%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU724^1 : 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 : n025.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 09:36:41 AM UTC 2026
% Result : Theorem 2.23s 1.72s
% Output : Refutation 2.23s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 6
% Syntax : Number of formulae : 34 ( 9 unt; 0 typ; 0 def)
% Number of atoms : 4811 (1332 equ; 0 cnn)
% Maximal formula atoms : 246 ( 141 avg)
% Number of connectives : 2404 ( 69 ~; 28 |; 772 &; 294 @)
% ( 4 <=>;1237 =>; 0 <=; 0 <~>)
% Maximal formula depth : 254 ( 63 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 438 ( 435 usr; 249 con; 0-4 aty)
% Number of variables : 81 ( 0 ^; 54 !; 27 ?; 81 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_259,type,
if: $i > $o > $i > $i > $i ).
thf(func_def_292,type,
sP0: $i > $i > $i > $o > $o ).
thf(func_def_296,type,
sP4: $i > $i > $o ).
thf(func_def_297,type,
sP5: $i > $o ).
thf(func_def_298,type,
sP6: $i > $i > $o ).
thf(func_def_299,type,
sP7: $i > $i > $o ).
thf(func_def_301,type,
sP9: $i > $i > $i > $o ).
thf(func_def_302,type,
sP10: $i > $i > $i > $i > $o ).
thf(func_def_310,type,
sK18: $i > $o ).
thf(func_def_311,type,
sK19: $i > $o ).
thf(func_def_312,type,
sK20: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_313,type,
sK21: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_321,type,
sK29: ( $i > $i ) > $i > $i > $i ).
thf(func_def_324,type,
sK32: $i > $i ).
thf(func_def_356,type,
sK64: $i > $o ).
thf(func_def_358,type,
sK66: $i > $i > $i ).
thf(func_def_378,type,
sK86: $i > $o ).
thf(func_def_380,type,
sK88: $i > ( $i > $o ) > $i ).
thf(func_def_382,type,
sK90: $i > $o ).
thf(func_def_384,type,
sK92: ( $i > $o ) > $i > $i ).
thf(func_def_392,type,
sK100: $i > $i > $i > $i > $i ).
thf(func_def_396,type,
sK104: $i > $o ).
thf(func_def_402,type,
sK110: $i > $o ).
thf(func_def_411,type,
sK119: $i > $o ).
thf(func_def_413,type,
sK121: ( $i > $o ) > $i ).
thf(func_def_414,type,
sK122: ( $i > $o ) > $i ).
thf(func_def_427,type,
sK135: $i > $i > $i > $i > $i ).
thf(func_def_428,type,
sK136: $i > $i > $o ).
thf(func_def_430,type,
sK138: $i > $i ).
thf(func_def_431,type,
sK139: $i > $i ).
thf(func_def_432,type,
sK140: $i > ( $i > $i > $o ) > $i ).
thf(func_def_433,type,
sK141: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_434,type,
sK142: $i > ( $i > $i > $o ) > $i ).
thf(func_def_448,type,
sK156: $i > $o ).
thf(func_def_455,type,
sK163: $i > $o ).
thf(func_def_456,type,
sK164: $i > $o ).
thf(func_def_457,type,
sK165: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_458,type,
sK166: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_464,type,
sK172: $i > $o ).
thf(func_def_465,type,
sK173: ( $i > $o ) > $i > $i ).
thf(func_def_474,type,
sK182: $i > $i > $o ).
thf(func_def_479,type,
sK187: $i > $o ).
thf(func_def_481,type,
sK189: $i > ( $i > $o ) > $i > $i ).
thf(func_def_482,type,
sK190: ( $i > $o ) > $i ).
thf(func_def_483,type,
sK191: $i > $o ).
thf(func_def_490,type,
sK198: $i > $i > $i > $i > $i ).
thf(func_def_491,type,
sK199: $i > $i > $i > $i > $i ).
thf(func_def_504,type,
sK212: $i > $i > $o ).
thf(func_def_513,type,
sK221: $i > $i ).
thf(func_def_516,type,
sK224: $i > $o ).
thf(func_def_526,type,
sK234: $i > $i > $i ).
thf(func_def_527,type,
sK235: $i > $i > $i ).
thf(func_def_538,type,
sK246: $i > $o ).
thf(func_def_541,type,
sK249: $i > ( $i > $o ) > $i ).
thf(func_def_542,type,
sK250: $i > $i > $i ).
thf(func_def_553,type,
sK261: $i > $i > $i > $i ).
thf(func_def_554,type,
sK262: $i > $i > $i > $i ).
thf(func_def_565,type,
sK273: $i > $i ).
thf(func_def_566,type,
sK274: ( $i > $i ) > $i > $i > $i ).
thf(func_def_567,type,
sK275: $i > $i > $o ).
thf(func_def_593,type,
sK301: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_594,type,
sK302: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_598,type,
sK306: $i > $o ).
thf(func_def_610,type,
sK318: $i > $i ).
thf(func_def_612,type,
sK320: ( $i > $o ) > $i > $i ).
thf(func_def_614,type,
sK322: $i > $o ).
thf(func_def_619,type,
sK327: $i > $i > ( $i > $i ) > $i ).
thf(func_def_620,type,
sK328: $i > $i ).
thf(func_def_637,type,
sK345: $i > $i > $i > $i ).
thf(func_def_645,type,
sK353: $i > $i > $o ).
thf(func_def_676,type,
sK384: $i > $i ).
thf(func_def_683,type,
sK391: $i > $i > $i ).
thf(func_def_686,type,
sK394: $i > $o ).
thf(func_def_699,type,
sK407: $i > $i > $i ).
thf(func_def_701,type,
sK409: $i > $o ).
thf(func_def_711,type,
sK419: ( $i > $o ) > $i > $i ).
thf(func_def_713,type,
sK421: $i > $o ).
thf(func_def_721,type,
sK429: $i > $o ).
thf(func_def_729,type,
sK437: $i > $i > $i > $i > $i ).
thf(func_def_730,type,
sK438: $i > $o ).
thf(func_def_733,type,
sK441: $i > ( $i > $o ) > $i ).
thf(func_def_734,type,
sK442: $i > ( $i > $o ) > $i ).
thf(func_def_740,type,
sK448: $i > $i > $o ).
thf(func_def_745,type,
sK453: $i > $i > $i > $i ).
thf(func_def_750,type,
sK458: $i > $i > $i ).
thf(func_def_753,type,
sK461: $i > $o ).
thf(func_def_755,type,
sK463: $i > ( $i > $o ) > $i ).
thf(func_def_756,type,
sK464: $i > $i > $i ).
thf(func_def_757,type,
sK465: $i > $i > $i ).
thf(func_def_758,type,
sK466: $i > $i > $i ).
thf(func_def_759,type,
sK467: $i > $i > $i ).
thf(func_def_760,type,
sK468: $i > $i > $i > $i ).
thf(func_def_761,type,
sK469: $i > $i ).
thf(func_def_762,type,
sK470: $i > $i ).
thf(func_def_763,type,
sK471: $i > $i ).
thf(func_def_764,type,
sK472: $i > $i ).
thf(func_def_765,type,
sK473: $i > $i > $i ).
thf(func_def_766,type,
sK474: $i > $i > $i > $i ).
thf(func_def_767,type,
sK475: $i > $i > $i > $i ).
thf(func_def_768,type,
sK476: $i > $i > $i ).
thf(func_def_769,type,
sK477: $i > $i > $i ).
thf(func_def_770,type,
sK478: $i > $i ).
thf(func_def_771,type,
sK479: $i > $i > $i ).
thf(func_def_773,type,
sK481: $i > $i ).
thf(func_def_774,type,
sK482: $i > $i ).
thf(func_def_775,type,
sK483: $i > $i > $o ).
thf(func_def_785,type,
sK493: $i > $o ).
thf(func_def_787,type,
sK495: ( $i > $o ) > $i > $i ).
thf(func_def_788,type,
sK496: ( $i > $o ) > $i > $i ).
thf(func_def_791,type,
sK499: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_792,type,
sK500: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_793,type,
sK501: $i > $o ).
thf(func_def_794,type,
sK502: $i > $o ).
thf(func_def_795,type,
sK503: $i > $i ).
thf(func_def_797,type,
sK505: $i > $i ).
thf(func_def_839,type,
sK547: $i > $o ).
thf(func_def_842,type,
sK550: $i > $i ).
thf(func_def_856,type,
sK564: $i > $i > $i > $i ).
thf(func_def_857,type,
sK565: $i > $i > $i > $i ).
thf(func_def_863,type,
sK571: $i > $o ).
thf(func_def_878,type,
sK586: ( $i > $o ) > $i > $i ).
thf(func_def_880,type,
sK588: $i > $o ).
thf(func_def_887,type,
sK595: $i > $i > $i ).
thf(func_def_892,type,
sK600: $i > $o ).
thf(func_def_894,type,
sK602: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_895,type,
sK603: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_899,type,
sK607: $i > $i > $o ).
thf(func_def_914,type,
sK622: $i > $i > $i > $i > $i ).
thf(func_def_915,type,
sK623: $i > $i > $i > $i > $i ).
thf(func_def_916,type,
sK624: $i > $i > $i > $i > $i ).
thf(func_def_917,type,
sK625: $i > $i > $i > $i > $i ).
thf(func_def_959,type,
sK667: $i > $i ).
thf(func_def_962,type,
sK670: $i > $i > ( $i > $i ) > $i ).
thf(func_def_965,type,
sK673: $i > $o ).
thf(func_def_968,type,
sK676: $i > $i ).
thf(func_def_971,type,
sK679: $i > $i > ( $i > $i ) > $i ).
thf(func_def_972,type,
sK680: $i > $o ).
thf(func_def_975,type,
sK683: $i > $o ).
thf(func_def_976,type,
sK684: $i > $i ).
thf(func_def_977,type,
sK685: ( $i > $o ) > $i ).
thf(func_def_992,type,
sK700: $i > $o ).
thf(func_def_998,type,
sK706: $i > $i > $i ).
thf(func_def_1003,type,
sK711: $o > $i > $i > $i ).
thf(func_def_1018,type,
sK726: $i > $i ).
thf(func_def_1020,type,
sK728: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1025,type,
sK733: $i > $i > $i ).
thf(func_def_1028,type,
sK736: $i > $o ).
thf(func_def_1029,type,
sK737: $i > $i ).
thf(func_def_1030,type,
sK738: ( $i > $o ) > $i ).
thf(func_def_1045,type,
sK753: $i > ( $i > $o ) > $i ).
thf(func_def_1046,type,
sK754: $i > $o ).
thf(func_def_1050,type,
sK758: $i > $i ).
thf(func_def_1056,type,
ph764:
!>[X0: $tType] : X0 ).
thf(f7043,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f7032]) ).
thf(f7032,plain,
$true != $true,
inference(superposition,[status(thm)],[f3827,f3293]) ).
thf(f3293,plain,
( ( in @ sK659 @ sK656 )
= $true ),
inference(cnf_transformation,[status(thm)],[f2058]) ).
thf(f2058,plain,
( ( setadjoinOr = $true )
& ( emptyinPowerset = $true )
& ( iftrueProp1 = $true )
& ( binunionRsub = $true )
& ( eqinunit = $true )
& ( funcGraphProp4 = $true )
& ( binintersectSubset3 = $true )
& ( eqimpsubset2 = $true )
& ( brelnall2 = $true )
& ( binintersectT_lem = $true )
& ( setukpairIR = $true )
& ( exuI1 = $true )
& ( setminusSubset1 = $true )
& ( binintersectSubset1 = $true )
& ( exuE3u = $true )
& ( iffalseProp1 = $true )
& ( bs114d = $true )
& ( subsetRefl = $true )
& ( subsetI2 = $true )
& ( omegaSAx = $true )
& ( dsetconstrI = $true )
& ( binunionEcases = $true )
& ( powerset__Cong = $true )
& ( notinsingleton = $true )
& ( kfstpairEq = $true )
& ( secondinupair = $true )
& ( dpsetconstrSub = $true )
& ( setukpairinjL = $true )
& ( uniqinunit = $true )
& ( binintersectLsub = $true )
& ( setukpairinjR = $true )
& ( emptysetsubset = $true )
& ( $true
= ( in @ sK656 @ ( powerset @ sK657 ) ) )
& ( $true
= ( in @ sK658 @ ( powerset @ sK657 ) ) )
& ( $true
= ( in @ sK659 @ sK657 ) )
& ( $true
= ( subset @ sK656 @ sK658 ) )
& ( ( in @ sK659 @ sK658 )
!= $true )
& ( ( in @ sK659 @ sK656 )
= $true )
& ( brelnall1 = $true )
& ( cartprodmempair1 = $true )
& ( eqbreln = $true )
& ( cartprodsndin = $true )
& ( binunionIR = $true )
& ( app = $true )
& ( complementT_lem = $true )
& ( setadjoinSub2 = $true )
& ( ksndpairEq = $true )
& ( setextsub = $true )
& ( emptyinunitempty = $true )
& ( setunionsingleton = $true )
& ( dpsetconstrER = $true )
& ( quantDeMorgan1 = $true )
& ( cartprodpairin = $true )
& ( ap2p = $true )
& ( binintersectSubset5 = $true )
& ( notequalI2 = $true )
& ( setext = $true )
& ( nonemptyI1 = $true )
& ( notsubsetI = $true )
& ( lam2p = $true )
& ( setukpairinjR12 = $true )
& ( beta2 = $true )
& ( replAx = $true )
& ( ap2apEq2 = $true )
& ( subsetemptysetimpeq = $true )
& ( eta2 = $true )
& ( setukpairIL = $true )
& ( dsetconstrER = $true )
& ( setadjoinIL = $true )
& ( notdexE = $true )
& ( setoftrueEq = $true )
& ( funcGraphProp2 = $true )
& ( upairset2IR = $true )
& ( subsetTI = $true )
& ( setadjoinSub = $true )
& ( binunionIL = $true )
& ( kfstsingleton = $true )
& ( funcImageSingleton = $true )
& ( binunionE = $true )
& ( notdallE = $true )
& ( subsetE = $true )
& ( singletonprop = $true )
& ( setminusLsub = $true )
& ( ifp = $true )
& ( subsetE2 = $true )
& ( exuI3 = $true )
& ( setminusSubset2 = $true )
& ( symdiffIneg2 = $true )
& ( upairset2E = $true )
& ( subPowSU = $true )
& ( singletoninpowunion = $true )
& ( quantDeMorgan2 = $true )
& ( setextAx = $true )
& ( binintersectSubset4 = $true )
& ( setOfPairsIsBReln = $true )
& ( setminusI = $true )
& ( setminusEL = $true )
& ( setunion__Cong = $true )
& ( emptyInPowerset = $true )
& ( setminusIRneg = $true )
& ( iftrue = $true )
& ( symdiffE = $true )
& ( exuE3e = $true )
& ( setunionE2 = $true )
& ( exuE2 = $true )
& ( setunionI = $true )
& ( ap2apEq1 = $true )
& ( contrasubsetT = $true )
& ( cartprodpairmemER = $true )
& ( prop2setI = $true )
& ( setadjoin__Cong = $true )
& ( exuEu = $true )
& ( setunionsingleton2 = $true )
& ( cartprodmempair = $true )
& ( eqimpsubset1 = $true )
& ( emptysetE = $true )
& ( noeltsimpempty = $true )
& ( cartprodfstin = $true )
& ( eta1 = $true )
& ( subsetI1 = $true )
& ( iftrueProp2 = $true )
& ( powersetI = $true )
& ( cartprodpairsurjEq = $true )
& ( binunionT_lem = $true )
& ( setadjoinAx = $true )
& ( complementTI1 = $true )
& ( emptyI = $true )
& ( kpairiskpair = $true )
& ( emptyE1 = $true )
& ( exuI2 = $true )
& ( setextT = $true )
& ( theprop = $true )
& ( emptysetAx = $true )
& ( ex1I2 = $true )
& ( funcextLem = $true )
& ( cartprodsndpairEq = $true )
& ( notequalI1 = $true )
& ( powersetI1 = $true )
& ( singletonsswitch = $true )
& ( in__Cong = $true )
& ( dpsetconstrEL2 = $true )
& ( omegaIndAx = $true )
& ( dpsetconstrERa = $true )
& ( ksndsingleton = $true )
& ( binintersectRsub = $true )
& ( quantDeMorgan3 = $true )
& ( apProp = $true )
& ( theeq = $true )
& ( setukpairinjR1 = $true )
& ( lamProp = $true )
& ( cartprodfstpairEq = $true )
& ( descrp = $true )
& ( binintersectEL = $true )
& ( sepInPowerset = $true )
& ( setunionAx = $true )
& ( upairsetIL = $true )
& ( lam2lamEq = $true )
& ( cartprodmempaircEq = $true )
& ( dsetconstrEL = $true )
& ( notinemptyset = $true )
& ( omega0Ax = $true )
& ( setukpairinjL1 = $true )
& ( funcGraphProp3 = $true )
& ( setminusILneg = $true )
& ( upairsetE = $true )
& ( omega__Cong = $true )
& ( ifSingleton = $true )
& ( sepSubset = $true )
& ( setminusERneg = $true )
& ( inCongP = $true )
& ( quantDeMorgan4 = $true )
& ( binintersectER = $true )
& ( binintersectTELcontra = $true )
& ( symdiffI1 = $true )
& ( ubforcartprodlem1 = $true )
& ( nonemptyImpWitness = $true )
& ( lamp = $true )
& ( disjointsetsI1 = $true )
& ( funcGraphProp1 = $true )
& ( setadjoinE = $true )
& ( exuE1 = $true )
& ( cartprodpairmemEL = $true )
& ( nonemptyE1 = $true )
& ( complementTE1 = $true )
& ( singletoninpowerset = $true )
& ( setunionE = $true )
& ( emptyset__Cong = $true )
& ( setminusELneg = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( infuncsetfunc = $true )
& ( nonemptyI = $true )
& ( iffalseProp2 = $true )
& ( powersetTI1 = $true )
& ( setminusER = $true )
& ( ex1E1 = $true )
& ( powersetE1 = $true )
& ( iftrueorfalse = $true )
& ( setukpairinjR11 = $true )
& ( binintersectTERcontra = $true )
& ( foundationAx = $true )
& ( inPowerset = $true )
& ( setbeta = $true )
& ( setukpairinjR2 = $true )
& ( emptysetimpfalse = $true )
& ( setunionsingleton1 = $true )
& ( binintersectI = $true )
& ( setminusT_lem = $true )
& ( ubforcartprodlem2 = $true )
& ( ex1E2 = $true )
& ( exu__Cong = $true )
& ( powersetTE1 = $true )
& ( powersetT_lem = $true )
& ( dpsetconstrEL1 = $true )
& ( singletonsubset = $true )
& ( binintersectSubset2 = $true )
& ( setadjoinIR = $true )
& ( kpairp = $true )
& ( symdiffIneg1 = $true )
& ( wellorderingAx = $true )
& ( binunionLsub = $true )
& ( upairsetIR = $true )
& ( dpsetconstrI = $true )
& ( funcext = $true )
& ( powersetE = $true )
& ( subsetTrans = $true )
& ( dsetconstr__Cong = $true )
& ( ubforcartprodlem3 = $true )
& ( prop2set2propI = $true )
& ( setukpairinjL2 = $true )
& ( powersetAx = $true )
& ( powersetsubset = $true )
& ( funcext2 = $true )
& ( ex1I = $true )
& ( singletonsuniq = $true )
& ( symdiffI2 = $true )
& ( funcinfuncset = $true )
& ( subbreln = $true )
& ( descr__Cong = $true )
& ( kpairsurjEq = $true )
& ( upairequniteq = $true )
& ( prop2setE = $true )
& ( iffalse = $true )
& ( vacuousDall = $true )
& ( subset2powerset = $true )
& ( beta1 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK656,sK657,sK658,sK659])],[f881,f2057,f2056,f2055]) ).
thf(f2055,plain,
( ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
& ( ( subset @ X0 @ X2 )
= $true )
& ( $true
!= ( in @ X3 @ X2 ) )
& ( $true
= ( in @ X3 @ X0 ) ) ) ) )
=> ( ( $true
= ( in @ sK656 @ ( powerset @ sK657 ) ) )
& ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ sK657 ) ) )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK657 ) )
& ( ( subset @ sK656 @ X2 )
= $true )
& ( $true
!= ( in @ X3 @ X2 ) )
& ( $true
= ( in @ X3 @ sK656 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2056,plain,
( ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ sK657 ) ) )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK657 ) )
& ( ( subset @ sK656 @ X2 )
= $true )
& ( $true
!= ( in @ X3 @ X2 ) )
& ( $true
= ( in @ X3 @ sK656 ) ) ) )
=> ( ( $true
= ( in @ sK658 @ ( powerset @ sK657 ) ) )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK657 ) )
& ( $true
= ( subset @ sK656 @ sK658 ) )
& ( $true
!= ( in @ X3 @ sK658 ) )
& ( $true
= ( in @ X3 @ sK656 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2057,plain,
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ sK657 ) )
& ( $true
= ( subset @ sK656 @ sK658 ) )
& ( $true
!= ( in @ X3 @ sK658 ) )
& ( $true
= ( in @ X3 @ sK656 ) ) )
=> ( ( $true
= ( in @ sK659 @ sK657 ) )
& ( $true
= ( subset @ sK656 @ sK658 ) )
& ( ( in @ sK659 @ sK658 )
!= $true )
& ( ( in @ sK659 @ sK656 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f881,plain,
( ( setadjoinOr = $true )
& ( emptyinPowerset = $true )
& ( iftrueProp1 = $true )
& ( binunionRsub = $true )
& ( eqinunit = $true )
& ( funcGraphProp4 = $true )
& ( binintersectSubset3 = $true )
& ( eqimpsubset2 = $true )
& ( brelnall2 = $true )
& ( binintersectT_lem = $true )
& ( setukpairIR = $true )
& ( exuI1 = $true )
& ( setminusSubset1 = $true )
& ( binintersectSubset1 = $true )
& ( exuE3u = $true )
& ( iffalseProp1 = $true )
& ( bs114d = $true )
& ( subsetRefl = $true )
& ( subsetI2 = $true )
& ( omegaSAx = $true )
& ( dsetconstrI = $true )
& ( binunionEcases = $true )
& ( powerset__Cong = $true )
& ( notinsingleton = $true )
& ( kfstpairEq = $true )
& ( secondinupair = $true )
& ( dpsetconstrSub = $true )
& ( setukpairinjL = $true )
& ( uniqinunit = $true )
& ( binintersectLsub = $true )
& ( setukpairinjR = $true )
& ( emptysetsubset = $true )
& ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
& ( ( subset @ X0 @ X2 )
= $true )
& ( $true
!= ( in @ X3 @ X2 ) )
& ( $true
= ( in @ X3 @ X0 ) ) ) ) )
& ( brelnall1 = $true )
& ( cartprodmempair1 = $true )
& ( eqbreln = $true )
& ( cartprodsndin = $true )
& ( binunionIR = $true )
& ( app = $true )
& ( complementT_lem = $true )
& ( setadjoinSub2 = $true )
& ( ksndpairEq = $true )
& ( setextsub = $true )
& ( emptyinunitempty = $true )
& ( setunionsingleton = $true )
& ( dpsetconstrER = $true )
& ( quantDeMorgan1 = $true )
& ( cartprodpairin = $true )
& ( ap2p = $true )
& ( binintersectSubset5 = $true )
& ( notequalI2 = $true )
& ( setext = $true )
& ( nonemptyI1 = $true )
& ( notsubsetI = $true )
& ( lam2p = $true )
& ( setukpairinjR12 = $true )
& ( beta2 = $true )
& ( replAx = $true )
& ( ap2apEq2 = $true )
& ( subsetemptysetimpeq = $true )
& ( eta2 = $true )
& ( setukpairIL = $true )
& ( dsetconstrER = $true )
& ( setadjoinIL = $true )
& ( notdexE = $true )
& ( setoftrueEq = $true )
& ( funcGraphProp2 = $true )
& ( upairset2IR = $true )
& ( subsetTI = $true )
& ( setadjoinSub = $true )
& ( binunionIL = $true )
& ( kfstsingleton = $true )
& ( funcImageSingleton = $true )
& ( binunionE = $true )
& ( notdallE = $true )
& ( subsetE = $true )
& ( singletonprop = $true )
& ( setminusLsub = $true )
& ( ifp = $true )
& ( subsetE2 = $true )
& ( exuI3 = $true )
& ( setminusSubset2 = $true )
& ( symdiffIneg2 = $true )
& ( upairset2E = $true )
& ( subPowSU = $true )
& ( singletoninpowunion = $true )
& ( quantDeMorgan2 = $true )
& ( setextAx = $true )
& ( binintersectSubset4 = $true )
& ( setOfPairsIsBReln = $true )
& ( setminusI = $true )
& ( setminusEL = $true )
& ( setunion__Cong = $true )
& ( emptyInPowerset = $true )
& ( setminusIRneg = $true )
& ( iftrue = $true )
& ( symdiffE = $true )
& ( exuE3e = $true )
& ( setunionE2 = $true )
& ( exuE2 = $true )
& ( setunionI = $true )
& ( ap2apEq1 = $true )
& ( contrasubsetT = $true )
& ( cartprodpairmemER = $true )
& ( prop2setI = $true )
& ( setadjoin__Cong = $true )
& ( exuEu = $true )
& ( setunionsingleton2 = $true )
& ( cartprodmempair = $true )
& ( eqimpsubset1 = $true )
& ( emptysetE = $true )
& ( noeltsimpempty = $true )
& ( cartprodfstin = $true )
& ( eta1 = $true )
& ( subsetI1 = $true )
& ( iftrueProp2 = $true )
& ( powersetI = $true )
& ( cartprodpairsurjEq = $true )
& ( binunionT_lem = $true )
& ( setadjoinAx = $true )
& ( complementTI1 = $true )
& ( emptyI = $true )
& ( kpairiskpair = $true )
& ( emptyE1 = $true )
& ( exuI2 = $true )
& ( setextT = $true )
& ( theprop = $true )
& ( emptysetAx = $true )
& ( ex1I2 = $true )
& ( funcextLem = $true )
& ( cartprodsndpairEq = $true )
& ( notequalI1 = $true )
& ( powersetI1 = $true )
& ( singletonsswitch = $true )
& ( in__Cong = $true )
& ( dpsetconstrEL2 = $true )
& ( omegaIndAx = $true )
& ( dpsetconstrERa = $true )
& ( ksndsingleton = $true )
& ( binintersectRsub = $true )
& ( quantDeMorgan3 = $true )
& ( apProp = $true )
& ( theeq = $true )
& ( setukpairinjR1 = $true )
& ( lamProp = $true )
& ( cartprodfstpairEq = $true )
& ( descrp = $true )
& ( binintersectEL = $true )
& ( sepInPowerset = $true )
& ( setunionAx = $true )
& ( upairsetIL = $true )
& ( lam2lamEq = $true )
& ( cartprodmempaircEq = $true )
& ( dsetconstrEL = $true )
& ( notinemptyset = $true )
& ( omega0Ax = $true )
& ( setukpairinjL1 = $true )
& ( funcGraphProp3 = $true )
& ( setminusILneg = $true )
& ( upairsetE = $true )
& ( omega__Cong = $true )
& ( ifSingleton = $true )
& ( sepSubset = $true )
& ( setminusERneg = $true )
& ( inCongP = $true )
& ( quantDeMorgan4 = $true )
& ( binintersectER = $true )
& ( binintersectTELcontra = $true )
& ( symdiffI1 = $true )
& ( ubforcartprodlem1 = $true )
& ( nonemptyImpWitness = $true )
& ( lamp = $true )
& ( disjointsetsI1 = $true )
& ( funcGraphProp1 = $true )
& ( setadjoinE = $true )
& ( exuE1 = $true )
& ( cartprodpairmemEL = $true )
& ( nonemptyE1 = $true )
& ( complementTE1 = $true )
& ( singletoninpowerset = $true )
& ( setunionE = $true )
& ( emptyset__Cong = $true )
& ( setminusELneg = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( infuncsetfunc = $true )
& ( nonemptyI = $true )
& ( iffalseProp2 = $true )
& ( powersetTI1 = $true )
& ( setminusER = $true )
& ( ex1E1 = $true )
& ( powersetE1 = $true )
& ( iftrueorfalse = $true )
& ( setukpairinjR11 = $true )
& ( binintersectTERcontra = $true )
& ( foundationAx = $true )
& ( inPowerset = $true )
& ( setbeta = $true )
& ( setukpairinjR2 = $true )
& ( emptysetimpfalse = $true )
& ( setunionsingleton1 = $true )
& ( binintersectI = $true )
& ( setminusT_lem = $true )
& ( ubforcartprodlem2 = $true )
& ( ex1E2 = $true )
& ( exu__Cong = $true )
& ( powersetTE1 = $true )
& ( powersetT_lem = $true )
& ( dpsetconstrEL1 = $true )
& ( singletonsubset = $true )
& ( binintersectSubset2 = $true )
& ( setadjoinIR = $true )
& ( kpairp = $true )
& ( symdiffIneg1 = $true )
& ( wellorderingAx = $true )
& ( binunionLsub = $true )
& ( upairsetIR = $true )
& ( dpsetconstrI = $true )
& ( funcext = $true )
& ( powersetE = $true )
& ( subsetTrans = $true )
& ( dsetconstr__Cong = $true )
& ( ubforcartprodlem3 = $true )
& ( prop2set2propI = $true )
& ( setukpairinjL2 = $true )
& ( powersetAx = $true )
& ( powersetsubset = $true )
& ( funcext2 = $true )
& ( ex1I = $true )
& ( singletonsuniq = $true )
& ( symdiffI2 = $true )
& ( funcinfuncset = $true )
& ( subbreln = $true )
& ( descr__Cong = $true )
& ( kpairsurjEq = $true )
& ( upairequniteq = $true )
& ( prop2setE = $true )
& ( iffalse = $true )
& ( vacuousDall = $true )
& ( subset2powerset = $true )
& ( beta1 = $true ) ),
inference(flattening,[status(thm)],[f880]) ).
thf(f880,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
& ( $true
!= ( in @ X3 @ X2 ) )
& ( ( subset @ X0 @ X2 )
= $true )
& ( $true
= ( in @ X3 @ X1 ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) ) )
& ( ( in @ X0 @ ( powerset @ X1 ) )
= $true ) )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f756]) ).
thf(f756,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
=> ( ( ( subset @ X0 @ X2 )
= $true )
=> ( ( $true
!= ( in @ X3 @ X2 ) )
=> ( $true
!= ( in @ X3 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f522]) ).
thf(f522,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
=> ( ( ( subset @ X0 @ X2 )
= $true )
=> ( ( $true
!= ( in @ X3 @ X2 ) )
=> ( $true
!= ( in @ X3 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f521]) ).
thf(f521,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( powerset @ X1 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( subset @ X0 @ X2 )
=> ( ~ ( in @ X3 @ X2 )
=> ~ ( in @ X3 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f246]) ).
thf(f246,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( subset @ X11 @ X16 )
=> ( ~ ( in @ X1 @ X16 )
=> ~ ( in @ X1 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f245]) ).
thf(f245,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( subset @ X11 @ X16 )
=> ( ~ ( in @ X1 @ X16 )
=> ~ ( in @ X1 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.IFvb0yMMZy/DTF2THF_26337.p',contrasubsetT1) ).
thf(f3827,plain,
( ( in @ sK659 @ sK656 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f3800]) ).
thf(f3800,plain,
( ( ( in @ sK659 @ sK656 )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f3661,f3295]) ).
thf(f3295,plain,
( $true
= ( subset @ sK656 @ sK658 ) ),
inference(cnf_transformation,[status(thm)],[f2058]) ).
thf(f3661,plain,
! [X0: $i] :
( ( $true
!= ( subset @ X0 @ sK658 ) )
| ( ( in @ sK659 @ X0 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3660]) ).
thf(f3660,plain,
! [X0: $i] :
( ( $true
!= ( subset @ X0 @ sK658 ) )
| ( ( in @ sK659 @ X0 )
!= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3658,f3246]) ).
thf(f3246,plain,
subsetE2 = $true,
inference(cnf_transformation,[status(thm)],[f2058]) ).
thf(f3658,plain,
! [X0: $i] :
( ( ( in @ sK659 @ X0 )
!= $true )
| ( $true
!= ( subset @ X0 @ sK658 ) )
| ( subsetE2 != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3653]) ).
thf(f3653,plain,
! [X0: $i] :
( ( $true != $true )
| ( subsetE2 != $true )
| ( ( in @ sK659 @ X0 )
!= $true )
| ( $true
!= ( subset @ X0 @ sK658 ) ) ),
inference(superposition,[status(thm)],[f3294,f2921]) ).
thf(f2921,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ X2 ) )
| ( $true
!= ( subset @ X1 @ X2 ) )
| ( subsetE2 != $true )
| ( $true
!= ( in @ X0 @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f1870]) ).
thf(f1870,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ X2 ) )
| ( $true
!= ( subset @ X1 @ X2 ) ) )
| ( subsetE2 != $true ) )
& ( ( subsetE2 = $true )
| ( ( $true
= ( in @ sK535 @ sK536 ) )
& ( $true
!= ( in @ sK535 @ sK537 ) )
& ( $true
= ( subset @ sK536 @ sK537 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK535,sK536,sK537])],[f1868,f1869]) ).
thf(f1869,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X3 @ X4 ) )
& ( ( in @ X3 @ X5 )
!= $true )
& ( ( subset @ X4 @ X5 )
= $true ) )
=> ( ( $true
= ( in @ sK535 @ sK536 ) )
& ( $true
!= ( in @ sK535 @ sK537 ) )
& ( $true
= ( subset @ sK536 @ sK537 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1868,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ X2 ) )
| ( $true
!= ( subset @ X1 @ X2 ) ) )
| ( subsetE2 != $true ) )
& ( ( subsetE2 = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X3 @ X4 ) )
& ( ( in @ X3 @ X5 )
!= $true )
& ( ( subset @ X4 @ X5 )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f1867]) ).
thf(f1867,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ X2 ) )
| ( $true
!= ( subset @ X1 @ X2 ) ) )
| ( subsetE2 != $true ) )
& ( ( subsetE2 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
& ( $true
!= ( in @ X0 @ X2 ) )
& ( $true
= ( subset @ X1 @ X2 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1045]) ).
thf(f1045,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ X2 ) )
| ( $true
!= ( subset @ X1 @ X2 ) ) )
<=> ( subsetE2 = $true ) ),
inference(flattening,[status(thm)],[f1044]) ).
thf(f1044,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ X2 ) )
| ( $true
!= ( subset @ X1 @ X2 ) ) )
<=> ( subsetE2 = $true ) ),
inference(ennf_transformation,[status(thm)],[f732]) ).
thf(f732,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( subset @ X1 @ X2 ) )
=> ( ( $true
!= ( in @ X0 @ X2 ) )
=> ( $true
!= ( in @ X0 @ X1 ) ) ) )
<=> ( subsetE2 = $true ) ),
inference(flattening,[status(thm)],[f251]) ).
thf(f251,plain,
( ( subsetE2 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( subset @ X1 @ X2 ) )
=> ( ( $true
!= ( in @ X0 @ X2 ) )
=> ( $true
!= ( in @ X0 @ X1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f250]) ).
thf(f250,plain,
( subsetE2
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( subset @ X1 @ X2 )
=> ( ~ ( in @ X0 @ X2 )
=> ~ ( in @ X0 @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f84]) ).
thf(f84,axiom,
( subsetE2
= ( ! [X1: $i,X3: $i,X4: $i] :
( ( subset @ X3 @ X4 )
=> ( ~ ( in @ X1 @ X4 )
=> ~ ( in @ X1 @ X3 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.IFvb0yMMZy/DTF2THF_26337.p',subsetE2) ).
thf(f3294,plain,
( ( in @ sK659 @ sK658 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f2058]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU724^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.12 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.13/0.32 % Computer : n025.cluster.edu
% 0.13/0.32 % Model : x86_64 x86_64
% 0.13/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.32 % Memory : 8042.1875MB
% 0.13/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.32 % CPULimit : 300
% 0.13/0.32 % WCLimit : 300
% 0.13/0.32 % DateTime : Tue Feb 24 12:32:12 EST 2026
% 0.13/0.32 % CPUTime :
% 0.13/0.32 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.44 ---- Original DTF file ---
% 0.15/0.44 thf(spec,logic,$$dhol).
% 0.15/0.44 %------------------------------------------------------------------------------
% 0.15/0.44 % File : SEU724^1 : TPTP v9.2.1. Released v3.7.0.
% 0.15/0.44 % Domain : Set Theory
% 0.15/0.44 % Problem : Typed Set Theory - Laws for Typed Sets
% 0.15/0.44 % Version : Especial.
% 0.15/0.44 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.15/0.44 % (! x:i.in x A -> subset X Y -> ~(in x Y) -> ~(in x X))))
% 0.15/0.44
% 0.15/0.44 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.15/0.44 % Source : [Bro08]
% 0.15/0.44 % Names : ZFC226g [Bro08]
% 0.15/0.44
% 0.15/0.44 % Status : Theorem
% 0.15/0.44 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 0.00 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v6.0.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.15/0.44 % Syntax : Number of formulae : 523 ( 244 unt; 278 typ; 244 def)
% 0.15/0.44 % Number of atoms : 1396 ( 369 equ; 0 cnn)
% 0.15/0.44 % Maximal formula atoms : 246 ( 5 avg)
% 0.15/0.44 % Number of connectives : 2749 ( 64 ~; 12 |; 44 &;1942 @)
% 0.15/0.44 % ( 14 <=>; 673 =>; 0 <=; 0 <~>)
% 0.15/0.44 % Maximal formula depth : 253 ( 2 avg)
% 0.15/0.44 % Number of types : 2 ( 0 usr)
% 0.15/0.44 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.15/0.44 % Number of symbols : 281 ( 278 usr; 244 con; 0-4 aty)
% 0.15/0.44 % Number of variables : 824 ( 74 ^; 715 !; 35 ?; 824 :)
% 0.15/0.44 % SPC : TH0_THM_EQU_NAR
% 0.15/0.44
% 0.15/0.44 % Comments : http://mathgate.info/detsetitem.php?id=284
% 0.15/0.44 %------------------------------------------------------------------------------
% 0.15/0.44 thf(in_type,type,
% 0.15/0.44 in: $i > $i > $o ).
% 0.15/0.44
% 0.15/0.44 thf(exu_type,type,
% 0.15/0.44 exu: ( $i > $o ) > $o ).
% 0.15/0.44
% 0.15/0.44 thf(exu,definition,
% 0.15/0.44 ( exu
% 0.15/0.44 = ( ^ [Xphi: $i > $o] :
% 0.15/0.44 ? [Xx: $i] :
% 0.15/0.44 ( ( Xphi @ Xx )
% 0.15/0.44 & ! [Xy: $i] :
% 0.15/0.44 ( ( Xphi @ Xy )
% 0.15/0.44 => ( Xx = Xy ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setextAx_type,type,
% 0.15/0.44 setextAx: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setextAx,definition,
% 0.15/0.44 ( setextAx
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 <=> ( in @ Xx @ B ) )
% 0.15/0.44 => ( A = B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(emptyset_type,type,
% 0.15/0.44 emptyset: $i ).
% 0.15/0.44
% 0.15/0.44 thf(emptysetAx_type,type,
% 0.15/0.44 emptysetAx: $o ).
% 0.15/0.44
% 0.15/0.44 thf(emptysetAx,definition,
% 0.15/0.44 ( emptysetAx
% 0.15/0.44 = ( ! [Xx: $i] :
% 0.15/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoin_type,type,
% 0.15/0.44 setadjoin: $i > $i > $i ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoinAx_type,type,
% 0.15/0.44 setadjoinAx: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoinAx,definition,
% 0.15/0.44 ( setadjoinAx
% 0.15/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.15/0.44 <=> ( ( Xy = Xx )
% 0.15/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(powerset_type,type,
% 0.15/0.44 powerset: $i > $i ).
% 0.15/0.44
% 0.15/0.44 thf(powersetAx_type,type,
% 0.15/0.44 powersetAx: $o ).
% 0.15/0.44
% 0.15/0.44 thf(powersetAx,definition,
% 0.15/0.44 ( powersetAx
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.15/0.44 <=> ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ B )
% 0.15/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setunion_type,type,
% 0.15/0.44 setunion: $i > $i ).
% 0.15/0.44
% 0.15/0.44 thf(setunionAx_type,type,
% 0.15/0.44 setunionAx: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setunionAx,definition,
% 0.15/0.44 ( setunionAx
% 0.15/0.44 = ( ! [A: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.15/0.44 <=> ? [B: $i] :
% 0.15/0.44 ( ( in @ Xx @ B )
% 0.15/0.44 & ( in @ B @ A ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(omega_type,type,
% 0.15/0.44 omega: $i ).
% 0.15/0.44
% 0.15/0.44 thf(omega0Ax_type,type,
% 0.15/0.44 omega0Ax: $o ).
% 0.15/0.44
% 0.15/0.44 thf(omega0Ax,definition,
% 0.15/0.44 ( omega0Ax
% 0.15/0.44 = ( in @ emptyset @ omega ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(omegaSAx_type,type,
% 0.15/0.44 omegaSAx: $o ).
% 0.15/0.44
% 0.15/0.44 thf(omegaSAx,definition,
% 0.15/0.44 ( omegaSAx
% 0.15/0.44 = ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ omega )
% 0.15/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(omegaIndAx_type,type,
% 0.15/0.44 omegaIndAx: $o ).
% 0.15/0.44
% 0.15/0.44 thf(omegaIndAx,definition,
% 0.15/0.44 ( omegaIndAx
% 0.15/0.44 = ( ! [A: $i] :
% 0.15/0.44 ( ( ( in @ emptyset @ A )
% 0.15/0.44 & ! [Xx: $i] :
% 0.15/0.44 ( ( ( in @ Xx @ omega )
% 0.15/0.44 & ( in @ Xx @ A ) )
% 0.15/0.44 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.15/0.44 => ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ omega )
% 0.15/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(replAx_type,type,
% 0.15/0.44 replAx: $o ).
% 0.15/0.44
% 0.15/0.44 thf(replAx,definition,
% 0.15/0.44 ( replAx
% 0.15/0.44 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( exu
% 0.15/0.44 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.15/0.44 => ? [B: $i] :
% 0.15/0.44 ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ B )
% 0.15/0.44 <=> ? [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ A )
% 0.15/0.44 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(foundationAx_type,type,
% 0.15/0.44 foundationAx: $o ).
% 0.15/0.44
% 0.15/0.44 thf(foundationAx,definition,
% 0.15/0.44 ( foundationAx
% 0.15/0.44 = ( ! [A: $i] :
% 0.15/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.15/0.44 => ? [B: $i] :
% 0.15/0.44 ( ( in @ B @ A )
% 0.15/0.44 & ~ ? [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ B )
% 0.15/0.44 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(wellorderingAx_type,type,
% 0.15/0.44 wellorderingAx: $o ).
% 0.15/0.44
% 0.15/0.44 thf(wellorderingAx,definition,
% 0.15/0.44 ( wellorderingAx
% 0.15/0.44 = ( ! [A: $i] :
% 0.15/0.44 ? [B: $i] :
% 0.15/0.44 ( ! [C: $i] :
% 0.15/0.44 ( ( in @ C @ B )
% 0.15/0.44 => ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ C )
% 0.15/0.44 => ( in @ Xx @ A ) ) )
% 0.15/0.44 & ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( ( in @ Xx @ A )
% 0.15/0.44 & ( in @ Xy @ A ) )
% 0.15/0.44 => ( ! [C: $i] :
% 0.15/0.44 ( ( in @ C @ B )
% 0.15/0.44 => ( ( in @ Xx @ C )
% 0.15/0.44 <=> ( in @ Xy @ C ) ) )
% 0.15/0.44 => ( Xx = Xy ) ) )
% 0.15/0.44 & ! [C: $i,D: $i] :
% 0.15/0.44 ( ( ( in @ C @ B )
% 0.15/0.44 & ( in @ D @ B ) )
% 0.15/0.44 => ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ C )
% 0.15/0.44 => ( in @ Xx @ D ) )
% 0.15/0.44 | ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ D )
% 0.15/0.44 => ( in @ Xx @ C ) ) ) )
% 0.15/0.44 & ! [C: $i] :
% 0.15/0.44 ( ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ C )
% 0.15/0.44 => ( in @ Xx @ A ) )
% 0.15/0.44 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.15/0.44 => ? [D: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ D @ B )
% 0.15/0.44 & ( in @ Xx @ C )
% 0.15/0.44 & ~ ? [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ D )
% 0.15/0.44 & ( in @ Xy @ C ) )
% 0.15/0.44 & ! [E: $i] :
% 0.15/0.44 ( ( in @ E @ B )
% 0.15/0.44 => ( ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ E )
% 0.15/0.44 => ( in @ Xy @ D ) )
% 0.15/0.44 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(descr_type,type,
% 0.15/0.44 descr: ( $i > $o ) > $i ).
% 0.15/0.44
% 0.15/0.44 thf(descrp_type,type,
% 0.15/0.44 descrp: $o ).
% 0.15/0.44
% 0.15/0.44 thf(descrp,definition,
% 0.15/0.44 ( descrp
% 0.15/0.44 = ( ! [Xphi: $i > $o] :
% 0.15/0.44 ( ( exu
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.44 => ( Xphi
% 0.15/0.44 @ ( descr
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(dsetconstr_type,type,
% 0.15/0.44 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.15/0.44
% 0.15/0.44 thf(dsetconstrI_type,type,
% 0.15/0.44 dsetconstrI: $o ).
% 0.15/0.44
% 0.15/0.44 thf(dsetconstrI,definition,
% 0.15/0.44 ( dsetconstrI
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( ( Xphi @ Xx )
% 0.15/0.44 => ( in @ Xx
% 0.15/0.44 @ ( dsetconstr @ A
% 0.15/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(dsetconstrEL_type,type,
% 0.15/0.44 dsetconstrEL: $o ).
% 0.15/0.44
% 0.15/0.44 thf(dsetconstrEL,definition,
% 0.15/0.44 ( dsetconstrEL
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx
% 0.15/0.44 @ ( dsetconstr @ A
% 0.15/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.15/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(dsetconstrER_type,type,
% 0.15/0.44 dsetconstrER: $o ).
% 0.15/0.44
% 0.15/0.44 thf(dsetconstrER,definition,
% 0.15/0.44 ( dsetconstrER
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx
% 0.15/0.44 @ ( dsetconstr @ A
% 0.15/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.15/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(exuE1_type,type,
% 0.15/0.44 exuE1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(exuE1,definition,
% 0.15/0.44 ( exuE1
% 0.15/0.44 = ( ! [Xphi: $i > $o] :
% 0.15/0.44 ( ( exu
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.44 => ? [Xx: $i] :
% 0.15/0.44 ( ( Xphi @ Xx )
% 0.15/0.44 & ! [Xy: $i] :
% 0.15/0.44 ( ( Xphi @ Xy )
% 0.15/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(prop2set_type,type,
% 0.15/0.44 prop2set: $o > $i ).
% 0.15/0.44
% 0.15/0.44 thf(prop2setE_type,type,
% 0.15/0.44 prop2setE: $o ).
% 0.15/0.44
% 0.15/0.44 thf(prop2setE,definition,
% 0.15/0.44 ( prop2setE
% 0.15/0.44 = ( ! [Xphi: $o,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.15/0.44 => Xphi ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(emptysetE_type,type,
% 0.15/0.44 emptysetE: $o ).
% 0.15/0.44
% 0.15/0.44 thf(emptysetE,definition,
% 0.15/0.44 ( emptysetE
% 0.15/0.44 = ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ emptyset )
% 0.15/0.44 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(emptysetimpfalse_type,type,
% 0.15/0.44 emptysetimpfalse: $o ).
% 0.15/0.44
% 0.15/0.44 thf(emptysetimpfalse,definition,
% 0.15/0.44 ( emptysetimpfalse
% 0.15/0.44 = ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ emptyset )
% 0.15/0.44 => $false ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(notinemptyset_type,type,
% 0.15/0.44 notinemptyset: $o ).
% 0.15/0.44
% 0.15/0.44 thf(notinemptyset,definition,
% 0.15/0.44 ( notinemptyset
% 0.15/0.44 = ( ! [Xx: $i] :
% 0.15/0.44 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(exuE3e_type,type,
% 0.15/0.44 exuE3e: $o ).
% 0.15/0.44
% 0.15/0.44 thf(exuE3e,definition,
% 0.15/0.44 ( exuE3e
% 0.15/0.44 = ( ! [Xphi: $i > $o] :
% 0.15/0.44 ( ( exu
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.44 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setext_type,type,
% 0.15/0.44 setext: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setext,definition,
% 0.15/0.44 ( setext
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ Xx @ B ) )
% 0.15/0.44 => ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ B )
% 0.15/0.44 => ( in @ Xx @ A ) )
% 0.15/0.44 => ( A = B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(emptyI_type,type,
% 0.15/0.44 emptyI: $o ).
% 0.15/0.44
% 0.15/0.44 thf(emptyI,definition,
% 0.15/0.44 ( emptyI
% 0.15/0.44 = ( ! [A: $i] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ~ ( in @ Xx @ A )
% 0.15/0.44 => ( A = emptyset ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(noeltsimpempty_type,type,
% 0.15/0.44 noeltsimpempty: $o ).
% 0.15/0.44
% 0.15/0.44 thf(noeltsimpempty,definition,
% 0.15/0.44 ( noeltsimpempty
% 0.15/0.44 = ( ! [A: $i] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ~ ( in @ Xx @ A )
% 0.15/0.44 => ( A = emptyset ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setbeta_type,type,
% 0.15/0.44 setbeta: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setbeta,definition,
% 0.15/0.44 ( setbeta
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( ( in @ Xx
% 0.15/0.44 @ ( dsetconstr @ A
% 0.15/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.15/0.44 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(nonempty_type,type,
% 0.15/0.44 nonempty: $i > $o ).
% 0.15/0.44
% 0.15/0.44 thf(nonempty,definition,
% 0.15/0.44 ( nonempty
% 0.15/0.44 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(nonemptyE1_type,type,
% 0.15/0.44 nonemptyE1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(nonemptyE1,definition,
% 0.15/0.44 ( nonemptyE1
% 0.15/0.44 = ( ! [A: $i] :
% 0.15/0.44 ( ( nonempty @ A )
% 0.15/0.44 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(nonemptyI_type,type,
% 0.15/0.44 nonemptyI: $o ).
% 0.15/0.44
% 0.15/0.44 thf(nonemptyI,definition,
% 0.15/0.44 ( nonemptyI
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( ( Xphi @ Xx )
% 0.15/0.44 => ( nonempty
% 0.15/0.44 @ ( dsetconstr @ A
% 0.15/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(nonemptyI1_type,type,
% 0.15/0.44 nonemptyI1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(nonemptyI1,definition,
% 0.15/0.44 ( nonemptyI1
% 0.15/0.44 = ( ! [A: $i] :
% 0.15/0.44 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.15/0.44 => ( nonempty @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoinIL_type,type,
% 0.15/0.44 setadjoinIL: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoinIL,definition,
% 0.15/0.44 ( setadjoinIL
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(emptyinunitempty_type,type,
% 0.15/0.44 emptyinunitempty: $o ).
% 0.15/0.44
% 0.15/0.44 thf(emptyinunitempty,definition,
% 0.15/0.44 ( emptyinunitempty
% 0.15/0.44 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoinIR_type,type,
% 0.15/0.44 setadjoinIR: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoinIR,definition,
% 0.15/0.44 ( setadjoinIR
% 0.15/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ A )
% 0.15/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoinE_type,type,
% 0.15/0.44 setadjoinE: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoinE,definition,
% 0.15/0.44 ( setadjoinE
% 0.15/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.15/0.44 => ! [Xphi: $o] :
% 0.15/0.44 ( ( ( Xy = Xx )
% 0.15/0.44 => Xphi )
% 0.15/0.44 => ( ( ( in @ Xy @ A )
% 0.15/0.44 => Xphi )
% 0.15/0.44 => Xphi ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoinOr_type,type,
% 0.15/0.44 setadjoinOr: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoinOr,definition,
% 0.15/0.44 ( setadjoinOr
% 0.15/0.44 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.15/0.44 => ( ( Xy = Xx )
% 0.15/0.44 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setoftrueEq_type,type,
% 0.15/0.44 setoftrueEq: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setoftrueEq,definition,
% 0.15/0.44 ( setoftrueEq
% 0.15/0.44 = ( ! [A: $i] :
% 0.15/0.44 ( ( dsetconstr @ A
% 0.15/0.44 @ ^ [Xx: $i] : $true )
% 0.15/0.44 = A ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(powersetI_type,type,
% 0.15/0.44 powersetI: $o ).
% 0.15/0.44
% 0.15/0.44 thf(powersetI,definition,
% 0.15/0.44 ( powersetI
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ B )
% 0.15/0.44 => ( in @ Xx @ A ) )
% 0.15/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(emptyinPowerset_type,type,
% 0.15/0.44 emptyinPowerset: $o ).
% 0.15/0.44
% 0.15/0.44 thf(emptyinPowerset,definition,
% 0.15/0.44 ( emptyinPowerset
% 0.15/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(emptyInPowerset_type,type,
% 0.15/0.44 emptyInPowerset: $o ).
% 0.15/0.44
% 0.15/0.44 thf(emptyInPowerset,definition,
% 0.15/0.44 ( emptyInPowerset
% 0.15/0.44 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(powersetE_type,type,
% 0.15/0.44 powersetE: $o ).
% 0.15/0.44
% 0.15/0.44 thf(powersetE,definition,
% 0.15/0.44 ( powersetE
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.15/0.44 => ( ( in @ Xx @ B )
% 0.15/0.44 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setunionI_type,type,
% 0.15/0.44 setunionI: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setunionI,definition,
% 0.15/0.44 ( setunionI
% 0.15/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.15/0.44 ( ( in @ Xx @ B )
% 0.15/0.44 => ( ( in @ B @ A )
% 0.15/0.44 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setunionE_type,type,
% 0.15/0.44 setunionE: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setunionE,definition,
% 0.15/0.44 ( setunionE
% 0.15/0.44 = ( ! [A: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.15/0.44 => ! [Xphi: $o] :
% 0.15/0.44 ( ! [B: $i] :
% 0.15/0.44 ( ( in @ Xx @ B )
% 0.15/0.44 => ( ( in @ B @ A )
% 0.15/0.44 => Xphi ) )
% 0.15/0.44 => Xphi ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(subPowSU_type,type,
% 0.15/0.44 subPowSU: $o ).
% 0.15/0.44
% 0.15/0.44 thf(subPowSU,definition,
% 0.15/0.44 ( subPowSU
% 0.15/0.44 = ( ! [A: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(exuE2_type,type,
% 0.15/0.44 exuE2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(exuE2,definition,
% 0.15/0.44 ( exuE2
% 0.15/0.44 = ( ! [Xphi: $i > $o] :
% 0.15/0.44 ( ( exu
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.44 => ? [Xx: $i] :
% 0.15/0.44 ! [Xy: $i] :
% 0.15/0.44 ( ( Xphi @ Xy )
% 0.15/0.44 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(nonemptyImpWitness_type,type,
% 0.15/0.44 nonemptyImpWitness: $o ).
% 0.15/0.44
% 0.15/0.44 thf(nonemptyImpWitness,definition,
% 0.15/0.44 ( nonemptyImpWitness
% 0.15/0.44 = ( ! [A: $i] :
% 0.15/0.44 ( ( nonempty @ A )
% 0.15/0.44 => ? [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 & $true ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(uniqinunit_type,type,
% 0.15/0.44 uniqinunit: $o ).
% 0.15/0.44
% 0.15/0.44 thf(uniqinunit,definition,
% 0.15/0.44 ( uniqinunit
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.15/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(notinsingleton_type,type,
% 0.15/0.44 notinsingleton: $o ).
% 0.15/0.44
% 0.15/0.44 thf(notinsingleton,definition,
% 0.15/0.44 ( notinsingleton
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( Xx != Xy )
% 0.15/0.44 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(eqinunit_type,type,
% 0.15/0.44 eqinunit: $o ).
% 0.15/0.44
% 0.15/0.44 thf(eqinunit,definition,
% 0.15/0.44 ( eqinunit
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( Xx = Xy )
% 0.15/0.44 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(singletonsswitch_type,type,
% 0.15/0.44 singletonsswitch: $o ).
% 0.15/0.44
% 0.15/0.44 thf(singletonsswitch,definition,
% 0.15/0.44 ( singletonsswitch
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.15/0.44 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(upairsetE_type,type,
% 0.15/0.44 upairsetE: $o ).
% 0.15/0.44
% 0.15/0.44 thf(upairsetE,definition,
% 0.15/0.44 ( upairsetE
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.15/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.15/0.44 => ( ( Xz = Xx )
% 0.15/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(upairsetIL_type,type,
% 0.15/0.44 upairsetIL: $o ).
% 0.15/0.44
% 0.15/0.44 thf(upairsetIL,definition,
% 0.15/0.44 ( upairsetIL
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(upairsetIR_type,type,
% 0.15/0.44 upairsetIR: $o ).
% 0.15/0.44
% 0.15/0.44 thf(upairsetIR,definition,
% 0.15/0.44 ( upairsetIR
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(emptyE1_type,type,
% 0.15/0.44 emptyE1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(emptyE1,definition,
% 0.15/0.44 ( emptyE1
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.44 ( ? [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 & ( Xphi @ Xx ) )
% 0.15/0.44 => ( ( ( dsetconstr @ A
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.44 = emptyset )
% 0.15/0.44 => $false ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(vacuousDall_type,type,
% 0.15/0.44 vacuousDall: $o ).
% 0.15/0.44
% 0.15/0.44 thf(vacuousDall,definition,
% 0.15/0.44 ( vacuousDall
% 0.15/0.44 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ emptyset )
% 0.15/0.44 => ( Xphi @ Xx ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(quantDeMorgan1_type,type,
% 0.15/0.44 quantDeMorgan1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(quantDeMorgan1,definition,
% 0.15/0.44 ( quantDeMorgan1
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.44 ( ~ ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( Xphi @ Xx ) )
% 0.15/0.44 => ? [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(quantDeMorgan2_type,type,
% 0.15/0.44 quantDeMorgan2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(quantDeMorgan2,definition,
% 0.15/0.44 ( quantDeMorgan2
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ~ ( Xphi @ Xx ) )
% 0.15/0.44 => ~ ? [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(quantDeMorgan3_type,type,
% 0.15/0.44 quantDeMorgan3: $o ).
% 0.15/0.44
% 0.15/0.44 thf(quantDeMorgan3,definition,
% 0.15/0.44 ( quantDeMorgan3
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.44 ( ~ ? [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 & ( Xphi @ Xx ) )
% 0.15/0.44 => ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(quantDeMorgan4_type,type,
% 0.15/0.44 quantDeMorgan4: $o ).
% 0.15/0.44
% 0.15/0.44 thf(quantDeMorgan4,definition,
% 0.15/0.44 ( quantDeMorgan4
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.44 ( ? [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 & ~ ( Xphi @ Xx ) )
% 0.15/0.44 => ~ ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(prop2setI_type,type,
% 0.15/0.44 prop2setI: $o ).
% 0.15/0.44
% 0.15/0.44 thf(prop2setI,definition,
% 0.15/0.44 ( prop2setI
% 0.15/0.44 = ( ! [Xphi: $o] :
% 0.15/0.44 ( Xphi
% 0.15/0.44 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(set2prop_type,type,
% 0.15/0.44 set2prop: $i > $o ).
% 0.15/0.44
% 0.15/0.44 thf(prop2set2propI_type,type,
% 0.15/0.44 prop2set2propI: $o ).
% 0.15/0.44
% 0.15/0.44 thf(prop2set2propI,definition,
% 0.15/0.44 ( prop2set2propI
% 0.15/0.44 = ( ! [Xphi: $o] :
% 0.15/0.44 ( Xphi
% 0.15/0.44 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(notdexE_type,type,
% 0.15/0.44 notdexE: $o ).
% 0.15/0.44
% 0.15/0.44 thf(notdexE,definition,
% 0.15/0.44 ( notdexE
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.44 ( ~ ? [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 & ( Xphi @ Xx ) )
% 0.15/0.44 => ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(notdallE_type,type,
% 0.15/0.44 notdallE: $o ).
% 0.15/0.44
% 0.15/0.44 thf(notdallE,definition,
% 0.15/0.44 ( notdallE
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.44 ( ~ ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( Xphi @ Xx ) )
% 0.15/0.44 => ? [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(exuI1_type,type,
% 0.15/0.44 exuI1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(exuI1,definition,
% 0.15/0.44 ( exuI1
% 0.15/0.44 = ( ! [Xphi: $i > $o] :
% 0.15/0.44 ( ? [Xx: $i] :
% 0.15/0.44 ( ( Xphi @ Xx )
% 0.15/0.44 & ! [Xy: $i] :
% 0.15/0.44 ( ( Xphi @ Xy )
% 0.15/0.44 => ( Xx = Xy ) ) )
% 0.15/0.44 => ( exu
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(exuI3_type,type,
% 0.15/0.44 exuI3: $o ).
% 0.15/0.44
% 0.15/0.44 thf(exuI3,definition,
% 0.15/0.44 ( exuI3
% 0.15/0.44 = ( ! [Xphi: $i > $o] :
% 0.15/0.44 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.15/0.44 => ( ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( Xphi @ Xx )
% 0.15/0.44 => ( ( Xphi @ Xy )
% 0.15/0.44 => ( Xx = Xy ) ) )
% 0.15/0.44 => ( exu
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(exuI2_type,type,
% 0.15/0.44 exuI2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(exuI2,definition,
% 0.15/0.44 ( exuI2
% 0.15/0.44 = ( ! [Xphi: $i > $o] :
% 0.15/0.44 ( ? [Xx: $i] :
% 0.15/0.44 ! [Xy: $i] :
% 0.15/0.44 ( ( Xphi @ Xy )
% 0.15/0.44 <=> ( Xy = Xx ) )
% 0.15/0.44 => ( exu
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(inCongP_type,type,
% 0.15/0.44 inCongP: $o ).
% 0.15/0.44
% 0.15/0.44 thf(inCongP,definition,
% 0.15/0.44 ( inCongP
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( A = B )
% 0.15/0.44 => ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( Xx = Xy )
% 0.15/0.44 => ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(in__Cong_type,type,
% 0.15/0.44 in__Cong: $o ).
% 0.15/0.44
% 0.15/0.44 thf(in__Cong,definition,
% 0.15/0.44 ( in__Cong
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( A = B )
% 0.15/0.44 => ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( Xx = Xy )
% 0.15/0.44 => ( ( in @ Xx @ A )
% 0.15/0.44 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(exuE3u_type,type,
% 0.15/0.44 exuE3u: $o ).
% 0.15/0.44
% 0.15/0.44 thf(exuE3u,definition,
% 0.15/0.44 ( exuE3u
% 0.15/0.44 = ( ! [Xphi: $i > $o] :
% 0.15/0.44 ( ( exu
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.44 => ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( Xphi @ Xx )
% 0.15/0.44 => ( ( Xphi @ Xy )
% 0.15/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(exu__Cong_type,type,
% 0.15/0.44 exu__Cong: $o ).
% 0.15/0.44
% 0.15/0.44 thf(exu__Cong,definition,
% 0.15/0.44 ( exu__Cong
% 0.15/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.15/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( Xx = Xy )
% 0.15/0.44 => ( ( Xphi @ Xx )
% 0.15/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.15/0.44 => ( ( exu
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.44 <=> ( exu
% 0.15/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(emptyset__Cong_type,type,
% 0.15/0.44 emptyset__Cong: $o ).
% 0.15/0.44
% 0.15/0.44 thf(emptyset__Cong,definition,
% 0.15/0.44 ( emptyset__Cong
% 0.15/0.44 = ( emptyset = emptyset ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoin__Cong_type,type,
% 0.15/0.44 setadjoin__Cong: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoin__Cong,definition,
% 0.15/0.44 ( setadjoin__Cong
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( Xx = Xy )
% 0.15/0.44 => ! [Xz: $i,Xu: $i] :
% 0.15/0.44 ( ( Xz = Xu )
% 0.15/0.44 => ( ( setadjoin @ Xx @ Xz )
% 0.15/0.44 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(powerset__Cong_type,type,
% 0.15/0.44 powerset__Cong: $o ).
% 0.15/0.44
% 0.15/0.44 thf(powerset__Cong,definition,
% 0.15/0.44 ( powerset__Cong
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( A = B )
% 0.15/0.44 => ( ( powerset @ A )
% 0.15/0.44 = ( powerset @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setunion__Cong_type,type,
% 0.15/0.44 setunion__Cong: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setunion__Cong,definition,
% 0.15/0.44 ( setunion__Cong
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( A = B )
% 0.15/0.44 => ( ( setunion @ A )
% 0.15/0.44 = ( setunion @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(omega__Cong_type,type,
% 0.15/0.44 omega__Cong: $o ).
% 0.15/0.44
% 0.15/0.44 thf(omega__Cong,definition,
% 0.15/0.44 ( omega__Cong
% 0.15/0.44 = ( omega = omega ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(exuEu_type,type,
% 0.15/0.44 exuEu: $o ).
% 0.15/0.44
% 0.15/0.44 thf(exuEu,definition,
% 0.15/0.44 ( exuEu
% 0.15/0.44 = ( ! [Xphi: $i > $o] :
% 0.15/0.44 ( ( exu
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.44 => ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( Xphi @ Xx )
% 0.15/0.44 => ( ( Xphi @ Xy )
% 0.15/0.44 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(descr__Cong_type,type,
% 0.15/0.44 descr__Cong: $o ).
% 0.15/0.44
% 0.15/0.44 thf(descr__Cong,definition,
% 0.15/0.44 ( descr__Cong
% 0.15/0.44 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.15/0.44 ( ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( Xx = Xy )
% 0.15/0.44 => ( ( Xphi @ Xx )
% 0.15/0.44 <=> ( Xpsi @ Xy ) ) )
% 0.15/0.44 => ( ( exu
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.44 => ( ( exu
% 0.15/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.15/0.44 => ( ( descr
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.44 = ( descr
% 0.15/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(dsetconstr__Cong_type,type,
% 0.15/0.44 dsetconstr__Cong: $o ).
% 0.15/0.44
% 0.15/0.44 thf(dsetconstr__Cong,definition,
% 0.15/0.44 ( dsetconstr__Cong
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( A = B )
% 0.15/0.44 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 => ( ( Xx = Xy )
% 0.15/0.44 => ( ( Xphi @ Xx )
% 0.15/0.44 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.15/0.44 => ( ( dsetconstr @ A
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.44 = ( dsetconstr @ B
% 0.15/0.44 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(subset_type,type,
% 0.15/0.44 subset: $i > $i > $o ).
% 0.15/0.44
% 0.15/0.44 thf(disjoint_type,type,
% 0.15/0.44 disjoint: $i > $i > $o ).
% 0.15/0.44
% 0.15/0.44 thf(setsmeet_type,type,
% 0.15/0.44 setsmeet: $i > $i > $o ).
% 0.15/0.44
% 0.15/0.44 thf(subsetI1_type,type,
% 0.15/0.44 subsetI1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(subsetI1,definition,
% 0.15/0.44 ( subsetI1
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ Xx @ B ) )
% 0.15/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(eqimpsubset2_type,type,
% 0.15/0.44 eqimpsubset2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(eqimpsubset2,definition,
% 0.15/0.44 ( eqimpsubset2
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( A = B )
% 0.15/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(eqimpsubset1_type,type,
% 0.15/0.44 eqimpsubset1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(eqimpsubset1,definition,
% 0.15/0.44 ( eqimpsubset1
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( A = B )
% 0.15/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(subsetI2_type,type,
% 0.15/0.44 subsetI2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(subsetI2,definition,
% 0.15/0.44 ( subsetI2
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ Xx @ B ) )
% 0.15/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(emptysetsubset_type,type,
% 0.15/0.44 emptysetsubset: $o ).
% 0.15/0.44
% 0.15/0.44 thf(emptysetsubset,definition,
% 0.15/0.44 ( emptysetsubset
% 0.15/0.44 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(subsetE_type,type,
% 0.15/0.44 subsetE: $o ).
% 0.15/0.44
% 0.15/0.44 thf(subsetE,definition,
% 0.15/0.44 ( subsetE
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( subset @ A @ B )
% 0.15/0.44 => ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(subsetE2_type,type,
% 0.15/0.44 subsetE2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(subsetE2,definition,
% 0.15/0.44 ( subsetE2
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( subset @ A @ B )
% 0.15/0.44 => ( ~ ( in @ Xx @ B )
% 0.15/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(notsubsetI_type,type,
% 0.15/0.44 notsubsetI: $o ).
% 0.15/0.44
% 0.15/0.44 thf(notsubsetI,definition,
% 0.15/0.44 ( notsubsetI
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( ~ ( in @ Xx @ B )
% 0.15/0.44 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(notequalI1_type,type,
% 0.15/0.44 notequalI1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(notequalI1,definition,
% 0.15/0.44 ( notequalI1
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ~ ( subset @ A @ B )
% 0.15/0.44 => ( A != B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(notequalI2_type,type,
% 0.15/0.44 notequalI2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(notequalI2,definition,
% 0.15/0.44 ( notequalI2
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( ~ ( in @ Xx @ B )
% 0.15/0.44 => ( A != B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(subsetRefl_type,type,
% 0.15/0.44 subsetRefl: $o ).
% 0.15/0.44
% 0.15/0.44 thf(subsetRefl,definition,
% 0.15/0.44 ( subsetRefl
% 0.15/0.44 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(subsetTrans_type,type,
% 0.15/0.44 subsetTrans: $o ).
% 0.15/0.44
% 0.15/0.44 thf(subsetTrans,definition,
% 0.15/0.44 ( subsetTrans
% 0.15/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.15/0.44 ( ( subset @ A @ B )
% 0.15/0.44 => ( ( subset @ B @ C )
% 0.15/0.44 => ( subset @ A @ C ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoinSub_type,type,
% 0.15/0.44 setadjoinSub: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoinSub,definition,
% 0.15/0.44 ( setadjoinSub
% 0.15/0.44 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoinSub2_type,type,
% 0.15/0.44 setadjoinSub2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setadjoinSub2,definition,
% 0.15/0.44 ( setadjoinSub2
% 0.15/0.44 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.15/0.44 ( ( subset @ A @ B )
% 0.15/0.44 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(subset2powerset_type,type,
% 0.15/0.44 subset2powerset: $o ).
% 0.15/0.44
% 0.15/0.44 thf(subset2powerset,definition,
% 0.15/0.44 ( subset2powerset
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( subset @ A @ B )
% 0.15/0.44 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setextsub_type,type,
% 0.15/0.44 setextsub: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setextsub,definition,
% 0.15/0.44 ( setextsub
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( subset @ A @ B )
% 0.15/0.44 => ( ( subset @ B @ A )
% 0.15/0.44 => ( A = B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(subsetemptysetimpeq_type,type,
% 0.15/0.44 subsetemptysetimpeq: $o ).
% 0.15/0.44
% 0.15/0.44 thf(subsetemptysetimpeq,definition,
% 0.15/0.44 ( subsetemptysetimpeq
% 0.15/0.44 = ( ! [A: $i] :
% 0.15/0.44 ( ( subset @ A @ emptyset )
% 0.15/0.44 => ( A = emptyset ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(powersetI1_type,type,
% 0.15/0.44 powersetI1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(powersetI1,definition,
% 0.15/0.44 ( powersetI1
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( subset @ B @ A )
% 0.15/0.44 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(powersetE1_type,type,
% 0.15/0.44 powersetE1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(powersetE1,definition,
% 0.15/0.44 ( powersetE1
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( in @ B @ ( powerset @ A ) )
% 0.15/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(inPowerset_type,type,
% 0.15/0.44 inPowerset: $o ).
% 0.15/0.44
% 0.15/0.44 thf(inPowerset,definition,
% 0.15/0.44 ( inPowerset
% 0.15/0.44 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(powersetsubset_type,type,
% 0.15/0.44 powersetsubset: $o ).
% 0.15/0.44
% 0.15/0.44 thf(powersetsubset,definition,
% 0.15/0.44 ( powersetsubset
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( subset @ A @ B )
% 0.15/0.44 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(sepInPowerset_type,type,
% 0.15/0.44 sepInPowerset: $o ).
% 0.15/0.44
% 0.15/0.44 thf(sepInPowerset,definition,
% 0.15/0.44 ( sepInPowerset
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.44 ( in
% 0.15/0.44 @ ( dsetconstr @ A
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.44 @ ( powerset @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(sepSubset_type,type,
% 0.15/0.44 sepSubset: $o ).
% 0.15/0.44
% 0.15/0.44 thf(sepSubset,definition,
% 0.15/0.44 ( sepSubset
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.44 ( subset
% 0.15/0.44 @ ( dsetconstr @ A
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.44 @ A ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(binunion_type,type,
% 0.15/0.44 binunion: $i > $i > $i ).
% 0.15/0.44
% 0.15/0.44 thf(binunionIL_type,type,
% 0.15/0.44 binunionIL: $o ).
% 0.15/0.44
% 0.15/0.44 thf(binunionIL,definition,
% 0.15/0.44 ( binunionIL
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(upairset2IR_type,type,
% 0.15/0.44 upairset2IR: $o ).
% 0.15/0.44
% 0.15/0.44 thf(upairset2IR,definition,
% 0.15/0.44 ( upairset2IR
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(binunionIR_type,type,
% 0.15/0.44 binunionIR: $o ).
% 0.15/0.44
% 0.15/0.44 thf(binunionIR,definition,
% 0.15/0.44 ( binunionIR
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ B )
% 0.15/0.44 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(binunionEcases_type,type,
% 0.15/0.44 binunionEcases: $o ).
% 0.15/0.44
% 0.15/0.44 thf(binunionEcases,definition,
% 0.15/0.44 ( binunionEcases
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.15/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.15/0.44 => ( ( ( in @ Xx @ A )
% 0.15/0.44 => Xphi )
% 0.15/0.44 => ( ( ( in @ Xx @ B )
% 0.15/0.44 => Xphi )
% 0.15/0.44 => Xphi ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(binunionE_type,type,
% 0.15/0.44 binunionE: $o ).
% 0.15/0.44
% 0.15/0.44 thf(binunionE,definition,
% 0.15/0.44 ( binunionE
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.15/0.44 => ( ( in @ Xx @ A )
% 0.15/0.44 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(binunionLsub_type,type,
% 0.15/0.44 binunionLsub: $o ).
% 0.15/0.44
% 0.15/0.44 thf(binunionLsub,definition,
% 0.15/0.44 ( binunionLsub
% 0.15/0.44 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(binunionRsub_type,type,
% 0.15/0.44 binunionRsub: $o ).
% 0.15/0.44
% 0.15/0.44 thf(binunionRsub,definition,
% 0.15/0.44 ( binunionRsub
% 0.15/0.44 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(binintersect_type,type,
% 0.15/0.44 binintersect: $i > $i > $i ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectI_type,type,
% 0.15/0.44 binintersectI: $o ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectI,definition,
% 0.15/0.44 ( binintersectI
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( ( in @ Xx @ B )
% 0.15/0.44 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectSubset5_type,type,
% 0.15/0.44 binintersectSubset5: $o ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectSubset5,definition,
% 0.15/0.44 ( binintersectSubset5
% 0.15/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.15/0.44 ( ( subset @ C @ A )
% 0.15/0.44 => ( ( subset @ C @ B )
% 0.15/0.44 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectEL_type,type,
% 0.15/0.44 binintersectEL: $o ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectEL,definition,
% 0.15/0.44 ( binintersectEL
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.15/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectLsub_type,type,
% 0.15/0.44 binintersectLsub: $o ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectLsub,definition,
% 0.15/0.44 ( binintersectLsub
% 0.15/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectSubset2_type,type,
% 0.15/0.44 binintersectSubset2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectSubset2,definition,
% 0.15/0.44 ( binintersectSubset2
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( subset @ A @ B )
% 0.15/0.44 => ( ( binintersect @ A @ B )
% 0.15/0.44 = A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectSubset3_type,type,
% 0.15/0.44 binintersectSubset3: $o ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectSubset3,definition,
% 0.15/0.44 ( binintersectSubset3
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( ( binintersect @ A @ B )
% 0.15/0.44 = B )
% 0.15/0.44 => ( subset @ B @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectER_type,type,
% 0.15/0.44 binintersectER: $o ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectER,definition,
% 0.15/0.44 ( binintersectER
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.15/0.44 => ( in @ Xx @ B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(disjointsetsI1_type,type,
% 0.15/0.44 disjointsetsI1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(disjointsetsI1,definition,
% 0.15/0.44 ( disjointsetsI1
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ~ ? [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 & ( in @ Xx @ B ) )
% 0.15/0.44 => ( ( binintersect @ A @ B )
% 0.15/0.44 = emptyset ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectRsub_type,type,
% 0.15/0.44 binintersectRsub: $o ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectRsub,definition,
% 0.15/0.44 ( binintersectRsub
% 0.15/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectSubset4_type,type,
% 0.15/0.44 binintersectSubset4: $o ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectSubset4,definition,
% 0.15/0.44 ( binintersectSubset4
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( subset @ B @ A )
% 0.15/0.44 => ( ( binintersect @ A @ B )
% 0.15/0.44 = B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectSubset1_type,type,
% 0.15/0.44 binintersectSubset1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(binintersectSubset1,definition,
% 0.15/0.44 ( binintersectSubset1
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( ( binintersect @ A @ B )
% 0.15/0.44 = A )
% 0.15/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(bs114d_type,type,
% 0.15/0.44 bs114d: $o ).
% 0.15/0.44
% 0.15/0.44 thf(bs114d,definition,
% 0.15/0.44 ( bs114d
% 0.15/0.44 = ( ! [A: $i,B: $i,C: $i] :
% 0.15/0.44 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.15/0.44 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(regular_type,type,
% 0.15/0.44 regular: $i > $o ).
% 0.15/0.44
% 0.15/0.44 thf(setminus_type,type,
% 0.15/0.44 setminus: $i > $i > $i ).
% 0.15/0.44
% 0.15/0.44 thf(setminusI_type,type,
% 0.15/0.44 setminusI: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setminusI,definition,
% 0.15/0.44 ( setminusI
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( ~ ( in @ Xx @ B )
% 0.15/0.44 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setminusEL_type,type,
% 0.15/0.44 setminusEL: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setminusEL,definition,
% 0.15/0.44 ( setminusEL
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.15/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setminusER_type,type,
% 0.15/0.44 setminusER: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setminusER,definition,
% 0.15/0.44 ( setminusER
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.15/0.44 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setminusSubset2_type,type,
% 0.15/0.44 setminusSubset2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setminusSubset2,definition,
% 0.15/0.44 ( setminusSubset2
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( subset @ A @ B )
% 0.15/0.44 => ( ( setminus @ A @ B )
% 0.15/0.44 = emptyset ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setminusERneg_type,type,
% 0.15/0.44 setminusERneg: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setminusERneg,definition,
% 0.15/0.44 ( setminusERneg
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.15/0.44 => ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setminusELneg_type,type,
% 0.15/0.44 setminusELneg: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setminusELneg,definition,
% 0.15/0.44 ( setminusELneg
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.15/0.44 => ( ~ ( in @ Xx @ B )
% 0.15/0.44 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setminusILneg_type,type,
% 0.15/0.44 setminusILneg: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setminusILneg,definition,
% 0.15/0.44 ( setminusILneg
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ~ ( in @ Xx @ A )
% 0.15/0.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setminusIRneg_type,type,
% 0.15/0.44 setminusIRneg: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setminusIRneg,definition,
% 0.15/0.44 ( setminusIRneg
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ B )
% 0.15/0.44 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setminusLsub_type,type,
% 0.15/0.44 setminusLsub: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setminusLsub,definition,
% 0.15/0.44 ( setminusLsub
% 0.15/0.44 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setminusSubset1_type,type,
% 0.15/0.44 setminusSubset1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setminusSubset1,definition,
% 0.15/0.44 ( setminusSubset1
% 0.15/0.44 = ( ! [A: $i,B: $i] :
% 0.15/0.44 ( ( ( setminus @ A @ B )
% 0.15/0.44 = emptyset )
% 0.15/0.44 => ( subset @ A @ B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(symdiff_type,type,
% 0.15/0.44 symdiff: $i > $i > $i ).
% 0.15/0.44
% 0.15/0.44 thf(symdiffE_type,type,
% 0.15/0.44 symdiffE: $o ).
% 0.15/0.44
% 0.15/0.44 thf(symdiffE,definition,
% 0.15/0.44 ( symdiffE
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.15/0.44 => ! [Xphi: $o] :
% 0.15/0.44 ( ( ( in @ Xx @ A )
% 0.15/0.44 => ( ~ ( in @ Xx @ B )
% 0.15/0.44 => Xphi ) )
% 0.15/0.44 => ( ( ~ ( in @ Xx @ A )
% 0.15/0.44 => ( ( in @ Xx @ B )
% 0.15/0.44 => Xphi ) )
% 0.15/0.44 => Xphi ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(symdiffI1_type,type,
% 0.15/0.44 symdiffI1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(symdiffI1,definition,
% 0.15/0.44 ( symdiffI1
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( ~ ( in @ Xx @ B )
% 0.15/0.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(symdiffI2_type,type,
% 0.15/0.44 symdiffI2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(symdiffI2,definition,
% 0.15/0.44 ( symdiffI2
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ~ ( in @ Xx @ A )
% 0.15/0.44 => ( ( in @ Xx @ B )
% 0.15/0.44 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(symdiffIneg1_type,type,
% 0.15/0.44 symdiffIneg1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(symdiffIneg1,definition,
% 0.15/0.44 ( symdiffIneg1
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( ( in @ Xx @ B )
% 0.15/0.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(symdiffIneg2_type,type,
% 0.15/0.44 symdiffIneg2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(symdiffIneg2,definition,
% 0.15/0.44 ( symdiffIneg2
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ~ ( in @ Xx @ A )
% 0.15/0.44 => ( ~ ( in @ Xx @ B )
% 0.15/0.44 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(iskpair_type,type,
% 0.15/0.44 iskpair: $i > $o ).
% 0.15/0.44
% 0.15/0.44 thf(secondinupair_type,type,
% 0.15/0.44 secondinupair: $o ).
% 0.15/0.44
% 0.15/0.44 thf(secondinupair,definition,
% 0.15/0.44 ( secondinupair
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairIL_type,type,
% 0.15/0.44 setukpairIL: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairIL,definition,
% 0.15/0.44 ( setukpairIL
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairIR_type,type,
% 0.15/0.44 setukpairIR: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairIR,definition,
% 0.15/0.44 ( setukpairIR
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(kpairiskpair_type,type,
% 0.15/0.44 kpairiskpair: $o ).
% 0.15/0.44
% 0.15/0.44 thf(kpairiskpair,definition,
% 0.15/0.44 ( kpairiskpair
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(kpair_type,type,
% 0.15/0.44 kpair: $i > $i > $i ).
% 0.15/0.44
% 0.15/0.44 thf(kpairp_type,type,
% 0.15/0.44 kpairp: $o ).
% 0.15/0.44
% 0.15/0.44 thf(kpairp,definition,
% 0.15/0.44 ( kpairp
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(cartprod_type,type,
% 0.15/0.44 cartprod: $i > $i > $i ).
% 0.15/0.44
% 0.15/0.44 thf(singletonsubset_type,type,
% 0.15/0.44 singletonsubset: $o ).
% 0.15/0.44
% 0.15/0.44 thf(singletonsubset,definition,
% 0.15/0.44 ( singletonsubset
% 0.15/0.44 = ( ! [A: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(singletoninpowerset_type,type,
% 0.15/0.44 singletoninpowerset: $o ).
% 0.15/0.44
% 0.15/0.44 thf(singletoninpowerset,definition,
% 0.15/0.44 ( singletoninpowerset
% 0.15/0.44 = ( ! [A: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(singletoninpowunion_type,type,
% 0.15/0.44 singletoninpowunion: $o ).
% 0.15/0.44
% 0.15/0.44 thf(singletoninpowunion,definition,
% 0.15/0.44 ( singletoninpowunion
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(upairset2E_type,type,
% 0.15/0.44 upairset2E: $o ).
% 0.15/0.44
% 0.15/0.44 thf(upairset2E,definition,
% 0.15/0.44 ( upairset2E
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.15/0.44 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.15/0.44 => ( ( Xz = Xx )
% 0.15/0.44 | ( Xz = Xy ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(upairsubunion_type,type,
% 0.15/0.44 upairsubunion: $o ).
% 0.15/0.44
% 0.15/0.44 thf(upairsubunion,definition,
% 0.15/0.44 ( upairsubunion
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(upairinpowunion_type,type,
% 0.15/0.44 upairinpowunion: $o ).
% 0.15/0.44
% 0.15/0.44 thf(upairinpowunion,definition,
% 0.15/0.44 ( upairinpowunion
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(ubforcartprodlem1_type,type,
% 0.15/0.44 ubforcartprodlem1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(ubforcartprodlem1,definition,
% 0.15/0.44 ( ubforcartprodlem1
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(ubforcartprodlem2_type,type,
% 0.15/0.44 ubforcartprodlem2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(ubforcartprodlem2,definition,
% 0.15/0.44 ( ubforcartprodlem2
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(ubforcartprodlem3_type,type,
% 0.15/0.44 ubforcartprodlem3: $o ).
% 0.15/0.44
% 0.15/0.44 thf(ubforcartprodlem3,definition,
% 0.15/0.44 ( ubforcartprodlem3
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodpairin_type,type,
% 0.15/0.44 cartprodpairin: $o ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodpairin,definition,
% 0.15/0.44 ( cartprodpairin
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodmempair1_type,type,
% 0.15/0.44 cartprodmempair1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodmempair1,definition,
% 0.15/0.44 ( cartprodmempair1
% 0.15/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.15/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.15/0.44 => ? [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 & ? [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 & ( Xu
% 0.15/0.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodmempair_type,type,
% 0.15/0.44 cartprodmempair: $o ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodmempair,definition,
% 0.15/0.44 ( cartprodmempair
% 0.15/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.15/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.15/0.44 => ( iskpair @ Xu ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setunionE2_type,type,
% 0.15/0.44 setunionE2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setunionE2,definition,
% 0.15/0.44 ( setunionE2
% 0.15/0.44 = ( ! [A: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ ( setunion @ A ) )
% 0.15/0.44 => ? [X: $i] :
% 0.15/0.44 ( ( in @ X @ A )
% 0.15/0.44 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setunionsingleton1_type,type,
% 0.15/0.44 setunionsingleton1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setunionsingleton1,definition,
% 0.15/0.44 ( setunionsingleton1
% 0.15/0.44 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setunionsingleton2_type,type,
% 0.15/0.44 setunionsingleton2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setunionsingleton2,definition,
% 0.15/0.44 ( setunionsingleton2
% 0.15/0.44 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setunionsingleton_type,type,
% 0.15/0.44 setunionsingleton: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setunionsingleton,definition,
% 0.15/0.44 ( setunionsingleton
% 0.15/0.44 = ( ! [Xx: $i] :
% 0.15/0.44 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.15/0.44 = Xx ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(singleton_type,type,
% 0.15/0.44 singleton: $i > $o ).
% 0.15/0.44
% 0.15/0.44 thf(singletonprop_type,type,
% 0.15/0.44 singletonprop: $o ).
% 0.15/0.44
% 0.15/0.44 thf(singletonprop,definition,
% 0.15/0.44 ( singletonprop
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ A )
% 0.15/0.44 => ( ( Xphi @ Xx )
% 0.15/0.44 => ( ( Xphi @ Xy )
% 0.15/0.44 => ( Xx = Xy ) ) ) ) )
% 0.15/0.44 => ( ? [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 & ( Xphi @ Xx ) )
% 0.15/0.44 => ( singleton
% 0.15/0.44 @ ( dsetconstr @ A
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(ex1_type,type,
% 0.15/0.44 ex1: $i > ( $i > $o ) > $o ).
% 0.15/0.44
% 0.15/0.44 thf(ex1,definition,
% 0.15/0.44 ( ex1
% 0.15/0.44 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.15/0.44 ( singleton
% 0.15/0.44 @ ( dsetconstr @ A
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(ex1E1_type,type,
% 0.15/0.44 ex1E1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(ex1E1,definition,
% 0.15/0.44 ( ex1E1
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.44 ( ( ex1 @ A
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.44 => ? [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(ex1I_type,type,
% 0.15/0.44 ex1I: $o ).
% 0.15/0.44
% 0.15/0.44 thf(ex1I,definition,
% 0.15/0.44 ( ex1I
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( ( Xphi @ Xx )
% 0.15/0.44 => ( ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ A )
% 0.15/0.44 => ( ( Xphi @ Xy )
% 0.15/0.44 => ( Xy = Xx ) ) )
% 0.15/0.44 => ( ex1 @ A
% 0.15/0.44 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(ex1I2_type,type,
% 0.15/0.44 ex1I2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(ex1I2,definition,
% 0.15/0.44 ( ex1I2
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ A )
% 0.15/0.44 => ( ( Xphi @ Xx )
% 0.15/0.44 => ( ( Xphi @ Xy )
% 0.15/0.44 => ( Xx = Xy ) ) ) ) )
% 0.15/0.44 => ( ? [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 & ( Xphi @ Xx ) )
% 0.15/0.44 => ( ex1 @ A
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(singletonsuniq_type,type,
% 0.15/0.44 singletonsuniq: $o ).
% 0.15/0.44
% 0.15/0.44 thf(singletonsuniq,definition,
% 0.15/0.44 ( singletonsuniq
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( ( setadjoin @ Xx @ emptyset )
% 0.15/0.44 = ( setadjoin @ Xy @ emptyset ) )
% 0.15/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(atmost1p_type,type,
% 0.15/0.44 atmost1p: $i > $o ).
% 0.15/0.44
% 0.15/0.44 thf(atleast2p_type,type,
% 0.15/0.44 atleast2p: $i > $o ).
% 0.15/0.44
% 0.15/0.44 thf(atmost2p_type,type,
% 0.15/0.44 atmost2p: $i > $o ).
% 0.15/0.44
% 0.15/0.44 thf(upairsetp_type,type,
% 0.15/0.44 upairsetp: $i > $o ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairinjL1_type,type,
% 0.15/0.44 setukpairinjL1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairinjL1,definition,
% 0.15/0.44 ( setukpairinjL1
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.15/0.44 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.15/0.44 => ( Xx = Xz ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(kfstsingleton_type,type,
% 0.15/0.44 kfstsingleton: $o ).
% 0.15/0.44
% 0.15/0.44 thf(kfstsingleton,definition,
% 0.15/0.44 ( kfstsingleton
% 0.15/0.44 = ( ! [Xu: $i] :
% 0.15/0.44 ( ( iskpair @ Xu )
% 0.15/0.44 => ( singleton
% 0.15/0.44 @ ( dsetconstr @ ( setunion @ Xu )
% 0.15/0.44 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(theprop_type,type,
% 0.15/0.44 theprop: $o ).
% 0.15/0.44
% 0.15/0.44 thf(theprop,definition,
% 0.15/0.44 ( theprop
% 0.15/0.44 = ( ! [X: $i] :
% 0.15/0.44 ( ( singleton @ X )
% 0.15/0.44 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(kfst_type,type,
% 0.15/0.44 kfst: $i > $i ).
% 0.15/0.44
% 0.15/0.44 thf(kfstpairEq_type,type,
% 0.15/0.44 kfstpairEq: $o ).
% 0.15/0.44
% 0.15/0.44 thf(kfstpairEq,definition,
% 0.15/0.44 ( kfstpairEq
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.15/0.44 = Xx ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodfstin_type,type,
% 0.15/0.44 cartprodfstin: $o ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodfstin,definition,
% 0.15/0.44 ( cartprodfstin
% 0.15/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.15/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.15/0.44 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairinjL2_type,type,
% 0.15/0.44 setukpairinjL2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairinjL2,definition,
% 0.15/0.44 ( setukpairinjL2
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.15/0.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.15/0.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.15/0.44 => ( Xx = Xz ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairinjL_type,type,
% 0.15/0.44 setukpairinjL: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairinjL,definition,
% 0.15/0.44 ( setukpairinjL
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.15/0.44 ( ( ( kpair @ Xx @ Xy )
% 0.15/0.44 = ( kpair @ Xz @ Xu ) )
% 0.15/0.44 => ( Xx = Xz ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairinjR11_type,type,
% 0.15/0.44 setukpairinjR11: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairinjR11,definition,
% 0.15/0.44 ( setukpairinjR11
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( Xx = Xy )
% 0.15/0.44 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.15/0.44 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairinjR12_type,type,
% 0.15/0.44 setukpairinjR12: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairinjR12,definition,
% 0.15/0.44 ( setukpairinjR12
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( Xx = Xy )
% 0.15/0.44 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.15/0.44 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairinjR1_type,type,
% 0.15/0.44 setukpairinjR1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairinjR1,definition,
% 0.15/0.44 ( setukpairinjR1
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.15/0.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.15/0.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.15/0.44 => ( ( Xz = Xu )
% 0.15/0.44 => ( Xy = Xu ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(upairequniteq_type,type,
% 0.15/0.44 upairequniteq: $o ).
% 0.15/0.44
% 0.15/0.44 thf(upairequniteq,definition,
% 0.15/0.44 ( upairequniteq
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.15/0.44 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.15/0.44 = ( setadjoin @ Xz @ emptyset ) )
% 0.15/0.44 => ( Xx = Xy ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairinjR2_type,type,
% 0.15/0.44 setukpairinjR2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairinjR2,definition,
% 0.15/0.44 ( setukpairinjR2
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.15/0.44 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.15/0.44 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.15/0.44 => ( Xy = Xu ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairinjR_type,type,
% 0.15/0.44 setukpairinjR: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setukpairinjR,definition,
% 0.15/0.44 ( setukpairinjR
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.15/0.44 ( ( ( kpair @ Xx @ Xy )
% 0.15/0.44 = ( kpair @ Xz @ Xu ) )
% 0.15/0.44 => ( Xy = Xu ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(ksndsingleton_type,type,
% 0.15/0.44 ksndsingleton: $o ).
% 0.15/0.44
% 0.15/0.44 thf(ksndsingleton,definition,
% 0.15/0.44 ( ksndsingleton
% 0.15/0.44 = ( ! [Xu: $i] :
% 0.15/0.44 ( ( iskpair @ Xu )
% 0.15/0.44 => ( singleton
% 0.15/0.44 @ ( dsetconstr @ ( setunion @ Xu )
% 0.15/0.44 @ ^ [Xx: $i] :
% 0.15/0.44 ( Xu
% 0.15/0.44 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(ksnd_type,type,
% 0.15/0.44 ksnd: $i > $i ).
% 0.15/0.44
% 0.15/0.44 thf(ksndpairEq_type,type,
% 0.15/0.44 ksndpairEq: $o ).
% 0.15/0.44
% 0.15/0.44 thf(ksndpairEq,definition,
% 0.15/0.44 ( ksndpairEq
% 0.15/0.44 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.15/0.44 = Xy ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(kpairsurjEq_type,type,
% 0.15/0.44 kpairsurjEq: $o ).
% 0.15/0.44
% 0.15/0.44 thf(kpairsurjEq,definition,
% 0.15/0.44 ( kpairsurjEq
% 0.15/0.44 = ( ! [Xu: $i] :
% 0.15/0.44 ( ( iskpair @ Xu )
% 0.15/0.44 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.15/0.44 = Xu ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodsndin_type,type,
% 0.15/0.44 cartprodsndin: $o ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodsndin,definition,
% 0.15/0.44 ( cartprodsndin
% 0.15/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.15/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.15/0.44 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodpairmemEL_type,type,
% 0.15/0.44 cartprodpairmemEL: $o ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodpairmemEL,definition,
% 0.15/0.44 ( cartprodpairmemEL
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.15/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodpairmemER_type,type,
% 0.15/0.44 cartprodpairmemER: $o ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodpairmemER,definition,
% 0.15/0.44 ( cartprodpairmemER
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.15/0.44 => ( in @ Xy @ B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodmempaircEq_type,type,
% 0.15/0.44 cartprodmempaircEq: $o ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodmempaircEq,definition,
% 0.15/0.44 ( cartprodmempaircEq
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 => ( ( kpair @ Xx @ Xy )
% 0.15/0.44 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodfstpairEq_type,type,
% 0.15/0.44 cartprodfstpairEq: $o ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodfstpairEq,definition,
% 0.15/0.44 ( cartprodfstpairEq
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.15/0.44 = Xx ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodsndpairEq_type,type,
% 0.15/0.44 cartprodsndpairEq: $o ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodsndpairEq,definition,
% 0.15/0.44 ( cartprodsndpairEq
% 0.15/0.44 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.15/0.44 = Xy ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodpairsurjEq_type,type,
% 0.15/0.44 cartprodpairsurjEq: $o ).
% 0.15/0.44
% 0.15/0.44 thf(cartprodpairsurjEq,definition,
% 0.15/0.44 ( cartprodpairsurjEq
% 0.15/0.44 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.15/0.44 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.15/0.44 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.15/0.44 = Xu ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(breln_type,type,
% 0.15/0.44 breln: $i > $i > $i > $o ).
% 0.15/0.44
% 0.15/0.44 thf(breln,definition,
% 0.15/0.44 ( breln
% 0.15/0.44 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(dpsetconstr_type,type,
% 0.15/0.44 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.15/0.44
% 0.15/0.44 thf(dpsetconstrI_type,type,
% 0.15/0.44 dpsetconstrI: $o ).
% 0.15/0.44
% 0.15/0.44 thf(dpsetconstrI,definition,
% 0.15/0.44 ( dpsetconstrI
% 0.15/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 => ( ( Xphi @ Xx @ Xy )
% 0.15/0.44 => ( in @ ( kpair @ Xx @ Xy )
% 0.15/0.44 @ ( dpsetconstr @ A @ B
% 0.15/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(dpsetconstrSub_type,type,
% 0.15/0.44 dpsetconstrSub: $o ).
% 0.15/0.44
% 0.15/0.44 thf(dpsetconstrSub,definition,
% 0.15/0.44 ( dpsetconstrSub
% 0.15/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.15/0.44 ( subset
% 0.15/0.44 @ ( dpsetconstr @ A @ B
% 0.15/0.44 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.15/0.44 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(setOfPairsIsBReln_type,type,
% 0.15/0.44 setOfPairsIsBReln: $o ).
% 0.15/0.44
% 0.15/0.44 thf(setOfPairsIsBReln,definition,
% 0.15/0.44 ( setOfPairsIsBReln
% 0.15/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.15/0.44 ( breln @ A @ B
% 0.15/0.44 @ ( dpsetconstr @ A @ B
% 0.15/0.44 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(dpsetconstrERa_type,type,
% 0.15/0.44 dpsetconstrERa: $o ).
% 0.15/0.44
% 0.15/0.44 thf(dpsetconstrERa,definition,
% 0.15/0.44 ( dpsetconstrERa
% 0.15/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.15/0.44 @ ( dpsetconstr @ A @ B
% 0.15/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.15/0.44 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(dpsetconstrEL1_type,type,
% 0.15/0.44 dpsetconstrEL1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(dpsetconstrEL1,definition,
% 0.15/0.44 ( dpsetconstrEL1
% 0.15/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( in @ ( kpair @ Xx @ Xy )
% 0.15/0.44 @ ( dpsetconstr @ A @ B
% 0.15/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.15/0.44 => ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(dpsetconstrEL2_type,type,
% 0.15/0.44 dpsetconstrEL2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(dpsetconstrEL2,definition,
% 0.15/0.44 ( dpsetconstrEL2
% 0.15/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( in @ ( kpair @ Xx @ Xy )
% 0.15/0.44 @ ( dpsetconstr @ A @ B
% 0.15/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.15/0.44 => ( in @ Xy @ B ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(dpsetconstrER_type,type,
% 0.15/0.44 dpsetconstrER: $o ).
% 0.15/0.44
% 0.15/0.44 thf(dpsetconstrER,definition,
% 0.15/0.44 ( dpsetconstrER
% 0.15/0.44 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( in @ ( kpair @ Xx @ Xy )
% 0.15/0.44 @ ( dpsetconstr @ A @ B
% 0.15/0.44 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.15/0.44 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(func_type,type,
% 0.15/0.44 func: $i > $i > $i > $o ).
% 0.15/0.44
% 0.15/0.44 thf(funcSet_type,type,
% 0.15/0.44 funcSet: $i > $i > $i ).
% 0.15/0.44
% 0.15/0.44 thf(funcImageSingleton_type,type,
% 0.15/0.44 funcImageSingleton: $o ).
% 0.15/0.44
% 0.15/0.44 thf(funcImageSingleton,definition,
% 0.15/0.44 ( funcImageSingleton
% 0.15/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.44 ( ( func @ A @ B @ Xf )
% 0.15/0.44 => ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( singleton
% 0.15/0.44 @ ( dsetconstr @ B
% 0.15/0.44 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(apProp_type,type,
% 0.15/0.44 apProp: $o ).
% 0.15/0.44
% 0.15/0.44 thf(apProp,definition,
% 0.15/0.44 ( apProp
% 0.15/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.44 ( ( func @ A @ B @ Xf )
% 0.15/0.44 => ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( in
% 0.15/0.44 @ ( setunion
% 0.15/0.44 @ ( dsetconstr @ B
% 0.15/0.44 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.15/0.44 @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(ap_type,type,
% 0.15/0.44 ap: $i > $i > $i > $i > $i ).
% 0.15/0.44
% 0.15/0.44 thf(app_type,type,
% 0.15/0.44 app: $o ).
% 0.15/0.44
% 0.15/0.44 thf(app,definition,
% 0.15/0.44 ( app
% 0.15/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.44 ( ( func @ A @ B @ Xf )
% 0.15/0.44 => ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(infuncsetfunc_type,type,
% 0.15/0.44 infuncsetfunc: $o ).
% 0.15/0.44
% 0.15/0.44 thf(infuncsetfunc,definition,
% 0.15/0.44 ( infuncsetfunc
% 0.15/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.44 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(ap2p_type,type,
% 0.15/0.44 ap2p: $o ).
% 0.15/0.44
% 0.15/0.44 thf(ap2p,definition,
% 0.15/0.44 ( ap2p
% 0.15/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.44 => ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(funcinfuncset_type,type,
% 0.15/0.44 funcinfuncset: $o ).
% 0.15/0.44
% 0.15/0.44 thf(funcinfuncset,definition,
% 0.15/0.44 ( funcinfuncset
% 0.15/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.44 ( ( func @ A @ B @ Xf )
% 0.15/0.44 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(lamProp_type,type,
% 0.15/0.44 lamProp: $o ).
% 0.15/0.44
% 0.15/0.44 thf(lamProp,definition,
% 0.15/0.44 ( lamProp
% 0.15/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.44 => ( func @ A @ B
% 0.15/0.44 @ ( dpsetconstr @ A @ B
% 0.15/0.44 @ ^ [Xx: $i,Xy: $i] :
% 0.15/0.44 ( ( Xf @ Xx )
% 0.15/0.44 = Xy ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(lam_type,type,
% 0.15/0.44 lam: $i > $i > ( $i > $i ) > $i ).
% 0.15/0.44
% 0.15/0.44 thf(lamp_type,type,
% 0.15/0.44 lamp: $o ).
% 0.15/0.44
% 0.15/0.44 thf(lamp,definition,
% 0.15/0.44 ( lamp
% 0.15/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.44 => ( func @ A @ B
% 0.15/0.44 @ ( lam @ A @ B
% 0.15/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(lam2p_type,type,
% 0.15/0.44 lam2p: $o ).
% 0.15/0.44
% 0.15/0.44 thf(lam2p,definition,
% 0.15/0.44 ( lam2p
% 0.15/0.44 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.44 => ( in
% 0.15/0.44 @ ( lam @ A @ B
% 0.15/0.44 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.15/0.44 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(brelnall1_type,type,
% 0.15/0.44 brelnall1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(brelnall1,definition,
% 0.15/0.44 ( brelnall1
% 0.15/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.15/0.44 ( ( breln @ A @ B @ R )
% 0.15/0.44 => ! [Xphi: $i > $o] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.44 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.15/0.44 => ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ R )
% 0.15/0.44 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(brelnall2_type,type,
% 0.15/0.44 brelnall2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(brelnall2,definition,
% 0.15/0.44 ( brelnall2
% 0.15/0.44 = ( ! [A: $i,B: $i,R: $i] :
% 0.15/0.44 ( ( breln @ A @ B @ R )
% 0.15/0.44 => ! [Xphi: $i > $o] :
% 0.15/0.44 ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.44 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.15/0.44 => ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ R )
% 0.15/0.44 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(ex1E2_type,type,
% 0.15/0.44 ex1E2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(ex1E2,definition,
% 0.15/0.44 ( ex1E2
% 0.15/0.44 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.44 ( ( ex1 @ A
% 0.15/0.44 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.44 => ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ A )
% 0.15/0.44 => ( ( Xphi @ Xx )
% 0.15/0.44 => ( ( Xphi @ Xy )
% 0.15/0.44 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(funcGraphProp1_type,type,
% 0.15/0.44 funcGraphProp1: $o ).
% 0.15/0.44
% 0.15/0.44 thf(funcGraphProp1,definition,
% 0.15/0.44 ( funcGraphProp1
% 0.15/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.44 ( ( func @ A @ B @ Xf )
% 0.15/0.44 => ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(funcGraphProp3_type,type,
% 0.15/0.44 funcGraphProp3: $o ).
% 0.15/0.44
% 0.15/0.44 thf(funcGraphProp3,definition,
% 0.15/0.44 ( funcGraphProp3
% 0.15/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.44 => ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(funcGraphProp2_type,type,
% 0.15/0.44 funcGraphProp2: $o ).
% 0.15/0.44
% 0.15/0.44 thf(funcGraphProp2,definition,
% 0.15/0.44 ( funcGraphProp2
% 0.15/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.44 ( ( func @ A @ B @ Xf )
% 0.15/0.44 => ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.15/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.44 = Xy ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(funcextLem_type,type,
% 0.15/0.44 funcextLem: $o ).
% 0.15/0.44
% 0.15/0.44 thf(funcextLem,definition,
% 0.15/0.44 ( funcextLem
% 0.15/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.44 ( ( func @ A @ B @ Xf )
% 0.15/0.44 => ! [Xg: $i] :
% 0.15/0.44 ( ( func @ A @ B @ Xg )
% 0.15/0.44 => ( ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.44 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.15/0.44 => ! [Xx: $i] :
% 0.15/0.44 ( ( in @ Xx @ A )
% 0.15/0.44 => ! [Xy: $i] :
% 0.15/0.44 ( ( in @ Xy @ B )
% 0.15/0.44 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.15/0.44 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.15/0.44
% 0.15/0.44 thf(funcGraphProp4_type,type,
% 0.15/0.44 funcGraphProp4: $o ).
% 0.15/0.44
% 0.15/0.44 thf(funcGraphProp4,definition,
% 0.15/0.44 ( funcGraphProp4
% 0.15/0.44 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.44 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.44 => ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ! [Xy: $i] :
% 0.15/0.45 ( ( in @ Xy @ B )
% 0.15/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.15/0.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.45 = Xy ) ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(subbreln_type,type,
% 0.15/0.45 subbreln: $o ).
% 0.15/0.45
% 0.15/0.45 thf(subbreln,definition,
% 0.15/0.45 ( subbreln
% 0.15/0.45 = ( ! [A: $i,B: $i,R: $i] :
% 0.15/0.45 ( ( breln @ A @ B @ R )
% 0.15/0.45 => ! [S: $i] :
% 0.15/0.45 ( ( breln @ A @ B @ S )
% 0.15/0.45 => ( ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ! [Xy: $i] :
% 0.15/0.45 ( ( in @ Xy @ B )
% 0.15/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.15/0.45 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(eqbreln_type,type,
% 0.15/0.45 eqbreln: $o ).
% 0.15/0.45
% 0.15/0.45 thf(eqbreln,definition,
% 0.15/0.45 ( eqbreln
% 0.15/0.45 = ( ! [A: $i,B: $i,R: $i] :
% 0.15/0.45 ( ( breln @ A @ B @ R )
% 0.15/0.45 => ! [S: $i] :
% 0.15/0.45 ( ( breln @ A @ B @ S )
% 0.15/0.45 => ( ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ! [Xy: $i] :
% 0.15/0.45 ( ( in @ Xy @ B )
% 0.15/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.15/0.45 => ( ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ! [Xy: $i] :
% 0.15/0.45 ( ( in @ Xy @ B )
% 0.15/0.45 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.15/0.45 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.15/0.45 => ( R = S ) ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(funcext_type,type,
% 0.15/0.45 funcext: $o ).
% 0.15/0.45
% 0.15/0.45 thf(funcext,definition,
% 0.15/0.45 ( funcext
% 0.15/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.45 ( ( func @ A @ B @ Xf )
% 0.15/0.45 => ! [Xg: $i] :
% 0.15/0.45 ( ( func @ A @ B @ Xg )
% 0.15/0.45 => ( ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.45 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.15/0.45 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(funcext2_type,type,
% 0.15/0.45 funcext2: $o ).
% 0.15/0.45
% 0.15/0.45 thf(funcext2,definition,
% 0.15/0.45 ( funcext2
% 0.15/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.45 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.45 => ! [Xg: $i] :
% 0.15/0.45 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.15/0.45 => ( ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.45 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.15/0.45 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(ap2apEq1_type,type,
% 0.15/0.45 ap2apEq1: $o ).
% 0.15/0.45
% 0.15/0.45 thf(ap2apEq1,definition,
% 0.15/0.45 ( ap2apEq1
% 0.15/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.45 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.45 => ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.45 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(ap2apEq2_type,type,
% 0.15/0.45 ap2apEq2: $o ).
% 0.15/0.45
% 0.15/0.45 thf(ap2apEq2,definition,
% 0.15/0.45 ( ap2apEq2
% 0.15/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.45 ( ( func @ A @ B @ Xf )
% 0.15/0.45 => ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.45 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(beta1_type,type,
% 0.15/0.45 beta1: $o ).
% 0.15/0.45
% 0.15/0.45 thf(beta1,definition,
% 0.15/0.45 ( beta1
% 0.15/0.45 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.45 ( ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.45 => ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ( ap @ A @ B
% 0.15/0.45 @ ( lam @ A @ B
% 0.15/0.45 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.15/0.45 @ Xx )
% 0.15/0.45 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(eta1_type,type,
% 0.15/0.45 eta1: $o ).
% 0.15/0.45
% 0.15/0.45 thf(eta1,definition,
% 0.15/0.45 ( eta1
% 0.15/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.45 ( ( func @ A @ B @ Xf )
% 0.15/0.45 => ( ( lam @ A @ B
% 0.15/0.45 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.15/0.45 = Xf ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(lam2lamEq_type,type,
% 0.15/0.45 lam2lamEq: $o ).
% 0.15/0.45
% 0.15/0.45 thf(lam2lamEq,definition,
% 0.15/0.45 ( lam2lamEq
% 0.15/0.45 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.45 ( ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.45 => ( ( lam @ A @ B
% 0.15/0.45 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.15/0.45 = ( lam @ A @ B
% 0.15/0.45 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(beta2_type,type,
% 0.15/0.45 beta2: $o ).
% 0.15/0.45
% 0.15/0.45 thf(beta2,definition,
% 0.15/0.45 ( beta2
% 0.15/0.45 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.45 ( ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.45 => ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ( ap @ A @ B
% 0.15/0.45 @ ( lam @ A @ B
% 0.15/0.45 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.15/0.45 @ Xx )
% 0.15/0.45 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(eta2_type,type,
% 0.15/0.45 eta2: $o ).
% 0.15/0.45
% 0.15/0.45 thf(eta2,definition,
% 0.15/0.45 ( eta2
% 0.15/0.45 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.45 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.45 => ( ( lam @ A @ B
% 0.15/0.45 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.15/0.45 = Xf ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(iffalseProp1_type,type,
% 0.15/0.45 iffalseProp1: $o ).
% 0.15/0.45
% 0.15/0.45 thf(iffalseProp1,definition,
% 0.15/0.45 ( iffalseProp1
% 0.15/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ! [Xy: $i] :
% 0.15/0.45 ( ( in @ Xy @ A )
% 0.15/0.45 => ( ~ Xphi
% 0.15/0.45 => ( in @ Xy
% 0.15/0.45 @ ( dsetconstr @ A
% 0.15/0.45 @ ^ [Xz: $i] :
% 0.15/0.45 ( ( Xphi
% 0.15/0.45 & ( Xz = Xx ) )
% 0.15/0.45 | ( ~ Xphi
% 0.15/0.45 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(iffalseProp2_type,type,
% 0.15/0.45 iffalseProp2: $o ).
% 0.15/0.45
% 0.15/0.45 thf(iffalseProp2,definition,
% 0.15/0.45 ( iffalseProp2
% 0.15/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ! [Xy: $i] :
% 0.15/0.45 ( ( in @ Xy @ A )
% 0.15/0.45 => ( ~ Xphi
% 0.15/0.45 => ( ( dsetconstr @ A
% 0.15/0.45 @ ^ [Xz: $i] :
% 0.15/0.45 ( ( Xphi
% 0.15/0.45 & ( Xz = Xx ) )
% 0.15/0.45 | ( ~ Xphi
% 0.15/0.45 & ( Xz = Xy ) ) ) )
% 0.15/0.45 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(iftrueProp1_type,type,
% 0.15/0.45 iftrueProp1: $o ).
% 0.15/0.45
% 0.15/0.45 thf(iftrueProp1,definition,
% 0.15/0.45 ( iftrueProp1
% 0.15/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ! [Xy: $i] :
% 0.15/0.45 ( ( in @ Xy @ A )
% 0.15/0.45 => ( Xphi
% 0.15/0.45 => ( in @ Xx
% 0.15/0.45 @ ( dsetconstr @ A
% 0.15/0.45 @ ^ [Xz: $i] :
% 0.15/0.45 ( ( Xphi
% 0.15/0.45 & ( Xz = Xx ) )
% 0.15/0.45 | ( ~ Xphi
% 0.15/0.45 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(iftrueProp2_type,type,
% 0.15/0.45 iftrueProp2: $o ).
% 0.15/0.45
% 0.15/0.45 thf(iftrueProp2,definition,
% 0.15/0.45 ( iftrueProp2
% 0.15/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ! [Xy: $i] :
% 0.15/0.45 ( ( in @ Xy @ A )
% 0.15/0.45 => ( Xphi
% 0.15/0.45 => ( ( dsetconstr @ A
% 0.15/0.45 @ ^ [Xz: $i] :
% 0.15/0.45 ( ( Xphi
% 0.15/0.45 & ( Xz = Xx ) )
% 0.15/0.45 | ( ~ Xphi
% 0.15/0.45 & ( Xz = Xy ) ) ) )
% 0.15/0.45 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(ifSingleton_type,type,
% 0.15/0.45 ifSingleton: $o ).
% 0.15/0.45
% 0.15/0.45 thf(ifSingleton,definition,
% 0.15/0.45 ( ifSingleton
% 0.15/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ! [Xy: $i] :
% 0.15/0.45 ( ( in @ Xy @ A )
% 0.15/0.45 => ( singleton
% 0.15/0.45 @ ( dsetconstr @ A
% 0.15/0.45 @ ^ [Xz: $i] :
% 0.15/0.45 ( ( Xphi
% 0.15/0.45 & ( Xz = Xx ) )
% 0.15/0.45 | ( ~ Xphi
% 0.15/0.45 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(if_type,type,
% 0.15/0.45 if: $i > $o > $i > $i > $i ).
% 0.15/0.45
% 0.15/0.45 thf(ifp_type,type,
% 0.15/0.45 ifp: $o ).
% 0.15/0.45
% 0.15/0.45 thf(ifp,definition,
% 0.15/0.45 ( ifp
% 0.15/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ! [Xy: $i] :
% 0.15/0.45 ( ( in @ Xy @ A )
% 0.15/0.45 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(theeq_type,type,
% 0.15/0.45 theeq: $o ).
% 0.15/0.45
% 0.15/0.45 thf(theeq,definition,
% 0.15/0.45 ( theeq
% 0.15/0.45 = ( ! [X: $i] :
% 0.15/0.45 ( ( singleton @ X )
% 0.15/0.45 => ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ X )
% 0.15/0.45 => ( ( setunion @ X )
% 0.15/0.45 = Xx ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(iftrue_type,type,
% 0.15/0.45 iftrue: $o ).
% 0.15/0.45
% 0.15/0.45 thf(iftrue,definition,
% 0.15/0.45 ( iftrue
% 0.15/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ! [Xy: $i] :
% 0.15/0.45 ( ( in @ Xy @ A )
% 0.15/0.45 => ( Xphi
% 0.15/0.45 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.15/0.45 = Xx ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(iffalse_type,type,
% 0.15/0.45 iffalse: $o ).
% 0.15/0.45
% 0.15/0.45 thf(iffalse,definition,
% 0.15/0.45 ( iffalse
% 0.15/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ! [Xy: $i] :
% 0.15/0.45 ( ( in @ Xy @ A )
% 0.15/0.45 => ( ~ Xphi
% 0.15/0.45 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.15/0.45 = Xy ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(iftrueorfalse_type,type,
% 0.15/0.45 iftrueorfalse: $o ).
% 0.15/0.45
% 0.15/0.45 thf(iftrueorfalse,definition,
% 0.15/0.45 ( iftrueorfalse
% 0.15/0.45 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ! [Xy: $i] :
% 0.15/0.45 ( ( in @ Xy @ A )
% 0.15/0.45 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(binintersectT_lem_type,type,
% 0.15/0.45 binintersectT_lem: $o ).
% 0.15/0.45
% 0.15/0.45 thf(binintersectT_lem,definition,
% 0.15/0.45 ( binintersectT_lem
% 0.15/0.45 = ( ! [A: $i,X: $i] :
% 0.15/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.45 => ! [Y: $i] :
% 0.15/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.45 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(binunionT_lem_type,type,
% 0.15/0.45 binunionT_lem: $o ).
% 0.15/0.45
% 0.15/0.45 thf(binunionT_lem,definition,
% 0.15/0.45 ( binunionT_lem
% 0.15/0.45 = ( ! [A: $i,X: $i] :
% 0.15/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.45 => ! [Y: $i] :
% 0.15/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.45 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(powersetT_lem_type,type,
% 0.15/0.45 powersetT_lem: $o ).
% 0.15/0.45
% 0.15/0.45 thf(powersetT_lem,definition,
% 0.15/0.45 ( powersetT_lem
% 0.15/0.45 = ( ! [A: $i,X: $i] :
% 0.15/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.45 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(setminusT_lem_type,type,
% 0.15/0.45 setminusT_lem: $o ).
% 0.15/0.45
% 0.15/0.45 thf(setminusT_lem,definition,
% 0.15/0.45 ( setminusT_lem
% 0.15/0.45 = ( ! [A: $i,X: $i] :
% 0.15/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.45 => ! [Y: $i] :
% 0.15/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.45 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(complementT_lem_type,type,
% 0.15/0.45 complementT_lem: $o ).
% 0.15/0.45
% 0.15/0.45 thf(complementT_lem,definition,
% 0.15/0.45 ( complementT_lem
% 0.15/0.45 = ( ! [A: $i,X: $i] :
% 0.15/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.45 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(setextT_type,type,
% 0.15/0.45 setextT: $o ).
% 0.15/0.45
% 0.15/0.45 thf(setextT,definition,
% 0.15/0.45 ( setextT
% 0.15/0.45 = ( ! [A: $i,X: $i] :
% 0.15/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.45 => ! [Y: $i] :
% 0.15/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.45 => ( ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ( in @ Xx @ X )
% 0.15/0.45 => ( in @ Xx @ Y ) ) )
% 0.15/0.45 => ( ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ( in @ Xx @ Y )
% 0.15/0.45 => ( in @ Xx @ X ) ) )
% 0.15/0.45 => ( X = Y ) ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(subsetTI_type,type,
% 0.15/0.45 subsetTI: $o ).
% 0.15/0.45
% 0.15/0.45 thf(subsetTI,definition,
% 0.15/0.45 ( subsetTI
% 0.15/0.45 = ( ! [A: $i,X: $i] :
% 0.15/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.45 => ! [Y: $i] :
% 0.15/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.45 => ( ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ( in @ Xx @ X )
% 0.15/0.45 => ( in @ Xx @ Y ) ) )
% 0.15/0.45 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(powersetTI1_type,type,
% 0.15/0.45 powersetTI1: $o ).
% 0.15/0.45
% 0.15/0.45 thf(powersetTI1,definition,
% 0.15/0.45 ( powersetTI1
% 0.15/0.45 = ( ! [A: $i,X: $i] :
% 0.15/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.45 => ! [Y: $i] :
% 0.15/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.45 => ( ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ( in @ Xx @ X )
% 0.15/0.45 => ( in @ Xx @ Y ) ) )
% 0.15/0.45 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(powersetTE1_type,type,
% 0.15/0.45 powersetTE1: $o ).
% 0.15/0.45
% 0.15/0.45 thf(powersetTE1,definition,
% 0.15/0.45 ( powersetTE1
% 0.15/0.45 = ( ! [A: $i,X: $i] :
% 0.15/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.45 => ! [Y: $i] :
% 0.15/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.45 => ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ( in @ X @ ( powerset @ Y ) )
% 0.15/0.45 => ( ( in @ Xx @ X )
% 0.15/0.45 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(complementTI1_type,type,
% 0.15/0.45 complementTI1: $o ).
% 0.15/0.45
% 0.15/0.45 thf(complementTI1,definition,
% 0.15/0.45 ( complementTI1
% 0.15/0.45 = ( ! [A: $i,X: $i] :
% 0.15/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.45 => ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ( in @ Xx @ X )
% 0.15/0.45 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(complementTE1_type,type,
% 0.15/0.45 complementTE1: $o ).
% 0.15/0.45
% 0.15/0.45 thf(complementTE1,definition,
% 0.15/0.45 ( complementTE1
% 0.15/0.45 = ( ! [A: $i,X: $i] :
% 0.15/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.45 => ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.15/0.45 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(binintersectTELcontra_type,type,
% 0.15/0.45 binintersectTELcontra: $o ).
% 0.15/0.45
% 0.15/0.45 thf(binintersectTELcontra,definition,
% 0.15/0.45 ( binintersectTELcontra
% 0.15/0.45 = ( ! [A: $i,X: $i] :
% 0.15/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.45 => ! [Y: $i] :
% 0.15/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.45 => ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ~ ( in @ Xx @ X )
% 0.15/0.45 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(binintersectTERcontra_type,type,
% 0.15/0.45 binintersectTERcontra: $o ).
% 0.15/0.45
% 0.15/0.45 thf(binintersectTERcontra,definition,
% 0.15/0.45 ( binintersectTERcontra
% 0.15/0.45 = ( ! [A: $i,X: $i] :
% 0.15/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.45 => ! [Y: $i] :
% 0.15/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.45 => ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ~ ( in @ Xx @ Y )
% 0.15/0.45 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(contrasubsetT_type,type,
% 0.15/0.45 contrasubsetT: $o ).
% 0.15/0.45
% 0.15/0.45 thf(contrasubsetT,definition,
% 0.15/0.45 ( contrasubsetT
% 0.15/0.45 = ( ! [A: $i,X: $i] :
% 0.15/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.45 => ! [Y: $i] :
% 0.15/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.45 => ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.15/0.45 => ( ( in @ Xx @ Y )
% 0.15/0.45 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 thf(contrasubsetT1,conjecture,
% 0.15/0.45 ( setextAx
% 0.15/0.45 => ( emptysetAx
% 0.15/0.45 => ( setadjoinAx
% 0.15/0.45 => ( powersetAx
% 0.15/0.45 => ( setunionAx
% 0.15/0.45 => ( omega0Ax
% 0.15/0.45 => ( omegaSAx
% 0.15/0.45 => ( omegaIndAx
% 0.15/0.45 => ( replAx
% 0.15/0.45 => ( foundationAx
% 0.15/0.45 => ( wellorderingAx
% 0.15/0.45 => ( descrp
% 0.15/0.45 => ( dsetconstrI
% 0.15/0.45 => ( dsetconstrEL
% 0.15/0.45 => ( dsetconstrER
% 0.15/0.45 => ( exuE1
% 0.15/0.45 => ( prop2setE
% 0.15/0.45 => ( emptysetE
% 0.15/0.45 => ( emptysetimpfalse
% 0.15/0.45 => ( notinemptyset
% 0.15/0.45 => ( exuE3e
% 0.15/0.45 => ( setext
% 0.15/0.45 => ( emptyI
% 0.15/0.45 => ( noeltsimpempty
% 0.15/0.45 => ( setbeta
% 0.15/0.45 => ( nonemptyE1
% 0.15/0.45 => ( nonemptyI
% 0.15/0.45 => ( nonemptyI1
% 0.15/0.45 => ( setadjoinIL
% 0.15/0.45 => ( emptyinunitempty
% 0.15/0.45 => ( setadjoinIR
% 0.15/0.45 => ( setadjoinE
% 0.15/0.45 => ( setadjoinOr
% 0.15/0.45 => ( setoftrueEq
% 0.15/0.45 => ( powersetI
% 0.15/0.45 => ( emptyinPowerset
% 0.15/0.45 => ( emptyInPowerset
% 0.15/0.45 => ( powersetE
% 0.15/0.45 => ( setunionI
% 0.15/0.45 => ( setunionE
% 0.15/0.45 => ( subPowSU
% 0.15/0.45 => ( exuE2
% 0.15/0.45 => ( nonemptyImpWitness
% 0.15/0.45 => ( uniqinunit
% 0.15/0.45 => ( notinsingleton
% 0.15/0.45 => ( eqinunit
% 0.15/0.45 => ( singletonsswitch
% 0.15/0.45 => ( upairsetE
% 0.15/0.45 => ( upairsetIL
% 0.15/0.45 => ( upairsetIR
% 0.15/0.45 => ( emptyE1
% 0.15/0.45 => ( vacuousDall
% 0.15/0.45 => ( quantDeMorgan1
% 0.15/0.45 => ( quantDeMorgan2
% 0.15/0.45 => ( quantDeMorgan3
% 0.15/0.45 => ( quantDeMorgan4
% 0.15/0.45 => ( prop2setI
% 0.15/0.45 => ( prop2set2propI
% 0.15/0.45 => ( notdexE
% 0.15/0.45 => ( notdallE
% 0.15/0.45 => ( exuI1
% 0.15/0.45 => ( exuI3
% 0.15/0.45 => ( exuI2
% 0.15/0.45 => ( inCongP
% 0.15/0.45 => ( in__Cong
% 0.15/0.45 => ( exuE3u
% 0.15/0.45 => ( exu__Cong
% 0.15/0.45 => ( emptyset__Cong
% 0.15/0.45 => ( setadjoin__Cong
% 0.15/0.45 => ( powerset__Cong
% 0.15/0.45 => ( setunion__Cong
% 0.15/0.45 => ( omega__Cong
% 0.15/0.45 => ( exuEu
% 0.15/0.45 => ( descr__Cong
% 0.15/0.45 => ( dsetconstr__Cong
% 0.15/0.45 => ( subsetI1
% 0.15/0.45 => ( eqimpsubset2
% 0.15/0.45 => ( eqimpsubset1
% 0.15/0.45 => ( subsetI2
% 0.15/0.45 => ( emptysetsubset
% 0.15/0.45 => ( subsetE
% 0.15/0.45 => ( subsetE2
% 0.15/0.45 => ( notsubsetI
% 0.15/0.45 => ( notequalI1
% 0.15/0.45 => ( notequalI2
% 0.15/0.45 => ( subsetRefl
% 0.15/0.45 => ( subsetTrans
% 0.15/0.45 => ( setadjoinSub
% 0.15/0.45 => ( setadjoinSub2
% 0.15/0.45 => ( subset2powerset
% 0.15/0.45 => ( setextsub
% 0.15/0.45 => ( subsetemptysetimpeq
% 0.15/0.45 => ( powersetI1
% 0.15/0.45 => ( powersetE1
% 0.15/0.45 => ( inPowerset
% 0.15/0.45 => ( powersetsubset
% 0.15/0.45 => ( sepInPowerset
% 0.15/0.45 => ( sepSubset
% 0.15/0.45 => ( binunionIL
% 0.15/0.45 => ( upairset2IR
% 0.15/0.45 => ( binunionIR
% 0.15/0.45 => ( binunionEcases
% 0.15/0.45 => ( binunionE
% 0.15/0.45 => ( binunionLsub
% 0.15/0.45 => ( binunionRsub
% 0.15/0.45 => ( binintersectI
% 0.15/0.45 => ( binintersectSubset5
% 0.15/0.45 => ( binintersectEL
% 0.15/0.45 => ( binintersectLsub
% 0.15/0.45 => ( binintersectSubset2
% 0.15/0.45 => ( binintersectSubset3
% 0.15/0.45 => ( binintersectER
% 0.15/0.45 => ( disjointsetsI1
% 0.15/0.45 => ( binintersectRsub
% 0.15/0.45 => ( binintersectSubset4
% 0.15/0.45 => ( binintersectSubset1
% 0.15/0.45 => ( bs114d
% 0.15/0.45 => ( setminusI
% 0.15/0.45 => ( setminusEL
% 0.15/0.45 => ( setminusER
% 0.15/0.45 => ( setminusSubset2
% 0.15/0.45 => ( setminusERneg
% 0.15/0.45 => ( setminusELneg
% 0.15/0.45 => ( setminusILneg
% 0.15/0.45 => ( setminusIRneg
% 0.15/0.45 => ( setminusLsub
% 0.15/0.45 => ( setminusSubset1
% 0.15/0.45 => ( symdiffE
% 0.15/0.45 => ( symdiffI1
% 0.15/0.45 => ( symdiffI2
% 0.15/0.45 => ( symdiffIneg1
% 0.15/0.45 => ( symdiffIneg2
% 0.15/0.45 => ( secondinupair
% 0.15/0.45 => ( setukpairIL
% 0.15/0.45 => ( setukpairIR
% 0.15/0.45 => ( kpairiskpair
% 0.15/0.45 => ( kpairp
% 0.15/0.45 => ( singletonsubset
% 0.15/0.45 => ( singletoninpowerset
% 0.15/0.45 => ( singletoninpowunion
% 0.15/0.45 => ( upairset2E
% 0.15/0.45 => ( upairsubunion
% 0.15/0.45 => ( upairinpowunion
% 0.15/0.45 => ( ubforcartprodlem1
% 0.15/0.45 => ( ubforcartprodlem2
% 0.15/0.45 => ( ubforcartprodlem3
% 0.15/0.45 => ( cartprodpairin
% 0.15/0.45 => ( cartprodmempair1
% 0.15/0.45 => ( cartprodmempair
% 0.15/0.45 => ( setunionE2
% 0.15/0.45 => ( setunionsingleton1
% 0.15/0.45 => ( setunionsingleton2
% 0.15/0.45 => ( setunionsingleton
% 0.15/0.45 => ( singletonprop
% 0.15/0.45 => ( ex1E1
% 0.15/0.45 => ( ex1I
% 0.15/0.45 => ( ex1I2
% 0.15/0.45 => ( singletonsuniq
% 0.15/0.45 => ( setukpairinjL1
% 0.15/0.45 => ( kfstsingleton
% 0.15/0.45 => ( theprop
% 0.15/0.45 => ( kfstpairEq
% 0.15/0.45 => ( cartprodfstin
% 0.15/0.45 => ( setukpairinjL2
% 0.15/0.45 => ( setukpairinjL
% 0.15/0.45 => ( setukpairinjR11
% 0.15/0.45 => ( setukpairinjR12
% 0.15/0.45 => ( setukpairinjR1
% 0.15/0.45 => ( upairequniteq
% 0.15/0.45 => ( setukpairinjR2
% 0.15/0.45 => ( setukpairinjR
% 0.15/0.45 => ( ksndsingleton
% 0.15/0.45 => ( ksndpairEq
% 0.15/0.45 => ( kpairsurjEq
% 0.15/0.45 => ( cartprodsndin
% 0.15/0.45 => ( cartprodpairmemEL
% 0.15/0.45 => ( cartprodpairmemER
% 0.15/0.45 => ( cartprodmempaircEq
% 0.15/0.45 => ( cartprodfstpairEq
% 0.15/0.45 => ( cartprodsndpairEq
% 0.15/0.45 => ( cartprodpairsurjEq
% 0.15/0.45 => ( dpsetconstrI
% 0.15/0.45 => ( dpsetconstrSub
% 0.15/0.45 => ( setOfPairsIsBReln
% 0.15/0.45 => ( dpsetconstrERa
% 0.15/0.45 => ( dpsetconstrEL1
% 0.15/0.45 => ( dpsetconstrEL2
% 0.15/0.45 => ( dpsetconstrER
% 0.15/0.45 => ( funcImageSingleton
% 0.15/0.45 => ( apProp
% 0.15/0.45 => ( app
% 0.15/0.45 => ( infuncsetfunc
% 0.15/0.45 => ( ap2p
% 0.15/0.45 => ( funcinfuncset
% 0.15/0.45 => ( lamProp
% 0.15/0.45 => ( lamp
% 0.15/0.45 => ( lam2p
% 0.15/0.45 => ( brelnall1
% 0.15/0.45 => ( brelnall2
% 0.15/0.45 => ( ex1E2
% 0.15/0.45 => ( funcGraphProp1
% 0.15/0.45 => ( funcGraphProp3
% 0.15/0.45 => ( funcGraphProp2
% 0.15/0.45 => ( funcextLem
% 0.15/0.45 => ( funcGraphProp4
% 0.15/0.45 => ( subbreln
% 0.15/0.45 => ( eqbreln
% 0.15/0.45 => ( funcext
% 0.15/0.45 => ( funcext2
% 0.15/0.45 => ( ap2apEq1
% 0.15/0.45 => ( ap2apEq2
% 0.15/0.45 => ( beta1
% 0.15/0.45 => ( eta1
% 0.15/0.45 => ( lam2lamEq
% 0.15/0.45 => ( beta2
% 0.15/0.45 => ( eta2
% 0.15/0.45 => ( iffalseProp1
% 0.15/0.45 => ( iffalseProp2
% 0.15/0.45 => ( iftrueProp1
% 0.15/0.45 => ( iftrueProp2
% 0.15/0.45 => ( ifSingleton
% 0.15/0.45 => ( ifp
% 0.15/0.45 => ( theeq
% 0.15/0.45 => ( iftrue
% 0.15/0.45 => ( iffalse
% 0.15/0.45 => ( iftrueorfalse
% 0.15/0.45 => ( binintersectT_lem
% 0.15/0.45 => ( binunionT_lem
% 0.15/0.45 => ( powersetT_lem
% 0.15/0.45 => ( setminusT_lem
% 0.15/0.45 => ( complementT_lem
% 0.15/0.45 => ( setextT
% 0.15/0.45 => ( subsetTI
% 0.15/0.45 => ( powersetTI1
% 0.15/0.45 => ( powersetTE1
% 0.15/0.45 => ( complementTI1
% 0.15/0.45 => ( complementTE1
% 0.15/0.45 => ( binintersectTELcontra
% 0.15/0.45 => ( binintersectTERcontra
% 0.15/0.45 => ( contrasubsetT
% 0.15/0.45 => ! [A: $i,X: $i] :
% 0.15/0.45 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.45 => ! [Y: $i] :
% 0.15/0.45 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.45 => ! [Xx: $i] :
% 0.15/0.45 ( ( in @ Xx @ A )
% 0.15/0.45 => ( ( subset @ X @ Y )
% 0.15/0.45 => ( ~ ( in @ Xx @ Y )
% 0.15/0.45 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.45
% 0.15/0.45 %------------------------------------------------------------------------------
% 0.15/0.45 ------------------------
% 2.07/1.35 ---- Embedded in THF ---
% 2.07/1.35 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.07/1.35 %%% Generated on Tue Feb 24 12:32:13 EST 2026
% 2.07/1.35 %%% using '$$dhol' embedding, version 1.3.0.
% 2.07/1.35 %%% Logic specification used:
% 2.07/1.35 %%% thf(spec, logic, $$dhol).
% 2.07/1.35
% 2.07/1.35 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.IFvb0yMMZy/DTF2DTF26337.p
% See solution above
% 2.07/1.35 ------------------------
% 2.18/1.40 ---- Cleaned THF ---
% 2.18/1.40 thf(in_type,type,
% 2.18/1.40 in: $i > $i > $o ).
% 2.18/1.40
% 2.18/1.40 thf(exu_type,type,
% 2.18/1.40 exu: ( $i > $o ) > $o ).
% 2.18/1.40
% 2.18/1.40 thf(exu,definition,
% 2.18/1.40 ( exu
% 2.18/1.40 = ( ^ [Xphi: $i > $o] :
% 2.18/1.40 ? [Xx: $i] :
% 2.18/1.40 ( ( Xphi @ Xx )
% 2.18/1.40 & ! [Xy: $i] :
% 2.18/1.40 ( ( Xphi @ Xy )
% 2.18/1.40 => ( Xx = Xy ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setextAx_type,type,
% 2.18/1.40 setextAx: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setextAx,definition,
% 2.18/1.40 ( setextAx
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 <=> ( in @ Xx @ B ) )
% 2.18/1.40 => ( A = B ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(emptyset_type,type,
% 2.18/1.40 emptyset: $i ).
% 2.18/1.40
% 2.18/1.40 thf(emptysetAx_type,type,
% 2.18/1.40 emptysetAx: $o ).
% 2.18/1.40
% 2.18/1.40 thf(emptysetAx,definition,
% 2.18/1.40 ( emptysetAx
% 2.18/1.40 = ( ! [Xx: $i] :
% 2.18/1.40 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoin_type,type,
% 2.18/1.40 setadjoin: $i > $i > $i ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoinAx_type,type,
% 2.18/1.40 setadjoinAx: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoinAx,definition,
% 2.18/1.40 ( setadjoinAx
% 2.18/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.18/1.40 <=> ( ( Xy = Xx )
% 2.18/1.40 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(powerset_type,type,
% 2.18/1.40 powerset: $i > $i ).
% 2.18/1.40
% 2.18/1.40 thf(powersetAx_type,type,
% 2.18/1.40 powersetAx: $o ).
% 2.18/1.40
% 2.18/1.40 thf(powersetAx,definition,
% 2.18/1.40 ( powersetAx
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( in @ B @ ( powerset @ A ) )
% 2.18/1.40 <=> ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ B )
% 2.18/1.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setunion_type,type,
% 2.18/1.40 setunion: $i > $i ).
% 2.18/1.40
% 2.18/1.40 thf(setunionAx_type,type,
% 2.18/1.40 setunionAx: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setunionAx,definition,
% 2.18/1.40 ( setunionAx
% 2.18/1.40 = ( ! [A: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ ( setunion @ A ) )
% 2.18/1.40 <=> ? [B: $i] :
% 2.18/1.40 ( ( in @ Xx @ B )
% 2.18/1.40 & ( in @ B @ A ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(omega_type,type,
% 2.18/1.40 omega: $i ).
% 2.18/1.40
% 2.18/1.40 thf(omega0Ax_type,type,
% 2.18/1.40 omega0Ax: $o ).
% 2.18/1.40
% 2.18/1.40 thf(omega0Ax,definition,
% 2.18/1.40 ( omega0Ax
% 2.18/1.40 = ( in @ emptyset @ omega ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(omegaSAx_type,type,
% 2.18/1.40 omegaSAx: $o ).
% 2.18/1.40
% 2.18/1.40 thf(omegaSAx,definition,
% 2.18/1.40 ( omegaSAx
% 2.18/1.40 = ( ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ omega )
% 2.18/1.40 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(omegaIndAx_type,type,
% 2.18/1.40 omegaIndAx: $o ).
% 2.18/1.40
% 2.18/1.40 thf(omegaIndAx,definition,
% 2.18/1.40 ( omegaIndAx
% 2.18/1.40 = ( ! [A: $i] :
% 2.18/1.40 ( ( ( in @ emptyset @ A )
% 2.18/1.40 & ! [Xx: $i] :
% 2.18/1.40 ( ( ( in @ Xx @ omega )
% 2.18/1.40 & ( in @ Xx @ A ) )
% 2.18/1.40 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.18/1.40 => ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ omega )
% 2.18/1.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(replAx_type,type,
% 2.18/1.40 replAx: $o ).
% 2.18/1.40
% 2.18/1.40 thf(replAx,definition,
% 2.18/1.40 ( replAx
% 2.18/1.40 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.18/1.40 ( ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( exu
% 2.18/1.40 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.18/1.40 => ? [B: $i] :
% 2.18/1.40 ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ B )
% 2.18/1.40 <=> ? [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ A )
% 2.18/1.40 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(foundationAx_type,type,
% 2.18/1.40 foundationAx: $o ).
% 2.18/1.40
% 2.18/1.40 thf(foundationAx,definition,
% 2.18/1.40 ( foundationAx
% 2.18/1.40 = ( ! [A: $i] :
% 2.18/1.40 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.18/1.40 => ? [B: $i] :
% 2.18/1.40 ( ( in @ B @ A )
% 2.18/1.40 & ~ ? [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ B )
% 2.18/1.40 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(wellorderingAx_type,type,
% 2.18/1.40 wellorderingAx: $o ).
% 2.18/1.40
% 2.18/1.40 thf(wellorderingAx,definition,
% 2.18/1.40 ( wellorderingAx
% 2.18/1.40 = ( ! [A: $i] :
% 2.18/1.40 ? [B: $i] :
% 2.18/1.40 ( ! [C: $i] :
% 2.18/1.40 ( ( in @ C @ B )
% 2.18/1.40 => ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ C )
% 2.18/1.40 => ( in @ Xx @ A ) ) )
% 2.18/1.40 & ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( ( in @ Xx @ A )
% 2.18/1.40 & ( in @ Xy @ A ) )
% 2.18/1.40 => ( ! [C: $i] :
% 2.18/1.40 ( ( in @ C @ B )
% 2.18/1.40 => ( ( in @ Xx @ C )
% 2.18/1.40 <=> ( in @ Xy @ C ) ) )
% 2.18/1.40 => ( Xx = Xy ) ) )
% 2.18/1.40 & ! [C: $i,D: $i] :
% 2.18/1.40 ( ( ( in @ C @ B )
% 2.18/1.40 & ( in @ D @ B ) )
% 2.18/1.40 => ( ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ C )
% 2.18/1.40 => ( in @ Xx @ D ) )
% 2.18/1.40 | ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ D )
% 2.18/1.40 => ( in @ Xx @ C ) ) ) )
% 2.18/1.40 & ! [C: $i] :
% 2.18/1.40 ( ( ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ C )
% 2.18/1.40 => ( in @ Xx @ A ) )
% 2.18/1.40 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.18/1.40 => ? [D: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ D @ B )
% 2.18/1.40 & ( in @ Xx @ C )
% 2.18/1.40 & ~ ? [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ D )
% 2.18/1.40 & ( in @ Xy @ C ) )
% 2.18/1.40 & ! [E: $i] :
% 2.18/1.40 ( ( in @ E @ B )
% 2.18/1.40 => ( ! [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ E )
% 2.18/1.40 => ( in @ Xy @ D ) )
% 2.18/1.40 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(descr_type,type,
% 2.18/1.40 descr: ( $i > $o ) > $i ).
% 2.18/1.40
% 2.18/1.40 thf(descrp_type,type,
% 2.18/1.40 descrp: $o ).
% 2.18/1.40
% 2.18/1.40 thf(descrp,definition,
% 2.18/1.40 ( descrp
% 2.18/1.40 = ( ! [Xphi: $i > $o] :
% 2.18/1.40 ( ( exu
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.40 => ( Xphi
% 2.18/1.40 @ ( descr
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(dsetconstr_type,type,
% 2.18/1.40 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.18/1.40
% 2.18/1.40 thf(dsetconstrI_type,type,
% 2.18/1.40 dsetconstrI: $o ).
% 2.18/1.40
% 2.18/1.40 thf(dsetconstrI,definition,
% 2.18/1.40 ( dsetconstrI
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( ( Xphi @ Xx )
% 2.18/1.40 => ( in @ Xx
% 2.18/1.40 @ ( dsetconstr @ A
% 2.18/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(dsetconstrEL_type,type,
% 2.18/1.40 dsetconstrEL: $o ).
% 2.18/1.40
% 2.18/1.40 thf(dsetconstrEL,definition,
% 2.18/1.40 ( dsetconstrEL
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx
% 2.18/1.40 @ ( dsetconstr @ A
% 2.18/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.18/1.40 => ( in @ Xx @ A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(dsetconstrER_type,type,
% 2.18/1.40 dsetconstrER: $o ).
% 2.18/1.40
% 2.18/1.40 thf(dsetconstrER,definition,
% 2.18/1.40 ( dsetconstrER
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx
% 2.18/1.40 @ ( dsetconstr @ A
% 2.18/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.18/1.40 => ( Xphi @ Xx ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(exuE1_type,type,
% 2.18/1.40 exuE1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(exuE1,definition,
% 2.18/1.40 ( exuE1
% 2.18/1.40 = ( ! [Xphi: $i > $o] :
% 2.18/1.40 ( ( exu
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.40 => ? [Xx: $i] :
% 2.18/1.40 ( ( Xphi @ Xx )
% 2.18/1.40 & ! [Xy: $i] :
% 2.18/1.40 ( ( Xphi @ Xy )
% 2.18/1.40 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(prop2set_type,type,
% 2.18/1.40 prop2set: $o > $i ).
% 2.18/1.40
% 2.18/1.40 thf(prop2setE_type,type,
% 2.18/1.40 prop2setE: $o ).
% 2.18/1.40
% 2.18/1.40 thf(prop2setE,definition,
% 2.18/1.40 ( prop2setE
% 2.18/1.40 = ( ! [Xphi: $o,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.18/1.40 => Xphi ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(emptysetE_type,type,
% 2.18/1.40 emptysetE: $o ).
% 2.18/1.40
% 2.18/1.40 thf(emptysetE,definition,
% 2.18/1.40 ( emptysetE
% 2.18/1.40 = ( ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ emptyset )
% 2.18/1.40 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(emptysetimpfalse_type,type,
% 2.18/1.40 emptysetimpfalse: $o ).
% 2.18/1.40
% 2.18/1.40 thf(emptysetimpfalse,definition,
% 2.18/1.40 ( emptysetimpfalse
% 2.18/1.40 = ( ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ emptyset )
% 2.18/1.40 => $false ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(notinemptyset_type,type,
% 2.18/1.40 notinemptyset: $o ).
% 2.18/1.40
% 2.18/1.40 thf(notinemptyset,definition,
% 2.18/1.40 ( notinemptyset
% 2.18/1.40 = ( ! [Xx: $i] :
% 2.18/1.40 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(exuE3e_type,type,
% 2.18/1.40 exuE3e: $o ).
% 2.18/1.40
% 2.18/1.40 thf(exuE3e,definition,
% 2.18/1.40 ( exuE3e
% 2.18/1.40 = ( ! [Xphi: $i > $o] :
% 2.18/1.40 ( ( exu
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.40 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setext_type,type,
% 2.18/1.40 setext: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setext,definition,
% 2.18/1.40 ( setext
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( in @ Xx @ B ) )
% 2.18/1.40 => ( ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ B )
% 2.18/1.40 => ( in @ Xx @ A ) )
% 2.18/1.40 => ( A = B ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(emptyI_type,type,
% 2.18/1.40 emptyI: $o ).
% 2.18/1.40
% 2.18/1.40 thf(emptyI,definition,
% 2.18/1.40 ( emptyI
% 2.18/1.40 = ( ! [A: $i] :
% 2.18/1.40 ( ! [Xx: $i] :
% 2.18/1.40 ~ ( in @ Xx @ A )
% 2.18/1.40 => ( A = emptyset ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(noeltsimpempty_type,type,
% 2.18/1.40 noeltsimpempty: $o ).
% 2.18/1.40
% 2.18/1.40 thf(noeltsimpempty,definition,
% 2.18/1.40 ( noeltsimpempty
% 2.18/1.40 = ( ! [A: $i] :
% 2.18/1.40 ( ! [Xx: $i] :
% 2.18/1.40 ~ ( in @ Xx @ A )
% 2.18/1.40 => ( A = emptyset ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setbeta_type,type,
% 2.18/1.40 setbeta: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setbeta,definition,
% 2.18/1.40 ( setbeta
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( ( in @ Xx
% 2.18/1.40 @ ( dsetconstr @ A
% 2.18/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.18/1.40 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(nonempty_type,type,
% 2.18/1.40 nonempty: $i > $o ).
% 2.18/1.40
% 2.18/1.40 thf(nonempty,definition,
% 2.18/1.40 ( nonempty
% 2.18/1.40 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(nonemptyE1_type,type,
% 2.18/1.40 nonemptyE1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(nonemptyE1,definition,
% 2.18/1.40 ( nonemptyE1
% 2.18/1.40 = ( ! [A: $i] :
% 2.18/1.40 ( ( nonempty @ A )
% 2.18/1.40 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(nonemptyI_type,type,
% 2.18/1.40 nonemptyI: $o ).
% 2.18/1.40
% 2.18/1.40 thf(nonemptyI,definition,
% 2.18/1.40 ( nonemptyI
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( ( Xphi @ Xx )
% 2.18/1.40 => ( nonempty
% 2.18/1.40 @ ( dsetconstr @ A
% 2.18/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(nonemptyI1_type,type,
% 2.18/1.40 nonemptyI1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(nonemptyI1,definition,
% 2.18/1.40 ( nonemptyI1
% 2.18/1.40 = ( ! [A: $i] :
% 2.18/1.40 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.18/1.40 => ( nonempty @ A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoinIL_type,type,
% 2.18/1.40 setadjoinIL: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoinIL,definition,
% 2.18/1.40 ( setadjoinIL
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(emptyinunitempty_type,type,
% 2.18/1.40 emptyinunitempty: $o ).
% 2.18/1.40
% 2.18/1.40 thf(emptyinunitempty,definition,
% 2.18/1.40 ( emptyinunitempty
% 2.18/1.40 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoinIR_type,type,
% 2.18/1.40 setadjoinIR: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoinIR,definition,
% 2.18/1.40 ( setadjoinIR
% 2.18/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ A )
% 2.18/1.40 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoinE_type,type,
% 2.18/1.40 setadjoinE: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoinE,definition,
% 2.18/1.40 ( setadjoinE
% 2.18/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.18/1.40 => ! [Xphi: $o] :
% 2.18/1.40 ( ( ( Xy = Xx )
% 2.18/1.40 => Xphi )
% 2.18/1.40 => ( ( ( in @ Xy @ A )
% 2.18/1.40 => Xphi )
% 2.18/1.40 => Xphi ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoinOr_type,type,
% 2.18/1.40 setadjoinOr: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoinOr,definition,
% 2.18/1.40 ( setadjoinOr
% 2.18/1.40 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.18/1.40 => ( ( Xy = Xx )
% 2.18/1.40 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setoftrueEq_type,type,
% 2.18/1.40 setoftrueEq: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setoftrueEq,definition,
% 2.18/1.40 ( setoftrueEq
% 2.18/1.40 = ( ! [A: $i] :
% 2.18/1.40 ( ( dsetconstr @ A
% 2.18/1.40 @ ^ [Xx: $i] : $true )
% 2.18/1.40 = A ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(powersetI_type,type,
% 2.18/1.40 powersetI: $o ).
% 2.18/1.40
% 2.18/1.40 thf(powersetI,definition,
% 2.18/1.40 ( powersetI
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ B )
% 2.18/1.40 => ( in @ Xx @ A ) )
% 2.18/1.40 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(emptyinPowerset_type,type,
% 2.18/1.40 emptyinPowerset: $o ).
% 2.18/1.40
% 2.18/1.40 thf(emptyinPowerset,definition,
% 2.18/1.40 ( emptyinPowerset
% 2.18/1.40 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(emptyInPowerset_type,type,
% 2.18/1.40 emptyInPowerset: $o ).
% 2.18/1.40
% 2.18/1.40 thf(emptyInPowerset,definition,
% 2.18/1.40 ( emptyInPowerset
% 2.18/1.40 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(powersetE_type,type,
% 2.18/1.40 powersetE: $o ).
% 2.18/1.40
% 2.18/1.40 thf(powersetE,definition,
% 2.18/1.40 ( powersetE
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ B @ ( powerset @ A ) )
% 2.18/1.40 => ( ( in @ Xx @ B )
% 2.18/1.40 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setunionI_type,type,
% 2.18/1.40 setunionI: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setunionI,definition,
% 2.18/1.40 ( setunionI
% 2.18/1.40 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.18/1.40 ( ( in @ Xx @ B )
% 2.18/1.40 => ( ( in @ B @ A )
% 2.18/1.40 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setunionE_type,type,
% 2.18/1.40 setunionE: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setunionE,definition,
% 2.18/1.40 ( setunionE
% 2.18/1.40 = ( ! [A: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ ( setunion @ A ) )
% 2.18/1.40 => ! [Xphi: $o] :
% 2.18/1.40 ( ! [B: $i] :
% 2.18/1.40 ( ( in @ Xx @ B )
% 2.18/1.40 => ( ( in @ B @ A )
% 2.18/1.40 => Xphi ) )
% 2.18/1.40 => Xphi ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(subPowSU_type,type,
% 2.18/1.40 subPowSU: $o ).
% 2.18/1.40
% 2.18/1.40 thf(subPowSU,definition,
% 2.18/1.40 ( subPowSU
% 2.18/1.40 = ( ! [A: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(exuE2_type,type,
% 2.18/1.40 exuE2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(exuE2,definition,
% 2.18/1.40 ( exuE2
% 2.18/1.40 = ( ! [Xphi: $i > $o] :
% 2.18/1.40 ( ( exu
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.40 => ? [Xx: $i] :
% 2.18/1.40 ! [Xy: $i] :
% 2.18/1.40 ( ( Xphi @ Xy )
% 2.18/1.40 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(nonemptyImpWitness_type,type,
% 2.18/1.40 nonemptyImpWitness: $o ).
% 2.18/1.40
% 2.18/1.40 thf(nonemptyImpWitness,definition,
% 2.18/1.40 ( nonemptyImpWitness
% 2.18/1.40 = ( ! [A: $i] :
% 2.18/1.40 ( ( nonempty @ A )
% 2.18/1.40 => ? [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 & $true ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(uniqinunit_type,type,
% 2.18/1.40 uniqinunit: $o ).
% 2.18/1.40
% 2.18/1.40 thf(uniqinunit,definition,
% 2.18/1.40 ( uniqinunit
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.18/1.40 => ( Xx = Xy ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(notinsingleton_type,type,
% 2.18/1.40 notinsingleton: $o ).
% 2.18/1.40
% 2.18/1.40 thf(notinsingleton,definition,
% 2.18/1.40 ( notinsingleton
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( Xx != Xy )
% 2.18/1.40 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(eqinunit_type,type,
% 2.18/1.40 eqinunit: $o ).
% 2.18/1.40
% 2.18/1.40 thf(eqinunit,definition,
% 2.18/1.40 ( eqinunit
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( Xx = Xy )
% 2.18/1.40 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(singletonsswitch_type,type,
% 2.18/1.40 singletonsswitch: $o ).
% 2.18/1.40
% 2.18/1.40 thf(singletonsswitch,definition,
% 2.18/1.40 ( singletonsswitch
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.18/1.40 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(upairsetE_type,type,
% 2.18/1.40 upairsetE: $o ).
% 2.18/1.40
% 2.18/1.40 thf(upairsetE,definition,
% 2.18/1.40 ( upairsetE
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.18/1.40 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.18/1.40 => ( ( Xz = Xx )
% 2.18/1.40 | ( Xz = Xy ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(upairsetIL_type,type,
% 2.18/1.40 upairsetIL: $o ).
% 2.18/1.40
% 2.18/1.40 thf(upairsetIL,definition,
% 2.18/1.40 ( upairsetIL
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(upairsetIR_type,type,
% 2.18/1.40 upairsetIR: $o ).
% 2.18/1.40
% 2.18/1.40 thf(upairsetIR,definition,
% 2.18/1.40 ( upairsetIR
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(emptyE1_type,type,
% 2.18/1.40 emptyE1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(emptyE1,definition,
% 2.18/1.40 ( emptyE1
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.40 ( ? [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 & ( Xphi @ Xx ) )
% 2.18/1.40 => ( ( ( dsetconstr @ A
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.40 = emptyset )
% 2.18/1.40 => $false ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(vacuousDall_type,type,
% 2.18/1.40 vacuousDall: $o ).
% 2.18/1.40
% 2.18/1.40 thf(vacuousDall,definition,
% 2.18/1.40 ( vacuousDall
% 2.18/1.40 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ emptyset )
% 2.18/1.40 => ( Xphi @ Xx ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(quantDeMorgan1_type,type,
% 2.18/1.40 quantDeMorgan1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(quantDeMorgan1,definition,
% 2.18/1.40 ( quantDeMorgan1
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.40 ( ~ ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( Xphi @ Xx ) )
% 2.18/1.40 => ? [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(quantDeMorgan2_type,type,
% 2.18/1.40 quantDeMorgan2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(quantDeMorgan2,definition,
% 2.18/1.40 ( quantDeMorgan2
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.40 ( ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ~ ( Xphi @ Xx ) )
% 2.18/1.40 => ~ ? [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(quantDeMorgan3_type,type,
% 2.18/1.40 quantDeMorgan3: $o ).
% 2.18/1.40
% 2.18/1.40 thf(quantDeMorgan3,definition,
% 2.18/1.40 ( quantDeMorgan3
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.40 ( ~ ? [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 & ( Xphi @ Xx ) )
% 2.18/1.40 => ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(quantDeMorgan4_type,type,
% 2.18/1.40 quantDeMorgan4: $o ).
% 2.18/1.40
% 2.18/1.40 thf(quantDeMorgan4,definition,
% 2.18/1.40 ( quantDeMorgan4
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.40 ( ? [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 & ~ ( Xphi @ Xx ) )
% 2.18/1.40 => ~ ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(prop2setI_type,type,
% 2.18/1.40 prop2setI: $o ).
% 2.18/1.40
% 2.18/1.40 thf(prop2setI,definition,
% 2.18/1.40 ( prop2setI
% 2.18/1.40 = ( ! [Xphi: $o] :
% 2.18/1.40 ( Xphi
% 2.18/1.40 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(set2prop_type,type,
% 2.18/1.40 set2prop: $i > $o ).
% 2.18/1.40
% 2.18/1.40 thf(prop2set2propI_type,type,
% 2.18/1.40 prop2set2propI: $o ).
% 2.18/1.40
% 2.18/1.40 thf(prop2set2propI,definition,
% 2.18/1.40 ( prop2set2propI
% 2.18/1.40 = ( ! [Xphi: $o] :
% 2.18/1.40 ( Xphi
% 2.18/1.40 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(notdexE_type,type,
% 2.18/1.40 notdexE: $o ).
% 2.18/1.40
% 2.18/1.40 thf(notdexE,definition,
% 2.18/1.40 ( notdexE
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.40 ( ~ ? [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 & ( Xphi @ Xx ) )
% 2.18/1.40 => ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(notdallE_type,type,
% 2.18/1.40 notdallE: $o ).
% 2.18/1.40
% 2.18/1.40 thf(notdallE,definition,
% 2.18/1.40 ( notdallE
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.40 ( ~ ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( Xphi @ Xx ) )
% 2.18/1.40 => ? [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(exuI1_type,type,
% 2.18/1.40 exuI1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(exuI1,definition,
% 2.18/1.40 ( exuI1
% 2.18/1.40 = ( ! [Xphi: $i > $o] :
% 2.18/1.40 ( ? [Xx: $i] :
% 2.18/1.40 ( ( Xphi @ Xx )
% 2.18/1.40 & ! [Xy: $i] :
% 2.18/1.40 ( ( Xphi @ Xy )
% 2.18/1.40 => ( Xx = Xy ) ) )
% 2.18/1.40 => ( exu
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(exuI3_type,type,
% 2.18/1.40 exuI3: $o ).
% 2.18/1.40
% 2.18/1.40 thf(exuI3,definition,
% 2.18/1.40 ( exuI3
% 2.18/1.40 = ( ! [Xphi: $i > $o] :
% 2.18/1.40 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.18/1.40 => ( ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( Xphi @ Xx )
% 2.18/1.40 => ( ( Xphi @ Xy )
% 2.18/1.40 => ( Xx = Xy ) ) )
% 2.18/1.40 => ( exu
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(exuI2_type,type,
% 2.18/1.40 exuI2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(exuI2,definition,
% 2.18/1.40 ( exuI2
% 2.18/1.40 = ( ! [Xphi: $i > $o] :
% 2.18/1.40 ( ? [Xx: $i] :
% 2.18/1.40 ! [Xy: $i] :
% 2.18/1.40 ( ( Xphi @ Xy )
% 2.18/1.40 <=> ( Xy = Xx ) )
% 2.18/1.40 => ( exu
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(inCongP_type,type,
% 2.18/1.40 inCongP: $o ).
% 2.18/1.40
% 2.18/1.40 thf(inCongP,definition,
% 2.18/1.40 ( inCongP
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( A = B )
% 2.18/1.40 => ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( Xx = Xy )
% 2.18/1.40 => ( ( in @ Xx @ A )
% 2.18/1.40 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(in__Cong_type,type,
% 2.18/1.40 in__Cong: $o ).
% 2.18/1.40
% 2.18/1.40 thf(in__Cong,definition,
% 2.18/1.40 ( in__Cong
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( A = B )
% 2.18/1.40 => ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( Xx = Xy )
% 2.18/1.40 => ( ( in @ Xx @ A )
% 2.18/1.40 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(exuE3u_type,type,
% 2.18/1.40 exuE3u: $o ).
% 2.18/1.40
% 2.18/1.40 thf(exuE3u,definition,
% 2.18/1.40 ( exuE3u
% 2.18/1.40 = ( ! [Xphi: $i > $o] :
% 2.18/1.40 ( ( exu
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.40 => ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( Xphi @ Xx )
% 2.18/1.40 => ( ( Xphi @ Xy )
% 2.18/1.40 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(exu__Cong_type,type,
% 2.18/1.40 exu__Cong: $o ).
% 2.18/1.40
% 2.18/1.40 thf(exu__Cong,definition,
% 2.18/1.40 ( exu__Cong
% 2.18/1.40 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.18/1.40 ( ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( Xx = Xy )
% 2.18/1.40 => ( ( Xphi @ Xx )
% 2.18/1.40 <=> ( Xpsi @ Xy ) ) )
% 2.18/1.40 => ( ( exu
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.40 <=> ( exu
% 2.18/1.40 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(emptyset__Cong_type,type,
% 2.18/1.40 emptyset__Cong: $o ).
% 2.18/1.40
% 2.18/1.40 thf(emptyset__Cong,definition,
% 2.18/1.40 ( emptyset__Cong
% 2.18/1.40 = ( emptyset = emptyset ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoin__Cong_type,type,
% 2.18/1.40 setadjoin__Cong: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoin__Cong,definition,
% 2.18/1.40 ( setadjoin__Cong
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( Xx = Xy )
% 2.18/1.40 => ! [Xz: $i,Xu: $i] :
% 2.18/1.40 ( ( Xz = Xu )
% 2.18/1.40 => ( ( setadjoin @ Xx @ Xz )
% 2.18/1.40 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(powerset__Cong_type,type,
% 2.18/1.40 powerset__Cong: $o ).
% 2.18/1.40
% 2.18/1.40 thf(powerset__Cong,definition,
% 2.18/1.40 ( powerset__Cong
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( A = B )
% 2.18/1.40 => ( ( powerset @ A )
% 2.18/1.40 = ( powerset @ B ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setunion__Cong_type,type,
% 2.18/1.40 setunion__Cong: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setunion__Cong,definition,
% 2.18/1.40 ( setunion__Cong
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( A = B )
% 2.18/1.40 => ( ( setunion @ A )
% 2.18/1.40 = ( setunion @ B ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(omega__Cong_type,type,
% 2.18/1.40 omega__Cong: $o ).
% 2.18/1.40
% 2.18/1.40 thf(omega__Cong,definition,
% 2.18/1.40 ( omega__Cong
% 2.18/1.40 = ( omega = omega ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(exuEu_type,type,
% 2.18/1.40 exuEu: $o ).
% 2.18/1.40
% 2.18/1.40 thf(exuEu,definition,
% 2.18/1.40 ( exuEu
% 2.18/1.40 = ( ! [Xphi: $i > $o] :
% 2.18/1.40 ( ( exu
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.40 => ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( Xphi @ Xx )
% 2.18/1.40 => ( ( Xphi @ Xy )
% 2.18/1.40 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(descr__Cong_type,type,
% 2.18/1.40 descr__Cong: $o ).
% 2.18/1.40
% 2.18/1.40 thf(descr__Cong,definition,
% 2.18/1.40 ( descr__Cong
% 2.18/1.40 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.18/1.40 ( ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( Xx = Xy )
% 2.18/1.40 => ( ( Xphi @ Xx )
% 2.18/1.40 <=> ( Xpsi @ Xy ) ) )
% 2.18/1.40 => ( ( exu
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.40 => ( ( exu
% 2.18/1.40 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.18/1.40 => ( ( descr
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.40 = ( descr
% 2.18/1.40 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(dsetconstr__Cong_type,type,
% 2.18/1.40 dsetconstr__Cong: $o ).
% 2.18/1.40
% 2.18/1.40 thf(dsetconstr__Cong,definition,
% 2.18/1.40 ( dsetconstr__Cong
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( A = B )
% 2.18/1.40 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.18/1.40 ( ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ! [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ B )
% 2.18/1.40 => ( ( Xx = Xy )
% 2.18/1.40 => ( ( Xphi @ Xx )
% 2.18/1.40 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.18/1.40 => ( ( dsetconstr @ A
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.40 = ( dsetconstr @ B
% 2.18/1.40 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(subset_type,type,
% 2.18/1.40 subset: $i > $i > $o ).
% 2.18/1.40
% 2.18/1.40 thf(disjoint_type,type,
% 2.18/1.40 disjoint: $i > $i > $o ).
% 2.18/1.40
% 2.18/1.40 thf(setsmeet_type,type,
% 2.18/1.40 setsmeet: $i > $i > $o ).
% 2.18/1.40
% 2.18/1.40 thf(subsetI1_type,type,
% 2.18/1.40 subsetI1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(subsetI1,definition,
% 2.18/1.40 ( subsetI1
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( in @ Xx @ B ) )
% 2.18/1.40 => ( subset @ A @ B ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(eqimpsubset2_type,type,
% 2.18/1.40 eqimpsubset2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(eqimpsubset2,definition,
% 2.18/1.40 ( eqimpsubset2
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( A = B )
% 2.18/1.40 => ( subset @ B @ A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(eqimpsubset1_type,type,
% 2.18/1.40 eqimpsubset1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(eqimpsubset1,definition,
% 2.18/1.40 ( eqimpsubset1
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( A = B )
% 2.18/1.40 => ( subset @ A @ B ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(subsetI2_type,type,
% 2.18/1.40 subsetI2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(subsetI2,definition,
% 2.18/1.40 ( subsetI2
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( in @ Xx @ B ) )
% 2.18/1.40 => ( subset @ A @ B ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(emptysetsubset_type,type,
% 2.18/1.40 emptysetsubset: $o ).
% 2.18/1.40
% 2.18/1.40 thf(emptysetsubset,definition,
% 2.18/1.40 ( emptysetsubset
% 2.18/1.40 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(subsetE_type,type,
% 2.18/1.40 subsetE: $o ).
% 2.18/1.40
% 2.18/1.40 thf(subsetE,definition,
% 2.18/1.40 ( subsetE
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( subset @ A @ B )
% 2.18/1.40 => ( ( in @ Xx @ A )
% 2.18/1.40 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(subsetE2_type,type,
% 2.18/1.40 subsetE2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(subsetE2,definition,
% 2.18/1.40 ( subsetE2
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( subset @ A @ B )
% 2.18/1.40 => ( ~ ( in @ Xx @ B )
% 2.18/1.40 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(notsubsetI_type,type,
% 2.18/1.40 notsubsetI: $o ).
% 2.18/1.40
% 2.18/1.40 thf(notsubsetI,definition,
% 2.18/1.40 ( notsubsetI
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( ~ ( in @ Xx @ B )
% 2.18/1.40 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(notequalI1_type,type,
% 2.18/1.40 notequalI1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(notequalI1,definition,
% 2.18/1.40 ( notequalI1
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ~ ( subset @ A @ B )
% 2.18/1.40 => ( A != B ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(notequalI2_type,type,
% 2.18/1.40 notequalI2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(notequalI2,definition,
% 2.18/1.40 ( notequalI2
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( ~ ( in @ Xx @ B )
% 2.18/1.40 => ( A != B ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(subsetRefl_type,type,
% 2.18/1.40 subsetRefl: $o ).
% 2.18/1.40
% 2.18/1.40 thf(subsetRefl,definition,
% 2.18/1.40 ( subsetRefl
% 2.18/1.40 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(subsetTrans_type,type,
% 2.18/1.40 subsetTrans: $o ).
% 2.18/1.40
% 2.18/1.40 thf(subsetTrans,definition,
% 2.18/1.40 ( subsetTrans
% 2.18/1.40 = ( ! [A: $i,B: $i,C: $i] :
% 2.18/1.40 ( ( subset @ A @ B )
% 2.18/1.40 => ( ( subset @ B @ C )
% 2.18/1.40 => ( subset @ A @ C ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoinSub_type,type,
% 2.18/1.40 setadjoinSub: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoinSub,definition,
% 2.18/1.40 ( setadjoinSub
% 2.18/1.40 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoinSub2_type,type,
% 2.18/1.40 setadjoinSub2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setadjoinSub2,definition,
% 2.18/1.40 ( setadjoinSub2
% 2.18/1.40 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.18/1.40 ( ( subset @ A @ B )
% 2.18/1.40 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(subset2powerset_type,type,
% 2.18/1.40 subset2powerset: $o ).
% 2.18/1.40
% 2.18/1.40 thf(subset2powerset,definition,
% 2.18/1.40 ( subset2powerset
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( subset @ A @ B )
% 2.18/1.40 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setextsub_type,type,
% 2.18/1.40 setextsub: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setextsub,definition,
% 2.18/1.40 ( setextsub
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( subset @ A @ B )
% 2.18/1.40 => ( ( subset @ B @ A )
% 2.18/1.40 => ( A = B ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(subsetemptysetimpeq_type,type,
% 2.18/1.40 subsetemptysetimpeq: $o ).
% 2.18/1.40
% 2.18/1.40 thf(subsetemptysetimpeq,definition,
% 2.18/1.40 ( subsetemptysetimpeq
% 2.18/1.40 = ( ! [A: $i] :
% 2.18/1.40 ( ( subset @ A @ emptyset )
% 2.18/1.40 => ( A = emptyset ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(powersetI1_type,type,
% 2.18/1.40 powersetI1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(powersetI1,definition,
% 2.18/1.40 ( powersetI1
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( subset @ B @ A )
% 2.18/1.40 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(powersetE1_type,type,
% 2.18/1.40 powersetE1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(powersetE1,definition,
% 2.18/1.40 ( powersetE1
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( in @ B @ ( powerset @ A ) )
% 2.18/1.40 => ( subset @ B @ A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(inPowerset_type,type,
% 2.18/1.40 inPowerset: $o ).
% 2.18/1.40
% 2.18/1.40 thf(inPowerset,definition,
% 2.18/1.40 ( inPowerset
% 2.18/1.40 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(powersetsubset_type,type,
% 2.18/1.40 powersetsubset: $o ).
% 2.18/1.40
% 2.18/1.40 thf(powersetsubset,definition,
% 2.18/1.40 ( powersetsubset
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( subset @ A @ B )
% 2.18/1.40 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(sepInPowerset_type,type,
% 2.18/1.40 sepInPowerset: $o ).
% 2.18/1.40
% 2.18/1.40 thf(sepInPowerset,definition,
% 2.18/1.40 ( sepInPowerset
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.40 ( in
% 2.18/1.40 @ ( dsetconstr @ A
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.40 @ ( powerset @ A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(sepSubset_type,type,
% 2.18/1.40 sepSubset: $o ).
% 2.18/1.40
% 2.18/1.40 thf(sepSubset,definition,
% 2.18/1.40 ( sepSubset
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.40 ( subset
% 2.18/1.40 @ ( dsetconstr @ A
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.40 @ A ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(binunion_type,type,
% 2.18/1.40 binunion: $i > $i > $i ).
% 2.18/1.40
% 2.18/1.40 thf(binunionIL_type,type,
% 2.18/1.40 binunionIL: $o ).
% 2.18/1.40
% 2.18/1.40 thf(binunionIL,definition,
% 2.18/1.40 ( binunionIL
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(upairset2IR_type,type,
% 2.18/1.40 upairset2IR: $o ).
% 2.18/1.40
% 2.18/1.40 thf(upairset2IR,definition,
% 2.18/1.40 ( upairset2IR
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(binunionIR_type,type,
% 2.18/1.40 binunionIR: $o ).
% 2.18/1.40
% 2.18/1.40 thf(binunionIR,definition,
% 2.18/1.40 ( binunionIR
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ B )
% 2.18/1.40 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(binunionEcases_type,type,
% 2.18/1.40 binunionEcases: $o ).
% 2.18/1.40
% 2.18/1.40 thf(binunionEcases,definition,
% 2.18/1.40 ( binunionEcases
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.18/1.40 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.18/1.40 => ( ( ( in @ Xx @ A )
% 2.18/1.40 => Xphi )
% 2.18/1.40 => ( ( ( in @ Xx @ B )
% 2.18/1.40 => Xphi )
% 2.18/1.40 => Xphi ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(binunionE_type,type,
% 2.18/1.40 binunionE: $o ).
% 2.18/1.40
% 2.18/1.40 thf(binunionE,definition,
% 2.18/1.40 ( binunionE
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.18/1.40 => ( ( in @ Xx @ A )
% 2.18/1.40 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(binunionLsub_type,type,
% 2.18/1.40 binunionLsub: $o ).
% 2.18/1.40
% 2.18/1.40 thf(binunionLsub,definition,
% 2.18/1.40 ( binunionLsub
% 2.18/1.40 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(binunionRsub_type,type,
% 2.18/1.40 binunionRsub: $o ).
% 2.18/1.40
% 2.18/1.40 thf(binunionRsub,definition,
% 2.18/1.40 ( binunionRsub
% 2.18/1.40 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(binintersect_type,type,
% 2.18/1.40 binintersect: $i > $i > $i ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectI_type,type,
% 2.18/1.40 binintersectI: $o ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectI,definition,
% 2.18/1.40 ( binintersectI
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( ( in @ Xx @ B )
% 2.18/1.40 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectSubset5_type,type,
% 2.18/1.40 binintersectSubset5: $o ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectSubset5,definition,
% 2.18/1.40 ( binintersectSubset5
% 2.18/1.40 = ( ! [A: $i,B: $i,C: $i] :
% 2.18/1.40 ( ( subset @ C @ A )
% 2.18/1.40 => ( ( subset @ C @ B )
% 2.18/1.40 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectEL_type,type,
% 2.18/1.40 binintersectEL: $o ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectEL,definition,
% 2.18/1.40 ( binintersectEL
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.18/1.40 => ( in @ Xx @ A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectLsub_type,type,
% 2.18/1.40 binintersectLsub: $o ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectLsub,definition,
% 2.18/1.40 ( binintersectLsub
% 2.18/1.40 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectSubset2_type,type,
% 2.18/1.40 binintersectSubset2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectSubset2,definition,
% 2.18/1.40 ( binintersectSubset2
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( subset @ A @ B )
% 2.18/1.40 => ( ( binintersect @ A @ B )
% 2.18/1.40 = A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectSubset3_type,type,
% 2.18/1.40 binintersectSubset3: $o ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectSubset3,definition,
% 2.18/1.40 ( binintersectSubset3
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( ( binintersect @ A @ B )
% 2.18/1.40 = B )
% 2.18/1.40 => ( subset @ B @ A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectER_type,type,
% 2.18/1.40 binintersectER: $o ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectER,definition,
% 2.18/1.40 ( binintersectER
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.18/1.40 => ( in @ Xx @ B ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(disjointsetsI1_type,type,
% 2.18/1.40 disjointsetsI1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(disjointsetsI1,definition,
% 2.18/1.40 ( disjointsetsI1
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ~ ? [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 & ( in @ Xx @ B ) )
% 2.18/1.40 => ( ( binintersect @ A @ B )
% 2.18/1.40 = emptyset ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectRsub_type,type,
% 2.18/1.40 binintersectRsub: $o ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectRsub,definition,
% 2.18/1.40 ( binintersectRsub
% 2.18/1.40 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectSubset4_type,type,
% 2.18/1.40 binintersectSubset4: $o ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectSubset4,definition,
% 2.18/1.40 ( binintersectSubset4
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( subset @ B @ A )
% 2.18/1.40 => ( ( binintersect @ A @ B )
% 2.18/1.40 = B ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectSubset1_type,type,
% 2.18/1.40 binintersectSubset1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(binintersectSubset1,definition,
% 2.18/1.40 ( binintersectSubset1
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( ( binintersect @ A @ B )
% 2.18/1.40 = A )
% 2.18/1.40 => ( subset @ A @ B ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(bs114d_type,type,
% 2.18/1.40 bs114d: $o ).
% 2.18/1.40
% 2.18/1.40 thf(bs114d,definition,
% 2.18/1.40 ( bs114d
% 2.18/1.40 = ( ! [A: $i,B: $i,C: $i] :
% 2.18/1.40 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.18/1.40 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(regular_type,type,
% 2.18/1.40 regular: $i > $o ).
% 2.18/1.40
% 2.18/1.40 thf(setminus_type,type,
% 2.18/1.40 setminus: $i > $i > $i ).
% 2.18/1.40
% 2.18/1.40 thf(setminusI_type,type,
% 2.18/1.40 setminusI: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setminusI,definition,
% 2.18/1.40 ( setminusI
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( ~ ( in @ Xx @ B )
% 2.18/1.40 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setminusEL_type,type,
% 2.18/1.40 setminusEL: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setminusEL,definition,
% 2.18/1.40 ( setminusEL
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.18/1.40 => ( in @ Xx @ A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setminusER_type,type,
% 2.18/1.40 setminusER: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setminusER,definition,
% 2.18/1.40 ( setminusER
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.18/1.40 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setminusSubset2_type,type,
% 2.18/1.40 setminusSubset2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setminusSubset2,definition,
% 2.18/1.40 ( setminusSubset2
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( subset @ A @ B )
% 2.18/1.40 => ( ( setminus @ A @ B )
% 2.18/1.40 = emptyset ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setminusERneg_type,type,
% 2.18/1.40 setminusERneg: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setminusERneg,definition,
% 2.18/1.40 ( setminusERneg
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.18/1.40 => ( ( in @ Xx @ A )
% 2.18/1.40 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setminusELneg_type,type,
% 2.18/1.40 setminusELneg: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setminusELneg,definition,
% 2.18/1.40 ( setminusELneg
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.18/1.40 => ( ~ ( in @ Xx @ B )
% 2.18/1.40 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setminusILneg_type,type,
% 2.18/1.40 setminusILneg: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setminusILneg,definition,
% 2.18/1.40 ( setminusILneg
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ~ ( in @ Xx @ A )
% 2.18/1.40 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setminusIRneg_type,type,
% 2.18/1.40 setminusIRneg: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setminusIRneg,definition,
% 2.18/1.40 ( setminusIRneg
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ B )
% 2.18/1.40 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setminusLsub_type,type,
% 2.18/1.40 setminusLsub: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setminusLsub,definition,
% 2.18/1.40 ( setminusLsub
% 2.18/1.40 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setminusSubset1_type,type,
% 2.18/1.40 setminusSubset1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setminusSubset1,definition,
% 2.18/1.40 ( setminusSubset1
% 2.18/1.40 = ( ! [A: $i,B: $i] :
% 2.18/1.40 ( ( ( setminus @ A @ B )
% 2.18/1.40 = emptyset )
% 2.18/1.40 => ( subset @ A @ B ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(symdiff_type,type,
% 2.18/1.40 symdiff: $i > $i > $i ).
% 2.18/1.40
% 2.18/1.40 thf(symdiffE_type,type,
% 2.18/1.40 symdiffE: $o ).
% 2.18/1.40
% 2.18/1.40 thf(symdiffE,definition,
% 2.18/1.40 ( symdiffE
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.18/1.40 => ! [Xphi: $o] :
% 2.18/1.40 ( ( ( in @ Xx @ A )
% 2.18/1.40 => ( ~ ( in @ Xx @ B )
% 2.18/1.40 => Xphi ) )
% 2.18/1.40 => ( ( ~ ( in @ Xx @ A )
% 2.18/1.40 => ( ( in @ Xx @ B )
% 2.18/1.40 => Xphi ) )
% 2.18/1.40 => Xphi ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(symdiffI1_type,type,
% 2.18/1.40 symdiffI1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(symdiffI1,definition,
% 2.18/1.40 ( symdiffI1
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( ~ ( in @ Xx @ B )
% 2.18/1.40 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(symdiffI2_type,type,
% 2.18/1.40 symdiffI2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(symdiffI2,definition,
% 2.18/1.40 ( symdiffI2
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ~ ( in @ Xx @ A )
% 2.18/1.40 => ( ( in @ Xx @ B )
% 2.18/1.40 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(symdiffIneg1_type,type,
% 2.18/1.40 symdiffIneg1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(symdiffIneg1,definition,
% 2.18/1.40 ( symdiffIneg1
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( ( in @ Xx @ B )
% 2.18/1.40 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(symdiffIneg2_type,type,
% 2.18/1.40 symdiffIneg2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(symdiffIneg2,definition,
% 2.18/1.40 ( symdiffIneg2
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ~ ( in @ Xx @ A )
% 2.18/1.40 => ( ~ ( in @ Xx @ B )
% 2.18/1.40 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(iskpair_type,type,
% 2.18/1.40 iskpair: $i > $o ).
% 2.18/1.40
% 2.18/1.40 thf(secondinupair_type,type,
% 2.18/1.40 secondinupair: $o ).
% 2.18/1.40
% 2.18/1.40 thf(secondinupair,definition,
% 2.18/1.40 ( secondinupair
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairIL_type,type,
% 2.18/1.40 setukpairIL: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairIL,definition,
% 2.18/1.40 ( setukpairIL
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairIR_type,type,
% 2.18/1.40 setukpairIR: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairIR,definition,
% 2.18/1.40 ( setukpairIR
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(kpairiskpair_type,type,
% 2.18/1.40 kpairiskpair: $o ).
% 2.18/1.40
% 2.18/1.40 thf(kpairiskpair,definition,
% 2.18/1.40 ( kpairiskpair
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(kpair_type,type,
% 2.18/1.40 kpair: $i > $i > $i ).
% 2.18/1.40
% 2.18/1.40 thf(kpairp_type,type,
% 2.18/1.40 kpairp: $o ).
% 2.18/1.40
% 2.18/1.40 thf(kpairp,definition,
% 2.18/1.40 ( kpairp
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(cartprod_type,type,
% 2.18/1.40 cartprod: $i > $i > $i ).
% 2.18/1.40
% 2.18/1.40 thf(singletonsubset_type,type,
% 2.18/1.40 singletonsubset: $o ).
% 2.18/1.40
% 2.18/1.40 thf(singletonsubset,definition,
% 2.18/1.40 ( singletonsubset
% 2.18/1.40 = ( ! [A: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(singletoninpowerset_type,type,
% 2.18/1.40 singletoninpowerset: $o ).
% 2.18/1.40
% 2.18/1.40 thf(singletoninpowerset,definition,
% 2.18/1.40 ( singletoninpowerset
% 2.18/1.40 = ( ! [A: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(singletoninpowunion_type,type,
% 2.18/1.40 singletoninpowunion: $o ).
% 2.18/1.40
% 2.18/1.40 thf(singletoninpowunion,definition,
% 2.18/1.40 ( singletoninpowunion
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(upairset2E_type,type,
% 2.18/1.40 upairset2E: $o ).
% 2.18/1.40
% 2.18/1.40 thf(upairset2E,definition,
% 2.18/1.40 ( upairset2E
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.18/1.40 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.18/1.40 => ( ( Xz = Xx )
% 2.18/1.40 | ( Xz = Xy ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(upairsubunion_type,type,
% 2.18/1.40 upairsubunion: $o ).
% 2.18/1.40
% 2.18/1.40 thf(upairsubunion,definition,
% 2.18/1.40 ( upairsubunion
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ! [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ B )
% 2.18/1.40 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(upairinpowunion_type,type,
% 2.18/1.40 upairinpowunion: $o ).
% 2.18/1.40
% 2.18/1.40 thf(upairinpowunion,definition,
% 2.18/1.40 ( upairinpowunion
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ! [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ B )
% 2.18/1.40 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(ubforcartprodlem1_type,type,
% 2.18/1.40 ubforcartprodlem1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(ubforcartprodlem1,definition,
% 2.18/1.40 ( ubforcartprodlem1
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ! [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ B )
% 2.18/1.40 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(ubforcartprodlem2_type,type,
% 2.18/1.40 ubforcartprodlem2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(ubforcartprodlem2,definition,
% 2.18/1.40 ( ubforcartprodlem2
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ! [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ B )
% 2.18/1.40 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(ubforcartprodlem3_type,type,
% 2.18/1.40 ubforcartprodlem3: $o ).
% 2.18/1.40
% 2.18/1.40 thf(ubforcartprodlem3,definition,
% 2.18/1.40 ( ubforcartprodlem3
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ! [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ B )
% 2.18/1.40 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodpairin_type,type,
% 2.18/1.40 cartprodpairin: $o ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodpairin,definition,
% 2.18/1.40 ( cartprodpairin
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ! [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ B )
% 2.18/1.40 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodmempair1_type,type,
% 2.18/1.40 cartprodmempair1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodmempair1,definition,
% 2.18/1.40 ( cartprodmempair1
% 2.18/1.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.18/1.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.18/1.40 => ? [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 & ? [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ B )
% 2.18/1.40 & ( Xu
% 2.18/1.40 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodmempair_type,type,
% 2.18/1.40 cartprodmempair: $o ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodmempair,definition,
% 2.18/1.40 ( cartprodmempair
% 2.18/1.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.18/1.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.18/1.40 => ( iskpair @ Xu ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setunionE2_type,type,
% 2.18/1.40 setunionE2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setunionE2,definition,
% 2.18/1.40 ( setunionE2
% 2.18/1.40 = ( ! [A: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ ( setunion @ A ) )
% 2.18/1.40 => ? [X: $i] :
% 2.18/1.40 ( ( in @ X @ A )
% 2.18/1.40 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setunionsingleton1_type,type,
% 2.18/1.40 setunionsingleton1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setunionsingleton1,definition,
% 2.18/1.40 ( setunionsingleton1
% 2.18/1.40 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setunionsingleton2_type,type,
% 2.18/1.40 setunionsingleton2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setunionsingleton2,definition,
% 2.18/1.40 ( setunionsingleton2
% 2.18/1.40 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setunionsingleton_type,type,
% 2.18/1.40 setunionsingleton: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setunionsingleton,definition,
% 2.18/1.40 ( setunionsingleton
% 2.18/1.40 = ( ! [Xx: $i] :
% 2.18/1.40 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.18/1.40 = Xx ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(singleton_type,type,
% 2.18/1.40 singleton: $i > $o ).
% 2.18/1.40
% 2.18/1.40 thf(singletonprop_type,type,
% 2.18/1.40 singletonprop: $o ).
% 2.18/1.40
% 2.18/1.40 thf(singletonprop,definition,
% 2.18/1.40 ( singletonprop
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.40 ( ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ! [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ A )
% 2.18/1.40 => ( ( Xphi @ Xx )
% 2.18/1.40 => ( ( Xphi @ Xy )
% 2.18/1.40 => ( Xx = Xy ) ) ) ) )
% 2.18/1.40 => ( ? [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 & ( Xphi @ Xx ) )
% 2.18/1.40 => ( singleton
% 2.18/1.40 @ ( dsetconstr @ A
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(ex1_type,type,
% 2.18/1.40 ex1: $i > ( $i > $o ) > $o ).
% 2.18/1.40
% 2.18/1.40 thf(ex1,definition,
% 2.18/1.40 ( ex1
% 2.18/1.40 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.18/1.40 ( singleton
% 2.18/1.40 @ ( dsetconstr @ A
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(ex1E1_type,type,
% 2.18/1.40 ex1E1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(ex1E1,definition,
% 2.18/1.40 ( ex1E1
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.40 ( ( ex1 @ A
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.40 => ? [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(ex1I_type,type,
% 2.18/1.40 ex1I: $o ).
% 2.18/1.40
% 2.18/1.40 thf(ex1I,definition,
% 2.18/1.40 ( ex1I
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ( ( Xphi @ Xx )
% 2.18/1.40 => ( ! [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ A )
% 2.18/1.40 => ( ( Xphi @ Xy )
% 2.18/1.40 => ( Xy = Xx ) ) )
% 2.18/1.40 => ( ex1 @ A
% 2.18/1.40 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(ex1I2_type,type,
% 2.18/1.40 ex1I2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(ex1I2,definition,
% 2.18/1.40 ( ex1I2
% 2.18/1.40 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.40 ( ! [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ! [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ A )
% 2.18/1.40 => ( ( Xphi @ Xx )
% 2.18/1.40 => ( ( Xphi @ Xy )
% 2.18/1.40 => ( Xx = Xy ) ) ) ) )
% 2.18/1.40 => ( ? [Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 & ( Xphi @ Xx ) )
% 2.18/1.40 => ( ex1 @ A
% 2.18/1.40 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(singletonsuniq_type,type,
% 2.18/1.40 singletonsuniq: $o ).
% 2.18/1.40
% 2.18/1.40 thf(singletonsuniq,definition,
% 2.18/1.40 ( singletonsuniq
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( ( setadjoin @ Xx @ emptyset )
% 2.18/1.40 = ( setadjoin @ Xy @ emptyset ) )
% 2.18/1.40 => ( Xx = Xy ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(atmost1p_type,type,
% 2.18/1.40 atmost1p: $i > $o ).
% 2.18/1.40
% 2.18/1.40 thf(atleast2p_type,type,
% 2.18/1.40 atleast2p: $i > $o ).
% 2.18/1.40
% 2.18/1.40 thf(atmost2p_type,type,
% 2.18/1.40 atmost2p: $i > $o ).
% 2.18/1.40
% 2.18/1.40 thf(upairsetp_type,type,
% 2.18/1.40 upairsetp: $i > $o ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairinjL1_type,type,
% 2.18/1.40 setukpairinjL1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairinjL1,definition,
% 2.18/1.40 ( setukpairinjL1
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.18/1.40 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.18/1.40 => ( Xx = Xz ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(kfstsingleton_type,type,
% 2.18/1.40 kfstsingleton: $o ).
% 2.18/1.40
% 2.18/1.40 thf(kfstsingleton,definition,
% 2.18/1.40 ( kfstsingleton
% 2.18/1.40 = ( ! [Xu: $i] :
% 2.18/1.40 ( ( iskpair @ Xu )
% 2.18/1.40 => ( singleton
% 2.18/1.40 @ ( dsetconstr @ ( setunion @ Xu )
% 2.18/1.40 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(theprop_type,type,
% 2.18/1.40 theprop: $o ).
% 2.18/1.40
% 2.18/1.40 thf(theprop,definition,
% 2.18/1.40 ( theprop
% 2.18/1.40 = ( ! [X: $i] :
% 2.18/1.40 ( ( singleton @ X )
% 2.18/1.40 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(kfst_type,type,
% 2.18/1.40 kfst: $i > $i ).
% 2.18/1.40
% 2.18/1.40 thf(kfstpairEq_type,type,
% 2.18/1.40 kfstpairEq: $o ).
% 2.18/1.40
% 2.18/1.40 thf(kfstpairEq,definition,
% 2.18/1.40 ( kfstpairEq
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.18/1.40 = Xx ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodfstin_type,type,
% 2.18/1.40 cartprodfstin: $o ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodfstin,definition,
% 2.18/1.40 ( cartprodfstin
% 2.18/1.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.18/1.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.18/1.40 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairinjL2_type,type,
% 2.18/1.40 setukpairinjL2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairinjL2,definition,
% 2.18/1.40 ( setukpairinjL2
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.18/1.40 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.18/1.40 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.18/1.40 => ( Xx = Xz ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairinjL_type,type,
% 2.18/1.40 setukpairinjL: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairinjL,definition,
% 2.18/1.40 ( setukpairinjL
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.18/1.40 ( ( ( kpair @ Xx @ Xy )
% 2.18/1.40 = ( kpair @ Xz @ Xu ) )
% 2.18/1.40 => ( Xx = Xz ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairinjR11_type,type,
% 2.18/1.40 setukpairinjR11: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairinjR11,definition,
% 2.18/1.40 ( setukpairinjR11
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( Xx = Xy )
% 2.18/1.40 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.18/1.40 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairinjR12_type,type,
% 2.18/1.40 setukpairinjR12: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairinjR12,definition,
% 2.18/1.40 ( setukpairinjR12
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( Xx = Xy )
% 2.18/1.40 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.18/1.40 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairinjR1_type,type,
% 2.18/1.40 setukpairinjR1: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairinjR1,definition,
% 2.18/1.40 ( setukpairinjR1
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.18/1.40 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.18/1.40 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.18/1.40 => ( ( Xz = Xu )
% 2.18/1.40 => ( Xy = Xu ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(upairequniteq_type,type,
% 2.18/1.40 upairequniteq: $o ).
% 2.18/1.40
% 2.18/1.40 thf(upairequniteq,definition,
% 2.18/1.40 ( upairequniteq
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.18/1.40 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.18/1.40 = ( setadjoin @ Xz @ emptyset ) )
% 2.18/1.40 => ( Xx = Xy ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairinjR2_type,type,
% 2.18/1.40 setukpairinjR2: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairinjR2,definition,
% 2.18/1.40 ( setukpairinjR2
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.18/1.40 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.18/1.40 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.18/1.40 => ( Xy = Xu ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairinjR_type,type,
% 2.18/1.40 setukpairinjR: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setukpairinjR,definition,
% 2.18/1.40 ( setukpairinjR
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.18/1.40 ( ( ( kpair @ Xx @ Xy )
% 2.18/1.40 = ( kpair @ Xz @ Xu ) )
% 2.18/1.40 => ( Xy = Xu ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(ksndsingleton_type,type,
% 2.18/1.40 ksndsingleton: $o ).
% 2.18/1.40
% 2.18/1.40 thf(ksndsingleton,definition,
% 2.18/1.40 ( ksndsingleton
% 2.18/1.40 = ( ! [Xu: $i] :
% 2.18/1.40 ( ( iskpair @ Xu )
% 2.18/1.40 => ( singleton
% 2.18/1.40 @ ( dsetconstr @ ( setunion @ Xu )
% 2.18/1.40 @ ^ [Xx: $i] :
% 2.18/1.40 ( Xu
% 2.18/1.40 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(ksnd_type,type,
% 2.18/1.40 ksnd: $i > $i ).
% 2.18/1.40
% 2.18/1.40 thf(ksndpairEq_type,type,
% 2.18/1.40 ksndpairEq: $o ).
% 2.18/1.40
% 2.18/1.40 thf(ksndpairEq,definition,
% 2.18/1.40 ( ksndpairEq
% 2.18/1.40 = ( ! [Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.18/1.40 = Xy ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(kpairsurjEq_type,type,
% 2.18/1.40 kpairsurjEq: $o ).
% 2.18/1.40
% 2.18/1.40 thf(kpairsurjEq,definition,
% 2.18/1.40 ( kpairsurjEq
% 2.18/1.40 = ( ! [Xu: $i] :
% 2.18/1.40 ( ( iskpair @ Xu )
% 2.18/1.40 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.18/1.40 = Xu ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodsndin_type,type,
% 2.18/1.40 cartprodsndin: $o ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodsndin,definition,
% 2.18/1.40 ( cartprodsndin
% 2.18/1.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.18/1.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.18/1.40 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodpairmemEL_type,type,
% 2.18/1.40 cartprodpairmemEL: $o ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodpairmemEL,definition,
% 2.18/1.40 ( cartprodpairmemEL
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.18/1.40 => ( in @ Xx @ A ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodpairmemER_type,type,
% 2.18/1.40 cartprodpairmemER: $o ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodpairmemER,definition,
% 2.18/1.40 ( cartprodpairmemER
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.18/1.40 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.18/1.40 => ( in @ Xy @ B ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodmempaircEq_type,type,
% 2.18/1.40 cartprodmempaircEq: $o ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodmempaircEq,definition,
% 2.18/1.40 ( cartprodmempaircEq
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ! [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ B )
% 2.18/1.40 => ( ( kpair @ Xx @ Xy )
% 2.18/1.40 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodfstpairEq_type,type,
% 2.18/1.40 cartprodfstpairEq: $o ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodfstpairEq,definition,
% 2.18/1.40 ( cartprodfstpairEq
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ! [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ B )
% 2.18/1.40 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.18/1.40 = Xx ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodsndpairEq_type,type,
% 2.18/1.40 cartprodsndpairEq: $o ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodsndpairEq,definition,
% 2.18/1.40 ( cartprodsndpairEq
% 2.18/1.40 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ! [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ B )
% 2.18/1.40 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.18/1.40 = Xy ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodpairsurjEq_type,type,
% 2.18/1.40 cartprodpairsurjEq: $o ).
% 2.18/1.40
% 2.18/1.40 thf(cartprodpairsurjEq,definition,
% 2.18/1.40 ( cartprodpairsurjEq
% 2.18/1.40 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.18/1.40 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.18/1.40 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.18/1.40 = Xu ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(breln_type,type,
% 2.18/1.40 breln: $i > $i > $i > $o ).
% 2.18/1.40
% 2.18/1.40 thf(breln,definition,
% 2.18/1.40 ( breln
% 2.18/1.40 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(dpsetconstr_type,type,
% 2.18/1.40 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.18/1.40
% 2.18/1.40 thf(dpsetconstrI_type,type,
% 2.18/1.40 dpsetconstrI: $o ).
% 2.18/1.40
% 2.18/1.40 thf(dpsetconstrI,definition,
% 2.18/1.40 ( dpsetconstrI
% 2.18/1.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ! [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ B )
% 2.18/1.40 => ( ( Xphi @ Xx @ Xy )
% 2.18/1.40 => ( in @ ( kpair @ Xx @ Xy )
% 2.18/1.40 @ ( dpsetconstr @ A @ B
% 2.18/1.40 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(dpsetconstrSub_type,type,
% 2.18/1.40 dpsetconstrSub: $o ).
% 2.18/1.40
% 2.18/1.40 thf(dpsetconstrSub,definition,
% 2.18/1.40 ( dpsetconstrSub
% 2.18/1.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.18/1.40 ( subset
% 2.18/1.40 @ ( dpsetconstr @ A @ B
% 2.18/1.40 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.18/1.40 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(setOfPairsIsBReln_type,type,
% 2.18/1.40 setOfPairsIsBReln: $o ).
% 2.18/1.40
% 2.18/1.40 thf(setOfPairsIsBReln,definition,
% 2.18/1.40 ( setOfPairsIsBReln
% 2.18/1.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.18/1.40 ( breln @ A @ B
% 2.18/1.40 @ ( dpsetconstr @ A @ B
% 2.18/1.40 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.18/1.40
% 2.18/1.40 thf(dpsetconstrERa_type,type,
% 2.18/1.40 dpsetconstrERa: $o ).
% 2.18/1.40
% 2.18/1.40 thf(dpsetconstrERa,definition,
% 2.18/1.40 ( dpsetconstrERa
% 2.18/1.40 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.18/1.40 ( ( in @ Xx @ A )
% 2.18/1.40 => ! [Xy: $i] :
% 2.18/1.40 ( ( in @ Xy @ B )
% 2.18/1.40 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.18/1.40 @ ( dpsetconstr @ A @ B
% 2.18/1.40 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.18/1.40 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(dpsetconstrEL1_type,type,
% 2.18/1.41 dpsetconstrEL1: $o ).
% 2.18/1.41
% 2.18/1.41 thf(dpsetconstrEL1,definition,
% 2.18/1.41 ( dpsetconstrEL1
% 2.18/1.41 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.18/1.41 ( ( in @ ( kpair @ Xx @ Xy )
% 2.18/1.41 @ ( dpsetconstr @ A @ B
% 2.18/1.41 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.18/1.41 => ( in @ Xx @ A ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(dpsetconstrEL2_type,type,
% 2.18/1.41 dpsetconstrEL2: $o ).
% 2.18/1.41
% 2.18/1.41 thf(dpsetconstrEL2,definition,
% 2.18/1.41 ( dpsetconstrEL2
% 2.18/1.41 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.18/1.41 ( ( in @ ( kpair @ Xx @ Xy )
% 2.18/1.41 @ ( dpsetconstr @ A @ B
% 2.18/1.41 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.18/1.41 => ( in @ Xy @ B ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(dpsetconstrER_type,type,
% 2.18/1.41 dpsetconstrER: $o ).
% 2.18/1.41
% 2.18/1.41 thf(dpsetconstrER,definition,
% 2.18/1.41 ( dpsetconstrER
% 2.18/1.41 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.18/1.41 ( ( in @ ( kpair @ Xx @ Xy )
% 2.18/1.41 @ ( dpsetconstr @ A @ B
% 2.18/1.41 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.18/1.41 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(func_type,type,
% 2.18/1.41 func: $i > $i > $i > $o ).
% 2.18/1.41
% 2.18/1.41 thf(funcSet_type,type,
% 2.18/1.41 funcSet: $i > $i > $i ).
% 2.18/1.41
% 2.18/1.41 thf(funcImageSingleton_type,type,
% 2.18/1.41 funcImageSingleton: $o ).
% 2.18/1.41
% 2.18/1.41 thf(funcImageSingleton,definition,
% 2.18/1.41 ( funcImageSingleton
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( func @ A @ B @ Xf )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( singleton
% 2.18/1.41 @ ( dsetconstr @ B
% 2.18/1.41 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(apProp_type,type,
% 2.18/1.41 apProp: $o ).
% 2.18/1.41
% 2.18/1.41 thf(apProp,definition,
% 2.18/1.41 ( apProp
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( func @ A @ B @ Xf )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( in
% 2.18/1.41 @ ( setunion
% 2.18/1.41 @ ( dsetconstr @ B
% 2.18/1.41 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.18/1.41 @ B ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(ap_type,type,
% 2.18/1.41 ap: $i > $i > $i > $i > $i ).
% 2.18/1.41
% 2.18/1.41 thf(app_type,type,
% 2.18/1.41 app: $o ).
% 2.18/1.41
% 2.18/1.41 thf(app,definition,
% 2.18/1.41 ( app
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( func @ A @ B @ Xf )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(infuncsetfunc_type,type,
% 2.18/1.41 infuncsetfunc: $o ).
% 2.18/1.41
% 2.18/1.41 thf(infuncsetfunc,definition,
% 2.18/1.41 ( infuncsetfunc
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.18/1.41 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(ap2p_type,type,
% 2.18/1.41 ap2p: $o ).
% 2.18/1.41
% 2.18/1.41 thf(ap2p,definition,
% 2.18/1.41 ( ap2p
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(funcinfuncset_type,type,
% 2.18/1.41 funcinfuncset: $o ).
% 2.18/1.41
% 2.18/1.41 thf(funcinfuncset,definition,
% 2.18/1.41 ( funcinfuncset
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( func @ A @ B @ Xf )
% 2.18/1.41 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(lamProp_type,type,
% 2.18/1.41 lamProp: $o ).
% 2.18/1.41
% 2.18/1.41 thf(lamProp,definition,
% 2.18/1.41 ( lamProp
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.18/1.41 ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.18/1.41 => ( func @ A @ B
% 2.18/1.41 @ ( dpsetconstr @ A @ B
% 2.18/1.41 @ ^ [Xx: $i,Xy: $i] :
% 2.18/1.41 ( ( Xf @ Xx )
% 2.18/1.41 = Xy ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(lam_type,type,
% 2.18/1.41 lam: $i > $i > ( $i > $i ) > $i ).
% 2.18/1.41
% 2.18/1.41 thf(lamp_type,type,
% 2.18/1.41 lamp: $o ).
% 2.18/1.41
% 2.18/1.41 thf(lamp,definition,
% 2.18/1.41 ( lamp
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.18/1.41 ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.18/1.41 => ( func @ A @ B
% 2.18/1.41 @ ( lam @ A @ B
% 2.18/1.41 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(lam2p_type,type,
% 2.18/1.41 lam2p: $o ).
% 2.18/1.41
% 2.18/1.41 thf(lam2p,definition,
% 2.18/1.41 ( lam2p
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.18/1.41 ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.18/1.41 => ( in
% 2.18/1.41 @ ( lam @ A @ B
% 2.18/1.41 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.18/1.41 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(brelnall1_type,type,
% 2.18/1.41 brelnall1: $o ).
% 2.18/1.41
% 2.18/1.41 thf(brelnall1,definition,
% 2.18/1.41 ( brelnall1
% 2.18/1.41 = ( ! [A: $i,B: $i,R: $i] :
% 2.18/1.41 ( ( breln @ A @ B @ R )
% 2.18/1.41 => ! [Xphi: $i > $o] :
% 2.18/1.41 ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ B )
% 2.18/1.41 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.18/1.41 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ R )
% 2.18/1.41 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(brelnall2_type,type,
% 2.18/1.41 brelnall2: $o ).
% 2.18/1.41
% 2.18/1.41 thf(brelnall2,definition,
% 2.18/1.41 ( brelnall2
% 2.18/1.41 = ( ! [A: $i,B: $i,R: $i] :
% 2.18/1.41 ( ( breln @ A @ B @ R )
% 2.18/1.41 => ! [Xphi: $i > $o] :
% 2.18/1.41 ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ B )
% 2.18/1.41 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.18/1.41 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ R )
% 2.18/1.41 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(ex1E2_type,type,
% 2.18/1.41 ex1E2: $o ).
% 2.18/1.41
% 2.18/1.41 thf(ex1E2,definition,
% 2.18/1.41 ( ex1E2
% 2.18/1.41 = ( ! [A: $i,Xphi: $i > $o] :
% 2.18/1.41 ( ( ex1 @ A
% 2.18/1.41 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ A )
% 2.18/1.41 => ( ( Xphi @ Xx )
% 2.18/1.41 => ( ( Xphi @ Xy )
% 2.18/1.41 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(funcGraphProp1_type,type,
% 2.18/1.41 funcGraphProp1: $o ).
% 2.18/1.41
% 2.18/1.41 thf(funcGraphProp1,definition,
% 2.18/1.41 ( funcGraphProp1
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( func @ A @ B @ Xf )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(funcGraphProp3_type,type,
% 2.18/1.41 funcGraphProp3: $o ).
% 2.18/1.41
% 2.18/1.41 thf(funcGraphProp3,definition,
% 2.18/1.41 ( funcGraphProp3
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(funcGraphProp2_type,type,
% 2.18/1.41 funcGraphProp2: $o ).
% 2.18/1.41
% 2.18/1.41 thf(funcGraphProp2,definition,
% 2.18/1.41 ( funcGraphProp2
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( func @ A @ B @ Xf )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ B )
% 2.18/1.41 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.18/1.41 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.18/1.41 = Xy ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(funcextLem_type,type,
% 2.18/1.41 funcextLem: $o ).
% 2.18/1.41
% 2.18/1.41 thf(funcextLem,definition,
% 2.18/1.41 ( funcextLem
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( func @ A @ B @ Xf )
% 2.18/1.41 => ! [Xg: $i] :
% 2.18/1.41 ( ( func @ A @ B @ Xg )
% 2.18/1.41 => ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.18/1.41 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ B )
% 2.18/1.41 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.18/1.41 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(funcGraphProp4_type,type,
% 2.18/1.41 funcGraphProp4: $o ).
% 2.18/1.41
% 2.18/1.41 thf(funcGraphProp4,definition,
% 2.18/1.41 ( funcGraphProp4
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ B )
% 2.18/1.41 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.18/1.41 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.18/1.41 = Xy ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(subbreln_type,type,
% 2.18/1.41 subbreln: $o ).
% 2.18/1.41
% 2.18/1.41 thf(subbreln,definition,
% 2.18/1.41 ( subbreln
% 2.18/1.41 = ( ! [A: $i,B: $i,R: $i] :
% 2.18/1.41 ( ( breln @ A @ B @ R )
% 2.18/1.41 => ! [S: $i] :
% 2.18/1.41 ( ( breln @ A @ B @ S )
% 2.18/1.41 => ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ B )
% 2.18/1.41 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.18/1.41 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.18/1.41 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(eqbreln_type,type,
% 2.18/1.41 eqbreln: $o ).
% 2.18/1.41
% 2.18/1.41 thf(eqbreln,definition,
% 2.18/1.41 ( eqbreln
% 2.18/1.41 = ( ! [A: $i,B: $i,R: $i] :
% 2.18/1.41 ( ( breln @ A @ B @ R )
% 2.18/1.41 => ! [S: $i] :
% 2.18/1.41 ( ( breln @ A @ B @ S )
% 2.18/1.41 => ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ B )
% 2.18/1.41 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.18/1.41 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.18/1.41 => ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ B )
% 2.18/1.41 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.18/1.41 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.18/1.41 => ( R = S ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(funcext_type,type,
% 2.18/1.41 funcext: $o ).
% 2.18/1.41
% 2.18/1.41 thf(funcext,definition,
% 2.18/1.41 ( funcext
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( func @ A @ B @ Xf )
% 2.18/1.41 => ! [Xg: $i] :
% 2.18/1.41 ( ( func @ A @ B @ Xg )
% 2.18/1.41 => ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.18/1.41 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.18/1.41 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(funcext2_type,type,
% 2.18/1.41 funcext2: $o ).
% 2.18/1.41
% 2.18/1.41 thf(funcext2,definition,
% 2.18/1.41 ( funcext2
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.18/1.41 => ! [Xg: $i] :
% 2.18/1.41 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.18/1.41 => ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.18/1.41 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.18/1.41 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(ap2apEq1_type,type,
% 2.18/1.41 ap2apEq1: $o ).
% 2.18/1.41
% 2.18/1.41 thf(ap2apEq1,definition,
% 2.18/1.41 ( ap2apEq1
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.18/1.41 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(ap2apEq2_type,type,
% 2.18/1.41 ap2apEq2: $o ).
% 2.18/1.41
% 2.18/1.41 thf(ap2apEq2,definition,
% 2.18/1.41 ( ap2apEq2
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( func @ A @ B @ Xf )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.18/1.41 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(beta1_type,type,
% 2.18/1.41 beta1: $o ).
% 2.18/1.41
% 2.18/1.41 thf(beta1,definition,
% 2.18/1.41 ( beta1
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.18/1.41 ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ( ap @ A @ B
% 2.18/1.41 @ ( lam @ A @ B
% 2.18/1.41 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.18/1.41 @ Xx )
% 2.18/1.41 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(eta1_type,type,
% 2.18/1.41 eta1: $o ).
% 2.18/1.41
% 2.18/1.41 thf(eta1,definition,
% 2.18/1.41 ( eta1
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( func @ A @ B @ Xf )
% 2.18/1.41 => ( ( lam @ A @ B
% 2.18/1.41 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.18/1.41 = Xf ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(lam2lamEq_type,type,
% 2.18/1.41 lam2lamEq: $o ).
% 2.18/1.41
% 2.18/1.41 thf(lam2lamEq,definition,
% 2.18/1.41 ( lam2lamEq
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.18/1.41 ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.18/1.41 => ( ( lam @ A @ B
% 2.18/1.41 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.18/1.41 = ( lam @ A @ B
% 2.18/1.41 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(beta2_type,type,
% 2.18/1.41 beta2: $o ).
% 2.18/1.41
% 2.18/1.41 thf(beta2,definition,
% 2.18/1.41 ( beta2
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.18/1.41 ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ( ap @ A @ B
% 2.18/1.41 @ ( lam @ A @ B
% 2.18/1.41 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.18/1.41 @ Xx )
% 2.18/1.41 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(eta2_type,type,
% 2.18/1.41 eta2: $o ).
% 2.18/1.41
% 2.18/1.41 thf(eta2,definition,
% 2.18/1.41 ( eta2
% 2.18/1.41 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.18/1.41 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.18/1.41 => ( ( lam @ A @ B
% 2.18/1.41 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.18/1.41 = Xf ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(iffalseProp1_type,type,
% 2.18/1.41 iffalseProp1: $o ).
% 2.18/1.41
% 2.18/1.41 thf(iffalseProp1,definition,
% 2.18/1.41 ( iffalseProp1
% 2.18/1.41 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ A )
% 2.18/1.41 => ( ~ Xphi
% 2.18/1.41 => ( in @ Xy
% 2.18/1.41 @ ( dsetconstr @ A
% 2.18/1.41 @ ^ [Xz: $i] :
% 2.18/1.41 ( ( Xphi
% 2.18/1.41 & ( Xz = Xx ) )
% 2.18/1.41 | ( ~ Xphi
% 2.18/1.41 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(iffalseProp2_type,type,
% 2.18/1.41 iffalseProp2: $o ).
% 2.18/1.41
% 2.18/1.41 thf(iffalseProp2,definition,
% 2.18/1.41 ( iffalseProp2
% 2.18/1.41 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ A )
% 2.18/1.41 => ( ~ Xphi
% 2.18/1.41 => ( ( dsetconstr @ A
% 2.18/1.41 @ ^ [Xz: $i] :
% 2.18/1.41 ( ( Xphi
% 2.18/1.41 & ( Xz = Xx ) )
% 2.18/1.41 | ( ~ Xphi
% 2.18/1.41 & ( Xz = Xy ) ) ) )
% 2.18/1.41 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(iftrueProp1_type,type,
% 2.18/1.41 iftrueProp1: $o ).
% 2.18/1.41
% 2.18/1.41 thf(iftrueProp1,definition,
% 2.18/1.41 ( iftrueProp1
% 2.18/1.41 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ A )
% 2.18/1.41 => ( Xphi
% 2.18/1.41 => ( in @ Xx
% 2.18/1.41 @ ( dsetconstr @ A
% 2.18/1.41 @ ^ [Xz: $i] :
% 2.18/1.41 ( ( Xphi
% 2.18/1.41 & ( Xz = Xx ) )
% 2.18/1.41 | ( ~ Xphi
% 2.18/1.41 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(iftrueProp2_type,type,
% 2.18/1.41 iftrueProp2: $o ).
% 2.18/1.41
% 2.18/1.41 thf(iftrueProp2,definition,
% 2.18/1.41 ( iftrueProp2
% 2.18/1.41 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ A )
% 2.18/1.41 => ( Xphi
% 2.18/1.41 => ( ( dsetconstr @ A
% 2.18/1.41 @ ^ [Xz: $i] :
% 2.18/1.41 ( ( Xphi
% 2.18/1.41 & ( Xz = Xx ) )
% 2.18/1.41 | ( ~ Xphi
% 2.18/1.41 & ( Xz = Xy ) ) ) )
% 2.18/1.41 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(ifSingleton_type,type,
% 2.18/1.41 ifSingleton: $o ).
% 2.18/1.41
% 2.18/1.41 thf(ifSingleton,definition,
% 2.18/1.41 ( ifSingleton
% 2.18/1.41 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ A )
% 2.18/1.41 => ( singleton
% 2.18/1.41 @ ( dsetconstr @ A
% 2.18/1.41 @ ^ [Xz: $i] :
% 2.18/1.41 ( ( Xphi
% 2.18/1.41 & ( Xz = Xx ) )
% 2.18/1.41 | ( ~ Xphi
% 2.18/1.41 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(if_type,type,
% 2.18/1.41 if: $i > $o > $i > $i > $i ).
% 2.18/1.41
% 2.18/1.41 thf(ifp_type,type,
% 2.18/1.41 ifp: $o ).
% 2.18/1.41
% 2.18/1.41 thf(ifp,definition,
% 2.18/1.41 ( ifp
% 2.18/1.41 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ A )
% 2.18/1.41 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(theeq_type,type,
% 2.18/1.41 theeq: $o ).
% 2.18/1.41
% 2.18/1.41 thf(theeq,definition,
% 2.18/1.41 ( theeq
% 2.18/1.41 = ( ! [X: $i] :
% 2.18/1.41 ( ( singleton @ X )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ X )
% 2.18/1.41 => ( ( setunion @ X )
% 2.18/1.41 = Xx ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(iftrue_type,type,
% 2.18/1.41 iftrue: $o ).
% 2.18/1.41
% 2.18/1.41 thf(iftrue,definition,
% 2.18/1.41 ( iftrue
% 2.18/1.41 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ A )
% 2.18/1.41 => ( Xphi
% 2.18/1.41 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.18/1.41 = Xx ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(iffalse_type,type,
% 2.18/1.41 iffalse: $o ).
% 2.18/1.41
% 2.18/1.41 thf(iffalse,definition,
% 2.18/1.41 ( iffalse
% 2.18/1.41 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ A )
% 2.18/1.41 => ( ~ Xphi
% 2.18/1.41 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.18/1.41 = Xy ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(iftrueorfalse_type,type,
% 2.18/1.41 iftrueorfalse: $o ).
% 2.18/1.41
% 2.18/1.41 thf(iftrueorfalse,definition,
% 2.18/1.41 ( iftrueorfalse
% 2.18/1.41 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ! [Xy: $i] :
% 2.18/1.41 ( ( in @ Xy @ A )
% 2.18/1.41 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(binintersectT_lem_type,type,
% 2.18/1.41 binintersectT_lem: $o ).
% 2.18/1.41
% 2.18/1.41 thf(binintersectT_lem,definition,
% 2.18/1.41 ( binintersectT_lem
% 2.18/1.41 = ( ! [A: $i,X: $i] :
% 2.18/1.41 ( ( in @ X @ ( powerset @ A ) )
% 2.18/1.41 => ! [Y: $i] :
% 2.18/1.41 ( ( in @ Y @ ( powerset @ A ) )
% 2.18/1.41 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(binunionT_lem_type,type,
% 2.18/1.41 binunionT_lem: $o ).
% 2.18/1.41
% 2.18/1.41 thf(binunionT_lem,definition,
% 2.18/1.41 ( binunionT_lem
% 2.18/1.41 = ( ! [A: $i,X: $i] :
% 2.18/1.41 ( ( in @ X @ ( powerset @ A ) )
% 2.18/1.41 => ! [Y: $i] :
% 2.18/1.41 ( ( in @ Y @ ( powerset @ A ) )
% 2.18/1.41 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(powersetT_lem_type,type,
% 2.18/1.41 powersetT_lem: $o ).
% 2.18/1.41
% 2.18/1.41 thf(powersetT_lem,definition,
% 2.18/1.41 ( powersetT_lem
% 2.18/1.41 = ( ! [A: $i,X: $i] :
% 2.18/1.41 ( ( in @ X @ ( powerset @ A ) )
% 2.18/1.41 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(setminusT_lem_type,type,
% 2.18/1.41 setminusT_lem: $o ).
% 2.18/1.41
% 2.18/1.41 thf(setminusT_lem,definition,
% 2.18/1.41 ( setminusT_lem
% 2.18/1.41 = ( ! [A: $i,X: $i] :
% 2.18/1.41 ( ( in @ X @ ( powerset @ A ) )
% 2.18/1.41 => ! [Y: $i] :
% 2.18/1.41 ( ( in @ Y @ ( powerset @ A ) )
% 2.18/1.41 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(complementT_lem_type,type,
% 2.18/1.41 complementT_lem: $o ).
% 2.18/1.41
% 2.18/1.41 thf(complementT_lem,definition,
% 2.18/1.41 ( complementT_lem
% 2.18/1.41 = ( ! [A: $i,X: $i] :
% 2.18/1.41 ( ( in @ X @ ( powerset @ A ) )
% 2.18/1.41 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(setextT_type,type,
% 2.18/1.41 setextT: $o ).
% 2.18/1.41
% 2.18/1.41 thf(setextT,definition,
% 2.18/1.41 ( setextT
% 2.18/1.41 = ( ! [A: $i,X: $i] :
% 2.18/1.41 ( ( in @ X @ ( powerset @ A ) )
% 2.18/1.41 => ! [Y: $i] :
% 2.18/1.41 ( ( in @ Y @ ( powerset @ A ) )
% 2.18/1.41 => ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ( in @ Xx @ X )
% 2.18/1.41 => ( in @ Xx @ Y ) ) )
% 2.18/1.41 => ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ( in @ Xx @ Y )
% 2.18/1.41 => ( in @ Xx @ X ) ) )
% 2.18/1.41 => ( X = Y ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(subsetTI_type,type,
% 2.18/1.41 subsetTI: $o ).
% 2.18/1.41
% 2.18/1.41 thf(subsetTI,definition,
% 2.18/1.41 ( subsetTI
% 2.18/1.41 = ( ! [A: $i,X: $i] :
% 2.18/1.41 ( ( in @ X @ ( powerset @ A ) )
% 2.18/1.41 => ! [Y: $i] :
% 2.18/1.41 ( ( in @ Y @ ( powerset @ A ) )
% 2.18/1.41 => ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ( in @ Xx @ X )
% 2.18/1.41 => ( in @ Xx @ Y ) ) )
% 2.18/1.41 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(powersetTI1_type,type,
% 2.18/1.41 powersetTI1: $o ).
% 2.18/1.41
% 2.18/1.41 thf(powersetTI1,definition,
% 2.18/1.41 ( powersetTI1
% 2.18/1.41 = ( ! [A: $i,X: $i] :
% 2.18/1.41 ( ( in @ X @ ( powerset @ A ) )
% 2.18/1.41 => ! [Y: $i] :
% 2.18/1.41 ( ( in @ Y @ ( powerset @ A ) )
% 2.18/1.41 => ( ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ( in @ Xx @ X )
% 2.18/1.41 => ( in @ Xx @ Y ) ) )
% 2.18/1.41 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(powersetTE1_type,type,
% 2.18/1.41 powersetTE1: $o ).
% 2.18/1.41
% 2.18/1.41 thf(powersetTE1,definition,
% 2.18/1.41 ( powersetTE1
% 2.18/1.41 = ( ! [A: $i,X: $i] :
% 2.18/1.41 ( ( in @ X @ ( powerset @ A ) )
% 2.18/1.41 => ! [Y: $i] :
% 2.18/1.41 ( ( in @ Y @ ( powerset @ A ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ( in @ X @ ( powerset @ Y ) )
% 2.18/1.41 => ( ( in @ Xx @ X )
% 2.18/1.41 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(complementTI1_type,type,
% 2.18/1.41 complementTI1: $o ).
% 2.18/1.41
% 2.18/1.41 thf(complementTI1,definition,
% 2.18/1.41 ( complementTI1
% 2.18/1.41 = ( ! [A: $i,X: $i] :
% 2.18/1.41 ( ( in @ X @ ( powerset @ A ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ( in @ Xx @ X )
% 2.18/1.41 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(complementTE1_type,type,
% 2.18/1.41 complementTE1: $o ).
% 2.18/1.41
% 2.18/1.41 thf(complementTE1,definition,
% 2.18/1.41 ( complementTE1
% 2.18/1.41 = ( ! [A: $i,X: $i] :
% 2.18/1.41 ( ( in @ X @ ( powerset @ A ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.18/1.41 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(binintersectTELcontra_type,type,
% 2.18/1.41 binintersectTELcontra: $o ).
% 2.18/1.41
% 2.18/1.41 thf(binintersectTELcontra,definition,
% 2.18/1.41 ( binintersectTELcontra
% 2.18/1.41 = ( ! [A: $i,X: $i] :
% 2.18/1.41 ( ( in @ X @ ( powerset @ A ) )
% 2.18/1.41 => ! [Y: $i] :
% 2.18/1.41 ( ( in @ Y @ ( powerset @ A ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ~ ( in @ Xx @ X )
% 2.18/1.41 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(binintersectTERcontra_type,type,
% 2.18/1.41 binintersectTERcontra: $o ).
% 2.18/1.41
% 2.18/1.41 thf(binintersectTERcontra,definition,
% 2.18/1.41 ( binintersectTERcontra
% 2.18/1.41 = ( ! [A: $i,X: $i] :
% 2.18/1.41 ( ( in @ X @ ( powerset @ A ) )
% 2.18/1.41 => ! [Y: $i] :
% 2.18/1.41 ( ( in @ Y @ ( powerset @ A ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ~ ( in @ Xx @ Y )
% 2.18/1.41 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(contrasubsetT_type,type,
% 2.18/1.41 contrasubsetT: $o ).
% 2.18/1.41
% 2.18/1.41 thf(contrasubsetT,definition,
% 2.18/1.41 ( contrasubsetT
% 2.18/1.41 = ( ! [A: $i,X: $i] :
% 2.18/1.41 ( ( in @ X @ ( powerset @ A ) )
% 2.18/1.41 => ! [Y: $i] :
% 2.18/1.41 ( ( in @ Y @ ( powerset @ A ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.18/1.41 => ( ( in @ Xx @ Y )
% 2.18/1.41 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.18/1.41
% 2.18/1.41 thf(contrasubsetT1,conjecture,
% 2.18/1.41 ( setextAx
% 2.18/1.41 => ( emptysetAx
% 2.18/1.41 => ( setadjoinAx
% 2.18/1.41 => ( powersetAx
% 2.18/1.41 => ( setunionAx
% 2.18/1.41 => ( omega0Ax
% 2.18/1.41 => ( omegaSAx
% 2.18/1.41 => ( omegaIndAx
% 2.18/1.41 => ( replAx
% 2.18/1.41 => ( foundationAx
% 2.18/1.41 => ( wellorderingAx
% 2.18/1.41 => ( descrp
% 2.18/1.41 => ( dsetconstrI
% 2.18/1.41 => ( dsetconstrEL
% 2.18/1.41 => ( dsetconstrER
% 2.18/1.41 => ( exuE1
% 2.18/1.41 => ( prop2setE
% 2.18/1.41 => ( emptysetE
% 2.18/1.41 => ( emptysetimpfalse
% 2.18/1.41 => ( notinemptyset
% 2.18/1.41 => ( exuE3e
% 2.18/1.41 => ( setext
% 2.18/1.41 => ( emptyI
% 2.18/1.41 => ( noeltsimpempty
% 2.18/1.41 => ( setbeta
% 2.18/1.41 => ( nonemptyE1
% 2.18/1.41 => ( nonemptyI
% 2.18/1.41 => ( nonemptyI1
% 2.18/1.41 => ( setadjoinIL
% 2.18/1.41 => ( emptyinunitempty
% 2.18/1.41 => ( setadjoinIR
% 2.18/1.41 => ( setadjoinE
% 2.18/1.41 => ( setadjoinOr
% 2.18/1.41 => ( setoftrueEq
% 2.18/1.41 => ( powersetI
% 2.18/1.41 => ( emptyinPowerset
% 2.18/1.41 => ( emptyInPowerset
% 2.18/1.41 => ( powersetE
% 2.18/1.41 => ( setunionI
% 2.18/1.41 => ( setunionE
% 2.18/1.41 => ( subPowSU
% 2.18/1.41 => ( exuE2
% 2.18/1.41 => ( nonemptyImpWitness
% 2.18/1.41 => ( uniqinunit
% 2.18/1.41 => ( notinsingleton
% 2.18/1.41 => ( eqinunit
% 2.18/1.41 => ( singletonsswitch
% 2.18/1.41 => ( upairsetE
% 2.18/1.41 => ( upairsetIL
% 2.18/1.41 => ( upairsetIR
% 2.18/1.41 => ( emptyE1
% 2.18/1.41 => ( vacuousDall
% 2.18/1.41 => ( quantDeMorgan1
% 2.18/1.41 => ( quantDeMorgan2
% 2.18/1.41 => ( quantDeMorgan3
% 2.18/1.41 => ( quantDeMorgan4
% 2.18/1.41 => ( prop2setI
% 2.18/1.41 => ( prop2set2propI
% 2.18/1.41 => ( notdexE
% 2.18/1.41 => ( notdallE
% 2.18/1.41 => ( exuI1
% 2.18/1.41 => ( exuI3
% 2.18/1.41 => ( exuI2
% 2.18/1.41 => ( inCongP
% 2.18/1.41 => ( in__Cong
% 2.18/1.41 => ( exuE3u
% 2.18/1.41 => ( exu__Cong
% 2.18/1.41 => ( emptyset__Cong
% 2.18/1.41 => ( setadjoin__Cong
% 2.18/1.41 => ( powerset__Cong
% 2.18/1.41 => ( setunion__Cong
% 2.18/1.41 => ( omega__Cong
% 2.18/1.41 => ( exuEu
% 2.18/1.41 => ( descr__Cong
% 2.18/1.41 => ( dsetconstr__Cong
% 2.18/1.41 => ( subsetI1
% 2.18/1.41 => ( eqimpsubset2
% 2.18/1.41 => ( eqimpsubset1
% 2.18/1.41 => ( subsetI2
% 2.18/1.41 => ( emptysetsubset
% 2.18/1.41 => ( subsetE
% 2.18/1.41 => ( subsetE2
% 2.18/1.41 => ( notsubsetI
% 2.18/1.41 => ( notequalI1
% 2.18/1.41 => ( notequalI2
% 2.18/1.41 => ( subsetRefl
% 2.18/1.41 => ( subsetTrans
% 2.18/1.41 => ( setadjoinSub
% 2.18/1.41 => ( setadjoinSub2
% 2.18/1.41 => ( subset2powerset
% 2.18/1.41 => ( setextsub
% 2.18/1.41 => ( subsetemptysetimpeq
% 2.18/1.41 => ( powersetI1
% 2.18/1.41 => ( powersetE1
% 2.18/1.41 => ( inPowerset
% 2.18/1.41 => ( powersetsubset
% 2.18/1.41 => ( sepInPowerset
% 2.18/1.41 => ( sepSubset
% 2.18/1.41 => ( binunionIL
% 2.18/1.41 => ( upairset2IR
% 2.18/1.41 => ( binunionIR
% 2.18/1.41 => ( binunionEcases
% 2.18/1.41 => ( binunionE
% 2.18/1.41 => ( binunionLsub
% 2.18/1.41 => ( binunionRsub
% 2.18/1.41 => ( binintersectI
% 2.18/1.41 => ( binintersectSubset5
% 2.18/1.41 => ( binintersectEL
% 2.18/1.41 => ( binintersectLsub
% 2.18/1.41 => ( binintersectSubset2
% 2.18/1.41 => ( binintersectSubset3
% 2.18/1.41 => ( binintersectER
% 2.18/1.41 => ( disjointsetsI1
% 2.18/1.41 => ( binintersectRsub
% 2.18/1.41 => ( binintersectSubset4
% 2.18/1.41 => ( binintersectSubset1
% 2.18/1.41 => ( bs114d
% 2.18/1.41 => ( setminusI
% 2.18/1.41 => ( setminusEL
% 2.18/1.41 => ( setminusER
% 2.18/1.41 => ( setminusSubset2
% 2.18/1.41 => ( setminusERneg
% 2.18/1.41 => ( setminusELneg
% 2.18/1.41 => ( setminusILneg
% 2.18/1.41 => ( setminusIRneg
% 2.18/1.41 => ( setminusLsub
% 2.18/1.41 => ( setminusSubset1
% 2.18/1.41 => ( symdiffE
% 2.18/1.41 => ( symdiffI1
% 2.18/1.41 => ( symdiffI2
% 2.18/1.41 => ( symdiffIneg1
% 2.18/1.41 => ( symdiffIneg2
% 2.18/1.41 => ( secondinupair
% 2.18/1.41 => ( setukpairIL
% 2.18/1.41 => ( setukpairIR
% 2.18/1.41 => ( kpairiskpair
% 2.18/1.41 => ( kpairp
% 2.18/1.41 => ( singletonsubset
% 2.18/1.41 => ( singletoninpowerset
% 2.18/1.41 => ( singletoninpowunion
% 2.18/1.41 => ( upairset2E
% 2.18/1.41 => ( upairsubunion
% 2.18/1.41 => ( upairinpowunion
% 2.18/1.41 => ( ubforcartprodlem1
% 2.18/1.41 => ( ubforcartprodlem2
% 2.18/1.41 => ( ubforcartprodlem3
% 2.18/1.41 => ( cartprodpairin
% 2.18/1.41 => ( cartprodmempair1
% 2.18/1.41 => ( cartprodmempair
% 2.18/1.41 => ( setunionE2
% 2.18/1.41 => ( setunionsingleton1
% 2.18/1.41 => ( setunionsingleton2
% 2.18/1.41 => ( setunionsingleton
% 2.18/1.41 => ( singletonprop
% 2.18/1.41 => ( ex1E1
% 2.18/1.41 => ( ex1I
% 2.18/1.41 => ( ex1I2
% 2.18/1.41 => ( singletonsuniq
% 2.18/1.41 => ( setukpairinjL1
% 2.18/1.41 => ( kfstsingleton
% 2.18/1.41 => ( theprop
% 2.18/1.41 => ( kfstpairEq
% 2.18/1.41 => ( cartprodfstin
% 2.18/1.41 => ( setukpairinjL2
% 2.18/1.41 => ( setukpairinjL
% 2.18/1.41 => ( setukpairinjR11
% 2.18/1.41 => ( setukpairinjR12
% 2.18/1.41 => ( setukpairinjR1
% 2.18/1.41 => ( upairequniteq
% 2.18/1.41 => ( setukpairinjR2
% 2.18/1.41 => ( setukpairinjR
% 2.18/1.41 => ( ksndsingleton
% 2.18/1.41 => ( ksndpairEq
% 2.18/1.41 => ( kpairsurjEq
% 2.18/1.41 => ( cartprodsndin
% 2.18/1.41 => ( cartprodpairmemEL
% 2.18/1.41 => ( cartprodpairmemER
% 2.18/1.41 => ( cartprodmempaircEq
% 2.18/1.41 => ( cartprodfstpairEq
% 2.18/1.41 => ( cartprodsndpairEq
% 2.18/1.41 => ( cartprodpairsurjEq
% 2.18/1.41 => ( dpsetconstrI
% 2.18/1.41 => ( dpsetconstrSub
% 2.18/1.41 => ( setOfPairsIsBReln
% 2.18/1.41 => ( dpsetconstrERa
% 2.18/1.41 => ( dpsetconstrEL1
% 2.18/1.41 => ( dpsetconstrEL2
% 2.18/1.41 => ( dpsetconstrER
% 2.18/1.41 => ( funcImageSingleton
% 2.18/1.41 => ( apProp
% 2.18/1.41 => ( app
% 2.18/1.41 => ( infuncsetfunc
% 2.18/1.41 => ( ap2p
% 2.18/1.41 => ( funcinfuncset
% 2.18/1.41 => ( lamProp
% 2.18/1.41 => ( lamp
% 2.18/1.41 => ( lam2p
% 2.18/1.41 => ( brelnall1
% 2.18/1.41 => ( brelnall2
% 2.18/1.41 => ( ex1E2
% 2.18/1.41 => ( funcGraphProp1
% 2.18/1.41 => ( funcGraphProp3
% 2.18/1.41 => ( funcGraphProp2
% 2.18/1.41 => ( funcextLem
% 2.18/1.41 => ( funcGraphProp4
% 2.18/1.41 => ( subbreln
% 2.18/1.41 => ( eqbreln
% 2.18/1.41 => ( funcext
% 2.18/1.41 => ( funcext2
% 2.18/1.41 => ( ap2apEq1
% 2.18/1.41 => ( ap2apEq2
% 2.18/1.41 => ( beta1
% 2.18/1.41 => ( eta1
% 2.18/1.41 => ( lam2lamEq
% 2.18/1.41 => ( beta2
% 2.18/1.41 => ( eta2
% 2.18/1.41 => ( iffalseProp1
% 2.18/1.41 => ( iffalseProp2
% 2.18/1.41 => ( iftrueProp1
% 2.18/1.41 => ( iftrueProp2
% 2.18/1.41 => ( ifSingleton
% 2.18/1.41 => ( ifp
% 2.18/1.41 => ( theeq
% 2.18/1.41 => ( iftrue
% 2.18/1.41 => ( iffalse
% 2.18/1.41 => ( iftrueorfalse
% 2.18/1.41 => ( binintersectT_lem
% 2.18/1.41 => ( binunionT_lem
% 2.18/1.41 => ( powersetT_lem
% 2.18/1.41 => ( setminusT_lem
% 2.18/1.41 => ( complementT_lem
% 2.18/1.41 => ( setextT
% 2.18/1.41 => ( subsetTI
% 2.18/1.41 => ( powersetTI1
% 2.18/1.41 => ( powersetTE1
% 2.18/1.41 => ( complementTI1
% 2.18/1.41 => ( complementTE1
% 2.18/1.41 => ( binintersectTELcontra
% 2.18/1.41 => ( binintersectTERcontra
% 2.18/1.41 => ( contrasubsetT
% 2.18/1.41 => ! [A: $i,X: $i] :
% 2.18/1.41 ( ( in @ X @ ( powerset @ A ) )
% 2.18/1.41 => ! [Y: $i] :
% 2.18/1.41 ( ( in @ Y @ ( powerset @ A ) )
% 2.18/1.41 => ! [Xx: $i] :
% 2.18/1.41 ( ( in @ Xx @ A )
% 2.18/1.41 => ( ( subset @ X @ Y )
% 2.18/1.41 => ( ~ ( in @ Xx @ Y )
% 2.18/1.41 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.18/1.41 ------------------------
% 2.18/1.43 % (26600)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_26337 for (2999ds/18Mi)
% 2.18/1.43 % (26595)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_26337 for (2999ds/4Mi)
% 2.18/1.43 % (26595)Instruction limit reached!
% 2.18/1.43 % (26595)------------------------------
% 2.18/1.43 % (26595)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.43 % (26595)Termination reason: Unknown
% 2.18/1.43 % (26595)Termination phase: shuffling
% 2.18/1.43
% 2.18/1.43 % (26595)Memory used [KB]: 1535
% 2.18/1.43 % (26595)Time elapsed: 0.003 s
% 2.18/1.43 % (26595)Instructions burned: 7 (million)
% 2.18/1.43 % (26595)------------------------------
% 2.18/1.43 % (26595)------------------------------
% 2.18/1.43 % (26594)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_26337 for (2999ds/183Mi)
% 2.18/1.43 % (26597)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_26337 for (2999ds/2Mi)
% 2.18/1.44 % (26597)Instruction limit reached!
% 2.18/1.44 % (26597)------------------------------
% 2.18/1.44 % (26597)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.44 % (26597)Termination reason: Unknown
% 2.18/1.44 % (26597)Termination phase: shuffling
% 2.18/1.44
% 2.18/1.44 % (26597)Memory used [KB]: 1407
% 2.18/1.44 % (26597)Time elapsed: 0.003 s
% 2.18/1.44 % (26597)Instructions burned: 3 (million)
% 2.18/1.44 % (26597)------------------------------
% 2.18/1.44 % (26597)------------------------------
% 2.18/1.44 % (26600)Instruction limit reached!
% 2.18/1.44 % (26600)------------------------------
% 2.18/1.44 % (26600)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.44 % (26600)Termination reason: Unknown
% 2.18/1.44 % (26600)Termination phase: shuffling
% 2.18/1.44
% 2.18/1.44 % (26600)Memory used [KB]: 1791
% 2.18/1.44 % (26600)Time elapsed: 0.008 s
% 2.18/1.44 % (26600)Instructions burned: 20 (million)
% 2.18/1.44 % (26600)------------------------------
% 2.18/1.44 % (26600)------------------------------
% 2.18/1.44 % (26596)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_26337 for (2999ds/27Mi)
% 2.18/1.44 % (26598)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_26337 for (2999ds/2Mi)
% 2.18/1.44 % (26599)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_26337 for (2999ds/275Mi)
% 2.18/1.44 % (26598)Instruction limit reached!
% 2.18/1.44 % (26598)------------------------------
% 2.18/1.44 % (26598)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.44 % (26598)Termination reason: Unknown
% 2.18/1.44 % (26598)Termination phase: shuffling
% 2.18/1.44
% 2.18/1.44 % (26598)Memory used [KB]: 1407
% 2.18/1.44 % (26598)Time elapsed: 0.003 s
% 2.18/1.44 % (26598)Instructions burned: 3 (million)
% 2.18/1.44 % (26598)------------------------------
% 2.18/1.44 % (26598)------------------------------
% 2.18/1.44 % (26602)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_26337 for (2999ds/3Mi)
% 2.18/1.44 % (26602)Instruction limit reached!
% 2.18/1.44 % (26602)------------------------------
% 2.18/1.44 % (26602)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.44 % (26602)Termination reason: Unknown
% 2.18/1.44 % (26602)Termination phase: shuffling
% 2.18/1.44
% 2.18/1.44 % (26602)Memory used [KB]: 1407
% 2.18/1.44 % (26602)Time elapsed: 0.002 s
% 2.18/1.44 % (26602)Instructions burned: 3 (million)
% 2.18/1.44 % (26602)------------------------------
% 2.18/1.44 % (26602)------------------------------
% 2.18/1.45 % (26607)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_26337 for (2999ds/15Mi)
% 2.18/1.45 % (26607)Instruction limit reached!
% 2.18/1.45 % (26607)------------------------------
% 2.18/1.45 % (26607)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.45 % (26607)Termination reason: Unknown
% 2.18/1.45 % (26607)Termination phase: shuffling
% 2.18/1.45
% 2.18/1.45 % (26607)Memory used [KB]: 1663
% 2.18/1.45 % (26607)Time elapsed: 0.006 s
% 2.18/1.45 % (26607)Instructions burned: 15 (million)
% 2.18/1.45 % (26607)------------------------------
% 2.18/1.45 % (26607)------------------------------
% 2.18/1.45 % (26606)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_26337 for (2999ds/37Mi)
% 2.18/1.45 % (26611)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_26337 for (2999ds/1041Mi)
% 2.18/1.45 % (26596)Instruction limit reached!
% 2.18/1.45 % (26596)------------------------------
% 2.18/1.45 % (26596)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.45 % (26596)Termination reason: Unknown
% 2.18/1.45 % (26596)Termination phase: shuffling
% 2.18/1.45
% 2.18/1.45 % (26596)Memory used [KB]: 1918
% 2.18/1.45 % (26596)Time elapsed: 0.016 s
% 2.18/1.45 % (26596)Instructions burned: 28 (million)
% 2.18/1.45 % (26596)------------------------------
% 2.18/1.45 % (26596)------------------------------
% 2.18/1.46 % (26609)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_26337 for (2999ds/3Mi)
% 2.18/1.46 % (26609)Instruction limit reached!
% 2.18/1.46 % (26609)------------------------------
% 2.18/1.46 % (26609)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.46 % (26609)Termination reason: Unknown
% 2.18/1.46 % (26609)Termination phase: shuffling
% 2.18/1.46
% 2.18/1.46 % (26609)Memory used [KB]: 1407
% 2.18/1.46 % (26609)Time elapsed: 0.004 s
% 2.18/1.46 % (26609)Instructions burned: 3 (million)
% 2.18/1.46 % (26609)------------------------------
% 2.18/1.46 % (26609)------------------------------
% 2.18/1.46 % (26616)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_26337 for (2999ds/7Mi)
% 2.18/1.46 % (26616)Instruction limit reached!
% 2.18/1.46 % (26616)------------------------------
% 2.18/1.46 % (26616)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.46 % (26616)Termination reason: Unknown
% 2.18/1.46 % (26616)Termination phase: shuffling
% 2.18/1.46
% 2.18/1.46 % (26616)Memory used [KB]: 1535
% 2.18/1.46 % (26616)Time elapsed: 0.003 s
% 2.18/1.46 % (26616)Instructions burned: 7 (million)
% 2.18/1.46 % (26616)------------------------------
% 2.18/1.46 % (26616)------------------------------
% 2.18/1.46 % (26606)Instruction limit reached!
% 2.18/1.46 % (26606)------------------------------
% 2.18/1.46 % (26606)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.46 % (26606)Termination reason: Unknown
% 2.18/1.46 % (26606)Termination phase: shuffling
% 2.18/1.46
% 2.18/1.46 % (26606)Memory used [KB]: 2174
% 2.18/1.46 % (26606)Time elapsed: 0.013 s
% 2.18/1.46 % (26606)Instructions burned: 37 (million)
% 2.18/1.46 % (26606)------------------------------
% 2.18/1.46 % (26606)------------------------------
% 2.18/1.47 % (26618)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_26337 for (2999ds/16Mi)
% 2.18/1.47 % (26622)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_26337 for (2999ds/3Mi)
% 2.18/1.47 % (26623)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_26337 for (2999ds/7Mi)
% 2.18/1.47 % (26622)Instruction limit reached!
% 2.18/1.47 % (26622)------------------------------
% 2.18/1.47 % (26622)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.47 % (26622)Termination reason: Unknown
% 2.18/1.47 % (26622)Termination phase: shuffling
% 2.18/1.47
% 2.18/1.47 % (26622)Memory used [KB]: 1535
% 2.18/1.47 % (26622)Time elapsed: 0.003 s
% 2.18/1.47 % (26622)Instructions burned: 6 (million)
% 2.18/1.47 % (26622)------------------------------
% 2.18/1.47 % (26622)------------------------------
% 2.18/1.47 % (26623)Instruction limit reached!
% 2.18/1.47 % (26623)------------------------------
% 2.18/1.47 % (26623)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.47 % (26623)Termination reason: Unknown
% 2.18/1.47 % (26623)Termination phase: shuffling
% 2.18/1.47
% 2.18/1.47 % (26623)Memory used [KB]: 1535
% 2.18/1.47 % (26623)Time elapsed: 0.004 s
% 2.18/1.47 % (26623)Instructions burned: 10 (million)
% 2.18/1.47 % (26623)------------------------------
% 2.18/1.47 % (26623)------------------------------
% 2.18/1.48 % (26620)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_26337 for (2999ds/3Mi)
% 2.18/1.48 % (26620)Instruction limit reached!
% 2.18/1.48 % (26620)------------------------------
% 2.18/1.48 % (26620)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.48 % (26620)Termination reason: Unknown
% 2.18/1.48 % (26620)Termination phase: shuffling
% 2.18/1.48
% 2.18/1.48 % (26620)Memory used [KB]: 1407
% 2.18/1.48 % (26620)Time elapsed: 0.004 s
% 2.18/1.48 % (26620)Instructions burned: 4 (million)
% 2.18/1.48 % (26620)------------------------------
% 2.18/1.48 % (26620)------------------------------
% 2.18/1.48 % (26618)Instruction limit reached!
% 2.18/1.48 % (26618)------------------------------
% 2.18/1.48 % (26618)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.48 % (26618)Termination reason: Unknown
% 2.18/1.48 % (26618)Termination phase: shuffling
% 2.18/1.48
% 2.18/1.48 % (26618)Memory used [KB]: 1791
% 2.18/1.48 % (26618)Time elapsed: 0.011 s
% 2.18/1.48 % (26618)Instructions burned: 17 (million)
% 2.18/1.48 % (26618)------------------------------
% 2.18/1.48 % (26618)------------------------------
% 2.18/1.48 % (26627)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_26337 for (2999ds/3Mi)
% 2.18/1.48 % (26627)Instruction limit reached!
% 2.18/1.48 % (26627)------------------------------
% 2.18/1.48 % (26627)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.48 % (26627)Termination reason: Unknown
% 2.18/1.48 % (26627)Termination phase: shuffling
% 2.18/1.48
% 2.18/1.48 % (26627)Memory used [KB]: 1407
% 2.18/1.48 % (26627)Time elapsed: 0.002 s
% 2.18/1.48 % (26627)Instructions burned: 3 (million)
% 2.18/1.48 % (26627)------------------------------
% 2.18/1.48 % (26627)------------------------------
% 2.18/1.48 % (26629)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_26337 for (2999ds/4Mi)
% 2.18/1.49 % (26629)Instruction limit reached!
% 2.18/1.49 % (26629)------------------------------
% 2.18/1.49 % (26629)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.49 % (26629)Termination reason: Unknown
% 2.18/1.49 % (26629)Termination phase: shuffling
% 2.18/1.49
% 2.18/1.49 % (26629)Memory used [KB]: 1535
% 2.18/1.49 % (26629)Time elapsed: 0.003 s
% 2.18/1.49 % (26629)Instructions burned: 7 (million)
% 2.18/1.49 % (26629)------------------------------
% 2.18/1.49 % (26629)------------------------------
% 2.18/1.49 % (26631)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_26337 for (2999ds/18Mi)
% 2.18/1.49 % (26594)Instruction limit reached!
% 2.18/1.49 % (26594)------------------------------
% 2.18/1.49 % (26594)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.49 % (26594)Termination reason: Unknown
% 2.18/1.49 % (26594)Termination phase: Saturation
% 2.18/1.49
% 2.18/1.49 % (26594)Memory used [KB]: 7547
% 2.18/1.49 % (26594)Time elapsed: 0.059 s
% 2.18/1.49 % (26594)Instructions burned: 185 (million)
% 2.18/1.49 % (26594)------------------------------
% 2.18/1.49 % (26594)------------------------------
% 2.18/1.49 % (26634)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_26337 for (2999ds/6Mi)
% 2.18/1.49 % (26634)Instruction limit reached!
% 2.18/1.49 % (26634)------------------------------
% 2.18/1.49 % (26634)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.49 % (26634)Termination reason: Unknown
% 2.18/1.49 % (26634)Termination phase: shuffling
% 2.18/1.49
% 2.18/1.49 % (26634)Memory used [KB]: 1535
% 2.18/1.49 % (26634)Time elapsed: 0.003 s
% 2.18/1.49 % (26634)Instructions burned: 7 (million)
% 2.18/1.49 % (26634)------------------------------
% 2.18/1.49 % (26634)------------------------------
% 2.18/1.50 % (26631)Instruction limit reached!
% 2.18/1.50 % (26631)------------------------------
% 2.18/1.50 % (26631)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.18/1.50 % (26631)Termination reason: Unknown
% 2.18/1.50 % (26631)Termination phase: shuffling
% 2.18/1.50
% 2.18/1.50 % (26631)Memory used [KB]: 1791
% 2.18/1.50 % (26631)Time elapsed: 0.006 s
% 2.18/1.50 % (26631)Instructions burned: 18 (million)
% 2.18/1.50 % (26631)------------------------------
% 2.18/1.50 % (26631)------------------------------
% 2.18/1.50 % (26632)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_26337 for (2999ds/710Mi)
% 2.18/1.50 % (26636)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_26337 for (2999ds/902Mi)
% 2.23/1.50 % (26639)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_26337 for (2999ds/21Mi)
% 2.23/1.50 % (26640)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_26337 for (2999ds/5Mi)
% 2.23/1.50 % (26641)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_26337 for (2999ds/6Mi)
% 2.23/1.51 % (26640)Instruction limit reached!
% 2.23/1.51 % (26640)------------------------------
% 2.23/1.51 % (26640)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.51 % (26640)Termination reason: Unknown
% 2.23/1.51 % (26640)Termination phase: shuffling
% 2.23/1.51
% 2.23/1.51 % (26640)Memory used [KB]: 1535
% 2.23/1.51 % (26640)Time elapsed: 0.003 s
% 2.23/1.51 % (26640)Instructions burned: 6 (million)
% 2.23/1.51 % (26640)------------------------------
% 2.23/1.51 % (26640)------------------------------
% 2.23/1.51 % (26641)Instruction limit reached!
% 2.23/1.51 % (26641)------------------------------
% 2.23/1.51 % (26641)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.51 % (26641)Termination reason: Unknown
% 2.23/1.51 % (26641)Termination phase: shuffling
% 2.23/1.51
% 2.23/1.51 % (26641)Memory used [KB]: 1535
% 2.23/1.51 % (26641)Time elapsed: 0.004 s
% 2.23/1.51 % (26641)Instructions burned: 9 (million)
% 2.23/1.51 % (26641)------------------------------
% 2.23/1.51 % (26641)------------------------------
% 2.23/1.51 % (26639)Instruction limit reached!
% 2.23/1.51 % (26639)------------------------------
% 2.23/1.51 % (26639)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.51 % (26639)Termination reason: Unknown
% 2.23/1.51 % (26639)Termination phase: shuffling
% 2.23/1.51
% 2.23/1.51 % (26639)Memory used [KB]: 1918
% 2.23/1.51 % (26639)Time elapsed: 0.008 s
% 2.23/1.51 % (26639)Instructions burned: 23 (million)
% 2.23/1.51 % (26639)------------------------------
% 2.23/1.51 % (26639)------------------------------
% 2.23/1.52 % (26645)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_26337 for (2999ds/377Mi)
% 2.23/1.52 % (26646)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_26337 for (2999ds/779Mi)
% 2.23/1.52 % (26648)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_26337 for (2999ds/19Mi)
% 2.23/1.52 % (26648)Instruction limit reached!
% 2.23/1.52 % (26648)------------------------------
% 2.23/1.52 % (26648)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.52 % (26648)Termination reason: Unknown
% 2.23/1.52 % (26648)Termination phase: shuffling
% 2.23/1.52
% 2.23/1.52 % (26648)Memory used [KB]: 1791
% 2.23/1.52 % (26648)Time elapsed: 0.007 s
% 2.23/1.52 % (26648)Instructions burned: 22 (million)
% 2.23/1.52 % (26648)------------------------------
% 2.23/1.52 % (26648)------------------------------
% 2.23/1.53 % (26599)Instruction limit reached!
% 2.23/1.53 % (26599)------------------------------
% 2.23/1.53 % (26599)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.53 % (26599)Termination reason: Unknown
% 2.23/1.53 % (26599)Termination phase: Saturation
% 2.23/1.53
% 2.23/1.53 % (26599)Memory used [KB]: 9466
% 2.23/1.53 % (26599)Time elapsed: 0.093 s
% 2.23/1.53 % (26599)Instructions burned: 276 (million)
% 2.23/1.53 % (26599)------------------------------
% 2.23/1.53 % (26599)------------------------------
% 2.23/1.53 % (26656)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_26337 for (2999ds/879Mi)
% 2.23/1.54 % (26661)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_26337 for (2998ds/17Mi)
% 2.23/1.54 % (26661)Instruction limit reached!
% 2.23/1.54 % (26661)------------------------------
% 2.23/1.54 % (26661)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.54 % (26661)Termination reason: Unknown
% 2.23/1.54 % (26661)Termination phase: shuffling
% 2.23/1.54
% 2.23/1.54 % (26661)Memory used [KB]: 1791
% 2.23/1.54 % (26661)Time elapsed: 0.006 s
% 2.23/1.54 % (26661)Instructions burned: 17 (million)
% 2.23/1.54 % (26661)------------------------------
% 2.23/1.54 % (26661)------------------------------
% 2.23/1.55 % (26665)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_26337 for (2998ds/3Mi)
% 2.23/1.55 % (26665)Instruction limit reached!
% 2.23/1.55 % (26665)------------------------------
% 2.23/1.55 % (26665)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.55 % (26665)Termination reason: Unknown
% 2.23/1.55 % (26665)Termination phase: shuffling
% 2.23/1.55
% 2.23/1.55 % (26665)Memory used [KB]: 1407
% 2.23/1.55 % (26665)Time elapsed: 0.002 s
% 2.23/1.55 % (26665)Instructions burned: 3 (million)
% 2.23/1.55 % (26665)------------------------------
% 2.23/1.55 % (26665)------------------------------
% 2.23/1.56 % (26668)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_26337 for (2998ds/30Mi)
% 2.23/1.57 % (26668)Instruction limit reached!
% 2.23/1.57 % (26668)------------------------------
% 2.23/1.57 % (26668)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.57 % (26668)Termination reason: Unknown
% 2.23/1.57 % (26668)Termination phase: shuffling
% 2.23/1.57
% 2.23/1.57 % (26668)Memory used [KB]: 2046
% 2.23/1.57 % (26668)Time elapsed: 0.010 s
% 2.23/1.57 % (26668)Instructions burned: 33 (million)
% 2.23/1.57 % (26668)------------------------------
% 2.23/1.57 % (26668)------------------------------
% 2.23/1.58 % (26682)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_26337 for (2998ds/127Mi)
% 2.23/1.61 % (26682)Instruction limit reached!
% 2.23/1.61 % (26682)------------------------------
% 2.23/1.61 % (26682)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.61 % (26682)Termination reason: Unknown
% 2.23/1.61 % (26682)Termination phase: Property scanning
% 2.23/1.61
% 2.23/1.61 % (26682)Memory used [KB]: 2558
% 2.23/1.61 % (26682)Time elapsed: 0.034 s
% 2.23/1.61 % (26682)Instructions burned: 130 (million)
% 2.23/1.61 % (26682)------------------------------
% 2.23/1.61 % (26682)------------------------------
% 2.23/1.62 % (26645)Instruction limit reached!
% 2.23/1.62 % (26645)------------------------------
% 2.23/1.62 % (26645)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.62 % (26645)Termination reason: Unknown
% 2.23/1.62 % (26645)Termination phase: Saturation
% 2.23/1.62
% 2.23/1.62 % (26645)Memory used [KB]: 9338
% 2.23/1.62 % (26645)Time elapsed: 0.106 s
% 2.23/1.62 % (26645)Instructions burned: 378 (million)
% 2.23/1.62 % (26645)------------------------------
% 2.23/1.62 % (26645)------------------------------
% 2.23/1.62 % (26706)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_26337 for (2997ds/100Mi)
% 2.23/1.63 % (26710)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_26337 for (2997ds/3Mi)
% 2.23/1.63 % (26710)Instruction limit reached!
% 2.23/1.63 % (26710)------------------------------
% 2.23/1.63 % (26710)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.63 % (26710)Termination reason: Unknown
% 2.23/1.63 % (26710)Termination phase: shuffling
% 2.23/1.63
% 2.23/1.63 % (26710)Memory used [KB]: 1407
% 2.23/1.63 % (26710)Time elapsed: 0.002 s
% 2.23/1.63 % (26710)Instructions burned: 3 (million)
% 2.23/1.63 % (26710)------------------------------
% 2.23/1.63 % (26710)------------------------------
% 2.23/1.64 % (26711)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_26337 for (2997ds/20Mi)
% 2.23/1.65 % (26711)Instruction limit reached!
% 2.23/1.65 % (26711)------------------------------
% 2.23/1.65 % (26711)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.65 % (26711)Termination reason: Unknown
% 2.23/1.65 % (26711)Termination phase: shuffling
% 2.23/1.65
% 2.23/1.65 % (26711)Memory used [KB]: 1791
% 2.23/1.65 % (26711)Time elapsed: 0.008 s
% 2.23/1.65 % (26711)Instructions burned: 22 (million)
% 2.23/1.65 % (26711)------------------------------
% 2.23/1.65 % (26711)------------------------------
% 2.23/1.65 % (26706)Instruction limit reached!
% 2.23/1.65 % (26706)------------------------------
% 2.23/1.65 % (26706)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.65 % (26706)Termination reason: Unknown
% 2.23/1.65 % (26706)Termination phase: Function definition elimination
% 2.23/1.65
% 2.23/1.65 % (26706)Memory used [KB]: 2558
% 2.23/1.65 % (26706)Time elapsed: 0.027 s
% 2.23/1.65 % (26706)Instructions burned: 101 (million)
% 2.23/1.65 % (26706)------------------------------
% 2.23/1.65 % (26706)------------------------------
% 2.23/1.66 % (26712)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_26337 for (2997ds/86Mi)
% 2.23/1.66 % (26713)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_26337 for (2997ds/107Mi)
% 2.23/1.68 % (26712)Instruction limit reached!
% 2.23/1.68 % (26712)------------------------------
% 2.23/1.68 % (26712)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.68 % (26712)Termination reason: Unknown
% 2.23/1.68 % (26712)Termination phase: Property scanning
% 2.23/1.68
% 2.23/1.68 % (26712)Memory used [KB]: 2430
% 2.23/1.68 % (26712)Time elapsed: 0.024 s
% 2.23/1.68 % (26712)Instructions burned: 89 (million)
% 2.23/1.68 % (26712)------------------------------
% 2.23/1.68 % (26712)------------------------------
% 2.23/1.69 % (26713)Instruction limit reached!
% 2.23/1.69 % (26713)------------------------------
% 2.23/1.69 % (26713)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.69 % (26713)Termination reason: Unknown
% 2.23/1.69 % (26713)Termination phase: Property scanning
% 2.23/1.69
% 2.23/1.69 % (26713)Memory used [KB]: 2430
% 2.23/1.69 % (26713)Time elapsed: 0.028 s
% 2.23/1.69 % (26713)Instructions burned: 107 (million)
% 2.23/1.69 % (26713)------------------------------
% 2.23/1.69 % (26713)------------------------------
% 2.23/1.69 % (26714)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_26337 for (2997ds/39Mi)
% 2.23/1.70 % (26715)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_26337 for (2997ds/448Mi)
% 2.23/1.70 % (26714)Instruction limit reached!
% 2.23/1.70 % (26714)------------------------------
% 2.23/1.70 % (26714)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.70 % (26714)Termination reason: Unknown
% 2.23/1.70 % (26714)Termination phase: shuffling
% 2.23/1.70
% 2.23/1.70 % (26714)Memory used [KB]: 2302
% 2.23/1.70 % (26714)Time elapsed: 0.012 s
% 2.23/1.70 % (26714)Instructions burned: 42 (million)
% 2.23/1.70 % (26714)------------------------------
% 2.23/1.70 % (26714)------------------------------
% 2.23/1.71 % (26716)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_26337 for (2997ds/46Mi)
% 2.23/1.71 % (26646)Instruction limit reached!
% 2.23/1.71 % (26646)------------------------------
% 2.23/1.71 % (26646)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.71 % (26646)Termination reason: Unknown
% 2.23/1.71 % (26646)Termination phase: Saturation
% 2.23/1.71
% 2.23/1.71 % (26646)Memory used [KB]: 7419
% 2.23/1.71 % (26646)Time elapsed: 0.195 s
% 2.23/1.71 % (26646)Instructions burned: 780 (million)
% 2.23/1.71 % (26646)------------------------------
% 2.23/1.71 % (26646)------------------------------
% 2.23/1.71 % (26636)First to succeed.
% 2.23/1.72 % (26636)Refutation found. Thanks to Tanya!
% 2.23/1.72 % SZS status Theorem for DTF2THF_26337
% 2.23/1.72 % SZS output start Proof for DTF2THF_26337
% 2.23/1.72 thf(func_def_0, type, in: $i > $i > $o).
% 2.23/1.72 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.23/1.72 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.23/1.72 thf(func_def_8, type, powerset: $i > $i).
% 2.23/1.72 thf(func_def_10, type, setunion: $i > $i).
% 2.23/1.72 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.23/1.72 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.23/1.72 thf(func_def_26, type, prop2set: $o > $i).
% 2.23/1.72 thf(func_def_36, type, nonempty: $i > $o).
% 2.23/1.72 thf(func_def_69, type, set2prop: $i > $o).
% 2.23/1.72 thf(func_def_88, type, subset: $i > $i > $o).
% 2.23/1.72 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.23/1.72 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.23/1.72 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.23/1.72 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.23/1.72 thf(func_def_135, type, regular: $i > $o).
% 2.23/1.72 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.23/1.72 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.23/1.72 thf(func_def_153, type, iskpair: $i > $o).
% 2.23/1.72 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.23/1.72 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.23/1.72 thf(func_def_177, type, singleton: $i > $o).
% 2.23/1.72 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.23/1.72 thf(func_def_184, type, atmost1p: $i > $o).
% 2.23/1.72 thf(func_def_185, type, atleast2p: $i > $o).
% 2.23/1.72 thf(func_def_186, type, atmost2p: $i > $o).
% 2.23/1.72 thf(func_def_187, type, upairsetp: $i > $o).
% 2.23/1.72 thf(func_def_191, type, kfst: $i > $i).
% 2.23/1.72 thf(func_def_203, type, ksnd: $i > $i).
% 2.23/1.72 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.23/1.72 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.23/1.72 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.23/1.72 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.23/1.72 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.23/1.72 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.23/1.72 thf(func_def_292, type, sP0: $i > $i > $i > $o > $o).
% 2.23/1.72 thf(func_def_296, type, sP4: $i > $i > $o).
% 2.23/1.72 thf(func_def_297, type, sP5: $i > $o).
% 2.23/1.72 thf(func_def_298, type, sP6: $i > $i > $o).
% 2.23/1.72 thf(func_def_299, type, sP7: $i > $i > $o).
% 2.23/1.72 thf(func_def_301, type, sP9: $i > $i > $i > $o).
% 2.23/1.72 thf(func_def_302, type, sP10: $i > $i > $i > $i > $o).
% 2.23/1.72 thf(func_def_310, type, sK18: $i > $o).
% 2.23/1.72 thf(func_def_311, type, sK19: $i > $o).
% 2.23/1.72 thf(func_def_312, type, sK20: ($i > $o) > ($i > $o) > $i).
% 2.23/1.72 thf(func_def_313, type, sK21: ($i > $o) > ($i > $o) > $i).
% 2.23/1.72 thf(func_def_321, type, sK29: ($i > $i) > $i > $i > $i).
% 2.23/1.72 thf(func_def_324, type, sK32: $i > $i).
% 2.23/1.72 thf(func_def_356, type, sK64: $i > $o).
% 2.23/1.72 thf(func_def_358, type, sK66: $i > $i > $i).
% 2.23/1.72 thf(func_def_378, type, sK86: $i > $o).
% 2.23/1.72 thf(func_def_380, type, sK88: $i > ($i > $o) > $i).
% 2.23/1.72 thf(func_def_382, type, sK90: $i > $o).
% 2.23/1.72 thf(func_def_384, type, sK92: ($i > $o) > $i > $i).
% 2.23/1.72 thf(func_def_392, type, sK100: $i > $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_396, type, sK104: $i > $o).
% 2.23/1.72 thf(func_def_402, type, sK110: $i > $o).
% 2.23/1.72 thf(func_def_411, type, sK119: $i > $o).
% 2.23/1.72 thf(func_def_413, type, sK121: ($i > $o) > $i).
% 2.23/1.72 thf(func_def_414, type, sK122: ($i > $o) > $i).
% 2.23/1.72 thf(func_def_427, type, sK135: $i > $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_428, type, sK136: $i > $i > $o).
% 2.23/1.72 thf(func_def_430, type, sK138: $i > $i).
% 2.23/1.72 thf(func_def_431, type, sK139: $i > $i).
% 2.23/1.72 thf(func_def_432, type, sK140: $i > ($i > $i > $o) > $i).
% 2.23/1.72 thf(func_def_433, type, sK141: $i > $i > ($i > $i > $o) > $i).
% 2.23/1.72 thf(func_def_434, type, sK142: $i > ($i > $i > $o) > $i).
% 2.23/1.72 thf(func_def_448, type, sK156: $i > $o).
% 2.23/1.72 thf(func_def_455, type, sK163: $i > $o).
% 2.23/1.72 thf(func_def_456, type, sK164: $i > $o).
% 2.23/1.72 thf(func_def_457, type, sK165: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.23/1.72 thf(func_def_458, type, sK166: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.23/1.72 thf(func_def_464, type, sK172: $i > $o).
% 2.23/1.72 thf(func_def_465, type, sK173: ($i > $o) > $i > $i).
% 2.23/1.72 thf(func_def_474, type, sK182: $i > $i > $o).
% 2.23/1.72 thf(func_def_479, type, sK187: $i > $o).
% 2.23/1.72 thf(func_def_481, type, sK189: $i > ($i > $o) > $i > $i).
% 2.23/1.72 thf(func_def_482, type, sK190: ($i > $o) > $i).
% 2.23/1.72 thf(func_def_483, type, sK191: $i > $o).
% 2.23/1.72 thf(func_def_490, type, sK198: $i > $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_491, type, sK199: $i > $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_504, type, sK212: $i > $i > $o).
% 2.23/1.72 thf(func_def_513, type, sK221: $i > $i).
% 2.23/1.72 thf(func_def_516, type, sK224: $i > $o).
% 2.23/1.72 thf(func_def_526, type, sK234: $i > $i > $i).
% 2.23/1.72 thf(func_def_527, type, sK235: $i > $i > $i).
% 2.23/1.72 thf(func_def_538, type, sK246: $i > $o).
% 2.23/1.72 thf(func_def_541, type, sK249: $i > ($i > $o) > $i).
% 2.23/1.72 thf(func_def_542, type, sK250: $i > $i > $i).
% 2.23/1.72 thf(func_def_553, type, sK261: $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_554, type, sK262: $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_565, type, sK273: $i > $i).
% 2.23/1.72 thf(func_def_566, type, sK274: ($i > $i) > $i > $i > $i).
% 2.23/1.72 thf(func_def_567, type, sK275: $i > $i > $o).
% 2.23/1.72 thf(func_def_593, type, sK301: ($i > $o) > $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_594, type, sK302: ($i > $o) > $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_598, type, sK306: $i > $o).
% 2.23/1.72 thf(func_def_610, type, sK318: $i > $i).
% 2.23/1.72 thf(func_def_612, type, sK320: ($i > $o) > $i > $i).
% 2.23/1.72 thf(func_def_614, type, sK322: $i > $o).
% 2.23/1.72 thf(func_def_619, type, sK327: $i > $i > ($i > $i) > $i).
% 2.23/1.72 thf(func_def_620, type, sK328: $i > $i).
% 2.23/1.72 thf(func_def_637, type, sK345: $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_645, type, sK353: $i > $i > $o).
% 2.23/1.72 thf(func_def_676, type, sK384: $i > $i).
% 2.23/1.72 thf(func_def_683, type, sK391: $i > $i > $i).
% 2.23/1.72 thf(func_def_686, type, sK394: $i > $o).
% 2.23/1.72 thf(func_def_699, type, sK407: $i > $i > $i).
% 2.23/1.72 thf(func_def_701, type, sK409: $i > $o).
% 2.23/1.72 thf(func_def_711, type, sK419: ($i > $o) > $i > $i).
% 2.23/1.72 thf(func_def_713, type, sK421: $i > $o).
% 2.23/1.72 thf(func_def_721, type, sK429: $i > $o).
% 2.23/1.72 thf(func_def_729, type, sK437: $i > $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_730, type, sK438: $i > $o).
% 2.23/1.72 thf(func_def_733, type, sK441: $i > ($i > $o) > $i).
% 2.23/1.72 thf(func_def_734, type, sK442: $i > ($i > $o) > $i).
% 2.23/1.72 thf(func_def_740, type, sK448: $i > $i > $o).
% 2.23/1.72 thf(func_def_745, type, sK453: $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_750, type, sK458: $i > $i > $i).
% 2.23/1.72 thf(func_def_753, type, sK461: $i > $o).
% 2.23/1.72 thf(func_def_755, type, sK463: $i > ($i > $o) > $i).
% 2.23/1.72 thf(func_def_756, type, sK464: $i > $i > $i).
% 2.23/1.72 thf(func_def_757, type, sK465: $i > $i > $i).
% 2.23/1.72 thf(func_def_758, type, sK466: $i > $i > $i).
% 2.23/1.72 thf(func_def_759, type, sK467: $i > $i > $i).
% 2.23/1.72 thf(func_def_760, type, sK468: $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_761, type, sK469: $i > $i).
% 2.23/1.72 thf(func_def_762, type, sK470: $i > $i).
% 2.23/1.72 thf(func_def_763, type, sK471: $i > $i).
% 2.23/1.72 thf(func_def_764, type, sK472: $i > $i).
% 2.23/1.72 thf(func_def_765, type, sK473: $i > $i > $i).
% 2.23/1.72 thf(func_def_766, type, sK474: $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_767, type, sK475: $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_768, type, sK476: $i > $i > $i).
% 2.23/1.72 thf(func_def_769, type, sK477: $i > $i > $i).
% 2.23/1.72 thf(func_def_770, type, sK478: $i > $i).
% 2.23/1.72 thf(func_def_771, type, sK479: $i > $i > $i).
% 2.23/1.72 thf(func_def_773, type, sK481: $i > $i).
% 2.23/1.72 thf(func_def_774, type, sK482: $i > $i).
% 2.23/1.72 thf(func_def_775, type, sK483: $i > $i > $o).
% 2.23/1.72 thf(func_def_785, type, sK493: $i > $o).
% 2.23/1.72 thf(func_def_787, type, sK495: ($i > $o) > $i > $i).
% 2.23/1.72 thf(func_def_788, type, sK496: ($i > $o) > $i > $i).
% 2.23/1.72 thf(func_def_791, type, sK499: ($i > $o) > ($i > $o) > $i).
% 2.23/1.72 thf(func_def_792, type, sK500: ($i > $o) > ($i > $o) > $i).
% 2.23/1.72 thf(func_def_793, type, sK501: $i > $o).
% 2.23/1.72 thf(func_def_794, type, sK502: $i > $o).
% 2.23/1.72 thf(func_def_795, type, sK503: $i > $i).
% 2.23/1.72 thf(func_def_797, type, sK505: $i > $i).
% 2.23/1.72 thf(func_def_839, type, sK547: $i > $o).
% 2.23/1.72 thf(func_def_842, type, sK550: $i > $i).
% 2.23/1.72 thf(func_def_856, type, sK564: $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_857, type, sK565: $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_863, type, sK571: $i > $o).
% 2.23/1.72 thf(func_def_878, type, sK586: ($i > $o) > $i > $i).
% 2.23/1.72 thf(func_def_880, type, sK588: $i > $o).
% 2.23/1.72 thf(func_def_887, type, sK595: $i > $i > $i).
% 2.23/1.72 thf(func_def_892, type, sK600: $i > $o).
% 2.23/1.72 thf(func_def_894, type, sK602: ($i > $o) > $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_895, type, sK603: ($i > $o) > $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_899, type, sK607: $i > $i > $o).
% 2.23/1.72 thf(func_def_914, type, sK622: $i > $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_915, type, sK623: $i > $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_916, type, sK624: $i > $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_917, type, sK625: $i > $i > $i > $i > $i).
% 2.23/1.72 thf(func_def_959, type, sK667: $i > $i).
% 2.23/1.72 thf(func_def_962, type, sK670: $i > $i > ($i > $i) > $i).
% 2.23/1.72 thf(func_def_965, type, sK673: $i > $o).
% 2.23/1.72 thf(func_def_968, type, sK676: $i > $i).
% 2.23/1.72 thf(func_def_971, type, sK679: $i > $i > ($i > $i) > $i).
% 2.23/1.72 thf(func_def_972, type, sK680: $i > $o).
% 2.23/1.72 thf(func_def_975, type, sK683: $i > $o).
% 2.23/1.72 thf(func_def_976, type, sK684: $i > $i).
% 2.23/1.72 thf(func_def_977, type, sK685: ($i > $o) > $i).
% 2.23/1.72 thf(func_def_992, type, sK700: $i > $o).
% 2.23/1.72 thf(func_def_998, type, sK706: $i > $i > $i).
% 2.23/1.72 thf(func_def_1003, type, sK711: $o > $i > $i > $i).
% 2.23/1.72 thf(func_def_1018, type, sK726: $i > $i).
% 2.23/1.72 thf(func_def_1020, type, sK728: $i > ($i > $i) > $i > $i).
% 2.23/1.72 thf(func_def_1025, type, sK733: $i > $i > $i).
% 2.23/1.72 thf(func_def_1028, type, sK736: $i > $o).
% 2.23/1.72 thf(func_def_1029, type, sK737: $i > $i).
% 2.23/1.72 thf(func_def_1030, type, sK738: ($i > $o) > $i).
% 2.23/1.72 thf(func_def_1045, type, sK753: $i > ($i > $o) > $i).
% 2.23/1.72 thf(func_def_1046, type, sK754: $i > $o).
% 2.23/1.72 thf(func_def_1050, type, sK758: $i > $i).
% 2.23/1.72 thf(func_def_1056, type, ph764: !>[X0: $tType]:(X0)).
% 2.23/1.72 thf(f7043,plain,(
% 2.23/1.72 $false),
% 2.23/1.72 inference(trivial_inequality_removal,[status(thm)],[f7032])).
% 2.23/1.72 thf(f7032,plain,(
% 2.23/1.72 ($true != $true)),
% 2.23/1.72 inference(superposition,[status(thm)],[f3827,f3293])).
% 2.23/1.72 thf(f3293,plain,(
% 2.23/1.72 ((in @ sK659 @ sK656) = $true)),
% 2.23/1.72 inference(cnf_transformation,[status(thm)],[f2058])).
% 2.23/1.72 thf(f2058,plain,(
% 2.23/1.72 (setadjoinOr = $true) & (emptyinPowerset = $true) & (iftrueProp1 = $true) & (binunionRsub = $true) & (eqinunit = $true) & (funcGraphProp4 = $true) & (binintersectSubset3 = $true) & (eqimpsubset2 = $true) & (brelnall2 = $true) & (binintersectT_lem = $true) & (setukpairIR = $true) & (exuI1 = $true) & (setminusSubset1 = $true) & (binintersectSubset1 = $true) & (exuE3u = $true) & (iffalseProp1 = $true) & (bs114d = $true) & (subsetRefl = $true) & (subsetI2 = $true) & (omegaSAx = $true) & (dsetconstrI = $true) & (binunionEcases = $true) & (powerset__Cong = $true) & (notinsingleton = $true) & (kfstpairEq = $true) & (secondinupair = $true) & (dpsetconstrSub = $true) & (setukpairinjL = $true) & (uniqinunit = $true) & (binintersectLsub = $true) & (setukpairinjR = $true) & (emptysetsubset = $true) & (($true = (in @ sK656 @ (powerset @ sK657))) & (($true = (in @ sK658 @ (powerset @ sK657))) & (($true = (in @ sK659 @ sK657)) & ($true = (subset @ sK656 @ sK658)) & ((in @ sK659 @ sK658) != $true) & ((in @ sK659 @ sK656) = $true)))) & (brelnall1 = $true) & (cartprodmempair1 = $true) & (eqbreln = $true) & (cartprodsndin = $true) & (binunionIR = $true) & (app = $true) & (complementT_lem = $true) & (setadjoinSub2 = $true) & (ksndpairEq = $true) & (setextsub = $true) & (emptyinunitempty = $true) & (setunionsingleton = $true) & (dpsetconstrER = $true) & (quantDeMorgan1 = $true) & (cartprodpairin = $true) & (ap2p = $true) & (binintersectSubset5 = $true) & (notequalI2 = $true) & (setext = $true) & (nonemptyI1 = $true) & (notsubsetI = $true) & (lam2p = $true) & (setukpairinjR12 = $true) & (beta2 = $true) & (replAx = $true) & (ap2apEq2 = $true) & (subsetemptysetimpeq = $true) & (eta2 = $true) & (setukpairIL = $true) & (dsetconstrER = $true) & (setadjoinIL = $true) & (notdexE = $true) & (setoftrueEq = $true) & (funcGraphProp2 = $true) & (upairset2IR = $true) & (subsetTI = $true) & (setadjoinSub = $true) & (binunionIL = $true) & (kfstsingleton = $true) & (funcImageSingleton = $true) & (binunionE = $true) & (notdallE = $true) & (subsetE = $true) & (singletonprop = $true) & (setminusLsub = $true) & (ifp = $true) & (subsetE2 = $true) & (exuI3 = $true) & (setminusSubset2 = $true) & (symdiffIneg2 = $true) & (upairset2E = $true) & (subPowSU = $true) & (singletoninpowunion = $true) & (quantDeMorgan2 = $true) & (setextAx = $true) & (binintersectSubset4 = $true) & (setOfPairsIsBReln = $true) & (setminusI = $true) & (setminusEL = $true) & (setunion__Cong = $true) & (emptyInPowerset = $true) & (setminusIRneg = $true) & (iftrue = $true) & (symdiffE = $true) & (exuE3e = $true) & (setunionE2 = $true) & (exuE2 = $true) & (setunionI = $true) & (ap2apEq1 = $true) & (contrasubsetT = $true) & (cartprodpairmemER = $true) & (prop2setI = $true) & (setadjoin__Cong = $true) & (exuEu = $true) & (setunionsingleton2 = $true) & (cartprodmempair = $true) & (eqimpsubset1 = $true) & (emptysetE = $true) & (noeltsimpempty = $true) & (cartprodfstin = $true) & (eta1 = $true) & (subsetI1 = $true) & (iftrueProp2 = $true) & (powersetI = $true) & (cartprodpairsurjEq = $true) & (binunionT_lem = $true) & (setadjoinAx = $true) & (complementTI1 = $true) & (emptyI = $true) & (kpairiskpair = $true) & (emptyE1 = $true) & (exuI2 = $true) & (setextT = $true) & (theprop = $true) & (emptysetAx = $true) & (ex1I2 = $true) & (funcextLem = $true) & (cartprodsndpairEq = $true) & (notequalI1 = $true) & (powersetI1 = $true) & (singletonsswitch = $true) & (in__Cong = $true) & (dpsetconstrEL2 = $true) & (omegaIndAx = $true) & (dpsetconstrERa = $true) & (ksndsingleton = $true) & (binintersectRsub = $true) & (quantDeMorgan3 = $true) & (apProp = $true) & (theeq = $true) & (setukpairinjR1 = $true) & (lamProp = $true) & (cartprodfstpairEq = $true) & (descrp = $true) & (binintersectEL = $true) & (sepInPowerset = $true) & (setunionAx = $true) & (upairsetIL = $true) & (lam2lamEq = $true) & (cartprodmempaircEq = $true) & (dsetconstrEL = $true) & (notinemptyset = $true) & (omega0Ax = $true) & (setukpairinjL1 = $true) & (funcGraphProp3 = $true) & (setminusILneg = $true) & (upairsetE = $true) & (omega__Cong = $true) & (ifSingleton = $true) & (sepSubset = $true) & (setminusERneg = $true) & (inCongP = $true) & (quantDeMorgan4 = $true) & (binintersectER = $true) & (binintersectTELcontra = $true) & (symdiffI1 = $true) & (ubforcartprodlem1 = $true) & (nonemptyImpWitness = $true) & (lamp = $true) & (disjointsetsI1 = $true) & (funcGraphProp1 = $true) & (setadjoinE = $true) & (exuE1 = $true) & (cartprodpairmemEL = $true) & (nonemptyE1 = $true) & (complementTE1 = $true) & (singletoninpowerset = $true) & (setunionE = $true) & (emptyset__Cong = $true) & (setminusELneg = $true) & (upairinpowunion = $true) & (upairsubunion = $true) & (infuncsetfunc = $true) & (nonemptyI = $true) & (iffalseProp2 = $true) & (powersetTI1 = $true) & (setminusER = $true) & (ex1E1 = $true) & (powersetE1 = $true) & (iftrueorfalse = $true) & (setukpairinjR11 = $true) & (binintersectTERcontra = $true) & (foundationAx = $true) & (inPowerset = $true) & (setbeta = $true) & (setukpairinjR2 = $true) & (emptysetimpfalse = $true) & (setunionsingleton1 = $true) & (binintersectI = $true) & (setminusT_lem = $true) & (ubforcartprodlem2 = $true) & (ex1E2 = $true) & (exu__Cong = $true) & (powersetTE1 = $true) & (powersetT_lem = $true) & (dpsetconstrEL1 = $true) & (singletonsubset = $true) & (binintersectSubset2 = $true) & (setadjoinIR = $true) & (kpairp = $true) & (symdiffIneg1 = $true) & (wellorderingAx = $true) & (binunionLsub = $true) & (upairsetIR = $true) & (dpsetconstrI = $true) & (funcext = $true) & (powersetE = $true) & (subsetTrans = $true) & (dsetconstr__Cong = $true) & (ubforcartprodlem3 = $true) & (prop2set2propI = $true) & (setukpairinjL2 = $true) & (powersetAx = $true) & (powersetsubset = $true) & (funcext2 = $true) & (ex1I = $true) & (singletonsuniq = $true) & (symdiffI2 = $true) & (funcinfuncset = $true) & (subbreln = $true) & (descr__Cong = $true) & (kpairsurjEq = $true) & (upairequniteq = $true) & (prop2setE = $true) & (iffalse = $true) & (vacuousDall = $true) & (subset2powerset = $true) & (beta1 = $true)),
% 2.23/1.72 inference(skolemisation,[status(esa),new_symbols(skolem,[sK656,sK657,sK658,sK659])],[f881,f2057,f2056,f2055])).
% 2.23/1.72 thf(f2055,plain,(
% 2.23/1.72 ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ? [X2 : $i] : (($true = (in @ X2 @ (powerset @ X1))) & ? [X3 : $i] : (($true = (in @ X3 @ X1)) & ((subset @ X0 @ X2) = $true) & ($true != (in @ X3 @ X2)) & ($true = (in @ X3 @ X0))))) => (($true = (in @ sK656 @ (powerset @ sK657))) & ? [X2 : $i] : (($true = (in @ X2 @ (powerset @ sK657))) & ? [X3 : $i] : (($true = (in @ X3 @ sK657)) & ((subset @ sK656 @ X2) = $true) & ($true != (in @ X3 @ X2)) & ($true = (in @ X3 @ sK656)))))),
% 2.23/1.72 introduced(definition,[],[choice_axiom])).
% 2.23/1.72 thf(f2056,plain,(
% 2.23/1.72 ? [X2 : $i] : (($true = (in @ X2 @ (powerset @ sK657))) & ? [X3 : $i] : (($true = (in @ X3 @ sK657)) & ((subset @ sK656 @ X2) = $true) & ($true != (in @ X3 @ X2)) & ($true = (in @ X3 @ sK656)))) => (($true = (in @ sK658 @ (powerset @ sK657))) & ? [X3 : $i] : (($true = (in @ X3 @ sK657)) & ($true = (subset @ sK656 @ sK658)) & ($true != (in @ X3 @ sK658)) & ($true = (in @ X3 @ sK656))))),
% 2.23/1.72 introduced(definition,[],[choice_axiom])).
% 2.23/1.72 thf(f2057,plain,(
% 2.23/1.72 ? [X3 : $i] : (($true = (in @ X3 @ sK657)) & ($true = (subset @ sK656 @ sK658)) & ($true != (in @ X3 @ sK658)) & ($true = (in @ X3 @ sK656))) => (($true = (in @ sK659 @ sK657)) & ($true = (subset @ sK656 @ sK658)) & ((in @ sK659 @ sK658) != $true) & ((in @ sK659 @ sK656) = $true))),
% 2.23/1.72 introduced(definition,[],[choice_axiom])).
% 2.23/1.72 thf(f881,plain,(
% 2.23/1.72 (setadjoinOr = $true) & (emptyinPowerset = $true) & (iftrueProp1 = $true) & (binunionRsub = $true) & (eqinunit = $true) & (funcGraphProp4 = $true) & (binintersectSubset3 = $true) & (eqimpsubset2 = $true) & (brelnall2 = $true) & (binintersectT_lem = $true) & (setukpairIR = $true) & (exuI1 = $true) & (setminusSubset1 = $true) & (binintersectSubset1 = $true) & (exuE3u = $true) & (iffalseProp1 = $true) & (bs114d = $true) & (subsetRefl = $true) & (subsetI2 = $true) & (omegaSAx = $true) & (dsetconstrI = $true) & (binunionEcases = $true) & (powerset__Cong = $true) & (notinsingleton = $true) & (kfstpairEq = $true) & (secondinupair = $true) & (dpsetconstrSub = $true) & (setukpairinjL = $true) & (uniqinunit = $true) & (binintersectLsub = $true) & (setukpairinjR = $true) & (emptysetsubset = $true) & ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ? [X2 : $i] : (($true = (in @ X2 @ (powerset @ X1))) & ? [X3 : $i] : (($true = (in @ X3 @ X1)) & ((subset @ X0 @ X2) = $true) & ($true != (in @ X3 @ X2)) & ($true = (in @ X3 @ X0))))) & (brelnall1 = $true) & (cartprodmempair1 = $true) & (eqbreln = $true) & (cartprodsndin = $true) & (binunionIR = $true) & (app = $true) & (complementT_lem = $true) & (setadjoinSub2 = $true) & (ksndpairEq = $true) & (setextsub = $true) & (emptyinunitempty = $true) & (setunionsingleton = $true) & (dpsetconstrER = $true) & (quantDeMorgan1 = $true) & (cartprodpairin = $true) & (ap2p = $true) & (binintersectSubset5 = $true) & (notequalI2 = $true) & (setext = $true) & (nonemptyI1 = $true) & (notsubsetI = $true) & (lam2p = $true) & (setukpairinjR12 = $true) & (beta2 = $true) & (replAx = $true) & (ap2apEq2 = $true) & (subsetemptysetimpeq = $true) & (eta2 = $true) & (setukpairIL = $true) & (dsetconstrER = $true) & (setadjoinIL = $true) & (notdexE = $true) & (setoftrueEq = $true) & (funcGraphProp2 = $true) & (upairset2IR = $true) & (subsetTI = $true) & (setadjoinSub = $true) & (binunionIL = $true) & (kfstsingleton = $true) & (funcImageSingleton = $true) & (binunionE = $true) & (notdallE = $true) & (subsetE = $true) & (singletonprop = $true) & (setminusLsub = $true) & (ifp = $true) & (subsetE2 = $true) & (exuI3 = $true) & (setminusSubset2 = $true) & (symdiffIneg2 = $true) & (upairset2E = $true) & (subPowSU = $true) & (singletoninpowunion = $true) & (quantDeMorgan2 = $true) & (setextAx = $true) & (binintersectSubset4 = $true) & (setOfPairsIsBReln = $true) & (setminusI = $true) & (setminusEL = $true) & (setunion__Cong = $true) & (emptyInPowerset = $true) & (setminusIRneg = $true) & (iftrue = $true) & (symdiffE = $true) & (exuE3e = $true) & (setunionE2 = $true) & (exuE2 = $true) & (setunionI = $true) & (ap2apEq1 = $true) & (contrasubsetT = $true) & (cartprodpairmemER = $true) & (prop2setI = $true) & (setadjoin__Cong = $true) & (exuEu = $true) & (setunionsingleton2 = $true) & (cartprodmempair = $true) & (eqimpsubset1 = $true) & (emptysetE = $true) & (noeltsimpempty = $true) & (cartprodfstin = $true) & (eta1 = $true) & (subsetI1 = $true) & (iftrueProp2 = $true) & (powersetI = $true) & (cartprodpairsurjEq = $true) & (binunionT_lem = $true) & (setadjoinAx = $true) & (complementTI1 = $true) & (emptyI = $true) & (kpairiskpair = $true) & (emptyE1 = $true) & (exuI2 = $true) & (setextT = $true) & (theprop = $true) & (emptysetAx = $true) & (ex1I2 = $true) & (funcextLem = $true) & (cartprodsndpairEq = $true) & (notequalI1 = $true) & (powersetI1 = $true) & (singletonsswitch = $true) & (in__Cong = $true) & (dpsetconstrEL2 = $true) & (omegaIndAx = $true) & (dpsetconstrERa = $true) & (ksndsingleton = $true) & (binintersectRsub = $true) & (quantDeMorgan3 = $true) & (apProp = $true) & (theeq = $true) & (setukpairinjR1 = $true) & (lamProp = $true) & (cartprodfstpairEq = $true) & (descrp = $true) & (binintersectEL = $true) & (sepInPowerset = $true) & (setunionAx = $true) & (upairsetIL = $true) & (lam2lamEq = $true) & (cartprodmempaircEq = $true) & (dsetconstrEL = $true) & (notinemptyset = $true) & (omega0Ax = $true) & (setukpairinjL1 = $true) & (funcGraphProp3 = $true) & (setminusILneg = $true) & (upairsetE = $true) & (omega__Cong = $true) & (ifSingleton = $true) & (sepSubset = $true) & (setminusERneg = $true) & (inCongP = $true) & (quantDeMorgan4 = $true) & (binintersectER = $true) & (binintersectTELcontra = $true) & (symdiffI1 = $true) & (ubforcartprodlem1 = $true) & (nonemptyImpWitness = $true) & (lamp = $true) & (disjointsetsI1 = $true) & (funcGraphProp1 = $true) & (setadjoinE = $true) & (exuE1 = $true) & (cartprodpairmemEL = $true) & (nonemptyE1 = $true) & (complementTE1 = $true) & (singletoninpowerset = $true) & (setunionE = $true) & (emptyset__Cong = $true) & (setminusELneg = $true) & (upairinpowunion = $true) & (upairsubunion = $true) & (infuncsetfunc = $true) & (nonemptyI = $true) & (iffalseProp2 = $true) & (powersetTI1 = $true) & (setminusER = $true) & (ex1E1 = $true) & (powersetE1 = $true) & (iftrueorfalse = $true) & (setukpairinjR11 = $true) & (binintersectTERcontra = $true) & (foundationAx = $true) & (inPowerset = $true) & (setbeta = $true) & (setukpairinjR2 = $true) & (emptysetimpfalse = $true) & (setunionsingleton1 = $true) & (binintersectI = $true) & (setminusT_lem = $true) & (ubforcartprodlem2 = $true) & (ex1E2 = $true) & (exu__Cong = $true) & (powersetTE1 = $true) & (powersetT_lem = $true) & (dpsetconstrEL1 = $true) & (singletonsubset = $true) & (binintersectSubset2 = $true) & (setadjoinIR = $true) & (kpairp = $true) & (symdiffIneg1 = $true) & (wellorderingAx = $true) & (binunionLsub = $true) & (upairsetIR = $true) & (dpsetconstrI = $true) & (funcext = $true) & (powersetE = $true) & (subsetTrans = $true) & (dsetconstr__Cong = $true) & (ubforcartprodlem3 = $true) & (prop2set2propI = $true) & (setukpairinjL2 = $true) & (powersetAx = $true) & (powersetsubset = $true) & (funcext2 = $true) & (ex1I = $true) & (singletonsuniq = $true) & (symdiffI2 = $true) & (funcinfuncset = $true) & (subbreln = $true) & (descr__Cong = $true) & (kpairsurjEq = $true) & (upairequniteq = $true) & (prop2setE = $true) & (iffalse = $true) & (vacuousDall = $true) & (subset2powerset = $true) & (beta1 = $true)),
% 2.23/1.72 inference(flattening,[status(thm)],[f880])).
% 2.23/1.72 thf(f880,plain,(
% 2.23/1.72 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (((($true = (in @ X3 @ X0)) & ($true != (in @ X3 @ X2))) & ((subset @ X0 @ X2) = $true)) & ($true = (in @ X3 @ X1))) & ($true = (in @ X2 @ (powerset @ X1)))) & ((in @ X0 @ (powerset @ X1)) = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.23/1.72 inference(ennf_transformation,[status(thm)],[f756])).
% 2.23/1.72 thf(f756,plain,(
% 2.23/1.72 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X1))) => ! [X3 : $i] : (($true = (in @ X3 @ X1)) => (((subset @ X0 @ X2) = $true) => (($true != (in @ X3 @ X2)) => ($true != (in @ X3 @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.23/1.72 inference(flattening,[status(thm)],[f522])).
% 2.23/1.72 thf(f522,plain,(
% 2.23/1.72 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X1))) => ! [X3 : $i] : (($true = (in @ X3 @ X1)) => (((subset @ X0 @ X2) = $true) => (~($true = (in @ X3 @ X2)) => ~($true = (in @ X3 @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.23/1.72 inference(fool_elimination,[status(thm)],[f521])).
% 2.23/1.72 thf(f521,plain,(
% 2.23/1.72 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => ! [X0 : $i,X1 : $i] : ((in @ X0 @ (powerset @ X1)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X1)) => ! [X3 : $i] : ((in @ X3 @ X1) => ((subset @ X0 @ X2) => (~(in @ X3 @ X2) => ~(in @ X3 @ X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.23/1.72 inference(rectify,[status(thm)],[f246])).
% 2.23/1.72 thf(f246,negated_conjecture,(
% 2.23/1.72 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((subset @ X11 @ X16) => (~(in @ X1 @ X16) => ~(in @ X1 @ X11))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.23/1.72 inference(negated_conjecture,[status(cth)],[f245])).
% 2.23/1.72 thf(f245,conjecture,(
% 2.23/1.72 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((subset @ X11 @ X16) => (~(in @ X1 @ X16) => ~(in @ X1 @ X11)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.23/1.72 file('/export/starexec/sandbox/tmp/tmp.IFvb0yMMZy/DTF2THF_26337.p',contrasubsetT1)).
% 2.23/1.72 thf(f3827,plain,(
% 2.23/1.72 ((in @ sK659 @ sK656) != $true)),
% 2.23/1.72 inference(trivial_inequality_removal,[status(thm)],[f3800])).
% 2.23/1.72 thf(f3800,plain,(
% 2.23/1.72 ((in @ sK659 @ sK656) != $true) | ($true != $true)),
% 2.23/1.72 inference(superposition,[status(thm)],[f3661,f3295])).
% 2.23/1.72 thf(f3295,plain,(
% 2.23/1.72 ($true = (subset @ sK656 @ sK658))),
% 2.23/1.72 inference(cnf_transformation,[status(thm)],[f2058])).
% 2.23/1.72 thf(f3661,plain,(
% 2.23/1.72 ( ! [X0 : $i] : (($true != (subset @ X0 @ sK658)) | ((in @ sK659 @ X0) != $true)) )),
% 2.23/1.72 inference(trivial_inequality_removal,[status(thm)],[f3660])).
% 2.23/1.72 thf(f3660,plain,(
% 2.23/1.72 ( ! [X0 : $i] : (($true != (subset @ X0 @ sK658)) | ((in @ sK659 @ X0) != $true) | ($true != $true)) )),
% 2.23/1.72 inference(forward_demodulation,[status(thm)],[f3658,f3246])).
% 2.23/1.72 thf(f3246,plain,(
% 2.23/1.72 (subsetE2 = $true)),
% 2.23/1.72 inference(cnf_transformation,[status(thm)],[f2058])).
% 2.23/1.72 thf(f3658,plain,(
% 2.23/1.72 ( ! [X0 : $i] : (((in @ sK659 @ X0) != $true) | ($true != (subset @ X0 @ sK658)) | (subsetE2 != $true)) )),
% 2.23/1.72 inference(trivial_inequality_removal,[status(thm)],[f3653])).
% 2.23/1.72 thf(f3653,plain,(
% 2.23/1.72 ( ! [X0 : $i] : (($true != $true) | (subsetE2 != $true) | ((in @ sK659 @ X0) != $true) | ($true != (subset @ X0 @ sK658))) )),
% 2.23/1.72 inference(superposition,[status(thm)],[f3294,f2921])).
% 2.23/1.72 thf(f2921,plain,(
% 2.23/1.72 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true = (in @ X0 @ X2)) | ($true != (subset @ X1 @ X2)) | (subsetE2 != $true) | ($true != (in @ X0 @ X1))) )),
% 2.23/1.72 inference(cnf_transformation,[status(thm)],[f1870])).
% 2.23/1.72 thf(f1870,plain,(
% 2.23/1.72 (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ X1)) | ($true = (in @ X0 @ X2)) | ($true != (subset @ X1 @ X2))) | (subsetE2 != $true)) & ((subsetE2 = $true) | (($true = (in @ sK535 @ sK536)) & ($true != (in @ sK535 @ sK537)) & ($true = (subset @ sK536 @ sK537))))),
% 2.23/1.72 inference(skolemisation,[status(esa),new_symbols(skolem,[sK535,sK536,sK537])],[f1868,f1869])).
% 2.23/1.72 thf(f1869,plain,(
% 2.23/1.72 ? [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X3 @ X4)) & ((in @ X3 @ X5) != $true) & ((subset @ X4 @ X5) = $true)) => (($true = (in @ sK535 @ sK536)) & ($true != (in @ sK535 @ sK537)) & ($true = (subset @ sK536 @ sK537)))),
% 2.23/1.72 introduced(definition,[],[choice_axiom])).
% 2.23/1.72 thf(f1868,plain,(
% 2.23/1.72 (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ X1)) | ($true = (in @ X0 @ X2)) | ($true != (subset @ X1 @ X2))) | (subsetE2 != $true)) & ((subsetE2 = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X3 @ X4)) & ((in @ X3 @ X5) != $true) & ((subset @ X4 @ X5) = $true)))),
% 2.23/1.72 inference(rectify,[status(thm)],[f1867])).
% 2.23/1.72 thf(f1867,plain,(
% 2.23/1.72 (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ X1)) | ($true = (in @ X0 @ X2)) | ($true != (subset @ X1 @ X2))) | (subsetE2 != $true)) & ((subsetE2 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X1)) & ($true != (in @ X0 @ X2)) & ($true = (subset @ X1 @ X2))))),
% 2.23/1.72 inference(nnf_transformation,[status(thm)],[f1045])).
% 2.23/1.72 thf(f1045,plain,(
% 2.23/1.72 ! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ X1)) | ($true = (in @ X0 @ X2)) | ($true != (subset @ X1 @ X2))) <=> (subsetE2 = $true)),
% 2.23/1.72 inference(flattening,[status(thm)],[f1044])).
% 2.23/1.72 thf(f1044,plain,(
% 2.23/1.72 ! [X0 : $i,X1 : $i,X2 : $i] : ((($true != (in @ X0 @ X1)) | ($true = (in @ X0 @ X2))) | ($true != (subset @ X1 @ X2))) <=> (subsetE2 = $true)),
% 2.23/1.72 inference(ennf_transformation,[status(thm)],[f732])).
% 2.23/1.72 thf(f732,plain,(
% 2.23/1.72 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (subset @ X1 @ X2)) => (($true != (in @ X0 @ X2)) => ($true != (in @ X0 @ X1)))) <=> (subsetE2 = $true)),
% 2.23/1.72 inference(flattening,[status(thm)],[f251])).
% 2.23/1.72 thf(f251,plain,(
% 2.23/1.72 (subsetE2 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (subset @ X1 @ X2)) => (~($true = (in @ X0 @ X2)) => ~($true = (in @ X0 @ X1))))),
% 2.23/1.72 inference(fool_elimination,[status(thm)],[f250])).
% 2.23/1.72 thf(f250,plain,(
% 2.23/1.72 (subsetE2 = ! [X0 : $i,X1 : $i,X2 : $i] : ((subset @ X1 @ X2) => (~(in @ X0 @ X2) => ~(in @ X0 @ X1))))),
% 2.23/1.72 inference(rectify,[status(thm)],[f84])).
% 2.23/1.72 thf(f84,axiom,(
% 2.23/1.72 (subsetE2 = ! [X1 : $i,X3 : $i,X4 : $i] : ((subset @ X3 @ X4) => (~(in @ X1 @ X4) => ~(in @ X1 @ X3))))),
% 2.23/1.72 file('/export/starexec/sandbox/tmp/tmp.IFvb0yMMZy/DTF2THF_26337.p',subsetE2)).
% 2.23/1.72 thf(f3294,plain,(
% 2.23/1.72 ((in @ sK659 @ sK658) != $true)),
% 2.23/1.72 inference(cnf_transformation,[status(thm)],[f2058])).
% 2.23/1.72 % SZS output end Proof for DTF2THF_26337
% 2.23/1.72 % (26636)------------------------------
% 2.23/1.72 % (26636)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.72 % (26636)Termination reason: Refutation
% 2.23/1.72
% 2.23/1.72 % (26636)Memory used [KB]: 11769
% 2.23/1.72 % (26636)Time elapsed: 0.220 s
% 2.23/1.72 % (26636)Instructions burned: 730 (million)
% 2.23/1.72 % (26636)------------------------------
% 2.23/1.72 % (26636)------------------------------
% 2.23/1.72 % (26587)Success in time 0.306 s
%------------------------------------------------------------------------------