%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU675^1 : 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 : n023.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 09:36:11 AM UTC 2026
% Result : Theorem 2.56s 1.99s
% Output : Refutation 2.56s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 5
% Syntax : Number of formulae : 32 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 3090 ( 828 equ; 0 cnn)
% Maximal formula atoms : 193 ( 96 avg)
% Number of connectives : 1630 ( 28 ~; 17 |; 586 &; 224 @)
% ( 2 <=>; 773 =>; 0 <=; 0 <~>)
% Maximal formula depth : 200 ( 46 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 14 ( 14 >; 0 *; 0 +; 0 <<)
% Number of symbols : 396 ( 393 usr; 198 con; 0-4 aty)
% Number of variables : 93 ( 30 ^; 45 !; 18 ?; 93 :)
% 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_241,type,
sP0: ( $i > $o ) > $i > $o ).
thf(func_def_243,type,
sP2: $i > $i > $o ).
thf(func_def_246,type,
sP5: ( $i > $o ) > $i > $o ).
thf(func_def_248,type,
sP7: ( $i > $o ) > $o ).
thf(func_def_250,type,
sP9: $i > $i > $o > $o ).
thf(func_def_252,type,
sP11: $i > ( $i > $i > $o ) > $i > $o ).
thf(func_def_253,type,
sP12: $i > ( $i > $i > $o ) > $o ).
thf(func_def_255,type,
sP14: ( $i > $o ) > $o ).
thf(func_def_256,type,
sP15: $o > $i > $i > $i > $o ).
thf(func_def_257,type,
sP16: $i > $o > $i > $i > $o ).
thf(func_def_259,type,
sP18: ( $i > $o ) > $o ).
thf(func_def_260,type,
sP19: $i > ( $i > $o ) > ( $i > $o ) > $i > $o ).
thf(func_def_261,type,
sP20: $i > $o ).
thf(func_def_262,type,
sP21: $i > $o ).
thf(func_def_263,type,
sP22: $i > $i > $o ).
thf(func_def_265,type,
sP24: ( $i > $o ) > $i > $o ).
thf(func_def_267,type,
sP26: $o > $i > $i > $o ).
thf(func_def_268,type,
sP27: $i > $i > $i > $o ).
thf(func_def_269,type,
sP28: $i > $i > $i > $o ).
thf(func_def_271,type,
sP30: $i > ( $i > $o ) > $o ).
thf(func_def_272,type,
sP31: ( $i > $o ) > $i > $o ).
thf(func_def_273,type,
sP32: ( $i > $o ) > $i > $o ).
thf(func_def_275,type,
sP34: $i > $i > $o ).
thf(func_def_277,type,
sP36: ( $i > $o ) > $i > $o ).
thf(func_def_279,type,
sP38: ( $i > $o ) > $i > $o ).
thf(func_def_282,type,
sP41: ( $i > $o ) > $i > $o ).
thf(func_def_283,type,
sP42: ( $i > $o ) > $i > $o ).
thf(func_def_285,type,
sP44: $i > $i > $i > $o ).
thf(func_def_286,type,
sP45: $i > $i > $o ).
thf(func_def_287,type,
sP46: $i > $i > $o ).
thf(func_def_288,type,
sP47: $i > $i > $o ).
thf(func_def_289,type,
sP48: $i > $i > $o ).
thf(func_def_290,type,
sP49: $i > $o ).
thf(func_def_291,type,
sP50: $i > $i > $i > $o ).
thf(func_def_292,type,
sP51: ( $i > $o ) > $o ).
thf(func_def_293,type,
sP52: $i > ( $i > $o ) > $i > $o ).
thf(func_def_294,type,
sP53: $i > $o > $i > $o ).
thf(func_def_297,type,
sP56: ( $i > $o ) > $o ).
thf(func_def_299,type,
sP58: $i > $o ).
thf(func_def_306,type,
sK65: $i > $o ).
thf(func_def_321,type,
sK80: $i > $o ).
thf(func_def_323,type,
sK82: $i > ( $i > $o ) > $i ).
thf(func_def_324,type,
sK83: ( $i > $o ) > $i > $i ).
thf(func_def_327,type,
sK86: $i > $i > $i ).
thf(func_def_328,type,
sK87: $i > $i > $i ).
thf(func_def_344,type,
sK103: $i > $i ).
thf(func_def_362,type,
sK121: $i > $o ).
thf(func_def_364,type,
sK123: $i > ( $i > $o ) > $i ).
thf(func_def_369,type,
sK128: $i > $o ).
thf(func_def_370,type,
sK129: ( $i > $o ) > $i > $i ).
thf(func_def_371,type,
sK130: ( $i > $o ) > $i > $i ).
thf(func_def_376,type,
sK135: $i > $o ).
thf(func_def_380,type,
sK139: $i > $i > $o ).
thf(func_def_392,type,
sK151: ( $i > $o ) > $i ).
thf(func_def_393,type,
sK152: $i > ( $i > $o ) > $i ).
thf(func_def_394,type,
sK153: $i > $o ).
thf(func_def_404,type,
sK163: $i > $i > $i ).
thf(func_def_409,type,
sK168: $i > $i > $o ).
thf(func_def_423,type,
sK182: $i > $i > $o ).
thf(func_def_424,type,
sK183: ( $i > $i > $o ) > $i > $i ).
thf(func_def_425,type,
sK184: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_426,type,
sK185: $i > ( $i > $i > $o ) > $i ).
thf(func_def_427,type,
sK186: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_429,type,
sK188: $i > ( $i > $o ) > $i ).
thf(func_def_430,type,
sK189: ( $i > $o ) > $i ).
thf(func_def_431,type,
sK190: $i > $o ).
thf(func_def_436,type,
sK195: $i > $o ).
thf(func_def_440,type,
sK199: $i > $o ).
thf(func_def_443,type,
sK202: $i > $o ).
thf(func_def_460,type,
sK219: $i > $o ).
thf(func_def_461,type,
sK220: $i > $o ).
thf(func_def_462,type,
sK221: $i > $o ).
thf(func_def_463,type,
sK222: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_464,type,
sK223: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_479,type,
sK238: ( $i > $o ) > $i ).
thf(func_def_480,type,
sK239: $i > ( $i > $o ) > $i ).
thf(func_def_481,type,
sK240: $i > $o ).
thf(func_def_482,type,
sK241: $i > $o ).
thf(func_def_490,type,
sK249: $i > $i > $o ).
thf(func_def_491,type,
sK250: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_492,type,
sK251: $i > ( $i > $o ) > ( $i > $o ) > $i > $i ).
thf(func_def_495,type,
sK254: $i > $o ).
thf(func_def_496,type,
sK255: $i > $o ).
thf(func_def_500,type,
sK259: $i > $i ).
thf(func_def_501,type,
sK260: $i > $i ).
thf(func_def_503,type,
sK262: $i > $o ).
thf(func_def_520,type,
sK279: $i > $i > $i ).
thf(func_def_537,type,
sK296: $i > $o ).
thf(func_def_538,type,
sK297: ( $i > $o ) > $i > $i ).
thf(func_def_539,type,
sK298: ( $i > $o ) > $i > $i ).
thf(func_def_550,type,
sK309: $i > $i ).
thf(func_def_552,type,
sK311: $i > $i > $i > $o ).
thf(func_def_574,type,
sK333: ( $i > $o ) > $i > $i ).
thf(func_def_575,type,
sK334: ( $i > $o ) > $i > $i ).
thf(func_def_576,type,
sK335: $i > ( $i > $o ) > $i ).
thf(func_def_578,type,
sK337: $i > $o ).
thf(func_def_580,type,
sK339: $i > $o ).
thf(func_def_581,type,
sK340: ( $i > $o ) > $i > $i ).
thf(func_def_582,type,
sK341: ( $i > $o ) > $i > $i ).
thf(func_def_591,type,
sK350: $i > $i > $o ).
thf(func_def_600,type,
sK359: $i > $i > $i ).
thf(func_def_601,type,
sK360: $i > $i ).
thf(func_def_609,type,
sK368: $i > $o ).
thf(func_def_610,type,
sK369: ( $i > $o ) > $i > $i ).
thf(func_def_611,type,
sK370: ( $i > $o ) > $i > $i ).
thf(func_def_612,type,
sK371: $i > $o ).
thf(func_def_614,type,
sK373: $i > ( $i > $o ) > $i ).
thf(func_def_615,type,
sK374: ( $i > $o ) > $i > $i ).
thf(func_def_617,type,
sK376: $i > $o ).
thf(func_def_625,type,
sK384: $i > $i > $i ).
thf(func_def_629,type,
sK388: $i > $o ).
thf(func_def_635,type,
sK394: ( $i > $o ) > $i > $i ).
thf(func_def_636,type,
sK395: ( $i > $o ) > $i > $i ).
thf(func_def_637,type,
sK396: ( $i > $o ) > $i > $i ).
thf(func_def_639,type,
sK398: $i > $o ).
thf(func_def_666,type,
sK425: $i > $o ).
thf(func_def_667,type,
sK426: $i > $o ).
thf(func_def_668,type,
sK427: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_669,type,
sK428: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_670,type,
sK429: $i > $i ).
thf(func_def_671,type,
sK430: $i > $i ).
thf(func_def_672,type,
sK431: $i > $i ).
thf(func_def_673,type,
sK432: $i > $i ).
thf(func_def_674,type,
sK433: $i > $i > $i ).
thf(func_def_675,type,
sK434: $i > $i > $i ).
thf(func_def_676,type,
sK435: $i > $i > $i ).
thf(func_def_677,type,
sK436: $i > $i > $i ).
thf(func_def_678,type,
sK437: $i > $i > $i ).
thf(func_def_679,type,
sK438: $i > $i > $i > $i ).
thf(func_def_680,type,
sK439: $i > $i > $i ).
thf(func_def_681,type,
sK440: $i > $i > $i ).
thf(func_def_682,type,
sK441: $i > $i > $i ).
thf(func_def_683,type,
sK442: $i > $i > $i ).
thf(func_def_684,type,
sK443: $i > $i > $i ).
thf(func_def_685,type,
sK444: $i > $i > $i > $i ).
thf(func_def_686,type,
sK445: $i > $i > $i > $i ).
thf(func_def_687,type,
sK446: $i > $i ).
thf(func_def_699,type,
sK458: $i > $o ).
thf(func_def_700,type,
sK459: ( $i > $o ) > $i ).
thf(func_def_715,type,
sK474: $i > $i ).
thf(func_def_733,type,
sK492: $i > $i > $i ).
thf(func_def_742,type,
sK501: $i > $o ).
thf(func_def_746,type,
sK505: $i > $i > $i > $i ).
thf(func_def_747,type,
sK506: $i > $i > $i > $i ).
thf(func_def_751,type,
sK510: $i > ( $i > $o ) > $i ).
thf(func_def_752,type,
sK511: ( $i > $o ) > $i ).
thf(func_def_753,type,
sK512: $i > $o ).
thf(func_def_757,type,
sK516: $i > ( $i > $o ) > $i > $i ).
thf(func_def_758,type,
sK517: $i > $o ).
thf(func_def_763,type,
sK522: $i > $o > $i > $i ).
thf(func_def_771,type,
sK530: $i > $i > $i ).
thf(func_def_782,type,
sK541: $i > $i > $i ).
thf(func_def_785,type,
sK544: $i > $i > $o ).
thf(func_def_790,type,
sK549: ( $i > $o ) > $i ).
thf(func_def_791,type,
sK550: ( $i > $o ) > $i ).
thf(func_def_792,type,
sK551: $i > $o ).
thf(func_def_799,type,
sK558: $i > $i > $o ).
thf(func_def_806,type,
sK565: $i > $i > $i ).
thf(func_def_807,type,
sK566: $i > $o ).
thf(func_def_815,type,
sK574: $i > $i > $i ).
thf(func_def_816,type,
sK575: $i > $i ).
thf(func_def_833,type,
sK592: $i > $i > $o ).
thf(func_def_845,type,
ph604:
!>[X0: $tType] : X0 ).
thf(f5709,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f5708,f2166]) ).
thf(f2166,plain,
( $true
!= ( func @ sK172 @ sK173 @ sK174 ) ),
inference(cnf_transformation,[status(thm)],[f1125]) ).
thf(f1125,plain,
( ( quantDeMorgan4 = $true )
& ( setukpairinjR2 = $true )
& ( ex1I2 = $true )
& ( setunionsingleton = $true )
& ( prop2set2propI = $true )
& ( notdallE = $true )
& ( inPowerset = $true )
& ( binunionEcases = $true )
& ( upairsetIL = $true )
& ( cartprodpairmemER = $true )
& ( prop2setE = $true )
& ( setukpairinjL2 = $true )
& ( dsetconstrEL = $true )
& ( ubforcartprodlem2 = $true )
& ( singletonsubset = $true )
& ( in__Cong = $true )
& ( setunionE2 = $true )
& ( subsetemptysetimpeq = $true )
& ( kfstpairEq = $true )
& ( ex1I = $true )
& ( dpsetconstrEL1 = $true )
& ( powersetE1 = $true )
& ( symdiffI2 = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setext = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( nonemptyI1 = $true )
& ( setminusERneg = $true )
& ( powersetAx = $true )
& ( setadjoinIL = $true )
& ( app = $true )
& ( dpsetconstrEL2 = $true )
& ( omega0Ax = $true )
& ( quantDeMorgan2 = $true )
& ( omegaSAx = $true )
& ( dpsetconstrER = $true )
& ( setextsub = $true )
& ( ksndpairEq = $true )
& ( setminusSubset1 = $true )
& ( upairsubunion = $true )
& ( upairset2IR = $true )
& ( binintersectSubset3 = $true )
& ( emptysetsubset = $true )
& ( binintersectSubset1 = $true )
& ( setbeta = $true )
& ( dpsetconstrI = $true )
& ( notequalI1 = $true )
& ( exuI3 = $true )
& ( upairset2E = $true )
& ( dsetconstr__Cong = $true )
& ( emptyinunitempty = $true )
& ( setukpairinjR11 = $true )
& ( binunionLsub = $true )
& ( singletoninpowunion = $true )
& ( cartprodsndin = $true )
& ( ubforcartprodlem1 = $true )
& ( kfstsingleton = $true )
& ( setminusIRneg = $true )
& ( exuE1 = $true )
& ( subsetRefl = $true )
& ( subsetE = $true )
& ( cartprodsndpairEq = $true )
& ( binunionIR = $true )
& ( replAx = $true )
& ( eqimpsubset1 = $true )
& ( notequalI2 = $true )
& ( setunionI = $true )
& ( singletonsuniq = $true )
& ( exu__Cong = $true )
& ( emptyset__Cong = $true )
& ( setadjoinSub2 = $true )
& ( notdexE = $true )
& ( exuE3e = $true )
& ( upairinpowunion = $true )
& ( nonemptyImpWitness = $true )
& ( cartprodmempaircEq = $true )
& ( exuEu = $true )
& ( foundationAx = $true )
& ( subPowSU = $true )
& ( binintersectSubset2 = $true )
& ( binintersectRsub = $true )
& ( setOfPairsIsBReln = $true )
& ( subset2powerset = $true )
& ( nonemptyE1 = $true )
& ( binintersectEL = $true )
& ( nonemptyI = $true )
& ( upairsetIR = $true )
& ( exuI1 = $true )
& ( bs114d = $true )
& ( setadjoinSub = $true )
& ( subsetI1 = $true )
& ( quantDeMorgan3 = $true )
& ( binintersectLsub = $true )
& ( setukpairIR = $true )
& ( quantDeMorgan1 = $true )
& ( eqimpsubset2 = $true )
& ( sepSubset = $true )
& ( omega__Cong = $true )
& ( setminusSubset2 = $true )
& ( subsetI2 = $true )
& ( ( in @ sK174 @ ( funcSet @ sK172 @ sK173 ) )
= $true )
& ( $true
!= ( func @ sK172 @ sK173 @ sK174 ) )
& ( upairequniteq = $true )
& ( subsetE2 = $true )
& ( setminusI = $true )
& ( apProp = $true )
& ( notsubsetI = $true )
& ( omegaIndAx = $true )
& ( kpairiskpair = $true )
& ( descr__Cong = $true )
& ( ksndsingleton = $true )
& ( powersetI1 = $true )
& ( exuI2 = $true )
& ( powersetI = $true )
& ( inCongP = $true )
& ( funcImageSingleton = $true )
& ( binintersectSubset4 = $true )
& ( subsetTrans = $true )
& ( binunionE = $true )
& ( setukpairinjR = $true )
& ( powersetsubset = $true )
& ( singletonsswitch = $true )
& ( setukpairIL = $true )
& ( emptyI = $true )
& ( dsetconstrER = $true )
& ( kpairp = $true )
& ( noeltsimpempty = $true )
& ( cartprodmempair = $true )
& ( symdiffI1 = $true )
& ( setminusELneg = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( cartprodfstin = $true )
& ( sepInPowerset = $true )
& ( theprop = $true )
& ( setminusEL = $true )
& ( setunion__Cong = $true )
& ( setadjoinE = $true )
& ( symdiffE = $true )
& ( ex1E1 = $true )
& ( dpsetconstrSub = $true )
& ( descrp = $true )
& ( upairsetE = $true )
& ( exuE2 = $true )
& ( secondinupair = $true )
& ( notinsingleton = $true )
& ( setoftrueEq = $true )
& ( setadjoinAx = $true )
& ( kpairsurjEq = $true )
& ( dpsetconstrERa = $true )
& ( emptyinPowerset = $true )
& ( ubforcartprodlem3 = $true )
& ( powerset__Cong = $true )
& ( cartprodpairmemEL = $true )
& ( powersetE = $true )
& ( notinemptyset = $true )
& ( binintersectER = $true )
& ( cartprodpairsurjEq = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjL = $true )
& ( uniqinunit = $true )
& ( symdiffIneg2 = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempair1 = $true )
& ( setunionE = $true )
& ( symdiffIneg1 = $true )
& ( singletonprop = $true )
& ( setminusLsub = $true )
& ( binunionIL = $true )
& ( setadjoinIR = $true )
& ( dsetconstrI = $true )
& ( emptysetimpfalse = $true )
& ( cartprodpairin = $true )
& ( exuE3u = $true )
& ( wellorderingAx = $true )
& ( setukpairinjL1 = $true )
& ( setukpairinjR1 = $true )
& ( eqinunit = $true )
& ( setunionAx = $true )
& ( setminusER = $true )
& ( setminusILneg = $true )
& ( setextAx = $true )
& ( singletoninpowerset = $true )
& ( emptysetAx = $true )
& ( binintersectSubset5 = $true )
& ( emptyInPowerset = $true )
& ( prop2setI = $true )
& ( emptysetE = $true )
& ( disjointsetsI1 = $true )
& ( setadjoinOr = $true )
& ( setadjoin__Cong = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK172,sK173,sK174])],[f792,f1124]) ).
thf(f1124,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( funcSet @ X0 @ X1 ) ) )
& ( $true
!= ( func @ X0 @ X1 @ X2 ) ) )
=> ( ( ( in @ sK174 @ ( funcSet @ sK172 @ sK173 ) )
= $true )
& ( $true
!= ( func @ sK172 @ sK173 @ sK174 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f792,plain,
( ( quantDeMorgan4 = $true )
& ( setukpairinjR2 = $true )
& ( ex1I2 = $true )
& ( setunionsingleton = $true )
& ( prop2set2propI = $true )
& ( notdallE = $true )
& ( inPowerset = $true )
& ( binunionEcases = $true )
& ( upairsetIL = $true )
& ( cartprodpairmemER = $true )
& ( prop2setE = $true )
& ( setukpairinjL2 = $true )
& ( dsetconstrEL = $true )
& ( ubforcartprodlem2 = $true )
& ( singletonsubset = $true )
& ( in__Cong = $true )
& ( setunionE2 = $true )
& ( subsetemptysetimpeq = $true )
& ( kfstpairEq = $true )
& ( ex1I = $true )
& ( dpsetconstrEL1 = $true )
& ( powersetE1 = $true )
& ( symdiffI2 = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setext = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( nonemptyI1 = $true )
& ( setminusERneg = $true )
& ( powersetAx = $true )
& ( setadjoinIL = $true )
& ( app = $true )
& ( dpsetconstrEL2 = $true )
& ( omega0Ax = $true )
& ( quantDeMorgan2 = $true )
& ( omegaSAx = $true )
& ( dpsetconstrER = $true )
& ( setextsub = $true )
& ( ksndpairEq = $true )
& ( setminusSubset1 = $true )
& ( upairsubunion = $true )
& ( upairset2IR = $true )
& ( binintersectSubset3 = $true )
& ( emptysetsubset = $true )
& ( binintersectSubset1 = $true )
& ( setbeta = $true )
& ( dpsetconstrI = $true )
& ( notequalI1 = $true )
& ( exuI3 = $true )
& ( upairset2E = $true )
& ( dsetconstr__Cong = $true )
& ( emptyinunitempty = $true )
& ( setukpairinjR11 = $true )
& ( binunionLsub = $true )
& ( singletoninpowunion = $true )
& ( cartprodsndin = $true )
& ( ubforcartprodlem1 = $true )
& ( kfstsingleton = $true )
& ( setminusIRneg = $true )
& ( exuE1 = $true )
& ( subsetRefl = $true )
& ( subsetE = $true )
& ( cartprodsndpairEq = $true )
& ( binunionIR = $true )
& ( replAx = $true )
& ( eqimpsubset1 = $true )
& ( notequalI2 = $true )
& ( setunionI = $true )
& ( singletonsuniq = $true )
& ( exu__Cong = $true )
& ( emptyset__Cong = $true )
& ( setadjoinSub2 = $true )
& ( notdexE = $true )
& ( exuE3e = $true )
& ( upairinpowunion = $true )
& ( nonemptyImpWitness = $true )
& ( cartprodmempaircEq = $true )
& ( exuEu = $true )
& ( foundationAx = $true )
& ( subPowSU = $true )
& ( binintersectSubset2 = $true )
& ( binintersectRsub = $true )
& ( setOfPairsIsBReln = $true )
& ( subset2powerset = $true )
& ( nonemptyE1 = $true )
& ( binintersectEL = $true )
& ( nonemptyI = $true )
& ( upairsetIR = $true )
& ( exuI1 = $true )
& ( bs114d = $true )
& ( setadjoinSub = $true )
& ( subsetI1 = $true )
& ( quantDeMorgan3 = $true )
& ( binintersectLsub = $true )
& ( setukpairIR = $true )
& ( quantDeMorgan1 = $true )
& ( eqimpsubset2 = $true )
& ( sepSubset = $true )
& ( omega__Cong = $true )
& ( setminusSubset2 = $true )
& ( subsetI2 = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( funcSet @ X0 @ X1 ) ) )
& ( $true
!= ( func @ X0 @ X1 @ X2 ) ) )
& ( upairequniteq = $true )
& ( subsetE2 = $true )
& ( setminusI = $true )
& ( apProp = $true )
& ( notsubsetI = $true )
& ( omegaIndAx = $true )
& ( kpairiskpair = $true )
& ( descr__Cong = $true )
& ( ksndsingleton = $true )
& ( powersetI1 = $true )
& ( exuI2 = $true )
& ( powersetI = $true )
& ( inCongP = $true )
& ( funcImageSingleton = $true )
& ( binintersectSubset4 = $true )
& ( subsetTrans = $true )
& ( binunionE = $true )
& ( setukpairinjR = $true )
& ( powersetsubset = $true )
& ( singletonsswitch = $true )
& ( setukpairIL = $true )
& ( emptyI = $true )
& ( dsetconstrER = $true )
& ( kpairp = $true )
& ( noeltsimpempty = $true )
& ( cartprodmempair = $true )
& ( symdiffI1 = $true )
& ( setminusELneg = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( cartprodfstin = $true )
& ( sepInPowerset = $true )
& ( theprop = $true )
& ( setminusEL = $true )
& ( setunion__Cong = $true )
& ( setadjoinE = $true )
& ( symdiffE = $true )
& ( ex1E1 = $true )
& ( dpsetconstrSub = $true )
& ( descrp = $true )
& ( upairsetE = $true )
& ( exuE2 = $true )
& ( secondinupair = $true )
& ( notinsingleton = $true )
& ( setoftrueEq = $true )
& ( setadjoinAx = $true )
& ( kpairsurjEq = $true )
& ( dpsetconstrERa = $true )
& ( emptyinPowerset = $true )
& ( ubforcartprodlem3 = $true )
& ( powerset__Cong = $true )
& ( cartprodpairmemEL = $true )
& ( powersetE = $true )
& ( notinemptyset = $true )
& ( binintersectER = $true )
& ( cartprodpairsurjEq = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjL = $true )
& ( uniqinunit = $true )
& ( symdiffIneg2 = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempair1 = $true )
& ( setunionE = $true )
& ( symdiffIneg1 = $true )
& ( singletonprop = $true )
& ( setminusLsub = $true )
& ( binunionIL = $true )
& ( setadjoinIR = $true )
& ( dsetconstrI = $true )
& ( emptysetimpfalse = $true )
& ( cartprodpairin = $true )
& ( exuE3u = $true )
& ( wellorderingAx = $true )
& ( setukpairinjL1 = $true )
& ( setukpairinjR1 = $true )
& ( eqinunit = $true )
& ( setunionAx = $true )
& ( setminusER = $true )
& ( setminusILneg = $true )
& ( setextAx = $true )
& ( singletoninpowerset = $true )
& ( emptysetAx = $true )
& ( binintersectSubset5 = $true )
& ( emptyInPowerset = $true )
& ( prop2setI = $true )
& ( emptysetE = $true )
& ( disjointsetsI1 = $true )
& ( setadjoinOr = $true )
& ( setadjoin__Cong = $true ) ),
inference(flattening,[status(thm)],[f791]) ).
thf(f791,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( funcSet @ X0 @ X1 ) ) )
& ( $true
!= ( func @ X0 @ X1 @ X2 ) ) )
& ( 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)],[f208]) ).
thf(f208,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 )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( funcSet @ X0 @ X1 ) ) )
=> ( $true
= ( func @ X0 @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f207]) ).
thf(f207,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
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( funcSet @ X0 @ X1 ) )
=> ( func @ X0 @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f201]) ).
thf(f201,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
=> ! [X3: $i,X4: $i,X13: $i] :
( ( in @ X13 @ ( funcSet @ X3 @ X4 ) )
=> ( func @ X3 @ X4 @ X13 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f200]) ).
thf(f200,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
=> ! [X3: $i,X4: $i,X13: $i] :
( ( in @ X13 @ ( funcSet @ X3 @ X4 ) )
=> ( func @ X3 @ X4 @ X13 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.T6NCxVKBVZ/DTF2THF_2187.p',infuncsetfunc) ).
thf(f5708,plain,
( $true
= ( func @ sK172 @ sK173 @ sK174 ) ),
inference(trivial_inequality_removal,[status(thm)],[f5696]) ).
thf(f5696,plain,
( ( $true
= ( func @ sK172 @ sK173 @ sK174 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f3640,f3625]) ).
thf(f3625,plain,
( $true
= ( in @ sK174 @ ( dsetconstr @ ( powerset @ ( cartprod @ sK172 @ sK173 ) ) @ ( func @ sK172 @ sK173 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f3051]) ).
thf(f3051,plain,
( $true
= ( in @ sK174
@ ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( powerset @ ( cartprod @ Y0 @ Y1 ) )
@ ^ [Y2: $i] : ( func @ Y0 @ Y1 @ Y2 ) )
@ sK172
@ sK173 ) ) ),
inference(definition_unfolding,[status(thm)],[f2167,f2359]) ).
thf(f2359,plain,
( funcSet
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( powerset @ ( cartprod @ Y0 @ Y1 ) )
@ ^ [Y2: $i] : ( func @ Y0 @ Y1 @ Y2 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f472]) ).
thf(f472,plain,
( funcSet
= ( ^ [Y0: $i,Y1: $i] :
( dsetconstr @ ( powerset @ ( cartprod @ Y0 @ Y1 ) )
@ ^ [Y2: $i] : ( func @ Y0 @ Y1 @ Y2 ) ) ) ),
inference(fool_elimination,[status(thm)],[f471]) ).
thf(f471,plain,
( funcSet
= ( ^ [X0: $i,X1: $i] :
( dsetconstr @ ( powerset @ ( cartprod @ X0 @ X1 ) )
@ ^ [X2: $i] : ( func @ X0 @ X1 @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f195]) ).
thf(f195,axiom,
( funcSet
= ( ^ [X3: $i,X4: $i] :
( dsetconstr @ ( powerset @ ( cartprod @ X3 @ X4 ) )
@ ^ [X13: $i] : ( func @ X3 @ X4 @ X13 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.T6NCxVKBVZ/DTF2THF_2187.p',funcSet) ).
thf(f2167,plain,
( ( in @ sK174 @ ( funcSet @ sK172 @ sK173 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1125]) ).
thf(f3640,plain,
! [X3: $i > $o,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4 @ ( dsetconstr @ X5 @ X3 ) ) )
| ( $true
= ( X3 @ X4 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f3639]) ).
thf(f3639,plain,
! [X3: $i > $o,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X3 @ Y0 ) ) ) )
| ( $true
= ( X3 @ X4 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3073]) ).
thf(f3073,plain,
! [X3: $i > $o,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X3 @ Y0 ) ) ) )
| ( $true
= ( X3 @ X4 ) )
| ( $true != $true ) ),
inference(definition_unfolding,[status(thm)],[f2313,f2143]) ).
thf(f2143,plain,
dsetconstrER = $true,
inference(cnf_transformation,[status(thm)],[f1125]) ).
thf(f2313,plain,
! [X3: $i > $o,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X3 @ Y0 ) ) ) )
| ( $true
= ( X3 @ X4 ) )
| ( dsetconstrER != $true ) ),
inference(cnf_transformation,[status(thm)],[f1181]) ).
thf(f1181,plain,
( ( ( dsetconstrER = $true )
| ( ( $true
= ( in @ sK200
@ ( dsetconstr @ sK201
@ ^ [Y0: $i] : ( sK199 @ Y0 ) ) ) )
& ( ( sK199 @ sK200 )
!= $true ) ) )
& ( ! [X3: $i > $o,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X3 @ Y0 ) ) ) )
| ( $true
= ( X3 @ X4 ) ) )
| ( dsetconstrER != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK199,sK200,sK201])],[f1179,f1180]) ).
thf(f1180,plain,
( ? [X0: $i > $o,X1: $i,X2: $i] :
( ( $true
= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) )
& ( ( X0 @ X1 )
!= $true ) )
=> ( ( $true
= ( in @ sK200
@ ( dsetconstr @ sK201
@ ^ [Y0: $i] : ( sK199 @ Y0 ) ) ) )
& ( ( sK199 @ sK200 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1179,plain,
( ( ( dsetconstrER = $true )
| ? [X0: $i > $o,X1: $i,X2: $i] :
( ( $true
= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) )
& ( ( X0 @ X1 )
!= $true ) ) )
& ( ! [X3: $i > $o,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4
@ ( dsetconstr @ X5
@ ^ [Y0: $i] : ( X3 @ Y0 ) ) ) )
| ( $true
= ( X3 @ X4 ) ) )
| ( dsetconstrER != $true ) ) ),
inference(rectify,[status(thm)],[f1178]) ).
thf(f1178,plain,
( ( ( dsetconstrER = $true )
| ? [X0: $i > $o,X1: $i,X2: $i] :
( ( $true
= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) )
& ( ( X0 @ X1 )
!= $true ) ) )
& ( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) )
| ( ( X0 @ X1 )
= $true ) )
| ( dsetconstrER != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f774]) ).
thf(f774,plain,
( ( dsetconstrER = $true )
<=> ! [X0: $i > $o,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) )
| ( ( X0 @ X1 )
= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f546]) ).
thf(f546,plain,
( ( dsetconstrER = $true )
<=> ! [X0: $i > $o,X1: $i,X2: $i] :
( ( $true
= ( in @ X1
@ ( dsetconstr @ X2
@ ^ [Y0: $i] : ( X0 @ Y0 ) ) ) )
=> ( ( X0 @ X1 )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f545]) ).
thf(f545,plain,
( dsetconstrER
= ( ! [X0: $i > $o,X1: $i,X2: $i] :
( ( in @ X1
@ ( dsetconstr @ X2
@ ^ [X3: $i] : ( X0 @ X3 ) ) )
=> ( X0 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f16]) ).
thf(f16,axiom,
( dsetconstrER
= ( ! [X0: $i > $o,X1: $i,X3: $i] :
( ( in @ X1
@ ( dsetconstr @ X3
@ ^ [X2: $i] : ( X0 @ X2 ) ) )
=> ( X0 @ X1 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.T6NCxVKBVZ/DTF2THF_2187.p',dsetconstrER) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU675^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.18/0.34 % Computer : n023.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Tue Feb 24 12:31:19 EST 2026
% 0.18/0.34 % CPUTime :
% 0.18/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.32/0.47 ---- Original DTF file ---
% 0.32/0.47 thf(spec,logic,$$dhol).
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 % File : SEU675^1 : TPTP v9.2.1. Released v3.7.0.
% 0.32/0.47 % Domain : Set Theory
% 0.32/0.47 % Problem : Functions - Application
% 0.32/0.47 % Version : Especial.
% 0.32/0.47 % English : (! A:i.! B:i.! f:i.in f (funcSet A B) -> func A B f)
% 0.32/0.47
% 0.32/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.32/0.47 % Source : [Bro08]
% 0.32/0.47 % Names : ZFC177g [Bro08]
% 0.32/0.47
% 0.32/0.47 % Status : Theorem
% 0.32/0.47 % Rating : 0.33 v9.1.0, 0.38 v9.0.0, 0.40 v8.2.0, 0.69 v8.1.0, 0.64 v7.5.0, 0.57 v7.4.0, 0.56 v7.2.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.57 v5.5.0, 0.50 v5.4.0, 0.60 v4.1.0, 1.00 v3.7.0
% 0.32/0.47 % Syntax : Number of formulae : 427 ( 199 unt; 227 typ; 199 def)
% 0.32/0.47 % Number of atoms : 1051 ( 292 equ; 0 cnn)
% 0.32/0.47 % Maximal formula atoms : 193 ( 5 avg)
% 0.32/0.47 % Number of connectives : 1901 ( 47 ~; 7 |; 36 &;1336 @)
% 0.32/0.47 % ( 14 <=>; 461 =>; 0 <=; 0 <~>)
% 0.32/0.47 % Maximal formula depth : 199 ( 2 avg)
% 0.32/0.47 % Number of types : 2 ( 0 usr)
% 0.32/0.47 % Number of type conns : 115 ( 115 >; 0 *; 0 +; 0 <<)
% 0.32/0.47 % Number of symbols : 230 ( 227 usr; 195 con; 0-4 aty)
% 0.32/0.47 % Number of variables : 619 ( 71 ^; 512 !; 36 ?; 619 :)
% 0.32/0.47 % SPC : TH0_THM_EQU_NAR
% 0.32/0.47
% 0.32/0.47 % Comments : http://mathgate.info/detsetitem.php?id=227
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 thf(in_type,type,
% 0.32/0.47 in: $i > $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(exu_type,type,
% 0.32/0.47 exu: ( $i > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(exu,definition,
% 0.32/0.47 ( exu
% 0.32/0.47 = ( ^ [Xphi: $i > $o] :
% 0.32/0.47 ? [Xx: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 & ! [Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setextAx_type,type,
% 0.32/0.47 setextAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setextAx,definition,
% 0.32/0.47 ( setextAx
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 <=> ( in @ Xx @ B ) )
% 0.32/0.47 => ( A = B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyset_type,type,
% 0.32/0.47 emptyset: $i ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetAx_type,type,
% 0.32/0.47 emptysetAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetAx,definition,
% 0.32/0.47 ( emptysetAx
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoin_type,type,
% 0.32/0.47 setadjoin: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinAx_type,type,
% 0.32/0.47 setadjoinAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinAx,definition,
% 0.32/0.47 ( setadjoinAx
% 0.32/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/0.47 <=> ( ( Xy = Xx )
% 0.32/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powerset_type,type,
% 0.32/0.47 powerset: $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(powersetAx_type,type,
% 0.32/0.47 powersetAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powersetAx,definition,
% 0.32/0.47 ( powersetAx
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.32/0.47 <=> ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunion_type,type,
% 0.32/0.47 setunion: $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(setunionAx_type,type,
% 0.32/0.47 setunionAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionAx,definition,
% 0.32/0.47 ( setunionAx
% 0.32/0.47 = ( ! [A: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.32/0.47 <=> ? [B: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(omega_type,type,
% 0.32/0.47 omega: $i ).
% 0.32/0.47
% 0.32/0.47 thf(omega0Ax_type,type,
% 0.32/0.47 omega0Ax: $o ).
% 0.32/0.47
% 0.32/0.47 thf(omega0Ax,definition,
% 0.32/0.47 ( omega0Ax
% 0.32/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(omegaSAx_type,type,
% 0.32/0.47 omegaSAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(omegaSAx,definition,
% 0.32/0.47 ( omegaSAx
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ omega )
% 0.32/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(omegaIndAx_type,type,
% 0.32/0.47 omegaIndAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(omegaIndAx,definition,
% 0.32/0.47 ( omegaIndAx
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ( ( in @ emptyset @ A )
% 0.32/0.47 & ! [Xx: $i] :
% 0.32/0.47 ( ( ( in @ Xx @ omega )
% 0.32/0.47 & ( in @ Xx @ A ) )
% 0.32/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ omega )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(replAx_type,type,
% 0.32/0.47 replAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(replAx,definition,
% 0.32/0.47 ( replAx
% 0.32/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( exu
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.32/0.47 => ? [B: $i] :
% 0.32/0.47 ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 <=> ? [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ A )
% 0.32/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(foundationAx_type,type,
% 0.32/0.47 foundationAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(foundationAx,definition,
% 0.32/0.47 ( foundationAx
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.32/0.47 => ? [B: $i] :
% 0.32/0.47 ( ( in @ B @ A )
% 0.32/0.47 & ~ ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(wellorderingAx_type,type,
% 0.32/0.47 wellorderingAx: $o ).
% 0.32/0.47
% 0.32/0.47 thf(wellorderingAx,definition,
% 0.32/0.47 ( wellorderingAx
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ? [B: $i] :
% 0.32/0.47 ( ! [C: $i] :
% 0.32/0.47 ( ( in @ C @ B )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ C )
% 0.32/0.47 => ( in @ Xx @ A ) ) )
% 0.32/0.47 & ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( ( in @ Xx @ A )
% 0.32/0.47 & ( in @ Xy @ A ) )
% 0.32/0.47 => ( ! [C: $i] :
% 0.32/0.47 ( ( in @ C @ B )
% 0.32/0.47 => ( ( in @ Xx @ C )
% 0.32/0.47 <=> ( in @ Xy @ C ) ) )
% 0.32/0.47 => ( Xx = Xy ) ) )
% 0.32/0.47 & ! [C: $i,D: $i] :
% 0.32/0.47 ( ( ( in @ C @ B )
% 0.32/0.47 & ( in @ D @ B ) )
% 0.32/0.47 => ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ C )
% 0.32/0.47 => ( in @ Xx @ D ) )
% 0.32/0.47 | ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ D )
% 0.32/0.47 => ( in @ Xx @ C ) ) ) )
% 0.32/0.47 & ! [C: $i] :
% 0.32/0.47 ( ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ C )
% 0.32/0.47 => ( in @ Xx @ A ) )
% 0.32/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.32/0.47 => ? [D: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ D @ B )
% 0.32/0.47 & ( in @ Xx @ C )
% 0.32/0.47 & ~ ? [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ D )
% 0.32/0.47 & ( in @ Xy @ C ) )
% 0.32/0.47 & ! [E: $i] :
% 0.32/0.47 ( ( in @ E @ B )
% 0.32/0.47 => ( ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ E )
% 0.32/0.47 => ( in @ Xy @ D ) )
% 0.32/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(descr_type,type,
% 0.32/0.47 descr: ( $i > $o ) > $i ).
% 0.32/0.47
% 0.32/0.47 thf(descrp_type,type,
% 0.32/0.47 descrp: $o ).
% 0.32/0.47
% 0.32/0.47 thf(descrp,definition,
% 0.32/0.47 ( descrp
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ( Xphi
% 0.32/0.47 @ ( descr
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstr_type,type,
% 0.32/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrI_type,type,
% 0.32/0.47 dsetconstrI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrI,definition,
% 0.32/0.47 ( dsetconstrI
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 => ( in @ Xx
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrEL_type,type,
% 0.32/0.47 dsetconstrEL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrEL,definition,
% 0.32/0.47 ( dsetconstrEL
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrER_type,type,
% 0.32/0.47 dsetconstrER: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstrER,definition,
% 0.32/0.47 ( dsetconstrER
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.32/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuE1_type,type,
% 0.32/0.47 exuE1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuE1,definition,
% 0.32/0.47 ( exuE1
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 & ! [Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(prop2set_type,type,
% 0.32/0.47 prop2set: $o > $i ).
% 0.32/0.47
% 0.32/0.47 thf(prop2setE_type,type,
% 0.32/0.47 prop2setE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(prop2setE,definition,
% 0.32/0.47 ( prop2setE
% 0.32/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.32/0.47 => Xphi ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetE_type,type,
% 0.32/0.47 emptysetE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetE,definition,
% 0.32/0.47 ( emptysetE
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ emptyset )
% 0.32/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetimpfalse_type,type,
% 0.32/0.47 emptysetimpfalse: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetimpfalse,definition,
% 0.32/0.47 ( emptysetimpfalse
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ emptyset )
% 0.32/0.47 => $false ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notinemptyset_type,type,
% 0.32/0.47 notinemptyset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notinemptyset,definition,
% 0.32/0.47 ( notinemptyset
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuE3e_type,type,
% 0.32/0.47 exuE3e: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuE3e,definition,
% 0.32/0.47 ( exuE3e
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setext_type,type,
% 0.32/0.47 setext: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setext,definition,
% 0.32/0.47 ( setext
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ B ) )
% 0.32/0.47 => ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ A ) )
% 0.32/0.47 => ( A = B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyI_type,type,
% 0.32/0.47 emptyI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyI,definition,
% 0.32/0.47 ( emptyI
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ~ ( in @ Xx @ A )
% 0.32/0.47 => ( A = emptyset ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(noeltsimpempty_type,type,
% 0.32/0.47 noeltsimpempty: $o ).
% 0.32/0.47
% 0.32/0.47 thf(noeltsimpempty,definition,
% 0.32/0.47 ( noeltsimpempty
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ~ ( in @ Xx @ A )
% 0.32/0.47 => ( A = emptyset ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setbeta_type,type,
% 0.32/0.47 setbeta: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setbeta,definition,
% 0.32/0.47 ( setbeta
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( in @ Xx
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.32/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(nonempty_type,type,
% 0.32/0.47 nonempty: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(nonempty,definition,
% 0.32/0.47 ( nonempty
% 0.32/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyE1_type,type,
% 0.32/0.47 nonemptyE1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyE1,definition,
% 0.32/0.47 ( nonemptyE1
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ( nonempty @ A )
% 0.32/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyI_type,type,
% 0.32/0.47 nonemptyI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyI,definition,
% 0.32/0.47 ( nonemptyI
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 => ( nonempty
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyI1_type,type,
% 0.32/0.47 nonemptyI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyI1,definition,
% 0.32/0.47 ( nonemptyI1
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.32/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinIL_type,type,
% 0.32/0.47 setadjoinIL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinIL,definition,
% 0.32/0.47 ( setadjoinIL
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyinunitempty_type,type,
% 0.32/0.47 emptyinunitempty: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyinunitempty,definition,
% 0.32/0.47 ( emptyinunitempty
% 0.32/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinIR_type,type,
% 0.32/0.47 setadjoinIR: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinIR,definition,
% 0.32/0.47 ( setadjoinIR
% 0.32/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ A )
% 0.32/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinE_type,type,
% 0.32/0.47 setadjoinE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinE,definition,
% 0.32/0.47 ( setadjoinE
% 0.32/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/0.47 => ! [Xphi: $o] :
% 0.32/0.47 ( ( ( Xy = Xx )
% 0.32/0.47 => Xphi )
% 0.32/0.47 => ( ( ( in @ Xy @ A )
% 0.32/0.47 => Xphi )
% 0.32/0.47 => Xphi ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinOr_type,type,
% 0.32/0.47 setadjoinOr: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinOr,definition,
% 0.32/0.47 ( setadjoinOr
% 0.32/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/0.47 => ( ( Xy = Xx )
% 0.32/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setoftrueEq_type,type,
% 0.32/0.47 setoftrueEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setoftrueEq,definition,
% 0.32/0.47 ( setoftrueEq
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xx: $i] : $true )
% 0.32/0.47 = A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powersetI_type,type,
% 0.32/0.47 powersetI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powersetI,definition,
% 0.32/0.47 ( powersetI
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ A ) )
% 0.32/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyinPowerset_type,type,
% 0.32/0.47 emptyinPowerset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyinPowerset,definition,
% 0.32/0.47 ( emptyinPowerset
% 0.32/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyInPowerset_type,type,
% 0.32/0.47 emptyInPowerset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyInPowerset,definition,
% 0.32/0.47 ( emptyInPowerset
% 0.32/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powersetE_type,type,
% 0.32/0.47 powersetE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powersetE,definition,
% 0.32/0.47 ( powersetE
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.32/0.47 => ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunionI_type,type,
% 0.32/0.47 setunionI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionI,definition,
% 0.32/0.47 ( setunionI
% 0.32/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( ( in @ B @ A )
% 0.32/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunionE_type,type,
% 0.32/0.47 setunionE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionE,definition,
% 0.32/0.47 ( setunionE
% 0.32/0.47 = ( ! [A: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.32/0.47 => ! [Xphi: $o] :
% 0.32/0.47 ( ! [B: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( ( in @ B @ A )
% 0.32/0.47 => Xphi ) )
% 0.32/0.47 => Xphi ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subPowSU_type,type,
% 0.32/0.47 subPowSU: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subPowSU,definition,
% 0.32/0.47 ( subPowSU
% 0.32/0.47 = ( ! [A: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuE2_type,type,
% 0.32/0.47 exuE2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuE2,definition,
% 0.32/0.47 ( exuE2
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ! [Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xy )
% 0.32/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyImpWitness_type,type,
% 0.32/0.47 nonemptyImpWitness: $o ).
% 0.32/0.47
% 0.32/0.47 thf(nonemptyImpWitness,definition,
% 0.32/0.47 ( nonemptyImpWitness
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ( nonempty @ A )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & $true ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(uniqinunit_type,type,
% 0.32/0.47 uniqinunit: $o ).
% 0.32/0.47
% 0.32/0.47 thf(uniqinunit,definition,
% 0.32/0.47 ( uniqinunit
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notinsingleton_type,type,
% 0.32/0.47 notinsingleton: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notinsingleton,definition,
% 0.32/0.47 ( notinsingleton
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx != Xy )
% 0.32/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(eqinunit_type,type,
% 0.32/0.47 eqinunit: $o ).
% 0.32/0.47
% 0.32/0.47 thf(eqinunit,definition,
% 0.32/0.47 ( eqinunit
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(singletonsswitch_type,type,
% 0.32/0.47 singletonsswitch: $o ).
% 0.32/0.47
% 0.32/0.47 thf(singletonsswitch,definition,
% 0.32/0.47 ( singletonsswitch
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetE_type,type,
% 0.32/0.47 upairsetE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetE,definition,
% 0.32/0.47 ( upairsetE
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.32/0.47 => ( ( Xz = Xx )
% 0.32/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetIL_type,type,
% 0.32/0.47 upairsetIL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetIL,definition,
% 0.32/0.47 ( upairsetIL
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetIR_type,type,
% 0.32/0.47 upairsetIR: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetIR,definition,
% 0.32/0.47 ( upairsetIR
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyE1_type,type,
% 0.32/0.47 emptyE1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyE1,definition,
% 0.32/0.47 ( emptyE1
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) )
% 0.32/0.47 => ( ( ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 = emptyset )
% 0.32/0.47 => $false ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(vacuousDall_type,type,
% 0.32/0.47 vacuousDall: $o ).
% 0.32/0.47
% 0.32/0.47 thf(vacuousDall,definition,
% 0.32/0.47 ( vacuousDall
% 0.32/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ emptyset )
% 0.32/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan1_type,type,
% 0.32/0.47 quantDeMorgan1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan1,definition,
% 0.32/0.47 ( quantDeMorgan1
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ~ ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan2_type,type,
% 0.32/0.47 quantDeMorgan2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan2,definition,
% 0.32/0.47 ( quantDeMorgan2
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ~ ( Xphi @ Xx ) )
% 0.32/0.47 => ~ ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan3_type,type,
% 0.32/0.47 quantDeMorgan3: $o ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan3,definition,
% 0.32/0.47 ( quantDeMorgan3
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ~ ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan4_type,type,
% 0.32/0.47 quantDeMorgan4: $o ).
% 0.32/0.47
% 0.32/0.47 thf(quantDeMorgan4,definition,
% 0.32/0.47 ( quantDeMorgan4
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ~ ( Xphi @ Xx ) )
% 0.32/0.47 => ~ ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(prop2setI_type,type,
% 0.32/0.47 prop2setI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(prop2setI,definition,
% 0.32/0.47 ( prop2setI
% 0.32/0.47 = ( ! [Xphi: $o] :
% 0.32/0.47 ( Xphi
% 0.32/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(set2prop_type,type,
% 0.32/0.47 set2prop: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(prop2set2propI_type,type,
% 0.32/0.47 prop2set2propI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(prop2set2propI,definition,
% 0.32/0.47 ( prop2set2propI
% 0.32/0.47 = ( ! [Xphi: $o] :
% 0.32/0.47 ( Xphi
% 0.32/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notdexE_type,type,
% 0.32/0.47 notdexE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notdexE,definition,
% 0.32/0.47 ( notdexE
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ~ ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notdallE_type,type,
% 0.32/0.47 notdallE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notdallE,definition,
% 0.32/0.47 ( notdallE
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ~ ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuI1_type,type,
% 0.32/0.47 exuI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuI1,definition,
% 0.32/0.47 ( exuI1
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ? [Xx: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 & ! [Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) )
% 0.32/0.47 => ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuI3_type,type,
% 0.32/0.47 exuI3: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuI3,definition,
% 0.32/0.47 ( exuI3
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.32/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 => ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) )
% 0.32/0.47 => ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuI2_type,type,
% 0.32/0.47 exuI2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuI2,definition,
% 0.32/0.47 ( exuI2
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ? [Xx: $i] :
% 0.32/0.47 ! [Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xy )
% 0.32/0.47 <=> ( Xy = Xx ) )
% 0.32/0.47 => ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(inCongP_type,type,
% 0.32/0.47 inCongP: $o ).
% 0.32/0.47
% 0.32/0.47 thf(inCongP,definition,
% 0.32/0.47 ( inCongP
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(in__Cong_type,type,
% 0.32/0.47 in__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(in__Cong,definition,
% 0.32/0.47 ( in__Cong
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( ( in @ Xx @ A )
% 0.32/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuE3u_type,type,
% 0.32/0.47 exuE3u: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuE3u,definition,
% 0.32/0.47 ( exuE3u
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 => ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exu__Cong_type,type,
% 0.32/0.47 exu__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exu__Cong,definition,
% 0.32/0.47 ( exu__Cong
% 0.32/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.32/0.47 => ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 <=> ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptyset__Cong_type,type,
% 0.32/0.47 emptyset__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptyset__Cong,definition,
% 0.32/0.47 ( emptyset__Cong
% 0.32/0.47 = ( emptyset = emptyset ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoin__Cong_type,type,
% 0.32/0.47 setadjoin__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoin__Cong,definition,
% 0.32/0.47 ( setadjoin__Cong
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ! [Xz: $i,Xu: $i] :
% 0.32/0.47 ( ( Xz = Xu )
% 0.32/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.32/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powerset__Cong_type,type,
% 0.32/0.47 powerset__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powerset__Cong,definition,
% 0.32/0.47 ( powerset__Cong
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ( ( powerset @ A )
% 0.32/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunion__Cong_type,type,
% 0.32/0.47 setunion__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunion__Cong,definition,
% 0.32/0.47 ( setunion__Cong
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ( ( setunion @ A )
% 0.32/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(omega__Cong_type,type,
% 0.32/0.47 omega__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(omega__Cong,definition,
% 0.32/0.47 ( omega__Cong
% 0.32/0.47 = ( omega = omega ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(exuEu_type,type,
% 0.32/0.47 exuEu: $o ).
% 0.32/0.47
% 0.32/0.47 thf(exuEu,definition,
% 0.32/0.47 ( exuEu
% 0.32/0.47 = ( ! [Xphi: $i > $o] :
% 0.32/0.47 ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xphi @ Xx )
% 0.32/0.47 => ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(descr__Cong_type,type,
% 0.32/0.47 descr__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(descr__Cong,definition,
% 0.32/0.47 ( descr__Cong
% 0.32/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.32/0.47 => ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ( ( exu
% 0.32/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.32/0.47 => ( ( descr
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 = ( descr
% 0.32/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstr__Cong_type,type,
% 0.32/0.47 dsetconstr__Cong: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dsetconstr__Cong,definition,
% 0.32/0.47 ( dsetconstr__Cong
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( Xx = Xy )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.32/0.47 => ( ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 = ( dsetconstr @ B
% 0.32/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subset_type,type,
% 0.32/0.47 subset: $i > $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(disjoint_type,type,
% 0.32/0.47 disjoint: $i > $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(setsmeet_type,type,
% 0.32/0.47 setsmeet: $i > $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetI1_type,type,
% 0.32/0.47 subsetI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetI1,definition,
% 0.32/0.47 ( subsetI1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ B ) )
% 0.32/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(eqimpsubset2_type,type,
% 0.32/0.47 eqimpsubset2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(eqimpsubset2,definition,
% 0.32/0.47 ( eqimpsubset2
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(eqimpsubset1_type,type,
% 0.32/0.47 eqimpsubset1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(eqimpsubset1,definition,
% 0.32/0.47 ( eqimpsubset1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( A = B )
% 0.32/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetI2_type,type,
% 0.32/0.47 subsetI2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetI2,definition,
% 0.32/0.47 ( subsetI2
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ B ) )
% 0.32/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetsubset_type,type,
% 0.32/0.47 emptysetsubset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(emptysetsubset,definition,
% 0.32/0.47 ( emptysetsubset
% 0.32/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetE_type,type,
% 0.32/0.47 subsetE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetE,definition,
% 0.32/0.47 ( subsetE
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetE2_type,type,
% 0.32/0.47 subsetE2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetE2,definition,
% 0.32/0.47 ( subsetE2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notsubsetI_type,type,
% 0.32/0.47 notsubsetI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notsubsetI,definition,
% 0.32/0.47 ( notsubsetI
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notequalI1_type,type,
% 0.32/0.47 notequalI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notequalI1,definition,
% 0.32/0.47 ( notequalI1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ~ ( subset @ A @ B )
% 0.32/0.47 => ( A != B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(notequalI2_type,type,
% 0.32/0.47 notequalI2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(notequalI2,definition,
% 0.32/0.47 ( notequalI2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ( A != B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetRefl_type,type,
% 0.32/0.47 subsetRefl: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetRefl,definition,
% 0.32/0.47 ( subsetRefl
% 0.32/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetTrans_type,type,
% 0.32/0.47 subsetTrans: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetTrans,definition,
% 0.32/0.47 ( subsetTrans
% 0.32/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( ( subset @ B @ C )
% 0.32/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinSub_type,type,
% 0.32/0.47 setadjoinSub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinSub,definition,
% 0.32/0.47 ( setadjoinSub
% 0.32/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinSub2_type,type,
% 0.32/0.47 setadjoinSub2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setadjoinSub2,definition,
% 0.32/0.47 ( setadjoinSub2
% 0.32/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subset2powerset_type,type,
% 0.32/0.47 subset2powerset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subset2powerset,definition,
% 0.32/0.47 ( subset2powerset
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setextsub_type,type,
% 0.32/0.47 setextsub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setextsub,definition,
% 0.32/0.47 ( setextsub
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( ( subset @ B @ A )
% 0.32/0.47 => ( A = B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(subsetemptysetimpeq_type,type,
% 0.32/0.47 subsetemptysetimpeq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(subsetemptysetimpeq,definition,
% 0.32/0.47 ( subsetemptysetimpeq
% 0.32/0.47 = ( ! [A: $i] :
% 0.32/0.47 ( ( subset @ A @ emptyset )
% 0.32/0.47 => ( A = emptyset ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powersetI1_type,type,
% 0.32/0.47 powersetI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powersetI1,definition,
% 0.32/0.47 ( powersetI1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( subset @ B @ A )
% 0.32/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powersetE1_type,type,
% 0.32/0.47 powersetE1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powersetE1,definition,
% 0.32/0.47 ( powersetE1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.32/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(inPowerset_type,type,
% 0.32/0.47 inPowerset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(inPowerset,definition,
% 0.32/0.47 ( inPowerset
% 0.32/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(powersetsubset_type,type,
% 0.32/0.47 powersetsubset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(powersetsubset,definition,
% 0.32/0.47 ( powersetsubset
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sepInPowerset_type,type,
% 0.32/0.47 sepInPowerset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(sepInPowerset,definition,
% 0.32/0.47 ( sepInPowerset
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( in
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(sepSubset_type,type,
% 0.32/0.47 sepSubset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(sepSubset,definition,
% 0.32/0.47 ( sepSubset
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( subset
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binunion_type,type,
% 0.32/0.47 binunion: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(binunionIL_type,type,
% 0.32/0.47 binunionIL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binunionIL,definition,
% 0.32/0.47 ( binunionIL
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairset2IR_type,type,
% 0.32/0.47 upairset2IR: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairset2IR,definition,
% 0.32/0.47 ( upairset2IR
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binunionIR_type,type,
% 0.32/0.47 binunionIR: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binunionIR,definition,
% 0.32/0.47 ( binunionIR
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binunionEcases_type,type,
% 0.32/0.47 binunionEcases: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binunionEcases,definition,
% 0.32/0.47 ( binunionEcases
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.32/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.32/0.47 => ( ( ( in @ Xx @ A )
% 0.32/0.47 => Xphi )
% 0.32/0.47 => ( ( ( in @ Xx @ B )
% 0.32/0.47 => Xphi )
% 0.32/0.47 => Xphi ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binunionE_type,type,
% 0.32/0.47 binunionE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binunionE,definition,
% 0.32/0.47 ( binunionE
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.32/0.47 => ( ( in @ Xx @ A )
% 0.32/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binunionLsub_type,type,
% 0.32/0.47 binunionLsub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binunionLsub,definition,
% 0.32/0.47 ( binunionLsub
% 0.32/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binunionRsub_type,type,
% 0.32/0.47 binunionRsub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binunionRsub,definition,
% 0.32/0.47 ( binunionRsub
% 0.32/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersect_type,type,
% 0.32/0.47 binintersect: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectI_type,type,
% 0.32/0.47 binintersectI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectI,definition,
% 0.32/0.47 ( binintersectI
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset5_type,type,
% 0.32/0.47 binintersectSubset5: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset5,definition,
% 0.32/0.47 ( binintersectSubset5
% 0.32/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.32/0.47 ( ( subset @ C @ A )
% 0.32/0.47 => ( ( subset @ C @ B )
% 0.32/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectEL_type,type,
% 0.32/0.47 binintersectEL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectEL,definition,
% 0.32/0.47 ( binintersectEL
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectLsub_type,type,
% 0.32/0.47 binintersectLsub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectLsub,definition,
% 0.32/0.47 ( binintersectLsub
% 0.32/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset2_type,type,
% 0.32/0.47 binintersectSubset2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset2,definition,
% 0.32/0.47 ( binintersectSubset2
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( ( binintersect @ A @ B )
% 0.32/0.47 = A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset3_type,type,
% 0.32/0.47 binintersectSubset3: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset3,definition,
% 0.32/0.47 ( binintersectSubset3
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( ( binintersect @ A @ B )
% 0.32/0.47 = B )
% 0.32/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectER_type,type,
% 0.32/0.47 binintersectER: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectER,definition,
% 0.32/0.47 ( binintersectER
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.32/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(disjointsetsI1_type,type,
% 0.32/0.47 disjointsetsI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(disjointsetsI1,definition,
% 0.32/0.47 ( disjointsetsI1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ~ ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( in @ Xx @ B ) )
% 0.32/0.47 => ( ( binintersect @ A @ B )
% 0.32/0.47 = emptyset ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectRsub_type,type,
% 0.32/0.47 binintersectRsub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectRsub,definition,
% 0.32/0.47 ( binintersectRsub
% 0.32/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset4_type,type,
% 0.32/0.47 binintersectSubset4: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset4,definition,
% 0.32/0.47 ( binintersectSubset4
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( subset @ B @ A )
% 0.32/0.47 => ( ( binintersect @ A @ B )
% 0.32/0.47 = B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset1_type,type,
% 0.32/0.47 binintersectSubset1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(binintersectSubset1,definition,
% 0.32/0.47 ( binintersectSubset1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( ( binintersect @ A @ B )
% 0.32/0.47 = A )
% 0.32/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(bs114d_type,type,
% 0.32/0.47 bs114d: $o ).
% 0.32/0.47
% 0.32/0.47 thf(bs114d,definition,
% 0.32/0.47 ( bs114d
% 0.32/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.32/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.32/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(regular_type,type,
% 0.32/0.47 regular: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminus_type,type,
% 0.32/0.47 setminus: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(setminusI_type,type,
% 0.32/0.47 setminusI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusI,definition,
% 0.32/0.47 ( setminusI
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusEL_type,type,
% 0.32/0.47 setminusEL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusEL,definition,
% 0.32/0.47 ( setminusEL
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusER_type,type,
% 0.32/0.47 setminusER: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusER,definition,
% 0.32/0.47 ( setminusER
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusSubset2_type,type,
% 0.32/0.47 setminusSubset2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusSubset2,definition,
% 0.32/0.47 ( setminusSubset2
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( subset @ A @ B )
% 0.32/0.47 => ( ( setminus @ A @ B )
% 0.32/0.47 = emptyset ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusERneg_type,type,
% 0.32/0.47 setminusERneg: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusERneg,definition,
% 0.32/0.47 ( setminusERneg
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/0.47 => ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusELneg_type,type,
% 0.32/0.47 setminusELneg: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusELneg,definition,
% 0.32/0.47 ( setminusELneg
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusILneg_type,type,
% 0.32/0.47 setminusILneg: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusILneg,definition,
% 0.32/0.47 ( setminusILneg
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ~ ( in @ Xx @ A )
% 0.32/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusIRneg_type,type,
% 0.32/0.47 setminusIRneg: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusIRneg,definition,
% 0.32/0.47 ( setminusIRneg
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ B )
% 0.32/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusLsub_type,type,
% 0.32/0.47 setminusLsub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusLsub,definition,
% 0.32/0.47 ( setminusLsub
% 0.32/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setminusSubset1_type,type,
% 0.32/0.47 setminusSubset1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setminusSubset1,definition,
% 0.32/0.47 ( setminusSubset1
% 0.32/0.47 = ( ! [A: $i,B: $i] :
% 0.32/0.47 ( ( ( setminus @ A @ B )
% 0.32/0.47 = emptyset )
% 0.32/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(symdiff_type,type,
% 0.32/0.47 symdiff: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffE_type,type,
% 0.32/0.47 symdiffE: $o ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffE,definition,
% 0.32/0.47 ( symdiffE
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.32/0.47 => ! [Xphi: $o] :
% 0.32/0.47 ( ( ( in @ Xx @ A )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => Xphi ) )
% 0.32/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.32/0.47 => ( ( in @ Xx @ B )
% 0.32/0.47 => Xphi ) )
% 0.32/0.47 => Xphi ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffI1_type,type,
% 0.32/0.47 symdiffI1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffI1,definition,
% 0.32/0.47 ( symdiffI1
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffI2_type,type,
% 0.32/0.47 symdiffI2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffI2,definition,
% 0.32/0.47 ( symdiffI2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ~ ( in @ Xx @ A )
% 0.32/0.47 => ( ( in @ Xx @ B )
% 0.32/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffIneg1_type,type,
% 0.32/0.47 symdiffIneg1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffIneg1,definition,
% 0.32/0.47 ( symdiffIneg1
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( in @ Xx @ B )
% 0.32/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffIneg2_type,type,
% 0.32/0.47 symdiffIneg2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(symdiffIneg2,definition,
% 0.32/0.47 ( symdiffIneg2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ~ ( in @ Xx @ A )
% 0.32/0.47 => ( ~ ( in @ Xx @ B )
% 0.32/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(iskpair_type,type,
% 0.32/0.47 iskpair: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(secondinupair_type,type,
% 0.32/0.47 secondinupair: $o ).
% 0.32/0.47
% 0.32/0.47 thf(secondinupair,definition,
% 0.32/0.47 ( secondinupair
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairIL_type,type,
% 0.32/0.47 setukpairIL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairIL,definition,
% 0.32/0.47 ( setukpairIL
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairIR_type,type,
% 0.32/0.47 setukpairIR: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairIR,definition,
% 0.32/0.47 ( setukpairIR
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(kpairiskpair_type,type,
% 0.32/0.47 kpairiskpair: $o ).
% 0.32/0.47
% 0.32/0.47 thf(kpairiskpair,definition,
% 0.32/0.47 ( kpairiskpair
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(kpair_type,type,
% 0.32/0.47 kpair: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(kpairp_type,type,
% 0.32/0.47 kpairp: $o ).
% 0.32/0.47
% 0.32/0.47 thf(kpairp,definition,
% 0.32/0.47 ( kpairp
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprod_type,type,
% 0.32/0.47 cartprod: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(singletonsubset_type,type,
% 0.32/0.47 singletonsubset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(singletonsubset,definition,
% 0.32/0.47 ( singletonsubset
% 0.32/0.47 = ( ! [A: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(singletoninpowerset_type,type,
% 0.32/0.47 singletoninpowerset: $o ).
% 0.32/0.47
% 0.32/0.47 thf(singletoninpowerset,definition,
% 0.32/0.47 ( singletoninpowerset
% 0.32/0.47 = ( ! [A: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(singletoninpowunion_type,type,
% 0.32/0.47 singletoninpowunion: $o ).
% 0.32/0.47
% 0.32/0.47 thf(singletoninpowunion,definition,
% 0.32/0.47 ( singletoninpowunion
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairset2E_type,type,
% 0.32/0.47 upairset2E: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairset2E,definition,
% 0.32/0.47 ( upairset2E
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.32/0.47 => ( ( Xz = Xx )
% 0.32/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairsubunion_type,type,
% 0.32/0.47 upairsubunion: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairsubunion,definition,
% 0.32/0.47 ( upairsubunion
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairinpowunion_type,type,
% 0.32/0.47 upairinpowunion: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairinpowunion,definition,
% 0.32/0.47 ( upairinpowunion
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ubforcartprodlem1_type,type,
% 0.32/0.47 ubforcartprodlem1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ubforcartprodlem1,definition,
% 0.32/0.47 ( ubforcartprodlem1
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ubforcartprodlem2_type,type,
% 0.32/0.47 ubforcartprodlem2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ubforcartprodlem2,definition,
% 0.32/0.47 ( ubforcartprodlem2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ubforcartprodlem3_type,type,
% 0.32/0.47 ubforcartprodlem3: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ubforcartprodlem3,definition,
% 0.32/0.47 ( ubforcartprodlem3
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodpairin_type,type,
% 0.32/0.47 cartprodpairin: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodpairin,definition,
% 0.32/0.47 ( cartprodpairin
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodmempair1_type,type,
% 0.32/0.47 cartprodmempair1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodmempair1,definition,
% 0.32/0.47 ( cartprodmempair1
% 0.32/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ? [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 & ( Xu
% 0.32/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodmempair_type,type,
% 0.32/0.47 cartprodmempair: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodmempair,definition,
% 0.32/0.47 ( cartprodmempair
% 0.32/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.47 => ( iskpair @ Xu ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunionE2_type,type,
% 0.32/0.47 setunionE2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionE2,definition,
% 0.32/0.47 ( setunionE2
% 0.32/0.47 = ( ! [A: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.32/0.47 => ? [X: $i] :
% 0.32/0.47 ( ( in @ X @ A )
% 0.32/0.47 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunionsingleton1_type,type,
% 0.32/0.47 setunionsingleton1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionsingleton1,definition,
% 0.32/0.47 ( setunionsingleton1
% 0.32/0.47 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunionsingleton2_type,type,
% 0.32/0.47 setunionsingleton2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionsingleton2,definition,
% 0.32/0.47 ( setunionsingleton2
% 0.32/0.47 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setunionsingleton_type,type,
% 0.32/0.47 setunionsingleton: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setunionsingleton,definition,
% 0.32/0.47 ( setunionsingleton
% 0.32/0.47 = ( ! [Xx: $i] :
% 0.32/0.47 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.32/0.47 = Xx ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(singleton_type,type,
% 0.32/0.47 singleton: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(singleton,definition,
% 0.32/0.47 ( singleton
% 0.32/0.47 = ( ^ [A: $i] :
% 0.32/0.47 ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( A
% 0.32/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(singletonprop_type,type,
% 0.32/0.47 singletonprop: $o ).
% 0.32/0.47
% 0.32/0.47 thf(singletonprop,definition,
% 0.32/0.47 ( singletonprop
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ A )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 => ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) )
% 0.32/0.47 => ( ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) )
% 0.32/0.47 => ( singleton
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ex1_type,type,
% 0.32/0.47 ex1: $i > ( $i > $o ) > $o ).
% 0.32/0.47
% 0.32/0.47 thf(ex1,definition,
% 0.32/0.47 ( ex1
% 0.32/0.47 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( singleton
% 0.32/0.47 @ ( dsetconstr @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ex1E1_type,type,
% 0.32/0.47 ex1E1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ex1E1,definition,
% 0.32/0.47 ( ex1E1
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ( ex1 @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.47 => ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ex1I_type,type,
% 0.32/0.47 ex1I: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ex1I,definition,
% 0.32/0.47 ( ex1I
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 => ( ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ A )
% 0.32/0.47 => ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xy = Xx ) ) )
% 0.32/0.47 => ( ex1 @ A
% 0.32/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ex1I2_type,type,
% 0.32/0.47 ex1I2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ex1I2,definition,
% 0.32/0.47 ( ex1I2
% 0.32/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.47 ( ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ A )
% 0.32/0.47 => ( ( Xphi @ Xx )
% 0.32/0.47 => ( ( Xphi @ Xy )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) )
% 0.32/0.47 => ( ? [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 & ( Xphi @ Xx ) )
% 0.32/0.47 => ( ex1 @ A
% 0.32/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(singletonsuniq_type,type,
% 0.32/0.47 singletonsuniq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(singletonsuniq,definition,
% 0.32/0.47 ( singletonsuniq
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( ( setadjoin @ Xx @ emptyset )
% 0.32/0.47 = ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(atmost1p_type,type,
% 0.32/0.47 atmost1p: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(atleast2p_type,type,
% 0.32/0.47 atleast2p: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(atmost2p_type,type,
% 0.32/0.47 atmost2p: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairsetp_type,type,
% 0.32/0.47 upairsetp: $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjL1_type,type,
% 0.32/0.47 setukpairinjL1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjL1,definition,
% 0.32/0.47 ( setukpairinjL1
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.47 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.32/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(kfstsingleton_type,type,
% 0.32/0.47 kfstsingleton: $o ).
% 0.32/0.47
% 0.32/0.47 thf(kfstsingleton,definition,
% 0.32/0.47 ( kfstsingleton
% 0.32/0.47 = ( ! [Xu: $i] :
% 0.32/0.47 ( ( iskpair @ Xu )
% 0.32/0.47 => ( singleton
% 0.32/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.32/0.47 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(theprop_type,type,
% 0.32/0.47 theprop: $o ).
% 0.32/0.47
% 0.32/0.47 thf(theprop,definition,
% 0.32/0.47 ( theprop
% 0.32/0.47 = ( ! [X: $i] :
% 0.32/0.47 ( ( singleton @ X )
% 0.32/0.47 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(kfst_type,type,
% 0.32/0.47 kfst: $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(kfstpairEq_type,type,
% 0.32/0.47 kfstpairEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(kfstpairEq,definition,
% 0.32/0.47 ( kfstpairEq
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.32/0.47 = Xx ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodfstin_type,type,
% 0.32/0.47 cartprodfstin: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodfstin,definition,
% 0.32/0.47 ( cartprodfstin
% 0.32/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.47 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjL2_type,type,
% 0.32/0.47 setukpairinjL2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjL2,definition,
% 0.32/0.47 ( setukpairinjL2
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.32/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjL_type,type,
% 0.32/0.47 setukpairinjL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjL,definition,
% 0.32/0.47 ( setukpairinjL
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.32/0.47 = ( kpair @ Xz @ Xu ) )
% 0.32/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR11_type,type,
% 0.32/0.47 setukpairinjR11: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR11,definition,
% 0.32/0.47 ( setukpairinjR11
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR12_type,type,
% 0.32/0.47 setukpairinjR12: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR12,definition,
% 0.32/0.47 ( setukpairinjR12
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( Xx = Xy )
% 0.32/0.47 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR1_type,type,
% 0.32/0.47 setukpairinjR1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR1,definition,
% 0.32/0.47 ( setukpairinjR1
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.32/0.47 => ( ( Xz = Xu )
% 0.32/0.47 => ( Xy = Xu ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(upairequniteq_type,type,
% 0.32/0.47 upairequniteq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(upairequniteq,definition,
% 0.32/0.47 ( upairequniteq
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.47 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.47 = ( setadjoin @ Xz @ emptyset ) )
% 0.32/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR2_type,type,
% 0.32/0.47 setukpairinjR2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR2,definition,
% 0.32/0.47 ( setukpairinjR2
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.32/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR_type,type,
% 0.32/0.47 setukpairinjR: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setukpairinjR,definition,
% 0.32/0.47 ( setukpairinjR
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.32/0.47 = ( kpair @ Xz @ Xu ) )
% 0.32/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ksndsingleton_type,type,
% 0.32/0.47 ksndsingleton: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ksndsingleton,definition,
% 0.32/0.47 ( ksndsingleton
% 0.32/0.47 = ( ! [Xu: $i] :
% 0.32/0.47 ( ( iskpair @ Xu )
% 0.32/0.47 => ( singleton
% 0.32/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.32/0.47 @ ^ [Xx: $i] :
% 0.32/0.47 ( Xu
% 0.32/0.47 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ksnd_type,type,
% 0.32/0.47 ksnd: $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(ksndpairEq_type,type,
% 0.32/0.47 ksndpairEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(ksndpairEq,definition,
% 0.32/0.47 ( ksndpairEq
% 0.32/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.32/0.47 = Xy ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(kpairsurjEq_type,type,
% 0.32/0.47 kpairsurjEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(kpairsurjEq,definition,
% 0.32/0.47 ( kpairsurjEq
% 0.32/0.47 = ( ! [Xu: $i] :
% 0.32/0.47 ( ( iskpair @ Xu )
% 0.32/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.32/0.47 = Xu ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodsndin_type,type,
% 0.32/0.47 cartprodsndin: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodsndin,definition,
% 0.32/0.47 ( cartprodsndin
% 0.32/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.47 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodpairmemEL_type,type,
% 0.32/0.47 cartprodpairmemEL: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodpairmemEL,definition,
% 0.32/0.47 ( cartprodpairmemEL
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodpairmemER_type,type,
% 0.32/0.47 cartprodpairmemER: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodpairmemER,definition,
% 0.32/0.47 ( cartprodpairmemER
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.32/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodmempaircEq_type,type,
% 0.32/0.47 cartprodmempaircEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodmempaircEq,definition,
% 0.32/0.47 ( cartprodmempaircEq
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( kpair @ Xx @ Xy )
% 0.32/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodfstpairEq_type,type,
% 0.32/0.47 cartprodfstpairEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodfstpairEq,definition,
% 0.32/0.47 ( cartprodfstpairEq
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.32/0.47 = Xx ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodsndpairEq_type,type,
% 0.32/0.47 cartprodsndpairEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodsndpairEq,definition,
% 0.32/0.47 ( cartprodsndpairEq
% 0.32/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.32/0.47 = Xy ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodpairsurjEq_type,type,
% 0.32/0.47 cartprodpairsurjEq: $o ).
% 0.32/0.47
% 0.32/0.47 thf(cartprodpairsurjEq,definition,
% 0.32/0.47 ( cartprodpairsurjEq
% 0.32/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.32/0.47 = Xu ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(breln_type,type,
% 0.32/0.47 breln: $i > $i > $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(breln,definition,
% 0.32/0.47 ( breln
% 0.32/0.47 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstr_type,type,
% 0.32/0.47 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrI_type,type,
% 0.32/0.47 dpsetconstrI: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrI,definition,
% 0.32/0.47 ( dpsetconstrI
% 0.32/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( Xphi @ Xx @ Xy )
% 0.32/0.47 => ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.47 @ ( dpsetconstr @ A @ B
% 0.32/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrSub_type,type,
% 0.32/0.47 dpsetconstrSub: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrSub,definition,
% 0.32/0.47 ( dpsetconstrSub
% 0.32/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.32/0.47 ( subset
% 0.32/0.47 @ ( dpsetconstr @ A @ B
% 0.32/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.32/0.47 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(setOfPairsIsBReln_type,type,
% 0.32/0.47 setOfPairsIsBReln: $o ).
% 0.32/0.47
% 0.32/0.47 thf(setOfPairsIsBReln,definition,
% 0.32/0.47 ( setOfPairsIsBReln
% 0.32/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.32/0.47 ( breln @ A @ B
% 0.32/0.47 @ ( dpsetconstr @ A @ B
% 0.32/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrERa_type,type,
% 0.32/0.47 dpsetconstrERa: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrERa,definition,
% 0.32/0.47 ( dpsetconstrERa
% 0.32/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ! [Xy: $i] :
% 0.32/0.47 ( ( in @ Xy @ B )
% 0.32/0.47 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.47 @ ( dpsetconstr @ A @ B
% 0.32/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.32/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrEL1_type,type,
% 0.32/0.47 dpsetconstrEL1: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrEL1,definition,
% 0.32/0.47 ( dpsetconstrEL1
% 0.32/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.47 @ ( dpsetconstr @ A @ B
% 0.32/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.32/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrEL2_type,type,
% 0.32/0.47 dpsetconstrEL2: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrEL2,definition,
% 0.32/0.47 ( dpsetconstrEL2
% 0.32/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.47 @ ( dpsetconstr @ A @ B
% 0.32/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.32/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrER_type,type,
% 0.32/0.47 dpsetconstrER: $o ).
% 0.32/0.47
% 0.32/0.47 thf(dpsetconstrER,definition,
% 0.32/0.47 ( dpsetconstrER
% 0.32/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.32/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.47 @ ( dpsetconstr @ A @ B
% 0.32/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.32/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(func_type,type,
% 0.32/0.47 func: $i > $i > $i > $o ).
% 0.32/0.47
% 0.32/0.47 thf(func,definition,
% 0.32/0.47 ( func
% 0.32/0.47 = ( ^ [A: $i,B: $i,R: $i] :
% 0.32/0.47 ( ( breln @ A @ B @ R )
% 0.32/0.47 & ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( ex1 @ B
% 0.32/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(funcSet_type,type,
% 0.32/0.47 funcSet: $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(funcSet,definition,
% 0.32/0.47 ( funcSet
% 0.32/0.47 = ( ^ [A: $i,B: $i] :
% 0.32/0.47 ( dsetconstr @ ( powerset @ ( cartprod @ A @ B ) )
% 0.32/0.47 @ ^ [Xf: $i] : ( func @ A @ B @ Xf ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(funcImageSingleton_type,type,
% 0.32/0.47 funcImageSingleton: $o ).
% 0.32/0.47
% 0.32/0.47 thf(funcImageSingleton,definition,
% 0.32/0.47 ( funcImageSingleton
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( func @ A @ B @ Xf )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( singleton
% 0.32/0.47 @ ( dsetconstr @ B
% 0.32/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(apProp_type,type,
% 0.32/0.47 apProp: $o ).
% 0.32/0.47
% 0.32/0.47 thf(apProp,definition,
% 0.32/0.47 ( apProp
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( func @ A @ B @ Xf )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in
% 0.32/0.47 @ ( setunion
% 0.32/0.47 @ ( dsetconstr @ B
% 0.32/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.32/0.47 @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(ap_type,type,
% 0.32/0.47 ap: $i > $i > $i > $i > $i ).
% 0.32/0.47
% 0.32/0.47 thf(ap,definition,
% 0.32/0.47 ( ap
% 0.32/0.47 = ( ^ [A: $i,B: $i,Xf: $i,Xx: $i] :
% 0.32/0.47 ( setunion
% 0.32/0.47 @ ( dsetconstr @ B
% 0.32/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(app_type,type,
% 0.32/0.47 app: $o ).
% 0.32/0.47
% 0.32/0.47 thf(app,definition,
% 0.32/0.47 ( app
% 0.32/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( func @ A @ B @ Xf )
% 0.32/0.47 => ! [Xx: $i] :
% 0.32/0.47 ( ( in @ Xx @ A )
% 0.32/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(infuncsetfunc,conjecture,
% 0.32/0.47 ( setextAx
% 0.32/0.47 => ( emptysetAx
% 0.32/0.47 => ( setadjoinAx
% 0.32/0.47 => ( powersetAx
% 0.32/0.47 => ( setunionAx
% 0.32/0.47 => ( omega0Ax
% 0.32/0.47 => ( omegaSAx
% 0.32/0.47 => ( omegaIndAx
% 0.32/0.47 => ( replAx
% 0.32/0.47 => ( foundationAx
% 0.32/0.47 => ( wellorderingAx
% 0.32/0.47 => ( descrp
% 0.32/0.47 => ( dsetconstrI
% 0.32/0.47 => ( dsetconstrEL
% 0.32/0.47 => ( dsetconstrER
% 0.32/0.47 => ( exuE1
% 0.32/0.47 => ( prop2setE
% 0.32/0.47 => ( emptysetE
% 0.32/0.47 => ( emptysetimpfalse
% 0.32/0.47 => ( notinemptyset
% 0.32/0.47 => ( exuE3e
% 0.32/0.47 => ( setext
% 0.32/0.47 => ( emptyI
% 0.32/0.47 => ( noeltsimpempty
% 0.32/0.47 => ( setbeta
% 0.32/0.47 => ( nonemptyE1
% 0.32/0.47 => ( nonemptyI
% 0.32/0.47 => ( nonemptyI1
% 0.32/0.47 => ( setadjoinIL
% 0.32/0.47 => ( emptyinunitempty
% 0.32/0.47 => ( setadjoinIR
% 0.32/0.47 => ( setadjoinE
% 0.32/0.47 => ( setadjoinOr
% 0.32/0.47 => ( setoftrueEq
% 0.32/0.47 => ( powersetI
% 0.32/0.47 => ( emptyinPowerset
% 0.32/0.47 => ( emptyInPowerset
% 0.32/0.47 => ( powersetE
% 0.32/0.47 => ( setunionI
% 0.32/0.47 => ( setunionE
% 0.32/0.47 => ( subPowSU
% 0.32/0.47 => ( exuE2
% 0.32/0.47 => ( nonemptyImpWitness
% 0.32/0.47 => ( uniqinunit
% 0.32/0.47 => ( notinsingleton
% 0.32/0.47 => ( eqinunit
% 0.32/0.47 => ( singletonsswitch
% 0.32/0.47 => ( upairsetE
% 0.32/0.47 => ( upairsetIL
% 0.32/0.47 => ( upairsetIR
% 0.32/0.47 => ( emptyE1
% 0.32/0.47 => ( vacuousDall
% 0.32/0.47 => ( quantDeMorgan1
% 0.32/0.47 => ( quantDeMorgan2
% 0.32/0.47 => ( quantDeMorgan3
% 0.32/0.47 => ( quantDeMorgan4
% 0.32/0.47 => ( prop2setI
% 0.32/0.47 => ( prop2set2propI
% 0.32/0.47 => ( notdexE
% 0.32/0.47 => ( notdallE
% 0.32/0.47 => ( exuI1
% 0.32/0.47 => ( exuI3
% 0.32/0.47 => ( exuI2
% 0.32/0.47 => ( inCongP
% 0.32/0.47 => ( in__Cong
% 0.32/0.47 => ( exuE3u
% 0.32/0.47 => ( exu__Cong
% 0.32/0.47 => ( emptyset__Cong
% 0.32/0.47 => ( setadjoin__Cong
% 0.32/0.47 => ( powerset__Cong
% 0.32/0.47 => ( setunion__Cong
% 0.32/0.47 => ( omega__Cong
% 0.32/0.47 => ( exuEu
% 0.32/0.47 => ( descr__Cong
% 0.32/0.47 => ( dsetconstr__Cong
% 0.32/0.47 => ( subsetI1
% 0.32/0.47 => ( eqimpsubset2
% 0.32/0.47 => ( eqimpsubset1
% 0.32/0.47 => ( subsetI2
% 0.32/0.47 => ( emptysetsubset
% 0.32/0.47 => ( subsetE
% 0.32/0.47 => ( subsetE2
% 0.32/0.47 => ( notsubsetI
% 0.32/0.47 => ( notequalI1
% 0.32/0.47 => ( notequalI2
% 0.32/0.47 => ( subsetRefl
% 0.32/0.47 => ( subsetTrans
% 0.32/0.47 => ( setadjoinSub
% 0.32/0.47 => ( setadjoinSub2
% 0.32/0.47 => ( subset2powerset
% 0.32/0.47 => ( setextsub
% 0.32/0.47 => ( subsetemptysetimpeq
% 0.32/0.47 => ( powersetI1
% 0.32/0.47 => ( powersetE1
% 0.32/0.47 => ( inPowerset
% 0.32/0.47 => ( powersetsubset
% 0.32/0.47 => ( sepInPowerset
% 0.32/0.47 => ( sepSubset
% 0.32/0.47 => ( binunionIL
% 0.32/0.47 => ( upairset2IR
% 0.32/0.47 => ( binunionIR
% 0.32/0.47 => ( binunionEcases
% 0.32/0.47 => ( binunionE
% 0.32/0.47 => ( binunionLsub
% 0.32/0.47 => ( binunionRsub
% 0.32/0.47 => ( binintersectI
% 0.32/0.47 => ( binintersectSubset5
% 0.32/0.47 => ( binintersectEL
% 0.32/0.47 => ( binintersectLsub
% 0.32/0.47 => ( binintersectSubset2
% 0.32/0.47 => ( binintersectSubset3
% 0.32/0.47 => ( binintersectER
% 0.32/0.47 => ( disjointsetsI1
% 0.32/0.47 => ( binintersectRsub
% 0.32/0.47 => ( binintersectSubset4
% 0.32/0.47 => ( binintersectSubset1
% 0.32/0.47 => ( bs114d
% 0.32/0.47 => ( setminusI
% 0.32/0.47 => ( setminusEL
% 0.32/0.47 => ( setminusER
% 0.32/0.47 => ( setminusSubset2
% 0.32/0.47 => ( setminusERneg
% 0.32/0.47 => ( setminusELneg
% 0.32/0.47 => ( setminusILneg
% 0.32/0.47 => ( setminusIRneg
% 0.32/0.47 => ( setminusLsub
% 0.32/0.47 => ( setminusSubset1
% 0.32/0.47 => ( symdiffE
% 0.32/0.47 => ( symdiffI1
% 0.32/0.47 => ( symdiffI2
% 0.32/0.47 => ( symdiffIneg1
% 0.32/0.47 => ( symdiffIneg2
% 0.32/0.47 => ( secondinupair
% 0.32/0.47 => ( setukpairIL
% 0.32/0.47 => ( setukpairIR
% 0.32/0.47 => ( kpairiskpair
% 0.32/0.47 => ( kpairp
% 0.32/0.47 => ( singletonsubset
% 0.32/0.47 => ( singletoninpowerset
% 0.32/0.47 => ( singletoninpowunion
% 0.32/0.47 => ( upairset2E
% 0.32/0.47 => ( upairsubunion
% 0.32/0.47 => ( upairinpowunion
% 0.32/0.47 => ( ubforcartprodlem1
% 0.32/0.47 => ( ubforcartprodlem2
% 0.32/0.47 => ( ubforcartprodlem3
% 0.32/0.47 => ( cartprodpairin
% 0.32/0.47 => ( cartprodmempair1
% 0.32/0.47 => ( cartprodmempair
% 0.32/0.47 => ( setunionE2
% 0.32/0.47 => ( setunionsingleton1
% 0.32/0.47 => ( setunionsingleton2
% 0.32/0.47 => ( setunionsingleton
% 0.32/0.47 => ( singletonprop
% 0.32/0.47 => ( ex1E1
% 0.32/0.47 => ( ex1I
% 0.32/0.47 => ( ex1I2
% 0.32/0.47 => ( singletonsuniq
% 0.32/0.47 => ( setukpairinjL1
% 0.32/0.47 => ( kfstsingleton
% 0.32/0.47 => ( theprop
% 0.32/0.47 => ( kfstpairEq
% 0.32/0.47 => ( cartprodfstin
% 0.32/0.47 => ( setukpairinjL2
% 0.32/0.47 => ( setukpairinjL
% 0.32/0.47 => ( setukpairinjR11
% 0.32/0.47 => ( setukpairinjR12
% 0.32/0.47 => ( setukpairinjR1
% 0.32/0.47 => ( upairequniteq
% 0.32/0.47 => ( setukpairinjR2
% 0.32/0.47 => ( setukpairinjR
% 0.32/0.47 => ( ksndsingleton
% 0.32/0.47 => ( ksndpairEq
% 0.32/0.47 => ( kpairsurjEq
% 0.32/0.47 => ( cartprodsndin
% 0.32/0.47 => ( cartprodpairmemEL
% 0.32/0.47 => ( cartprodpairmemER
% 0.32/0.47 => ( cartprodmempaircEq
% 0.32/0.47 => ( cartprodfstpairEq
% 0.32/0.47 => ( cartprodsndpairEq
% 0.32/0.47 => ( cartprodpairsurjEq
% 0.32/0.47 => ( dpsetconstrI
% 0.32/0.47 => ( dpsetconstrSub
% 0.32/0.47 => ( setOfPairsIsBReln
% 0.32/0.47 => ( dpsetconstrERa
% 0.32/0.47 => ( dpsetconstrEL1
% 0.32/0.47 => ( dpsetconstrEL2
% 0.32/0.47 => ( dpsetconstrER
% 0.32/0.47 => ( funcImageSingleton
% 0.32/0.47 => ( apProp
% 0.32/0.47 => ( app
% 0.32/0.47 => ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.47 => ( func @ A @ B @ Xf ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 ------------------------
% 2.37/1.44 ---- Embedded in THF ---
% 2.37/1.45 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.37/1.45 %%% Generated on Tue Feb 24 12:31:20 EST 2026
% 2.37/1.45 %%% using '$$dhol' embedding, version 1.3.0.
% 2.37/1.45 %%% Logic specification used:
% 2.37/1.45 %%% thf(spec, logic, $$dhol).
% 2.37/1.45
% 2.37/1.45 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.T6NCxVKBVZ/DTF2DTF2187.p
% See solution above
% 2.37/1.45 ------------------------
% 2.37/1.47 ---- Cleaned THF ---
% 2.37/1.47 thf(in_type,type,
% 2.37/1.47 in: $i > $i > $o ).
% 2.37/1.47
% 2.37/1.47 thf(exu_type,type,
% 2.37/1.47 exu: ( $i > $o ) > $o ).
% 2.37/1.47
% 2.37/1.47 thf(exu,definition,
% 2.37/1.47 ( exu
% 2.37/1.47 = ( ^ [Xphi: $i > $o] :
% 2.37/1.47 ? [Xx: $i] :
% 2.37/1.47 ( ( Xphi @ Xx )
% 2.37/1.47 & ! [Xy: $i] :
% 2.37/1.47 ( ( Xphi @ Xy )
% 2.37/1.47 => ( Xx = Xy ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setextAx_type,type,
% 2.37/1.47 setextAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setextAx,definition,
% 2.37/1.47 ( setextAx
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 <=> ( in @ Xx @ B ) )
% 2.37/1.47 => ( A = B ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptyset_type,type,
% 2.37/1.47 emptyset: $i ).
% 2.37/1.47
% 2.37/1.47 thf(emptysetAx_type,type,
% 2.37/1.47 emptysetAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptysetAx,definition,
% 2.37/1.47 ( emptysetAx
% 2.37/1.47 = ( ! [Xx: $i] :
% 2.37/1.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoin_type,type,
% 2.37/1.47 setadjoin: $i > $i > $i ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinAx_type,type,
% 2.37/1.47 setadjoinAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinAx,definition,
% 2.37/1.47 ( setadjoinAx
% 2.37/1.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.37/1.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.37/1.47 <=> ( ( Xy = Xx )
% 2.37/1.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(powerset_type,type,
% 2.37/1.47 powerset: $i > $i ).
% 2.37/1.47
% 2.37/1.47 thf(powersetAx_type,type,
% 2.37/1.47 powersetAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(powersetAx,definition,
% 2.37/1.47 ( powersetAx
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( in @ B @ ( powerset @ A ) )
% 2.37/1.47 <=> ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setunion_type,type,
% 2.37/1.47 setunion: $i > $i ).
% 2.37/1.47
% 2.37/1.47 thf(setunionAx_type,type,
% 2.37/1.47 setunionAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setunionAx,definition,
% 2.37/1.47 ( setunionAx
% 2.37/1.47 = ( ! [A: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( setunion @ A ) )
% 2.37/1.47 <=> ? [B: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 & ( in @ B @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(omega_type,type,
% 2.37/1.47 omega: $i ).
% 2.37/1.47
% 2.37/1.47 thf(omega0Ax_type,type,
% 2.37/1.47 omega0Ax: $o ).
% 2.37/1.47
% 2.37/1.47 thf(omega0Ax,definition,
% 2.37/1.47 ( omega0Ax
% 2.37/1.47 = ( in @ emptyset @ omega ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(omegaSAx_type,type,
% 2.37/1.47 omegaSAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(omegaSAx,definition,
% 2.37/1.47 ( omegaSAx
% 2.37/1.47 = ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ omega )
% 2.37/1.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(omegaIndAx_type,type,
% 2.37/1.47 omegaIndAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(omegaIndAx,definition,
% 2.37/1.47 ( omegaIndAx
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ( ( in @ emptyset @ A )
% 2.37/1.47 & ! [Xx: $i] :
% 2.37/1.47 ( ( ( in @ Xx @ omega )
% 2.37/1.47 & ( in @ Xx @ A ) )
% 2.37/1.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.37/1.47 => ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ omega )
% 2.37/1.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(replAx_type,type,
% 2.37/1.47 replAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(replAx,definition,
% 2.37/1.47 ( replAx
% 2.37/1.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( exu
% 2.37/1.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.37/1.47 => ? [B: $i] :
% 2.37/1.47 ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 <=> ? [Xy: $i] :
% 2.37/1.47 ( ( in @ Xy @ A )
% 2.37/1.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(foundationAx_type,type,
% 2.37/1.47 foundationAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(foundationAx,definition,
% 2.37/1.47 ( foundationAx
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.37/1.47 => ? [B: $i] :
% 2.37/1.47 ( ( in @ B @ A )
% 2.37/1.47 & ~ ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(wellorderingAx_type,type,
% 2.37/1.47 wellorderingAx: $o ).
% 2.37/1.47
% 2.37/1.47 thf(wellorderingAx,definition,
% 2.37/1.47 ( wellorderingAx
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ? [B: $i] :
% 2.37/1.47 ( ! [C: $i] :
% 2.37/1.47 ( ( in @ C @ B )
% 2.37/1.47 => ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ C )
% 2.37/1.47 => ( in @ Xx @ A ) ) )
% 2.37/1.47 & ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( ( in @ Xx @ A )
% 2.37/1.47 & ( in @ Xy @ A ) )
% 2.37/1.47 => ( ! [C: $i] :
% 2.37/1.47 ( ( in @ C @ B )
% 2.37/1.47 => ( ( in @ Xx @ C )
% 2.37/1.47 <=> ( in @ Xy @ C ) ) )
% 2.37/1.47 => ( Xx = Xy ) ) )
% 2.37/1.47 & ! [C: $i,D: $i] :
% 2.37/1.47 ( ( ( in @ C @ B )
% 2.37/1.47 & ( in @ D @ B ) )
% 2.37/1.47 => ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ C )
% 2.37/1.47 => ( in @ Xx @ D ) )
% 2.37/1.47 | ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ D )
% 2.37/1.47 => ( in @ Xx @ C ) ) ) )
% 2.37/1.47 & ! [C: $i] :
% 2.37/1.47 ( ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ C )
% 2.37/1.47 => ( in @ Xx @ A ) )
% 2.37/1.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.37/1.47 => ? [D: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ D @ B )
% 2.37/1.47 & ( in @ Xx @ C )
% 2.37/1.47 & ~ ? [Xy: $i] :
% 2.37/1.47 ( ( in @ Xy @ D )
% 2.37/1.47 & ( in @ Xy @ C ) )
% 2.37/1.47 & ! [E: $i] :
% 2.37/1.47 ( ( in @ E @ B )
% 2.37/1.47 => ( ! [Xy: $i] :
% 2.37/1.47 ( ( in @ Xy @ E )
% 2.37/1.47 => ( in @ Xy @ D ) )
% 2.37/1.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(descr_type,type,
% 2.37/1.47 descr: ( $i > $o ) > $i ).
% 2.37/1.47
% 2.37/1.47 thf(descrp_type,type,
% 2.37/1.47 descrp: $o ).
% 2.37/1.47
% 2.37/1.47 thf(descrp,definition,
% 2.37/1.47 ( descrp
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 => ( Xphi
% 2.37/1.47 @ ( descr
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(dsetconstr_type,type,
% 2.37/1.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.37/1.47
% 2.37/1.47 thf(dsetconstrI_type,type,
% 2.37/1.47 dsetconstrI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(dsetconstrI,definition,
% 2.37/1.47 ( dsetconstrI
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( ( Xphi @ Xx )
% 2.37/1.47 => ( in @ Xx
% 2.37/1.47 @ ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(dsetconstrEL_type,type,
% 2.37/1.47 dsetconstrEL: $o ).
% 2.37/1.47
% 2.37/1.47 thf(dsetconstrEL,definition,
% 2.37/1.47 ( dsetconstrEL
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx
% 2.37/1.47 @ ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.37/1.47 => ( in @ Xx @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(dsetconstrER_type,type,
% 2.37/1.47 dsetconstrER: $o ).
% 2.37/1.47
% 2.37/1.47 thf(dsetconstrER,definition,
% 2.37/1.47 ( dsetconstrER
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx
% 2.37/1.47 @ ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.37/1.47 => ( Xphi @ Xx ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exuE1_type,type,
% 2.37/1.47 exuE1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exuE1,definition,
% 2.37/1.47 ( exuE1
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 => ? [Xx: $i] :
% 2.37/1.47 ( ( Xphi @ Xx )
% 2.37/1.47 & ! [Xy: $i] :
% 2.37/1.47 ( ( Xphi @ Xy )
% 2.37/1.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(prop2set_type,type,
% 2.37/1.47 prop2set: $o > $i ).
% 2.37/1.47
% 2.37/1.47 thf(prop2setE_type,type,
% 2.37/1.47 prop2setE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(prop2setE,definition,
% 2.37/1.47 ( prop2setE
% 2.37/1.47 = ( ! [Xphi: $o,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.37/1.47 => Xphi ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptysetE_type,type,
% 2.37/1.47 emptysetE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptysetE,definition,
% 2.37/1.47 ( emptysetE
% 2.37/1.47 = ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ emptyset )
% 2.37/1.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptysetimpfalse_type,type,
% 2.37/1.47 emptysetimpfalse: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptysetimpfalse,definition,
% 2.37/1.47 ( emptysetimpfalse
% 2.37/1.47 = ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ emptyset )
% 2.37/1.47 => $false ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(notinemptyset_type,type,
% 2.37/1.47 notinemptyset: $o ).
% 2.37/1.47
% 2.37/1.47 thf(notinemptyset,definition,
% 2.37/1.47 ( notinemptyset
% 2.37/1.47 = ( ! [Xx: $i] :
% 2.37/1.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exuE3e_type,type,
% 2.37/1.47 exuE3e: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exuE3e,definition,
% 2.37/1.47 ( exuE3e
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setext_type,type,
% 2.37/1.47 setext: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setext,definition,
% 2.37/1.47 ( setext
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( in @ Xx @ B ) )
% 2.37/1.47 => ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 => ( in @ Xx @ A ) )
% 2.37/1.47 => ( A = B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptyI_type,type,
% 2.37/1.47 emptyI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptyI,definition,
% 2.37/1.47 ( emptyI
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ~ ( in @ Xx @ A )
% 2.37/1.47 => ( A = emptyset ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(noeltsimpempty_type,type,
% 2.37/1.47 noeltsimpempty: $o ).
% 2.37/1.47
% 2.37/1.47 thf(noeltsimpempty,definition,
% 2.37/1.47 ( noeltsimpempty
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ~ ( in @ Xx @ A )
% 2.37/1.47 => ( A = emptyset ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setbeta_type,type,
% 2.37/1.47 setbeta: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setbeta,definition,
% 2.37/1.47 ( setbeta
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( ( in @ Xx
% 2.37/1.47 @ ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.37/1.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(nonempty_type,type,
% 2.37/1.47 nonempty: $i > $o ).
% 2.37/1.47
% 2.37/1.47 thf(nonempty,definition,
% 2.37/1.47 ( nonempty
% 2.37/1.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(nonemptyE1_type,type,
% 2.37/1.47 nonemptyE1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(nonemptyE1,definition,
% 2.37/1.47 ( nonemptyE1
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ( nonempty @ A )
% 2.37/1.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(nonemptyI_type,type,
% 2.37/1.47 nonemptyI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(nonemptyI,definition,
% 2.37/1.47 ( nonemptyI
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( ( Xphi @ Xx )
% 2.37/1.47 => ( nonempty
% 2.37/1.47 @ ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(nonemptyI1_type,type,
% 2.37/1.47 nonemptyI1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(nonemptyI1,definition,
% 2.37/1.47 ( nonemptyI1
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.37/1.47 => ( nonempty @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinIL_type,type,
% 2.37/1.47 setadjoinIL: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinIL,definition,
% 2.37/1.47 ( setadjoinIL
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptyinunitempty_type,type,
% 2.37/1.47 emptyinunitempty: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptyinunitempty,definition,
% 2.37/1.47 ( emptyinunitempty
% 2.37/1.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinIR_type,type,
% 2.37/1.47 setadjoinIR: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinIR,definition,
% 2.37/1.47 ( setadjoinIR
% 2.37/1.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.37/1.47 ( ( in @ Xy @ A )
% 2.37/1.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinE_type,type,
% 2.37/1.47 setadjoinE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinE,definition,
% 2.37/1.47 ( setadjoinE
% 2.37/1.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.37/1.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.37/1.47 => ! [Xphi: $o] :
% 2.37/1.47 ( ( ( Xy = Xx )
% 2.37/1.47 => Xphi )
% 2.37/1.47 => ( ( ( in @ Xy @ A )
% 2.37/1.47 => Xphi )
% 2.37/1.47 => Xphi ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinOr_type,type,
% 2.37/1.47 setadjoinOr: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoinOr,definition,
% 2.37/1.47 ( setadjoinOr
% 2.37/1.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.37/1.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.37/1.47 => ( ( Xy = Xx )
% 2.37/1.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setoftrueEq_type,type,
% 2.37/1.47 setoftrueEq: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setoftrueEq,definition,
% 2.37/1.47 ( setoftrueEq
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xx: $i] : $true )
% 2.37/1.47 = A ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(powersetI_type,type,
% 2.37/1.47 powersetI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(powersetI,definition,
% 2.37/1.47 ( powersetI
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 => ( in @ Xx @ A ) )
% 2.37/1.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptyinPowerset_type,type,
% 2.37/1.47 emptyinPowerset: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptyinPowerset,definition,
% 2.37/1.47 ( emptyinPowerset
% 2.37/1.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptyInPowerset_type,type,
% 2.37/1.47 emptyInPowerset: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptyInPowerset,definition,
% 2.37/1.47 ( emptyInPowerset
% 2.37/1.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(powersetE_type,type,
% 2.37/1.47 powersetE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(powersetE,definition,
% 2.37/1.47 ( powersetE
% 2.37/1.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ B @ ( powerset @ A ) )
% 2.37/1.47 => ( ( in @ Xx @ B )
% 2.37/1.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setunionI_type,type,
% 2.37/1.47 setunionI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setunionI,definition,
% 2.37/1.47 ( setunionI
% 2.37/1.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 => ( ( in @ B @ A )
% 2.37/1.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setunionE_type,type,
% 2.37/1.47 setunionE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setunionE,definition,
% 2.37/1.47 ( setunionE
% 2.37/1.47 = ( ! [A: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( setunion @ A ) )
% 2.37/1.47 => ! [Xphi: $o] :
% 2.37/1.47 ( ! [B: $i] :
% 2.37/1.47 ( ( in @ Xx @ B )
% 2.37/1.47 => ( ( in @ B @ A )
% 2.37/1.47 => Xphi ) )
% 2.37/1.47 => Xphi ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(subPowSU_type,type,
% 2.37/1.47 subPowSU: $o ).
% 2.37/1.47
% 2.37/1.47 thf(subPowSU,definition,
% 2.37/1.47 ( subPowSU
% 2.37/1.47 = ( ! [A: $i,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exuE2_type,type,
% 2.37/1.47 exuE2: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exuE2,definition,
% 2.37/1.47 ( exuE2
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 => ? [Xx: $i] :
% 2.37/1.47 ! [Xy: $i] :
% 2.37/1.47 ( ( Xphi @ Xy )
% 2.37/1.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(nonemptyImpWitness_type,type,
% 2.37/1.47 nonemptyImpWitness: $o ).
% 2.37/1.47
% 2.37/1.47 thf(nonemptyImpWitness,definition,
% 2.37/1.47 ( nonemptyImpWitness
% 2.37/1.47 = ( ! [A: $i] :
% 2.37/1.47 ( ( nonempty @ A )
% 2.37/1.47 => ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & $true ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(uniqinunit_type,type,
% 2.37/1.47 uniqinunit: $o ).
% 2.37/1.47
% 2.37/1.47 thf(uniqinunit,definition,
% 2.37/1.47 ( uniqinunit
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.37/1.47 => ( Xx = Xy ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(notinsingleton_type,type,
% 2.37/1.47 notinsingleton: $o ).
% 2.37/1.47
% 2.37/1.47 thf(notinsingleton,definition,
% 2.37/1.47 ( notinsingleton
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xx != Xy )
% 2.37/1.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(eqinunit_type,type,
% 2.37/1.47 eqinunit: $o ).
% 2.37/1.47
% 2.37/1.47 thf(eqinunit,definition,
% 2.37/1.47 ( eqinunit
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xx = Xy )
% 2.37/1.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(singletonsswitch_type,type,
% 2.37/1.47 singletonsswitch: $o ).
% 2.37/1.47
% 2.37/1.47 thf(singletonsswitch,definition,
% 2.37/1.47 ( singletonsswitch
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.37/1.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(upairsetE_type,type,
% 2.37/1.47 upairsetE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(upairsetE,definition,
% 2.37/1.47 ( upairsetE
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.37/1.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.37/1.47 => ( ( Xz = Xx )
% 2.37/1.47 | ( Xz = Xy ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(upairsetIL_type,type,
% 2.37/1.47 upairsetIL: $o ).
% 2.37/1.47
% 2.37/1.47 thf(upairsetIL,definition,
% 2.37/1.47 ( upairsetIL
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(upairsetIR_type,type,
% 2.37/1.47 upairsetIR: $o ).
% 2.37/1.47
% 2.37/1.47 thf(upairsetIR,definition,
% 2.37/1.47 ( upairsetIR
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptyE1_type,type,
% 2.37/1.47 emptyE1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptyE1,definition,
% 2.37/1.47 ( emptyE1
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.47 ( ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & ( Xphi @ Xx ) )
% 2.37/1.47 => ( ( ( dsetconstr @ A
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 = emptyset )
% 2.37/1.47 => $false ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(vacuousDall_type,type,
% 2.37/1.47 vacuousDall: $o ).
% 2.37/1.47
% 2.37/1.47 thf(vacuousDall,definition,
% 2.37/1.47 ( vacuousDall
% 2.37/1.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ emptyset )
% 2.37/1.47 => ( Xphi @ Xx ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(quantDeMorgan1_type,type,
% 2.37/1.47 quantDeMorgan1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(quantDeMorgan1,definition,
% 2.37/1.47 ( quantDeMorgan1
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.47 ( ~ ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( Xphi @ Xx ) )
% 2.37/1.47 => ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(quantDeMorgan2_type,type,
% 2.37/1.47 quantDeMorgan2: $o ).
% 2.37/1.47
% 2.37/1.47 thf(quantDeMorgan2,definition,
% 2.37/1.47 ( quantDeMorgan2
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.47 ( ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ~ ( Xphi @ Xx ) )
% 2.37/1.47 => ~ ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(quantDeMorgan3_type,type,
% 2.37/1.47 quantDeMorgan3: $o ).
% 2.37/1.47
% 2.37/1.47 thf(quantDeMorgan3,definition,
% 2.37/1.47 ( quantDeMorgan3
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.47 ( ~ ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & ( Xphi @ Xx ) )
% 2.37/1.47 => ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(quantDeMorgan4_type,type,
% 2.37/1.47 quantDeMorgan4: $o ).
% 2.37/1.47
% 2.37/1.47 thf(quantDeMorgan4,definition,
% 2.37/1.47 ( quantDeMorgan4
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.47 ( ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & ~ ( Xphi @ Xx ) )
% 2.37/1.47 => ~ ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(prop2setI_type,type,
% 2.37/1.47 prop2setI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(prop2setI,definition,
% 2.37/1.47 ( prop2setI
% 2.37/1.47 = ( ! [Xphi: $o] :
% 2.37/1.47 ( Xphi
% 2.37/1.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(set2prop_type,type,
% 2.37/1.47 set2prop: $i > $o ).
% 2.37/1.47
% 2.37/1.47 thf(prop2set2propI_type,type,
% 2.37/1.47 prop2set2propI: $o ).
% 2.37/1.47
% 2.37/1.47 thf(prop2set2propI,definition,
% 2.37/1.47 ( prop2set2propI
% 2.37/1.47 = ( ! [Xphi: $o] :
% 2.37/1.47 ( Xphi
% 2.37/1.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(notdexE_type,type,
% 2.37/1.47 notdexE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(notdexE,definition,
% 2.37/1.47 ( notdexE
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.47 ( ~ ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & ( Xphi @ Xx ) )
% 2.37/1.47 => ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(notdallE_type,type,
% 2.37/1.47 notdallE: $o ).
% 2.37/1.47
% 2.37/1.47 thf(notdallE,definition,
% 2.37/1.47 ( notdallE
% 2.37/1.47 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.47 ( ~ ! [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 => ( Xphi @ Xx ) )
% 2.37/1.47 => ? [Xx: $i] :
% 2.37/1.47 ( ( in @ Xx @ A )
% 2.37/1.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exuI1_type,type,
% 2.37/1.47 exuI1: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exuI1,definition,
% 2.37/1.47 ( exuI1
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ? [Xx: $i] :
% 2.37/1.47 ( ( Xphi @ Xx )
% 2.37/1.47 & ! [Xy: $i] :
% 2.37/1.47 ( ( Xphi @ Xy )
% 2.37/1.47 => ( Xx = Xy ) ) )
% 2.37/1.47 => ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exuI3_type,type,
% 2.37/1.47 exuI3: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exuI3,definition,
% 2.37/1.47 ( exuI3
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.37/1.47 => ( ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xphi @ Xx )
% 2.37/1.47 => ( ( Xphi @ Xy )
% 2.37/1.47 => ( Xx = Xy ) ) )
% 2.37/1.47 => ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exuI2_type,type,
% 2.37/1.47 exuI2: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exuI2,definition,
% 2.37/1.47 ( exuI2
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ? [Xx: $i] :
% 2.37/1.47 ! [Xy: $i] :
% 2.37/1.47 ( ( Xphi @ Xy )
% 2.37/1.47 <=> ( Xy = Xx ) )
% 2.37/1.47 => ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(inCongP_type,type,
% 2.37/1.47 inCongP: $o ).
% 2.37/1.47
% 2.37/1.47 thf(inCongP,definition,
% 2.37/1.47 ( inCongP
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( A = B )
% 2.37/1.47 => ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xx = Xy )
% 2.37/1.47 => ( ( in @ Xx @ A )
% 2.37/1.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(in__Cong_type,type,
% 2.37/1.47 in__Cong: $o ).
% 2.37/1.47
% 2.37/1.47 thf(in__Cong,definition,
% 2.37/1.47 ( in__Cong
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( A = B )
% 2.37/1.47 => ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xx = Xy )
% 2.37/1.47 => ( ( in @ Xx @ A )
% 2.37/1.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exuE3u_type,type,
% 2.37/1.47 exuE3u: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exuE3u,definition,
% 2.37/1.47 ( exuE3u
% 2.37/1.47 = ( ! [Xphi: $i > $o] :
% 2.37/1.47 ( ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 => ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xphi @ Xx )
% 2.37/1.47 => ( ( Xphi @ Xy )
% 2.37/1.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(exu__Cong_type,type,
% 2.37/1.47 exu__Cong: $o ).
% 2.37/1.47
% 2.37/1.47 thf(exu__Cong,definition,
% 2.37/1.47 ( exu__Cong
% 2.37/1.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.37/1.47 ( ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xx = Xy )
% 2.37/1.47 => ( ( Xphi @ Xx )
% 2.37/1.47 <=> ( Xpsi @ Xy ) ) )
% 2.37/1.47 => ( ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.47 <=> ( exu
% 2.37/1.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(emptyset__Cong_type,type,
% 2.37/1.47 emptyset__Cong: $o ).
% 2.37/1.47
% 2.37/1.47 thf(emptyset__Cong,definition,
% 2.37/1.47 ( emptyset__Cong
% 2.37/1.47 = ( emptyset = emptyset ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoin__Cong_type,type,
% 2.37/1.47 setadjoin__Cong: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setadjoin__Cong,definition,
% 2.37/1.47 ( setadjoin__Cong
% 2.37/1.47 = ( ! [Xx: $i,Xy: $i] :
% 2.37/1.47 ( ( Xx = Xy )
% 2.37/1.47 => ! [Xz: $i,Xu: $i] :
% 2.37/1.47 ( ( Xz = Xu )
% 2.37/1.47 => ( ( setadjoin @ Xx @ Xz )
% 2.37/1.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(powerset__Cong_type,type,
% 2.37/1.47 powerset__Cong: $o ).
% 2.37/1.47
% 2.37/1.47 thf(powerset__Cong,definition,
% 2.37/1.47 ( powerset__Cong
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( A = B )
% 2.37/1.47 => ( ( powerset @ A )
% 2.37/1.47 = ( powerset @ B ) ) ) ) ) ).
% 2.37/1.47
% 2.37/1.47 thf(setunion__Cong_type,type,
% 2.37/1.47 setunion__Cong: $o ).
% 2.37/1.47
% 2.37/1.47 thf(setunion__Cong,definition,
% 2.37/1.47 ( setunion__Cong
% 2.37/1.47 = ( ! [A: $i,B: $i] :
% 2.37/1.47 ( ( A = B )
% 2.37/1.47 => ( ( setunion @ A )
% 2.37/1.48 = ( setunion @ B ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(omega__Cong_type,type,
% 2.37/1.48 omega__Cong: $o ).
% 2.37/1.48
% 2.37/1.48 thf(omega__Cong,definition,
% 2.37/1.48 ( omega__Cong
% 2.37/1.48 = ( omega = omega ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(exuEu_type,type,
% 2.37/1.48 exuEu: $o ).
% 2.37/1.48
% 2.37/1.48 thf(exuEu,definition,
% 2.37/1.48 ( exuEu
% 2.37/1.48 = ( ! [Xphi: $i > $o] :
% 2.37/1.48 ( ( exu
% 2.37/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.48 => ! [Xx: $i,Xy: $i] :
% 2.37/1.48 ( ( Xphi @ Xx )
% 2.37/1.48 => ( ( Xphi @ Xy )
% 2.37/1.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(descr__Cong_type,type,
% 2.37/1.48 descr__Cong: $o ).
% 2.37/1.48
% 2.37/1.48 thf(descr__Cong,definition,
% 2.37/1.48 ( descr__Cong
% 2.37/1.48 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.37/1.48 ( ! [Xx: $i,Xy: $i] :
% 2.37/1.48 ( ( Xx = Xy )
% 2.37/1.48 => ( ( Xphi @ Xx )
% 2.37/1.48 <=> ( Xpsi @ Xy ) ) )
% 2.37/1.48 => ( ( exu
% 2.37/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.48 => ( ( exu
% 2.37/1.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.37/1.48 => ( ( descr
% 2.37/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.48 = ( descr
% 2.37/1.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(dsetconstr__Cong_type,type,
% 2.37/1.48 dsetconstr__Cong: $o ).
% 2.37/1.48
% 2.37/1.48 thf(dsetconstr__Cong,definition,
% 2.37/1.48 ( dsetconstr__Cong
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ( A = B )
% 2.37/1.48 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.37/1.48 ( ! [Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ! [Xy: $i] :
% 2.37/1.48 ( ( in @ Xy @ B )
% 2.37/1.48 => ( ( Xx = Xy )
% 2.37/1.48 => ( ( Xphi @ Xx )
% 2.37/1.48 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.37/1.48 => ( ( dsetconstr @ A
% 2.37/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.48 = ( dsetconstr @ B
% 2.37/1.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(subset_type,type,
% 2.37/1.48 subset: $i > $i > $o ).
% 2.37/1.48
% 2.37/1.48 thf(disjoint_type,type,
% 2.37/1.48 disjoint: $i > $i > $o ).
% 2.37/1.48
% 2.37/1.48 thf(setsmeet_type,type,
% 2.37/1.48 setsmeet: $i > $i > $o ).
% 2.37/1.48
% 2.37/1.48 thf(subsetI1_type,type,
% 2.37/1.48 subsetI1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(subsetI1,definition,
% 2.37/1.48 ( subsetI1
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ! [Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( in @ Xx @ B ) )
% 2.37/1.48 => ( subset @ A @ B ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(eqimpsubset2_type,type,
% 2.37/1.48 eqimpsubset2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(eqimpsubset2,definition,
% 2.37/1.48 ( eqimpsubset2
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ( A = B )
% 2.37/1.48 => ( subset @ B @ A ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(eqimpsubset1_type,type,
% 2.37/1.48 eqimpsubset1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(eqimpsubset1,definition,
% 2.37/1.48 ( eqimpsubset1
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ( A = B )
% 2.37/1.48 => ( subset @ A @ B ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(subsetI2_type,type,
% 2.37/1.48 subsetI2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(subsetI2,definition,
% 2.37/1.48 ( subsetI2
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ! [Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( in @ Xx @ B ) )
% 2.37/1.48 => ( subset @ A @ B ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(emptysetsubset_type,type,
% 2.37/1.48 emptysetsubset: $o ).
% 2.37/1.48
% 2.37/1.48 thf(emptysetsubset,definition,
% 2.37/1.48 ( emptysetsubset
% 2.37/1.48 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(subsetE_type,type,
% 2.37/1.48 subsetE: $o ).
% 2.37/1.48
% 2.37/1.48 thf(subsetE,definition,
% 2.37/1.48 ( subsetE
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( subset @ A @ B )
% 2.37/1.48 => ( ( in @ Xx @ A )
% 2.37/1.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(subsetE2_type,type,
% 2.37/1.48 subsetE2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(subsetE2,definition,
% 2.37/1.48 ( subsetE2
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( subset @ A @ B )
% 2.37/1.48 => ( ~ ( in @ Xx @ B )
% 2.37/1.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(notsubsetI_type,type,
% 2.37/1.48 notsubsetI: $o ).
% 2.37/1.48
% 2.37/1.48 thf(notsubsetI,definition,
% 2.37/1.48 ( notsubsetI
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( ~ ( in @ Xx @ B )
% 2.37/1.48 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(notequalI1_type,type,
% 2.37/1.48 notequalI1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(notequalI1,definition,
% 2.37/1.48 ( notequalI1
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ~ ( subset @ A @ B )
% 2.37/1.48 => ( A != B ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(notequalI2_type,type,
% 2.37/1.48 notequalI2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(notequalI2,definition,
% 2.37/1.48 ( notequalI2
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( ~ ( in @ Xx @ B )
% 2.37/1.48 => ( A != B ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(subsetRefl_type,type,
% 2.37/1.48 subsetRefl: $o ).
% 2.37/1.48
% 2.37/1.48 thf(subsetRefl,definition,
% 2.37/1.48 ( subsetRefl
% 2.37/1.48 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(subsetTrans_type,type,
% 2.37/1.48 subsetTrans: $o ).
% 2.37/1.48
% 2.37/1.48 thf(subsetTrans,definition,
% 2.37/1.48 ( subsetTrans
% 2.37/1.48 = ( ! [A: $i,B: $i,C: $i] :
% 2.37/1.48 ( ( subset @ A @ B )
% 2.37/1.48 => ( ( subset @ B @ C )
% 2.37/1.48 => ( subset @ A @ C ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setadjoinSub_type,type,
% 2.37/1.48 setadjoinSub: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setadjoinSub,definition,
% 2.37/1.48 ( setadjoinSub
% 2.37/1.48 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setadjoinSub2_type,type,
% 2.37/1.48 setadjoinSub2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setadjoinSub2,definition,
% 2.37/1.48 ( setadjoinSub2
% 2.37/1.48 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.37/1.48 ( ( subset @ A @ B )
% 2.37/1.48 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(subset2powerset_type,type,
% 2.37/1.48 subset2powerset: $o ).
% 2.37/1.48
% 2.37/1.48 thf(subset2powerset,definition,
% 2.37/1.48 ( subset2powerset
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ( subset @ A @ B )
% 2.37/1.48 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setextsub_type,type,
% 2.37/1.48 setextsub: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setextsub,definition,
% 2.37/1.48 ( setextsub
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ( subset @ A @ B )
% 2.37/1.48 => ( ( subset @ B @ A )
% 2.37/1.48 => ( A = B ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(subsetemptysetimpeq_type,type,
% 2.37/1.48 subsetemptysetimpeq: $o ).
% 2.37/1.48
% 2.37/1.48 thf(subsetemptysetimpeq,definition,
% 2.37/1.48 ( subsetemptysetimpeq
% 2.37/1.48 = ( ! [A: $i] :
% 2.37/1.48 ( ( subset @ A @ emptyset )
% 2.37/1.48 => ( A = emptyset ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(powersetI1_type,type,
% 2.37/1.48 powersetI1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(powersetI1,definition,
% 2.37/1.48 ( powersetI1
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ( subset @ B @ A )
% 2.37/1.48 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(powersetE1_type,type,
% 2.37/1.48 powersetE1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(powersetE1,definition,
% 2.37/1.48 ( powersetE1
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ( in @ B @ ( powerset @ A ) )
% 2.37/1.48 => ( subset @ B @ A ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(inPowerset_type,type,
% 2.37/1.48 inPowerset: $o ).
% 2.37/1.48
% 2.37/1.48 thf(inPowerset,definition,
% 2.37/1.48 ( inPowerset
% 2.37/1.48 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(powersetsubset_type,type,
% 2.37/1.48 powersetsubset: $o ).
% 2.37/1.48
% 2.37/1.48 thf(powersetsubset,definition,
% 2.37/1.48 ( powersetsubset
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ( subset @ A @ B )
% 2.37/1.48 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(sepInPowerset_type,type,
% 2.37/1.48 sepInPowerset: $o ).
% 2.37/1.48
% 2.37/1.48 thf(sepInPowerset,definition,
% 2.37/1.48 ( sepInPowerset
% 2.37/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.48 ( in
% 2.37/1.48 @ ( dsetconstr @ A
% 2.37/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.48 @ ( powerset @ A ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(sepSubset_type,type,
% 2.37/1.48 sepSubset: $o ).
% 2.37/1.48
% 2.37/1.48 thf(sepSubset,definition,
% 2.37/1.48 ( sepSubset
% 2.37/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.48 ( subset
% 2.37/1.48 @ ( dsetconstr @ A
% 2.37/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.48 @ A ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(binunion_type,type,
% 2.37/1.48 binunion: $i > $i > $i ).
% 2.37/1.48
% 2.37/1.48 thf(binunionIL_type,type,
% 2.37/1.48 binunionIL: $o ).
% 2.37/1.48
% 2.37/1.48 thf(binunionIL,definition,
% 2.37/1.48 ( binunionIL
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(upairset2IR_type,type,
% 2.37/1.48 upairset2IR: $o ).
% 2.37/1.48
% 2.37/1.48 thf(upairset2IR,definition,
% 2.37/1.48 ( upairset2IR
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(binunionIR_type,type,
% 2.37/1.48 binunionIR: $o ).
% 2.37/1.48
% 2.37/1.48 thf(binunionIR,definition,
% 2.37/1.48 ( binunionIR
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ B )
% 2.37/1.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(binunionEcases_type,type,
% 2.37/1.48 binunionEcases: $o ).
% 2.37/1.48
% 2.37/1.48 thf(binunionEcases,definition,
% 2.37/1.48 ( binunionEcases
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.37/1.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.37/1.48 => ( ( ( in @ Xx @ A )
% 2.37/1.48 => Xphi )
% 2.37/1.48 => ( ( ( in @ Xx @ B )
% 2.37/1.48 => Xphi )
% 2.37/1.48 => Xphi ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(binunionE_type,type,
% 2.37/1.48 binunionE: $o ).
% 2.37/1.48
% 2.37/1.48 thf(binunionE,definition,
% 2.37/1.48 ( binunionE
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.37/1.48 => ( ( in @ Xx @ A )
% 2.37/1.48 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(binunionLsub_type,type,
% 2.37/1.48 binunionLsub: $o ).
% 2.37/1.48
% 2.37/1.48 thf(binunionLsub,definition,
% 2.37/1.48 ( binunionLsub
% 2.37/1.48 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(binunionRsub_type,type,
% 2.37/1.48 binunionRsub: $o ).
% 2.37/1.48
% 2.37/1.48 thf(binunionRsub,definition,
% 2.37/1.48 ( binunionRsub
% 2.37/1.48 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(binintersect_type,type,
% 2.37/1.48 binintersect: $i > $i > $i ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectI_type,type,
% 2.37/1.48 binintersectI: $o ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectI,definition,
% 2.37/1.48 ( binintersectI
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( ( in @ Xx @ B )
% 2.37/1.48 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectSubset5_type,type,
% 2.37/1.48 binintersectSubset5: $o ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectSubset5,definition,
% 2.37/1.48 ( binintersectSubset5
% 2.37/1.48 = ( ! [A: $i,B: $i,C: $i] :
% 2.37/1.48 ( ( subset @ C @ A )
% 2.37/1.48 => ( ( subset @ C @ B )
% 2.37/1.48 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectEL_type,type,
% 2.37/1.48 binintersectEL: $o ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectEL,definition,
% 2.37/1.48 ( binintersectEL
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.37/1.48 => ( in @ Xx @ A ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectLsub_type,type,
% 2.37/1.48 binintersectLsub: $o ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectLsub,definition,
% 2.37/1.48 ( binintersectLsub
% 2.37/1.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectSubset2_type,type,
% 2.37/1.48 binintersectSubset2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectSubset2,definition,
% 2.37/1.48 ( binintersectSubset2
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ( subset @ A @ B )
% 2.37/1.48 => ( ( binintersect @ A @ B )
% 2.37/1.48 = A ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectSubset3_type,type,
% 2.37/1.48 binintersectSubset3: $o ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectSubset3,definition,
% 2.37/1.48 ( binintersectSubset3
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ( ( binintersect @ A @ B )
% 2.37/1.48 = B )
% 2.37/1.48 => ( subset @ B @ A ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectER_type,type,
% 2.37/1.48 binintersectER: $o ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectER,definition,
% 2.37/1.48 ( binintersectER
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.37/1.48 => ( in @ Xx @ B ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(disjointsetsI1_type,type,
% 2.37/1.48 disjointsetsI1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(disjointsetsI1,definition,
% 2.37/1.48 ( disjointsetsI1
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ~ ? [Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 & ( in @ Xx @ B ) )
% 2.37/1.48 => ( ( binintersect @ A @ B )
% 2.37/1.48 = emptyset ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectRsub_type,type,
% 2.37/1.48 binintersectRsub: $o ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectRsub,definition,
% 2.37/1.48 ( binintersectRsub
% 2.37/1.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectSubset4_type,type,
% 2.37/1.48 binintersectSubset4: $o ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectSubset4,definition,
% 2.37/1.48 ( binintersectSubset4
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ( subset @ B @ A )
% 2.37/1.48 => ( ( binintersect @ A @ B )
% 2.37/1.48 = B ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectSubset1_type,type,
% 2.37/1.48 binintersectSubset1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(binintersectSubset1,definition,
% 2.37/1.48 ( binintersectSubset1
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ( ( binintersect @ A @ B )
% 2.37/1.48 = A )
% 2.37/1.48 => ( subset @ A @ B ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(bs114d_type,type,
% 2.37/1.48 bs114d: $o ).
% 2.37/1.48
% 2.37/1.48 thf(bs114d,definition,
% 2.37/1.48 ( bs114d
% 2.37/1.48 = ( ! [A: $i,B: $i,C: $i] :
% 2.37/1.48 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.37/1.48 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(regular_type,type,
% 2.37/1.48 regular: $i > $o ).
% 2.37/1.48
% 2.37/1.48 thf(setminus_type,type,
% 2.37/1.48 setminus: $i > $i > $i ).
% 2.37/1.48
% 2.37/1.48 thf(setminusI_type,type,
% 2.37/1.48 setminusI: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setminusI,definition,
% 2.37/1.48 ( setminusI
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( ~ ( in @ Xx @ B )
% 2.37/1.48 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setminusEL_type,type,
% 2.37/1.48 setminusEL: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setminusEL,definition,
% 2.37/1.48 ( setminusEL
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.37/1.48 => ( in @ Xx @ A ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setminusER_type,type,
% 2.37/1.48 setminusER: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setminusER,definition,
% 2.37/1.48 ( setminusER
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.37/1.48 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setminusSubset2_type,type,
% 2.37/1.48 setminusSubset2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setminusSubset2,definition,
% 2.37/1.48 ( setminusSubset2
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ( subset @ A @ B )
% 2.37/1.48 => ( ( setminus @ A @ B )
% 2.37/1.48 = emptyset ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setminusERneg_type,type,
% 2.37/1.48 setminusERneg: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setminusERneg,definition,
% 2.37/1.48 ( setminusERneg
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.37/1.48 => ( ( in @ Xx @ A )
% 2.37/1.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setminusELneg_type,type,
% 2.37/1.48 setminusELneg: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setminusELneg,definition,
% 2.37/1.48 ( setminusELneg
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.37/1.48 => ( ~ ( in @ Xx @ B )
% 2.37/1.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setminusILneg_type,type,
% 2.37/1.48 setminusILneg: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setminusILneg,definition,
% 2.37/1.48 ( setminusILneg
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ~ ( in @ Xx @ A )
% 2.37/1.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setminusIRneg_type,type,
% 2.37/1.48 setminusIRneg: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setminusIRneg,definition,
% 2.37/1.48 ( setminusIRneg
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ B )
% 2.37/1.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setminusLsub_type,type,
% 2.37/1.48 setminusLsub: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setminusLsub,definition,
% 2.37/1.48 ( setminusLsub
% 2.37/1.48 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setminusSubset1_type,type,
% 2.37/1.48 setminusSubset1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setminusSubset1,definition,
% 2.37/1.48 ( setminusSubset1
% 2.37/1.48 = ( ! [A: $i,B: $i] :
% 2.37/1.48 ( ( ( setminus @ A @ B )
% 2.37/1.48 = emptyset )
% 2.37/1.48 => ( subset @ A @ B ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(symdiff_type,type,
% 2.37/1.48 symdiff: $i > $i > $i ).
% 2.37/1.48
% 2.37/1.48 thf(symdiffE_type,type,
% 2.37/1.48 symdiffE: $o ).
% 2.37/1.48
% 2.37/1.48 thf(symdiffE,definition,
% 2.37/1.48 ( symdiffE
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.37/1.48 => ! [Xphi: $o] :
% 2.37/1.48 ( ( ( in @ Xx @ A )
% 2.37/1.48 => ( ~ ( in @ Xx @ B )
% 2.37/1.48 => Xphi ) )
% 2.37/1.48 => ( ( ~ ( in @ Xx @ A )
% 2.37/1.48 => ( ( in @ Xx @ B )
% 2.37/1.48 => Xphi ) )
% 2.37/1.48 => Xphi ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(symdiffI1_type,type,
% 2.37/1.48 symdiffI1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(symdiffI1,definition,
% 2.37/1.48 ( symdiffI1
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( ~ ( in @ Xx @ B )
% 2.37/1.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(symdiffI2_type,type,
% 2.37/1.48 symdiffI2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(symdiffI2,definition,
% 2.37/1.48 ( symdiffI2
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ~ ( in @ Xx @ A )
% 2.37/1.48 => ( ( in @ Xx @ B )
% 2.37/1.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(symdiffIneg1_type,type,
% 2.37/1.48 symdiffIneg1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(symdiffIneg1,definition,
% 2.37/1.48 ( symdiffIneg1
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( ( in @ Xx @ B )
% 2.37/1.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(symdiffIneg2_type,type,
% 2.37/1.48 symdiffIneg2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(symdiffIneg2,definition,
% 2.37/1.48 ( symdiffIneg2
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ~ ( in @ Xx @ A )
% 2.37/1.48 => ( ~ ( in @ Xx @ B )
% 2.37/1.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(iskpair_type,type,
% 2.37/1.48 iskpair: $i > $o ).
% 2.37/1.48
% 2.37/1.48 thf(secondinupair_type,type,
% 2.37/1.48 secondinupair: $o ).
% 2.37/1.48
% 2.37/1.48 thf(secondinupair,definition,
% 2.37/1.48 ( secondinupair
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairIL_type,type,
% 2.37/1.48 setukpairIL: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairIL,definition,
% 2.37/1.48 ( setukpairIL
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairIR_type,type,
% 2.37/1.48 setukpairIR: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairIR,definition,
% 2.37/1.48 ( setukpairIR
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(kpairiskpair_type,type,
% 2.37/1.48 kpairiskpair: $o ).
% 2.37/1.48
% 2.37/1.48 thf(kpairiskpair,definition,
% 2.37/1.48 ( kpairiskpair
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(kpair_type,type,
% 2.37/1.48 kpair: $i > $i > $i ).
% 2.37/1.48
% 2.37/1.48 thf(kpairp_type,type,
% 2.37/1.48 kpairp: $o ).
% 2.37/1.48
% 2.37/1.48 thf(kpairp,definition,
% 2.37/1.48 ( kpairp
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(cartprod_type,type,
% 2.37/1.48 cartprod: $i > $i > $i ).
% 2.37/1.48
% 2.37/1.48 thf(singletonsubset_type,type,
% 2.37/1.48 singletonsubset: $o ).
% 2.37/1.48
% 2.37/1.48 thf(singletonsubset,definition,
% 2.37/1.48 ( singletonsubset
% 2.37/1.48 = ( ! [A: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(singletoninpowerset_type,type,
% 2.37/1.48 singletoninpowerset: $o ).
% 2.37/1.48
% 2.37/1.48 thf(singletoninpowerset,definition,
% 2.37/1.48 ( singletoninpowerset
% 2.37/1.48 = ( ! [A: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(singletoninpowunion_type,type,
% 2.37/1.48 singletoninpowunion: $o ).
% 2.37/1.48
% 2.37/1.48 thf(singletoninpowunion,definition,
% 2.37/1.48 ( singletoninpowunion
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(upairset2E_type,type,
% 2.37/1.48 upairset2E: $o ).
% 2.37/1.48
% 2.37/1.48 thf(upairset2E,definition,
% 2.37/1.48 ( upairset2E
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.37/1.48 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.37/1.48 => ( ( Xz = Xx )
% 2.37/1.48 | ( Xz = Xy ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(upairsubunion_type,type,
% 2.37/1.48 upairsubunion: $o ).
% 2.37/1.48
% 2.37/1.48 thf(upairsubunion,definition,
% 2.37/1.48 ( upairsubunion
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ! [Xy: $i] :
% 2.37/1.48 ( ( in @ Xy @ B )
% 2.37/1.48 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(upairinpowunion_type,type,
% 2.37/1.48 upairinpowunion: $o ).
% 2.37/1.48
% 2.37/1.48 thf(upairinpowunion,definition,
% 2.37/1.48 ( upairinpowunion
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ! [Xy: $i] :
% 2.37/1.48 ( ( in @ Xy @ B )
% 2.37/1.48 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(ubforcartprodlem1_type,type,
% 2.37/1.48 ubforcartprodlem1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(ubforcartprodlem1,definition,
% 2.37/1.48 ( ubforcartprodlem1
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ! [Xy: $i] :
% 2.37/1.48 ( ( in @ Xy @ B )
% 2.37/1.48 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(ubforcartprodlem2_type,type,
% 2.37/1.48 ubforcartprodlem2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(ubforcartprodlem2,definition,
% 2.37/1.48 ( ubforcartprodlem2
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ! [Xy: $i] :
% 2.37/1.48 ( ( in @ Xy @ B )
% 2.37/1.48 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(ubforcartprodlem3_type,type,
% 2.37/1.48 ubforcartprodlem3: $o ).
% 2.37/1.48
% 2.37/1.48 thf(ubforcartprodlem3,definition,
% 2.37/1.48 ( ubforcartprodlem3
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ! [Xy: $i] :
% 2.37/1.48 ( ( in @ Xy @ B )
% 2.37/1.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodpairin_type,type,
% 2.37/1.48 cartprodpairin: $o ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodpairin,definition,
% 2.37/1.48 ( cartprodpairin
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ! [Xy: $i] :
% 2.37/1.48 ( ( in @ Xy @ B )
% 2.37/1.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodmempair1_type,type,
% 2.37/1.48 cartprodmempair1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodmempair1,definition,
% 2.37/1.48 ( cartprodmempair1
% 2.37/1.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.37/1.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.37/1.48 => ? [Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 & ? [Xy: $i] :
% 2.37/1.48 ( ( in @ Xy @ B )
% 2.37/1.48 & ( Xu
% 2.37/1.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodmempair_type,type,
% 2.37/1.48 cartprodmempair: $o ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodmempair,definition,
% 2.37/1.48 ( cartprodmempair
% 2.37/1.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.37/1.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.37/1.48 => ( iskpair @ Xu ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setunionE2_type,type,
% 2.37/1.48 setunionE2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setunionE2,definition,
% 2.37/1.48 ( setunionE2
% 2.37/1.48 = ( ! [A: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ ( setunion @ A ) )
% 2.37/1.48 => ? [X: $i] :
% 2.37/1.48 ( ( in @ X @ A )
% 2.37/1.48 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setunionsingleton1_type,type,
% 2.37/1.48 setunionsingleton1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setunionsingleton1,definition,
% 2.37/1.48 ( setunionsingleton1
% 2.37/1.48 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setunionsingleton2_type,type,
% 2.37/1.48 setunionsingleton2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setunionsingleton2,definition,
% 2.37/1.48 ( setunionsingleton2
% 2.37/1.48 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setunionsingleton_type,type,
% 2.37/1.48 setunionsingleton: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setunionsingleton,definition,
% 2.37/1.48 ( setunionsingleton
% 2.37/1.48 = ( ! [Xx: $i] :
% 2.37/1.48 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.37/1.48 = Xx ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(singleton_type,type,
% 2.37/1.48 singleton: $i > $o ).
% 2.37/1.48
% 2.37/1.48 thf(singleton,definition,
% 2.37/1.48 ( singleton
% 2.37/1.48 = ( ^ [A: $i] :
% 2.37/1.48 ? [Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 & ( A
% 2.37/1.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(singletonprop_type,type,
% 2.37/1.48 singletonprop: $o ).
% 2.37/1.48
% 2.37/1.48 thf(singletonprop,definition,
% 2.37/1.48 ( singletonprop
% 2.37/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.48 ( ! [Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ! [Xy: $i] :
% 2.37/1.48 ( ( in @ Xy @ A )
% 2.37/1.48 => ( ( Xphi @ Xx )
% 2.37/1.48 => ( ( Xphi @ Xy )
% 2.37/1.48 => ( Xx = Xy ) ) ) ) )
% 2.37/1.48 => ( ? [Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 & ( Xphi @ Xx ) )
% 2.37/1.48 => ( singleton
% 2.37/1.48 @ ( dsetconstr @ A
% 2.37/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(ex1_type,type,
% 2.37/1.48 ex1: $i > ( $i > $o ) > $o ).
% 2.37/1.48
% 2.37/1.48 thf(ex1,definition,
% 2.37/1.48 ( ex1
% 2.37/1.48 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.37/1.48 ( singleton
% 2.37/1.48 @ ( dsetconstr @ A
% 2.37/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(ex1E1_type,type,
% 2.37/1.48 ex1E1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(ex1E1,definition,
% 2.37/1.48 ( ex1E1
% 2.37/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.48 ( ( ex1 @ A
% 2.37/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.37/1.48 => ? [Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(ex1I_type,type,
% 2.37/1.48 ex1I: $o ).
% 2.37/1.48
% 2.37/1.48 thf(ex1I,definition,
% 2.37/1.48 ( ex1I
% 2.37/1.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( ( Xphi @ Xx )
% 2.37/1.48 => ( ! [Xy: $i] :
% 2.37/1.48 ( ( in @ Xy @ A )
% 2.37/1.48 => ( ( Xphi @ Xy )
% 2.37/1.48 => ( Xy = Xx ) ) )
% 2.37/1.48 => ( ex1 @ A
% 2.37/1.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(ex1I2_type,type,
% 2.37/1.48 ex1I2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(ex1I2,definition,
% 2.37/1.48 ( ex1I2
% 2.37/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.37/1.48 ( ! [Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ! [Xy: $i] :
% 2.37/1.48 ( ( in @ Xy @ A )
% 2.37/1.48 => ( ( Xphi @ Xx )
% 2.37/1.48 => ( ( Xphi @ Xy )
% 2.37/1.48 => ( Xx = Xy ) ) ) ) )
% 2.37/1.48 => ( ? [Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 & ( Xphi @ Xx ) )
% 2.37/1.48 => ( ex1 @ A
% 2.37/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(singletonsuniq_type,type,
% 2.37/1.48 singletonsuniq: $o ).
% 2.37/1.48
% 2.37/1.48 thf(singletonsuniq,definition,
% 2.37/1.48 ( singletonsuniq
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.37/1.48 ( ( ( setadjoin @ Xx @ emptyset )
% 2.37/1.48 = ( setadjoin @ Xy @ emptyset ) )
% 2.37/1.48 => ( Xx = Xy ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(atmost1p_type,type,
% 2.37/1.48 atmost1p: $i > $o ).
% 2.37/1.48
% 2.37/1.48 thf(atleast2p_type,type,
% 2.37/1.48 atleast2p: $i > $o ).
% 2.37/1.48
% 2.37/1.48 thf(atmost2p_type,type,
% 2.37/1.48 atmost2p: $i > $o ).
% 2.37/1.48
% 2.37/1.48 thf(upairsetp_type,type,
% 2.37/1.48 upairsetp: $i > $o ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairinjL1_type,type,
% 2.37/1.48 setukpairinjL1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairinjL1,definition,
% 2.37/1.48 ( setukpairinjL1
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.37/1.48 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.37/1.48 => ( Xx = Xz ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(kfstsingleton_type,type,
% 2.37/1.48 kfstsingleton: $o ).
% 2.37/1.48
% 2.37/1.48 thf(kfstsingleton,definition,
% 2.37/1.48 ( kfstsingleton
% 2.37/1.48 = ( ! [Xu: $i] :
% 2.37/1.48 ( ( iskpair @ Xu )
% 2.37/1.48 => ( singleton
% 2.37/1.48 @ ( dsetconstr @ ( setunion @ Xu )
% 2.37/1.48 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(theprop_type,type,
% 2.37/1.48 theprop: $o ).
% 2.37/1.48
% 2.37/1.48 thf(theprop,definition,
% 2.37/1.48 ( theprop
% 2.37/1.48 = ( ! [X: $i] :
% 2.37/1.48 ( ( singleton @ X )
% 2.37/1.48 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(kfst_type,type,
% 2.37/1.48 kfst: $i > $i ).
% 2.37/1.48
% 2.37/1.48 thf(kfstpairEq_type,type,
% 2.37/1.48 kfstpairEq: $o ).
% 2.37/1.48
% 2.37/1.48 thf(kfstpairEq,definition,
% 2.37/1.48 ( kfstpairEq
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.37/1.48 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.37/1.48 = Xx ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodfstin_type,type,
% 2.37/1.48 cartprodfstin: $o ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodfstin,definition,
% 2.37/1.48 ( cartprodfstin
% 2.37/1.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.37/1.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.37/1.48 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairinjL2_type,type,
% 2.37/1.48 setukpairinjL2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairinjL2,definition,
% 2.37/1.48 ( setukpairinjL2
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.37/1.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.37/1.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.37/1.48 => ( Xx = Xz ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairinjL_type,type,
% 2.37/1.48 setukpairinjL: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairinjL,definition,
% 2.37/1.48 ( setukpairinjL
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.37/1.48 ( ( ( kpair @ Xx @ Xy )
% 2.37/1.48 = ( kpair @ Xz @ Xu ) )
% 2.37/1.48 => ( Xx = Xz ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairinjR11_type,type,
% 2.37/1.48 setukpairinjR11: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairinjR11,definition,
% 2.37/1.48 ( setukpairinjR11
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.37/1.48 ( ( Xx = Xy )
% 2.37/1.48 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.37/1.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairinjR12_type,type,
% 2.37/1.48 setukpairinjR12: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairinjR12,definition,
% 2.37/1.48 ( setukpairinjR12
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.37/1.48 ( ( Xx = Xy )
% 2.37/1.48 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.37/1.48 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairinjR1_type,type,
% 2.37/1.48 setukpairinjR1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairinjR1,definition,
% 2.37/1.48 ( setukpairinjR1
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.37/1.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.37/1.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.37/1.48 => ( ( Xz = Xu )
% 2.37/1.48 => ( Xy = Xu ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(upairequniteq_type,type,
% 2.37/1.48 upairequniteq: $o ).
% 2.37/1.48
% 2.37/1.48 thf(upairequniteq,definition,
% 2.37/1.48 ( upairequniteq
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.37/1.48 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.37/1.48 = ( setadjoin @ Xz @ emptyset ) )
% 2.37/1.48 => ( Xx = Xy ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairinjR2_type,type,
% 2.37/1.48 setukpairinjR2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairinjR2,definition,
% 2.37/1.48 ( setukpairinjR2
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.37/1.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.37/1.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.37/1.48 => ( Xy = Xu ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairinjR_type,type,
% 2.37/1.48 setukpairinjR: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setukpairinjR,definition,
% 2.37/1.48 ( setukpairinjR
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.37/1.48 ( ( ( kpair @ Xx @ Xy )
% 2.37/1.48 = ( kpair @ Xz @ Xu ) )
% 2.37/1.48 => ( Xy = Xu ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(ksndsingleton_type,type,
% 2.37/1.48 ksndsingleton: $o ).
% 2.37/1.48
% 2.37/1.48 thf(ksndsingleton,definition,
% 2.37/1.48 ( ksndsingleton
% 2.37/1.48 = ( ! [Xu: $i] :
% 2.37/1.48 ( ( iskpair @ Xu )
% 2.37/1.48 => ( singleton
% 2.37/1.48 @ ( dsetconstr @ ( setunion @ Xu )
% 2.37/1.48 @ ^ [Xx: $i] :
% 2.37/1.48 ( Xu
% 2.37/1.48 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(ksnd_type,type,
% 2.37/1.48 ksnd: $i > $i ).
% 2.37/1.48
% 2.37/1.48 thf(ksndpairEq_type,type,
% 2.37/1.48 ksndpairEq: $o ).
% 2.37/1.48
% 2.37/1.48 thf(ksndpairEq,definition,
% 2.37/1.48 ( ksndpairEq
% 2.37/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.37/1.48 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.37/1.48 = Xy ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(kpairsurjEq_type,type,
% 2.37/1.48 kpairsurjEq: $o ).
% 2.37/1.48
% 2.37/1.48 thf(kpairsurjEq,definition,
% 2.37/1.48 ( kpairsurjEq
% 2.37/1.48 = ( ! [Xu: $i] :
% 2.37/1.48 ( ( iskpair @ Xu )
% 2.37/1.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.37/1.48 = Xu ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodsndin_type,type,
% 2.37/1.48 cartprodsndin: $o ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodsndin,definition,
% 2.37/1.48 ( cartprodsndin
% 2.37/1.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.37/1.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.37/1.48 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodpairmemEL_type,type,
% 2.37/1.48 cartprodpairmemEL: $o ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodpairmemEL,definition,
% 2.37/1.48 ( cartprodpairmemEL
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.37/1.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.37/1.48 => ( in @ Xx @ A ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodpairmemER_type,type,
% 2.37/1.48 cartprodpairmemER: $o ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodpairmemER,definition,
% 2.37/1.48 ( cartprodpairmemER
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.37/1.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.37/1.48 => ( in @ Xy @ B ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodmempaircEq_type,type,
% 2.37/1.48 cartprodmempaircEq: $o ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodmempaircEq,definition,
% 2.37/1.48 ( cartprodmempaircEq
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ! [Xy: $i] :
% 2.37/1.48 ( ( in @ Xy @ B )
% 2.37/1.48 => ( ( kpair @ Xx @ Xy )
% 2.37/1.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodfstpairEq_type,type,
% 2.37/1.48 cartprodfstpairEq: $o ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodfstpairEq,definition,
% 2.37/1.48 ( cartprodfstpairEq
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ! [Xy: $i] :
% 2.37/1.48 ( ( in @ Xy @ B )
% 2.37/1.48 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.37/1.48 = Xx ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodsndpairEq_type,type,
% 2.37/1.48 cartprodsndpairEq: $o ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodsndpairEq,definition,
% 2.37/1.48 ( cartprodsndpairEq
% 2.37/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ! [Xy: $i] :
% 2.37/1.48 ( ( in @ Xy @ B )
% 2.37/1.48 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.37/1.48 = Xy ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodpairsurjEq_type,type,
% 2.37/1.48 cartprodpairsurjEq: $o ).
% 2.37/1.48
% 2.37/1.48 thf(cartprodpairsurjEq,definition,
% 2.37/1.48 ( cartprodpairsurjEq
% 2.37/1.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.37/1.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.37/1.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.37/1.48 = Xu ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(breln_type,type,
% 2.37/1.48 breln: $i > $i > $i > $o ).
% 2.37/1.48
% 2.37/1.48 thf(breln,definition,
% 2.37/1.48 ( breln
% 2.37/1.48 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(dpsetconstr_type,type,
% 2.37/1.48 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.37/1.48
% 2.37/1.48 thf(dpsetconstrI_type,type,
% 2.37/1.48 dpsetconstrI: $o ).
% 2.37/1.48
% 2.37/1.48 thf(dpsetconstrI,definition,
% 2.37/1.48 ( dpsetconstrI
% 2.37/1.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ! [Xy: $i] :
% 2.37/1.48 ( ( in @ Xy @ B )
% 2.37/1.48 => ( ( Xphi @ Xx @ Xy )
% 2.37/1.48 => ( in @ ( kpair @ Xx @ Xy )
% 2.37/1.48 @ ( dpsetconstr @ A @ B
% 2.37/1.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(dpsetconstrSub_type,type,
% 2.37/1.48 dpsetconstrSub: $o ).
% 2.37/1.48
% 2.37/1.48 thf(dpsetconstrSub,definition,
% 2.37/1.48 ( dpsetconstrSub
% 2.37/1.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.37/1.48 ( subset
% 2.37/1.48 @ ( dpsetconstr @ A @ B
% 2.37/1.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.37/1.48 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(setOfPairsIsBReln_type,type,
% 2.37/1.48 setOfPairsIsBReln: $o ).
% 2.37/1.48
% 2.37/1.48 thf(setOfPairsIsBReln,definition,
% 2.37/1.48 ( setOfPairsIsBReln
% 2.37/1.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.37/1.48 ( breln @ A @ B
% 2.37/1.48 @ ( dpsetconstr @ A @ B
% 2.37/1.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(dpsetconstrERa_type,type,
% 2.37/1.48 dpsetconstrERa: $o ).
% 2.37/1.48
% 2.37/1.48 thf(dpsetconstrERa,definition,
% 2.37/1.48 ( dpsetconstrERa
% 2.37/1.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ! [Xy: $i] :
% 2.37/1.48 ( ( in @ Xy @ B )
% 2.37/1.48 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.37/1.48 @ ( dpsetconstr @ A @ B
% 2.37/1.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.37/1.48 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(dpsetconstrEL1_type,type,
% 2.37/1.48 dpsetconstrEL1: $o ).
% 2.37/1.48
% 2.37/1.48 thf(dpsetconstrEL1,definition,
% 2.37/1.48 ( dpsetconstrEL1
% 2.37/1.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.37/1.48 ( ( in @ ( kpair @ Xx @ Xy )
% 2.37/1.48 @ ( dpsetconstr @ A @ B
% 2.37/1.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.37/1.48 => ( in @ Xx @ A ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(dpsetconstrEL2_type,type,
% 2.37/1.48 dpsetconstrEL2: $o ).
% 2.37/1.48
% 2.37/1.48 thf(dpsetconstrEL2,definition,
% 2.37/1.48 ( dpsetconstrEL2
% 2.37/1.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.37/1.48 ( ( in @ ( kpair @ Xx @ Xy )
% 2.37/1.48 @ ( dpsetconstr @ A @ B
% 2.37/1.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.37/1.48 => ( in @ Xy @ B ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(dpsetconstrER_type,type,
% 2.37/1.48 dpsetconstrER: $o ).
% 2.37/1.48
% 2.37/1.48 thf(dpsetconstrER,definition,
% 2.37/1.48 ( dpsetconstrER
% 2.37/1.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.37/1.48 ( ( in @ ( kpair @ Xx @ Xy )
% 2.37/1.48 @ ( dpsetconstr @ A @ B
% 2.37/1.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.37/1.48 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(func_type,type,
% 2.37/1.48 func: $i > $i > $i > $o ).
% 2.37/1.48
% 2.37/1.48 thf(func,definition,
% 2.37/1.48 ( func
% 2.37/1.48 = ( ^ [A: $i,B: $i,R: $i] :
% 2.37/1.48 ( ( breln @ A @ B @ R )
% 2.37/1.48 & ! [Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( ex1 @ B
% 2.37/1.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(funcSet_type,type,
% 2.37/1.48 funcSet: $i > $i > $i ).
% 2.37/1.48
% 2.37/1.48 thf(funcSet,definition,
% 2.37/1.48 ( funcSet
% 2.37/1.48 = ( ^ [A: $i,B: $i] :
% 2.37/1.48 ( dsetconstr @ ( powerset @ ( cartprod @ A @ B ) )
% 2.37/1.48 @ ^ [Xf: $i] : ( func @ A @ B @ Xf ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(funcImageSingleton_type,type,
% 2.37/1.48 funcImageSingleton: $o ).
% 2.37/1.48
% 2.37/1.48 thf(funcImageSingleton,definition,
% 2.37/1.48 ( funcImageSingleton
% 2.37/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.37/1.48 ( ( func @ A @ B @ Xf )
% 2.37/1.48 => ! [Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( singleton
% 2.37/1.48 @ ( dsetconstr @ B
% 2.37/1.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(apProp_type,type,
% 2.37/1.48 apProp: $o ).
% 2.37/1.48
% 2.37/1.48 thf(apProp,definition,
% 2.37/1.48 ( apProp
% 2.37/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.37/1.48 ( ( func @ A @ B @ Xf )
% 2.37/1.48 => ! [Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( in
% 2.37/1.48 @ ( setunion
% 2.37/1.48 @ ( dsetconstr @ B
% 2.37/1.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.37/1.48 @ B ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(ap_type,type,
% 2.37/1.48 ap: $i > $i > $i > $i > $i ).
% 2.37/1.48
% 2.37/1.48 thf(ap,definition,
% 2.37/1.48 ( ap
% 2.37/1.48 = ( ^ [A: $i,B: $i,Xf: $i,Xx: $i] :
% 2.37/1.48 ( setunion
% 2.37/1.48 @ ( dsetconstr @ B
% 2.37/1.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(app_type,type,
% 2.37/1.48 app: $o ).
% 2.37/1.48
% 2.37/1.48 thf(app,definition,
% 2.37/1.48 ( app
% 2.37/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.37/1.48 ( ( func @ A @ B @ Xf )
% 2.37/1.48 => ! [Xx: $i] :
% 2.37/1.48 ( ( in @ Xx @ A )
% 2.37/1.48 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.37/1.48
% 2.37/1.48 thf(infuncsetfunc,conjecture,
% 2.37/1.48 ( setextAx
% 2.37/1.48 => ( emptysetAx
% 2.37/1.48 => ( setadjoinAx
% 2.37/1.48 => ( powersetAx
% 2.37/1.48 => ( setunionAx
% 2.37/1.48 => ( omega0Ax
% 2.37/1.48 => ( omegaSAx
% 2.37/1.48 => ( omegaIndAx
% 2.37/1.48 => ( replAx
% 2.37/1.48 => ( foundationAx
% 2.37/1.48 => ( wellorderingAx
% 2.37/1.48 => ( descrp
% 2.37/1.48 => ( dsetconstrI
% 2.37/1.48 => ( dsetconstrEL
% 2.37/1.48 => ( dsetconstrER
% 2.37/1.48 => ( exuE1
% 2.37/1.48 => ( prop2setE
% 2.37/1.48 => ( emptysetE
% 2.37/1.48 => ( emptysetimpfalse
% 2.37/1.48 => ( notinemptyset
% 2.37/1.48 => ( exuE3e
% 2.37/1.48 => ( setext
% 2.37/1.48 => ( emptyI
% 2.37/1.48 => ( noeltsimpempty
% 2.37/1.48 => ( setbeta
% 2.37/1.48 => ( nonemptyE1
% 2.37/1.48 => ( nonemptyI
% 2.37/1.48 => ( nonemptyI1
% 2.37/1.48 => ( setadjoinIL
% 2.37/1.48 => ( emptyinunitempty
% 2.37/1.48 => ( setadjoinIR
% 2.37/1.48 => ( setadjoinE
% 2.37/1.48 => ( setadjoinOr
% 2.37/1.48 => ( setoftrueEq
% 2.37/1.48 => ( powersetI
% 2.37/1.48 => ( emptyinPowerset
% 2.37/1.48 => ( emptyInPowerset
% 2.37/1.48 => ( powersetE
% 2.37/1.48 => ( setunionI
% 2.37/1.48 => ( setunionE
% 2.37/1.48 => ( subPowSU
% 2.37/1.48 => ( exuE2
% 2.37/1.48 => ( nonemptyImpWitness
% 2.37/1.48 => ( uniqinunit
% 2.37/1.48 => ( notinsingleton
% 2.37/1.48 => ( eqinunit
% 2.37/1.48 => ( singletonsswitch
% 2.37/1.48 => ( upairsetE
% 2.37/1.48 => ( upairsetIL
% 2.37/1.48 => ( upairsetIR
% 2.37/1.48 => ( emptyE1
% 2.37/1.48 => ( vacuousDall
% 2.37/1.48 => ( quantDeMorgan1
% 2.37/1.48 => ( quantDeMorgan2
% 2.37/1.48 => ( quantDeMorgan3
% 2.37/1.48 => ( quantDeMorgan4
% 2.37/1.48 => ( prop2setI
% 2.37/1.48 => ( prop2set2propI
% 2.37/1.48 => ( notdexE
% 2.37/1.48 => ( notdallE
% 2.37/1.48 => ( exuI1
% 2.37/1.48 => ( exuI3
% 2.37/1.48 => ( exuI2
% 2.37/1.48 => ( inCongP
% 2.37/1.48 => ( in__Cong
% 2.37/1.48 => ( exuE3u
% 2.37/1.48 => ( exu__Cong
% 2.37/1.48 => ( emptyset__Cong
% 2.37/1.48 => ( setadjoin__Cong
% 2.37/1.48 => ( powerset__Cong
% 2.37/1.48 => ( setunion__Cong
% 2.37/1.48 => ( omega__Cong
% 2.37/1.48 => ( exuEu
% 2.37/1.48 => ( descr__Cong
% 2.37/1.48 => ( dsetconstr__Cong
% 2.37/1.48 => ( subsetI1
% 2.37/1.48 => ( eqimpsubset2
% 2.37/1.48 => ( eqimpsubset1
% 2.37/1.48 => ( subsetI2
% 2.37/1.48 => ( emptysetsubset
% 2.37/1.48 => ( subsetE
% 2.37/1.48 => ( subsetE2
% 2.37/1.48 => ( notsubsetI
% 2.37/1.48 => ( notequalI1
% 2.37/1.48 => ( notequalI2
% 2.37/1.48 => ( subsetRefl
% 2.37/1.48 => ( subsetTrans
% 2.37/1.48 => ( setadjoinSub
% 2.37/1.48 => ( setadjoinSub2
% 2.37/1.48 => ( subset2powerset
% 2.37/1.48 => ( setextsub
% 2.37/1.48 => ( subsetemptysetimpeq
% 2.37/1.48 => ( powersetI1
% 2.37/1.48 => ( powersetE1
% 2.37/1.48 => ( inPowerset
% 2.37/1.48 => ( powersetsubset
% 2.37/1.48 => ( sepInPowerset
% 2.37/1.48 => ( sepSubset
% 2.37/1.48 => ( binunionIL
% 2.37/1.48 => ( upairset2IR
% 2.37/1.48 => ( binunionIR
% 2.37/1.48 => ( binunionEcases
% 2.37/1.48 => ( binunionE
% 2.37/1.48 => ( binunionLsub
% 2.37/1.48 => ( binunionRsub
% 2.37/1.48 => ( binintersectI
% 2.37/1.48 => ( binintersectSubset5
% 2.37/1.48 => ( binintersectEL
% 2.37/1.48 => ( binintersectLsub
% 2.37/1.48 => ( binintersectSubset2
% 2.37/1.48 => ( binintersectSubset3
% 2.37/1.48 => ( binintersectER
% 2.37/1.48 => ( disjointsetsI1
% 2.37/1.48 => ( binintersectRsub
% 2.37/1.48 => ( binintersectSubset4
% 2.37/1.48 => ( binintersectSubset1
% 2.37/1.48 => ( bs114d
% 2.37/1.48 => ( setminusI
% 2.37/1.48 => ( setminusEL
% 2.37/1.48 => ( setminusER
% 2.37/1.48 => ( setminusSubset2
% 2.37/1.48 => ( setminusERneg
% 2.37/1.48 => ( setminusELneg
% 2.37/1.48 => ( setminusILneg
% 2.37/1.48 => ( setminusIRneg
% 2.37/1.48 => ( setminusLsub
% 2.37/1.48 => ( setminusSubset1
% 2.37/1.48 => ( symdiffE
% 2.37/1.48 => ( symdiffI1
% 2.37/1.48 => ( symdiffI2
% 2.37/1.48 => ( symdiffIneg1
% 2.37/1.48 => ( symdiffIneg2
% 2.37/1.48 => ( secondinupair
% 2.37/1.48 => ( setukpairIL
% 2.37/1.48 => ( setukpairIR
% 2.37/1.48 => ( kpairiskpair
% 2.37/1.48 => ( kpairp
% 2.37/1.48 => ( singletonsubset
% 2.37/1.48 => ( singletoninpowerset
% 2.37/1.48 => ( singletoninpowunion
% 2.37/1.48 => ( upairset2E
% 2.37/1.48 => ( upairsubunion
% 2.37/1.48 => ( upairinpowunion
% 2.37/1.48 => ( ubforcartprodlem1
% 2.37/1.48 => ( ubforcartprodlem2
% 2.37/1.48 => ( ubforcartprodlem3
% 2.37/1.48 => ( cartprodpairin
% 2.37/1.48 => ( cartprodmempair1
% 2.37/1.48 => ( cartprodmempair
% 2.37/1.48 => ( setunionE2
% 2.37/1.48 => ( setunionsingleton1
% 2.37/1.48 => ( setunionsingleton2
% 2.37/1.48 => ( setunionsingleton
% 2.37/1.48 => ( singletonprop
% 2.37/1.48 => ( ex1E1
% 2.37/1.48 => ( ex1I
% 2.37/1.48 => ( ex1I2
% 2.37/1.48 => ( singletonsuniq
% 2.37/1.48 => ( setukpairinjL1
% 2.37/1.48 => ( kfstsingleton
% 2.37/1.48 => ( theprop
% 2.37/1.48 => ( kfstpairEq
% 2.37/1.48 => ( cartprodfstin
% 2.37/1.48 => ( setukpairinjL2
% 2.37/1.48 => ( setukpairinjL
% 2.37/1.48 => ( setukpairinjR11
% 2.37/1.48 => ( setukpairinjR12
% 2.37/1.48 => ( setukpairinjR1
% 2.37/1.48 => ( upairequniteq
% 2.37/1.48 => ( setukpairinjR2
% 2.37/1.48 => ( setukpairinjR
% 2.37/1.48 => ( ksndsingleton
% 2.37/1.48 => ( ksndpairEq
% 2.37/1.48 => ( kpairsurjEq
% 2.37/1.48 => ( cartprodsndin
% 2.37/1.48 => ( cartprodpairmemEL
% 2.37/1.48 => ( cartprodpairmemER
% 2.37/1.48 => ( cartprodmempaircEq
% 2.37/1.48 => ( cartprodfstpairEq
% 2.37/1.48 => ( cartprodsndpairEq
% 2.37/1.48 => ( cartprodpairsurjEq
% 2.37/1.48 => ( dpsetconstrI
% 2.37/1.48 => ( dpsetconstrSub
% 2.37/1.48 => ( setOfPairsIsBReln
% 2.37/1.48 => ( dpsetconstrERa
% 2.37/1.48 => ( dpsetconstrEL1
% 2.37/1.48 => ( dpsetconstrEL2
% 2.37/1.48 => ( dpsetconstrER
% 2.37/1.48 => ( funcImageSingleton
% 2.37/1.48 => ( apProp
% 2.37/1.48 => ( app
% 2.37/1.48 => ! [A: $i,B: $i,Xf: $i] :
% 2.37/1.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.37/1.48 => ( func @ A @ B @ Xf ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.37/1.48 ------------------------
% 2.56/1.51 % (2519)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_2187 for (2999ds/2Mi)
% 2.56/1.51 % (2519)Instruction limit reached!
% 2.56/1.51 % (2519)------------------------------
% 2.56/1.51 % (2519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.51 % (2519)Termination reason: Unknown
% 2.56/1.51 % (2519)Termination phase: shuffling
% 2.56/1.51
% 2.56/1.51 % (2519)Memory used [KB]: 1407
% 2.56/1.51 % (2519)Time elapsed: 0.002 s
% 2.56/1.51 % (2519)Instructions burned: 2 (million)
% 2.56/1.51 % (2519)------------------------------
% 2.56/1.51 % (2519)------------------------------
% 2.56/1.51 % (2517)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_2187 for (2999ds/4Mi)
% 2.56/1.51 % (2517)Instruction limit reached!
% 2.56/1.51 % (2517)------------------------------
% 2.56/1.51 % (2517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.51 % (2517)Termination reason: Unknown
% 2.56/1.51 % (2517)Termination phase: shuffling
% 2.56/1.51
% 2.56/1.51 % (2517)Memory used [KB]: 1407
% 2.56/1.51 % (2517)Time elapsed: 0.003 s
% 2.56/1.51 % (2517)Instructions burned: 6 (million)
% 2.56/1.51 % (2517)------------------------------
% 2.56/1.51 % (2517)------------------------------
% 2.56/1.51 % (2516)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_2187 for (2999ds/183Mi)
% 2.56/1.51 % (2518)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_2187 for (2999ds/27Mi)
% 2.56/1.51 % (2520)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_2187 for (2999ds/2Mi)
% 2.56/1.51 % (2522)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_2187 for (2999ds/18Mi)
% 2.56/1.51 % (2521)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_2187 for (2999ds/275Mi)
% 2.56/1.51 % (2520)Instruction limit reached!
% 2.56/1.51 % (2520)------------------------------
% 2.56/1.51 % (2520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.51 % (2520)Termination reason: Unknown
% 2.56/1.51 % (2520)Termination phase: shuffling
% 2.56/1.52
% 2.56/1.52 % (2520)Memory used [KB]: 1407
% 2.56/1.52 % (2520)Time elapsed: 0.004 s
% 2.56/1.52 % (2520)Instructions burned: 3 (million)
% 2.56/1.52 % (2520)------------------------------
% 2.56/1.52 % (2520)------------------------------
% 2.56/1.52 % (2523)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_2187 for (2999ds/3Mi)
% 2.56/1.52 % (2523)Instruction limit reached!
% 2.56/1.52 % (2523)------------------------------
% 2.56/1.52 % (2523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.52 % (2523)Termination reason: Unknown
% 2.56/1.52 % (2523)Termination phase: shuffling
% 2.56/1.52
% 2.56/1.52 % (2523)Memory used [KB]: 1407
% 2.56/1.52 % (2523)Time elapsed: 0.002 s
% 2.56/1.52 % (2523)Instructions burned: 3 (million)
% 2.56/1.52 % (2523)------------------------------
% 2.56/1.52 % (2523)------------------------------
% 2.56/1.52 % (2524)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_2187 for (2999ds/37Mi)
% 2.56/1.53 % (2522)Instruction limit reached!
% 2.56/1.53 % (2522)------------------------------
% 2.56/1.53 % (2522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.53 % (2522)Termination reason: Unknown
% 2.56/1.53 % (2522)Termination phase: Property scanning
% 2.56/1.53
% 2.56/1.53 % (2522)Memory used [KB]: 1791
% 2.56/1.53 % (2522)Time elapsed: 0.014 s
% 2.56/1.53 % (2522)Instructions burned: 19 (million)
% 2.56/1.53 % (2522)------------------------------
% 2.56/1.53 % (2522)------------------------------
% 2.56/1.53 % (2526)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_2187 for (2999ds/3Mi)
% 2.56/1.53 % (2526)Instruction limit reached!
% 2.56/1.53 % (2526)------------------------------
% 2.56/1.53 % (2526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.53 % (2526)Termination reason: Unknown
% 2.56/1.53 % (2526)Termination phase: shuffling
% 2.56/1.53
% 2.56/1.53 % (2526)Memory used [KB]: 1407
% 2.56/1.53 % (2526)Time elapsed: 0.002 s
% 2.56/1.53 % (2526)Instructions burned: 3 (million)
% 2.56/1.53 % (2526)------------------------------
% 2.56/1.53 % (2526)------------------------------
% 2.56/1.53 % (2518)Instruction limit reached!
% 2.56/1.53 % (2518)------------------------------
% 2.56/1.53 % (2518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.53 % (2518)Termination reason: Unknown
% 2.56/1.53 % (2518)Termination phase: shuffling
% 2.56/1.53
% 2.56/1.53 % (2518)Memory used [KB]: 1918
% 2.56/1.53 % (2518)Time elapsed: 0.017 s
% 2.56/1.53 % (2518)Instructions burned: 28 (million)
% 2.56/1.53 % (2518)------------------------------
% 2.56/1.53 % (2518)------------------------------
% 2.56/1.53 % (2524)Instruction limit reached!
% 2.56/1.53 % (2524)------------------------------
% 2.56/1.53 % (2524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.53 % (2524)Termination reason: Unknown
% 2.56/1.53 % (2524)Termination phase: Property scanning
% 2.56/1.53
% 2.56/1.53 % (2524)Memory used [KB]: 2046
% 2.56/1.53 % (2524)Time elapsed: 0.012 s
% 2.56/1.53 % (2524)Instructions burned: 40 (million)
% 2.56/1.53 % (2524)------------------------------
% 2.56/1.53 % (2524)------------------------------
% 2.56/1.53 % (2525)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_2187 for (2999ds/15Mi)
% 2.56/1.54 % (2528)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_2187 for (2999ds/7Mi)
% 2.56/1.54 % (2529)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_2187 for (2999ds/16Mi)
% 2.56/1.54 % (2528)Instruction limit reached!
% 2.56/1.54 % (2528)------------------------------
% 2.56/1.54 % (2528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.54 % (2528)Termination reason: Unknown
% 2.56/1.54 % (2528)Termination phase: shuffling
% 2.56/1.54
% 2.56/1.54 % (2528)Memory used [KB]: 1535
% 2.56/1.54 % (2528)Time elapsed: 0.004 s
% 2.56/1.54 % (2528)Instructions burned: 10 (million)
% 2.56/1.54 % (2528)------------------------------
% 2.56/1.54 % (2528)------------------------------
% 2.56/1.54 % (2525)Instruction limit reached!
% 2.56/1.54 % (2525)------------------------------
% 2.56/1.54 % (2525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.54 % (2525)Termination reason: Unknown
% 2.56/1.54 % (2525)Termination phase: shuffling
% 2.56/1.54
% 2.56/1.54 % (2525)Memory used [KB]: 1663
% 2.56/1.54 % (2525)Time elapsed: 0.011 s
% 2.56/1.54 % (2525)Instructions burned: 16 (million)
% 2.56/1.54 % (2525)------------------------------
% 2.56/1.54 % (2525)------------------------------
% 2.56/1.54 % (2529)Instruction limit reached!
% 2.56/1.54 % (2529)------------------------------
% 2.56/1.54 % (2529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.54 % (2529)Termination reason: Unknown
% 2.56/1.54 % (2529)Termination phase: shuffling
% 2.56/1.54
% 2.56/1.54 % (2529)Memory used [KB]: 1663
% 2.56/1.54 % (2529)Time elapsed: 0.006 s
% 2.56/1.54 % (2529)Instructions burned: 17 (million)
% 2.56/1.54 % (2529)------------------------------
% 2.56/1.54 % (2529)------------------------------
% 2.56/1.54 % (2527)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_2187 for (2999ds/1041Mi)
% 2.56/1.54 % (2530)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_2187 for (2999ds/3Mi)
% 2.56/1.55 % (2530)Instruction limit reached!
% 2.56/1.55 % (2530)------------------------------
% 2.56/1.55 % (2530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.55 % (2530)Termination reason: Unknown
% 2.56/1.55 % (2530)Termination phase: shuffling
% 2.56/1.55
% 2.56/1.55 % (2530)Memory used [KB]: 1407
% 2.56/1.55 % (2530)Time elapsed: 0.004 s
% 2.56/1.55 % (2530)Instructions burned: 4 (million)
% 2.56/1.55 % (2530)------------------------------
% 2.56/1.55 % (2530)------------------------------
% 2.56/1.55 % (2531)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_2187 for (2999ds/3Mi)
% 2.56/1.55 % (2531)Instruction limit reached!
% 2.56/1.55 % (2531)------------------------------
% 2.56/1.55 % (2531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.55 % (2531)Termination reason: Unknown
% 2.56/1.55 % (2531)Termination phase: shuffling
% 2.56/1.55
% 2.56/1.55 % (2531)Memory used [KB]: 1407
% 2.56/1.55 % (2531)Time elapsed: 0.002 s
% 2.56/1.55 % (2531)Instructions burned: 3 (million)
% 2.56/1.55 % (2531)------------------------------
% 2.56/1.55 % (2531)------------------------------
% 2.56/1.55 % (2533)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_2187 for (2999ds/3Mi)
% 2.56/1.55 % (2533)Instruction limit reached!
% 2.56/1.55 % (2533)------------------------------
% 2.56/1.55 % (2533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.55 % (2533)Termination reason: Unknown
% 2.56/1.55 % (2533)Termination phase: shuffling
% 2.56/1.55
% 2.56/1.55 % (2533)Memory used [KB]: 1407
% 2.56/1.55 % (2533)Time elapsed: 0.002 s
% 2.56/1.55 % (2533)Instructions burned: 4 (million)
% 2.56/1.55 % (2533)------------------------------
% 2.56/1.55 % (2533)------------------------------
% 2.56/1.56 % (2532)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_2187 for (2999ds/7Mi)
% 2.56/1.56 % (2532)Instruction limit reached!
% 2.56/1.56 % (2532)------------------------------
% 2.56/1.56 % (2532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.56 % (2532)Termination reason: Unknown
% 2.56/1.56 % (2532)Termination phase: shuffling
% 2.56/1.56
% 2.56/1.56 % (2532)Memory used [KB]: 1535
% 2.56/1.56 % (2532)Time elapsed: 0.004 s
% 2.56/1.56 % (2532)Instructions burned: 8 (million)
% 2.56/1.56 % (2532)------------------------------
% 2.56/1.56 % (2532)------------------------------
% 2.56/1.56 % (2535)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_2187 for (2999ds/18Mi)
% 2.56/1.56 % (2534)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_2187 for (2999ds/4Mi)
% 2.56/1.56 % (2534)Instruction limit reached!
% 2.56/1.56 % (2534)------------------------------
% 2.56/1.56 % (2534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.56 % (2534)Termination reason: Unknown
% 2.56/1.56 % (2534)Termination phase: shuffling
% 2.56/1.56
% 2.56/1.56 % (2534)Memory used [KB]: 1407
% 2.56/1.56 % (2534)Time elapsed: 0.002 s
% 2.56/1.56 % (2534)Instructions burned: 4 (million)
% 2.56/1.56 % (2534)------------------------------
% 2.56/1.56 % (2534)------------------------------
% 2.56/1.56 % (2536)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_2187 for (2999ds/710Mi)
% 2.56/1.56 % (2535)Instruction limit reached!
% 2.56/1.56 % (2535)------------------------------
% 2.56/1.56 % (2535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.56 % (2535)Termination reason: Unknown
% 2.56/1.56 % (2535)Termination phase: shuffling
% 2.56/1.56
% 2.56/1.56 % (2535)Memory used [KB]: 1791
% 2.56/1.56 % (2535)Time elapsed: 0.006 s
% 2.56/1.56 % (2535)Instructions burned: 18 (million)
% 2.56/1.56 % (2535)------------------------------
% 2.56/1.56 % (2535)------------------------------
% 2.56/1.57 % (2537)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_2187 for (2999ds/6Mi)
% 2.56/1.57 % (2538)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_2187 for (2999ds/902Mi)
% 2.56/1.57 % (2537)Instruction limit reached!
% 2.56/1.57 % (2537)------------------------------
% 2.56/1.57 % (2537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.57 % (2537)Termination reason: Unknown
% 2.56/1.57 % (2537)Termination phase: shuffling
% 2.56/1.57
% 2.56/1.57 % (2537)Memory used [KB]: 1407
% 2.56/1.57 % (2537)Time elapsed: 0.004 s
% 2.56/1.57 % (2537)Instructions burned: 7 (million)
% 2.56/1.57 % (2537)------------------------------
% 2.56/1.57 % (2537)------------------------------
% 2.56/1.57 % (2539)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_2187 for (2999ds/21Mi)
% 2.56/1.58 % (2539)Instruction limit reached!
% 2.56/1.58 % (2539)------------------------------
% 2.56/1.58 % (2539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.58 % (2539)Termination reason: Unknown
% 2.56/1.58 % (2539)Termination phase: shuffling
% 2.56/1.58
% 2.56/1.58 % (2539)Memory used [KB]: 1791
% 2.56/1.58 % (2539)Time elapsed: 0.007 s
% 2.56/1.58 % (2539)Instructions burned: 22 (million)
% 2.56/1.58 % (2539)------------------------------
% 2.56/1.58 % (2539)------------------------------
% 2.56/1.58 % (2540)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_2187 for (2999ds/5Mi)
% 2.56/1.58 % (2540)Instruction limit reached!
% 2.56/1.58 % (2540)------------------------------
% 2.56/1.58 % (2540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.58 % (2540)Termination reason: Unknown
% 2.56/1.58 % (2540)Termination phase: shuffling
% 2.56/1.58
% 2.56/1.58 % (2540)Memory used [KB]: 1535
% 2.56/1.58 % (2540)Time elapsed: 0.004 s
% 2.56/1.58 % (2540)Instructions burned: 8 (million)
% 2.56/1.58 % (2540)------------------------------
% 2.56/1.58 % (2540)------------------------------
% 2.56/1.59 % (2541)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_2187 for (2999ds/6Mi)
% 2.56/1.59 % (2541)Instruction limit reached!
% 2.56/1.59 % (2541)------------------------------
% 2.56/1.59 % (2541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.59 % (2541)Termination reason: Unknown
% 2.56/1.59 % (2541)Termination phase: shuffling
% 2.56/1.59
% 2.56/1.59 % (2541)Memory used [KB]: 1407
% 2.56/1.59 % (2541)Time elapsed: 0.003 s
% 2.56/1.59 % (2541)Instructions burned: 7 (million)
% 2.56/1.59 % (2541)------------------------------
% 2.56/1.59 % (2541)------------------------------
% 2.56/1.59 % (2542)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_2187 for (2999ds/377Mi)
% 2.56/1.60 % (2543)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_2187 for (2999ds/779Mi)
% 2.56/1.60 % (2516)Instruction limit reached!
% 2.56/1.60 % (2516)------------------------------
% 2.56/1.60 % (2516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.60 % (2516)Termination reason: Unknown
% 2.56/1.60 % (2516)Termination phase: Saturation
% 2.56/1.60
% 2.56/1.60 % (2516)Memory used [KB]: 7547
% 2.56/1.60 % (2516)Time elapsed: 0.092 s
% 2.56/1.60 % (2516)Instructions burned: 183 (million)
% 2.56/1.60 % (2516)------------------------------
% 2.56/1.60 % (2516)------------------------------
% 2.56/1.62 % (2544)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_2187 for (2998ds/19Mi)
% 2.56/1.62 % (2521)Instruction limit reached!
% 2.56/1.62 % (2521)------------------------------
% 2.56/1.62 % (2521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.62 % (2521)Termination reason: Unknown
% 2.56/1.62 % (2521)Termination phase: Saturation
% 2.56/1.62
% 2.56/1.62 % (2521)Memory used [KB]: 8443
% 2.56/1.62 % (2521)Time elapsed: 0.107 s
% 2.56/1.62 % (2521)Instructions burned: 275 (million)
% 2.56/1.62 % (2521)------------------------------
% 2.56/1.62 % (2521)------------------------------
% 2.56/1.62 % (2544)Instruction limit reached!
% 2.56/1.62 % (2544)------------------------------
% 2.56/1.62 % (2544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.62 % (2544)Termination reason: Unknown
% 2.56/1.62 % (2544)Termination phase: shuffling
% 2.56/1.62
% 2.56/1.62 % (2544)Memory used [KB]: 1791
% 2.56/1.62 % (2544)Time elapsed: 0.007 s
% 2.56/1.62 % (2544)Instructions burned: 20 (million)
% 2.56/1.62 % (2544)------------------------------
% 2.56/1.62 % (2544)------------------------------
% 2.56/1.63 % (2545)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_2187 for (2998ds/879Mi)
% 2.56/1.63 % (2546)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_2187 for (2998ds/17Mi)
% 2.56/1.64 % (2546)Instruction limit reached!
% 2.56/1.64 % (2546)------------------------------
% 2.56/1.64 % (2546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.64 % (2546)Termination reason: Unknown
% 2.56/1.64 % (2546)Termination phase: shuffling
% 2.56/1.64
% 2.56/1.64 % (2546)Memory used [KB]: 1791
% 2.56/1.64 % (2546)Time elapsed: 0.007 s
% 2.56/1.64 % (2546)Instructions burned: 18 (million)
% 2.56/1.64 % (2546)------------------------------
% 2.56/1.64 % (2546)------------------------------
% 2.56/1.64 % (2538)Refutation not found, incomplete strategy
% 2.56/1.64 % (2538)------------------------------
% 2.56/1.64 % (2538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.64 % (2538)Termination reason: Refutation not found, incomplete strategy
% 2.56/1.64
% 2.56/1.64
% 2.56/1.64 % (2538)Memory used [KB]: 8315
% 2.56/1.64 % (2538)Time elapsed: 0.070 s
% 2.56/1.64 % (2538)Instructions burned: 213 (million)
% 2.56/1.64 % (2538)------------------------------
% 2.56/1.64 % (2538)------------------------------
% 2.56/1.65 % (2547)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_2187 for (2998ds/3Mi)
% 2.56/1.65 % (2547)Instruction limit reached!
% 2.56/1.65 % (2547)------------------------------
% 2.56/1.65 % (2547)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.65 % (2547)Termination reason: Unknown
% 2.56/1.65 % (2547)Termination phase: shuffling
% 2.56/1.65
% 2.56/1.65 % (2547)Memory used [KB]: 1407
% 2.56/1.65 % (2547)Time elapsed: 0.002 s
% 2.56/1.65 % (2547)Instructions burned: 4 (million)
% 2.56/1.65 % (2547)------------------------------
% 2.56/1.65 % (2547)------------------------------
% 2.56/1.65 % (2548)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_2187 for (2998ds/30Mi)
% 2.56/1.66 % (2548)Instruction limit reached!
% 2.56/1.66 % (2548)------------------------------
% 2.56/1.66 % (2548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.66 % (2548)Termination reason: Unknown
% 2.56/1.66 % (2548)Termination phase: shuffling
% 2.56/1.66
% 2.56/1.66 % (2548)Memory used [KB]: 1918
% 2.56/1.66 % (2548)Time elapsed: 0.010 s
% 2.56/1.66 % (2548)Instructions burned: 31 (million)
% 2.56/1.66 % (2548)------------------------------
% 2.56/1.66 % (2548)------------------------------
% 2.56/1.66 % (2549)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_2187 for (2998ds/127Mi)
% 2.56/1.67 % (2550)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_2187 for (2998ds/100Mi)
% 2.56/1.69 % (2549)Instruction limit reached!
% 2.56/1.69 % (2549)------------------------------
% 2.56/1.69 % (2549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.69 % (2549)Termination reason: Unknown
% 2.56/1.69 % (2549)Termination phase: Saturation
% 2.56/1.69
% 2.56/1.69 % (2549)Memory used [KB]: 6908
% 2.56/1.69 % (2549)Time elapsed: 0.037 s
% 2.56/1.69 % (2549)Instructions burned: 127 (million)
% 2.56/1.69 % (2549)------------------------------
% 2.56/1.69 % (2549)------------------------------
% 2.56/1.70 % (2550)Instruction limit reached!
% 2.56/1.70 % (2550)------------------------------
% 2.56/1.70 % (2550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.70 % (2550)Termination reason: Unknown
% 2.56/1.70 % (2550)Termination phase: Saturation
% 2.56/1.70
% 2.56/1.70 % (2550)Memory used [KB]: 6780
% 2.56/1.70 % (2550)Time elapsed: 0.029 s
% 2.56/1.70 % (2550)Instructions burned: 100 (million)
% 2.56/1.70 % (2550)------------------------------
% 2.56/1.70 % (2550)------------------------------
% 2.56/1.70 % (2551)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_2187 for (2997ds/3Mi)
% 2.56/1.71 % (2551)Instruction limit reached!
% 2.56/1.71 % (2551)------------------------------
% 2.56/1.71 % (2551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.71 % (2551)Termination reason: Unknown
% 2.56/1.71 % (2551)Termination phase: shuffling
% 2.56/1.71
% 2.56/1.71 % (2551)Memory used [KB]: 1407
% 2.56/1.71 % (2551)Time elapsed: 0.003 s
% 2.56/1.71 % (2551)Instructions burned: 4 (million)
% 2.56/1.71 % (2551)------------------------------
% 2.56/1.71 % (2551)------------------------------
% 2.56/1.71 % (2552)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_2187 for (2997ds/20Mi)
% 2.56/1.71 % (2542)Instruction limit reached!
% 2.56/1.71 % (2542)------------------------------
% 2.56/1.71 % (2542)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.71 % (2542)Termination reason: Unknown
% 2.56/1.71 % (2542)Termination phase: Saturation
% 2.56/1.71
% 2.56/1.71 % (2542)Memory used [KB]: 8443
% 2.56/1.71 % (2542)Time elapsed: 0.120 s
% 2.56/1.71 % (2542)Instructions burned: 378 (million)
% 2.56/1.71 % (2542)------------------------------
% 2.56/1.71 % (2542)------------------------------
% 2.56/1.71 % (2552)Instruction limit reached!
% 2.56/1.71 % (2552)------------------------------
% 2.56/1.71 % (2552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.71 % (2552)Termination reason: Unknown
% 2.56/1.71 % (2552)Termination phase: Property scanning
% 2.56/1.71
% 2.56/1.71 % (2552)Memory used [KB]: 1663
% 2.56/1.71 % (2552)Time elapsed: 0.009 s
% 2.56/1.71 % (2552)Instructions burned: 22 (million)
% 2.56/1.71 % (2552)------------------------------
% 2.56/1.71 % (2552)------------------------------
% 2.56/1.72 % (2553)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_2187 for (2997ds/86Mi)
% 2.56/1.72 % (2554)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_2187 for (2997ds/107Mi)
% 2.56/1.72 % (2555)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_2187 for (2997ds/39Mi)
% 2.56/1.73 % (2555)Instruction limit reached!
% 2.56/1.73 % (2555)------------------------------
% 2.56/1.73 % (2555)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.73 % (2555)Termination reason: Unknown
% 2.56/1.73 % (2555)Termination phase: Property scanning
% 2.56/1.73
% 2.56/1.73 % (2555)Memory used [KB]: 2046
% 2.56/1.73 % (2555)Time elapsed: 0.012 s
% 2.56/1.73 % (2555)Instructions burned: 39 (million)
% 2.56/1.73 % (2555)------------------------------
% 2.56/1.73 % (2555)------------------------------
% 2.56/1.74 % (2553)Instruction limit reached!
% 2.56/1.74 % (2553)------------------------------
% 2.56/1.74 % (2553)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.74 % (2553)Termination reason: Unknown
% 2.56/1.74 % (2553)Termination phase: Property scanning
% 2.56/1.74
% 2.56/1.74 % (2553)Memory used [KB]: 2174
% 2.56/1.74 % (2553)Time elapsed: 0.026 s
% 2.56/1.74 % (2553)Instructions burned: 89 (million)
% 2.56/1.74 % (2553)------------------------------
% 2.56/1.74 % (2553)------------------------------
% 2.56/1.74 % (2556)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_2187 for (2997ds/448Mi)
% 2.56/1.75 % (2554)Instruction limit reached!
% 2.56/1.75 % (2554)------------------------------
% 2.56/1.75 % (2554)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.75 % (2554)Termination reason: Unknown
% 2.56/1.75 % (2554)Termination phase: Property scanning
% 2.56/1.75
% 2.56/1.75 % (2554)Memory used [KB]: 2174
% 2.56/1.75 % (2554)Time elapsed: 0.030 s
% 2.56/1.75 % (2554)Instructions burned: 108 (million)
% 2.56/1.75 % (2554)------------------------------
% 2.56/1.75 % (2554)------------------------------
% 2.56/1.75 % (2557)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_2187 for (2997ds/46Mi)
% 2.56/1.76 % (2558)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_2187 for (2997ds/98Mi)
% 2.56/1.76 % (2557)Instruction limit reached!
% 2.56/1.76 % (2557)------------------------------
% 2.56/1.76 % (2557)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.76 % (2557)Termination reason: Unknown
% 2.56/1.76 % (2557)Termination phase: Property scanning
% 2.56/1.76
% 2.56/1.76 % (2557)Memory used [KB]: 1918
% 2.56/1.76 % (2557)Time elapsed: 0.014 s
% 2.56/1.76 % (2557)Instructions burned: 47 (million)
% 2.56/1.76 % (2557)------------------------------
% 2.56/1.76 % (2557)------------------------------
% 2.56/1.77 % (2559)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_2187 for (2997ds/507Mi)
% 2.56/1.78 % (2536)Instruction limit reached!
% 2.56/1.78 % (2536)------------------------------
% 2.56/1.78 % (2536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.78 % (2536)Termination reason: Unknown
% 2.56/1.78 % (2536)Termination phase: Saturation
% 2.56/1.78
% 2.56/1.78 % (2536)Memory used [KB]: 9594
% 2.56/1.78 % (2536)Time elapsed: 0.222 s
% 2.56/1.78 % (2536)Instructions burned: 710 (million)
% 2.56/1.78 % (2536)------------------------------
% 2.56/1.78 % (2536)------------------------------
% 2.56/1.79 % (2558)Instruction limit reached!
% 2.56/1.79 % (2558)------------------------------
% 2.56/1.79 % (2558)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.79 % (2558)Termination reason: Unknown
% 2.56/1.79 % (2558)Termination phase: Property scanning
% 2.56/1.79
% 2.56/1.79 % (2558)Memory used [KB]: 2302
% 2.56/1.79 % (2558)Time elapsed: 0.028 s
% 2.56/1.79 % (2558)Instructions burned: 99 (million)
% 2.56/1.79 % (2558)------------------------------
% 2.56/1.79 % (2558)------------------------------
% 2.56/1.79 % (2543)Instruction limit reached!
% 2.56/1.79 % (2543)------------------------------
% 2.56/1.79 % (2543)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.79 % (2543)Termination reason: Unknown
% 2.56/1.79 % (2543)Termination phase: Saturation
% 2.56/1.79
% 2.56/1.79 % (2543)Memory used [KB]: 7036
% 2.56/1.79 % (2543)Time elapsed: 0.190 s
% 2.56/1.79 % (2543)Instructions burned: 782 (million)
% 2.56/1.79 % (2543)------------------------------
% 2.56/1.79 % (2543)------------------------------
% 2.56/1.79 % (2560)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_2187 for (2996ds/149Mi)
% 2.56/1.80 % (2561)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_2187 for (2996ds/39Mi)
% 2.56/1.80 % (2562)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_2187 for (2996ds/6Mi)
% 2.56/1.80 % (2562)Instruction limit reached!
% 2.56/1.80 % (2562)------------------------------
% 2.56/1.80 % (2562)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.80 % (2562)Termination reason: Unknown
% 2.56/1.80 % (2562)Termination phase: shuffling
% 2.56/1.80
% 2.56/1.80 % (2562)Memory used [KB]: 1407
% 2.56/1.80 % (2562)Time elapsed: 0.003 s
% 2.56/1.80 % (2562)Instructions burned: 7 (million)
% 2.56/1.80 % (2562)------------------------------
% 2.56/1.80 % (2562)------------------------------
% 2.56/1.81 % (2561)Instruction limit reached!
% 2.56/1.81 % (2561)------------------------------
% 2.56/1.81 % (2561)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.81 % (2561)Termination reason: Unknown
% 2.56/1.81 % (2561)Termination phase: Property scanning
% 2.56/1.81
% 2.56/1.81 % (2561)Memory used [KB]: 1918
% 2.56/1.81 % (2561)Time elapsed: 0.013 s
% 2.56/1.81 % (2561)Instructions burned: 39 (million)
% 2.56/1.81 % (2561)------------------------------
% 2.56/1.81 % (2561)------------------------------
% 2.56/1.81 % (2563)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_2187 for (2996ds/1087Mi)
% 2.56/1.82 % (2564)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_2187 for (2996ds/47Mi)
% 2.56/1.83 % (2560)Instruction limit reached!
% 2.56/1.83 % (2560)------------------------------
% 2.56/1.83 % (2560)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.83 % (2560)Termination reason: Unknown
% 2.56/1.83 % (2560)Termination phase: Saturation
% 2.56/1.83
% 2.56/1.83 % (2560)Memory used [KB]: 7291
% 2.56/1.83 % (2560)Time elapsed: 0.041 s
% 2.56/1.83 % (2564)Instruction limit reached!
% 2.56/1.83 % (2564)------------------------------
% 2.56/1.83 % (2564)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.83 % (2560)Instructions burned: 150 (million)
% 2.56/1.83 % (2564)Termination reason: Unknown
% 2.56/1.83 % (2560)------------------------------
% 2.56/1.83 % (2560)------------------------------
% 2.56/1.83 % (2564)Termination phase: Naming
% 2.56/1.83
% 2.56/1.83 % (2564)Memory used [KB]: 2046
% 2.56/1.83 % (2564)Time elapsed: 0.016 s
% 2.56/1.83 % (2564)Instructions burned: 50 (million)
% 2.56/1.83 % (2564)------------------------------
% 2.56/1.83 % (2564)------------------------------
% 2.56/1.84 % (2527)Instruction limit reached!
% 2.56/1.84 % (2527)------------------------------
% 2.56/1.84 % (2527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.84 % (2527)Termination reason: Unknown
% 2.56/1.84 % (2527)Termination phase: Saturation
% 2.56/1.84
% 2.56/1.84 % (2527)Memory used [KB]: 11257
% 2.56/1.84 % (2527)Time elapsed: 0.300 s
% 2.56/1.84 % (2527)Instructions burned: 1042 (million)
% 2.56/1.84 % (2527)------------------------------
% 2.56/1.84 % (2527)------------------------------
% 2.56/1.84 % (2566)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_2187 for (2996ds/432Mi)
% 2.56/1.84 % (2565)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_2187 for (2996ds/44Mi)
% 2.56/1.85 % (2567)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_2187 for (2996ds/62Mi)
% 2.56/1.85 % (2556)Instruction limit reached!
% 2.56/1.85 % (2556)------------------------------
% 2.56/1.85 % (2556)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.85 % (2556)Termination reason: Unknown
% 2.56/1.85 % (2556)Termination phase: Saturation
% 2.56/1.85
% 2.56/1.85 % (2556)Memory used [KB]: 7036
% 2.56/1.85 % (2556)Time elapsed: 0.110 s
% 2.56/1.85 % (2556)Instructions burned: 449 (million)
% 2.56/1.85 % (2556)------------------------------
% 2.56/1.85 % (2556)------------------------------
% 2.56/1.86 % (2565)Instruction limit reached!
% 2.56/1.86 % (2565)------------------------------
% 2.56/1.86 % (2565)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.86 % (2565)Termination reason: Unknown
% 2.56/1.86 % (2565)Termination phase: Property scanning
% 2.56/1.86
% 2.56/1.86 % (2565)Memory used [KB]: 2046
% 2.56/1.86 % (2565)Time elapsed: 0.014 s
% 2.56/1.86 % (2565)Instructions burned: 47 (million)
% 2.56/1.86 % (2565)------------------------------
% 2.56/1.86 % (2565)------------------------------
% 2.56/1.86 % (2568)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_2187 for (2996ds/31Mi)
% 2.56/1.87 % (2569)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_2187 for (2996ds/96Mi)
% 2.56/1.87 % (2567)Instruction limit reached!
% 2.56/1.87 % (2567)------------------------------
% 2.56/1.87 % (2567)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.87 % (2567)Termination reason: Unknown
% 2.56/1.87 % (2567)Termination phase: Property scanning
% 2.56/1.87
% 2.56/1.87 % (2567)Memory used [KB]: 2046
% 2.56/1.87 % (2567)Time elapsed: 0.018 s
% 2.56/1.87 % (2567)Instructions burned: 62 (million)
% 2.56/1.87 % (2567)------------------------------
% 2.56/1.87 % (2567)------------------------------
% 2.56/1.87 % (2568)Instruction limit reached!
% 2.56/1.87 % (2568)------------------------------
% 2.56/1.87 % (2568)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.87 % (2568)Termination reason: Unknown
% 2.56/1.87 % (2568)Termination phase: Property scanning
% 2.56/1.87
% 2.56/1.87 % (2568)Memory used [KB]: 1918
% 2.56/1.87 % (2568)Time elapsed: 0.010 s
% 2.56/1.87 % (2568)Instructions burned: 33 (million)
% 2.56/1.87 % (2568)------------------------------
% 2.56/1.87 % (2568)------------------------------
% 2.56/1.88 % (2570)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_2187 for (2996ds/453Mi)
% 2.56/1.88 % (2571)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_2187 for (2996ds/71Mi)
% 2.56/1.89 % (2569)Instruction limit reached!
% 2.56/1.89 % (2569)------------------------------
% 2.56/1.89 % (2569)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.89 % (2569)Termination reason: Unknown
% 2.56/1.89 % (2569)Termination phase: Property scanning
% 2.56/1.89
% 2.56/1.89 % (2569)Memory used [KB]: 2174
% 2.56/1.89 % (2569)Time elapsed: 0.028 s
% 2.56/1.89 % (2569)Instructions burned: 96 (million)
% 2.56/1.89 % (2569)------------------------------
% 2.56/1.89 % (2569)------------------------------
% 2.56/1.90 % (2571)Instruction limit reached!
% 2.56/1.90 % (2571)------------------------------
% 2.56/1.90 % (2571)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.90 % (2571)Termination reason: Unknown
% 2.56/1.90 % (2571)Termination phase: Function definition elimination
% 2.56/1.90
% 2.56/1.90 % (2571)Memory used [KB]: 2174
% 2.56/1.90 % (2571)Time elapsed: 0.020 s
% 2.56/1.90 % (2571)Instructions burned: 72 (million)
% 2.56/1.90 % (2571)------------------------------
% 2.56/1.90 % (2571)------------------------------
% 2.56/1.90 % (2572)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_2187 for (2995ds/83Mi)
% 2.56/1.91 % (2573)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_2187 for (2995ds/11Mi)
% 2.56/1.91 % (2573)Instruction limit reached!
% 2.56/1.91 % (2573)------------------------------
% 2.56/1.91 % (2573)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.91 % (2573)Termination reason: Unknown
% 2.56/1.91 % (2573)Termination phase: shuffling
% 2.56/1.91
% 2.56/1.91 % (2573)Memory used [KB]: 1535
% 2.56/1.91 % (2573)Time elapsed: 0.004 s
% 2.56/1.91 % (2573)Instructions burned: 12 (million)
% 2.56/1.91 % (2573)------------------------------
% 2.56/1.91 % (2573)------------------------------
% 2.56/1.92 % (2545)Instruction limit reached!
% 2.56/1.92 % (2545)------------------------------
% 2.56/1.92 % (2545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.92 % (2545)Termination reason: Unknown
% 2.56/1.92 % (2545)Termination phase: Saturation
% 2.56/1.92
% 2.56/1.92 % (2545)Memory used [KB]: 8699
% 2.56/1.92 % (2545)Time elapsed: 0.290 s
% 2.56/1.92 % (2545)Instructions burned: 880 (million)
% 2.56/1.92 % (2545)------------------------------
% 2.56/1.92 % (2545)------------------------------
% 2.56/1.92 % (2574)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_2187 for (2995ds/41Mi)
% 2.56/1.93 % (2572)Instruction limit reached!
% 2.56/1.93 % (2572)------------------------------
% 2.56/1.93 % (2572)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.93 % (2572)Termination reason: Unknown
% 2.56/1.93 % (2572)Termination phase: Property scanning
% 2.56/1.93
% 2.56/1.93 % (2572)Memory used [KB]: 2046
% 2.56/1.93 % (2572)Time elapsed: 0.025 s
% 2.56/1.93 % (2572)Instructions burned: 85 (million)
% 2.56/1.93 % (2572)------------------------------
% 2.56/1.93 % (2572)------------------------------
% 2.56/1.93 % (2575)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_2187 for (2995ds/6Mi)
% 2.56/1.93 % (2575)Instruction limit reached!
% 2.56/1.93 % (2575)------------------------------
% 2.56/1.93 % (2575)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.93 % (2575)Termination reason: Unknown
% 2.56/1.93 % (2575)Termination phase: shuffling
% 2.56/1.93
% 2.56/1.93 % (2575)Memory used [KB]: 1407
% 2.56/1.93 % (2575)Time elapsed: 0.003 s
% 2.56/1.93 % (2575)Instructions burned: 7 (million)
% 2.56/1.93 % (2575)------------------------------
% 2.56/1.93 % (2575)------------------------------
% 2.56/1.93 % (2574)Instruction limit reached!
% 2.56/1.93 % (2574)------------------------------
% 2.56/1.93 % (2574)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.93 % (2574)Termination reason: Unknown
% 2.56/1.93 % (2574)Termination phase: Property scanning
% 2.56/1.93
% 2.56/1.93 % (2574)Memory used [KB]: 2046
% 2.56/1.93 % (2574)Time elapsed: 0.012 s
% 2.56/1.93 % (2574)Instructions burned: 42 (million)
% 2.56/1.93 % (2574)------------------------------
% 2.56/1.93 % (2574)------------------------------
% 2.56/1.93 % (2559)Instruction limit reached!
% 2.56/1.93 % (2559)------------------------------
% 2.56/1.93 % (2559)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.93 % (2559)Termination reason: Unknown
% 2.56/1.93 % (2559)Termination phase: Saturation
% 2.56/1.93
% 2.56/1.93 % (2559)Memory used [KB]: 7547
% 2.56/1.93 % (2559)Time elapsed: 0.162 s
% 2.56/1.93 % (2559)Instructions burned: 510 (million)
% 2.56/1.93 % (2559)------------------------------
% 2.56/1.93 % (2559)------------------------------
% 2.56/1.94 % (2576)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_2187 for (2995ds/150Mi)
% 2.56/1.94 % (2578)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_2187 for (2995ds/109Mi)
% 2.56/1.94 % (2577)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_2187 for (2995ds/343Mi)
% 2.56/1.94 % (2579)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_2187 for (2995ds/253Mi)
% 2.56/1.95 % (2566)Instruction limit reached!
% 2.56/1.95 % (2566)------------------------------
% 2.56/1.95 % (2566)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.95 % (2566)Termination reason: Unknown
% 2.56/1.95 % (2566)Termination phase: Saturation
% 2.56/1.95
% 2.56/1.95 % (2566)Memory used [KB]: 3709
% 2.56/1.95 % (2566)Time elapsed: 0.109 s
% 2.56/1.95 % (2566)Instructions burned: 434 (million)
% 2.56/1.95 % (2566)------------------------------
% 2.56/1.95 % (2566)------------------------------
% 2.56/1.96 % (2580)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_2187 for (2995ds/140Mi)
% 2.56/1.97 % (2578)Instruction limit reached!
% 2.56/1.97 % (2578)------------------------------
% 2.56/1.97 % (2578)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.97 % (2578)Termination reason: Unknown
% 2.56/1.97 % (2578)Termination phase: Property scanning
% 2.56/1.97
% 2.56/1.97 % (2578)Memory used [KB]: 2174
% 2.56/1.97 % (2578)Time elapsed: 0.028 s
% 2.56/1.97 % (2578)Instructions burned: 109 (million)
% 2.56/1.97 % (2578)------------------------------
% 2.56/1.97 % (2578)------------------------------
% 2.56/1.98 % (2581)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_2187 for (2995ds/614Mi)
% 2.56/1.98 % (2576)Instruction limit reached!
% 2.56/1.98 % (2576)------------------------------
% 2.56/1.98 % (2576)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.98 % (2576)Termination reason: Unknown
% 2.56/1.98 % (2576)Termination phase: Saturation
% 2.56/1.98
% 2.56/1.98 % (2576)Memory used [KB]: 8187
% 2.56/1.98 % (2576)Time elapsed: 0.045 s
% 2.56/1.98 % (2576)Instructions burned: 152 (million)
% 2.56/1.98 % (2576)------------------------------
% 2.56/1.98 % (2576)------------------------------
% 2.56/1.99 % (2563)First to succeed.
% 2.56/1.99 % (2563)Refutation found. Thanks to Tanya!
% 2.56/1.99 % SZS status Theorem for DTF2THF_2187
% 2.56/1.99 % SZS output start Proof for DTF2THF_2187
% 2.56/1.99 thf(func_def_0, type, in: $i > $i > $o).
% 2.56/1.99 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.56/1.99 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.56/1.99 thf(func_def_8, type, powerset: $i > $i).
% 2.56/1.99 thf(func_def_10, type, setunion: $i > $i).
% 2.56/1.99 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.56/1.99 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.56/1.99 thf(func_def_26, type, prop2set: $o > $i).
% 2.56/1.99 thf(func_def_36, type, nonempty: $i > $o).
% 2.56/1.99 thf(func_def_69, type, set2prop: $i > $o).
% 2.56/1.99 thf(func_def_88, type, subset: $i > $i > $o).
% 2.56/1.99 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.56/1.99 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.56/1.99 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.56/1.99 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.56/1.99 thf(func_def_135, type, regular: $i > $o).
% 2.56/1.99 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.56/1.99 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.56/1.99 thf(func_def_153, type, iskpair: $i > $o).
% 2.56/1.99 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.56/1.99 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.56/1.99 thf(func_def_177, type, singleton: $i > $o).
% 2.56/1.99 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.56/1.99 thf(func_def_184, type, atmost1p: $i > $o).
% 2.56/1.99 thf(func_def_185, type, atleast2p: $i > $o).
% 2.56/1.99 thf(func_def_186, type, atmost2p: $i > $o).
% 2.56/1.99 thf(func_def_187, type, upairsetp: $i > $o).
% 2.56/1.99 thf(func_def_191, type, kfst: $i > $i).
% 2.56/1.99 thf(func_def_203, type, ksnd: $i > $i).
% 2.56/1.99 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.56/1.99 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.56/1.99 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.56/1.99 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.56/1.99 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.56/1.99 thf(func_def_241, type, sP0: ($i > $o) > $i > $o).
% 2.56/1.99 thf(func_def_243, type, sP2: $i > $i > $o).
% 2.56/1.99 thf(func_def_246, type, sP5: ($i > $o) > $i > $o).
% 2.56/1.99 thf(func_def_248, type, sP7: ($i > $o) > $o).
% 2.56/1.99 thf(func_def_250, type, sP9: $i > $i > $o > $o).
% 2.56/1.99 thf(func_def_252, type, sP11: $i > ($i > $i > $o) > $i > $o).
% 2.56/1.99 thf(func_def_253, type, sP12: $i > ($i > $i > $o) > $o).
% 2.56/1.99 thf(func_def_255, type, sP14: ($i > $o) > $o).
% 2.56/1.99 thf(func_def_256, type, sP15: $o > $i > $i > $i > $o).
% 2.56/1.99 thf(func_def_257, type, sP16: $i > $o > $i > $i > $o).
% 2.56/1.99 thf(func_def_259, type, sP18: ($i > $o) > $o).
% 2.56/1.99 thf(func_def_260, type, sP19: $i > ($i > $o) > ($i > $o) > $i > $o).
% 2.56/1.99 thf(func_def_261, type, sP20: $i > $o).
% 2.56/1.99 thf(func_def_262, type, sP21: $i > $o).
% 2.56/1.99 thf(func_def_263, type, sP22: $i > $i > $o).
% 2.56/1.99 thf(func_def_265, type, sP24: ($i > $o) > $i > $o).
% 2.56/1.99 thf(func_def_267, type, sP26: $o > $i > $i > $o).
% 2.56/1.99 thf(func_def_268, type, sP27: $i > $i > $i > $o).
% 2.56/1.99 thf(func_def_269, type, sP28: $i > $i > $i > $o).
% 2.56/1.99 thf(func_def_271, type, sP30: $i > ($i > $o) > $o).
% 2.56/1.99 thf(func_def_272, type, sP31: ($i > $o) > $i > $o).
% 2.56/1.99 thf(func_def_273, type, sP32: ($i > $o) > $i > $o).
% 2.56/1.99 thf(func_def_275, type, sP34: $i > $i > $o).
% 2.56/1.99 thf(func_def_277, type, sP36: ($i > $o) > $i > $o).
% 2.56/1.99 thf(func_def_279, type, sP38: ($i > $o) > $i > $o).
% 2.56/1.99 thf(func_def_282, type, sP41: ($i > $o) > $i > $o).
% 2.56/1.99 thf(func_def_283, type, sP42: ($i > $o) > $i > $o).
% 2.56/1.99 thf(func_def_285, type, sP44: $i > $i > $i > $o).
% 2.56/1.99 thf(func_def_286, type, sP45: $i > $i > $o).
% 2.56/1.99 thf(func_def_287, type, sP46: $i > $i > $o).
% 2.56/1.99 thf(func_def_288, type, sP47: $i > $i > $o).
% 2.56/1.99 thf(func_def_289, type, sP48: $i > $i > $o).
% 2.56/1.99 thf(func_def_290, type, sP49: $i > $o).
% 2.56/1.99 thf(func_def_291, type, sP50: $i > $i > $i > $o).
% 2.56/1.99 thf(func_def_292, type, sP51: ($i > $o) > $o).
% 2.56/1.99 thf(func_def_293, type, sP52: $i > ($i > $o) > $i > $o).
% 2.56/1.99 thf(func_def_294, type, sP53: $i > $o > $i > $o).
% 2.56/1.99 thf(func_def_297, type, sP56: ($i > $o) > $o).
% 2.56/1.99 thf(func_def_299, type, sP58: $i > $o).
% 2.56/1.99 thf(func_def_306, type, sK65: $i > $o).
% 2.56/1.99 thf(func_def_321, type, sK80: $i > $o).
% 2.56/1.99 thf(func_def_323, type, sK82: $i > ($i > $o) > $i).
% 2.56/1.99 thf(func_def_324, type, sK83: ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_327, type, sK86: $i > $i > $i).
% 2.56/1.99 thf(func_def_328, type, sK87: $i > $i > $i).
% 2.56/1.99 thf(func_def_344, type, sK103: $i > $i).
% 2.56/1.99 thf(func_def_362, type, sK121: $i > $o).
% 2.56/1.99 thf(func_def_364, type, sK123: $i > ($i > $o) > $i).
% 2.56/1.99 thf(func_def_369, type, sK128: $i > $o).
% 2.56/1.99 thf(func_def_370, type, sK129: ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_371, type, sK130: ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_376, type, sK135: $i > $o).
% 2.56/1.99 thf(func_def_380, type, sK139: $i > $i > $o).
% 2.56/1.99 thf(func_def_392, type, sK151: ($i > $o) > $i).
% 2.56/1.99 thf(func_def_393, type, sK152: $i > ($i > $o) > $i).
% 2.56/1.99 thf(func_def_394, type, sK153: $i > $o).
% 2.56/1.99 thf(func_def_404, type, sK163: $i > $i > $i).
% 2.56/1.99 thf(func_def_409, type, sK168: $i > $i > $o).
% 2.56/1.99 thf(func_def_423, type, sK182: $i > $i > $o).
% 2.56/1.99 thf(func_def_424, type, sK183: ($i > $i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_425, type, sK184: $i > $i > ($i > $i > $o) > $i).
% 2.56/1.99 thf(func_def_426, type, sK185: $i > ($i > $i > $o) > $i).
% 2.56/1.99 thf(func_def_427, type, sK186: $i > ($i > $i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_429, type, sK188: $i > ($i > $o) > $i).
% 2.56/1.99 thf(func_def_430, type, sK189: ($i > $o) > $i).
% 2.56/1.99 thf(func_def_431, type, sK190: $i > $o).
% 2.56/1.99 thf(func_def_436, type, sK195: $i > $o).
% 2.56/1.99 thf(func_def_440, type, sK199: $i > $o).
% 2.56/1.99 thf(func_def_443, type, sK202: $i > $o).
% 2.56/1.99 thf(func_def_460, type, sK219: $i > $o).
% 2.56/1.99 thf(func_def_461, type, sK220: $i > $o).
% 2.56/1.99 thf(func_def_462, type, sK221: $i > $o).
% 2.56/1.99 thf(func_def_463, type, sK222: ($i > $o) > ($i > $o) > $i).
% 2.56/1.99 thf(func_def_464, type, sK223: ($i > $o) > ($i > $o) > $i).
% 2.56/1.99 thf(func_def_479, type, sK238: ($i > $o) > $i).
% 2.56/1.99 thf(func_def_480, type, sK239: $i > ($i > $o) > $i).
% 2.56/1.99 thf(func_def_481, type, sK240: $i > $o).
% 2.56/1.99 thf(func_def_482, type, sK241: $i > $o).
% 2.56/1.99 thf(func_def_490, type, sK249: $i > $i > $o).
% 2.56/1.99 thf(func_def_491, type, sK250: $i > ($i > $o) > ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_492, type, sK251: $i > ($i > $o) > ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_495, type, sK254: $i > $o).
% 2.56/1.99 thf(func_def_496, type, sK255: $i > $o).
% 2.56/1.99 thf(func_def_500, type, sK259: $i > $i).
% 2.56/1.99 thf(func_def_501, type, sK260: $i > $i).
% 2.56/1.99 thf(func_def_503, type, sK262: $i > $o).
% 2.56/1.99 thf(func_def_520, type, sK279: $i > $i > $i).
% 2.56/1.99 thf(func_def_537, type, sK296: $i > $o).
% 2.56/1.99 thf(func_def_538, type, sK297: ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_539, type, sK298: ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_550, type, sK309: $i > $i).
% 2.56/1.99 thf(func_def_552, type, sK311: $i > $i > $i > $o).
% 2.56/1.99 thf(func_def_574, type, sK333: ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_575, type, sK334: ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_576, type, sK335: $i > ($i > $o) > $i).
% 2.56/1.99 thf(func_def_578, type, sK337: $i > $o).
% 2.56/1.99 thf(func_def_580, type, sK339: $i > $o).
% 2.56/1.99 thf(func_def_581, type, sK340: ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_582, type, sK341: ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_591, type, sK350: $i > $i > $o).
% 2.56/1.99 thf(func_def_600, type, sK359: $i > $i > $i).
% 2.56/1.99 thf(func_def_601, type, sK360: $i > $i).
% 2.56/1.99 thf(func_def_609, type, sK368: $i > $o).
% 2.56/1.99 thf(func_def_610, type, sK369: ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_611, type, sK370: ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_612, type, sK371: $i > $o).
% 2.56/1.99 thf(func_def_614, type, sK373: $i > ($i > $o) > $i).
% 2.56/1.99 thf(func_def_615, type, sK374: ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_617, type, sK376: $i > $o).
% 2.56/1.99 thf(func_def_625, type, sK384: $i > $i > $i).
% 2.56/1.99 thf(func_def_629, type, sK388: $i > $o).
% 2.56/1.99 thf(func_def_635, type, sK394: ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_636, type, sK395: ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_637, type, sK396: ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_639, type, sK398: $i > $o).
% 2.56/1.99 thf(func_def_666, type, sK425: $i > $o).
% 2.56/1.99 thf(func_def_667, type, sK426: $i > $o).
% 2.56/1.99 thf(func_def_668, type, sK427: ($i > $o) > ($i > $o) > $i).
% 2.56/1.99 thf(func_def_669, type, sK428: ($i > $o) > ($i > $o) > $i).
% 2.56/1.99 thf(func_def_670, type, sK429: $i > $i).
% 2.56/1.99 thf(func_def_671, type, sK430: $i > $i).
% 2.56/1.99 thf(func_def_672, type, sK431: $i > $i).
% 2.56/1.99 thf(func_def_673, type, sK432: $i > $i).
% 2.56/1.99 thf(func_def_674, type, sK433: $i > $i > $i).
% 2.56/1.99 thf(func_def_675, type, sK434: $i > $i > $i).
% 2.56/1.99 thf(func_def_676, type, sK435: $i > $i > $i).
% 2.56/1.99 thf(func_def_677, type, sK436: $i > $i > $i).
% 2.56/1.99 thf(func_def_678, type, sK437: $i > $i > $i).
% 2.56/1.99 thf(func_def_679, type, sK438: $i > $i > $i > $i).
% 2.56/1.99 thf(func_def_680, type, sK439: $i > $i > $i).
% 2.56/1.99 thf(func_def_681, type, sK440: $i > $i > $i).
% 2.56/1.99 thf(func_def_682, type, sK441: $i > $i > $i).
% 2.56/1.99 thf(func_def_683, type, sK442: $i > $i > $i).
% 2.56/1.99 thf(func_def_684, type, sK443: $i > $i > $i).
% 2.56/1.99 thf(func_def_685, type, sK444: $i > $i > $i > $i).
% 2.56/1.99 thf(func_def_686, type, sK445: $i > $i > $i > $i).
% 2.56/1.99 thf(func_def_687, type, sK446: $i > $i).
% 2.56/1.99 thf(func_def_699, type, sK458: $i > $o).
% 2.56/1.99 thf(func_def_700, type, sK459: ($i > $o) > $i).
% 2.56/1.99 thf(func_def_715, type, sK474: $i > $i).
% 2.56/1.99 thf(func_def_733, type, sK492: $i > $i > $i).
% 2.56/1.99 thf(func_def_742, type, sK501: $i > $o).
% 2.56/1.99 thf(func_def_746, type, sK505: $i > $i > $i > $i).
% 2.56/1.99 thf(func_def_747, type, sK506: $i > $i > $i > $i).
% 2.56/1.99 thf(func_def_751, type, sK510: $i > ($i > $o) > $i).
% 2.56/1.99 thf(func_def_752, type, sK511: ($i > $o) > $i).
% 2.56/1.99 thf(func_def_753, type, sK512: $i > $o).
% 2.56/1.99 thf(func_def_757, type, sK516: $i > ($i > $o) > $i > $i).
% 2.56/1.99 thf(func_def_758, type, sK517: $i > $o).
% 2.56/1.99 thf(func_def_763, type, sK522: $i > $o > $i > $i).
% 2.56/1.99 thf(func_def_771, type, sK530: $i > $i > $i).
% 2.56/1.99 thf(func_def_782, type, sK541: $i > $i > $i).
% 2.56/1.99 thf(func_def_785, type, sK544: $i > $i > $o).
% 2.56/1.99 thf(func_def_790, type, sK549: ($i > $o) > $i).
% 2.56/1.99 thf(func_def_791, type, sK550: ($i > $o) > $i).
% 2.56/1.99 thf(func_def_792, type, sK551: $i > $o).
% 2.56/1.99 thf(func_def_799, type, sK558: $i > $i > $o).
% 2.56/1.99 thf(func_def_806, type, sK565: $i > $i > $i).
% 2.56/1.99 thf(func_def_807, type, sK566: $i > $o).
% 2.56/1.99 thf(func_def_815, type, sK574: $i > $i > $i).
% 2.56/1.99 thf(func_def_816, type, sK575: $i > $i).
% 2.56/1.99 thf(func_def_833, type, sK592: $i > $i > $o).
% 2.56/1.99 thf(func_def_845, type, ph604: !>[X0: $tType]:(X0)).
% 2.56/1.99 thf(f5709,plain,(
% 2.56/1.99 $false),
% 2.56/1.99 inference(subsumption_resolution,[status(thm)],[f5708,f2166])).
% 2.56/1.99 thf(f2166,plain,(
% 2.56/1.99 ($true != (func @ sK172 @ sK173 @ sK174))),
% 2.56/1.99 inference(cnf_transformation,[status(thm)],[f1125])).
% 2.56/1.99 thf(f1125,plain,(
% 2.56/1.99 (quantDeMorgan4 = $true) & (setukpairinjR2 = $true) & (ex1I2 = $true) & (setunionsingleton = $true) & (prop2set2propI = $true) & (notdallE = $true) & (inPowerset = $true) & (binunionEcases = $true) & (upairsetIL = $true) & (cartprodpairmemER = $true) & (prop2setE = $true) & (setukpairinjL2 = $true) & (dsetconstrEL = $true) & (ubforcartprodlem2 = $true) & (singletonsubset = $true) & (in__Cong = $true) & (setunionE2 = $true) & (subsetemptysetimpeq = $true) & (kfstpairEq = $true) & (ex1I = $true) & (dpsetconstrEL1 = $true) & (powersetE1 = $true) & (symdiffI2 = $true) & (setunionsingleton2 = $true) & (setunionsingleton1 = $true) & (setext = $true) & (binintersectI = $true) & (binunionRsub = $true) & (nonemptyI1 = $true) & (setminusERneg = $true) & (powersetAx = $true) & (setadjoinIL = $true) & (app = $true) & (dpsetconstrEL2 = $true) & (omega0Ax = $true) & (quantDeMorgan2 = $true) & (omegaSAx = $true) & (dpsetconstrER = $true) & (setextsub = $true) & (ksndpairEq = $true) & (setminusSubset1 = $true) & (upairsubunion = $true) & (upairset2IR = $true) & (binintersectSubset3 = $true) & (emptysetsubset = $true) & (binintersectSubset1 = $true) & (setbeta = $true) & (dpsetconstrI = $true) & (notequalI1 = $true) & (exuI3 = $true) & (upairset2E = $true) & (dsetconstr__Cong = $true) & (emptyinunitempty = $true) & (setukpairinjR11 = $true) & (binunionLsub = $true) & (singletoninpowunion = $true) & (cartprodsndin = $true) & (ubforcartprodlem1 = $true) & (kfstsingleton = $true) & (setminusIRneg = $true) & (exuE1 = $true) & (subsetRefl = $true) & (subsetE = $true) & (cartprodsndpairEq = $true) & (binunionIR = $true) & (replAx = $true) & (eqimpsubset1 = $true) & (notequalI2 = $true) & (setunionI = $true) & (singletonsuniq = $true) & (exu__Cong = $true) & (emptyset__Cong = $true) & (setadjoinSub2 = $true) & (notdexE = $true) & (exuE3e = $true) & (upairinpowunion = $true) & (nonemptyImpWitness = $true) & (cartprodmempaircEq = $true) & (exuEu = $true) & (foundationAx = $true) & (subPowSU = $true) & (binintersectSubset2 = $true) & (binintersectRsub = $true) & (setOfPairsIsBReln = $true) & (subset2powerset = $true) & (nonemptyE1 = $true) & (binintersectEL = $true) & (nonemptyI = $true) & (upairsetIR = $true) & (exuI1 = $true) & (bs114d = $true) & (setadjoinSub = $true) & (subsetI1 = $true) & (quantDeMorgan3 = $true) & (binintersectLsub = $true) & (setukpairIR = $true) & (quantDeMorgan1 = $true) & (eqimpsubset2 = $true) & (sepSubset = $true) & (omega__Cong = $true) & (setminusSubset2 = $true) & (subsetI2 = $true) & (((in @ sK174 @ (funcSet @ sK172 @ sK173)) = $true) & ($true != (func @ sK172 @ sK173 @ sK174))) & (upairequniteq = $true) & (subsetE2 = $true) & (setminusI = $true) & (apProp = $true) & (notsubsetI = $true) & (omegaIndAx = $true) & (kpairiskpair = $true) & (descr__Cong = $true) & (ksndsingleton = $true) & (powersetI1 = $true) & (exuI2 = $true) & (powersetI = $true) & (inCongP = $true) & (funcImageSingleton = $true) & (binintersectSubset4 = $true) & (subsetTrans = $true) & (binunionE = $true) & (setukpairinjR = $true) & (powersetsubset = $true) & (singletonsswitch = $true) & (setukpairIL = $true) & (emptyI = $true) & (dsetconstrER = $true) & (kpairp = $true) & (noeltsimpempty = $true) & (cartprodmempair = $true) & (symdiffI1 = $true) & (setminusELneg = $true) & (vacuousDall = $true) & (emptyE1 = $true) & (cartprodfstin = $true) & (sepInPowerset = $true) & (theprop = $true) & (setminusEL = $true) & (setunion__Cong = $true) & (setadjoinE = $true) & (symdiffE = $true) & (ex1E1 = $true) & (dpsetconstrSub = $true) & (descrp = $true) & (upairsetE = $true) & (exuE2 = $true) & (secondinupair = $true) & (notinsingleton = $true) & (setoftrueEq = $true) & (setadjoinAx = $true) & (kpairsurjEq = $true) & (dpsetconstrERa = $true) & (emptyinPowerset = $true) & (ubforcartprodlem3 = $true) & (powerset__Cong = $true) & (cartprodpairmemEL = $true) & (powersetE = $true) & (notinemptyset = $true) & (binintersectER = $true) & (cartprodpairsurjEq = $true) & (setukpairinjR12 = $true) & (setukpairinjL = $true) & (uniqinunit = $true) & (symdiffIneg2 = $true) & (cartprodfstpairEq = $true) & (cartprodmempair1 = $true) & (setunionE = $true) & (symdiffIneg1 = $true) & (singletonprop = $true) & (setminusLsub = $true) & (binunionIL = $true) & (setadjoinIR = $true) & (dsetconstrI = $true) & (emptysetimpfalse = $true) & (cartprodpairin = $true) & (exuE3u = $true) & (wellorderingAx = $true) & (setukpairinjL1 = $true) & (setukpairinjR1 = $true) & (eqinunit = $true) & (setunionAx = $true) & (setminusER = $true) & (setminusILneg = $true) & (setextAx = $true) & (singletoninpowerset = $true) & (emptysetAx = $true) & (binintersectSubset5 = $true) & (emptyInPowerset = $true) & (prop2setI = $true) & (emptysetE = $true) & (disjointsetsI1 = $true) & (setadjoinOr = $true) & (setadjoin__Cong = $true)),
% 2.56/1.99 inference(skolemisation,[status(esa),new_symbols(skolem,[sK172,sK173,sK174])],[f792,f1124])).
% 2.56/1.99 thf(f1124,plain,(
% 2.56/1.99 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (funcSet @ X0 @ X1))) & ($true != (func @ X0 @ X1 @ X2))) => (((in @ sK174 @ (funcSet @ sK172 @ sK173)) = $true) & ($true != (func @ sK172 @ sK173 @ sK174)))),
% 2.56/1.99 introduced(definition,[],[choice_axiom])).
% 2.56/1.99 thf(f792,plain,(
% 2.56/1.99 (quantDeMorgan4 = $true) & (setukpairinjR2 = $true) & (ex1I2 = $true) & (setunionsingleton = $true) & (prop2set2propI = $true) & (notdallE = $true) & (inPowerset = $true) & (binunionEcases = $true) & (upairsetIL = $true) & (cartprodpairmemER = $true) & (prop2setE = $true) & (setukpairinjL2 = $true) & (dsetconstrEL = $true) & (ubforcartprodlem2 = $true) & (singletonsubset = $true) & (in__Cong = $true) & (setunionE2 = $true) & (subsetemptysetimpeq = $true) & (kfstpairEq = $true) & (ex1I = $true) & (dpsetconstrEL1 = $true) & (powersetE1 = $true) & (symdiffI2 = $true) & (setunionsingleton2 = $true) & (setunionsingleton1 = $true) & (setext = $true) & (binintersectI = $true) & (binunionRsub = $true) & (nonemptyI1 = $true) & (setminusERneg = $true) & (powersetAx = $true) & (setadjoinIL = $true) & (app = $true) & (dpsetconstrEL2 = $true) & (omega0Ax = $true) & (quantDeMorgan2 = $true) & (omegaSAx = $true) & (dpsetconstrER = $true) & (setextsub = $true) & (ksndpairEq = $true) & (setminusSubset1 = $true) & (upairsubunion = $true) & (upairset2IR = $true) & (binintersectSubset3 = $true) & (emptysetsubset = $true) & (binintersectSubset1 = $true) & (setbeta = $true) & (dpsetconstrI = $true) & (notequalI1 = $true) & (exuI3 = $true) & (upairset2E = $true) & (dsetconstr__Cong = $true) & (emptyinunitempty = $true) & (setukpairinjR11 = $true) & (binunionLsub = $true) & (singletoninpowunion = $true) & (cartprodsndin = $true) & (ubforcartprodlem1 = $true) & (kfstsingleton = $true) & (setminusIRneg = $true) & (exuE1 = $true) & (subsetRefl = $true) & (subsetE = $true) & (cartprodsndpairEq = $true) & (binunionIR = $true) & (replAx = $true) & (eqimpsubset1 = $true) & (notequalI2 = $true) & (setunionI = $true) & (singletonsuniq = $true) & (exu__Cong = $true) & (emptyset__Cong = $true) & (setadjoinSub2 = $true) & (notdexE = $true) & (exuE3e = $true) & (upairinpowunion = $true) & (nonemptyImpWitness = $true) & (cartprodmempaircEq = $true) & (exuEu = $true) & (foundationAx = $true) & (subPowSU = $true) & (binintersectSubset2 = $true) & (binintersectRsub = $true) & (setOfPairsIsBReln = $true) & (subset2powerset = $true) & (nonemptyE1 = $true) & (binintersectEL = $true) & (nonemptyI = $true) & (upairsetIR = $true) & (exuI1 = $true) & (bs114d = $true) & (setadjoinSub = $true) & (subsetI1 = $true) & (quantDeMorgan3 = $true) & (binintersectLsub = $true) & (setukpairIR = $true) & (quantDeMorgan1 = $true) & (eqimpsubset2 = $true) & (sepSubset = $true) & (omega__Cong = $true) & (setminusSubset2 = $true) & (subsetI2 = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (funcSet @ X0 @ X1))) & ($true != (func @ X0 @ X1 @ X2))) & (upairequniteq = $true) & (subsetE2 = $true) & (setminusI = $true) & (apProp = $true) & (notsubsetI = $true) & (omegaIndAx = $true) & (kpairiskpair = $true) & (descr__Cong = $true) & (ksndsingleton = $true) & (powersetI1 = $true) & (exuI2 = $true) & (powersetI = $true) & (inCongP = $true) & (funcImageSingleton = $true) & (binintersectSubset4 = $true) & (subsetTrans = $true) & (binunionE = $true) & (setukpairinjR = $true) & (powersetsubset = $true) & (singletonsswitch = $true) & (setukpairIL = $true) & (emptyI = $true) & (dsetconstrER = $true) & (kpairp = $true) & (noeltsimpempty = $true) & (cartprodmempair = $true) & (symdiffI1 = $true) & (setminusELneg = $true) & (vacuousDall = $true) & (emptyE1 = $true) & (cartprodfstin = $true) & (sepInPowerset = $true) & (theprop = $true) & (setminusEL = $true) & (setunion__Cong = $true) & (setadjoinE = $true) & (symdiffE = $true) & (ex1E1 = $true) & (dpsetconstrSub = $true) & (descrp = $true) & (upairsetE = $true) & (exuE2 = $true) & (secondinupair = $true) & (notinsingleton = $true) & (setoftrueEq = $true) & (setadjoinAx = $true) & (kpairsurjEq = $true) & (dpsetconstrERa = $true) & (emptyinPowerset = $true) & (ubforcartprodlem3 = $true) & (powerset__Cong = $true) & (cartprodpairmemEL = $true) & (powersetE = $true) & (notinemptyset = $true) & (binintersectER = $true) & (cartprodpairsurjEq = $true) & (setukpairinjR12 = $true) & (setukpairinjL = $true) & (uniqinunit = $true) & (symdiffIneg2 = $true) & (cartprodfstpairEq = $true) & (cartprodmempair1 = $true) & (setunionE = $true) & (symdiffIneg1 = $true) & (singletonprop = $true) & (setminusLsub = $true) & (binunionIL = $true) & (setadjoinIR = $true) & (dsetconstrI = $true) & (emptysetimpfalse = $true) & (cartprodpairin = $true) & (exuE3u = $true) & (wellorderingAx = $true) & (setukpairinjL1 = $true) & (setukpairinjR1 = $true) & (eqinunit = $true) & (setunionAx = $true) & (setminusER = $true) & (setminusILneg = $true) & (setextAx = $true) & (singletoninpowerset = $true) & (emptysetAx = $true) & (binintersectSubset5 = $true) & (emptyInPowerset = $true) & (prop2setI = $true) & (emptysetE = $true) & (disjointsetsI1 = $true) & (setadjoinOr = $true) & (setadjoin__Cong = $true)),
% 2.56/1.99 inference(flattening,[status(thm)],[f791])).
% 2.56/1.99 thf(f791,plain,(
% 2.56/1.99 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (funcSet @ X0 @ X1))) & ($true != (func @ X0 @ X1 @ X2))) & (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)),
% 2.56/1.99 inference(ennf_transformation,[status(thm)],[f208])).
% 2.56/1.99 thf(f208,plain,(
% 2.56/1.99 ~((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) => ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (funcSet @ X0 @ X1))) => ($true = (func @ X0 @ X1 @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.56/1.99 inference(fool_elimination,[status(thm)],[f207])).
% 2.56/1.99 thf(f207,plain,(
% 2.56/1.99 ~(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 => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (funcSet @ X0 @ X1)) => (func @ X0 @ X1 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.56/1.99 inference(rectify,[status(thm)],[f201])).
% 2.56/1.99 thf(f201,negated_conjecture,(
% 2.56/1.99 ~(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 => ! [X3 : $i,X4 : $i,X13 : $i] : ((in @ X13 @ (funcSet @ X3 @ X4)) => (func @ X3 @ X4 @ X13)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.56/1.99 inference(negated_conjecture,[status(cth)],[f200])).
% 2.56/1.99 thf(f200,conjecture,(
% 2.56/1.99 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 => ! [X3 : $i,X4 : $i,X13 : $i] : ((in @ X13 @ (funcSet @ X3 @ X4)) => (func @ X3 @ X4 @ X13))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.56/1.99 file('/export/starexec/sandbox/tmp/tmp.T6NCxVKBVZ/DTF2THF_2187.p',infuncsetfunc)).
% 2.56/1.99 thf(f5708,plain,(
% 2.56/1.99 ($true = (func @ sK172 @ sK173 @ sK174))),
% 2.56/1.99 inference(trivial_inequality_removal,[status(thm)],[f5696])).
% 2.56/1.99 thf(f5696,plain,(
% 2.56/1.99 ($true = (func @ sK172 @ sK173 @ sK174)) | ($true != $true)),
% 2.56/1.99 inference(superposition,[status(thm)],[f3640,f3625])).
% 2.56/1.99 thf(f3625,plain,(
% 2.56/1.99 ($true = (in @ sK174 @ (dsetconstr @ (powerset @ (cartprod @ sK172 @ sK173)) @ (func @ sK172 @ sK173))))),
% 2.56/1.99 inference(beta_eta_normalization,[status(thm)],[f3051])).
% 2.56/1.99 thf(f3051,plain,(
% 2.56/1.99 ($true = (in @ sK174 @ ((^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (powerset @ (cartprod @ Y0 @ Y1)) @ (^[Y2 : $i]: (func @ Y0 @ Y1 @ Y2)))))) @ sK172 @ sK173)))),
% 2.56/1.99 inference(definition_unfolding,[status(thm)],[f2167,f2359])).
% 2.56/1.99 thf(f2359,plain,(
% 2.56/1.99 (funcSet = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (powerset @ (cartprod @ Y0 @ Y1)) @ (^[Y2 : $i]: (func @ Y0 @ Y1 @ Y2)))))))),
% 2.56/1.99 inference(cnf_transformation,[status(thm)],[f472])).
% 2.56/1.99 thf(f472,plain,(
% 2.56/1.99 (funcSet = (^[Y0 : $i]: ((^[Y1 : $i]: (dsetconstr @ (powerset @ (cartprod @ Y0 @ Y1)) @ (^[Y2 : $i]: (func @ Y0 @ Y1 @ Y2)))))))),
% 2.56/1.99 inference(fool_elimination,[status(thm)],[f471])).
% 2.56/1.99 thf(f471,plain,(
% 2.56/1.99 (funcSet = (^[X0 : $i, X1 : $i] : (dsetconstr @ (powerset @ (cartprod @ X0 @ X1)) @ (^[X2 : $i] : (func @ X0 @ X1 @ X2)))))),
% 2.56/1.99 inference(rectify,[status(thm)],[f195])).
% 2.56/1.99 thf(f195,axiom,(
% 2.56/1.99 (funcSet = (^[X3 : $i, X4 : $i] : (dsetconstr @ (powerset @ (cartprod @ X3 @ X4)) @ (^[X13 : $i] : (func @ X3 @ X4 @ X13)))))),
% 2.56/1.99 file('/export/starexec/sandbox/tmp/tmp.T6NCxVKBVZ/DTF2THF_2187.p',funcSet)).
% 2.56/1.99 thf(f2167,plain,(
% 2.56/1.99 ((in @ sK174 @ (funcSet @ sK172 @ sK173)) = $true)),
% 2.56/1.99 inference(cnf_transformation,[status(thm)],[f1125])).
% 2.56/1.99 thf(f3640,plain,(
% 2.56/1.99 ( ! [X3 : $i > $o,X4 : $i,X5 : $i] : (($true != (in @ X4 @ (dsetconstr @ X5 @ X3))) | ($true = (X3 @ X4))) )),
% 2.56/1.99 inference(beta_eta_normalization,[status(thm)],[f3639])).
% 2.56/1.99 thf(f3639,plain,(
% 2.56/1.99 ( ! [X3 : $i > $o,X4 : $i,X5 : $i] : (($true != (in @ X4 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X3 @ Y0))))) | ($true = (X3 @ X4))) )),
% 2.56/1.99 inference(trivial_inequality_removal,[status(thm)],[f3073])).
% 2.56/1.99 thf(f3073,plain,(
% 2.56/1.99 ( ! [X3 : $i > $o,X4 : $i,X5 : $i] : (($true != (in @ X4 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X3 @ Y0))))) | ($true = (X3 @ X4)) | ($true != $true)) )),
% 2.56/1.99 inference(definition_unfolding,[status(thm)],[f2313,f2143])).
% 2.56/1.99 thf(f2143,plain,(
% 2.56/1.99 (dsetconstrER = $true)),
% 2.56/1.99 inference(cnf_transformation,[status(thm)],[f1125])).
% 2.56/1.99 thf(f2313,plain,(
% 2.56/1.99 ( ! [X3 : $i > $o,X4 : $i,X5 : $i] : (($true != (in @ X4 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X3 @ Y0))))) | ($true = (X3 @ X4)) | (dsetconstrER != $true)) )),
% 2.56/1.99 inference(cnf_transformation,[status(thm)],[f1181])).
% 2.56/1.99 thf(f1181,plain,(
% 2.56/1.99 ((dsetconstrER = $true) | (($true = (in @ sK200 @ (dsetconstr @ sK201 @ (^[Y0 : $i]: (sK199 @ Y0))))) & ((sK199 @ sK200) != $true))) & (! [X3 : $i > $o,X4 : $i,X5 : $i] : (($true != (in @ X4 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X3 @ Y0))))) | ($true = (X3 @ X4))) | (dsetconstrER != $true))),
% 2.56/1.99 inference(skolemisation,[status(esa),new_symbols(skolem,[sK199,sK200,sK201])],[f1179,f1180])).
% 2.56/1.99 thf(f1180,plain,(
% 2.56/1.99 ? [X0 : $i > $o,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0))))) & ((X0 @ X1) != $true)) => (($true = (in @ sK200 @ (dsetconstr @ sK201 @ (^[Y0 : $i]: (sK199 @ Y0))))) & ((sK199 @ sK200) != $true))),
% 2.56/1.99 introduced(definition,[],[choice_axiom])).
% 2.56/1.99 thf(f1179,plain,(
% 2.56/1.99 ((dsetconstrER = $true) | ? [X0 : $i > $o,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0))))) & ((X0 @ X1) != $true))) & (! [X3 : $i > $o,X4 : $i,X5 : $i] : (($true != (in @ X4 @ (dsetconstr @ X5 @ (^[Y0 : $i]: (X3 @ Y0))))) | ($true = (X3 @ X4))) | (dsetconstrER != $true))),
% 2.56/1.99 inference(rectify,[status(thm)],[f1178])).
% 2.56/1.99 thf(f1178,plain,(
% 2.56/1.99 ((dsetconstrER = $true) | ? [X0 : $i > $o,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0))))) & ((X0 @ X1) != $true))) & (! [X0 : $i > $o,X1 : $i,X2 : $i] : (($true != (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0))))) | ((X0 @ X1) = $true)) | (dsetconstrER != $true))),
% 2.56/1.99 inference(nnf_transformation,[status(thm)],[f774])).
% 2.56/1.99 thf(f774,plain,(
% 2.56/1.99 (dsetconstrER = $true) <=> ! [X0 : $i > $o,X1 : $i,X2 : $i] : (($true != (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0))))) | ((X0 @ X1) = $true))),
% 2.56/1.99 inference(ennf_transformation,[status(thm)],[f546])).
% 2.56/1.99 thf(f546,plain,(
% 2.56/1.99 (dsetconstrER = $true) <=> ! [X0 : $i > $o,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (dsetconstr @ X2 @ (^[Y0 : $i]: (X0 @ Y0))))) => ((X0 @ X1) = $true))),
% 2.56/1.99 inference(fool_elimination,[status(thm)],[f545])).
% 2.56/1.99 thf(f545,plain,(
% 2.56/1.99 (dsetconstrER = ! [X0 : $i > $o,X1 : $i,X2 : $i] : ((in @ X1 @ (dsetconstr @ X2 @ (^[X3 : $i] : (X0 @ X3)))) => (X0 @ X1)))),
% 2.56/1.99 inference(rectify,[status(thm)],[f16])).
% 2.56/1.99 thf(f16,axiom,(
% 2.56/1.99 (dsetconstrER = ! [X0 : $i > $o,X1 : $i,X3 : $i] : ((in @ X1 @ (dsetconstr @ X3 @ (^[X2 : $i] : (X0 @ X2)))) => (X0 @ X1)))),
% 2.56/1.99 file('/export/starexec/sandbox/tmp/tmp.T6NCxVKBVZ/DTF2THF_2187.p',dsetconstrER)).
% 2.56/1.99 % SZS output end Proof for DTF2THF_2187
% 2.56/1.99 % (2563)------------------------------
% 2.56/1.99 % (2563)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.56/1.99 % (2563)Termination reason: Refutation
% 2.56/1.99
% 2.56/1.99 % (2563)Memory used [KB]: 9722
% 2.56/1.99 % (2563)Time elapsed: 0.179 s
% 2.56/1.99 % (2563)Instructions burned: 559 (million)
% 2.56/1.99 % (2563)------------------------------
% 2.56/1.99 % (2563)------------------------------
% 2.56/1.99 % (2515)Success in time 0.501 s
%------------------------------------------------------------------------------