf(func_def_376, type, sK124: ($i > $o) > $i > $i > $i > $i).
% 2.87/1.88 thf(func_def_377, type, sK125: ($i > $o) > $i > $i > $i > $i).
% 2.87/1.88 thf(func_def_390, type, sK138: $i > $i > $i).
% 2.87/1.88 thf(func_def_410, type, sK158: $i > $i).
% 2.87/1.88 thf(func_def_411, type, sK159: ($i > $i) > $i > $i > $i).
% 2.87/1.88 thf(func_def_417, type, sK165: $i > $i > $i).
% 2.87/1.88 thf(func_def_420, type, sK168: $i > $i).
% 2.87/1.88 thf(func_def_428, type, sK176: $i > $o).
% 2.87/1.88 thf(func_def_429, type, sK177: $i > $o).
% 2.87/1.88 thf(func_def_430, type, sK178: ($i > $o) > ($i > $o) > $i).
% 2.87/1.88 thf(func_def_431, type, sK179: ($i > $o) > ($i > $o) > $i).
% 2.87/1.88 thf(func_def_437, type, sK185: $i > $i).
% 2.87/1.88 thf(func_def_445, type, sK193: $i > ($i > $o) > $i).
% 2.87/1.88 thf(func_def_446, type, sK194: $i > $o).
% 2.87/1.88 thf(func_def_448, type, sK196: $i > $o).
% 2.87/1.88 thf(func_def_454, type, sK202: $i > $i > $o).
% 2.87/1.88 thf(func_def_455, type, sK203: $i > $i).
% 2.87/1.88 thf(func_def_456, type, sK204: $i > $i).
% 2.87/1.88 thf(func_def_457, type, sK205: ($i > $i > $o) > $i > $i).
% 2.87/1.88 thf(func_def_458, type, sK206: $i > ($i > $i > $o) > $i > $i).
% 2.87/1.88 thf(func_def_459, type, sK207: ($i > $i > $o) > $i > $i).
% 2.87/1.88 thf(func_def_461, type, sK209: $i > $i > $o).
% 2.87/1.88 thf(func_def_468, type, sK216: $i > $o).
% 2.87/1.88 thf(func_def_469, type, sK217: $i > $o).
% 2.87/1.88 thf(func_def_470, type, sK218: ($i > $o) > ($i > $o) > $i).
% 2.87/1.88 thf(func_def_471, type, sK219: ($i > $o) > ($i > $o) > $i).
% 2.87/1.88 thf(func_def_480, type, sK228: $i > $o).
% 2.87/1.88 thf(func_def_482, type, sK230: $i > $o).
% 2.87/1.88 thf(func_def_484, type, sK232: $i > $o).
% 2.87/1.88 thf(func_def_503, type, sK251: $i > $o).
% 2.87/1.88 thf(func_def_505, type, sK253: $i > $i).
% 2.87/1.88 thf(func_def_508, type, sK256: $i > $i).
% 2.87/1.88 thf(func_def_510, type, sK258: $i > $i).
% 2.87/1.88 thf(func_def_512, type, sK260: $i > $o).
% 2.87/1.88 thf(func_def_514, type, sK262: ($i > $o) > $i > $i).
% 2.87/1.88 thf(func_def_515, type, sK263: ($i > $o) > $i > $i).
% 2.87/1.88 thf(func_def_524, type, sK272: $i > $i > $i).
% 2.87/1.88 thf(func_def_541, type, sK289: $i > $o).
% 2.87/1.88 thf(func_def_542, type, sK290: $i > $i).
% 2.87/1.88 thf(func_def_543, type, sK291: ($i > $o) > $i).
% 2.87/1.88 thf(func_def_547, type, sK295: $i > $i > $o).
% 2.87/1.88 thf(func_def_555, type, sK303: $i > $i > $o).
% 2.87/1.88 thf(func_def_560, type, sK308: $i > ($i > $o) > $i).
% 2.87/1.88 thf(func_def_561, type, sK309: $i > $o).
% 2.87/1.88 thf(func_def_564, type, sK312: $i > $i > $i > $i).
% 2.87/1.88 thf(func_def_565, type, sK313: $i > $i > $i > $i).
% 2.87/1.88 thf(func_def_571, type, sK319: $i > $o).
% 2.87/1.88 thf(func_def_577, type, sK325: $i > $i).
% 2.87/1.88 thf(func_def_578, type, sK326: ($i > $i) > $i > $i > $i).
% 2.87/1.88 thf(func_def_591, type, sK339: $i > $i > $i).
% 2.87/1.88 thf(func_def_624, type, sK372: $i > $i > $i).
% 2.87/1.88 thf(func_def_625, type, sK373: $i > $i > $i).
% 2.87/1.88 thf(func_def_626, type, sK374: $i > $i > $i).
% 2.87/1.88 thf(func_def_627, type, sK375: $i > $i > $i).
% 2.87/1.88 thf(func_def_628, type, sK376: $i > $i > $i > $i).
% 2.87/1.88 thf(func_def_629, type, sK377: $i > $i).
% 2.87/1.88 thf(func_def_630, type, sK378: $i > $i).
% 2.87/1.88 thf(func_def_631, type, sK379: $i > $i).
% 2.87/1.88 thf(func_def_632, type, sK380: $i > $i).
% 2.87/1.88 thf(func_def_633, type, sK381: $i > $i > $i).
% 2.87/1.88 thf(func_def_634, type, sK382: $i > $i > $i > $i).
% 2.87/1.88 thf(func_def_635, type, sK383: $i > $i > $i > $i).
% 2.87/1.88 thf(func_def_636, type, sK384: $i > $i > $i).
% 2.87/1.88 thf(func_def_637, type, sK385: $i > $i > $i).
% 2.87/1.88 thf(func_def_638, type, sK386: $i > $i).
% 2.87/1.88 thf(func_def_639, type, sK387: $i > $i > $i).
% 2.87/1.88 thf(func_def_641, type, sK389: $i > $i).
% 2.87/1.88 thf(func_def_642, type, sK390: $i > $i).
% 2.87/1.88 thf(func_def_650, type, sK398: $i > $o).
% 2.87/1.88 thf(func_def_664, type, sK412: $i > $o).
% 2.87/1.88 thf(func_def_666, type, sK414: ($i > $o) > $i > $i).
% 2.87/1.88 thf(func_def_667, type, sK415: ($i > $o) > $i > $i).
% 2.87/1.88 thf(func_def_673, type, sK421: $i > $o).
% 2.87/1.88 thf(func_def_674, type, sK422: $i > $i).
% 2.87/1.88 thf(func_def_675, type, sK423: ($i > $o) > $i).
% 2.87/1.88 thf(func_def_687, type, sK435: $i > $o).
% 2.87/1.88 thf(func_def_691, type, sK439: $i > $o).
% 2.87/1.88 thf(func_def_692, type, sK440: $i > $o).
% 2.87/1.88 thf(func_def_693, type, sK441: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.87/1.88 thf(func_def_694, type, sK442: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.87/1.88 thf(func_def_699, type, sK447: $i > $o).
% 2.87/1.88 thf(func_def_702, type, sK450: $i > ($i > $o) > $i).
% 2.87/1.88 thf(func_def_703, type, sK451: $i > $o).
% 2.87/1.88 thf(func_def_710, type, sK458: $i > $i).
% 2.87/1.88 thf(func_def_712, type, sK460: $i > ($i > $i) > $i > $i).
% 2.87/1.88 thf(func_def_714, type, sK462: $i > $i).
% 2.87/1.88 thf(func_def_717, type, sK465: $i > $o).
% 2.87/1.88 thf(func_def_723, type, sK471: $i > $i > $i).
% 2.87/1.88 thf(func_def_726, type, sK474: ($i > $o) > $i > $i > $i > $i).
% 2.87/1.88 thf(func_def_727, type, sK475: ($i > $o) > $i > $i > $i > $i).
% 2.87/1.88 thf(func_def_731, type, sK479: $i > $o).
% 2.87/1.88 thf(func_def_746, type, sK494: $i > $i > $i).
% 2.87/1.88 thf(func_def_748, type, sK496: $i > $o).
% 2.87/1.88 thf(func_def_750, type, sK498: $i > ($i > $o) > $i > $i).
% 2.87/1.88 thf(func_def_761, type, sK509: $i > $i > $i).
% 2.87/1.88 thf(func_def_762, type, sK510: $i > $i > $i).
% 2.87/1.88 thf(func_def_766, type, sK514: $i > $i > $i).
% 2.87/1.88 thf(func_def_786, type, sK534: $i > $o).
% 2.87/1.88 thf(func_def_788, type, sK536: ($i > $o) > $i > $i).
% 2.87/1.88 thf(func_def_794, type, sK542: $i > ($i > $o) > $i).
% 2.87/1.88 thf(func_def_795, type, sK543: $i > $o).
% 2.87/1.88 thf(func_def_830, type, sK578: $i > $i > $o).
% 2.87/1.88 thf(func_def_836, type, sK584: $i > $o).
% 2.87/1.88 thf(func_def_853, type, ph601: !>[X0: $tType]:(X0)).
% 2.87/1.88 thf(f2979,plain,(
% 2.87/1.88 $false),
% 2.87/1.88 inference(trivial_inequality_removal,[status(thm)],[f2978])).
% 2.87/1.88 thf(f2978,plain,(
% 2.87/1.88 ($true != $true)),
% 2.87/1.88 inference(superposition,[status(thm)],[f2977,f2955])).
% 2.87/1.88 thf(f2955,plain,(
% 2.87/1.88 ($true = (func @ sK129 @ sK130 @ sK131))),
% 2.87/1.88 inference(trivial_inequality_removal,[status(thm)],[f2954])).
% 2.87/1.88 thf(f2954,plain,(
% 2.87/1.88 ($true != $true) | ($true = (func @ sK129 @ sK130 @ sK131))),
% 2.87/1.88 inference(forward_demodulation,[status(thm)],[f2950,f2083])).
% 2.87/1.88 thf(f2083,plain,(
% 2.87/1.88 (infuncsetfunc = $true)),
% 2.87/1.88 inference(cnf_transformation,[status(thm)],[f1070])).
% 2.87/1.88 thf(f1070,plain,(
% 2.87/1.88 (setext = $true) & (kpairiskpair = $true) & (binunionIR = $true) & (setminusEL = $true) & (cartprodsndpairEq = $true) & (subsetI1 = $true) & (powersetsubset = $true) & (emptyinPowerset = $true) & (cartprodmempair = $true) & (exu__Cong = $true) & (disjointsetsI1 = $true) & (powersetAx = $true) & (prop2setE = $true) & (dsetconstrER = $true) & (binintersectSubset1 = $true) & (quantDeMorgan2 = $true) & (exuI2 = $true) & (omegaSAx = $true) & (quantDeMorgan1 = $true) & (ex1I2 = $true) & (eqimpsubset1 = $true) & (powerset__Cong = $true) & (setextsub = $true) & (setminusLsub = $true) & (setadjoinSub2 = $true) & (dpsetconstrSub = $true) & (setukpairinjL1 = $true) & (notdallE = $true) & (secondinupair = $true) & (cartprodfstin = $true) & (notinemptyset = $true) & (subsetE2 = $true) & (cartprodpairin = $true) & (brelnall1 = $true) & (omega__Cong = $true) & (cartprodpairmemEL = $true) & (setunionI = $true) & (($true = (in @ sK131 @ (funcSet @ sK129 @ sK130))) & (($true = (in @ sK132 @ sK129)) & ((in @ (kpair @ sK132 @ (ap @ sK129 @ sK130 @ sK131 @ sK132)) @ sK131) != $true))) & (sepSubset = $true) & (singletonsuniq = $true) & (setextAx = $true) & (cartprodpairmemER = $true) & (upairsetE = $true) & (cartprodmempaircEq = $true) & (singletoninpowunion = $true) & (upairset2IR = $true) & (cartprodsndin = $true) & (prop2setI = $true) & (kpairsurjEq = $true) & (eqimpsubset2 = $true) & (quantDeMorgan4 = $true) & (emptyInPowerset = $true) & (binunionIL = $true) & (dpsetconstrEL2 = $true) & (cartprodmempair1 = $true) & (notequalI1 = $true) & (binintersectI = $true) & (inPowerset = $true) & (theprop = $true) & (ubforcartprodlem1 = $true) & (setukpairinjR2 = $true) & (subsetemptysetimpeq = $true) & (ksndpairEq = $true) & (descr__Cong = $true) & (emptyinunitempty = $true) & (subset2powerset = $true) & (brelnall2 = $true) & (notdexE = $true) & (dsetconstrI = $true) & (notinsingleton = $true) & (dpsetconstrER = $true) & (symdiffIneg2 = $true) & (upairset2E = $true) & (emptysetsubset = $true) & (eqinunit = $true) & (ex1E2 = $true) & (setminusER = $true) & (vacuousDall = $true) & (binunionLsub = $true) & (cartprodfstpairEq = $true) & (binintersectSubset5 = $true) & (ex1I = $true) & (setunionAx = $true) & (singletonsubset = $true) & (setadjoinIL = $true) & (exuE1 = $true) & (subsetRefl = $true) & (binintersectSubset3 = $true) & (infuncsetfunc = $true) & (lam2p = $true) & (setadjoinIR = $true) & (powersetE1 = $true) & (setunionE = $true) & (foundationAx = $true) & (dpsetconstrERa = $true) & (emptysetAx = $true) & (lamProp = $true) & (singletonsswitch = $true) & (exuEu = $true) & (emptysetimpfalse = $true) & (setminusELneg = $true) & (setminusSubset2 = $true) & (powersetI = $true) & (symdiffIneg1 = $true) & (kfstpairEq = $true) & (kpairp = $true) & (funcGraphProp1 = $true) & (exuI3 = $true) & (binintersectER = $true) & (setOfPairsIsBReln = $true) & (binintersectEL = $true) & (nonemptyI1 = $true) & (singletonprop = $true) & (binunionEcases = $true) & (powersetE = $true) & (prop2set2propI = $true) & (noeltsimpempty = $true) & (setukpairinjR1 = $true) & (setunion__Cong = $true) & (replAx = $true) & (inCongP = $true) & (ksndsingleton = $true) & (ex1E1 = $true) & (setunionsingleton2 = $true) & (wellorderingAx = $true) & (upairinpowunion = $true) & (setminusI = $true) & (dpsetconstrEL1 = $true) & (ap2p = $true) & (setminusSubset1 = $true) & (setukpairIR = $true) & (subPowSU = $true) & (setukpairinjR12 = $true) & (upairequniteq = $true) & (funcImageSingleton = $true) & (setminusERneg = $true) & (subsetTrans = $true) & (setoftrueEq = $true) & (binunionE = $true) & (exuE3u = $true) & (nonemptyE1 = $true) & (exuE3e = $true) & (setadjoin__Cong = $true) & (setunionE2 = $true) & (binintersectSubset4 = $true) & (kfstsingleton = $true) & (omega0Ax = $true) & (sepInPowerset = $true) & (uniqinunit = $true) & (notequalI2 = $true) & (setadjoinE = $true) & (funcinfuncset = $true) & (omegaIndAx = $true) & (binunionRsub = $true) & (symdiffE = $true) & (setukpairinjL2 = $true) & (subsetI2 = $true) & (binintersectSubset2 = $true) & (ubforcartprodlem3 = $true) & (setunionsingleton = $true) & (cartprodpairsurjEq = $true) & (notsubsetI = $true) & (upairsubunion = $true) & (dsetconstr__Cong = $true) & (setunionsingleton1 = $true) & (apProp = $true) & (setukpairinjR11 = $true) & (setminusILneg = $true) & (binintersectRsub = $true) & (setadjoinOr = $true) & (upairsetIR = $true) & (exuE2 = $true) & (setukpairinjR = $true) & (singletoninpowerset = $true) & (dpsetconstrI = $true) & (emptyset__Cong = $true) & (emptyE1 = $true) & (ubforcartprodlem2 = $true) & (upairsetIL = $true) & (nonemptyI = $true) & (setadjoinSub = $true) & (setadjoinAx = $true) & (powersetI1 = $true) & (setminusIRneg = $true) & (emptysetE = $true) & (nonemptyImpWitness = $true) & (app = $true) & (quantDeMorgan3 = $true) & (setbeta = $true) & (in__Cong = $true) & (setukpairIL = $true) & (binintersectLsub = $true) & (symdiffI1 = $true) & (dsetconstrEL = $true) & (setukpairinjL = $true) & (subsetE = $true) & (descrp = $true) & (emptyI = $true) & (lamp = $true) & (exuI1 = $true) & (bs114d = $true) & (symdiffI2 = $true)),
% 2.87/1.88 inference(skolemisation,[status(esa),new_symbols(skolem,[sK129,sK130,sK131,sK132])],[f850,f1069,f1068])).
% 2.87/1.88 thf(f1068,plain,(
% 2.87/1.88 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (funcSet @ X0 @ X1))) & ? [X3 : $i] : (($true = (in @ X3 @ X0)) & ($true != (in @ (kpair @ X3 @ (ap @ X0 @ X1 @ X2 @ X3)) @ X2)))) => (($true = (in @ sK131 @ (funcSet @ sK129 @ sK130))) & ? [X3 : $i] : (((in @ X3 @ sK129) = $true) & ((in @ (kpair @ X3 @ (ap @ sK129 @ sK130 @ sK131 @ X3)) @ sK131) != $true)))),
% 2.87/1.88 introduced(definition,[],[choice_axiom])).
% 2.87/1.88 thf(f1069,plain,(
% 2.87/1.88 ? [X3 : $i] : (((in @ X3 @ sK129) = $true) & ((in @ (kpair @ X3 @ (ap @ sK129 @ sK130 @ sK131 @ X3)) @ sK131) != $true)) => (($true = (in @ sK132 @ sK129)) & ((in @ (kpair @ sK132 @ (ap @ sK129 @ sK130 @ sK131 @ sK132)) @ sK131) != $true))),
% 2.87/1.88 introduced(definition,[],[choice_axiom])).
% 2.87/1.88 thf(f850,plain,(
% 2.87/1.88 (setext = $true) & (kpairiskpair = $true) & (binunionIR = $true) & (setminusEL = $true) & (cartprodsndpairEq = $true) & (subsetI1 = $true) & (powersetsubset = $true) & (emptyinPowerset = $true) & (cartprodmempair = $true) & (exu__Cong = $true) & (disjointsetsI1 = $true) & (powersetAx = $true) & (prop2setE = $true) & (dsetconstrER = $true) & (binintersectSubset1 = $true) & (quantDeMorgan2 = $true) & (exuI2 = $true) & (omegaSAx = $true) & (quantDeMorgan1 = $true) & (ex1I2 = $true) & (eqimpsubset1 = $true) & (powerset__Cong = $true) & (setextsub = $true) & (setminusLsub = $true) & (setadjoinSub2 = $true) & (dpsetconstrSub = $true) & (setukpairinjL1 = $true) & (notdallE = $true) & (secondinupair = $true) & (cartprodfstin = $true) & (notinemptyset = $true) & (subsetE2 = $true) & (cartprodpairin = $true) & (brelnall1 = $true) & (omega__Cong = $true) & (cartprodpairmemEL = $true) & (setunionI = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (funcSet @ X0 @ X1))) & ? [X3 : $i] : (($true = (in @ X3 @ X0)) & ($true != (in @ (kpair @ X3 @ (ap @ X0 @ X1 @ X2 @ X3)) @ X2)))) & (sepSubset = $true) & (singletonsuniq = $true) & (setextAx = $true) & (cartprodpairmemER = $true) & (upairsetE = $true) & (cartprodmempaircEq = $true) & (singletoninpowunion = $true) & (upairset2IR = $true) & (cartprodsndin = $true) & (prop2setI = $true) & (kpairsurjEq = $true) & (eqimpsubset2 = $true) & (quantDeMorgan4 = $true) & (emptyInPowerset = $true) & (binunionIL = $true) & (dpsetconstrEL2 = $true) & (cartprodmempair1 = $true) & (notequalI1 = $true) & (binintersectI = $true) & (inPowerset = $true) & (theprop = $true) & (ubforcartprodlem1 = $true) & (setukpairinjR2 = $true) & (subsetemptysetimpeq = $true) & (ksndpairEq = $true) & (descr__Cong = $true) & (emptyinunitempty = $true) & (subset2powerset = $true) & (brelnall2 = $true) & (notdexE = $true) & (dsetconstrI = $true) & (notinsingleton = $true) & (dpsetconstrER = $true) & (symdiffIneg2 = $true) & (upairset2E = $true) & (emptysetsubset = $true) & (eqinunit = $true) & (ex1E2 = $true) & (setminusER = $true) & (vacuousDall = $true) & (binunionLsub = $true) & (cartprodfstpairEq = $true) & (binintersectSubset5 = $true) & (ex1I = $true) & (setunionAx = $true) & (singletonsubset = $true) & (setadjoinIL = $true) & (exuE1 = $true) & (subsetRefl = $true) & (binintersectSubset3 = $true) & (infuncsetfunc = $true) & (lam2p = $true) & (setadjoinIR = $true) & (powersetE1 = $true) & (setunionE = $true) & (foundationAx = $true) & (dpsetconstrERa = $true) & (emptysetAx = $true) & (lamProp = $true) & (singletonsswitch = $true) & (exuEu = $true) & (emptysetimpfalse = $true) & (setminusELneg = $true) & (setminusSubset2 = $true) & (powersetI = $true) & (symdiffIneg1 = $true) & (kfstpairEq = $true) & (kpairp = $true) & (funcGraphProp1 = $true) & (exuI3 = $true) & (binintersectER = $true) & (setOfPairsIsBReln = $true) & (binintersectEL = $true) & (nonemptyI1 = $true) & (singletonprop = $true) & (binunionEcases = $true) & (powersetE = $true) & (prop2set2propI = $true) & (noeltsimpempty = $true) & (setukpairinjR1 = $true) & (setunion__Cong = $true) & (replAx = $true) & (inCongP = $true) & (ksndsingleton = $true) & (ex1E1 = $true) & (setunionsingleton2 = $true) & (wellorderingAx = $true) & (upairinpowunion = $true) & (setminusI = $true) & (dpsetconstrEL1 = $true) & (ap2p = $true) & (setminusSubset1 = $true) & (setukpairIR = $true) & (subPowSU = $true) & (setukpairinjR12 = $true) & (upairequniteq = $true) & (funcImageSingleton = $true) & (setminusERneg = $true) & (subsetTrans = $true) & (setoftrueEq = $true) & (binunionE = $true) & (exuE3u = $true) & (nonemptyE1 = $true) & (exuE3e = $true) & (setadjoin__Cong = $true) & (setunionE2 = $true) & (binintersectSubset4 = $true) & (kfstsingleton = $true) & (omega0Ax = $true) & (sepInPowerset = $true) & (uniqinunit = $true) & (notequalI2 = $true) & (setadjoinE = $true) & (funcinfuncset = $true) & (omegaIndAx = $true) & (binunionRsub = $true) & (symdiffE = $true) & (setukpairinjL2 = $true) & (subsetI2 = $true) & (binintersectSubset2 = $true) & (ubforcartprodlem3 = $true) & (setunionsingleton = $true) & (cartprodpairsurjEq = $true) & (notsubsetI = $true) & (upairsubunion = $true) & (dsetconstr__Cong = $true) & (setunionsingleton1 = $true) & (apProp = $true) & (setukpairinjR11 = $true) & (setminusILneg = $true) & (binintersectRsub = $true) & (setadjoinOr = $true) & (upairsetIR = $true) & (exuE2 = $true) & (setukpairinjR = $true) & (singletoninpowerset = $true) & (dpsetconstrI = $true) & (emptyset__Cong = $true) & (emptyE1 = $true) & (ubforcartprodlem2 = $true) & (upairsetIL = $true) & (nonemptyI = $true) & (setadjoinSub = $true) & (setadjoinAx = $true) & (powersetI1 = $true) & (setminusIRneg = $true) & (emptysetE = $true) & (nonemptyImpWitness = $true) & (app = $true) & (quantDeMorgan3 = $true) & (setbeta = $true) & (in__Cong = $true) & (setukpairIL = $true) & (binintersectLsub = $true) & (symdiffI1 = $true) & (dsetconstrEL = $true) & (setukpairinjL = $true) & (subsetE = $true) & (descrp = $true) & (emptyI = $true) & (lamp = $true) & (exuI1 = $true) & (bs114d = $true) & (symdiffI2 = $true)),
% 2.87/1.88 inference(flattening,[status(thm)],[f849])).
% 2.87/1.88 thf(f849,plain,(
% 2.87/1.88 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (funcSet @ X0 @ X1))) & ? [X3 : $i] : (($true = (in @ X3 @ X0)) & ($true != (in @ (kpair @ X3 @ (ap @ X0 @ X1 @ X2 @ X3)) @ X2)))) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.87/1.88 inference(ennf_transformation,[status(thm)],[f434])).
% 2.87/1.88 thf(f434,plain,(
% 2.87/1.88 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (funcSet @ X0 @ X1))) => ! [X3 : $i] : (($true = (in @ X3 @ X0)) => ($true = (in @ (kpair @ X3 @ (ap @ X0 @ X1 @ X2 @ X3)) @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.87/1.88 inference(fool_elimination,[status(thm)],[f433])).
% 2.87/1.88 thf(f433,plain,(
% 2.87/1.88 ~(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 => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (funcSet @ X0 @ X1)) => ! [X3 : $i] : ((in @ X3 @ X0) => (in @ (kpair @ X3 @ (ap @ X0 @ X1 @ X2 @ X3)) @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.87/1.88 inference(rectify,[status(thm)],[f210])).
% 2.87/1.88 thf(f210,negated_conjecture,(
% 2.87/1.88 ~(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 => ! [X3 : $i,X4 : $i,X13 : $i] : ((in @ X13 @ (funcSet @ X3 @ X4)) => ! [X1 : $i] : ((in @ X1 @ X3) => (in @ (kpair @ X1 @ (ap @ X3 @ X4 @ X13 @ X1)) @ X13))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.87/1.88 inference(negated_conjecture,[status(cth)],[f209])).
% 2.87/1.88 thf(f209,conjecture,(
% 2.87/1.88 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 => ! [X3 : $i,X4 : $i,X13 : $i] : ((in @ X13 @ (funcSet @ X3 @ X4)) => ! [X1 : $i] : ((in @ X1 @ X3) => (in @ (kpair @ X1 @ (ap @ X3 @ X4 @ X13 @ X1)) @ X13)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.87/1.88 file('/export/starexec/sandbox/tmp/tmp.ClpnksWlLJ/DTF2THF_22405.p',funcGraphProp3)).
% 2.87/1.88 thf(f2950,plain,(
% 2.87/1.88 ($true = (func @ sK129 @ sK130 @ sK131)) | (infuncsetfunc != $true)),
% 2.87/1.88 inference(trivial_inequality_removal,[status(thm)],[f2948])).
% 2.87/1.88 thf(f2948,plain,(
% 2.87/1.88 (infuncsetfunc != $true) | ($true != $true) | ($true = (func @ sK129 @ sK130 @ sK131))),
% 2.87/1.88 inference(superposition,[status(thm)],[f2289,f2136])).
% 2.87/1.88 thf(f2136,plain,(
% 2.87/1.88 ($true = (in @ sK131 @ (funcSet @ sK129 @ sK130)))),
% 2.87/1.88 inference(cnf_transformation,[status(thm)],[f1070])).
% 2.87/1.88 thf(f2289,plain,(
% 2.87/1.88 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != (in @ X0 @ (funcSet @ X2 @ X1))) | ($true = (func @ X2 @ X1 @ X0)) | (infuncsetfunc != $true)) )),
% 2.87/1.88 inference(cnf_transformation,[status(thm)],[f1222])).
% 2.87/1.88 thf(f1222,plain,(
% 2.87/1.88 (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ (funcSet @ X2 @ X1))) | ($true = (func @ X2 @ X1 @ X0))) | (infuncsetfunc != $true)) & ((infuncsetfunc = $true) | (($true = (in @ sK213 @ (funcSet @ sK215 @ sK214))) & ((func @ sK215 @ sK214 @ sK213) != $true)))),
% 2.87/1.88 inference(skolemisation,[status(esa),new_symbols(skolem,[sK213,sK214,sK215])],[f1220,f1221])).
% 2.87/1.88 thf(f1221,plain,(
% 2.87/1.88 ? [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X3 @ (funcSet @ X5 @ X4))) & ($true != (func @ X5 @ X4 @ X3))) => (($true = (in @ sK213 @ (funcSet @ sK215 @ sK214))) & ((func @ sK215 @ sK214 @ sK213) != $true))),
% 2.87/1.88 introduced(definition,[],[choice_axiom])).
% 2.87/1.88 thf(f1220,plain,(
% 2.87/1.88 (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ (funcSet @ X2 @ X1))) | ($true = (func @ X2 @ X1 @ X0))) | (infuncsetfunc != $true)) & ((infuncsetfunc = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X3 @ (funcSet @ X5 @ X4))) & ($true != (func @ X5 @ X4 @ X3))))),
% 2.87/1.88 inference(rectify,[status(thm)],[f1219])).
% 2.87/1.88 thf(f1219,plain,(
% 2.87/1.88 (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ (funcSet @ X2 @ X1))) | ($true = (func @ X2 @ X1 @ X0))) | (infuncsetfunc != $true)) & ((infuncsetfunc = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (funcSet @ X2 @ X1))) & ($true != (func @ X2 @ X1 @ X0))))),
% 2.87/1.88 inference(nnf_transformation,[status(thm)],[f683])).
% 2.87/1.88 thf(f683,plain,(
% 2.87/1.88 ! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ (funcSet @ X2 @ X1))) | ($true = (func @ X2 @ X1 @ X0))) <=> (infuncsetfunc = $true)),
% 2.87/1.88 inference(ennf_transformation,[status(thm)],[f377])).
% 2.87/1.88 thf(f377,plain,(
% 2.87/1.88 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (funcSet @ X2 @ X1))) => ($true = (func @ X2 @ X1 @ X0))) <=> (infuncsetfunc = $true)),
% 2.87/1.88 inference(fool_elimination,[status(thm)],[f376])).
% 2.87/1.88 thf(f376,plain,(
% 2.87/1.88 (infuncsetfunc = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ (funcSet @ X2 @ X1)) => (func @ X2 @ X1 @ X0)))),
% 2.87/1.88 inference(rectify,[status(thm)],[f198])).
% 2.87/1.88 thf(f198,axiom,(
% 2.87/1.88 (infuncsetfunc = ! [X13 : $i,X4 : $i,X3 : $i] : ((in @ X13 @ (funcSet @ X3 @ X4)) => (func @ X3 @ X4 @ X13)))),
% 2.87/1.88 file('/export/starexec/sandbox/tmp/tmp.ClpnksWlLJ/DTF2THF_22405.p',infuncsetfunc)).
% 2.87/1.88 thf(f2977,plain,(
% 2.87/1.88 ($true != (func @ sK129 @ sK130 @ sK131))),
% 2.87/1.88 inference(trivial_inequality_removal,[status(thm)],[f2976])).
% 2.87/1.88 thf(f2976,plain,(
% 2.87/1.88 ($true != (func @ sK129 @ sK130 @ sK131)) | ($true != $true)),
% 2.87/1.88 inference(forward_demodulation,[status(thm)],[f2975,f2135])).
% 2.87/1.88 thf(f2135,plain,(
% 2.87/1.88 ($true = (in @ sK132 @ sK129))),
% 2.87/1.88 inference(cnf_transformation,[status(thm)],[f1070])).
% 2.87/1.88 thf(f2975,plain,(
% 2.87/1.88 ($true != (in @ sK132 @ sK129)) | ($true != (func @ sK129 @ sK130 @ sK131))),
% 2.87/1.88 inference(trivial_inequality_removal,[status(thm)],[f2974])).
% 2.87/1.88 thf(f2974,plain,(
% 2.87/1.88 ($true != (in @ sK132 @ sK129)) | ($true != (func @ sK129 @ sK130 @ sK131)) | ($true != $true)),
% 2.87/1.88 inference(forward_demodulation,[status(thm)],[f2962,f2065])).
% 2.87/1.88 thf(f2065,plain,(
% 2.87/1.88 (funcGraphProp1 = $true)),
% 2.87/1.88 inference(cnf_transformation,[status(thm)],[f1070])).
% 2.87/1.88 thf(f2962,plain,(
% 2.87/1.88 (funcGraphProp1 != $true) | ($true != (func @ sK129 @ sK130 @ sK131)) | ($true != (in @ sK132 @ sK129))),
% 2.87/1.88 inference(trivial_inequality_removal,[status(thm)],[f2956])).
% 2.87/1.88 thf(f2956,plain,(
% 2.87/1.88 ($true != (in @ sK132 @ sK129)) | ($true != $true) | (funcGraphProp1 != $true) | ($true != (func @ sK129 @ sK130 @ sK131))),
% 2.87/1.88 inference(superposition,[status(thm)],[f2134,f2733])).
% 2.87/1.88 thf(f2733,plain,(
% 2.87/1.88 ( ! [X6 : $i,X7 : $i,X4 : $i,X5 : $i] : (((in @ (kpair @ X7 @ (ap @ X6 @ X4 @ X5 @ X7)) @ X5) = $true) | ((func @ X6 @ X4 @ X5) != $true) | (funcGraphProp1 != $true) | ((in @ X7 @ X6) != $true)) )),
% 2.87/1.88 inference(cnf_transformation,[status(thm)],[f1745])).
% 2.87/1.88 thf(f1745,plain,(
% 2.87/1.88 ((funcGraphProp1 = $true) | (((func @ sK548 @ sK546 @ sK547) = $true) & (((in @ (kpair @ sK549 @ (ap @ sK548 @ sK546 @ sK547 @ sK549)) @ sK547) != $true) & ((in @ sK549 @ sK548) = $true)))) & (! [X4 : $i,X5 : $i,X6 : $i] : (((func @ X6 @ X4 @ X5) != $true) | ! [X7 : $i] : (((in @ (kpair @ X7 @ (ap @ X6 @ X4 @ X5 @ X7)) @ X5) = $true) | ((in @ X7 @ X6) != $true))) | (funcGraphProp1 != $true))),
% 2.87/1.88 inference(skolemisation,[status(esa),new_symbols(skolem,[sK546,sK547,sK548,sK549])],[f1742,f1744,f1743])).
% 2.87/1.88 thf(f1743,plain,(
% 2.87/1.88 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X2 @ X0 @ X1)) & ? [X3 : $i] : (((in @ (kpair @ X3 @ (ap @ X2 @ X0 @ X1 @ X3)) @ X1) != $true) & ($true = (in @ X3 @ X2)))) => (((func @ sK548 @ sK546 @ sK547) = $true) & ? [X3 : $i] : (($true != (in @ (kpair @ X3 @ (ap @ sK548 @ sK546 @ sK547 @ X3)) @ sK547)) & ($true = (in @ X3 @ sK548))))),
% 2.87/1.88 introduced(definition,[],[choice_axiom])).
% 2.87/1.88 thf(f1744,plain,(
% 2.87/1.88 ? [X3 : $i] : (($true != (in @ (kpair @ X3 @ (ap @ sK548 @ sK546 @ sK547 @ X3)) @ sK547)) & ($true = (in @ X3 @ sK548))) => (((in @ (kpair @ sK549 @ (ap @ sK548 @ sK546 @ sK547 @ sK549)) @ sK547) != $true) & ((in @ sK549 @ sK548) = $true))),
% 2.87/1.88 introduced(definition,[],[choice_axiom])).
% 2.87/1.88 thf(f1742,plain,(
% 2.87/1.88 ((funcGraphProp1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X2 @ X0 @ X1)) & ? [X3 : $i] : (((in @ (kpair @ X3 @ (ap @ X2 @ X0 @ X1 @ X3)) @ X1) != $true) & ($true = (in @ X3 @ X2))))) & (! [X4 : $i,X5 : $i,X6 : $i] : (((func @ X6 @ X4 @ X5) != $true) | ! [X7 : $i] : (((in @ (kpair @ X7 @ (ap @ X6 @ X4 @ X5 @ X7)) @ X5) = $true) | ((in @ X7 @ X6) != $true))) | (funcGraphProp1 != $true))),
% 2.87/1.88 inference(rectify,[status(thm)],[f1741])).
% 2.87/1.88 thf(f1741,plain,(
% 2.87/1.88 ((funcGraphProp1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X2 @ X0 @ X1)) & ? [X3 : $i] : (((in @ (kpair @ X3 @ (ap @ X2 @ X0 @ X1 @ X3)) @ X1) != $true) & ($true = (in @ X3 @ X2))))) & (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (func @ X2 @ X0 @ X1)) | ! [X3 : $i] : (((in @ (kpair @ X3 @ (ap @ X2 @ X0 @ X1 @ X3)) @ X1) = $true) | ($true != (in @ X3 @ X2)))) | (funcGraphProp1 != $true))),
% 2.87/1.88 inference(nnf_transformation,[status(thm)],[f782])).
% 2.87/1.88 thf(f782,plain,(
% 2.87/1.88 (funcGraphProp1 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true != (func @ X2 @ X0 @ X1)) | ! [X3 : $i] : (((in @ (kpair @ X3 @ (ap @ X2 @ X0 @ X1 @ X3)) @ X1) = $true) | ($true != (in @ X3 @ X2))))),
% 2.87/1.88 inference(ennf_transformation,[status(thm)],[f317])).
% 2.87/1.88 thf(f317,plain,(
% 2.87/1.88 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X2 @ X0 @ X1)) => ! [X3 : $i] : (($true = (in @ X3 @ X2)) => ((in @ (kpair @ X3 @ (ap @ X2 @ X0 @ X1 @ X3)) @ X1) = $true))) <=> (funcGraphProp1 = $true)),
% 2.87/1.88 inference(fool_elimination,[status(thm)],[f316])).
% 2.87/1.88 thf(f316,plain,(
% 2.87/1.88 (funcGraphProp1 = ! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X2 @ X0 @ X1) => ! [X3 : $i] : ((in @ X3 @ X2) => (in @ (kpair @ X3 @ (ap @ X2 @ X0 @ X1 @ X3)) @ X1))))),
% 2.87/1.88 inference(rectify,[status(thm)],[f208])).
% 2.87/1.88 thf(f208,axiom,(
% 2.87/1.88 (funcGraphProp1 = ! [X4 : $i,X13 : $i,X3 : $i] : ((func @ X3 @ X4 @ X13) => ! [X1 : $i] : ((in @ X1 @ X3) => (in @ (kpair @ X1 @ (ap @ X3 @ X4 @ X13 @ X1)) @ X13))))),
% 2.87/1.88 file('/export/starexec/sandbox/tmp/tmp.ClpnksWlLJ/DTF2THF_22405.p',funcGraphProp1)).
% 2.87/1.88 thf(f2134,plain,(
% 2.87/1.88 ((in @ (kpair @ sK132 @ (ap @ sK129 @ sK130 @ sK131 @ sK132)) @ sK131) != $true)),
% 2.87/1.88 inference(cnf_transformation,[status(thm)],[f1070])).
% 2.87/1.88 % SZS output end Proof for DTF2THF_22405
% 2.87/1.88 % (22542)------------------------------
% 2.87/1.88 % (22542)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.87/1.88 % (22542)Termination reason: Refutation
% 2.87/1.88
% 2.87/1.88 % (22542)Memory used [KB]: 8955
% 2.87/1.88 % (22542)Time elapsed: 0.128 s
% 2.87/1.88 % (22542)Instructions burned: 248 (million)
% 2.87/1.88 % (22542)------------------------------
% 2.87/1.88 % (22542)------------------------------
% 2.87/1.88 % (22518)Success in time 0.224 s
%------------------------------------------------------------------------------