%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU699^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n008.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 09:36:29 AM UTC 2026
% Result : Theorem 2.47s 1.59s
% Output : Refutation 2.47s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 6
% Syntax : Number of formulae : 42 ( 13 unt; 0 typ; 0 def)
% Number of atoms : 3592 ( 968 equ; 0 cnn)
% Maximal formula atoms : 217 ( 85 avg)
% Number of connectives : 2000 ( 53 ~; 34 |; 666 &; 370 @)
% ( 4 <=>; 873 =>; 0 <=; 0 <~>)
% Maximal formula depth : 224 ( 40 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 409 ( 406 usr; 226 con; 0-4 aty)
% Number of variables : 113 ( 23 ^; 63 !; 27 ?; 113 :)
% 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_268,type,
sP3: $i > $o > $i > $i > $o ).
thf(func_def_270,type,
sP5: $i > $i > $i > $i > $o ).
thf(func_def_272,type,
sP7: $i > $i > $o ).
thf(func_def_273,type,
sP8: $i > $o ).
thf(func_def_274,type,
sP9: $i > $i > $o ).
thf(func_def_275,type,
sP10: $i > $i > $o ).
thf(func_def_280,type,
sK15: $i > $o ).
thf(func_def_283,type,
sK18: $i > $i > $o ).
thf(func_def_286,type,
sK21: ( $i > $o ) > $i ).
thf(func_def_287,type,
sK22: $i > $o ).
thf(func_def_288,type,
sK23: $i > $i ).
thf(func_def_296,type,
sK31: $i > $o ).
thf(func_def_299,type,
sK34: $i > ( $i > $o ) > $i ).
thf(func_def_307,type,
sK42: $i > $o ).
thf(func_def_309,type,
sK44: $i > $o ).
thf(func_def_312,type,
sK47: $i > ( $i > $o ) > $i ).
thf(func_def_313,type,
sK48: $i > $o ).
thf(func_def_321,type,
sK56: $i > $o ).
thf(func_def_323,type,
sK58: ( $i > $o ) > $i ).
thf(func_def_324,type,
sK59: ( $i > $o ) > $i ).
thf(func_def_325,type,
sK60: $i > $o ).
thf(func_def_328,type,
sK63: $i > ( $i > $o ) > $i ).
thf(func_def_329,type,
sK64: $i > ( $i > $o ) > $i ).
thf(func_def_330,type,
sK65: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_331,type,
sK66: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_334,type,
sK69: $i > $o ).
thf(func_def_335,type,
sK70: $i > $o ).
thf(func_def_354,type,
sK89: $i > $o ).
thf(func_def_371,type,
sK106: $i > $i ).
thf(func_def_376,type,
sK111: $i > $o ).
thf(func_def_377,type,
sK112: $i > $o ).
thf(func_def_378,type,
sK113: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_379,type,
sK114: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_387,type,
sK122: $o > $i > $i > $i ).
thf(func_def_395,type,
sK130: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_396,type,
sK131: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_400,type,
sK135: $i > $o ).
thf(func_def_402,type,
sK137: $i > $i ).
thf(func_def_411,type,
sK146: $i > $i > $i ).
thf(func_def_417,type,
sK152: ( $i > $o ) > $i ).
thf(func_def_418,type,
sK153: $i > $o ).
thf(func_def_419,type,
sK154: $i > $i ).
thf(func_def_435,type,
sK170: $i > $i ).
thf(func_def_437,type,
sK172: $i > $i > $o ).
thf(func_def_447,type,
sK182: $i > $o ).
thf(func_def_466,type,
sK201: $i > $i > $i ).
thf(func_def_467,type,
sK202: $i > $i > $i ).
thf(func_def_473,type,
sK208: $i > $o ).
thf(func_def_475,type,
sK210: $i > ( $i > $o ) > $i ).
thf(func_def_479,type,
sK214: $i > $i ).
thf(func_def_481,type,
sK216: $i > ( $i > $i ) > $i > $i ).
thf(func_def_489,type,
sK224: $i > $i > $i ).
thf(func_def_507,type,
sK242: $i > $i > $o ).
thf(func_def_513,type,
sK248: $i > $i > $o ).
thf(func_def_517,type,
sK252: $i > $o ).
thf(func_def_528,type,
sK263: $i > $o ).
thf(func_def_533,type,
sK268: $i > $i > $i ).
thf(func_def_540,type,
sK275: $i > $i > $i > $i > $i ).
thf(func_def_542,type,
sK277: $i > $o ).
thf(func_def_544,type,
sK279: $i > $i ).
thf(func_def_553,type,
sK288: $i > $i > $i ).
thf(func_def_555,type,
sK290: $i > $i ).
thf(func_def_557,type,
sK292: $i > ( $i > $i ) > $i > $i ).
thf(func_def_562,type,
sK297: $i > $i > $i > $i > $i ).
thf(func_def_563,type,
sK298: $i > $i > $i > $i > $i ).
thf(func_def_585,type,
sK320: $i > $o ).
thf(func_def_597,type,
sK332: $i > $i > $o ).
thf(func_def_601,type,
sK336: ( $i > $o ) > $i ).
thf(func_def_602,type,
sK337: $i > $o ).
thf(func_def_615,type,
sK350: $i > $o ).
thf(func_def_626,type,
sK361: $i > $i > $i ).
thf(func_def_627,type,
sK362: $i > $i > $i ).
thf(func_def_635,type,
sK370: $i > $o ).
thf(func_def_637,type,
sK372: ( $i > $o ) > $i > $i ).
thf(func_def_638,type,
sK373: ( $i > $o ) > $i > $i ).
thf(func_def_642,type,
sK377: ( $i > $o ) > $i > $i ).
thf(func_def_644,type,
sK379: $i > $o ).
thf(func_def_646,type,
sK381: $i > $i ).
thf(func_def_650,type,
sK385: $i > $i > ( $i > $i ) > $i ).
thf(func_def_661,type,
sK396: $i > $i > $o ).
thf(func_def_669,type,
sK404: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_670,type,
sK405: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_671,type,
sK406: $i > $o ).
thf(func_def_672,type,
sK407: $i > $o ).
thf(func_def_673,type,
sK408: $i > $o ).
thf(func_def_676,type,
sK411: $i > ( $i > $o ) > $i ).
thf(func_def_696,type,
sK431: $i > $i > $o ).
thf(func_def_697,type,
sK432: $i > $i ).
thf(func_def_698,type,
sK433: $i > $i ).
thf(func_def_699,type,
sK434: ( $i > $i > $o ) > $i > $i ).
thf(func_def_700,type,
sK435: ( $i > $i > $o ) > $i > $i ).
thf(func_def_701,type,
sK436: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_707,type,
sK442: $i > $i ).
thf(func_def_710,type,
sK445: $i > $i > ( $i > $i ) > $i ).
thf(func_def_711,type,
sK446: $i > $i ).
thf(func_def_735,type,
sK470: $i > $i > $i > $i > $i ).
thf(func_def_736,type,
sK471: $i > $i > $i > $i > $i ).
thf(func_def_737,type,
sK472: $i > $i > $i > $i > $i ).
thf(func_def_738,type,
sK473: $i > $i > $i > $i > $i ).
thf(func_def_757,type,
sK492: $i > $i > $i ).
thf(func_def_761,type,
sK496: ( $i > $o ) > $i > $i ).
thf(func_def_763,type,
sK498: $i > $o ).
thf(func_def_768,type,
sK503: $i > $o ).
thf(func_def_776,type,
sK511: $i > $i ).
thf(func_def_779,type,
sK514: $i > $i > ( $i > $i ) > $i ).
thf(func_def_784,type,
sK519: $i > $i > $i > $i > $i ).
thf(func_def_803,type,
sK538: $i > $i > $i ).
thf(func_def_804,type,
sK539: $i > $i > $i ).
thf(func_def_805,type,
sK540: $i > $i > $i ).
thf(func_def_806,type,
sK541: $i > $i > $i ).
thf(func_def_807,type,
sK542: $i > $i > $i > $i ).
thf(func_def_808,type,
sK543: $i > $i ).
thf(func_def_809,type,
sK544: $i > $i ).
thf(func_def_810,type,
sK545: $i > $i ).
thf(func_def_811,type,
sK546: $i > $i ).
thf(func_def_812,type,
sK547: $i > $i > $i ).
thf(func_def_813,type,
sK548: $i > $i > $i > $i ).
thf(func_def_814,type,
sK549: $i > $i > $i > $i ).
thf(func_def_815,type,
sK550: $i > $i > $i ).
thf(func_def_816,type,
sK551: $i > $i > $i ).
thf(func_def_817,type,
sK552: $i > $i ).
thf(func_def_818,type,
sK553: $i > $i > $i ).
thf(func_def_820,type,
sK555: $i > $i ).
thf(func_def_821,type,
sK556: $i > $i ).
thf(func_def_822,type,
sK557: $i > ( $i > $o ) > $i ).
thf(func_def_823,type,
sK558: $i > $o ).
thf(func_def_830,type,
sK565: $i > $o ).
thf(func_def_832,type,
sK567: $i > ( $i > $o ) > $i > $i ).
thf(func_def_835,type,
sK570: ( $i > $o ) > $i > $i ).
thf(func_def_837,type,
sK572: $i > $o ).
thf(func_def_847,type,
sK582: $i > $o ).
thf(func_def_858,type,
sK593: $i > ( $i > $o ) > $i ).
thf(func_def_859,type,
sK594: $i > $o ).
thf(func_def_869,type,
sK604: $i > $i > $i > $i > $i ).
thf(func_def_870,type,
sK605: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_871,type,
sK606: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_875,type,
sK610: $i > $o ).
thf(func_def_879,type,
sK614: $i > $i ).
thf(func_def_880,type,
sK615: $i > $i ).
thf(func_def_888,type,
sK623: $i > $i > $i ).
thf(func_def_900,type,
sK635: $i > ( $i > $i ) > $i > $i ).
thf(func_def_902,type,
sK637: $i > $i ).
thf(func_def_911,type,
sK646: $i > $i > $i > $i ).
thf(func_def_912,type,
sK647: $i > $i > $i > $i ).
thf(func_def_922,type,
sK657: $i > $i > $o ).
thf(func_def_927,type,
sK662: $i > $o ).
thf(func_def_938,type,
ph673:
!>[X0: $tType] : X0 ).
thf(f3217,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f3216]) ).
thf(f3216,plain,
$true != $true,
inference(superposition,[status(thm)],[f3215,f3209]) ).
thf(f3209,plain,
( $true
= ( func @ sK341 @ sK343 @ sK342 ) ),
inference(trivial_inequality_removal,[status(thm)],[f3208]) ).
thf(f3208,plain,
( ( $true != $true )
| ( $true
= ( func @ sK341 @ sK343 @ sK342 ) ) ),
inference(forward_demodulation,[status(thm)],[f3207,f2544]) ).
thf(f2544,plain,
infuncsetfunc = $true,
inference(cnf_transformation,[status(thm)],[f1474]) ).
thf(f1474,plain,
( ( sepInPowerset = $true )
& ( binintersectEL = $true )
& ( notinsingleton = $true )
& ( singletonsswitch = $true )
& ( binunionIR = $true )
& ( eqinunit = $true )
& ( dpsetconstrI = $true )
& ( ( lam @ sK341 @ sK343
@ ^ [Y0: $i] : ( ap @ sK341 @ sK343 @ sK342 @ Y0 ) )
!= sK342 )
& ( $true
= ( in @ sK342 @ ( funcSet @ sK341 @ sK343 ) ) )
& ( funcGraphProp2 = $true )
& ( setOfPairsIsBReln = $true )
& ( setoftrueEq = $true )
& ( descrp = $true )
& ( setunionE = $true )
& ( funcGraphProp3 = $true )
& ( exuE3e = $true )
& ( nonemptyI1 = $true )
& ( setunionsingleton = $true )
& ( exuI2 = $true )
& ( ksndsingleton = $true )
& ( powersetI1 = $true )
& ( secondinupair = $true )
& ( in__Cong = $true )
& ( setminusERneg = $true )
& ( exuI3 = $true )
& ( setminusELneg = $true )
& ( singletonprop = $true )
& ( funcGraphProp1 = $true )
& ( binintersectER = $true )
& ( kpairsurjEq = $true )
& ( wellorderingAx = $true )
& ( cartprodmempair1 = $true )
& ( ex1E2 = $true )
& ( lamp = $true )
& ( binunionEcases = $true )
& ( emptyset__Cong = $true )
& ( upairinpowunion = $true )
& ( brelnall1 = $true )
& ( exuE2 = $true )
& ( symdiffIneg2 = $true )
& ( subsetI1 = $true )
& ( subsetRefl = $true )
& ( dpsetconstrER = $true )
& ( upairequniteq = $true )
& ( prop2setE = $true )
& ( symdiffIneg1 = $true )
& ( emptyinunitempty = $true )
& ( funcinfuncset = $true )
& ( symdiffE = $true )
& ( ap2apEq1 = $true )
& ( kpairp = $true )
& ( setminusEL = $true )
& ( kfstsingleton = $true )
& ( ex1I = $true )
& ( prop2setI = $true )
& ( symdiffI1 = $true )
& ( setadjoinAx = $true )
& ( upairsetE = $true )
& ( exu__Cong = $true )
& ( descr__Cong = $true )
& ( omegaIndAx = $true )
& ( ubforcartprodlem3 = $true )
& ( setadjoinSub = $true )
& ( setminusLsub = $true )
& ( infuncsetfunc = $true )
& ( dpsetconstrSub = $true )
& ( inPowerset = $true )
& ( funcGraphProp4 = $true )
& ( powersetI = $true )
& ( dpsetconstrERa = $true )
& ( notdallE = $true )
& ( subset2powerset = $true )
& ( eqbreln = $true )
& ( quantDeMorgan2 = $true )
& ( ksndpairEq = $true )
& ( powersetE = $true )
& ( exuE3u = $true )
& ( cartprodsndin = $true )
& ( setadjoinSub2 = $true )
& ( funcextLem = $true )
& ( subsetE2 = $true )
& ( upairset2IR = $true )
& ( exuI1 = $true )
& ( setextAx = $true )
& ( nonemptyI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodpairmemEL = $true )
& ( app = $true )
& ( setukpairinjR2 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( vacuousDall = $true )
& ( singletonsuniq = $true )
& ( funcext2 = $true )
& ( setukpairinjR1 = $true )
& ( lam2lamEq = $true )
& ( upairset2E = $true )
& ( uniqinunit = $true )
& ( powersetE1 = $true )
& ( notsubsetI = $true )
& ( setadjoinOr = $true )
& ( upairsubunion = $true )
& ( cartprodmempair = $true )
& ( theprop = $true )
& ( apProp = $true )
& ( ap2apEq2 = $true )
& ( singletoninpowunion = $true )
& ( brelnall2 = $true )
& ( binintersectSubset2 = $true )
& ( cartprodpairin = $true )
& ( nonemptyE1 = $true )
& ( ap2p = $true )
& ( cartprodmempaircEq = $true )
& ( setukpairinjL1 = $true )
& ( setext = $true )
& ( emptysetAx = $true )
& ( binintersectI = $true )
& ( beta1 = $true )
& ( setminusILneg = $true )
& ( symdiffI2 = $true )
& ( powerset__Cong = $true )
& ( lam2p = $true )
& ( subsetI2 = $true )
& ( setukpairinjL2 = $true )
& ( eta1 = $true )
& ( notequalI1 = $true )
& ( exuE1 = $true )
& ( setminusI = $true )
& ( emptyI = $true )
& ( omegaSAx = $true )
& ( setbeta = $true )
& ( omega0Ax = $true )
& ( setukpairinjR = $true )
& ( ex1I2 = $true )
& ( notinemptyset = $true )
& ( lamProp = $true )
& ( dsetconstrEL = $true )
& ( setminusSubset1 = $true )
& ( subsetE = $true )
& ( exuEu = $true )
& ( binintersectSubset4 = $true )
& ( emptyinPowerset = $true )
& ( subsetemptysetimpeq = $true )
& ( setadjoinIL = $true )
& ( quantDeMorgan4 = $true )
& ( sepSubset = $true )
& ( noeltsimpempty = $true )
& ( binunionE = $true )
& ( singletonsubset = $true )
& ( setukpairIR = $true )
& ( setadjoinIR = $true )
& ( foundationAx = $true )
& ( setminusER = $true )
& ( binintersectSubset3 = $true )
& ( replAx = $true )
& ( subPowSU = $true )
& ( singletoninpowerset = $true )
& ( upairsetIL = $true )
& ( binintersectRsub = $true )
& ( setunion__Cong = $true )
& ( notdexE = $true )
& ( emptyInPowerset = $true )
& ( prop2set2propI = $true )
& ( cartprodfstpairEq = $true )
& ( setunionI = $true )
& ( nonemptyImpWitness = $true )
& ( setminusIRneg = $true )
& ( dpsetconstrEL1 = $true )
& ( setadjoinE = $true )
& ( funcext = $true )
& ( cartprodpairmemER = $true )
& ( notequalI2 = $true )
& ( binintersectLsub = $true )
& ( emptysetsubset = $true )
& ( bs114d = $true )
& ( powersetsubset = $true )
& ( setminusSubset2 = $true )
& ( omega__Cong = $true )
& ( quantDeMorgan3 = $true )
& ( upairsetIR = $true )
& ( setukpairIL = $true )
& ( binunionLsub = $true )
& ( binintersectSubset1 = $true )
& ( eqimpsubset2 = $true )
& ( dsetconstrER = $true )
& ( setunionsingleton1 = $true )
& ( setadjoin__Cong = $true )
& ( cartprodfstin = $true )
& ( setukpairinjR12 = $true )
& ( kpairiskpair = $true )
& ( quantDeMorgan1 = $true )
& ( binunionRsub = $true )
& ( funcImageSingleton = $true )
& ( beta2 = $true )
& ( dsetconstr__Cong = $true )
& ( setextsub = $true )
& ( subsetTrans = $true )
& ( eqimpsubset1 = $true )
& ( disjointsetsI1 = $true )
& ( ubforcartprodlem2 = $true )
& ( emptysetE = $true )
& ( ubforcartprodlem1 = $true )
& ( emptysetimpfalse = $true )
& ( dsetconstrI = $true )
& ( binintersectSubset5 = $true )
& ( powersetAx = $true )
& ( binunionIL = $true )
& ( cartprodsndpairEq = $true )
& ( setunionAx = $true )
& ( kfstpairEq = $true )
& ( emptyE1 = $true )
& ( subbreln = $true )
& ( ex1E1 = $true )
& ( inCongP = $true )
& ( setunionE2 = $true )
& ( dpsetconstrEL2 = $true )
& ( setunionsingleton2 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK341,sK342,sK343])],[f754,f1473]) ).
thf(f1473,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( lam @ X0 @ X2
@ ^ [Y0: $i] : ( ap @ X0 @ X2 @ X1 @ Y0 ) )
!= X1 )
& ( $true
= ( in @ X1 @ ( funcSet @ X0 @ X2 ) ) ) )
=> ( ( ( lam @ sK341 @ sK343
@ ^ [Y0: $i] : ( ap @ sK341 @ sK343 @ sK342 @ Y0 ) )
!= sK342 )
& ( $true
= ( in @ sK342 @ ( funcSet @ sK341 @ sK343 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f754,plain,
( ( sepInPowerset = $true )
& ( binintersectEL = $true )
& ( notinsingleton = $true )
& ( singletonsswitch = $true )
& ( binunionIR = $true )
& ( eqinunit = $true )
& ( dpsetconstrI = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( ( lam @ X0 @ X2
@ ^ [Y0: $i] : ( ap @ X0 @ X2 @ X1 @ Y0 ) )
!= X1 )
& ( $true
= ( in @ X1 @ ( funcSet @ X0 @ X2 ) ) ) )
& ( funcGraphProp2 = $true )
& ( setOfPairsIsBReln = $true )
& ( setoftrueEq = $true )
& ( descrp = $true )
& ( setunionE = $true )
& ( funcGraphProp3 = $true )
& ( exuE3e = $true )
& ( nonemptyI1 = $true )
& ( setunionsingleton = $true )
& ( exuI2 = $true )
& ( ksndsingleton = $true )
& ( powersetI1 = $true )
& ( secondinupair = $true )
& ( in__Cong = $true )
& ( setminusERneg = $true )
& ( exuI3 = $true )
& ( setminusELneg = $true )
& ( singletonprop = $true )
& ( funcGraphProp1 = $true )
& ( binintersectER = $true )
& ( kpairsurjEq = $true )
& ( wellorderingAx = $true )
& ( cartprodmempair1 = $true )
& ( ex1E2 = $true )
& ( lamp = $true )
& ( binunionEcases = $true )
& ( emptyset__Cong = $true )
& ( upairinpowunion = $true )
& ( brelnall1 = $true )
& ( exuE2 = $true )
& ( symdiffIneg2 = $true )
& ( subsetI1 = $true )
& ( subsetRefl = $true )
& ( dpsetconstrER = $true )
& ( upairequniteq = $true )
& ( prop2setE = $true )
& ( symdiffIneg1 = $true )
& ( emptyinunitempty = $true )
& ( funcinfuncset = $true )
& ( symdiffE = $true )
& ( ap2apEq1 = $true )
& ( kpairp = $true )
& ( setminusEL = $true )
& ( kfstsingleton = $true )
& ( ex1I = $true )
& ( prop2setI = $true )
& ( symdiffI1 = $true )
& ( setadjoinAx = $true )
& ( upairsetE = $true )
& ( exu__Cong = $true )
& ( descr__Cong = $true )
& ( omegaIndAx = $true )
& ( ubforcartprodlem3 = $true )
& ( setadjoinSub = $true )
& ( setminusLsub = $true )
& ( infuncsetfunc = $true )
& ( dpsetconstrSub = $true )
& ( inPowerset = $true )
& ( funcGraphProp4 = $true )
& ( powersetI = $true )
& ( dpsetconstrERa = $true )
& ( notdallE = $true )
& ( subset2powerset = $true )
& ( eqbreln = $true )
& ( quantDeMorgan2 = $true )
& ( ksndpairEq = $true )
& ( powersetE = $true )
& ( exuE3u = $true )
& ( cartprodsndin = $true )
& ( setadjoinSub2 = $true )
& ( funcextLem = $true )
& ( subsetE2 = $true )
& ( upairset2IR = $true )
& ( exuI1 = $true )
& ( setextAx = $true )
& ( nonemptyI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodpairmemEL = $true )
& ( app = $true )
& ( setukpairinjR2 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( vacuousDall = $true )
& ( singletonsuniq = $true )
& ( funcext2 = $true )
& ( setukpairinjR1 = $true )
& ( lam2lamEq = $true )
& ( upairset2E = $true )
& ( uniqinunit = $true )
& ( powersetE1 = $true )
& ( notsubsetI = $true )
& ( setadjoinOr = $true )
& ( upairsubunion = $true )
& ( cartprodmempair = $true )
& ( theprop = $true )
& ( apProp = $true )
& ( ap2apEq2 = $true )
& ( singletoninpowunion = $true )
& ( brelnall2 = $true )
& ( binintersectSubset2 = $true )
& ( cartprodpairin = $true )
& ( nonemptyE1 = $true )
& ( ap2p = $true )
& ( cartprodmempaircEq = $true )
& ( setukpairinjL1 = $true )
& ( setext = $true )
& ( emptysetAx = $true )
& ( binintersectI = $true )
& ( beta1 = $true )
& ( setminusILneg = $true )
& ( symdiffI2 = $true )
& ( powerset__Cong = $true )
& ( lam2p = $true )
& ( subsetI2 = $true )
& ( setukpairinjL2 = $true )
& ( eta1 = $true )
& ( notequalI1 = $true )
& ( exuE1 = $true )
& ( setminusI = $true )
& ( emptyI = $true )
& ( omegaSAx = $true )
& ( setbeta = $true )
& ( omega0Ax = $true )
& ( setukpairinjR = $true )
& ( ex1I2 = $true )
& ( notinemptyset = $true )
& ( lamProp = $true )
& ( dsetconstrEL = $true )
& ( setminusSubset1 = $true )
& ( subsetE = $true )
& ( exuEu = $true )
& ( binintersectSubset4 = $true )
& ( emptyinPowerset = $true )
& ( subsetemptysetimpeq = $true )
& ( setadjoinIL = $true )
& ( quantDeMorgan4 = $true )
& ( sepSubset = $true )
& ( noeltsimpempty = $true )
& ( binunionE = $true )
& ( singletonsubset = $true )
& ( setukpairIR = $true )
& ( setadjoinIR = $true )
& ( foundationAx = $true )
& ( setminusER = $true )
& ( binintersectSubset3 = $true )
& ( replAx = $true )
& ( subPowSU = $true )
& ( singletoninpowerset = $true )
& ( upairsetIL = $true )
& ( binintersectRsub = $true )
& ( setunion__Cong = $true )
& ( notdexE = $true )
& ( emptyInPowerset = $true )
& ( prop2set2propI = $true )
& ( cartprodfstpairEq = $true )
& ( setunionI = $true )
& ( nonemptyImpWitness = $true )
& ( setminusIRneg = $true )
& ( dpsetconstrEL1 = $true )
& ( setadjoinE = $true )
& ( funcext = $true )
& ( cartprodpairmemER = $true )
& ( notequalI2 = $true )
& ( binintersectLsub = $true )
& ( emptysetsubset = $true )
& ( bs114d = $true )
& ( powersetsubset = $true )
& ( setminusSubset2 = $true )
& ( omega__Cong = $true )
& ( quantDeMorgan3 = $true )
& ( upairsetIR = $true )
& ( setukpairIL = $true )
& ( binunionLsub = $true )
& ( binintersectSubset1 = $true )
& ( eqimpsubset2 = $true )
& ( dsetconstrER = $true )
& ( setunionsingleton1 = $true )
& ( setadjoin__Cong = $true )
& ( cartprodfstin = $true )
& ( setukpairinjR12 = $true )
& ( kpairiskpair = $true )
& ( quantDeMorgan1 = $true )
& ( binunionRsub = $true )
& ( funcImageSingleton = $true )
& ( beta2 = $true )
& ( dsetconstr__Cong = $true )
& ( setextsub = $true )
& ( subsetTrans = $true )
& ( eqimpsubset1 = $true )
& ( disjointsetsI1 = $true )
& ( ubforcartprodlem2 = $true )
& ( emptysetE = $true )
& ( ubforcartprodlem1 = $true )
& ( emptysetimpfalse = $true )
& ( dsetconstrI = $true )
& ( binintersectSubset5 = $true )
& ( powersetAx = $true )
& ( binunionIL = $true )
& ( cartprodsndpairEq = $true )
& ( setunionAx = $true )
& ( kfstpairEq = $true )
& ( emptyE1 = $true )
& ( subbreln = $true )
& ( ex1E1 = $true )
& ( inCongP = $true )
& ( setunionE2 = $true )
& ( dpsetconstrEL2 = $true )
& ( setunionsingleton2 = $true ) ),
inference(flattening,[status(thm)],[f753]) ).
thf(f753,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( lam @ X0 @ X2
@ ^ [Y0: $i] : ( ap @ X0 @ X2 @ X1 @ Y0 ) )
!= X1 )
& ( $true
= ( in @ X1 @ ( funcSet @ X0 @ X2 ) ) ) )
& ( 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)],[f313]) ).
thf(f313,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 )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ ( funcSet @ X0 @ X2 ) ) )
=> ( ( lam @ X0 @ X2
@ ^ [Y0: $i] : ( ap @ X0 @ X2 @ X1 @ Y0 ) )
= X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f312]) ).
thf(f312,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
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X1 @ ( funcSet @ X0 @ X2 ) )
=> ( X1
= ( lam @ X0 @ X2
@ ^ [X3: $i] : ( ap @ X0 @ X2 @ X1 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f222]) ).
thf(f222,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
=> ! [X3: $i,X12: $i,X4: $i] :
( ( in @ X12 @ ( funcSet @ X3 @ X4 ) )
=> ( X12
= ( lam @ X3 @ X4
@ ^ [X1: $i] : ( ap @ X3 @ X4 @ X12 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f221]) ).
thf(f221,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
=> ! [X3: $i,X12: $i,X4: $i] :
( ( in @ X12 @ ( funcSet @ X3 @ X4 ) )
=> ( X12
= ( lam @ X3 @ X4
@ ^ [X1: $i] : ( ap @ X3 @ X4 @ X12 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.GXJcYPDlCU/DTF2THF_6599.p',eta2) ).
thf(f3207,plain,
( ( infuncsetfunc != $true )
| ( $true
= ( func @ sK341 @ sK343 @ sK342 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3204]) ).
thf(f3204,plain,
( ( $true != $true )
| ( $true
= ( func @ sK341 @ sK343 @ sK342 ) )
| ( infuncsetfunc != $true ) ),
inference(superposition,[status(thm)],[f2707,f2600]) ).
thf(f2600,plain,
( $true
= ( in @ sK342 @ ( funcSet @ sK341 @ sK343 ) ) ),
inference(cnf_transformation,[status(thm)],[f1474]) ).
thf(f2707,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
!= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) )
| ( infuncsetfunc != $true )
| ( $true
= ( func @ X2 @ X1 @ X0 ) ) ),
inference(cnf_transformation,[status(thm)],[f1591]) ).
thf(f1591,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X2 @ X1 @ X0 ) )
| ( $true
!= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) ) )
| ( infuncsetfunc != $true ) )
& ( ( infuncsetfunc = $true )
| ( ( $true
!= ( func @ sK425 @ sK424 @ sK423 ) )
& ( $true
= ( in @ sK423 @ ( funcSet @ sK425 @ sK424 ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK423,sK424,sK425])],[f1589,f1590]) ).
thf(f1590,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( func @ X5 @ X4 @ X3 ) )
& ( $true
= ( in @ X3 @ ( funcSet @ X5 @ X4 ) ) ) )
=> ( ( $true
!= ( func @ sK425 @ sK424 @ sK423 ) )
& ( $true
= ( in @ sK423 @ ( funcSet @ sK425 @ sK424 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1589,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X2 @ X1 @ X0 ) )
| ( $true
!= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) ) )
| ( infuncsetfunc != $true ) )
& ( ( infuncsetfunc = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( func @ X5 @ X4 @ X3 ) )
& ( $true
= ( in @ X3 @ ( funcSet @ X5 @ X4 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f1588]) ).
thf(f1588,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X2 @ X1 @ X0 ) )
| ( $true
!= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) ) )
| ( infuncsetfunc != $true ) )
& ( ( infuncsetfunc = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( func @ X2 @ X1 @ X0 ) )
& ( $true
= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f915]) ).
thf(f915,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X2 @ X1 @ X0 ) )
| ( $true
!= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) ) )
<=> ( infuncsetfunc = $true ) ),
inference(ennf_transformation,[status(thm)],[f548]) ).
thf(f548,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( funcSet @ X2 @ X1 ) ) )
=> ( $true
= ( func @ X2 @ X1 @ X0 ) ) )
<=> ( infuncsetfunc = $true ) ),
inference(fool_elimination,[status(thm)],[f547]) ).
thf(f547,plain,
( infuncsetfunc
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ ( funcSet @ X2 @ X1 ) )
=> ( func @ X2 @ X1 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f197]) ).
thf(f197,axiom,
( infuncsetfunc
= ( ! [X12: $i,X4: $i,X3: $i] :
( ( in @ X12 @ ( funcSet @ X3 @ X4 ) )
=> ( func @ X3 @ X4 @ X12 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.GXJcYPDlCU/DTF2THF_6599.p',infuncsetfunc) ).
thf(f3215,plain,
( $true
!= ( func @ sK341 @ sK343 @ sK342 ) ),
inference(trivial_inequality_removal,[status(thm)],[f3214]) ).
thf(f3214,plain,
( ( $true != $true )
| ( $true
!= ( func @ sK341 @ sK343 @ sK342 ) ) ),
inference(forward_demodulation,[status(thm)],[f3213,f2484]) ).
thf(f2484,plain,
eta1 = $true,
inference(cnf_transformation,[status(thm)],[f1474]) ).
thf(f3213,plain,
( ( $true
!= ( func @ sK341 @ sK343 @ sK342 ) )
| ( eta1 != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3212]) ).
thf(f3212,plain,
( ( eta1 != $true )
| ( sK342 != sK342 )
| ( $true
!= ( func @ sK341 @ sK343 @ sK342 ) ) ),
inference(superposition,[status(thm)],[f3203,f3189]) ).
thf(f3189,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( lam @ X4 @ X3 @ ( ap @ X4 @ X3 @ X5 ) )
= X5 )
| ( eta1 != $true )
| ( $true
!= ( func @ X4 @ X3 @ X5 ) ) ),
inference(beta_eta_normalization,[status(thm)],[f2747]) ).
thf(f2747,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( func @ X4 @ X3 @ X5 ) )
| ( eta1 != $true )
| ( ( lam @ X4 @ X3
@ ^ [Y0: $i] : ( ap @ X4 @ X3 @ X5 @ Y0 ) )
= X5 ) ),
inference(cnf_transformation,[status(thm)],[f1635]) ).
thf(f1635,plain,
( ( ( eta1 = $true )
| ( ( sK450
!= ( lam @ sK449 @ sK448
@ ^ [Y0: $i] : ( ap @ sK449 @ sK448 @ sK450 @ Y0 ) ) )
& ( $true
= ( func @ sK449 @ sK448 @ sK450 ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( lam @ X4 @ X3
@ ^ [Y0: $i] : ( ap @ X4 @ X3 @ X5 @ Y0 ) )
= X5 )
| ( $true
!= ( func @ X4 @ X3 @ X5 ) ) )
| ( eta1 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK448,sK449,sK450])],[f1633,f1634]) ).
thf(f1634,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( lam @ X1 @ X0
@ ^ [Y0: $i] : ( ap @ X1 @ X0 @ X2 @ Y0 ) )
!= X2 )
& ( $true
= ( func @ X1 @ X0 @ X2 ) ) )
=> ( ( sK450
!= ( lam @ sK449 @ sK448
@ ^ [Y0: $i] : ( ap @ sK449 @ sK448 @ sK450 @ Y0 ) ) )
& ( $true
= ( func @ sK449 @ sK448 @ sK450 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1633,plain,
( ( ( eta1 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( lam @ X1 @ X0
@ ^ [Y0: $i] : ( ap @ X1 @ X0 @ X2 @ Y0 ) )
!= X2 )
& ( $true
= ( func @ X1 @ X0 @ X2 ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( lam @ X4 @ X3
@ ^ [Y0: $i] : ( ap @ X4 @ X3 @ X5 @ Y0 ) )
= X5 )
| ( $true
!= ( func @ X4 @ X3 @ X5 ) ) )
| ( eta1 != $true ) ) ),
inference(rectify,[status(thm)],[f1632]) ).
thf(f1632,plain,
( ( ( eta1 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( lam @ X1 @ X0
@ ^ [Y0: $i] : ( ap @ X1 @ X0 @ X2 @ Y0 ) )
!= X2 )
& ( $true
= ( func @ X1 @ X0 @ X2 ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( lam @ X1 @ X0
@ ^ [Y0: $i] : ( ap @ X1 @ X0 @ X2 @ Y0 ) )
= X2 )
| ( $true
!= ( func @ X1 @ X0 @ X2 ) ) )
| ( eta1 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f802]) ).
thf(f802,plain,
( ( eta1 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( lam @ X1 @ X0
@ ^ [Y0: $i] : ( ap @ X1 @ X0 @ X2 @ Y0 ) )
= X2 )
| ( $true
!= ( func @ X1 @ X0 @ X2 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f617]) ).
thf(f617,plain,
( ( eta1 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( func @ X1 @ X0 @ X2 ) )
=> ( ( lam @ X1 @ X0
@ ^ [Y0: $i] : ( ap @ X1 @ X0 @ X2 @ Y0 ) )
= X2 ) ) ),
inference(fool_elimination,[status(thm)],[f616]) ).
thf(f616,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( func @ X1 @ X0 @ X2 )
=> ( ( lam @ X1 @ X0
@ ^ [X3: $i] : ( ap @ X1 @ X0 @ X2 @ X3 ) )
= X2 ) )
= eta1 ),
inference(rectify,[status(thm)],[f218]) ).
thf(f218,axiom,
! [X4: $i,X3: $i,X12: $i] :
( ( ( func @ X3 @ X4 @ X12 )
=> ( ( lam @ X3 @ X4
@ ^ [X1: $i] : ( ap @ X3 @ X4 @ X12 @ X1 ) )
= X12 ) )
= eta1 ),
file('/export/starexec/sandbox2/tmp/tmp.GXJcYPDlCU/DTF2THF_6599.p',eta1) ).
thf(f3203,plain,
( ( lam @ sK341 @ sK343 @ ( ap @ sK341 @ sK343 @ sK342 ) )
!= sK342 ),
inference(beta_eta_normalization,[status(thm)],[f2601]) ).
thf(f2601,plain,
( ( lam @ sK341 @ sK343
@ ^ [Y0: $i] : ( ap @ sK341 @ sK343 @ sK342 @ Y0 ) )
!= sK342 ),
inference(cnf_transformation,[status(thm)],[f1474]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU699^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.10 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.12/0.29 % Computer : n008.cluster.edu
% 0.12/0.29 % Model : x86_64 x86_64
% 0.12/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.29 % Memory : 8042.1875MB
% 0.12/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.29 % CPULimit : 300
% 0.12/0.29 % WCLimit : 300
% 0.12/0.29 % DateTime : Tue Feb 24 12:31:24 EST 2026
% 0.12/0.29 % CPUTime :
% 0.12/0.29 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.38 ---- Original DTF file ---
% 0.16/0.39 thf(spec,logic,$$dhol).
% 0.16/0.39 %------------------------------------------------------------------------------
% 0.16/0.39 % File : SEU699^1 : TPTP v9.2.1. Released v3.7.0.
% 0.16/0.39 % Domain : Set Theory
% 0.16/0.39 % Problem : Functions - Extensionality and Beta Reduction
% 0.16/0.39 % Version : Especial.
% 0.16/0.39 % English : (! A:i.! B:i.! f:i.in f (funcSet A B) ->
% 0.16/0.39 % lam A B (^ x:i.ap A B f x) = f)
% 0.16/0.39
% 0.16/0.39 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.16/0.39 % Source : [Bro08]
% 0.16/0.39 % Names : ZFC201g [Bro08]
% 0.16/0.39
% 0.16/0.39 % Status : Theorem
% 0.16/0.39 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.38 v8.1.0, 0.27 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 0.80 v4.1.0, 1.00 v3.7.0
% 0.16/0.39 % Syntax : Number of formulae : 473 ( 220 unt; 252 typ; 220 def)
% 0.16/0.39 % Number of atoms : 1215 ( 330 equ; 0 cnn)
% 0.16/0.39 % Maximal formula atoms : 217 ( 5 avg)
% 0.16/0.39 % Number of connectives : 2342 ( 47 ~; 7 |; 35 &;1672 @)
% 0.16/0.39 % ( 14 <=>; 567 =>; 0 <=; 0 <~>)
% 0.16/0.39 % Maximal formula depth : 223 ( 2 avg)
% 0.16/0.39 % Number of types : 2 ( 0 usr)
% 0.16/0.39 % Number of type conns : 128 ( 128 >; 0 *; 0 +; 0 <<)
% 0.16/0.39 % Number of symbols : 255 ( 252 usr; 219 con; 0-4 aty)
% 0.16/0.39 % Number of variables : 731 ( 70 ^; 625 !; 36 ?; 731 :)
% 0.16/0.39 % SPC : TH0_THM_EQU_NAR
% 0.16/0.39
% 0.16/0.39 % Comments : http://mathgate.info/detsetitem.php?id=251
% 0.16/0.39 %------------------------------------------------------------------------------
% 0.16/0.39 thf(in_type,type,
% 0.16/0.39 in: $i > $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(exu_type,type,
% 0.16/0.39 exu: ( $i > $o ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(exu,definition,
% 0.16/0.39 ( exu
% 0.16/0.39 = ( ^ [Xphi: $i > $o] :
% 0.16/0.39 ? [Xx: $i] :
% 0.16/0.39 ( ( Xphi @ Xx )
% 0.16/0.39 & ! [Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setextAx_type,type,
% 0.16/0.39 setextAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setextAx,definition,
% 0.16/0.39 ( setextAx
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 <=> ( in @ Xx @ B ) )
% 0.16/0.39 => ( A = B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyset_type,type,
% 0.16/0.39 emptyset: $i ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetAx_type,type,
% 0.16/0.39 emptysetAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetAx,definition,
% 0.16/0.39 ( emptysetAx
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoin_type,type,
% 0.16/0.39 setadjoin: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinAx_type,type,
% 0.16/0.39 setadjoinAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinAx,definition,
% 0.16/0.39 ( setadjoinAx
% 0.16/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.39 <=> ( ( Xy = Xx )
% 0.16/0.39 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powerset_type,type,
% 0.16/0.39 powerset: $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(powersetAx_type,type,
% 0.16/0.39 powersetAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powersetAx,definition,
% 0.16/0.39 ( powersetAx
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.39 <=> ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunion_type,type,
% 0.16/0.39 setunion: $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(setunionAx_type,type,
% 0.16/0.39 setunionAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionAx,definition,
% 0.16/0.39 ( setunionAx
% 0.16/0.39 = ( ! [A: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.39 <=> ? [B: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 & ( in @ B @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(omega_type,type,
% 0.16/0.39 omega: $i ).
% 0.16/0.39
% 0.16/0.39 thf(omega0Ax_type,type,
% 0.16/0.39 omega0Ax: $o ).
% 0.16/0.39
% 0.16/0.39 thf(omega0Ax,definition,
% 0.16/0.39 ( omega0Ax
% 0.16/0.39 = ( in @ emptyset @ omega ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(omegaSAx_type,type,
% 0.16/0.39 omegaSAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(omegaSAx,definition,
% 0.16/0.39 ( omegaSAx
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ omega )
% 0.16/0.39 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(omegaIndAx_type,type,
% 0.16/0.39 omegaIndAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(omegaIndAx,definition,
% 0.16/0.39 ( omegaIndAx
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ( ( in @ emptyset @ A )
% 0.16/0.39 & ! [Xx: $i] :
% 0.16/0.39 ( ( ( in @ Xx @ omega )
% 0.16/0.39 & ( in @ Xx @ A ) )
% 0.16/0.39 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ omega )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(replAx_type,type,
% 0.16/0.39 replAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(replAx,definition,
% 0.16/0.39 ( replAx
% 0.16/0.39 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( exu
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.16/0.39 => ? [B: $i] :
% 0.16/0.39 ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 <=> ? [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ A )
% 0.16/0.39 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(foundationAx_type,type,
% 0.16/0.39 foundationAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(foundationAx,definition,
% 0.16/0.39 ( foundationAx
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.39 => ? [B: $i] :
% 0.16/0.39 ( ( in @ B @ A )
% 0.16/0.39 & ~ ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(wellorderingAx_type,type,
% 0.16/0.39 wellorderingAx: $o ).
% 0.16/0.39
% 0.16/0.39 thf(wellorderingAx,definition,
% 0.16/0.39 ( wellorderingAx
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ? [B: $i] :
% 0.16/0.39 ( ! [C: $i] :
% 0.16/0.39 ( ( in @ C @ B )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ C )
% 0.16/0.39 => ( in @ Xx @ A ) ) )
% 0.16/0.39 & ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( ( in @ Xx @ A )
% 0.16/0.39 & ( in @ Xy @ A ) )
% 0.16/0.39 => ( ! [C: $i] :
% 0.16/0.39 ( ( in @ C @ B )
% 0.16/0.39 => ( ( in @ Xx @ C )
% 0.16/0.39 <=> ( in @ Xy @ C ) ) )
% 0.16/0.39 => ( Xx = Xy ) ) )
% 0.16/0.39 & ! [C: $i,D: $i] :
% 0.16/0.39 ( ( ( in @ C @ B )
% 0.16/0.39 & ( in @ D @ B ) )
% 0.16/0.39 => ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ C )
% 0.16/0.39 => ( in @ Xx @ D ) )
% 0.16/0.39 | ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ D )
% 0.16/0.39 => ( in @ Xx @ C ) ) ) )
% 0.16/0.39 & ! [C: $i] :
% 0.16/0.39 ( ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ C )
% 0.16/0.39 => ( in @ Xx @ A ) )
% 0.16/0.39 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.16/0.39 => ? [D: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ D @ B )
% 0.16/0.39 & ( in @ Xx @ C )
% 0.16/0.39 & ~ ? [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ D )
% 0.16/0.39 & ( in @ Xy @ C ) )
% 0.16/0.39 & ! [E: $i] :
% 0.16/0.39 ( ( in @ E @ B )
% 0.16/0.39 => ( ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ E )
% 0.16/0.39 => ( in @ Xy @ D ) )
% 0.16/0.39 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(descr_type,type,
% 0.16/0.39 descr: ( $i > $o ) > $i ).
% 0.16/0.39
% 0.16/0.39 thf(descrp_type,type,
% 0.16/0.39 descrp: $o ).
% 0.16/0.39
% 0.16/0.39 thf(descrp,definition,
% 0.16/0.39 ( descrp
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ( Xphi
% 0.16/0.39 @ ( descr
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstr_type,type,
% 0.16/0.39 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrI_type,type,
% 0.16/0.39 dsetconstrI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrI,definition,
% 0.16/0.39 ( dsetconstrI
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 => ( in @ Xx
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrEL_type,type,
% 0.16/0.39 dsetconstrEL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrEL,definition,
% 0.16/0.39 ( dsetconstrEL
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrER_type,type,
% 0.16/0.39 dsetconstrER: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstrER,definition,
% 0.16/0.39 ( dsetconstrER
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.39 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuE1_type,type,
% 0.16/0.39 exuE1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuE1,definition,
% 0.16/0.39 ( exuE1
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ( ( Xphi @ Xx )
% 0.16/0.39 & ! [Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(prop2set_type,type,
% 0.16/0.39 prop2set: $o > $i ).
% 0.16/0.39
% 0.16/0.39 thf(prop2setE_type,type,
% 0.16/0.39 prop2setE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(prop2setE,definition,
% 0.16/0.39 ( prop2setE
% 0.16/0.39 = ( ! [Xphi: $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.16/0.39 => Xphi ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetE_type,type,
% 0.16/0.39 emptysetE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetE,definition,
% 0.16/0.39 ( emptysetE
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ emptyset )
% 0.16/0.39 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetimpfalse_type,type,
% 0.16/0.39 emptysetimpfalse: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetimpfalse,definition,
% 0.16/0.39 ( emptysetimpfalse
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ emptyset )
% 0.16/0.39 => $false ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(notinemptyset_type,type,
% 0.16/0.39 notinemptyset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(notinemptyset,definition,
% 0.16/0.39 ( notinemptyset
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuE3e_type,type,
% 0.16/0.39 exuE3e: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuE3e,definition,
% 0.16/0.39 ( exuE3e
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setext_type,type,
% 0.16/0.39 setext: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setext,definition,
% 0.16/0.39 ( setext
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xx @ B ) )
% 0.16/0.39 => ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ A ) )
% 0.16/0.39 => ( A = B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyI_type,type,
% 0.16/0.39 emptyI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptyI,definition,
% 0.16/0.39 ( emptyI
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ~ ( in @ Xx @ A )
% 0.16/0.39 => ( A = emptyset ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(noeltsimpempty_type,type,
% 0.16/0.39 noeltsimpempty: $o ).
% 0.16/0.39
% 0.16/0.39 thf(noeltsimpempty,definition,
% 0.16/0.39 ( noeltsimpempty
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ~ ( in @ Xx @ A )
% 0.16/0.39 => ( A = emptyset ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setbeta_type,type,
% 0.16/0.39 setbeta: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setbeta,definition,
% 0.16/0.39 ( setbeta
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( in @ Xx
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.39 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(nonempty_type,type,
% 0.16/0.39 nonempty: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(nonempty,definition,
% 0.16/0.39 ( nonempty
% 0.16/0.39 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyE1_type,type,
% 0.16/0.39 nonemptyE1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyE1,definition,
% 0.16/0.39 ( nonemptyE1
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ( nonempty @ A )
% 0.16/0.39 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyI_type,type,
% 0.16/0.39 nonemptyI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyI,definition,
% 0.16/0.39 ( nonemptyI
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 => ( nonempty
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyI1_type,type,
% 0.16/0.39 nonemptyI1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyI1,definition,
% 0.16/0.39 ( nonemptyI1
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.39 => ( nonempty @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinIL_type,type,
% 0.16/0.39 setadjoinIL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinIL,definition,
% 0.16/0.39 ( setadjoinIL
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyinunitempty_type,type,
% 0.16/0.39 emptyinunitempty: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptyinunitempty,definition,
% 0.16/0.39 ( emptyinunitempty
% 0.16/0.39 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinIR_type,type,
% 0.16/0.39 setadjoinIR: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinIR,definition,
% 0.16/0.39 ( setadjoinIR
% 0.16/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ A )
% 0.16/0.39 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinE_type,type,
% 0.16/0.39 setadjoinE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinE,definition,
% 0.16/0.39 ( setadjoinE
% 0.16/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.39 => ! [Xphi: $o] :
% 0.16/0.39 ( ( ( Xy = Xx )
% 0.16/0.39 => Xphi )
% 0.16/0.39 => ( ( ( in @ Xy @ A )
% 0.16/0.39 => Xphi )
% 0.16/0.39 => Xphi ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinOr_type,type,
% 0.16/0.39 setadjoinOr: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinOr,definition,
% 0.16/0.39 ( setadjoinOr
% 0.16/0.39 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.39 => ( ( Xy = Xx )
% 0.16/0.39 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setoftrueEq_type,type,
% 0.16/0.39 setoftrueEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setoftrueEq,definition,
% 0.16/0.39 ( setoftrueEq
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xx: $i] : $true )
% 0.16/0.39 = A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powersetI_type,type,
% 0.16/0.39 powersetI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powersetI,definition,
% 0.16/0.39 ( powersetI
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ A ) )
% 0.16/0.39 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyinPowerset_type,type,
% 0.16/0.39 emptyinPowerset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptyinPowerset,definition,
% 0.16/0.39 ( emptyinPowerset
% 0.16/0.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyInPowerset_type,type,
% 0.16/0.39 emptyInPowerset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptyInPowerset,definition,
% 0.16/0.39 ( emptyInPowerset
% 0.16/0.39 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powersetE_type,type,
% 0.16/0.39 powersetE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powersetE,definition,
% 0.16/0.39 ( powersetE
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.39 => ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunionI_type,type,
% 0.16/0.39 setunionI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionI,definition,
% 0.16/0.39 ( setunionI
% 0.16/0.39 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( ( in @ B @ A )
% 0.16/0.39 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunionE_type,type,
% 0.16/0.39 setunionE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionE,definition,
% 0.16/0.39 ( setunionE
% 0.16/0.39 = ( ! [A: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.39 => ! [Xphi: $o] :
% 0.16/0.39 ( ! [B: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( ( in @ B @ A )
% 0.16/0.39 => Xphi ) )
% 0.16/0.39 => Xphi ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subPowSU_type,type,
% 0.16/0.39 subPowSU: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subPowSU,definition,
% 0.16/0.39 ( subPowSU
% 0.16/0.39 = ( ! [A: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuE2_type,type,
% 0.16/0.39 exuE2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuE2,definition,
% 0.16/0.39 ( exuE2
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ! [Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xy )
% 0.16/0.39 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyImpWitness_type,type,
% 0.16/0.39 nonemptyImpWitness: $o ).
% 0.16/0.39
% 0.16/0.39 thf(nonemptyImpWitness,definition,
% 0.16/0.39 ( nonemptyImpWitness
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ( nonempty @ A )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & $true ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(uniqinunit_type,type,
% 0.16/0.39 uniqinunit: $o ).
% 0.16/0.39
% 0.16/0.39 thf(uniqinunit,definition,
% 0.16/0.39 ( uniqinunit
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(notinsingleton_type,type,
% 0.16/0.39 notinsingleton: $o ).
% 0.16/0.39
% 0.16/0.39 thf(notinsingleton,definition,
% 0.16/0.39 ( notinsingleton
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx != Xy )
% 0.16/0.39 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(eqinunit_type,type,
% 0.16/0.39 eqinunit: $o ).
% 0.16/0.39
% 0.16/0.39 thf(eqinunit,definition,
% 0.16/0.39 ( eqinunit
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(singletonsswitch_type,type,
% 0.16/0.39 singletonsswitch: $o ).
% 0.16/0.39
% 0.16/0.39 thf(singletonsswitch,definition,
% 0.16/0.39 ( singletonsswitch
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.39 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(upairsetE_type,type,
% 0.16/0.39 upairsetE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairsetE,definition,
% 0.16/0.39 ( upairsetE
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.39 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.39 => ( ( Xz = Xx )
% 0.16/0.39 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(upairsetIL_type,type,
% 0.16/0.39 upairsetIL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairsetIL,definition,
% 0.16/0.39 ( upairsetIL
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(upairsetIR_type,type,
% 0.16/0.39 upairsetIR: $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairsetIR,definition,
% 0.16/0.39 ( upairsetIR
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyE1_type,type,
% 0.16/0.39 emptyE1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptyE1,definition,
% 0.16/0.39 ( emptyE1
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( Xphi @ Xx ) )
% 0.16/0.39 => ( ( ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 = emptyset )
% 0.16/0.39 => $false ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(vacuousDall_type,type,
% 0.16/0.39 vacuousDall: $o ).
% 0.16/0.39
% 0.16/0.39 thf(vacuousDall,definition,
% 0.16/0.39 ( vacuousDall
% 0.16/0.39 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ emptyset )
% 0.16/0.39 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(quantDeMorgan1_type,type,
% 0.16/0.39 quantDeMorgan1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(quantDeMorgan1,definition,
% 0.16/0.39 ( quantDeMorgan1
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ~ ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( Xphi @ Xx ) )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(quantDeMorgan2_type,type,
% 0.16/0.39 quantDeMorgan2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(quantDeMorgan2,definition,
% 0.16/0.39 ( quantDeMorgan2
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ~ ( Xphi @ Xx ) )
% 0.16/0.39 => ~ ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(quantDeMorgan3_type,type,
% 0.16/0.39 quantDeMorgan3: $o ).
% 0.16/0.39
% 0.16/0.39 thf(quantDeMorgan3,definition,
% 0.16/0.39 ( quantDeMorgan3
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ~ ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( Xphi @ Xx ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(quantDeMorgan4_type,type,
% 0.16/0.39 quantDeMorgan4: $o ).
% 0.16/0.39
% 0.16/0.39 thf(quantDeMorgan4,definition,
% 0.16/0.39 ( quantDeMorgan4
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ~ ( Xphi @ Xx ) )
% 0.16/0.39 => ~ ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(prop2setI_type,type,
% 0.16/0.39 prop2setI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(prop2setI,definition,
% 0.16/0.39 ( prop2setI
% 0.16/0.39 = ( ! [Xphi: $o] :
% 0.16/0.39 ( Xphi
% 0.16/0.39 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(set2prop_type,type,
% 0.16/0.39 set2prop: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(prop2set2propI_type,type,
% 0.16/0.39 prop2set2propI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(prop2set2propI,definition,
% 0.16/0.39 ( prop2set2propI
% 0.16/0.39 = ( ! [Xphi: $o] :
% 0.16/0.39 ( Xphi
% 0.16/0.39 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(notdexE_type,type,
% 0.16/0.39 notdexE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(notdexE,definition,
% 0.16/0.39 ( notdexE
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ~ ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( Xphi @ Xx ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(notdallE_type,type,
% 0.16/0.39 notdallE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(notdallE,definition,
% 0.16/0.39 ( notdallE
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ~ ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( Xphi @ Xx ) )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuI1_type,type,
% 0.16/0.39 exuI1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuI1,definition,
% 0.16/0.39 ( exuI1
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ? [Xx: $i] :
% 0.16/0.39 ( ( Xphi @ Xx )
% 0.16/0.39 & ! [Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) )
% 0.16/0.39 => ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuI3_type,type,
% 0.16/0.39 exuI3: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuI3,definition,
% 0.16/0.39 ( exuI3
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.16/0.39 => ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xx )
% 0.16/0.39 => ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) )
% 0.16/0.39 => ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuI2_type,type,
% 0.16/0.39 exuI2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuI2,definition,
% 0.16/0.39 ( exuI2
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ? [Xx: $i] :
% 0.16/0.39 ! [Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xy )
% 0.16/0.39 <=> ( Xy = Xx ) )
% 0.16/0.39 => ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(inCongP_type,type,
% 0.16/0.39 inCongP: $o ).
% 0.16/0.39
% 0.16/0.39 thf(inCongP,definition,
% 0.16/0.39 ( inCongP
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( A = B )
% 0.16/0.39 => ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(in__Cong_type,type,
% 0.16/0.39 in__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(in__Cong,definition,
% 0.16/0.39 ( in__Cong
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( A = B )
% 0.16/0.39 => ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ( ( in @ Xx @ A )
% 0.16/0.39 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuE3u_type,type,
% 0.16/0.39 exuE3u: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuE3u,definition,
% 0.16/0.39 ( exuE3u
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xx )
% 0.16/0.39 => ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exu__Cong_type,type,
% 0.16/0.39 exu__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exu__Cong,definition,
% 0.16/0.39 ( exu__Cong
% 0.16/0.39 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.39 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.39 => ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 <=> ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptyset__Cong_type,type,
% 0.16/0.39 emptyset__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptyset__Cong,definition,
% 0.16/0.39 ( emptyset__Cong
% 0.16/0.39 = ( emptyset = emptyset ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoin__Cong_type,type,
% 0.16/0.39 setadjoin__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoin__Cong,definition,
% 0.16/0.39 ( setadjoin__Cong
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ! [Xz: $i,Xu: $i] :
% 0.16/0.39 ( ( Xz = Xu )
% 0.16/0.39 => ( ( setadjoin @ Xx @ Xz )
% 0.16/0.39 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powerset__Cong_type,type,
% 0.16/0.39 powerset__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powerset__Cong,definition,
% 0.16/0.39 ( powerset__Cong
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( A = B )
% 0.16/0.39 => ( ( powerset @ A )
% 0.16/0.39 = ( powerset @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunion__Cong_type,type,
% 0.16/0.39 setunion__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunion__Cong,definition,
% 0.16/0.39 ( setunion__Cong
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( A = B )
% 0.16/0.39 => ( ( setunion @ A )
% 0.16/0.39 = ( setunion @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(omega__Cong_type,type,
% 0.16/0.39 omega__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(omega__Cong,definition,
% 0.16/0.39 ( omega__Cong
% 0.16/0.39 = ( omega = omega ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(exuEu_type,type,
% 0.16/0.39 exuEu: $o ).
% 0.16/0.39
% 0.16/0.39 thf(exuEu,definition,
% 0.16/0.39 ( exuEu
% 0.16/0.39 = ( ! [Xphi: $i > $o] :
% 0.16/0.39 ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xphi @ Xx )
% 0.16/0.39 => ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(descr__Cong_type,type,
% 0.16/0.39 descr__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(descr__Cong,definition,
% 0.16/0.39 ( descr__Cong
% 0.16/0.39 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.39 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.39 => ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ( ( exu
% 0.16/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.16/0.39 => ( ( descr
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 = ( descr
% 0.16/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstr__Cong_type,type,
% 0.16/0.39 dsetconstr__Cong: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dsetconstr__Cong,definition,
% 0.16/0.39 ( dsetconstr__Cong
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( A = B )
% 0.16/0.39 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( Xx = Xy )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.16/0.39 => ( ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 = ( dsetconstr @ B
% 0.16/0.39 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subset_type,type,
% 0.16/0.39 subset: $i > $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(disjoint_type,type,
% 0.16/0.39 disjoint: $i > $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(setsmeet_type,type,
% 0.16/0.39 setsmeet: $i > $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(subsetI1_type,type,
% 0.16/0.39 subsetI1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subsetI1,definition,
% 0.16/0.39 ( subsetI1
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xx @ B ) )
% 0.16/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(eqimpsubset2_type,type,
% 0.16/0.39 eqimpsubset2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(eqimpsubset2,definition,
% 0.16/0.39 ( eqimpsubset2
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( A = B )
% 0.16/0.39 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(eqimpsubset1_type,type,
% 0.16/0.39 eqimpsubset1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(eqimpsubset1,definition,
% 0.16/0.39 ( eqimpsubset1
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( A = B )
% 0.16/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subsetI2_type,type,
% 0.16/0.39 subsetI2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subsetI2,definition,
% 0.16/0.39 ( subsetI2
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xx @ B ) )
% 0.16/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetsubset_type,type,
% 0.16/0.39 emptysetsubset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(emptysetsubset,definition,
% 0.16/0.39 ( emptysetsubset
% 0.16/0.39 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subsetE_type,type,
% 0.16/0.39 subsetE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subsetE,definition,
% 0.16/0.39 ( subsetE
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subsetE2_type,type,
% 0.16/0.39 subsetE2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subsetE2,definition,
% 0.16/0.39 ( subsetE2
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( ~ ( in @ Xx @ B )
% 0.16/0.39 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(notsubsetI_type,type,
% 0.16/0.39 notsubsetI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(notsubsetI,definition,
% 0.16/0.39 ( notsubsetI
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ~ ( in @ Xx @ B )
% 0.16/0.39 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(notequalI1_type,type,
% 0.16/0.39 notequalI1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(notequalI1,definition,
% 0.16/0.39 ( notequalI1
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ~ ( subset @ A @ B )
% 0.16/0.39 => ( A != B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(notequalI2_type,type,
% 0.16/0.39 notequalI2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(notequalI2,definition,
% 0.16/0.39 ( notequalI2
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ~ ( in @ Xx @ B )
% 0.16/0.39 => ( A != B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subsetRefl_type,type,
% 0.16/0.39 subsetRefl: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subsetRefl,definition,
% 0.16/0.39 ( subsetRefl
% 0.16/0.39 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subsetTrans_type,type,
% 0.16/0.39 subsetTrans: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subsetTrans,definition,
% 0.16/0.39 ( subsetTrans
% 0.16/0.39 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( ( subset @ B @ C )
% 0.16/0.39 => ( subset @ A @ C ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinSub_type,type,
% 0.16/0.39 setadjoinSub: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinSub,definition,
% 0.16/0.39 ( setadjoinSub
% 0.16/0.39 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinSub2_type,type,
% 0.16/0.39 setadjoinSub2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setadjoinSub2,definition,
% 0.16/0.39 ( setadjoinSub2
% 0.16/0.39 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subset2powerset_type,type,
% 0.16/0.39 subset2powerset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subset2powerset,definition,
% 0.16/0.39 ( subset2powerset
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setextsub_type,type,
% 0.16/0.39 setextsub: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setextsub,definition,
% 0.16/0.39 ( setextsub
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( ( subset @ B @ A )
% 0.16/0.39 => ( A = B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subsetemptysetimpeq_type,type,
% 0.16/0.39 subsetemptysetimpeq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subsetemptysetimpeq,definition,
% 0.16/0.39 ( subsetemptysetimpeq
% 0.16/0.39 = ( ! [A: $i] :
% 0.16/0.39 ( ( subset @ A @ emptyset )
% 0.16/0.39 => ( A = emptyset ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powersetI1_type,type,
% 0.16/0.39 powersetI1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powersetI1,definition,
% 0.16/0.39 ( powersetI1
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( subset @ B @ A )
% 0.16/0.39 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powersetE1_type,type,
% 0.16/0.39 powersetE1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powersetE1,definition,
% 0.16/0.39 ( powersetE1
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.39 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(inPowerset_type,type,
% 0.16/0.39 inPowerset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(inPowerset,definition,
% 0.16/0.39 ( inPowerset
% 0.16/0.39 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(powersetsubset_type,type,
% 0.16/0.39 powersetsubset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(powersetsubset,definition,
% 0.16/0.39 ( powersetsubset
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(sepInPowerset_type,type,
% 0.16/0.39 sepInPowerset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(sepInPowerset,definition,
% 0.16/0.39 ( sepInPowerset
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( in
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 @ ( powerset @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(sepSubset_type,type,
% 0.16/0.39 sepSubset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(sepSubset,definition,
% 0.16/0.39 ( sepSubset
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( subset
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binunion_type,type,
% 0.16/0.39 binunion: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(binunionIL_type,type,
% 0.16/0.39 binunionIL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binunionIL,definition,
% 0.16/0.39 ( binunionIL
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(upairset2IR_type,type,
% 0.16/0.39 upairset2IR: $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairset2IR,definition,
% 0.16/0.39 ( upairset2IR
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binunionIR_type,type,
% 0.16/0.39 binunionIR: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binunionIR,definition,
% 0.16/0.39 ( binunionIR
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binunionEcases_type,type,
% 0.16/0.39 binunionEcases: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binunionEcases,definition,
% 0.16/0.39 ( binunionEcases
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.16/0.39 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.39 => ( ( ( in @ Xx @ A )
% 0.16/0.39 => Xphi )
% 0.16/0.39 => ( ( ( in @ Xx @ B )
% 0.16/0.39 => Xphi )
% 0.16/0.39 => Xphi ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binunionE_type,type,
% 0.16/0.39 binunionE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binunionE,definition,
% 0.16/0.39 ( binunionE
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.39 => ( ( in @ Xx @ A )
% 0.16/0.39 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binunionLsub_type,type,
% 0.16/0.39 binunionLsub: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binunionLsub,definition,
% 0.16/0.39 ( binunionLsub
% 0.16/0.39 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binunionRsub_type,type,
% 0.16/0.39 binunionRsub: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binunionRsub,definition,
% 0.16/0.39 ( binunionRsub
% 0.16/0.39 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersect_type,type,
% 0.16/0.39 binintersect: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectI_type,type,
% 0.16/0.39 binintersectI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectI,definition,
% 0.16/0.39 ( binintersectI
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset5_type,type,
% 0.16/0.39 binintersectSubset5: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset5,definition,
% 0.16/0.39 ( binintersectSubset5
% 0.16/0.39 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.39 ( ( subset @ C @ A )
% 0.16/0.39 => ( ( subset @ C @ B )
% 0.16/0.39 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectEL_type,type,
% 0.16/0.39 binintersectEL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectEL,definition,
% 0.16/0.39 ( binintersectEL
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectLsub_type,type,
% 0.16/0.39 binintersectLsub: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectLsub,definition,
% 0.16/0.39 ( binintersectLsub
% 0.16/0.39 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset2_type,type,
% 0.16/0.39 binintersectSubset2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset2,definition,
% 0.16/0.39 ( binintersectSubset2
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( ( binintersect @ A @ B )
% 0.16/0.39 = A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset3_type,type,
% 0.16/0.39 binintersectSubset3: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset3,definition,
% 0.16/0.39 ( binintersectSubset3
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( ( binintersect @ A @ B )
% 0.16/0.39 = B )
% 0.16/0.39 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectER_type,type,
% 0.16/0.39 binintersectER: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectER,definition,
% 0.16/0.39 ( binintersectER
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.39 => ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(disjointsetsI1_type,type,
% 0.16/0.39 disjointsetsI1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(disjointsetsI1,definition,
% 0.16/0.39 ( disjointsetsI1
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ~ ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( in @ Xx @ B ) )
% 0.16/0.39 => ( ( binintersect @ A @ B )
% 0.16/0.39 = emptyset ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectRsub_type,type,
% 0.16/0.39 binintersectRsub: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectRsub,definition,
% 0.16/0.39 ( binintersectRsub
% 0.16/0.39 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset4_type,type,
% 0.16/0.39 binintersectSubset4: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset4,definition,
% 0.16/0.39 ( binintersectSubset4
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( subset @ B @ A )
% 0.16/0.39 => ( ( binintersect @ A @ B )
% 0.16/0.39 = B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset1_type,type,
% 0.16/0.39 binintersectSubset1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(binintersectSubset1,definition,
% 0.16/0.39 ( binintersectSubset1
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( ( binintersect @ A @ B )
% 0.16/0.39 = A )
% 0.16/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(bs114d_type,type,
% 0.16/0.39 bs114d: $o ).
% 0.16/0.39
% 0.16/0.39 thf(bs114d,definition,
% 0.16/0.39 ( bs114d
% 0.16/0.39 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.39 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.16/0.39 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(regular_type,type,
% 0.16/0.39 regular: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminus_type,type,
% 0.16/0.39 setminus: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(setminusI_type,type,
% 0.16/0.39 setminusI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusI,definition,
% 0.16/0.39 ( setminusI
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ~ ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusEL_type,type,
% 0.16/0.39 setminusEL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusEL,definition,
% 0.16/0.39 ( setminusEL
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusER_type,type,
% 0.16/0.39 setminusER: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusER,definition,
% 0.16/0.39 ( setminusER
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.39 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusSubset2_type,type,
% 0.16/0.39 setminusSubset2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusSubset2,definition,
% 0.16/0.39 ( setminusSubset2
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( subset @ A @ B )
% 0.16/0.39 => ( ( setminus @ A @ B )
% 0.16/0.39 = emptyset ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusERneg_type,type,
% 0.16/0.39 setminusERneg: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusERneg,definition,
% 0.16/0.39 ( setminusERneg
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.39 => ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusELneg_type,type,
% 0.16/0.39 setminusELneg: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusELneg,definition,
% 0.16/0.39 ( setminusELneg
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.39 => ( ~ ( in @ Xx @ B )
% 0.16/0.39 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusILneg_type,type,
% 0.16/0.39 setminusILneg: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusILneg,definition,
% 0.16/0.39 ( setminusILneg
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ~ ( in @ Xx @ A )
% 0.16/0.39 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusIRneg_type,type,
% 0.16/0.39 setminusIRneg: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusIRneg,definition,
% 0.16/0.39 ( setminusIRneg
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ B )
% 0.16/0.39 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusLsub_type,type,
% 0.16/0.39 setminusLsub: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusLsub,definition,
% 0.16/0.39 ( setminusLsub
% 0.16/0.39 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setminusSubset1_type,type,
% 0.16/0.39 setminusSubset1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setminusSubset1,definition,
% 0.16/0.39 ( setminusSubset1
% 0.16/0.39 = ( ! [A: $i,B: $i] :
% 0.16/0.39 ( ( ( setminus @ A @ B )
% 0.16/0.39 = emptyset )
% 0.16/0.39 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(symdiff_type,type,
% 0.16/0.39 symdiff: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffE_type,type,
% 0.16/0.39 symdiffE: $o ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffE,definition,
% 0.16/0.39 ( symdiffE
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.16/0.39 => ! [Xphi: $o] :
% 0.16/0.39 ( ( ( in @ Xx @ A )
% 0.16/0.39 => ( ~ ( in @ Xx @ B )
% 0.16/0.39 => Xphi ) )
% 0.16/0.39 => ( ( ~ ( in @ Xx @ A )
% 0.16/0.39 => ( ( in @ Xx @ B )
% 0.16/0.39 => Xphi ) )
% 0.16/0.39 => Xphi ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffI1_type,type,
% 0.16/0.39 symdiffI1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffI1,definition,
% 0.16/0.39 ( symdiffI1
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ~ ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffI2_type,type,
% 0.16/0.39 symdiffI2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffI2,definition,
% 0.16/0.39 ( symdiffI2
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ~ ( in @ Xx @ A )
% 0.16/0.39 => ( ( in @ Xx @ B )
% 0.16/0.39 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffIneg1_type,type,
% 0.16/0.39 symdiffIneg1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffIneg1,definition,
% 0.16/0.39 ( symdiffIneg1
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( in @ Xx @ B )
% 0.16/0.39 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffIneg2_type,type,
% 0.16/0.39 symdiffIneg2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(symdiffIneg2,definition,
% 0.16/0.39 ( symdiffIneg2
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ~ ( in @ Xx @ A )
% 0.16/0.39 => ( ~ ( in @ Xx @ B )
% 0.16/0.39 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(iskpair_type,type,
% 0.16/0.39 iskpair: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(secondinupair_type,type,
% 0.16/0.39 secondinupair: $o ).
% 0.16/0.39
% 0.16/0.39 thf(secondinupair,definition,
% 0.16/0.39 ( secondinupair
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairIL_type,type,
% 0.16/0.39 setukpairIL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairIL,definition,
% 0.16/0.39 ( setukpairIL
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairIR_type,type,
% 0.16/0.39 setukpairIR: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairIR,definition,
% 0.16/0.39 ( setukpairIR
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(kpairiskpair_type,type,
% 0.16/0.39 kpairiskpair: $o ).
% 0.16/0.39
% 0.16/0.39 thf(kpairiskpair,definition,
% 0.16/0.39 ( kpairiskpair
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(kpair_type,type,
% 0.16/0.39 kpair: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(kpairp_type,type,
% 0.16/0.39 kpairp: $o ).
% 0.16/0.39
% 0.16/0.39 thf(kpairp,definition,
% 0.16/0.39 ( kpairp
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprod_type,type,
% 0.16/0.39 cartprod: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(singletonsubset_type,type,
% 0.16/0.39 singletonsubset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(singletonsubset,definition,
% 0.16/0.39 ( singletonsubset
% 0.16/0.39 = ( ! [A: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(singletoninpowerset_type,type,
% 0.16/0.39 singletoninpowerset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(singletoninpowerset,definition,
% 0.16/0.39 ( singletoninpowerset
% 0.16/0.39 = ( ! [A: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(singletoninpowunion_type,type,
% 0.16/0.39 singletoninpowunion: $o ).
% 0.16/0.39
% 0.16/0.39 thf(singletoninpowunion,definition,
% 0.16/0.39 ( singletoninpowunion
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(upairset2E_type,type,
% 0.16/0.39 upairset2E: $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairset2E,definition,
% 0.16/0.39 ( upairset2E
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.39 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.39 => ( ( Xz = Xx )
% 0.16/0.39 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(upairsubunion_type,type,
% 0.16/0.39 upairsubunion: $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairsubunion,definition,
% 0.16/0.39 ( upairsubunion
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(upairinpowunion_type,type,
% 0.16/0.39 upairinpowunion: $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairinpowunion,definition,
% 0.16/0.39 ( upairinpowunion
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ubforcartprodlem1_type,type,
% 0.16/0.39 ubforcartprodlem1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ubforcartprodlem1,definition,
% 0.16/0.39 ( ubforcartprodlem1
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ubforcartprodlem2_type,type,
% 0.16/0.39 ubforcartprodlem2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ubforcartprodlem2,definition,
% 0.16/0.39 ( ubforcartprodlem2
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ubforcartprodlem3_type,type,
% 0.16/0.39 ubforcartprodlem3: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ubforcartprodlem3,definition,
% 0.16/0.39 ( ubforcartprodlem3
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodpairin_type,type,
% 0.16/0.39 cartprodpairin: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodpairin,definition,
% 0.16/0.39 ( cartprodpairin
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodmempair1_type,type,
% 0.16/0.39 cartprodmempair1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodmempair1,definition,
% 0.16/0.39 ( cartprodmempair1
% 0.16/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ? [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 & ( Xu
% 0.16/0.39 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodmempair_type,type,
% 0.16/0.39 cartprodmempair: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodmempair,definition,
% 0.16/0.39 ( cartprodmempair
% 0.16/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.39 => ( iskpair @ Xu ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunionE2_type,type,
% 0.16/0.39 setunionE2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionE2,definition,
% 0.16/0.39 ( setunionE2
% 0.16/0.39 = ( ! [A: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.39 => ? [X: $i] :
% 0.16/0.39 ( ( in @ X @ A )
% 0.16/0.39 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunionsingleton1_type,type,
% 0.16/0.39 setunionsingleton1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionsingleton1,definition,
% 0.16/0.39 ( setunionsingleton1
% 0.16/0.39 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunionsingleton2_type,type,
% 0.16/0.39 setunionsingleton2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionsingleton2,definition,
% 0.16/0.39 ( setunionsingleton2
% 0.16/0.39 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setunionsingleton_type,type,
% 0.16/0.39 setunionsingleton: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setunionsingleton,definition,
% 0.16/0.39 ( setunionsingleton
% 0.16/0.39 = ( ! [Xx: $i] :
% 0.16/0.39 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.16/0.39 = Xx ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(singleton_type,type,
% 0.16/0.39 singleton: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(singleton,definition,
% 0.16/0.39 ( singleton
% 0.16/0.39 = ( ^ [A: $i] :
% 0.16/0.39 ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( A
% 0.16/0.39 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(singletonprop_type,type,
% 0.16/0.39 singletonprop: $o ).
% 0.16/0.39
% 0.16/0.39 thf(singletonprop,definition,
% 0.16/0.39 ( singletonprop
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ A )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 => ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) )
% 0.16/0.39 => ( ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( Xphi @ Xx ) )
% 0.16/0.39 => ( singleton
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ex1_type,type,
% 0.16/0.39 ex1: $i > ( $i > $o ) > $o ).
% 0.16/0.39
% 0.16/0.39 thf(ex1,definition,
% 0.16/0.39 ( ex1
% 0.16/0.39 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( singleton
% 0.16/0.39 @ ( dsetconstr @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ex1E1_type,type,
% 0.16/0.39 ex1E1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ex1E1,definition,
% 0.16/0.39 ( ex1E1
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ( ex1 @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ex1I_type,type,
% 0.16/0.39 ex1I: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ex1I,definition,
% 0.16/0.39 ( ex1I
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 => ( ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ A )
% 0.16/0.39 => ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xy = Xx ) ) )
% 0.16/0.39 => ( ex1 @ A
% 0.16/0.39 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ex1I2_type,type,
% 0.16/0.39 ex1I2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ex1I2,definition,
% 0.16/0.39 ( ex1I2
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ A )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 => ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) )
% 0.16/0.39 => ( ? [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 & ( Xphi @ Xx ) )
% 0.16/0.39 => ( ex1 @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(singletonsuniq_type,type,
% 0.16/0.39 singletonsuniq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(singletonsuniq,definition,
% 0.16/0.39 ( singletonsuniq
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( ( setadjoin @ Xx @ emptyset )
% 0.16/0.39 = ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(atmost1p_type,type,
% 0.16/0.39 atmost1p: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(atleast2p_type,type,
% 0.16/0.39 atleast2p: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(atmost2p_type,type,
% 0.16/0.39 atmost2p: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairsetp_type,type,
% 0.16/0.39 upairsetp: $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjL1_type,type,
% 0.16/0.39 setukpairinjL1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjL1,definition,
% 0.16/0.39 ( setukpairinjL1
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.39 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.39 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(kfstsingleton_type,type,
% 0.16/0.39 kfstsingleton: $o ).
% 0.16/0.39
% 0.16/0.39 thf(kfstsingleton,definition,
% 0.16/0.39 ( kfstsingleton
% 0.16/0.39 = ( ! [Xu: $i] :
% 0.16/0.39 ( ( iskpair @ Xu )
% 0.16/0.39 => ( singleton
% 0.16/0.39 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.39 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(theprop_type,type,
% 0.16/0.39 theprop: $o ).
% 0.16/0.39
% 0.16/0.39 thf(theprop,definition,
% 0.16/0.39 ( theprop
% 0.16/0.39 = ( ! [X: $i] :
% 0.16/0.39 ( ( singleton @ X )
% 0.16/0.39 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(kfst_type,type,
% 0.16/0.39 kfst: $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(kfstpairEq_type,type,
% 0.16/0.39 kfstpairEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(kfstpairEq,definition,
% 0.16/0.39 ( kfstpairEq
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.39 = Xx ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodfstin_type,type,
% 0.16/0.39 cartprodfstin: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodfstin,definition,
% 0.16/0.39 ( cartprodfstin
% 0.16/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.39 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjL2_type,type,
% 0.16/0.39 setukpairinjL2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjL2,definition,
% 0.16/0.39 ( setukpairinjL2
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.39 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.39 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.39 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjL_type,type,
% 0.16/0.39 setukpairinjL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjL,definition,
% 0.16/0.39 ( setukpairinjL
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.39 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.39 = ( kpair @ Xz @ Xu ) )
% 0.16/0.39 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR11_type,type,
% 0.16/0.39 setukpairinjR11: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR11,definition,
% 0.16/0.39 ( setukpairinjR11
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.39 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR12_type,type,
% 0.16/0.39 setukpairinjR12: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR12,definition,
% 0.16/0.39 ( setukpairinjR12
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xx = Xy )
% 0.16/0.39 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.39 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR1_type,type,
% 0.16/0.39 setukpairinjR1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR1,definition,
% 0.16/0.39 ( setukpairinjR1
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.39 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.39 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.39 => ( ( Xz = Xu )
% 0.16/0.39 => ( Xy = Xu ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(upairequniteq_type,type,
% 0.16/0.39 upairequniteq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(upairequniteq,definition,
% 0.16/0.39 ( upairequniteq
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.39 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.39 = ( setadjoin @ Xz @ emptyset ) )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR2_type,type,
% 0.16/0.39 setukpairinjR2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR2,definition,
% 0.16/0.39 ( setukpairinjR2
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.39 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.39 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.39 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR_type,type,
% 0.16/0.39 setukpairinjR: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setukpairinjR,definition,
% 0.16/0.39 ( setukpairinjR
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.39 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.39 = ( kpair @ Xz @ Xu ) )
% 0.16/0.39 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ksndsingleton_type,type,
% 0.16/0.39 ksndsingleton: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ksndsingleton,definition,
% 0.16/0.39 ( ksndsingleton
% 0.16/0.39 = ( ! [Xu: $i] :
% 0.16/0.39 ( ( iskpair @ Xu )
% 0.16/0.39 => ( singleton
% 0.16/0.39 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.39 @ ^ [Xx: $i] :
% 0.16/0.39 ( Xu
% 0.16/0.39 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ksnd_type,type,
% 0.16/0.39 ksnd: $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(ksndpairEq_type,type,
% 0.16/0.39 ksndpairEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ksndpairEq,definition,
% 0.16/0.39 ( ksndpairEq
% 0.16/0.39 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.39 = Xy ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(kpairsurjEq_type,type,
% 0.16/0.39 kpairsurjEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(kpairsurjEq,definition,
% 0.16/0.39 ( kpairsurjEq
% 0.16/0.39 = ( ! [Xu: $i] :
% 0.16/0.39 ( ( iskpair @ Xu )
% 0.16/0.39 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.39 = Xu ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodsndin_type,type,
% 0.16/0.39 cartprodsndin: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodsndin,definition,
% 0.16/0.39 ( cartprodsndin
% 0.16/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.39 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodpairmemEL_type,type,
% 0.16/0.39 cartprodpairmemEL: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodpairmemEL,definition,
% 0.16/0.39 ( cartprodpairmemEL
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodpairmemER_type,type,
% 0.16/0.39 cartprodpairmemER: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodpairmemER,definition,
% 0.16/0.39 ( cartprodpairmemER
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.39 => ( in @ Xy @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodmempaircEq_type,type,
% 0.16/0.39 cartprodmempaircEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodmempaircEq,definition,
% 0.16/0.39 ( cartprodmempaircEq
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( kpair @ Xx @ Xy )
% 0.16/0.39 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodfstpairEq_type,type,
% 0.16/0.39 cartprodfstpairEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodfstpairEq,definition,
% 0.16/0.39 ( cartprodfstpairEq
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.39 = Xx ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodsndpairEq_type,type,
% 0.16/0.39 cartprodsndpairEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodsndpairEq,definition,
% 0.16/0.39 ( cartprodsndpairEq
% 0.16/0.39 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.39 = Xy ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodpairsurjEq_type,type,
% 0.16/0.39 cartprodpairsurjEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(cartprodpairsurjEq,definition,
% 0.16/0.39 ( cartprodpairsurjEq
% 0.16/0.39 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.39 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.39 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.39 = Xu ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(breln_type,type,
% 0.16/0.39 breln: $i > $i > $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(breln,definition,
% 0.16/0.39 ( breln
% 0.16/0.39 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstr_type,type,
% 0.16/0.39 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrI_type,type,
% 0.16/0.39 dpsetconstrI: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrI,definition,
% 0.16/0.39 ( dpsetconstrI
% 0.16/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( Xphi @ Xx @ Xy )
% 0.16/0.39 => ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.39 @ ( dpsetconstr @ A @ B
% 0.16/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrSub_type,type,
% 0.16/0.39 dpsetconstrSub: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrSub,definition,
% 0.16/0.39 ( dpsetconstrSub
% 0.16/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.39 ( subset
% 0.16/0.39 @ ( dpsetconstr @ A @ B
% 0.16/0.39 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.16/0.39 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(setOfPairsIsBReln_type,type,
% 0.16/0.39 setOfPairsIsBReln: $o ).
% 0.16/0.39
% 0.16/0.39 thf(setOfPairsIsBReln,definition,
% 0.16/0.39 ( setOfPairsIsBReln
% 0.16/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.39 ( breln @ A @ B
% 0.16/0.39 @ ( dpsetconstr @ A @ B
% 0.16/0.39 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrERa_type,type,
% 0.16/0.39 dpsetconstrERa: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrERa,definition,
% 0.16/0.39 ( dpsetconstrERa
% 0.16/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.39 @ ( dpsetconstr @ A @ B
% 0.16/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.39 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrEL1_type,type,
% 0.16/0.39 dpsetconstrEL1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrEL1,definition,
% 0.16/0.39 ( dpsetconstrEL1
% 0.16/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.39 @ ( dpsetconstr @ A @ B
% 0.16/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.39 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrEL2_type,type,
% 0.16/0.39 dpsetconstrEL2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrEL2,definition,
% 0.16/0.39 ( dpsetconstrEL2
% 0.16/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.39 @ ( dpsetconstr @ A @ B
% 0.16/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.39 => ( in @ Xy @ B ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrER_type,type,
% 0.16/0.39 dpsetconstrER: $o ).
% 0.16/0.39
% 0.16/0.39 thf(dpsetconstrER,definition,
% 0.16/0.39 ( dpsetconstrER
% 0.16/0.39 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.39 @ ( dpsetconstr @ A @ B
% 0.16/0.39 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.39 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(func_type,type,
% 0.16/0.39 func: $i > $i > $i > $o ).
% 0.16/0.39
% 0.16/0.39 thf(funcSet_type,type,
% 0.16/0.39 funcSet: $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(funcImageSingleton_type,type,
% 0.16/0.39 funcImageSingleton: $o ).
% 0.16/0.39
% 0.16/0.39 thf(funcImageSingleton,definition,
% 0.16/0.39 ( funcImageSingleton
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( func @ A @ B @ Xf )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( singleton
% 0.16/0.39 @ ( dsetconstr @ B
% 0.16/0.39 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(apProp_type,type,
% 0.16/0.39 apProp: $o ).
% 0.16/0.39
% 0.16/0.39 thf(apProp,definition,
% 0.16/0.39 ( apProp
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( func @ A @ B @ Xf )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in
% 0.16/0.39 @ ( setunion
% 0.16/0.39 @ ( dsetconstr @ B
% 0.16/0.39 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.16/0.39 @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ap_type,type,
% 0.16/0.39 ap: $i > $i > $i > $i > $i ).
% 0.16/0.39
% 0.16/0.39 thf(app_type,type,
% 0.16/0.39 app: $o ).
% 0.16/0.39
% 0.16/0.39 thf(app,definition,
% 0.16/0.39 ( app
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( func @ A @ B @ Xf )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(infuncsetfunc_type,type,
% 0.16/0.39 infuncsetfunc: $o ).
% 0.16/0.39
% 0.16/0.39 thf(infuncsetfunc,definition,
% 0.16/0.39 ( infuncsetfunc
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.39 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ap2p_type,type,
% 0.16/0.39 ap2p: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ap2p,definition,
% 0.16/0.39 ( ap2p
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(funcinfuncset_type,type,
% 0.16/0.39 funcinfuncset: $o ).
% 0.16/0.39
% 0.16/0.39 thf(funcinfuncset,definition,
% 0.16/0.39 ( funcinfuncset
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( func @ A @ B @ Xf )
% 0.16/0.39 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(lamProp_type,type,
% 0.16/0.39 lamProp: $o ).
% 0.16/0.39
% 0.16/0.39 thf(lamProp,definition,
% 0.16/0.39 ( lamProp
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.39 => ( func @ A @ B
% 0.16/0.39 @ ( dpsetconstr @ A @ B
% 0.16/0.39 @ ^ [Xx: $i,Xy: $i] :
% 0.16/0.39 ( ( Xf @ Xx )
% 0.16/0.39 = Xy ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(lam_type,type,
% 0.16/0.39 lam: $i > $i > ( $i > $i ) > $i ).
% 0.16/0.39
% 0.16/0.39 thf(lamp_type,type,
% 0.16/0.39 lamp: $o ).
% 0.16/0.39
% 0.16/0.39 thf(lamp,definition,
% 0.16/0.39 ( lamp
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.39 => ( func @ A @ B
% 0.16/0.39 @ ( lam @ A @ B
% 0.16/0.39 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(lam2p_type,type,
% 0.16/0.39 lam2p: $o ).
% 0.16/0.39
% 0.16/0.39 thf(lam2p,definition,
% 0.16/0.39 ( lam2p
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.39 => ( in
% 0.16/0.39 @ ( lam @ A @ B
% 0.16/0.39 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.16/0.39 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(brelnall1_type,type,
% 0.16/0.39 brelnall1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(brelnall1,definition,
% 0.16/0.39 ( brelnall1
% 0.16/0.39 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.39 ( ( breln @ A @ B @ R )
% 0.16/0.39 => ! [Xphi: $i > $o] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.39 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ R )
% 0.16/0.39 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(brelnall2_type,type,
% 0.16/0.39 brelnall2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(brelnall2,definition,
% 0.16/0.39 ( brelnall2
% 0.16/0.39 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.39 ( ( breln @ A @ B @ R )
% 0.16/0.39 => ! [Xphi: $i > $o] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.39 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ R )
% 0.16/0.39 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ex1E2_type,type,
% 0.16/0.39 ex1E2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ex1E2,definition,
% 0.16/0.39 ( ex1E2
% 0.16/0.39 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.39 ( ( ex1 @ A
% 0.16/0.39 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ A )
% 0.16/0.39 => ( ( Xphi @ Xx )
% 0.16/0.39 => ( ( Xphi @ Xy )
% 0.16/0.39 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(funcGraphProp1_type,type,
% 0.16/0.39 funcGraphProp1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(funcGraphProp1,definition,
% 0.16/0.39 ( funcGraphProp1
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( func @ A @ B @ Xf )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(funcGraphProp3_type,type,
% 0.16/0.39 funcGraphProp3: $o ).
% 0.16/0.39
% 0.16/0.39 thf(funcGraphProp3,definition,
% 0.16/0.39 ( funcGraphProp3
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(funcGraphProp2_type,type,
% 0.16/0.39 funcGraphProp2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(funcGraphProp2,definition,
% 0.16/0.39 ( funcGraphProp2
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( func @ A @ B @ Xf )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.16/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.39 = Xy ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(funcextLem_type,type,
% 0.16/0.39 funcextLem: $o ).
% 0.16/0.39
% 0.16/0.39 thf(funcextLem,definition,
% 0.16/0.39 ( funcextLem
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( func @ A @ B @ Xf )
% 0.16/0.39 => ! [Xg: $i] :
% 0.16/0.39 ( ( func @ A @ B @ Xg )
% 0.16/0.39 => ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.39 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.16/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(funcGraphProp4_type,type,
% 0.16/0.39 funcGraphProp4: $o ).
% 0.16/0.39
% 0.16/0.39 thf(funcGraphProp4,definition,
% 0.16/0.39 ( funcGraphProp4
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.16/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.39 = Xy ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(subbreln_type,type,
% 0.16/0.39 subbreln: $o ).
% 0.16/0.39
% 0.16/0.39 thf(subbreln,definition,
% 0.16/0.39 ( subbreln
% 0.16/0.39 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.39 ( ( breln @ A @ B @ R )
% 0.16/0.39 => ! [S: $i] :
% 0.16/0.39 ( ( breln @ A @ B @ S )
% 0.16/0.39 => ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.16/0.39 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(eqbreln_type,type,
% 0.16/0.39 eqbreln: $o ).
% 0.16/0.39
% 0.16/0.39 thf(eqbreln,definition,
% 0.16/0.39 ( eqbreln
% 0.16/0.39 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.39 ( ( breln @ A @ B @ R )
% 0.16/0.39 => ! [S: $i] :
% 0.16/0.39 ( ( breln @ A @ B @ S )
% 0.16/0.39 => ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.16/0.39 => ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ! [Xy: $i] :
% 0.16/0.39 ( ( in @ Xy @ B )
% 0.16/0.39 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.16/0.39 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.16/0.39 => ( R = S ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(funcext_type,type,
% 0.16/0.39 funcext: $o ).
% 0.16/0.39
% 0.16/0.39 thf(funcext,definition,
% 0.16/0.39 ( funcext
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( func @ A @ B @ Xf )
% 0.16/0.39 => ! [Xg: $i] :
% 0.16/0.39 ( ( func @ A @ B @ Xg )
% 0.16/0.39 => ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.39 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.16/0.39 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(funcext2_type,type,
% 0.16/0.39 funcext2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(funcext2,definition,
% 0.16/0.39 ( funcext2
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.39 => ! [Xg: $i] :
% 0.16/0.39 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.16/0.39 => ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.39 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.16/0.39 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ap2apEq1_type,type,
% 0.16/0.39 ap2apEq1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ap2apEq1,definition,
% 0.16/0.39 ( ap2apEq1
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.39 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(ap2apEq2_type,type,
% 0.16/0.39 ap2apEq2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(ap2apEq2,definition,
% 0.16/0.39 ( ap2apEq2
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( func @ A @ B @ Xf )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.39 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(beta1_type,type,
% 0.16/0.39 beta1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(beta1,definition,
% 0.16/0.39 ( beta1
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( ap @ A @ B
% 0.16/0.39 @ ( lam @ A @ B
% 0.16/0.39 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.16/0.39 @ Xx )
% 0.16/0.39 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(eta1_type,type,
% 0.16/0.39 eta1: $o ).
% 0.16/0.39
% 0.16/0.39 thf(eta1,definition,
% 0.16/0.39 ( eta1
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( func @ A @ B @ Xf )
% 0.16/0.39 => ( ( lam @ A @ B
% 0.16/0.39 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.16/0.39 = Xf ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(lam2lamEq_type,type,
% 0.16/0.39 lam2lamEq: $o ).
% 0.16/0.39
% 0.16/0.39 thf(lam2lamEq,definition,
% 0.16/0.39 ( lam2lamEq
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.39 => ( ( lam @ A @ B
% 0.16/0.39 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.16/0.39 = ( lam @ A @ B
% 0.16/0.39 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(beta2_type,type,
% 0.16/0.39 beta2: $o ).
% 0.16/0.39
% 0.16/0.39 thf(beta2,definition,
% 0.16/0.39 ( beta2
% 0.16/0.39 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.39 ( ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.39 => ! [Xx: $i] :
% 0.16/0.39 ( ( in @ Xx @ A )
% 0.16/0.39 => ( ( ap @ A @ B
% 0.16/0.39 @ ( lam @ A @ B
% 0.16/0.39 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.16/0.39 @ Xx )
% 0.16/0.39 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 thf(eta2,conjecture,
% 0.16/0.39 ( setextAx
% 0.16/0.39 => ( emptysetAx
% 0.16/0.39 => ( setadjoinAx
% 0.16/0.39 => ( powersetAx
% 0.16/0.39 => ( setunionAx
% 0.16/0.39 => ( omega0Ax
% 0.16/0.39 => ( omegaSAx
% 0.16/0.39 => ( omegaIndAx
% 0.16/0.39 => ( replAx
% 0.16/0.39 => ( foundationAx
% 0.16/0.39 => ( wellorderingAx
% 0.16/0.39 => ( descrp
% 0.16/0.39 => ( dsetconstrI
% 0.16/0.39 => ( dsetconstrEL
% 0.16/0.39 => ( dsetconstrER
% 0.16/0.39 => ( exuE1
% 0.16/0.39 => ( prop2setE
% 0.16/0.39 => ( emptysetE
% 0.16/0.39 => ( emptysetimpfalse
% 0.16/0.39 => ( notinemptyset
% 0.16/0.39 => ( exuE3e
% 0.16/0.39 => ( setext
% 0.16/0.39 => ( emptyI
% 0.16/0.39 => ( noeltsimpempty
% 0.16/0.39 => ( setbeta
% 0.16/0.39 => ( nonemptyE1
% 0.16/0.39 => ( nonemptyI
% 0.16/0.39 => ( nonemptyI1
% 0.16/0.39 => ( setadjoinIL
% 0.16/0.39 => ( emptyinunitempty
% 0.16/0.39 => ( setadjoinIR
% 0.16/0.39 => ( setadjoinE
% 0.16/0.39 => ( setadjoinOr
% 0.16/0.39 => ( setoftrueEq
% 0.16/0.39 => ( powersetI
% 0.16/0.39 => ( emptyinPowerset
% 0.16/0.39 => ( emptyInPowerset
% 0.16/0.39 => ( powersetE
% 0.16/0.39 => ( setunionI
% 0.16/0.39 => ( setunionE
% 0.16/0.39 => ( subPowSU
% 0.16/0.39 => ( exuE2
% 0.16/0.39 => ( nonemptyImpWitness
% 0.16/0.39 => ( uniqinunit
% 0.16/0.39 => ( notinsingleton
% 0.16/0.39 => ( eqinunit
% 0.16/0.39 => ( singletonsswitch
% 0.16/0.39 => ( upairsetE
% 0.16/0.39 => ( upairsetIL
% 0.16/0.39 => ( upairsetIR
% 0.16/0.39 => ( emptyE1
% 0.16/0.39 => ( vacuousDall
% 0.16/0.39 => ( quantDeMorgan1
% 0.16/0.39 => ( quantDeMorgan2
% 0.16/0.39 => ( quantDeMorgan3
% 0.16/0.39 => ( quantDeMorgan4
% 0.16/0.39 => ( prop2setI
% 0.16/0.39 => ( prop2set2propI
% 0.16/0.39 => ( notdexE
% 0.16/0.39 => ( notdallE
% 0.16/0.39 => ( exuI1
% 0.16/0.39 => ( exuI3
% 0.16/0.39 => ( exuI2
% 0.16/0.39 => ( inCongP
% 0.16/0.39 => ( in__Cong
% 0.16/0.39 => ( exuE3u
% 0.16/0.39 => ( exu__Cong
% 0.16/0.39 => ( emptyset__Cong
% 0.16/0.39 => ( setadjoin__Cong
% 0.16/0.39 => ( powerset__Cong
% 0.16/0.39 => ( setunion__Cong
% 0.16/0.39 => ( omega__Cong
% 0.16/0.39 => ( exuEu
% 0.16/0.39 => ( descr__Cong
% 0.16/0.39 => ( dsetconstr__Cong
% 0.16/0.39 => ( subsetI1
% 0.16/0.39 => ( eqimpsubset2
% 0.16/0.39 => ( eqimpsubset1
% 0.16/0.39 => ( subsetI2
% 0.16/0.39 => ( emptysetsubset
% 0.16/0.39 => ( subsetE
% 0.16/0.39 => ( subsetE2
% 0.16/0.39 => ( notsubsetI
% 0.16/0.39 => ( notequalI1
% 0.16/0.39 => ( notequalI2
% 0.16/0.39 => ( subsetRefl
% 0.16/0.39 => ( subsetTrans
% 0.16/0.39 => ( setadjoinSub
% 0.16/0.39 => ( setadjoinSub2
% 0.16/0.39 => ( subset2powerset
% 0.16/0.39 => ( setextsub
% 0.16/0.39 => ( subsetemptysetimpeq
% 0.16/0.39 => ( powersetI1
% 0.16/0.39 => ( powersetE1
% 0.16/0.39 => ( inPowerset
% 0.16/0.39 => ( powersetsubset
% 0.16/0.39 => ( sepInPowerset
% 0.16/0.39 => ( sepSubset
% 0.16/0.39 => ( binunionIL
% 0.16/0.39 => ( upairset2IR
% 0.16/0.39 => ( binunionIR
% 0.16/0.39 => ( binunionEcases
% 0.16/0.39 => ( binunionE
% 0.16/0.39 => ( binunionLsub
% 0.16/0.39 => ( binunionRsub
% 0.16/0.39 => ( binintersectI
% 0.16/0.39 => ( binintersectSubset5
% 0.16/0.39 => ( binintersectEL
% 0.16/0.39 => ( binintersectLsub
% 0.16/0.39 => ( binintersectSubset2
% 0.16/0.39 => ( binintersectSubset3
% 0.16/0.39 => ( binintersectER
% 0.16/0.39 => ( disjointsetsI1
% 0.16/0.39 => ( binintersectRsub
% 0.16/0.39 => ( binintersectSubset4
% 0.16/0.39 => ( binintersectSubset1
% 0.16/0.39 => ( bs114d
% 0.16/0.39 => ( setminusI
% 0.16/0.39 => ( setminusEL
% 0.16/0.39 => ( setminusER
% 0.16/0.39 => ( setminusSubset2
% 0.16/0.39 => ( setminusERneg
% 0.16/0.39 => ( setminusELneg
% 0.16/0.39 => ( setminusILneg
% 0.16/0.39 => ( setminusIRneg
% 0.16/0.39 => ( setminusLsub
% 0.16/0.39 => ( setminusSubset1
% 0.16/0.39 => ( symdiffE
% 0.16/0.39 => ( symdiffI1
% 0.16/0.39 => ( symdiffI2
% 0.16/0.39 => ( symdiffIneg1
% 0.16/0.39 => ( symdiffIneg2
% 0.16/0.39 => ( secondinupair
% 0.16/0.39 => ( setukpairIL
% 0.16/0.39 => ( setukpairIR
% 0.16/0.39 => ( kpairiskpair
% 0.16/0.39 => ( kpairp
% 0.16/0.39 => ( singletonsubset
% 0.16/0.39 => ( singletoninpowerset
% 0.16/0.39 => ( singletoninpowunion
% 0.16/0.39 => ( upairset2E
% 0.16/0.39 => ( upairsubunion
% 0.16/0.39 => ( upairinpowunion
% 0.16/0.39 => ( ubforcartprodlem1
% 0.16/0.39 => ( ubforcartprodlem2
% 0.16/0.39 => ( ubforcartprodlem3
% 0.16/0.39 => ( cartprodpairin
% 0.16/0.39 => ( cartprodmempair1
% 0.16/0.39 => ( cartprodmempair
% 0.16/0.39 => ( setunionE2
% 0.16/0.39 => ( setunionsingleton1
% 0.16/0.39 => ( setunionsingleton2
% 0.16/0.39 => ( setunionsingleton
% 0.16/0.39 => ( singletonprop
% 0.16/0.39 => ( ex1E1
% 0.16/0.39 => ( ex1I
% 0.16/0.39 => ( ex1I2
% 0.16/0.39 => ( singletonsuniq
% 0.16/0.39 => ( setukpairinjL1
% 0.16/0.39 => ( kfstsingleton
% 0.16/0.39 => ( theprop
% 0.16/0.39 => ( kfstpairEq
% 0.16/0.39 => ( cartprodfstin
% 0.16/0.39 => ( setukpairinjL2
% 0.16/0.39 => ( setukpairinjL
% 0.16/0.39 => ( setukpairinjR11
% 0.16/0.39 => ( setukpairinjR12
% 0.16/0.39 => ( setukpairinjR1
% 0.16/0.39 => ( upairequniteq
% 0.16/0.39 => ( setukpairinjR2
% 0.16/0.39 => ( setukpairinjR
% 0.16/0.39 => ( ksndsingleton
% 0.16/0.39 => ( ksndpairEq
% 0.16/0.39 => ( kpairsurjEq
% 0.16/0.39 => ( cartprodsndin
% 0.16/0.39 => ( cartprodpairmemEL
% 0.16/0.39 => ( cartprodpairmemER
% 0.16/0.39 => ( cartprodmempaircEq
% 0.16/0.39 => ( cartprodfstpairEq
% 0.16/0.39 => ( cartprodsndpairEq
% 0.16/0.39 => ( cartprodpairsurjEq
% 0.16/0.39 => ( dpsetconstrI
% 0.16/0.39 => ( dpsetconstrSub
% 0.16/0.39 => ( setOfPairsIsBReln
% 0.16/0.39 => ( dpsetconstrERa
% 0.16/0.39 => ( dpsetconstrEL1
% 0.16/0.39 => ( dpsetconstrEL2
% 0.16/0.39 => ( dpsetconstrER
% 0.16/0.39 => ( funcImageSingleton
% 0.16/0.39 => ( apProp
% 0.16/0.39 => ( app
% 0.16/0.39 => ( infuncsetfunc
% 0.16/0.39 => ( ap2p
% 0.16/0.39 => ( funcinfuncset
% 0.16/0.39 => ( lamProp
% 0.16/0.39 => ( lamp
% 0.16/0.39 => ( lam2p
% 0.16/0.39 => ( brelnall1
% 0.16/0.39 => ( brelnall2
% 0.16/0.39 => ( ex1E2
% 0.16/0.39 => ( funcGraphProp1
% 0.16/0.39 => ( funcGraphProp3
% 0.16/0.39 => ( funcGraphProp2
% 0.16/0.39 => ( funcextLem
% 0.16/0.39 => ( funcGraphProp4
% 0.16/0.39 => ( subbreln
% 0.16/0.39 => ( eqbreln
% 0.16/0.39 => ( funcext
% 0.16/0.39 => ( funcext2
% 0.16/0.39 => ( ap2apEq1
% 0.16/0.39 => ( ap2apEq2
% 0.16/0.39 => ( beta1
% 0.16/0.39 => ( eta1
% 0.16/0.39 => ( lam2lamEq
% 0.16/0.39 => ( beta2
% 0.16/0.39 => ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.39 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.39 => ( ( lam @ A @ B
% 0.16/0.39 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.16/0.39 = Xf ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.39
% 0.16/0.39 %------------------------------------------------------------------------------
% 0.16/0.39 ------------------------
% 2.40/1.32 ---- Embedded in THF ---
% 2.40/1.32 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.40/1.32 %%% Generated on Tue Feb 24 12:31:25 EST 2026
% 2.40/1.32 %%% using '$$dhol' embedding, version 1.3.0.
% 2.40/1.32 %%% Logic specification used:
% 2.40/1.32 %%% thf(spec, logic, $$dhol).
% 2.40/1.32
% 2.40/1.32 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.GXJcYPDlCU/DTF2DTF6599.p
% See solution above
% 2.40/1.32 ------------------------
% 2.40/1.36 ---- Cleaned THF ---
% 2.40/1.36 thf(in_type,type,
% 2.40/1.36 in: $i > $i > $o ).
% 2.40/1.36
% 2.40/1.36 thf(exu_type,type,
% 2.40/1.36 exu: ( $i > $o ) > $o ).
% 2.40/1.36
% 2.40/1.36 thf(exu,definition,
% 2.40/1.36 ( exu
% 2.40/1.36 = ( ^ [Xphi: $i > $o] :
% 2.40/1.36 ? [Xx: $i] :
% 2.40/1.36 ( ( Xphi @ Xx )
% 2.40/1.36 & ! [Xy: $i] :
% 2.40/1.36 ( ( Xphi @ Xy )
% 2.40/1.36 => ( Xx = Xy ) ) ) ) ) ).
% 2.40/1.36
% 2.40/1.36 thf(setextAx_type,type,
% 2.40/1.36 setextAx: $o ).
% 2.40/1.36
% 2.40/1.36 thf(setextAx,definition,
% 2.40/1.36 ( setextAx
% 2.40/1.36 = ( ! [A: $i,B: $i] :
% 2.40/1.36 ( ! [Xx: $i] :
% 2.40/1.36 ( ( in @ Xx @ A )
% 2.40/1.36 <=> ( in @ Xx @ B ) )
% 2.40/1.36 => ( A = B ) ) ) ) ).
% 2.40/1.36
% 2.40/1.36 thf(emptyset_type,type,
% 2.40/1.36 emptyset: $i ).
% 2.40/1.36
% 2.40/1.36 thf(emptysetAx_type,type,
% 2.40/1.36 emptysetAx: $o ).
% 2.40/1.36
% 2.40/1.36 thf(emptysetAx,definition,
% 2.40/1.36 ( emptysetAx
% 2.40/1.36 = ( ! [Xx: $i] :
% 2.40/1.36 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.40/1.36
% 2.40/1.36 thf(setadjoin_type,type,
% 2.40/1.36 setadjoin: $i > $i > $i ).
% 2.40/1.36
% 2.40/1.36 thf(setadjoinAx_type,type,
% 2.40/1.36 setadjoinAx: $o ).
% 2.40/1.36
% 2.40/1.36 thf(setadjoinAx,definition,
% 2.40/1.36 ( setadjoinAx
% 2.40/1.36 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.40/1.36 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.40/1.36 <=> ( ( Xy = Xx )
% 2.40/1.36 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.40/1.36
% 2.40/1.36 thf(powerset_type,type,
% 2.40/1.36 powerset: $i > $i ).
% 2.40/1.36
% 2.40/1.36 thf(powersetAx_type,type,
% 2.40/1.36 powersetAx: $o ).
% 2.40/1.36
% 2.40/1.36 thf(powersetAx,definition,
% 2.40/1.36 ( powersetAx
% 2.40/1.36 = ( ! [A: $i,B: $i] :
% 2.40/1.36 ( ( in @ B @ ( powerset @ A ) )
% 2.40/1.36 <=> ! [Xx: $i] :
% 2.40/1.36 ( ( in @ Xx @ B )
% 2.40/1.36 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.40/1.36
% 2.40/1.36 thf(setunion_type,type,
% 2.40/1.36 setunion: $i > $i ).
% 2.40/1.36
% 2.40/1.36 thf(setunionAx_type,type,
% 2.40/1.36 setunionAx: $o ).
% 2.40/1.36
% 2.40/1.36 thf(setunionAx,definition,
% 2.40/1.36 ( setunionAx
% 2.40/1.36 = ( ! [A: $i,Xx: $i] :
% 2.40/1.36 ( ( in @ Xx @ ( setunion @ A ) )
% 2.40/1.36 <=> ? [B: $i] :
% 2.40/1.36 ( ( in @ Xx @ B )
% 2.40/1.36 & ( in @ B @ A ) ) ) ) ) ).
% 2.40/1.36
% 2.40/1.36 thf(omega_type,type,
% 2.40/1.36 omega: $i ).
% 2.40/1.36
% 2.40/1.36 thf(omega0Ax_type,type,
% 2.40/1.36 omega0Ax: $o ).
% 2.40/1.36
% 2.40/1.36 thf(omega0Ax,definition,
% 2.40/1.36 ( omega0Ax
% 2.40/1.36 = ( in @ emptyset @ omega ) ) ).
% 2.40/1.36
% 2.40/1.36 thf(omegaSAx_type,type,
% 2.40/1.36 omegaSAx: $o ).
% 2.40/1.36
% 2.40/1.36 thf(omegaSAx,definition,
% 2.40/1.36 ( omegaSAx
% 2.40/1.36 = ( ! [Xx: $i] :
% 2.40/1.36 ( ( in @ Xx @ omega )
% 2.40/1.36 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.40/1.36
% 2.40/1.36 thf(omegaIndAx_type,type,
% 2.40/1.36 omegaIndAx: $o ).
% 2.40/1.36
% 2.40/1.36 thf(omegaIndAx,definition,
% 2.40/1.36 ( omegaIndAx
% 2.40/1.36 = ( ! [A: $i] :
% 2.40/1.36 ( ( ( in @ emptyset @ A )
% 2.40/1.36 & ! [Xx: $i] :
% 2.40/1.36 ( ( ( in @ Xx @ omega )
% 2.40/1.36 & ( in @ Xx @ A ) )
% 2.40/1.36 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.40/1.36 => ! [Xx: $i] :
% 2.40/1.36 ( ( in @ Xx @ omega )
% 2.40/1.36 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.40/1.36
% 2.40/1.36 thf(replAx_type,type,
% 2.40/1.36 replAx: $o ).
% 2.40/1.36
% 2.40/1.36 thf(replAx,definition,
% 2.40/1.36 ( replAx
% 2.40/1.36 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.40/1.36 ( ! [Xx: $i] :
% 2.40/1.36 ( ( in @ Xx @ A )
% 2.40/1.36 => ( exu
% 2.40/1.36 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.40/1.36 => ? [B: $i] :
% 2.40/1.36 ! [Xx: $i] :
% 2.40/1.36 ( ( in @ Xx @ B )
% 2.40/1.36 <=> ? [Xy: $i] :
% 2.40/1.36 ( ( in @ Xy @ A )
% 2.40/1.36 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.40/1.36
% 2.40/1.36 thf(foundationAx_type,type,
% 2.40/1.36 foundationAx: $o ).
% 2.40/1.36
% 2.40/1.36 thf(foundationAx,definition,
% 2.40/1.36 ( foundationAx
% 2.40/1.36 = ( ! [A: $i] :
% 2.40/1.36 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.40/1.36 => ? [B: $i] :
% 2.40/1.36 ( ( in @ B @ A )
% 2.40/1.36 & ~ ? [Xx: $i] :
% 2.40/1.36 ( ( in @ Xx @ B )
% 2.40/1.36 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.40/1.36
% 2.40/1.36 thf(wellorderingAx_type,type,
% 2.40/1.36 wellorderingAx: $o ).
% 2.40/1.36
% 2.40/1.36 thf(wellorderingAx,definition,
% 2.40/1.36 ( wellorderingAx
% 2.40/1.36 = ( ! [A: $i] :
% 2.40/1.36 ? [B: $i] :
% 2.40/1.36 ( ! [C: $i] :
% 2.40/1.36 ( ( in @ C @ B )
% 2.40/1.36 => ! [Xx: $i] :
% 2.40/1.36 ( ( in @ Xx @ C )
% 2.40/1.36 => ( in @ Xx @ A ) ) )
% 2.40/1.36 & ! [Xx: $i,Xy: $i] :
% 2.40/1.36 ( ( ( in @ Xx @ A )
% 2.40/1.36 & ( in @ Xy @ A ) )
% 2.40/1.36 => ( ! [C: $i] :
% 2.40/1.36 ( ( in @ C @ B )
% 2.40/1.36 => ( ( in @ Xx @ C )
% 2.40/1.36 <=> ( in @ Xy @ C ) ) )
% 2.40/1.36 => ( Xx = Xy ) ) )
% 2.40/1.36 & ! [C: $i,D: $i] :
% 2.40/1.36 ( ( ( in @ C @ B )
% 2.40/1.36 & ( in @ D @ B ) )
% 2.40/1.36 => ( ! [Xx: $i] :
% 2.40/1.36 ( ( in @ Xx @ C )
% 2.40/1.36 => ( in @ Xx @ D ) )
% 2.40/1.36 | ! [Xx: $i] :
% 2.40/1.36 ( ( in @ Xx @ D )
% 2.40/1.36 => ( in @ Xx @ C ) ) ) )
% 2.40/1.36 & ! [C: $i] :
% 2.40/1.36 ( ( ! [Xx: $i] :
% 2.40/1.36 ( ( in @ Xx @ C )
% 2.40/1.36 => ( in @ Xx @ A ) )
% 2.40/1.36 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.40/1.36 => ? [D: $i,Xx: $i] :
% 2.40/1.36 ( ( in @ D @ B )
% 2.40/1.36 & ( in @ Xx @ C )
% 2.40/1.36 & ~ ? [Xy: $i] :
% 2.40/1.36 ( ( in @ Xy @ D )
% 2.40/1.36 & ( in @ Xy @ C ) )
% 2.40/1.36 & ! [E: $i] :
% 2.40/1.36 ( ( in @ E @ B )
% 2.40/1.36 => ( ! [Xy: $i] :
% 2.40/1.36 ( ( in @ Xy @ E )
% 2.40/1.36 => ( in @ Xy @ D ) )
% 2.40/1.36 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.40/1.36
% 2.40/1.36 thf(descr_type,type,
% 2.40/1.36 descr: ( $i > $o ) > $i ).
% 2.40/1.36
% 2.40/1.36 thf(descrp_type,type,
% 2.40/1.36 descrp: $o ).
% 2.40/1.36
% 2.40/1.36 thf(descrp,definition,
% 2.40/1.36 ( descrp
% 2.40/1.36 = ( ! [Xphi: $i > $o] :
% 2.40/1.36 ( ( exu
% 2.40/1.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.40/1.36 => ( Xphi
% 2.40/1.36 @ ( descr
% 2.40/1.36 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(dsetconstr_type,type,
% 2.40/1.37 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.40/1.37
% 2.40/1.37 thf(dsetconstrI_type,type,
% 2.40/1.37 dsetconstrI: $o ).
% 2.40/1.37
% 2.40/1.37 thf(dsetconstrI,definition,
% 2.40/1.37 ( dsetconstrI
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ( Xphi @ Xx )
% 2.40/1.37 => ( in @ Xx
% 2.40/1.37 @ ( dsetconstr @ A
% 2.40/1.37 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(dsetconstrEL_type,type,
% 2.40/1.37 dsetconstrEL: $o ).
% 2.40/1.37
% 2.40/1.37 thf(dsetconstrEL,definition,
% 2.40/1.37 ( dsetconstrEL
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx
% 2.40/1.37 @ ( dsetconstr @ A
% 2.40/1.37 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.40/1.37 => ( in @ Xx @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(dsetconstrER_type,type,
% 2.40/1.37 dsetconstrER: $o ).
% 2.40/1.37
% 2.40/1.37 thf(dsetconstrER,definition,
% 2.40/1.37 ( dsetconstrER
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx
% 2.40/1.37 @ ( dsetconstr @ A
% 2.40/1.37 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.40/1.37 => ( Xphi @ Xx ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(exuE1_type,type,
% 2.40/1.37 exuE1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(exuE1,definition,
% 2.40/1.37 ( exuE1
% 2.40/1.37 = ( ! [Xphi: $i > $o] :
% 2.40/1.37 ( ( exu
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.40/1.37 => ? [Xx: $i] :
% 2.40/1.37 ( ( Xphi @ Xx )
% 2.40/1.37 & ! [Xy: $i] :
% 2.40/1.37 ( ( Xphi @ Xy )
% 2.40/1.37 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(prop2set_type,type,
% 2.40/1.37 prop2set: $o > $i ).
% 2.40/1.37
% 2.40/1.37 thf(prop2setE_type,type,
% 2.40/1.37 prop2setE: $o ).
% 2.40/1.37
% 2.40/1.37 thf(prop2setE,definition,
% 2.40/1.37 ( prop2setE
% 2.40/1.37 = ( ! [Xphi: $o,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.40/1.37 => Xphi ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(emptysetE_type,type,
% 2.40/1.37 emptysetE: $o ).
% 2.40/1.37
% 2.40/1.37 thf(emptysetE,definition,
% 2.40/1.37 ( emptysetE
% 2.40/1.37 = ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ emptyset )
% 2.40/1.37 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(emptysetimpfalse_type,type,
% 2.40/1.37 emptysetimpfalse: $o ).
% 2.40/1.37
% 2.40/1.37 thf(emptysetimpfalse,definition,
% 2.40/1.37 ( emptysetimpfalse
% 2.40/1.37 = ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ emptyset )
% 2.40/1.37 => $false ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(notinemptyset_type,type,
% 2.40/1.37 notinemptyset: $o ).
% 2.40/1.37
% 2.40/1.37 thf(notinemptyset,definition,
% 2.40/1.37 ( notinemptyset
% 2.40/1.37 = ( ! [Xx: $i] :
% 2.40/1.37 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(exuE3e_type,type,
% 2.40/1.37 exuE3e: $o ).
% 2.40/1.37
% 2.40/1.37 thf(exuE3e,definition,
% 2.40/1.37 ( exuE3e
% 2.40/1.37 = ( ! [Xphi: $i > $o] :
% 2.40/1.37 ( ( exu
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.40/1.37 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setext_type,type,
% 2.40/1.37 setext: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setext,definition,
% 2.40/1.37 ( setext
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ Xx @ B ) )
% 2.40/1.37 => ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ B )
% 2.40/1.37 => ( in @ Xx @ A ) )
% 2.40/1.37 => ( A = B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(emptyI_type,type,
% 2.40/1.37 emptyI: $o ).
% 2.40/1.37
% 2.40/1.37 thf(emptyI,definition,
% 2.40/1.37 ( emptyI
% 2.40/1.37 = ( ! [A: $i] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ~ ( in @ Xx @ A )
% 2.40/1.37 => ( A = emptyset ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(noeltsimpempty_type,type,
% 2.40/1.37 noeltsimpempty: $o ).
% 2.40/1.37
% 2.40/1.37 thf(noeltsimpempty,definition,
% 2.40/1.37 ( noeltsimpempty
% 2.40/1.37 = ( ! [A: $i] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ~ ( in @ Xx @ A )
% 2.40/1.37 => ( A = emptyset ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setbeta_type,type,
% 2.40/1.37 setbeta: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setbeta,definition,
% 2.40/1.37 ( setbeta
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ( in @ Xx
% 2.40/1.37 @ ( dsetconstr @ A
% 2.40/1.37 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.40/1.37 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(nonempty_type,type,
% 2.40/1.37 nonempty: $i > $o ).
% 2.40/1.37
% 2.40/1.37 thf(nonempty,definition,
% 2.40/1.37 ( nonempty
% 2.40/1.37 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(nonemptyE1_type,type,
% 2.40/1.37 nonemptyE1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(nonemptyE1,definition,
% 2.40/1.37 ( nonemptyE1
% 2.40/1.37 = ( ! [A: $i] :
% 2.40/1.37 ( ( nonempty @ A )
% 2.40/1.37 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(nonemptyI_type,type,
% 2.40/1.37 nonemptyI: $o ).
% 2.40/1.37
% 2.40/1.37 thf(nonemptyI,definition,
% 2.40/1.37 ( nonemptyI
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ( Xphi @ Xx )
% 2.40/1.37 => ( nonempty
% 2.40/1.37 @ ( dsetconstr @ A
% 2.40/1.37 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(nonemptyI1_type,type,
% 2.40/1.37 nonemptyI1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(nonemptyI1,definition,
% 2.40/1.37 ( nonemptyI1
% 2.40/1.37 = ( ! [A: $i] :
% 2.40/1.37 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.40/1.37 => ( nonempty @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setadjoinIL_type,type,
% 2.40/1.37 setadjoinIL: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setadjoinIL,definition,
% 2.40/1.37 ( setadjoinIL
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(emptyinunitempty_type,type,
% 2.40/1.37 emptyinunitempty: $o ).
% 2.40/1.37
% 2.40/1.37 thf(emptyinunitempty,definition,
% 2.40/1.37 ( emptyinunitempty
% 2.40/1.37 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setadjoinIR_type,type,
% 2.40/1.37 setadjoinIR: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setadjoinIR,definition,
% 2.40/1.37 ( setadjoinIR
% 2.40/1.37 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ A )
% 2.40/1.37 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setadjoinE_type,type,
% 2.40/1.37 setadjoinE: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setadjoinE,definition,
% 2.40/1.37 ( setadjoinE
% 2.40/1.37 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.40/1.37 => ! [Xphi: $o] :
% 2.40/1.37 ( ( ( Xy = Xx )
% 2.40/1.37 => Xphi )
% 2.40/1.37 => ( ( ( in @ Xy @ A )
% 2.40/1.37 => Xphi )
% 2.40/1.37 => Xphi ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setadjoinOr_type,type,
% 2.40/1.37 setadjoinOr: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setadjoinOr,definition,
% 2.40/1.37 ( setadjoinOr
% 2.40/1.37 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.40/1.37 => ( ( Xy = Xx )
% 2.40/1.37 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setoftrueEq_type,type,
% 2.40/1.37 setoftrueEq: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setoftrueEq,definition,
% 2.40/1.37 ( setoftrueEq
% 2.40/1.37 = ( ! [A: $i] :
% 2.40/1.37 ( ( dsetconstr @ A
% 2.40/1.37 @ ^ [Xx: $i] : $true )
% 2.40/1.37 = A ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(powersetI_type,type,
% 2.40/1.37 powersetI: $o ).
% 2.40/1.37
% 2.40/1.37 thf(powersetI,definition,
% 2.40/1.37 ( powersetI
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ B )
% 2.40/1.37 => ( in @ Xx @ A ) )
% 2.40/1.37 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(emptyinPowerset_type,type,
% 2.40/1.37 emptyinPowerset: $o ).
% 2.40/1.37
% 2.40/1.37 thf(emptyinPowerset,definition,
% 2.40/1.37 ( emptyinPowerset
% 2.40/1.37 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(emptyInPowerset_type,type,
% 2.40/1.37 emptyInPowerset: $o ).
% 2.40/1.37
% 2.40/1.37 thf(emptyInPowerset,definition,
% 2.40/1.37 ( emptyInPowerset
% 2.40/1.37 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(powersetE_type,type,
% 2.40/1.37 powersetE: $o ).
% 2.40/1.37
% 2.40/1.37 thf(powersetE,definition,
% 2.40/1.37 ( powersetE
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ B @ ( powerset @ A ) )
% 2.40/1.37 => ( ( in @ Xx @ B )
% 2.40/1.37 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setunionI_type,type,
% 2.40/1.37 setunionI: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setunionI,definition,
% 2.40/1.37 ( setunionI
% 2.40/1.37 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.40/1.37 ( ( in @ Xx @ B )
% 2.40/1.37 => ( ( in @ B @ A )
% 2.40/1.37 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setunionE_type,type,
% 2.40/1.37 setunionE: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setunionE,definition,
% 2.40/1.37 ( setunionE
% 2.40/1.37 = ( ! [A: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ ( setunion @ A ) )
% 2.40/1.37 => ! [Xphi: $o] :
% 2.40/1.37 ( ! [B: $i] :
% 2.40/1.37 ( ( in @ Xx @ B )
% 2.40/1.37 => ( ( in @ B @ A )
% 2.40/1.37 => Xphi ) )
% 2.40/1.37 => Xphi ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(subPowSU_type,type,
% 2.40/1.37 subPowSU: $o ).
% 2.40/1.37
% 2.40/1.37 thf(subPowSU,definition,
% 2.40/1.37 ( subPowSU
% 2.40/1.37 = ( ! [A: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(exuE2_type,type,
% 2.40/1.37 exuE2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(exuE2,definition,
% 2.40/1.37 ( exuE2
% 2.40/1.37 = ( ! [Xphi: $i > $o] :
% 2.40/1.37 ( ( exu
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.40/1.37 => ? [Xx: $i] :
% 2.40/1.37 ! [Xy: $i] :
% 2.40/1.37 ( ( Xphi @ Xy )
% 2.40/1.37 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(nonemptyImpWitness_type,type,
% 2.40/1.37 nonemptyImpWitness: $o ).
% 2.40/1.37
% 2.40/1.37 thf(nonemptyImpWitness,definition,
% 2.40/1.37 ( nonemptyImpWitness
% 2.40/1.37 = ( ! [A: $i] :
% 2.40/1.37 ( ( nonempty @ A )
% 2.40/1.37 => ? [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 & $true ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(uniqinunit_type,type,
% 2.40/1.37 uniqinunit: $o ).
% 2.40/1.37
% 2.40/1.37 thf(uniqinunit,definition,
% 2.40/1.37 ( uniqinunit
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.40/1.37 => ( Xx = Xy ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(notinsingleton_type,type,
% 2.40/1.37 notinsingleton: $o ).
% 2.40/1.37
% 2.40/1.37 thf(notinsingleton,definition,
% 2.40/1.37 ( notinsingleton
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( Xx != Xy )
% 2.40/1.37 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(eqinunit_type,type,
% 2.40/1.37 eqinunit: $o ).
% 2.40/1.37
% 2.40/1.37 thf(eqinunit,definition,
% 2.40/1.37 ( eqinunit
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( Xx = Xy )
% 2.40/1.37 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(singletonsswitch_type,type,
% 2.40/1.37 singletonsswitch: $o ).
% 2.40/1.37
% 2.40/1.37 thf(singletonsswitch,definition,
% 2.40/1.37 ( singletonsswitch
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.40/1.37 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(upairsetE_type,type,
% 2.40/1.37 upairsetE: $o ).
% 2.40/1.37
% 2.40/1.37 thf(upairsetE,definition,
% 2.40/1.37 ( upairsetE
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.40/1.37 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.40/1.37 => ( ( Xz = Xx )
% 2.40/1.37 | ( Xz = Xy ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(upairsetIL_type,type,
% 2.40/1.37 upairsetIL: $o ).
% 2.40/1.37
% 2.40/1.37 thf(upairsetIL,definition,
% 2.40/1.37 ( upairsetIL
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(upairsetIR_type,type,
% 2.40/1.37 upairsetIR: $o ).
% 2.40/1.37
% 2.40/1.37 thf(upairsetIR,definition,
% 2.40/1.37 ( upairsetIR
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(emptyE1_type,type,
% 2.40/1.37 emptyE1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(emptyE1,definition,
% 2.40/1.37 ( emptyE1
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o] :
% 2.40/1.37 ( ? [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 & ( Xphi @ Xx ) )
% 2.40/1.37 => ( ( ( dsetconstr @ A
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.40/1.37 = emptyset )
% 2.40/1.37 => $false ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(vacuousDall_type,type,
% 2.40/1.37 vacuousDall: $o ).
% 2.40/1.37
% 2.40/1.37 thf(vacuousDall,definition,
% 2.40/1.37 ( vacuousDall
% 2.40/1.37 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ emptyset )
% 2.40/1.37 => ( Xphi @ Xx ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(quantDeMorgan1_type,type,
% 2.40/1.37 quantDeMorgan1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(quantDeMorgan1,definition,
% 2.40/1.37 ( quantDeMorgan1
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o] :
% 2.40/1.37 ( ~ ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( Xphi @ Xx ) )
% 2.40/1.37 => ? [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(quantDeMorgan2_type,type,
% 2.40/1.37 quantDeMorgan2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(quantDeMorgan2,definition,
% 2.40/1.37 ( quantDeMorgan2
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ~ ( Xphi @ Xx ) )
% 2.40/1.37 => ~ ? [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(quantDeMorgan3_type,type,
% 2.40/1.37 quantDeMorgan3: $o ).
% 2.40/1.37
% 2.40/1.37 thf(quantDeMorgan3,definition,
% 2.40/1.37 ( quantDeMorgan3
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o] :
% 2.40/1.37 ( ~ ? [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 & ( Xphi @ Xx ) )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(quantDeMorgan4_type,type,
% 2.40/1.37 quantDeMorgan4: $o ).
% 2.40/1.37
% 2.40/1.37 thf(quantDeMorgan4,definition,
% 2.40/1.37 ( quantDeMorgan4
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o] :
% 2.40/1.37 ( ? [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 & ~ ( Xphi @ Xx ) )
% 2.40/1.37 => ~ ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(prop2setI_type,type,
% 2.40/1.37 prop2setI: $o ).
% 2.40/1.37
% 2.40/1.37 thf(prop2setI,definition,
% 2.40/1.37 ( prop2setI
% 2.40/1.37 = ( ! [Xphi: $o] :
% 2.40/1.37 ( Xphi
% 2.40/1.37 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(set2prop_type,type,
% 2.40/1.37 set2prop: $i > $o ).
% 2.40/1.37
% 2.40/1.37 thf(prop2set2propI_type,type,
% 2.40/1.37 prop2set2propI: $o ).
% 2.40/1.37
% 2.40/1.37 thf(prop2set2propI,definition,
% 2.40/1.37 ( prop2set2propI
% 2.40/1.37 = ( ! [Xphi: $o] :
% 2.40/1.37 ( Xphi
% 2.40/1.37 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(notdexE_type,type,
% 2.40/1.37 notdexE: $o ).
% 2.40/1.37
% 2.40/1.37 thf(notdexE,definition,
% 2.40/1.37 ( notdexE
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o] :
% 2.40/1.37 ( ~ ? [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 & ( Xphi @ Xx ) )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(notdallE_type,type,
% 2.40/1.37 notdallE: $o ).
% 2.40/1.37
% 2.40/1.37 thf(notdallE,definition,
% 2.40/1.37 ( notdallE
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o] :
% 2.40/1.37 ( ~ ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( Xphi @ Xx ) )
% 2.40/1.37 => ? [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(exuI1_type,type,
% 2.40/1.37 exuI1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(exuI1,definition,
% 2.40/1.37 ( exuI1
% 2.40/1.37 = ( ! [Xphi: $i > $o] :
% 2.40/1.37 ( ? [Xx: $i] :
% 2.40/1.37 ( ( Xphi @ Xx )
% 2.40/1.37 & ! [Xy: $i] :
% 2.40/1.37 ( ( Xphi @ Xy )
% 2.40/1.37 => ( Xx = Xy ) ) )
% 2.40/1.37 => ( exu
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(exuI3_type,type,
% 2.40/1.37 exuI3: $o ).
% 2.40/1.37
% 2.40/1.37 thf(exuI3,definition,
% 2.40/1.37 ( exuI3
% 2.40/1.37 = ( ! [Xphi: $i > $o] :
% 2.40/1.37 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.40/1.37 => ( ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( Xphi @ Xx )
% 2.40/1.37 => ( ( Xphi @ Xy )
% 2.40/1.37 => ( Xx = Xy ) ) )
% 2.40/1.37 => ( exu
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(exuI2_type,type,
% 2.40/1.37 exuI2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(exuI2,definition,
% 2.40/1.37 ( exuI2
% 2.40/1.37 = ( ! [Xphi: $i > $o] :
% 2.40/1.37 ( ? [Xx: $i] :
% 2.40/1.37 ! [Xy: $i] :
% 2.40/1.37 ( ( Xphi @ Xy )
% 2.40/1.37 <=> ( Xy = Xx ) )
% 2.40/1.37 => ( exu
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(inCongP_type,type,
% 2.40/1.37 inCongP: $o ).
% 2.40/1.37
% 2.40/1.37 thf(inCongP,definition,
% 2.40/1.37 ( inCongP
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( A = B )
% 2.40/1.37 => ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( Xx = Xy )
% 2.40/1.37 => ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(in__Cong_type,type,
% 2.40/1.37 in__Cong: $o ).
% 2.40/1.37
% 2.40/1.37 thf(in__Cong,definition,
% 2.40/1.37 ( in__Cong
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( A = B )
% 2.40/1.37 => ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( Xx = Xy )
% 2.40/1.37 => ( ( in @ Xx @ A )
% 2.40/1.37 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(exuE3u_type,type,
% 2.40/1.37 exuE3u: $o ).
% 2.40/1.37
% 2.40/1.37 thf(exuE3u,definition,
% 2.40/1.37 ( exuE3u
% 2.40/1.37 = ( ! [Xphi: $i > $o] :
% 2.40/1.37 ( ( exu
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.40/1.37 => ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( Xphi @ Xx )
% 2.40/1.37 => ( ( Xphi @ Xy )
% 2.40/1.37 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(exu__Cong_type,type,
% 2.40/1.37 exu__Cong: $o ).
% 2.40/1.37
% 2.40/1.37 thf(exu__Cong,definition,
% 2.40/1.37 ( exu__Cong
% 2.40/1.37 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.40/1.37 ( ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( Xx = Xy )
% 2.40/1.37 => ( ( Xphi @ Xx )
% 2.40/1.37 <=> ( Xpsi @ Xy ) ) )
% 2.40/1.37 => ( ( exu
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.40/1.37 <=> ( exu
% 2.40/1.37 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(emptyset__Cong_type,type,
% 2.40/1.37 emptyset__Cong: $o ).
% 2.40/1.37
% 2.40/1.37 thf(emptyset__Cong,definition,
% 2.40/1.37 ( emptyset__Cong
% 2.40/1.37 = ( emptyset = emptyset ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setadjoin__Cong_type,type,
% 2.40/1.37 setadjoin__Cong: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setadjoin__Cong,definition,
% 2.40/1.37 ( setadjoin__Cong
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( Xx = Xy )
% 2.40/1.37 => ! [Xz: $i,Xu: $i] :
% 2.40/1.37 ( ( Xz = Xu )
% 2.40/1.37 => ( ( setadjoin @ Xx @ Xz )
% 2.40/1.37 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(powerset__Cong_type,type,
% 2.40/1.37 powerset__Cong: $o ).
% 2.40/1.37
% 2.40/1.37 thf(powerset__Cong,definition,
% 2.40/1.37 ( powerset__Cong
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( A = B )
% 2.40/1.37 => ( ( powerset @ A )
% 2.40/1.37 = ( powerset @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setunion__Cong_type,type,
% 2.40/1.37 setunion__Cong: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setunion__Cong,definition,
% 2.40/1.37 ( setunion__Cong
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( A = B )
% 2.40/1.37 => ( ( setunion @ A )
% 2.40/1.37 = ( setunion @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(omega__Cong_type,type,
% 2.40/1.37 omega__Cong: $o ).
% 2.40/1.37
% 2.40/1.37 thf(omega__Cong,definition,
% 2.40/1.37 ( omega__Cong
% 2.40/1.37 = ( omega = omega ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(exuEu_type,type,
% 2.40/1.37 exuEu: $o ).
% 2.40/1.37
% 2.40/1.37 thf(exuEu,definition,
% 2.40/1.37 ( exuEu
% 2.40/1.37 = ( ! [Xphi: $i > $o] :
% 2.40/1.37 ( ( exu
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.40/1.37 => ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( Xphi @ Xx )
% 2.40/1.37 => ( ( Xphi @ Xy )
% 2.40/1.37 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(descr__Cong_type,type,
% 2.40/1.37 descr__Cong: $o ).
% 2.40/1.37
% 2.40/1.37 thf(descr__Cong,definition,
% 2.40/1.37 ( descr__Cong
% 2.40/1.37 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.40/1.37 ( ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( Xx = Xy )
% 2.40/1.37 => ( ( Xphi @ Xx )
% 2.40/1.37 <=> ( Xpsi @ Xy ) ) )
% 2.40/1.37 => ( ( exu
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.40/1.37 => ( ( exu
% 2.40/1.37 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.40/1.37 => ( ( descr
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.40/1.37 = ( descr
% 2.40/1.37 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(dsetconstr__Cong_type,type,
% 2.40/1.37 dsetconstr__Cong: $o ).
% 2.40/1.37
% 2.40/1.37 thf(dsetconstr__Cong,definition,
% 2.40/1.37 ( dsetconstr__Cong
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( A = B )
% 2.40/1.37 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( ( Xx = Xy )
% 2.40/1.37 => ( ( Xphi @ Xx )
% 2.40/1.37 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.40/1.37 => ( ( dsetconstr @ A
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.40/1.37 = ( dsetconstr @ B
% 2.40/1.37 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(subset_type,type,
% 2.40/1.37 subset: $i > $i > $o ).
% 2.40/1.37
% 2.40/1.37 thf(disjoint_type,type,
% 2.40/1.37 disjoint: $i > $i > $o ).
% 2.40/1.37
% 2.40/1.37 thf(setsmeet_type,type,
% 2.40/1.37 setsmeet: $i > $i > $o ).
% 2.40/1.37
% 2.40/1.37 thf(subsetI1_type,type,
% 2.40/1.37 subsetI1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(subsetI1,definition,
% 2.40/1.37 ( subsetI1
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ Xx @ B ) )
% 2.40/1.37 => ( subset @ A @ B ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(eqimpsubset2_type,type,
% 2.40/1.37 eqimpsubset2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(eqimpsubset2,definition,
% 2.40/1.37 ( eqimpsubset2
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( A = B )
% 2.40/1.37 => ( subset @ B @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(eqimpsubset1_type,type,
% 2.40/1.37 eqimpsubset1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(eqimpsubset1,definition,
% 2.40/1.37 ( eqimpsubset1
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( A = B )
% 2.40/1.37 => ( subset @ A @ B ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(subsetI2_type,type,
% 2.40/1.37 subsetI2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(subsetI2,definition,
% 2.40/1.37 ( subsetI2
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ Xx @ B ) )
% 2.40/1.37 => ( subset @ A @ B ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(emptysetsubset_type,type,
% 2.40/1.37 emptysetsubset: $o ).
% 2.40/1.37
% 2.40/1.37 thf(emptysetsubset,definition,
% 2.40/1.37 ( emptysetsubset
% 2.40/1.37 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(subsetE_type,type,
% 2.40/1.37 subsetE: $o ).
% 2.40/1.37
% 2.40/1.37 thf(subsetE,definition,
% 2.40/1.37 ( subsetE
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( subset @ A @ B )
% 2.40/1.37 => ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(subsetE2_type,type,
% 2.40/1.37 subsetE2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(subsetE2,definition,
% 2.40/1.37 ( subsetE2
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( subset @ A @ B )
% 2.40/1.37 => ( ~ ( in @ Xx @ B )
% 2.40/1.37 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(notsubsetI_type,type,
% 2.40/1.37 notsubsetI: $o ).
% 2.40/1.37
% 2.40/1.37 thf(notsubsetI,definition,
% 2.40/1.37 ( notsubsetI
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ~ ( in @ Xx @ B )
% 2.40/1.37 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(notequalI1_type,type,
% 2.40/1.37 notequalI1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(notequalI1,definition,
% 2.40/1.37 ( notequalI1
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ~ ( subset @ A @ B )
% 2.40/1.37 => ( A != B ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(notequalI2_type,type,
% 2.40/1.37 notequalI2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(notequalI2,definition,
% 2.40/1.37 ( notequalI2
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ~ ( in @ Xx @ B )
% 2.40/1.37 => ( A != B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(subsetRefl_type,type,
% 2.40/1.37 subsetRefl: $o ).
% 2.40/1.37
% 2.40/1.37 thf(subsetRefl,definition,
% 2.40/1.37 ( subsetRefl
% 2.40/1.37 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(subsetTrans_type,type,
% 2.40/1.37 subsetTrans: $o ).
% 2.40/1.37
% 2.40/1.37 thf(subsetTrans,definition,
% 2.40/1.37 ( subsetTrans
% 2.40/1.37 = ( ! [A: $i,B: $i,C: $i] :
% 2.40/1.37 ( ( subset @ A @ B )
% 2.40/1.37 => ( ( subset @ B @ C )
% 2.40/1.37 => ( subset @ A @ C ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setadjoinSub_type,type,
% 2.40/1.37 setadjoinSub: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setadjoinSub,definition,
% 2.40/1.37 ( setadjoinSub
% 2.40/1.37 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setadjoinSub2_type,type,
% 2.40/1.37 setadjoinSub2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setadjoinSub2,definition,
% 2.40/1.37 ( setadjoinSub2
% 2.40/1.37 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.40/1.37 ( ( subset @ A @ B )
% 2.40/1.37 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(subset2powerset_type,type,
% 2.40/1.37 subset2powerset: $o ).
% 2.40/1.37
% 2.40/1.37 thf(subset2powerset,definition,
% 2.40/1.37 ( subset2powerset
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( subset @ A @ B )
% 2.40/1.37 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setextsub_type,type,
% 2.40/1.37 setextsub: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setextsub,definition,
% 2.40/1.37 ( setextsub
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( subset @ A @ B )
% 2.40/1.37 => ( ( subset @ B @ A )
% 2.40/1.37 => ( A = B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(subsetemptysetimpeq_type,type,
% 2.40/1.37 subsetemptysetimpeq: $o ).
% 2.40/1.37
% 2.40/1.37 thf(subsetemptysetimpeq,definition,
% 2.40/1.37 ( subsetemptysetimpeq
% 2.40/1.37 = ( ! [A: $i] :
% 2.40/1.37 ( ( subset @ A @ emptyset )
% 2.40/1.37 => ( A = emptyset ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(powersetI1_type,type,
% 2.40/1.37 powersetI1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(powersetI1,definition,
% 2.40/1.37 ( powersetI1
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( subset @ B @ A )
% 2.40/1.37 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(powersetE1_type,type,
% 2.40/1.37 powersetE1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(powersetE1,definition,
% 2.40/1.37 ( powersetE1
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( in @ B @ ( powerset @ A ) )
% 2.40/1.37 => ( subset @ B @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(inPowerset_type,type,
% 2.40/1.37 inPowerset: $o ).
% 2.40/1.37
% 2.40/1.37 thf(inPowerset,definition,
% 2.40/1.37 ( inPowerset
% 2.40/1.37 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(powersetsubset_type,type,
% 2.40/1.37 powersetsubset: $o ).
% 2.40/1.37
% 2.40/1.37 thf(powersetsubset,definition,
% 2.40/1.37 ( powersetsubset
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( subset @ A @ B )
% 2.40/1.37 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(sepInPowerset_type,type,
% 2.40/1.37 sepInPowerset: $o ).
% 2.40/1.37
% 2.40/1.37 thf(sepInPowerset,definition,
% 2.40/1.37 ( sepInPowerset
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o] :
% 2.40/1.37 ( in
% 2.40/1.37 @ ( dsetconstr @ A
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.40/1.37 @ ( powerset @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(sepSubset_type,type,
% 2.40/1.37 sepSubset: $o ).
% 2.40/1.37
% 2.40/1.37 thf(sepSubset,definition,
% 2.40/1.37 ( sepSubset
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o] :
% 2.40/1.37 ( subset
% 2.40/1.37 @ ( dsetconstr @ A
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.40/1.37 @ A ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(binunion_type,type,
% 2.40/1.37 binunion: $i > $i > $i ).
% 2.40/1.37
% 2.40/1.37 thf(binunionIL_type,type,
% 2.40/1.37 binunionIL: $o ).
% 2.40/1.37
% 2.40/1.37 thf(binunionIL,definition,
% 2.40/1.37 ( binunionIL
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(upairset2IR_type,type,
% 2.40/1.37 upairset2IR: $o ).
% 2.40/1.37
% 2.40/1.37 thf(upairset2IR,definition,
% 2.40/1.37 ( upairset2IR
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(binunionIR_type,type,
% 2.40/1.37 binunionIR: $o ).
% 2.40/1.37
% 2.40/1.37 thf(binunionIR,definition,
% 2.40/1.37 ( binunionIR
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ B )
% 2.40/1.37 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(binunionEcases_type,type,
% 2.40/1.37 binunionEcases: $o ).
% 2.40/1.37
% 2.40/1.37 thf(binunionEcases,definition,
% 2.40/1.37 ( binunionEcases
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.40/1.37 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.40/1.37 => ( ( ( in @ Xx @ A )
% 2.40/1.37 => Xphi )
% 2.40/1.37 => ( ( ( in @ Xx @ B )
% 2.40/1.37 => Xphi )
% 2.40/1.37 => Xphi ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(binunionE_type,type,
% 2.40/1.37 binunionE: $o ).
% 2.40/1.37
% 2.40/1.37 thf(binunionE,definition,
% 2.40/1.37 ( binunionE
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.40/1.37 => ( ( in @ Xx @ A )
% 2.40/1.37 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(binunionLsub_type,type,
% 2.40/1.37 binunionLsub: $o ).
% 2.40/1.37
% 2.40/1.37 thf(binunionLsub,definition,
% 2.40/1.37 ( binunionLsub
% 2.40/1.37 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(binunionRsub_type,type,
% 2.40/1.37 binunionRsub: $o ).
% 2.40/1.37
% 2.40/1.37 thf(binunionRsub,definition,
% 2.40/1.37 ( binunionRsub
% 2.40/1.37 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(binintersect_type,type,
% 2.40/1.37 binintersect: $i > $i > $i ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectI_type,type,
% 2.40/1.37 binintersectI: $o ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectI,definition,
% 2.40/1.37 ( binintersectI
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ( in @ Xx @ B )
% 2.40/1.37 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectSubset5_type,type,
% 2.40/1.37 binintersectSubset5: $o ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectSubset5,definition,
% 2.40/1.37 ( binintersectSubset5
% 2.40/1.37 = ( ! [A: $i,B: $i,C: $i] :
% 2.40/1.37 ( ( subset @ C @ A )
% 2.40/1.37 => ( ( subset @ C @ B )
% 2.40/1.37 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectEL_type,type,
% 2.40/1.37 binintersectEL: $o ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectEL,definition,
% 2.40/1.37 ( binintersectEL
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.40/1.37 => ( in @ Xx @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectLsub_type,type,
% 2.40/1.37 binintersectLsub: $o ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectLsub,definition,
% 2.40/1.37 ( binintersectLsub
% 2.40/1.37 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectSubset2_type,type,
% 2.40/1.37 binintersectSubset2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectSubset2,definition,
% 2.40/1.37 ( binintersectSubset2
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( subset @ A @ B )
% 2.40/1.37 => ( ( binintersect @ A @ B )
% 2.40/1.37 = A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectSubset3_type,type,
% 2.40/1.37 binintersectSubset3: $o ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectSubset3,definition,
% 2.40/1.37 ( binintersectSubset3
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( ( binintersect @ A @ B )
% 2.40/1.37 = B )
% 2.40/1.37 => ( subset @ B @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectER_type,type,
% 2.40/1.37 binintersectER: $o ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectER,definition,
% 2.40/1.37 ( binintersectER
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.40/1.37 => ( in @ Xx @ B ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(disjointsetsI1_type,type,
% 2.40/1.37 disjointsetsI1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(disjointsetsI1,definition,
% 2.40/1.37 ( disjointsetsI1
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ~ ? [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 & ( in @ Xx @ B ) )
% 2.40/1.37 => ( ( binintersect @ A @ B )
% 2.40/1.37 = emptyset ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectRsub_type,type,
% 2.40/1.37 binintersectRsub: $o ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectRsub,definition,
% 2.40/1.37 ( binintersectRsub
% 2.40/1.37 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectSubset4_type,type,
% 2.40/1.37 binintersectSubset4: $o ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectSubset4,definition,
% 2.40/1.37 ( binintersectSubset4
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( subset @ B @ A )
% 2.40/1.37 => ( ( binintersect @ A @ B )
% 2.40/1.37 = B ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectSubset1_type,type,
% 2.40/1.37 binintersectSubset1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(binintersectSubset1,definition,
% 2.40/1.37 ( binintersectSubset1
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( ( binintersect @ A @ B )
% 2.40/1.37 = A )
% 2.40/1.37 => ( subset @ A @ B ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(bs114d_type,type,
% 2.40/1.37 bs114d: $o ).
% 2.40/1.37
% 2.40/1.37 thf(bs114d,definition,
% 2.40/1.37 ( bs114d
% 2.40/1.37 = ( ! [A: $i,B: $i,C: $i] :
% 2.40/1.37 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.40/1.37 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(regular_type,type,
% 2.40/1.37 regular: $i > $o ).
% 2.40/1.37
% 2.40/1.37 thf(setminus_type,type,
% 2.40/1.37 setminus: $i > $i > $i ).
% 2.40/1.37
% 2.40/1.37 thf(setminusI_type,type,
% 2.40/1.37 setminusI: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setminusI,definition,
% 2.40/1.37 ( setminusI
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ~ ( in @ Xx @ B )
% 2.40/1.37 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setminusEL_type,type,
% 2.40/1.37 setminusEL: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setminusEL,definition,
% 2.40/1.37 ( setminusEL
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.40/1.37 => ( in @ Xx @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setminusER_type,type,
% 2.40/1.37 setminusER: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setminusER,definition,
% 2.40/1.37 ( setminusER
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.40/1.37 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setminusSubset2_type,type,
% 2.40/1.37 setminusSubset2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setminusSubset2,definition,
% 2.40/1.37 ( setminusSubset2
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( subset @ A @ B )
% 2.40/1.37 => ( ( setminus @ A @ B )
% 2.40/1.37 = emptyset ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setminusERneg_type,type,
% 2.40/1.37 setminusERneg: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setminusERneg,definition,
% 2.40/1.37 ( setminusERneg
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.40/1.37 => ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setminusELneg_type,type,
% 2.40/1.37 setminusELneg: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setminusELneg,definition,
% 2.40/1.37 ( setminusELneg
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.40/1.37 => ( ~ ( in @ Xx @ B )
% 2.40/1.37 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setminusILneg_type,type,
% 2.40/1.37 setminusILneg: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setminusILneg,definition,
% 2.40/1.37 ( setminusILneg
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ~ ( in @ Xx @ A )
% 2.40/1.37 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setminusIRneg_type,type,
% 2.40/1.37 setminusIRneg: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setminusIRneg,definition,
% 2.40/1.37 ( setminusIRneg
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ B )
% 2.40/1.37 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setminusLsub_type,type,
% 2.40/1.37 setminusLsub: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setminusLsub,definition,
% 2.40/1.37 ( setminusLsub
% 2.40/1.37 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setminusSubset1_type,type,
% 2.40/1.37 setminusSubset1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setminusSubset1,definition,
% 2.40/1.37 ( setminusSubset1
% 2.40/1.37 = ( ! [A: $i,B: $i] :
% 2.40/1.37 ( ( ( setminus @ A @ B )
% 2.40/1.37 = emptyset )
% 2.40/1.37 => ( subset @ A @ B ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(symdiff_type,type,
% 2.40/1.37 symdiff: $i > $i > $i ).
% 2.40/1.37
% 2.40/1.37 thf(symdiffE_type,type,
% 2.40/1.37 symdiffE: $o ).
% 2.40/1.37
% 2.40/1.37 thf(symdiffE,definition,
% 2.40/1.37 ( symdiffE
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.40/1.37 => ! [Xphi: $o] :
% 2.40/1.37 ( ( ( in @ Xx @ A )
% 2.40/1.37 => ( ~ ( in @ Xx @ B )
% 2.40/1.37 => Xphi ) )
% 2.40/1.37 => ( ( ~ ( in @ Xx @ A )
% 2.40/1.37 => ( ( in @ Xx @ B )
% 2.40/1.37 => Xphi ) )
% 2.40/1.37 => Xphi ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(symdiffI1_type,type,
% 2.40/1.37 symdiffI1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(symdiffI1,definition,
% 2.40/1.37 ( symdiffI1
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ~ ( in @ Xx @ B )
% 2.40/1.37 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(symdiffI2_type,type,
% 2.40/1.37 symdiffI2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(symdiffI2,definition,
% 2.40/1.37 ( symdiffI2
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ~ ( in @ Xx @ A )
% 2.40/1.37 => ( ( in @ Xx @ B )
% 2.40/1.37 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(symdiffIneg1_type,type,
% 2.40/1.37 symdiffIneg1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(symdiffIneg1,definition,
% 2.40/1.37 ( symdiffIneg1
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ( in @ Xx @ B )
% 2.40/1.37 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(symdiffIneg2_type,type,
% 2.40/1.37 symdiffIneg2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(symdiffIneg2,definition,
% 2.40/1.37 ( symdiffIneg2
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ~ ( in @ Xx @ A )
% 2.40/1.37 => ( ~ ( in @ Xx @ B )
% 2.40/1.37 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(iskpair_type,type,
% 2.40/1.37 iskpair: $i > $o ).
% 2.40/1.37
% 2.40/1.37 thf(secondinupair_type,type,
% 2.40/1.37 secondinupair: $o ).
% 2.40/1.37
% 2.40/1.37 thf(secondinupair,definition,
% 2.40/1.37 ( secondinupair
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairIL_type,type,
% 2.40/1.37 setukpairIL: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairIL,definition,
% 2.40/1.37 ( setukpairIL
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairIR_type,type,
% 2.40/1.37 setukpairIR: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairIR,definition,
% 2.40/1.37 ( setukpairIR
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(kpairiskpair_type,type,
% 2.40/1.37 kpairiskpair: $o ).
% 2.40/1.37
% 2.40/1.37 thf(kpairiskpair,definition,
% 2.40/1.37 ( kpairiskpair
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(kpair_type,type,
% 2.40/1.37 kpair: $i > $i > $i ).
% 2.40/1.37
% 2.40/1.37 thf(kpairp_type,type,
% 2.40/1.37 kpairp: $o ).
% 2.40/1.37
% 2.40/1.37 thf(kpairp,definition,
% 2.40/1.37 ( kpairp
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(cartprod_type,type,
% 2.40/1.37 cartprod: $i > $i > $i ).
% 2.40/1.37
% 2.40/1.37 thf(singletonsubset_type,type,
% 2.40/1.37 singletonsubset: $o ).
% 2.40/1.37
% 2.40/1.37 thf(singletonsubset,definition,
% 2.40/1.37 ( singletonsubset
% 2.40/1.37 = ( ! [A: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(singletoninpowerset_type,type,
% 2.40/1.37 singletoninpowerset: $o ).
% 2.40/1.37
% 2.40/1.37 thf(singletoninpowerset,definition,
% 2.40/1.37 ( singletoninpowerset
% 2.40/1.37 = ( ! [A: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(singletoninpowunion_type,type,
% 2.40/1.37 singletoninpowunion: $o ).
% 2.40/1.37
% 2.40/1.37 thf(singletoninpowunion,definition,
% 2.40/1.37 ( singletoninpowunion
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(upairset2E_type,type,
% 2.40/1.37 upairset2E: $o ).
% 2.40/1.37
% 2.40/1.37 thf(upairset2E,definition,
% 2.40/1.37 ( upairset2E
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.40/1.37 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.40/1.37 => ( ( Xz = Xx )
% 2.40/1.37 | ( Xz = Xy ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(upairsubunion_type,type,
% 2.40/1.37 upairsubunion: $o ).
% 2.40/1.37
% 2.40/1.37 thf(upairsubunion,definition,
% 2.40/1.37 ( upairsubunion
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(upairinpowunion_type,type,
% 2.40/1.37 upairinpowunion: $o ).
% 2.40/1.37
% 2.40/1.37 thf(upairinpowunion,definition,
% 2.40/1.37 ( upairinpowunion
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(ubforcartprodlem1_type,type,
% 2.40/1.37 ubforcartprodlem1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(ubforcartprodlem1,definition,
% 2.40/1.37 ( ubforcartprodlem1
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(ubforcartprodlem2_type,type,
% 2.40/1.37 ubforcartprodlem2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(ubforcartprodlem2,definition,
% 2.40/1.37 ( ubforcartprodlem2
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(ubforcartprodlem3_type,type,
% 2.40/1.37 ubforcartprodlem3: $o ).
% 2.40/1.37
% 2.40/1.37 thf(ubforcartprodlem3,definition,
% 2.40/1.37 ( ubforcartprodlem3
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodpairin_type,type,
% 2.40/1.37 cartprodpairin: $o ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodpairin,definition,
% 2.40/1.37 ( cartprodpairin
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodmempair1_type,type,
% 2.40/1.37 cartprodmempair1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodmempair1,definition,
% 2.40/1.37 ( cartprodmempair1
% 2.40/1.37 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.40/1.37 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.40/1.37 => ? [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 & ? [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 & ( Xu
% 2.40/1.37 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodmempair_type,type,
% 2.40/1.37 cartprodmempair: $o ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodmempair,definition,
% 2.40/1.37 ( cartprodmempair
% 2.40/1.37 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.40/1.37 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.40/1.37 => ( iskpair @ Xu ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setunionE2_type,type,
% 2.40/1.37 setunionE2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setunionE2,definition,
% 2.40/1.37 ( setunionE2
% 2.40/1.37 = ( ! [A: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ ( setunion @ A ) )
% 2.40/1.37 => ? [X: $i] :
% 2.40/1.37 ( ( in @ X @ A )
% 2.40/1.37 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setunionsingleton1_type,type,
% 2.40/1.37 setunionsingleton1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setunionsingleton1,definition,
% 2.40/1.37 ( setunionsingleton1
% 2.40/1.37 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setunionsingleton2_type,type,
% 2.40/1.37 setunionsingleton2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setunionsingleton2,definition,
% 2.40/1.37 ( setunionsingleton2
% 2.40/1.37 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setunionsingleton_type,type,
% 2.40/1.37 setunionsingleton: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setunionsingleton,definition,
% 2.40/1.37 ( setunionsingleton
% 2.40/1.37 = ( ! [Xx: $i] :
% 2.40/1.37 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.40/1.37 = Xx ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(singleton_type,type,
% 2.40/1.37 singleton: $i > $o ).
% 2.40/1.37
% 2.40/1.37 thf(singleton,definition,
% 2.40/1.37 ( singleton
% 2.40/1.37 = ( ^ [A: $i] :
% 2.40/1.37 ? [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 & ( A
% 2.40/1.37 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(singletonprop_type,type,
% 2.40/1.37 singletonprop: $o ).
% 2.40/1.37
% 2.40/1.37 thf(singletonprop,definition,
% 2.40/1.37 ( singletonprop
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ A )
% 2.40/1.37 => ( ( Xphi @ Xx )
% 2.40/1.37 => ( ( Xphi @ Xy )
% 2.40/1.37 => ( Xx = Xy ) ) ) ) )
% 2.40/1.37 => ( ? [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 & ( Xphi @ Xx ) )
% 2.40/1.37 => ( singleton
% 2.40/1.37 @ ( dsetconstr @ A
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(ex1_type,type,
% 2.40/1.37 ex1: $i > ( $i > $o ) > $o ).
% 2.40/1.37
% 2.40/1.37 thf(ex1,definition,
% 2.40/1.37 ( ex1
% 2.40/1.37 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.40/1.37 ( singleton
% 2.40/1.37 @ ( dsetconstr @ A
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(ex1E1_type,type,
% 2.40/1.37 ex1E1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(ex1E1,definition,
% 2.40/1.37 ( ex1E1
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o] :
% 2.40/1.37 ( ( ex1 @ A
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.40/1.37 => ? [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(ex1I_type,type,
% 2.40/1.37 ex1I: $o ).
% 2.40/1.37
% 2.40/1.37 thf(ex1I,definition,
% 2.40/1.37 ( ex1I
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ( Xphi @ Xx )
% 2.40/1.37 => ( ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ A )
% 2.40/1.37 => ( ( Xphi @ Xy )
% 2.40/1.37 => ( Xy = Xx ) ) )
% 2.40/1.37 => ( ex1 @ A
% 2.40/1.37 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(ex1I2_type,type,
% 2.40/1.37 ex1I2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(ex1I2,definition,
% 2.40/1.37 ( ex1I2
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ A )
% 2.40/1.37 => ( ( Xphi @ Xx )
% 2.40/1.37 => ( ( Xphi @ Xy )
% 2.40/1.37 => ( Xx = Xy ) ) ) ) )
% 2.40/1.37 => ( ? [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 & ( Xphi @ Xx ) )
% 2.40/1.37 => ( ex1 @ A
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(singletonsuniq_type,type,
% 2.40/1.37 singletonsuniq: $o ).
% 2.40/1.37
% 2.40/1.37 thf(singletonsuniq,definition,
% 2.40/1.37 ( singletonsuniq
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( ( setadjoin @ Xx @ emptyset )
% 2.40/1.37 = ( setadjoin @ Xy @ emptyset ) )
% 2.40/1.37 => ( Xx = Xy ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(atmost1p_type,type,
% 2.40/1.37 atmost1p: $i > $o ).
% 2.40/1.37
% 2.40/1.37 thf(atleast2p_type,type,
% 2.40/1.37 atleast2p: $i > $o ).
% 2.40/1.37
% 2.40/1.37 thf(atmost2p_type,type,
% 2.40/1.37 atmost2p: $i > $o ).
% 2.40/1.37
% 2.40/1.37 thf(upairsetp_type,type,
% 2.40/1.37 upairsetp: $i > $o ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairinjL1_type,type,
% 2.40/1.37 setukpairinjL1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairinjL1,definition,
% 2.40/1.37 ( setukpairinjL1
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.40/1.37 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.40/1.37 => ( Xx = Xz ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(kfstsingleton_type,type,
% 2.40/1.37 kfstsingleton: $o ).
% 2.40/1.37
% 2.40/1.37 thf(kfstsingleton,definition,
% 2.40/1.37 ( kfstsingleton
% 2.40/1.37 = ( ! [Xu: $i] :
% 2.40/1.37 ( ( iskpair @ Xu )
% 2.40/1.37 => ( singleton
% 2.40/1.37 @ ( dsetconstr @ ( setunion @ Xu )
% 2.40/1.37 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(theprop_type,type,
% 2.40/1.37 theprop: $o ).
% 2.40/1.37
% 2.40/1.37 thf(theprop,definition,
% 2.40/1.37 ( theprop
% 2.40/1.37 = ( ! [X: $i] :
% 2.40/1.37 ( ( singleton @ X )
% 2.40/1.37 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(kfst_type,type,
% 2.40/1.37 kfst: $i > $i ).
% 2.40/1.37
% 2.40/1.37 thf(kfstpairEq_type,type,
% 2.40/1.37 kfstpairEq: $o ).
% 2.40/1.37
% 2.40/1.37 thf(kfstpairEq,definition,
% 2.40/1.37 ( kfstpairEq
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.40/1.37 = Xx ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodfstin_type,type,
% 2.40/1.37 cartprodfstin: $o ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodfstin,definition,
% 2.40/1.37 ( cartprodfstin
% 2.40/1.37 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.40/1.37 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.40/1.37 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairinjL2_type,type,
% 2.40/1.37 setukpairinjL2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairinjL2,definition,
% 2.40/1.37 ( setukpairinjL2
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.40/1.37 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.40/1.37 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.40/1.37 => ( Xx = Xz ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairinjL_type,type,
% 2.40/1.37 setukpairinjL: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairinjL,definition,
% 2.40/1.37 ( setukpairinjL
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.40/1.37 ( ( ( kpair @ Xx @ Xy )
% 2.40/1.37 = ( kpair @ Xz @ Xu ) )
% 2.40/1.37 => ( Xx = Xz ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairinjR11_type,type,
% 2.40/1.37 setukpairinjR11: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairinjR11,definition,
% 2.40/1.37 ( setukpairinjR11
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( Xx = Xy )
% 2.40/1.37 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.40/1.37 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairinjR12_type,type,
% 2.40/1.37 setukpairinjR12: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairinjR12,definition,
% 2.40/1.37 ( setukpairinjR12
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( Xx = Xy )
% 2.40/1.37 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.40/1.37 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairinjR1_type,type,
% 2.40/1.37 setukpairinjR1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairinjR1,definition,
% 2.40/1.37 ( setukpairinjR1
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.40/1.37 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.40/1.37 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.40/1.37 => ( ( Xz = Xu )
% 2.40/1.37 => ( Xy = Xu ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(upairequniteq_type,type,
% 2.40/1.37 upairequniteq: $o ).
% 2.40/1.37
% 2.40/1.37 thf(upairequniteq,definition,
% 2.40/1.37 ( upairequniteq
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.40/1.37 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.40/1.37 = ( setadjoin @ Xz @ emptyset ) )
% 2.40/1.37 => ( Xx = Xy ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairinjR2_type,type,
% 2.40/1.37 setukpairinjR2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairinjR2,definition,
% 2.40/1.37 ( setukpairinjR2
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.40/1.37 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.40/1.37 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.40/1.37 => ( Xy = Xu ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairinjR_type,type,
% 2.40/1.37 setukpairinjR: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setukpairinjR,definition,
% 2.40/1.37 ( setukpairinjR
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.40/1.37 ( ( ( kpair @ Xx @ Xy )
% 2.40/1.37 = ( kpair @ Xz @ Xu ) )
% 2.40/1.37 => ( Xy = Xu ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(ksndsingleton_type,type,
% 2.40/1.37 ksndsingleton: $o ).
% 2.40/1.37
% 2.40/1.37 thf(ksndsingleton,definition,
% 2.40/1.37 ( ksndsingleton
% 2.40/1.37 = ( ! [Xu: $i] :
% 2.40/1.37 ( ( iskpair @ Xu )
% 2.40/1.37 => ( singleton
% 2.40/1.37 @ ( dsetconstr @ ( setunion @ Xu )
% 2.40/1.37 @ ^ [Xx: $i] :
% 2.40/1.37 ( Xu
% 2.40/1.37 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(ksnd_type,type,
% 2.40/1.37 ksnd: $i > $i ).
% 2.40/1.37
% 2.40/1.37 thf(ksndpairEq_type,type,
% 2.40/1.37 ksndpairEq: $o ).
% 2.40/1.37
% 2.40/1.37 thf(ksndpairEq,definition,
% 2.40/1.37 ( ksndpairEq
% 2.40/1.37 = ( ! [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.40/1.37 = Xy ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(kpairsurjEq_type,type,
% 2.40/1.37 kpairsurjEq: $o ).
% 2.40/1.37
% 2.40/1.37 thf(kpairsurjEq,definition,
% 2.40/1.37 ( kpairsurjEq
% 2.40/1.37 = ( ! [Xu: $i] :
% 2.40/1.37 ( ( iskpair @ Xu )
% 2.40/1.37 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.40/1.37 = Xu ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodsndin_type,type,
% 2.40/1.37 cartprodsndin: $o ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodsndin,definition,
% 2.40/1.37 ( cartprodsndin
% 2.40/1.37 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.40/1.37 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.40/1.37 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodpairmemEL_type,type,
% 2.40/1.37 cartprodpairmemEL: $o ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodpairmemEL,definition,
% 2.40/1.37 ( cartprodpairmemEL
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.40/1.37 => ( in @ Xx @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodpairmemER_type,type,
% 2.40/1.37 cartprodpairmemER: $o ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodpairmemER,definition,
% 2.40/1.37 ( cartprodpairmemER
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.40/1.37 => ( in @ Xy @ B ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodmempaircEq_type,type,
% 2.40/1.37 cartprodmempaircEq: $o ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodmempaircEq,definition,
% 2.40/1.37 ( cartprodmempaircEq
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( ( kpair @ Xx @ Xy )
% 2.40/1.37 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodfstpairEq_type,type,
% 2.40/1.37 cartprodfstpairEq: $o ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodfstpairEq,definition,
% 2.40/1.37 ( cartprodfstpairEq
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.40/1.37 = Xx ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodsndpairEq_type,type,
% 2.40/1.37 cartprodsndpairEq: $o ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodsndpairEq,definition,
% 2.40/1.37 ( cartprodsndpairEq
% 2.40/1.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.40/1.37 = Xy ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodpairsurjEq_type,type,
% 2.40/1.37 cartprodpairsurjEq: $o ).
% 2.40/1.37
% 2.40/1.37 thf(cartprodpairsurjEq,definition,
% 2.40/1.37 ( cartprodpairsurjEq
% 2.40/1.37 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.40/1.37 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.40/1.37 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.40/1.37 = Xu ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(breln_type,type,
% 2.40/1.37 breln: $i > $i > $i > $o ).
% 2.40/1.37
% 2.40/1.37 thf(breln,definition,
% 2.40/1.37 ( breln
% 2.40/1.37 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(dpsetconstr_type,type,
% 2.40/1.37 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.40/1.37
% 2.40/1.37 thf(dpsetconstrI_type,type,
% 2.40/1.37 dpsetconstrI: $o ).
% 2.40/1.37
% 2.40/1.37 thf(dpsetconstrI,definition,
% 2.40/1.37 ( dpsetconstrI
% 2.40/1.37 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( ( Xphi @ Xx @ Xy )
% 2.40/1.37 => ( in @ ( kpair @ Xx @ Xy )
% 2.40/1.37 @ ( dpsetconstr @ A @ B
% 2.40/1.37 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(dpsetconstrSub_type,type,
% 2.40/1.37 dpsetconstrSub: $o ).
% 2.40/1.37
% 2.40/1.37 thf(dpsetconstrSub,definition,
% 2.40/1.37 ( dpsetconstrSub
% 2.40/1.37 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.40/1.37 ( subset
% 2.40/1.37 @ ( dpsetconstr @ A @ B
% 2.40/1.37 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.40/1.37 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(setOfPairsIsBReln_type,type,
% 2.40/1.37 setOfPairsIsBReln: $o ).
% 2.40/1.37
% 2.40/1.37 thf(setOfPairsIsBReln,definition,
% 2.40/1.37 ( setOfPairsIsBReln
% 2.40/1.37 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.40/1.37 ( breln @ A @ B
% 2.40/1.37 @ ( dpsetconstr @ A @ B
% 2.40/1.37 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(dpsetconstrERa_type,type,
% 2.40/1.37 dpsetconstrERa: $o ).
% 2.40/1.37
% 2.40/1.37 thf(dpsetconstrERa,definition,
% 2.40/1.37 ( dpsetconstrERa
% 2.40/1.37 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.40/1.37 @ ( dpsetconstr @ A @ B
% 2.40/1.37 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.40/1.37 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(dpsetconstrEL1_type,type,
% 2.40/1.37 dpsetconstrEL1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(dpsetconstrEL1,definition,
% 2.40/1.37 ( dpsetconstrEL1
% 2.40/1.37 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( in @ ( kpair @ Xx @ Xy )
% 2.40/1.37 @ ( dpsetconstr @ A @ B
% 2.40/1.37 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.40/1.37 => ( in @ Xx @ A ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(dpsetconstrEL2_type,type,
% 2.40/1.37 dpsetconstrEL2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(dpsetconstrEL2,definition,
% 2.40/1.37 ( dpsetconstrEL2
% 2.40/1.37 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( in @ ( kpair @ Xx @ Xy )
% 2.40/1.37 @ ( dpsetconstr @ A @ B
% 2.40/1.37 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.40/1.37 => ( in @ Xy @ B ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(dpsetconstrER_type,type,
% 2.40/1.37 dpsetconstrER: $o ).
% 2.40/1.37
% 2.40/1.37 thf(dpsetconstrER,definition,
% 2.40/1.37 ( dpsetconstrER
% 2.40/1.37 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( in @ ( kpair @ Xx @ Xy )
% 2.40/1.37 @ ( dpsetconstr @ A @ B
% 2.40/1.37 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.40/1.37 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(func_type,type,
% 2.40/1.37 func: $i > $i > $i > $o ).
% 2.40/1.37
% 2.40/1.37 thf(funcSet_type,type,
% 2.40/1.37 funcSet: $i > $i > $i ).
% 2.40/1.37
% 2.40/1.37 thf(funcImageSingleton_type,type,
% 2.40/1.37 funcImageSingleton: $o ).
% 2.40/1.37
% 2.40/1.37 thf(funcImageSingleton,definition,
% 2.40/1.37 ( funcImageSingleton
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( func @ A @ B @ Xf )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( singleton
% 2.40/1.37 @ ( dsetconstr @ B
% 2.40/1.37 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(apProp_type,type,
% 2.40/1.37 apProp: $o ).
% 2.40/1.37
% 2.40/1.37 thf(apProp,definition,
% 2.40/1.37 ( apProp
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( func @ A @ B @ Xf )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in
% 2.40/1.37 @ ( setunion
% 2.40/1.37 @ ( dsetconstr @ B
% 2.40/1.37 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.40/1.37 @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(ap_type,type,
% 2.40/1.37 ap: $i > $i > $i > $i > $i ).
% 2.40/1.37
% 2.40/1.37 thf(app_type,type,
% 2.40/1.37 app: $o ).
% 2.40/1.37
% 2.40/1.37 thf(app,definition,
% 2.40/1.37 ( app
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( func @ A @ B @ Xf )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(infuncsetfunc_type,type,
% 2.40/1.37 infuncsetfunc: $o ).
% 2.40/1.37
% 2.40/1.37 thf(infuncsetfunc,definition,
% 2.40/1.37 ( infuncsetfunc
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.40/1.37 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(ap2p_type,type,
% 2.40/1.37 ap2p: $o ).
% 2.40/1.37
% 2.40/1.37 thf(ap2p,definition,
% 2.40/1.37 ( ap2p
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(funcinfuncset_type,type,
% 2.40/1.37 funcinfuncset: $o ).
% 2.40/1.37
% 2.40/1.37 thf(funcinfuncset,definition,
% 2.40/1.37 ( funcinfuncset
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( func @ A @ B @ Xf )
% 2.40/1.37 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(lamProp_type,type,
% 2.40/1.37 lamProp: $o ).
% 2.40/1.37
% 2.40/1.37 thf(lamProp,definition,
% 2.40/1.37 ( lamProp
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.40/1.37 => ( func @ A @ B
% 2.40/1.37 @ ( dpsetconstr @ A @ B
% 2.40/1.37 @ ^ [Xx: $i,Xy: $i] :
% 2.40/1.37 ( ( Xf @ Xx )
% 2.40/1.37 = Xy ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(lam_type,type,
% 2.40/1.37 lam: $i > $i > ( $i > $i ) > $i ).
% 2.40/1.37
% 2.40/1.37 thf(lamp_type,type,
% 2.40/1.37 lamp: $o ).
% 2.40/1.37
% 2.40/1.37 thf(lamp,definition,
% 2.40/1.37 ( lamp
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.40/1.37 => ( func @ A @ B
% 2.40/1.37 @ ( lam @ A @ B
% 2.40/1.37 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(lam2p_type,type,
% 2.40/1.37 lam2p: $o ).
% 2.40/1.37
% 2.40/1.37 thf(lam2p,definition,
% 2.40/1.37 ( lam2p
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.40/1.37 => ( in
% 2.40/1.37 @ ( lam @ A @ B
% 2.40/1.37 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.40/1.37 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(brelnall1_type,type,
% 2.40/1.37 brelnall1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(brelnall1,definition,
% 2.40/1.37 ( brelnall1
% 2.40/1.37 = ( ! [A: $i,B: $i,R: $i] :
% 2.40/1.37 ( ( breln @ A @ B @ R )
% 2.40/1.37 => ! [Xphi: $i > $o] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.40/1.37 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ R )
% 2.40/1.37 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(brelnall2_type,type,
% 2.40/1.37 brelnall2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(brelnall2,definition,
% 2.40/1.37 ( brelnall2
% 2.40/1.37 = ( ! [A: $i,B: $i,R: $i] :
% 2.40/1.37 ( ( breln @ A @ B @ R )
% 2.40/1.37 => ! [Xphi: $i > $o] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.40/1.37 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ R )
% 2.40/1.37 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(ex1E2_type,type,
% 2.40/1.37 ex1E2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(ex1E2,definition,
% 2.40/1.37 ( ex1E2
% 2.40/1.37 = ( ! [A: $i,Xphi: $i > $o] :
% 2.40/1.37 ( ( ex1 @ A
% 2.40/1.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ A )
% 2.40/1.37 => ( ( Xphi @ Xx )
% 2.40/1.37 => ( ( Xphi @ Xy )
% 2.40/1.37 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(funcGraphProp1_type,type,
% 2.40/1.37 funcGraphProp1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(funcGraphProp1,definition,
% 2.40/1.37 ( funcGraphProp1
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( func @ A @ B @ Xf )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(funcGraphProp3_type,type,
% 2.40/1.37 funcGraphProp3: $o ).
% 2.40/1.37
% 2.40/1.37 thf(funcGraphProp3,definition,
% 2.40/1.37 ( funcGraphProp3
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(funcGraphProp2_type,type,
% 2.40/1.37 funcGraphProp2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(funcGraphProp2,definition,
% 2.40/1.37 ( funcGraphProp2
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( func @ A @ B @ Xf )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.40/1.37 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.40/1.37 = Xy ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(funcextLem_type,type,
% 2.40/1.37 funcextLem: $o ).
% 2.40/1.37
% 2.40/1.37 thf(funcextLem,definition,
% 2.40/1.37 ( funcextLem
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( func @ A @ B @ Xf )
% 2.40/1.37 => ! [Xg: $i] :
% 2.40/1.37 ( ( func @ A @ B @ Xg )
% 2.40/1.37 => ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.40/1.37 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.40/1.37 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(funcGraphProp4_type,type,
% 2.40/1.37 funcGraphProp4: $o ).
% 2.40/1.37
% 2.40/1.37 thf(funcGraphProp4,definition,
% 2.40/1.37 ( funcGraphProp4
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.40/1.37 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.40/1.37 = Xy ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(subbreln_type,type,
% 2.40/1.37 subbreln: $o ).
% 2.40/1.37
% 2.40/1.37 thf(subbreln,definition,
% 2.40/1.37 ( subbreln
% 2.40/1.37 = ( ! [A: $i,B: $i,R: $i] :
% 2.40/1.37 ( ( breln @ A @ B @ R )
% 2.40/1.37 => ! [S: $i] :
% 2.40/1.37 ( ( breln @ A @ B @ S )
% 2.40/1.37 => ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.40/1.37 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.40/1.37 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(eqbreln_type,type,
% 2.40/1.37 eqbreln: $o ).
% 2.40/1.37
% 2.40/1.37 thf(eqbreln,definition,
% 2.40/1.37 ( eqbreln
% 2.40/1.37 = ( ! [A: $i,B: $i,R: $i] :
% 2.40/1.37 ( ( breln @ A @ B @ R )
% 2.40/1.37 => ! [S: $i] :
% 2.40/1.37 ( ( breln @ A @ B @ S )
% 2.40/1.37 => ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.40/1.37 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.40/1.37 => ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ! [Xy: $i] :
% 2.40/1.37 ( ( in @ Xy @ B )
% 2.40/1.37 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.40/1.37 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.40/1.37 => ( R = S ) ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(funcext_type,type,
% 2.40/1.37 funcext: $o ).
% 2.40/1.37
% 2.40/1.37 thf(funcext,definition,
% 2.40/1.37 ( funcext
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( func @ A @ B @ Xf )
% 2.40/1.37 => ! [Xg: $i] :
% 2.40/1.37 ( ( func @ A @ B @ Xg )
% 2.40/1.37 => ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.40/1.37 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.40/1.37 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(funcext2_type,type,
% 2.40/1.37 funcext2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(funcext2,definition,
% 2.40/1.37 ( funcext2
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.40/1.37 => ! [Xg: $i] :
% 2.40/1.37 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.40/1.37 => ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.40/1.37 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.40/1.37 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(ap2apEq1_type,type,
% 2.40/1.37 ap2apEq1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(ap2apEq1,definition,
% 2.40/1.37 ( ap2apEq1
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.40/1.37 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(ap2apEq2_type,type,
% 2.40/1.37 ap2apEq2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(ap2apEq2,definition,
% 2.40/1.37 ( ap2apEq2
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( func @ A @ B @ Xf )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.40/1.37 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(beta1_type,type,
% 2.40/1.37 beta1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(beta1,definition,
% 2.40/1.37 ( beta1
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ( ap @ A @ B
% 2.40/1.37 @ ( lam @ A @ B
% 2.40/1.37 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.40/1.37 @ Xx )
% 2.40/1.37 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(eta1_type,type,
% 2.40/1.37 eta1: $o ).
% 2.40/1.37
% 2.40/1.37 thf(eta1,definition,
% 2.40/1.37 ( eta1
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( func @ A @ B @ Xf )
% 2.40/1.37 => ( ( lam @ A @ B
% 2.40/1.37 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.40/1.37 = Xf ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(lam2lamEq_type,type,
% 2.40/1.37 lam2lamEq: $o ).
% 2.40/1.37
% 2.40/1.37 thf(lam2lamEq,definition,
% 2.40/1.37 ( lam2lamEq
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.40/1.37 => ( ( lam @ A @ B
% 2.40/1.37 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.40/1.37 = ( lam @ A @ B
% 2.40/1.37 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(beta2_type,type,
% 2.40/1.37 beta2: $o ).
% 2.40/1.37
% 2.40/1.37 thf(beta2,definition,
% 2.40/1.37 ( beta2
% 2.40/1.37 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.40/1.37 ( ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.40/1.37 => ! [Xx: $i] :
% 2.40/1.37 ( ( in @ Xx @ A )
% 2.40/1.37 => ( ( ap @ A @ B
% 2.40/1.37 @ ( lam @ A @ B
% 2.40/1.37 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.40/1.37 @ Xx )
% 2.40/1.37 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.40/1.37
% 2.40/1.37 thf(eta2,conjecture,
% 2.40/1.37 ( setextAx
% 2.40/1.37 => ( emptysetAx
% 2.40/1.37 => ( setadjoinAx
% 2.40/1.37 => ( powersetAx
% 2.40/1.37 => ( setunionAx
% 2.40/1.37 => ( omega0Ax
% 2.40/1.37 => ( omegaSAx
% 2.40/1.37 => ( omegaIndAx
% 2.40/1.37 => ( replAx
% 2.40/1.37 => ( foundationAx
% 2.40/1.37 => ( wellorderingAx
% 2.40/1.37 => ( descrp
% 2.40/1.37 => ( dsetconstrI
% 2.40/1.37 => ( dsetconstrEL
% 2.40/1.37 => ( dsetconstrER
% 2.40/1.37 => ( exuE1
% 2.40/1.37 => ( prop2setE
% 2.40/1.37 => ( emptysetE
% 2.40/1.37 => ( emptysetimpfalse
% 2.40/1.37 => ( notinemptyset
% 2.40/1.37 => ( exuE3e
% 2.40/1.37 => ( setext
% 2.40/1.37 => ( emptyI
% 2.40/1.37 => ( noeltsimpempty
% 2.40/1.37 => ( setbeta
% 2.40/1.37 => ( nonemptyE1
% 2.40/1.37 => ( nonemptyI
% 2.40/1.37 => ( nonemptyI1
% 2.40/1.37 => ( setadjoinIL
% 2.40/1.37 => ( emptyinunitempty
% 2.40/1.37 => ( setadjoinIR
% 2.40/1.37 => ( setadjoinE
% 2.40/1.37 => ( setadjoinOr
% 2.40/1.37 => ( setoftrueEq
% 2.40/1.37 => ( powersetI
% 2.40/1.37 => ( emptyinPowerset
% 2.40/1.37 => ( emptyInPowerset
% 2.40/1.37 => ( powersetE
% 2.40/1.37 => ( setunionI
% 2.40/1.37 => ( setunionE
% 2.40/1.37 => ( subPowSU
% 2.40/1.37 => ( exuE2
% 2.40/1.37 => ( nonemptyImpWitness
% 2.40/1.37 => ( uniqinunit
% 2.40/1.37 => ( notinsingleton
% 2.40/1.37 => ( eqinunit
% 2.40/1.37 => ( singletonsswitch
% 2.40/1.37 => ( upairsetE
% 2.40/1.37 => ( upairsetIL
% 2.40/1.37 => ( upairsetIR
% 2.40/1.37 => ( emptyE1
% 2.40/1.37 => ( vacuousDall
% 2.40/1.37 => ( quantDeMorgan1
% 2.40/1.37 => ( quantDeMorgan2
% 2.40/1.37 => ( quantDeMorgan3
% 2.40/1.37 => ( quantDeMorgan4
% 2.40/1.37 => ( prop2setI
% 2.40/1.37 => ( prop2set2propI
% 2.40/1.37 => ( notdexE
% 2.40/1.37 => ( notdallE
% 2.40/1.37 => ( exuI1
% 2.40/1.37 => ( exuI3
% 2.40/1.37 => ( exuI2
% 2.40/1.37 => ( inCongP
% 2.40/1.37 => ( in__Cong
% 2.40/1.37 => ( exuE3u
% 2.40/1.37 => ( exu__Cong
% 2.40/1.37 => ( emptyset__Cong
% 2.40/1.37 => ( setadjoin__Cong
% 2.40/1.37 => ( powerset__Cong
% 2.40/1.37 => ( setunion__Cong
% 2.40/1.37 => ( omega__Cong
% 2.40/1.37 => ( exuEu
% 2.40/1.37 => ( descr__Cong
% 2.40/1.37 => ( dsetconstr__Cong
% 2.40/1.37 => ( subsetI1
% 2.40/1.37 => ( eqimpsubset2
% 2.40/1.37 => ( eqimpsubset1
% 2.40/1.37 => ( subsetI2
% 2.40/1.37 => ( emptysetsubset
% 2.40/1.37 => ( subsetE
% 2.40/1.37 => ( subsetE2
% 2.40/1.37 => ( notsubsetI
% 2.40/1.37 => ( notequalI1
% 2.40/1.37 => ( notequalI2
% 2.40/1.37 => ( subsetRefl
% 2.40/1.37 => ( subsetTrans
% 2.40/1.37 => ( setadjoinSub
% 2.40/1.37 => ( setadjoinSub2
% 2.40/1.37 => ( subset2powerset
% 2.40/1.37 => ( setextsub
% 2.40/1.37 => ( subsetemptysetimpeq
% 2.40/1.37 => ( powersetI1
% 2.40/1.37 => ( powersetE1
% 2.40/1.37 => ( inPowerset
% 2.40/1.37 => ( powersetsubset
% 2.40/1.37 => ( sepInPowerset
% 2.40/1.37 => ( sepSubset
% 2.40/1.37 => ( binunionIL
% 2.40/1.37 => ( upairset2IR
% 2.40/1.37 => ( binunionIR
% 2.40/1.37 => ( binunionEcases
% 2.40/1.37 => ( binunionE
% 2.40/1.37 => ( binunionLsub
% 2.40/1.37 => ( binunionRsub
% 2.40/1.37 => ( binintersectI
% 2.40/1.37 => ( binintersectSubset5
% 2.40/1.37 => ( binintersectEL
% 2.40/1.37 => ( binintersectLsub
% 2.40/1.37 => ( binintersectSubset2
% 2.40/1.37 => ( binintersectSubset3
% 2.40/1.37 => ( binintersectER
% 2.40/1.37 => ( disjointsetsI1
% 2.40/1.37 => ( binintersectRsub
% 2.40/1.37 => ( binintersectSubset4
% 2.40/1.37 => ( binintersectSubset1
% 2.40/1.37 => ( bs114d
% 2.40/1.37 => ( setminusI
% 2.40/1.37 => ( setminusEL
% 2.40/1.37 => ( setminusER
% 2.40/1.37 => ( setminusSubset2
% 2.40/1.37 => ( setminusERneg
% 2.40/1.37 => ( setminusELneg
% 2.40/1.37 => ( setminusILneg
% 2.40/1.37 => ( setminusIRneg
% 2.40/1.37 => ( setminusLsub
% 2.40/1.37 => ( setminusSubset1
% 2.40/1.37 => ( symdiffE
% 2.40/1.37 => ( symdiffI1
% 2.40/1.37 => ( symdiffI2
% 2.40/1.37 => ( symdiffIneg1
% 2.40/1.37 => ( symdiffIneg2
% 2.40/1.37 => ( secondinupair
% 2.40/1.37 => ( setukpairIL
% 2.40/1.37 => ( setukpairIR
% 2.40/1.37 => ( kpairiskpair
% 2.40/1.37 => ( kpairp
% 2.40/1.37 => ( singletonsubset
% 2.40/1.37 => ( singletoninpowerset
% 2.40/1.37 => ( singletoninpowunion
% 2.40/1.37 => ( upairset2E
% 2.40/1.37 => ( upairsubunion
% 2.40/1.37 => ( upairinpowunion
% 2.40/1.37 => ( ubforcartprodlem1
% 2.40/1.37 => ( ubforcartprodlem2
% 2.40/1.37 => ( ubforcartprodlem3
% 2.40/1.37 => ( cartprodpairin
% 2.40/1.37 => ( cartprodmempair1
% 2.40/1.37 => ( cartprodmempair
% 2.40/1.37 => ( setunionE2
% 2.40/1.37 => ( setunionsingleton1
% 2.40/1.37 => ( setunionsingleton2
% 2.40/1.37 => ( setunionsingleton
% 2.40/1.37 => ( singletonprop
% 2.40/1.37 => ( ex1E1
% 2.40/1.37 => ( ex1I
% 2.40/1.37 => ( ex1I2
% 2.40/1.37 => ( singletonsuniq
% 2.40/1.37 => ( setukpairinjL1
% 2.40/1.37 => ( kfstsingleton
% 2.40/1.37 => ( theprop
% 2.40/1.37 => ( kfstpairEq
% 2.40/1.37 => ( cartprodfstin
% 2.40/1.37 => ( setukpairinjL2
% 2.40/1.37 => ( setukpairinjL
% 2.40/1.37 => ( setukpairinjR11
% 2.40/1.37 => ( setukpairinjR12
% 2.40/1.37 => ( setukpairinjR1
% 2.40/1.37 => ( upairequniteq
% 2.40/1.37 => ( setukpairinjR2
% 2.40/1.37 => ( setukpairinjR
% 2.40/1.37 => ( ksndsingleton
% 2.40/1.37 => ( ksndpairEq
% 2.40/1.37 => ( kpairsurjEq
% 2.40/1.37 => ( cartprodsndin
% 2.40/1.37 => ( cartprodpairmemEL
% 2.40/1.37 => ( cartprodpairmemER
% 2.40/1.37 => ( cartprodmempaircEq
% 2.40/1.37 => ( cartprodfstpairEq
% 2.40/1.37 => ( cartprodsndpairEq
% 2.40/1.37 => ( cartprodpairsurjEq
% 2.40/1.37 => ( dpsetconstrI
% 2.40/1.37 => ( dpsetconstrSub
% 2.40/1.37 => ( setOfPairsIsBReln
% 2.40/1.37 => ( dpsetconstrERa
% 2.40/1.37 => ( dpsetconstrEL1
% 2.40/1.37 => ( dpsetconstrEL2
% 2.40/1.37 => ( dpsetconstrER
% 2.40/1.37 => ( funcImageSingleton
% 2.40/1.37 => ( apProp
% 2.40/1.37 => ( app
% 2.40/1.37 => ( infuncsetfunc
% 2.40/1.37 => ( ap2p
% 2.40/1.37 => ( funcinfuncset
% 2.40/1.37 => ( lamProp
% 2.40/1.37 => ( lamp
% 2.40/1.37 => ( lam2p
% 2.40/1.37 => ( brelnall1
% 2.40/1.37 => ( brelnall2
% 2.40/1.37 => ( ex1E2
% 2.40/1.37 => ( funcGraphProp1
% 2.40/1.37 => ( funcGraphProp3
% 2.40/1.37 => ( funcGraphProp2
% 2.40/1.37 => ( funcextLem
% 2.40/1.37 => ( funcGraphProp4
% 2.40/1.37 => ( subbreln
% 2.40/1.37 => ( eqbreln
% 2.40/1.37 => ( funcext
% 2.40/1.37 => ( funcext2
% 2.40/1.37 => ( ap2apEq1
% 2.40/1.37 => ( ap2apEq2
% 2.40/1.37 => ( beta1
% 2.40/1.37 => ( eta1
% 2.40/1.37 => ( lam2lamEq
% 2.40/1.37 => ( beta2
% 2.40/1.37 => ! [A: $i,B: $i,Xf: $i] :
% 2.40/1.37 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.40/1.37 => ( ( lam @ A @ B
% 2.40/1.37 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.40/1.37 = Xf ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.40/1.37 ------------------------
% 2.40/1.39 % (6826)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_6599 for (2999ds/27Mi)
% 2.40/1.39 % (6829)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_6599 for (2999ds/275Mi)
% 2.47/1.40 % (6825)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_6599 for (2999ds/4Mi)
% 2.47/1.40 % (6824)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_6599 for (2999ds/183Mi)
% 2.47/1.40 % (6827)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_6599 for (2999ds/2Mi)
% 2.47/1.40 % (6828)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_6599 for (2999ds/2Mi)
% 2.47/1.40 % (6830)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_6599 for (2999ds/18Mi)
% 2.47/1.40 % (6826)Instruction limit reached!
% 2.47/1.40 % (6826)------------------------------
% 2.47/1.40 % (6826)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.40 % (6826)Termination reason: Unknown
% 2.47/1.40 % (6826)Termination phase: shuffling
% 2.47/1.40
% 2.47/1.40 % (6826)Memory used [KB]: 1918
% 2.47/1.40 % (6826)Time elapsed: 0.009 s
% 2.47/1.40 % (6826)Instructions burned: 28 (million)
% 2.47/1.40 % (6826)------------------------------
% 2.47/1.40 % (6826)------------------------------
% 2.47/1.40 % (6827)Instruction limit reached!
% 2.47/1.40 % (6827)------------------------------
% 2.47/1.40 % (6827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.40 % (6827)Termination reason: Unknown
% 2.47/1.40 % (6827)Termination phase: shuffling
% 2.47/1.40
% 2.47/1.40 % (6827)Memory used [KB]: 1407
% 2.47/1.40 % (6827)Time elapsed: 0.004 s
% 2.47/1.40 % (6827)Instructions burned: 3 (million)
% 2.47/1.40 % (6827)------------------------------
% 2.47/1.40 % (6827)------------------------------
% 2.47/1.40 % (6828)Instruction limit reached!
% 2.47/1.40 % (6828)------------------------------
% 2.47/1.40 % (6828)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.40 % (6828)Termination reason: Unknown
% 2.47/1.40 % (6828)Termination phase: shuffling
% 2.47/1.40
% 2.47/1.40 % (6828)Memory used [KB]: 1407
% 2.47/1.40 % (6828)Time elapsed: 0.003 s
% 2.47/1.40 % (6828)Instructions burned: 3 (million)
% 2.47/1.40 % (6828)------------------------------
% 2.47/1.40 % (6828)------------------------------
% 2.47/1.40 % (6825)Instruction limit reached!
% 2.47/1.40 % (6825)------------------------------
% 2.47/1.40 % (6825)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.40 % (6825)Termination reason: Unknown
% 2.47/1.40 % (6825)Termination phase: shuffling
% 2.47/1.40
% 2.47/1.40 % (6825)Memory used [KB]: 1407
% 2.47/1.40 % (6825)Time elapsed: 0.005 s
% 2.47/1.40 % (6825)Instructions burned: 5 (million)
% 2.47/1.40 % (6825)------------------------------
% 2.47/1.40 % (6825)------------------------------
% 2.47/1.41 % (6831)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_6599 for (2999ds/3Mi)
% 2.47/1.41 % (6831)Instruction limit reached!
% 2.47/1.41 % (6831)------------------------------
% 2.47/1.41 % (6831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.41 % (6831)Termination reason: Unknown
% 2.47/1.41 % (6831)Termination phase: shuffling
% 2.47/1.41
% 2.47/1.41 % (6831)Memory used [KB]: 1407
% 2.47/1.41 % (6831)Time elapsed: 0.002 s
% 2.47/1.41 % (6831)Instructions burned: 3 (million)
% 2.47/1.41 % (6831)------------------------------
% 2.47/1.41 % (6831)------------------------------
% 2.47/1.41 % (6830)Instruction limit reached!
% 2.47/1.41 % (6830)------------------------------
% 2.47/1.41 % (6830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.41 % (6830)Termination reason: Unknown
% 2.47/1.41 % (6830)Termination phase: shuffling
% 2.47/1.41
% 2.47/1.41 % (6830)Memory used [KB]: 1791
% 2.47/1.41 % (6830)Time elapsed: 0.014 s
% 2.47/1.41 % (6830)Instructions burned: 19 (million)
% 2.47/1.41 % (6830)------------------------------
% 2.47/1.41 % (6830)------------------------------
% 2.47/1.42 % (6833)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_6599 for (2999ds/15Mi)
% 2.47/1.42 % (6832)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_6599 for (2999ds/37Mi)
% 2.47/1.42 % (6834)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_6599 for (2999ds/3Mi)
% 2.47/1.42 % (6835)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_6599 for (2999ds/1041Mi)
% 2.47/1.42 % (6834)Instruction limit reached!
% 2.47/1.42 % (6834)------------------------------
% 2.47/1.42 % (6834)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.42 % (6834)Termination reason: Unknown
% 2.47/1.42 % (6834)Termination phase: shuffling
% 2.47/1.42
% 2.47/1.42 % (6834)Memory used [KB]: 1407
% 2.47/1.42 % (6834)Time elapsed: 0.003 s
% 2.47/1.42 % (6834)Instructions burned: 3 (million)
% 2.47/1.42 % (6834)------------------------------
% 2.47/1.42 % (6834)------------------------------
% 2.47/1.43 % (6833)Instruction limit reached!
% 2.47/1.43 % (6833)------------------------------
% 2.47/1.43 % (6833)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.43 % (6833)Termination reason: Unknown
% 2.47/1.43 % (6833)Termination phase: shuffling
% 2.47/1.43
% 2.47/1.43 % (6833)Memory used [KB]: 1663
% 2.47/1.43 % (6833)Time elapsed: 0.010 s
% 2.47/1.43 % (6833)Instructions burned: 15 (million)
% 2.47/1.43 % (6833)------------------------------
% 2.47/1.43 % (6833)------------------------------
% 2.47/1.43 % (6836)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_6599 for (2999ds/7Mi)
% 2.47/1.43 % (6836)Instruction limit reached!
% 2.47/1.43 % (6836)------------------------------
% 2.47/1.43 % (6836)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.43 % (6836)Termination reason: Unknown
% 2.47/1.43 % (6836)Termination phase: shuffling
% 2.47/1.43
% 2.47/1.43 % (6836)Memory used [KB]: 1535
% 2.47/1.43 % (6836)Time elapsed: 0.006 s
% 2.47/1.43 % (6836)Instructions burned: 7 (million)
% 2.47/1.43 % (6836)------------------------------
% 2.47/1.43 % (6836)------------------------------
% 2.47/1.44 % (6837)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_6599 for (2999ds/16Mi)
% 2.47/1.44 % (6832)Instruction limit reached!
% 2.47/1.44 % (6832)------------------------------
% 2.47/1.44 % (6832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.44 % (6832)Termination reason: Unknown
% 2.47/1.44 % (6832)Termination phase: shuffling
% 2.47/1.44
% 2.47/1.44 % (6832)Memory used [KB]: 2174
% 2.47/1.44 % (6832)Time elapsed: 0.021 s
% 2.47/1.44 % (6832)Instructions burned: 37 (million)
% 2.47/1.44 % (6832)------------------------------
% 2.47/1.44 % (6832)------------------------------
% 2.47/1.44 % (6838)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_6599 for (2999ds/3Mi)
% 2.47/1.44 % (6837)Instruction limit reached!
% 2.47/1.44 % (6837)------------------------------
% 2.47/1.44 % (6837)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.44 % (6837)Termination reason: Unknown
% 2.47/1.44 % (6837)Termination phase: shuffling
% 2.47/1.44
% 2.47/1.44 % (6837)Memory used [KB]: 1791
% 2.47/1.44 % (6837)Time elapsed: 0.007 s
% 2.47/1.44 % (6837)Instructions burned: 16 (million)
% 2.47/1.44 % (6837)------------------------------
% 2.47/1.44 % (6837)------------------------------
% 2.47/1.44 % (6838)Instruction limit reached!
% 2.47/1.44 % (6838)------------------------------
% 2.47/1.44 % (6838)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.44 % (6838)Termination reason: Unknown
% 2.47/1.44 % (6838)Termination phase: shuffling
% 2.47/1.44
% 2.47/1.44 % (6838)Memory used [KB]: 1407
% 2.47/1.44 % (6838)Time elapsed: 0.003 s
% 2.47/1.44 % (6838)Instructions burned: 4 (million)
% 2.47/1.44 % (6838)------------------------------
% 2.47/1.44 % (6838)------------------------------
% 2.47/1.45 % (6839)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_6599 for (2999ds/3Mi)
% 2.47/1.45 % (6842)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_6599 for (2999ds/4Mi)
% 2.47/1.45 % (6841)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_6599 for (2999ds/3Mi)
% 2.47/1.45 % (6839)Instruction limit reached!
% 2.47/1.45 % (6839)------------------------------
% 2.47/1.45 % (6839)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.45 % (6839)Termination reason: Unknown
% 2.47/1.45 % (6839)Termination phase: shuffling
% 2.47/1.45
% 2.47/1.45 % (6839)Memory used [KB]: 1407
% 2.47/1.45 % (6839)Time elapsed: 0.004 s
% 2.47/1.45 % (6839)Instructions burned: 4 (million)
% 2.47/1.45 % (6839)------------------------------
% 2.47/1.45 % (6839)------------------------------
% 2.47/1.46 % (6841)Instruction limit reached!
% 2.47/1.46 % (6841)------------------------------
% 2.47/1.46 % (6841)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.46 % (6841)Termination reason: Unknown
% 2.47/1.46 % (6841)Termination phase: shuffling
% 2.47/1.46
% 2.47/1.46 % (6841)Memory used [KB]: 1407
% 2.47/1.46 % (6841)Time elapsed: 0.002 s
% 2.47/1.46 % (6841)Instructions burned: 4 (million)
% 2.47/1.46 % (6841)------------------------------
% 2.47/1.46 % (6841)------------------------------
% 2.47/1.46 % (6842)Instruction limit reached!
% 2.47/1.46 % (6842)------------------------------
% 2.47/1.46 % (6842)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.46 % (6842)Termination reason: Unknown
% 2.47/1.46 % (6842)Termination phase: shuffling
% 2.47/1.46
% 2.47/1.46 % (6842)Memory used [KB]: 1407
% 2.47/1.46 % (6842)Time elapsed: 0.003 s
% 2.47/1.46 % (6842)Instructions burned: 5 (million)
% 2.47/1.46 % (6842)------------------------------
% 2.47/1.46 % (6842)------------------------------
% 2.47/1.46 % (6840)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_6599 for (2999ds/7Mi)
% 2.47/1.46 % (6840)Instruction limit reached!
% 2.47/1.46 % (6840)------------------------------
% 2.47/1.46 % (6840)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.46 % (6840)Termination reason: Unknown
% 2.47/1.46 % (6840)Termination phase: shuffling
% 2.47/1.46
% 2.47/1.46 % (6840)Memory used [KB]: 1535
% 2.47/1.46 % (6840)Time elapsed: 0.006 s
% 2.47/1.46 % (6840)Instructions burned: 8 (million)
% 2.47/1.46 % (6840)------------------------------
% 2.47/1.46 % (6840)------------------------------
% 2.47/1.46 % (6845)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_6599 for (2999ds/6Mi)
% 2.47/1.47 % (6844)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_6599 for (2999ds/710Mi)
% 2.47/1.47 % (6845)Instruction limit reached!
% 2.47/1.47 % (6845)------------------------------
% 2.47/1.47 % (6845)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.47 % (6845)Termination reason: Unknown
% 2.47/1.47 % (6845)Termination phase: shuffling
% 2.47/1.47
% 2.47/1.47 % (6845)Memory used [KB]: 1535
% 2.47/1.47 % (6845)Time elapsed: 0.003 s
% 2.47/1.47 % (6845)Instructions burned: 7 (million)
% 2.47/1.47 % (6845)------------------------------
% 2.47/1.47 % (6845)------------------------------
% 2.47/1.47 % (6843)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_6599 for (2999ds/18Mi)
% 2.47/1.48 % (6847)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_6599 for (2999ds/21Mi)
% 2.47/1.48 % (6846)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_6599 for (2999ds/902Mi)
% 2.47/1.48 % (6843)Instruction limit reached!
% 2.47/1.48 % (6843)------------------------------
% 2.47/1.48 % (6843)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.48 % (6843)Termination reason: Unknown
% 2.47/1.48 % (6843)Termination phase: shuffling
% 2.47/1.48
% 2.47/1.48 % (6843)Memory used [KB]: 1791
% 2.47/1.48 % (6843)Time elapsed: 0.011 s
% 2.47/1.48 % (6843)Instructions burned: 19 (million)
% 2.47/1.48 % (6843)------------------------------
% 2.47/1.48 % (6843)------------------------------
% 2.47/1.48 % (6829)Instruction limit reached!
% 2.47/1.48 % (6829)------------------------------
% 2.47/1.48 % (6829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.48 % (6829)Termination reason: Unknown
% 2.47/1.48 % (6829)Termination phase: Saturation
% 2.47/1.48
% 2.47/1.48 % (6829)Memory used [KB]: 8827
% 2.47/1.48 % (6829)Time elapsed: 0.089 s
% 2.47/1.48 % (6829)Instructions burned: 276 (million)
% 2.47/1.48 % (6829)------------------------------
% 2.47/1.48 % (6829)------------------------------
% 2.47/1.48 % (6847)Instruction limit reached!
% 2.47/1.48 % (6847)------------------------------
% 2.47/1.48 % (6847)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.48 % (6847)Termination reason: Unknown
% 2.47/1.48 % (6847)Termination phase: shuffling
% 2.47/1.48
% 2.47/1.48 % (6847)Memory used [KB]: 1791
% 2.47/1.48 % (6847)Time elapsed: 0.008 s
% 2.47/1.48 % (6847)Instructions burned: 24 (million)
% 2.47/1.48 % (6847)------------------------------
% 2.47/1.48 % (6847)------------------------------
% 2.47/1.49 % (6849)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_6599 for (2998ds/6Mi)
% 2.47/1.49 % (6850)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_6599 for (2998ds/377Mi)
% 2.47/1.49 % (6824)Instruction limit reached!
% 2.47/1.49 % (6824)------------------------------
% 2.47/1.49 % (6824)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.49 % (6824)Termination reason: Unknown
% 2.47/1.49 % (6824)Termination phase: Saturation
% 2.47/1.49
% 2.47/1.49 % (6824)Memory used [KB]: 7547
% 2.47/1.49 % (6824)Time elapsed: 0.093 s
% 2.47/1.49 % (6824)Instructions burned: 183 (million)
% 2.47/1.49 % (6824)------------------------------
% 2.47/1.49 % (6824)------------------------------
% 2.47/1.49 % (6848)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_6599 for (2999ds/5Mi)
% 2.47/1.49 % (6849)Instruction limit reached!
% 2.47/1.49 % (6849)------------------------------
% 2.47/1.49 % (6849)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.49 % (6849)Termination reason: Unknown
% 2.47/1.49 % (6849)Termination phase: shuffling
% 2.47/1.49
% 2.47/1.49 % (6849)Memory used [KB]: 1535
% 2.47/1.49 % (6849)Time elapsed: 0.003 s
% 2.47/1.49 % (6849)Instructions burned: 7 (million)
% 2.47/1.49 % (6849)------------------------------
% 2.47/1.49 % (6849)------------------------------
% 2.47/1.50 % (6848)Instruction limit reached!
% 2.47/1.50 % (6848)------------------------------
% 2.47/1.50 % (6848)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.50 % (6848)Termination reason: Unknown
% 2.47/1.50 % (6848)Termination phase: shuffling
% 2.47/1.50
% 2.47/1.50 % (6848)Memory used [KB]: 1407
% 2.47/1.50 % (6848)Time elapsed: 0.003 s
% 2.47/1.50 % (6848)Instructions burned: 6 (million)
% 2.47/1.50 % (6848)------------------------------
% 2.47/1.50 % (6848)------------------------------
% 2.47/1.50 % (6852)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_6599 for (2998ds/19Mi)
% 2.47/1.51 % (6853)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_6599 for (2998ds/879Mi)
% 2.47/1.51 % (6852)Instruction limit reached!
% 2.47/1.51 % (6852)------------------------------
% 2.47/1.51 % (6852)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.51 % (6852)Termination reason: Unknown
% 2.47/1.51 % (6852)Termination phase: shuffling
% 2.47/1.51
% 2.47/1.51 % (6852)Memory used [KB]: 1791
% 2.47/1.51 % (6852)Time elapsed: 0.006 s
% 2.47/1.51 % (6852)Instructions burned: 19 (million)
% 2.47/1.51 % (6852)------------------------------
% 2.47/1.51 % (6852)------------------------------
% 2.47/1.51 % (6851)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_6599 for (2998ds/779Mi)
% 2.47/1.52 % (6854)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_6599 for (2998ds/17Mi)
% 2.47/1.52 % (6854)Instruction limit reached!
% 2.47/1.52 % (6854)------------------------------
% 2.47/1.52 % (6854)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.52 % (6854)Termination reason: Unknown
% 2.47/1.52 % (6854)Termination phase: shuffling
% 2.47/1.52
% 2.47/1.52 % (6854)Memory used [KB]: 1791
% 2.47/1.52 % (6854)Time elapsed: 0.006 s
% 2.47/1.52 % (6854)Instructions burned: 18 (million)
% 2.47/1.52 % (6854)------------------------------
% 2.47/1.52 % (6854)------------------------------
% 2.47/1.53 % (6855)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_6599 for (2998ds/3Mi)
% 2.47/1.53 % (6855)Instruction limit reached!
% 2.47/1.53 % (6855)------------------------------
% 2.47/1.53 % (6855)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.53 % (6855)Termination reason: Unknown
% 2.47/1.53 % (6855)Termination phase: shuffling
% 2.47/1.53
% 2.47/1.53 % (6855)Memory used [KB]: 1407
% 2.47/1.53 % (6855)Time elapsed: 0.002 s
% 2.47/1.53 % (6855)Instructions burned: 3 (million)
% 2.47/1.53 % (6855)------------------------------
% 2.47/1.53 % (6855)------------------------------
% 2.47/1.54 % (6856)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_6599 for (2998ds/30Mi)
% 2.47/1.55 % (6856)Instruction limit reached!
% 2.47/1.55 % (6856)------------------------------
% 2.47/1.55 % (6856)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.55 % (6856)Termination reason: Unknown
% 2.47/1.55 % (6856)Termination phase: shuffling
% 2.47/1.55
% 2.47/1.55 % (6856)Memory used [KB]: 1918
% 2.47/1.55 % (6856)Time elapsed: 0.010 s
% 2.47/1.55 % (6856)Instructions burned: 31 (million)
% 2.47/1.55 % (6856)------------------------------
% 2.47/1.55 % (6856)------------------------------
% 2.47/1.56 % (6857)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_6599 for (2998ds/127Mi)
% 2.47/1.59 % (6846)First to succeed.
% 2.47/1.59 % (6846)Refutation found. Thanks to Tanya!
% 2.47/1.59 % SZS status Theorem for DTF2THF_6599
% 2.47/1.59 % SZS output start Proof for DTF2THF_6599
% 2.47/1.59 thf(func_def_0, type, in: $i > $i > $o).
% 2.47/1.59 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.47/1.59 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.47/1.59 thf(func_def_8, type, powerset: $i > $i).
% 2.47/1.59 thf(func_def_10, type, setunion: $i > $i).
% 2.47/1.59 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.47/1.59 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.47/1.59 thf(func_def_26, type, prop2set: $o > $i).
% 2.47/1.59 thf(func_def_36, type, nonempty: $i > $o).
% 2.47/1.59 thf(func_def_69, type, set2prop: $i > $o).
% 2.47/1.59 thf(func_def_88, type, subset: $i > $i > $o).
% 2.47/1.59 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.47/1.59 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.47/1.59 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.47/1.59 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.47/1.59 thf(func_def_135, type, regular: $i > $o).
% 2.47/1.59 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.47/1.59 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.47/1.59 thf(func_def_153, type, iskpair: $i > $o).
% 2.47/1.59 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.47/1.59 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.47/1.59 thf(func_def_177, type, singleton: $i > $o).
% 2.47/1.59 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.47/1.59 thf(func_def_184, type, atmost1p: $i > $o).
% 2.47/1.59 thf(func_def_185, type, atleast2p: $i > $o).
% 2.47/1.59 thf(func_def_186, type, atmost2p: $i > $o).
% 2.47/1.59 thf(func_def_187, type, upairsetp: $i > $o).
% 2.47/1.59 thf(func_def_191, type, kfst: $i > $i).
% 2.47/1.59 thf(func_def_203, type, ksnd: $i > $i).
% 2.47/1.59 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.47/1.59 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.47/1.59 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.47/1.59 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.47/1.59 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.47/1.59 thf(func_def_268, type, sP3: $i > $o > $i > $i > $o).
% 2.47/1.59 thf(func_def_270, type, sP5: $i > $i > $i > $i > $o).
% 2.47/1.59 thf(func_def_272, type, sP7: $i > $i > $o).
% 2.47/1.59 thf(func_def_273, type, sP8: $i > $o).
% 2.47/1.59 thf(func_def_274, type, sP9: $i > $i > $o).
% 2.47/1.59 thf(func_def_275, type, sP10: $i > $i > $o).
% 2.47/1.59 thf(func_def_280, type, sK15: $i > $o).
% 2.47/1.59 thf(func_def_283, type, sK18: $i > $i > $o).
% 2.47/1.59 thf(func_def_286, type, sK21: ($i > $o) > $i).
% 2.47/1.59 thf(func_def_287, type, sK22: $i > $o).
% 2.47/1.59 thf(func_def_288, type, sK23: $i > $i).
% 2.47/1.59 thf(func_def_296, type, sK31: $i > $o).
% 2.47/1.59 thf(func_def_299, type, sK34: $i > ($i > $o) > $i).
% 2.47/1.59 thf(func_def_307, type, sK42: $i > $o).
% 2.47/1.59 thf(func_def_309, type, sK44: $i > $o).
% 2.47/1.59 thf(func_def_312, type, sK47: $i > ($i > $o) > $i).
% 2.47/1.59 thf(func_def_313, type, sK48: $i > $o).
% 2.47/1.59 thf(func_def_321, type, sK56: $i > $o).
% 2.47/1.59 thf(func_def_323, type, sK58: ($i > $o) > $i).
% 2.47/1.59 thf(func_def_324, type, sK59: ($i > $o) > $i).
% 2.47/1.59 thf(func_def_325, type, sK60: $i > $o).
% 2.47/1.59 thf(func_def_328, type, sK63: $i > ($i > $o) > $i).
% 2.47/1.59 thf(func_def_329, type, sK64: $i > ($i > $o) > $i).
% 2.47/1.59 thf(func_def_330, type, sK65: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.47/1.59 thf(func_def_331, type, sK66: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.47/1.59 thf(func_def_334, type, sK69: $i > $o).
% 2.47/1.59 thf(func_def_335, type, sK70: $i > $o).
% 2.47/1.59 thf(func_def_354, type, sK89: $i > $o).
% 2.47/1.59 thf(func_def_371, type, sK106: $i > $i).
% 2.47/1.59 thf(func_def_376, type, sK111: $i > $o).
% 2.47/1.59 thf(func_def_377, type, sK112: $i > $o).
% 2.47/1.59 thf(func_def_378, type, sK113: ($i > $o) > ($i > $o) > $i).
% 2.47/1.59 thf(func_def_379, type, sK114: ($i > $o) > ($i > $o) > $i).
% 2.47/1.59 thf(func_def_387, type, sK122: $o > $i > $i > $i).
% 2.47/1.59 thf(func_def_395, type, sK130: ($i > $o) > $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_396, type, sK131: ($i > $o) > $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_400, type, sK135: $i > $o).
% 2.47/1.59 thf(func_def_402, type, sK137: $i > $i).
% 2.47/1.59 thf(func_def_411, type, sK146: $i > $i > $i).
% 2.47/1.59 thf(func_def_417, type, sK152: ($i > $o) > $i).
% 2.47/1.59 thf(func_def_418, type, sK153: $i > $o).
% 2.47/1.59 thf(func_def_419, type, sK154: $i > $i).
% 2.47/1.59 thf(func_def_435, type, sK170: $i > $i).
% 2.47/1.59 thf(func_def_437, type, sK172: $i > $i > $o).
% 2.47/1.59 thf(func_def_447, type, sK182: $i > $o).
% 2.47/1.59 thf(func_def_466, type, sK201: $i > $i > $i).
% 2.47/1.59 thf(func_def_467, type, sK202: $i > $i > $i).
% 2.47/1.59 thf(func_def_473, type, sK208: $i > $o).
% 2.47/1.59 thf(func_def_475, type, sK210: $i > ($i > $o) > $i).
% 2.47/1.59 thf(func_def_479, type, sK214: $i > $i).
% 2.47/1.59 thf(func_def_481, type, sK216: $i > ($i > $i) > $i > $i).
% 2.47/1.59 thf(func_def_489, type, sK224: $i > $i > $i).
% 2.47/1.59 thf(func_def_507, type, sK242: $i > $i > $o).
% 2.47/1.59 thf(func_def_513, type, sK248: $i > $i > $o).
% 2.47/1.59 thf(func_def_517, type, sK252: $i > $o).
% 2.47/1.59 thf(func_def_528, type, sK263: $i > $o).
% 2.47/1.59 thf(func_def_533, type, sK268: $i > $i > $i).
% 2.47/1.59 thf(func_def_540, type, sK275: $i > $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_542, type, sK277: $i > $o).
% 2.47/1.59 thf(func_def_544, type, sK279: $i > $i).
% 2.47/1.59 thf(func_def_553, type, sK288: $i > $i > $i).
% 2.47/1.59 thf(func_def_555, type, sK290: $i > $i).
% 2.47/1.59 thf(func_def_557, type, sK292: $i > ($i > $i) > $i > $i).
% 2.47/1.59 thf(func_def_562, type, sK297: $i > $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_563, type, sK298: $i > $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_585, type, sK320: $i > $o).
% 2.47/1.59 thf(func_def_597, type, sK332: $i > $i > $o).
% 2.47/1.59 thf(func_def_601, type, sK336: ($i > $o) > $i).
% 2.47/1.59 thf(func_def_602, type, sK337: $i > $o).
% 2.47/1.59 thf(func_def_615, type, sK350: $i > $o).
% 2.47/1.59 thf(func_def_626, type, sK361: $i > $i > $i).
% 2.47/1.59 thf(func_def_627, type, sK362: $i > $i > $i).
% 2.47/1.59 thf(func_def_635, type, sK370: $i > $o).
% 2.47/1.59 thf(func_def_637, type, sK372: ($i > $o) > $i > $i).
% 2.47/1.59 thf(func_def_638, type, sK373: ($i > $o) > $i > $i).
% 2.47/1.59 thf(func_def_642, type, sK377: ($i > $o) > $i > $i).
% 2.47/1.59 thf(func_def_644, type, sK379: $i > $o).
% 2.47/1.59 thf(func_def_646, type, sK381: $i > $i).
% 2.47/1.59 thf(func_def_650, type, sK385: $i > $i > ($i > $i) > $i).
% 2.47/1.59 thf(func_def_661, type, sK396: $i > $i > $o).
% 2.47/1.59 thf(func_def_669, type, sK404: ($i > $o) > ($i > $o) > $i).
% 2.47/1.59 thf(func_def_670, type, sK405: ($i > $o) > ($i > $o) > $i).
% 2.47/1.59 thf(func_def_671, type, sK406: $i > $o).
% 2.47/1.59 thf(func_def_672, type, sK407: $i > $o).
% 2.47/1.59 thf(func_def_673, type, sK408: $i > $o).
% 2.47/1.59 thf(func_def_676, type, sK411: $i > ($i > $o) > $i).
% 2.47/1.59 thf(func_def_696, type, sK431: $i > $i > $o).
% 2.47/1.59 thf(func_def_697, type, sK432: $i > $i).
% 2.47/1.59 thf(func_def_698, type, sK433: $i > $i).
% 2.47/1.59 thf(func_def_699, type, sK434: ($i > $i > $o) > $i > $i).
% 2.47/1.59 thf(func_def_700, type, sK435: ($i > $i > $o) > $i > $i).
% 2.47/1.59 thf(func_def_701, type, sK436: $i > ($i > $i > $o) > $i > $i).
% 2.47/1.59 thf(func_def_707, type, sK442: $i > $i).
% 2.47/1.59 thf(func_def_710, type, sK445: $i > $i > ($i > $i) > $i).
% 2.47/1.59 thf(func_def_711, type, sK446: $i > $i).
% 2.47/1.59 thf(func_def_735, type, sK470: $i > $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_736, type, sK471: $i > $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_737, type, sK472: $i > $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_738, type, sK473: $i > $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_757, type, sK492: $i > $i > $i).
% 2.47/1.59 thf(func_def_761, type, sK496: ($i > $o) > $i > $i).
% 2.47/1.59 thf(func_def_763, type, sK498: $i > $o).
% 2.47/1.59 thf(func_def_768, type, sK503: $i > $o).
% 2.47/1.59 thf(func_def_776, type, sK511: $i > $i).
% 2.47/1.59 thf(func_def_779, type, sK514: $i > $i > ($i > $i) > $i).
% 2.47/1.59 thf(func_def_784, type, sK519: $i > $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_803, type, sK538: $i > $i > $i).
% 2.47/1.59 thf(func_def_804, type, sK539: $i > $i > $i).
% 2.47/1.59 thf(func_def_805, type, sK540: $i > $i > $i).
% 2.47/1.59 thf(func_def_806, type, sK541: $i > $i > $i).
% 2.47/1.59 thf(func_def_807, type, sK542: $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_808, type, sK543: $i > $i).
% 2.47/1.59 thf(func_def_809, type, sK544: $i > $i).
% 2.47/1.59 thf(func_def_810, type, sK545: $i > $i).
% 2.47/1.59 thf(func_def_811, type, sK546: $i > $i).
% 2.47/1.59 thf(func_def_812, type, sK547: $i > $i > $i).
% 2.47/1.59 thf(func_def_813, type, sK548: $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_814, type, sK549: $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_815, type, sK550: $i > $i > $i).
% 2.47/1.59 thf(func_def_816, type, sK551: $i > $i > $i).
% 2.47/1.59 thf(func_def_817, type, sK552: $i > $i).
% 2.47/1.59 thf(func_def_818, type, sK553: $i > $i > $i).
% 2.47/1.59 thf(func_def_820, type, sK555: $i > $i).
% 2.47/1.59 thf(func_def_821, type, sK556: $i > $i).
% 2.47/1.59 thf(func_def_822, type, sK557: $i > ($i > $o) > $i).
% 2.47/1.59 thf(func_def_823, type, sK558: $i > $o).
% 2.47/1.59 thf(func_def_830, type, sK565: $i > $o).
% 2.47/1.59 thf(func_def_832, type, sK567: $i > ($i > $o) > $i > $i).
% 2.47/1.59 thf(func_def_835, type, sK570: ($i > $o) > $i > $i).
% 2.47/1.59 thf(func_def_837, type, sK572: $i > $o).
% 2.47/1.59 thf(func_def_847, type, sK582: $i > $o).
% 2.47/1.59 thf(func_def_858, type, sK593: $i > ($i > $o) > $i).
% 2.47/1.59 thf(func_def_859, type, sK594: $i > $o).
% 2.47/1.59 thf(func_def_869, type, sK604: $i > $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_870, type, sK605: ($i > $o) > $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_871, type, sK606: ($i > $o) > $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_875, type, sK610: $i > $o).
% 2.47/1.59 thf(func_def_879, type, sK614: $i > $i).
% 2.47/1.59 thf(func_def_880, type, sK615: $i > $i).
% 2.47/1.59 thf(func_def_888, type, sK623: $i > $i > $i).
% 2.47/1.59 thf(func_def_900, type, sK635: $i > ($i > $i) > $i > $i).
% 2.47/1.59 thf(func_def_902, type, sK637: $i > $i).
% 2.47/1.59 thf(func_def_911, type, sK646: $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_912, type, sK647: $i > $i > $i > $i).
% 2.47/1.59 thf(func_def_922, type, sK657: $i > $i > $o).
% 2.47/1.59 thf(func_def_927, type, sK662: $i > $o).
% 2.47/1.59 thf(func_def_938, type, ph673: !>[X0: $tType]:(X0)).
% 2.47/1.59 thf(f3217,plain,(
% 2.47/1.59 $false),
% 2.47/1.59 inference(trivial_inequality_removal,[status(thm)],[f3216])).
% 2.47/1.59 thf(f3216,plain,(
% 2.47/1.59 ($true != $true)),
% 2.47/1.59 inference(superposition,[status(thm)],[f3215,f3209])).
% 2.47/1.59 thf(f3209,plain,(
% 2.47/1.59 ($true = (func @ sK341 @ sK343 @ sK342))),
% 2.47/1.59 inference(trivial_inequality_removal,[status(thm)],[f3208])).
% 2.47/1.59 thf(f3208,plain,(
% 2.47/1.59 ($true != $true) | ($true = (func @ sK341 @ sK343 @ sK342))),
% 2.47/1.59 inference(forward_demodulation,[status(thm)],[f3207,f2544])).
% 2.47/1.59 thf(f2544,plain,(
% 2.47/1.59 (infuncsetfunc = $true)),
% 2.47/1.59 inference(cnf_transformation,[status(thm)],[f1474])).
% 2.47/1.59 thf(f1474,plain,(
% 2.47/1.59 (sepInPowerset = $true) & (binintersectEL = $true) & (notinsingleton = $true) & (singletonsswitch = $true) & (binunionIR = $true) & (eqinunit = $true) & (dpsetconstrI = $true) & (((lam @ sK341 @ sK343 @ (^[Y0 : $i]: (ap @ sK341 @ sK343 @ sK342 @ Y0))) != sK342) & ($true = (in @ sK342 @ (funcSet @ sK341 @ sK343)))) & (funcGraphProp2 = $true) & (setOfPairsIsBReln = $true) & (setoftrueEq = $true) & (descrp = $true) & (setunionE = $true) & (funcGraphProp3 = $true) & (exuE3e = $true) & (nonemptyI1 = $true) & (setunionsingleton = $true) & (exuI2 = $true) & (ksndsingleton = $true) & (powersetI1 = $true) & (secondinupair = $true) & (in__Cong = $true) & (setminusERneg = $true) & (exuI3 = $true) & (setminusELneg = $true) & (singletonprop = $true) & (funcGraphProp1 = $true) & (binintersectER = $true) & (kpairsurjEq = $true) & (wellorderingAx = $true) & (cartprodmempair1 = $true) & (ex1E2 = $true) & (lamp = $true) & (binunionEcases = $true) & (emptyset__Cong = $true) & (upairinpowunion = $true) & (brelnall1 = $true) & (exuE2 = $true) & (symdiffIneg2 = $true) & (subsetI1 = $true) & (subsetRefl = $true) & (dpsetconstrER = $true) & (upairequniteq = $true) & (prop2setE = $true) & (symdiffIneg1 = $true) & (emptyinunitempty = $true) & (funcinfuncset = $true) & (symdiffE = $true) & (ap2apEq1 = $true) & (kpairp = $true) & (setminusEL = $true) & (kfstsingleton = $true) & (ex1I = $true) & (prop2setI = $true) & (symdiffI1 = $true) & (setadjoinAx = $true) & (upairsetE = $true) & (exu__Cong = $true) & (descr__Cong = $true) & (omegaIndAx = $true) & (ubforcartprodlem3 = $true) & (setadjoinSub = $true) & (setminusLsub = $true) & (infuncsetfunc = $true) & (dpsetconstrSub = $true) & (inPowerset = $true) & (funcGraphProp4 = $true) & (powersetI = $true) & (dpsetconstrERa = $true) & (notdallE = $true) & (subset2powerset = $true) & (eqbreln = $true) & (quantDeMorgan2 = $true) & (ksndpairEq = $true) & (powersetE = $true) & (exuE3u = $true) & (cartprodsndin = $true) & (setadjoinSub2 = $true) & (funcextLem = $true) & (subsetE2 = $true) & (upairset2IR = $true) & (exuI1 = $true) & (setextAx = $true) & (nonemptyI = $true) & (cartprodpairsurjEq = $true) & (cartprodpairmemEL = $true) & (app = $true) & (setukpairinjR2 = $true) & (setukpairinjR11 = $true) & (setukpairinjL = $true) & (vacuousDall = $true) & (singletonsuniq = $true) & (funcext2 = $true) & (setukpairinjR1 = $true) & (lam2lamEq = $true) & (upairset2E = $true) & (uniqinunit = $true) & (powersetE1 = $true) & (notsubsetI = $true) & (setadjoinOr = $true) & (upairsubunion = $true) & (cartprodmempair = $true) & (theprop = $true) & (apProp = $true) & (ap2apEq2 = $true) & (singletoninpowunion = $true) & (brelnall2 = $true) & (binintersectSubset2 = $true) & (cartprodpairin = $true) & (nonemptyE1 = $true) & (ap2p = $true) & (cartprodmempaircEq = $true) & (setukpairinjL1 = $true) & (setext = $true) & (emptysetAx = $true) & (binintersectI = $true) & (beta1 = $true) & (setminusILneg = $true) & (symdiffI2 = $true) & (powerset__Cong = $true) & (lam2p = $true) & (subsetI2 = $true) & (setukpairinjL2 = $true) & (eta1 = $true) & (notequalI1 = $true) & (exuE1 = $true) & (setminusI = $true) & (emptyI = $true) & (omegaSAx = $true) & (setbeta = $true) & (omega0Ax = $true) & (setukpairinjR = $true) & (ex1I2 = $true) & (notinemptyset = $true) & (lamProp = $true) & (dsetconstrEL = $true) & (setminusSubset1 = $true) & (subsetE = $true) & (exuEu = $true) & (binintersectSubset4 = $true) & (emptyinPowerset = $true) & (subsetemptysetimpeq = $true) & (setadjoinIL = $true) & (quantDeMorgan4 = $true) & (sepSubset = $true) & (noeltsimpempty = $true) & (binunionE = $true) & (singletonsubset = $true) & (setukpairIR = $true) & (setadjoinIR = $true) & (foundationAx = $true) & (setminusER = $true) & (binintersectSubset3 = $true) & (replAx = $true) & (subPowSU = $true) & (singletoninpowerset = $true) & (upairsetIL = $true) & (binintersectRsub = $true) & (setunion__Cong = $true) & (notdexE = $true) & (emptyInPowerset = $true) & (prop2set2propI = $true) & (cartprodfstpairEq = $true) & (setunionI = $true) & (nonemptyImpWitness = $true) & (setminusIRneg = $true) & (dpsetconstrEL1 = $true) & (setadjoinE = $true) & (funcext = $true) & (cartprodpairmemER = $true) & (notequalI2 = $true) & (binintersectLsub = $true) & (emptysetsubset = $true) & (bs114d = $true) & (powersetsubset = $true) & (setminusSubset2 = $true) & (omega__Cong = $true) & (quantDeMorgan3 = $true) & (upairsetIR = $true) & (setukpairIL = $true) & (binunionLsub = $true) & (binintersectSubset1 = $true) & (eqimpsubset2 = $true) & (dsetconstrER = $true) & (setunionsingleton1 = $true) & (setadjoin__Cong = $true) & (cartprodfstin = $true) & (setukpairinjR12 = $true) & (kpairiskpair = $true) & (quantDeMorgan1 = $true) & (binunionRsub = $true) & (funcImageSingleton = $true) & (beta2 = $true) & (dsetconstr__Cong = $true) & (setextsub = $true) & (subsetTrans = $true) & (eqimpsubset1 = $true) & (disjointsetsI1 = $true) & (ubforcartprodlem2 = $true) & (emptysetE = $true) & (ubforcartprodlem1 = $true) & (emptysetimpfalse = $true) & (dsetconstrI = $true) & (binintersectSubset5 = $true) & (powersetAx = $true) & (binunionIL = $true) & (cartprodsndpairEq = $true) & (setunionAx = $true) & (kfstpairEq = $true) & (emptyE1 = $true) & (subbreln = $true) & (ex1E1 = $true) & (inCongP = $true) & (setunionE2 = $true) & (dpsetconstrEL2 = $true) & (setunionsingleton2 = $true)),
% 2.47/1.59 inference(skolemisation,[status(esa),new_symbols(skolem,[sK341,sK342,sK343])],[f754,f1473])).
% 2.47/1.59 thf(f1473,plain,(
% 2.47/1.59 ? [X0 : $i,X1 : $i,X2 : $i] : (((lam @ X0 @ X2 @ (^[Y0 : $i]: (ap @ X0 @ X2 @ X1 @ Y0))) != X1) & ($true = (in @ X1 @ (funcSet @ X0 @ X2)))) => (((lam @ sK341 @ sK343 @ (^[Y0 : $i]: (ap @ sK341 @ sK343 @ sK342 @ Y0))) != sK342) & ($true = (in @ sK342 @ (funcSet @ sK341 @ sK343))))),
% 2.47/1.59 introduced(definition,[],[choice_axiom])).
% 2.47/1.59 thf(f754,plain,(
% 2.47/1.59 (sepInPowerset = $true) & (binintersectEL = $true) & (notinsingleton = $true) & (singletonsswitch = $true) & (binunionIR = $true) & (eqinunit = $true) & (dpsetconstrI = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (((lam @ X0 @ X2 @ (^[Y0 : $i]: (ap @ X0 @ X2 @ X1 @ Y0))) != X1) & ($true = (in @ X1 @ (funcSet @ X0 @ X2)))) & (funcGraphProp2 = $true) & (setOfPairsIsBReln = $true) & (setoftrueEq = $true) & (descrp = $true) & (setunionE = $true) & (funcGraphProp3 = $true) & (exuE3e = $true) & (nonemptyI1 = $true) & (setunionsingleton = $true) & (exuI2 = $true) & (ksndsingleton = $true) & (powersetI1 = $true) & (secondinupair = $true) & (in__Cong = $true) & (setminusERneg = $true) & (exuI3 = $true) & (setminusELneg = $true) & (singletonprop = $true) & (funcGraphProp1 = $true) & (binintersectER = $true) & (kpairsurjEq = $true) & (wellorderingAx = $true) & (cartprodmempair1 = $true) & (ex1E2 = $true) & (lamp = $true) & (binunionEcases = $true) & (emptyset__Cong = $true) & (upairinpowunion = $true) & (brelnall1 = $true) & (exuE2 = $true) & (symdiffIneg2 = $true) & (subsetI1 = $true) & (subsetRefl = $true) & (dpsetconstrER = $true) & (upairequniteq = $true) & (prop2setE = $true) & (symdiffIneg1 = $true) & (emptyinunitempty = $true) & (funcinfuncset = $true) & (symdiffE = $true) & (ap2apEq1 = $true) & (kpairp = $true) & (setminusEL = $true) & (kfstsingleton = $true) & (ex1I = $true) & (prop2setI = $true) & (symdiffI1 = $true) & (setadjoinAx = $true) & (upairsetE = $true) & (exu__Cong = $true) & (descr__Cong = $true) & (omegaIndAx = $true) & (ubforcartprodlem3 = $true) & (setadjoinSub = $true) & (setminusLsub = $true) & (infuncsetfunc = $true) & (dpsetconstrSub = $true) & (inPowerset = $true) & (funcGraphProp4 = $true) & (powersetI = $true) & (dpsetconstrERa = $true) & (notdallE = $true) & (subset2powerset = $true) & (eqbreln = $true) & (quantDeMorgan2 = $true) & (ksndpairEq = $true) & (powersetE = $true) & (exuE3u = $true) & (cartprodsndin = $true) & (setadjoinSub2 = $true) & (funcextLem = $true) & (subsetE2 = $true) & (upairset2IR = $true) & (exuI1 = $true) & (setextAx = $true) & (nonemptyI = $true) & (cartprodpairsurjEq = $true) & (cartprodpairmemEL = $true) & (app = $true) & (setukpairinjR2 = $true) & (setukpairinjR11 = $true) & (setukpairinjL = $true) & (vacuousDall = $true) & (singletonsuniq = $true) & (funcext2 = $true) & (setukpairinjR1 = $true) & (lam2lamEq = $true) & (upairset2E = $true) & (uniqinunit = $true) & (powersetE1 = $true) & (notsubsetI = $true) & (setadjoinOr = $true) & (upairsubunion = $true) & (cartprodmempair = $true) & (theprop = $true) & (apProp = $true) & (ap2apEq2 = $true) & (singletoninpowunion = $true) & (brelnall2 = $true) & (binintersectSubset2 = $true) & (cartprodpairin = $true) & (nonemptyE1 = $true) & (ap2p = $true) & (cartprodmempaircEq = $true) & (setukpairinjL1 = $true) & (setext = $true) & (emptysetAx = $true) & (binintersectI = $true) & (beta1 = $true) & (setminusILneg = $true) & (symdiffI2 = $true) & (powerset__Cong = $true) & (lam2p = $true) & (subsetI2 = $true) & (setukpairinjL2 = $true) & (eta1 = $true) & (notequalI1 = $true) & (exuE1 = $true) & (setminusI = $true) & (emptyI = $true) & (omegaSAx = $true) & (setbeta = $true) & (omega0Ax = $true) & (setukpairinjR = $true) & (ex1I2 = $true) & (notinemptyset = $true) & (lamProp = $true) & (dsetconstrEL = $true) & (setminusSubset1 = $true) & (subsetE = $true) & (exuEu = $true) & (binintersectSubset4 = $true) & (emptyinPowerset = $true) & (subsetemptysetimpeq = $true) & (setadjoinIL = $true) & (quantDeMorgan4 = $true) & (sepSubset = $true) & (noeltsimpempty = $true) & (binunionE = $true) & (singletonsubset = $true) & (setukpairIR = $true) & (setadjoinIR = $true) & (foundationAx = $true) & (setminusER = $true) & (binintersectSubset3 = $true) & (replAx = $true) & (subPowSU = $true) & (singletoninpowerset = $true) & (upairsetIL = $true) & (binintersectRsub = $true) & (setunion__Cong = $true) & (notdexE = $true) & (emptyInPowerset = $true) & (prop2set2propI = $true) & (cartprodfstpairEq = $true) & (setunionI = $true) & (nonemptyImpWitness = $true) & (setminusIRneg = $true) & (dpsetconstrEL1 = $true) & (setadjoinE = $true) & (funcext = $true) & (cartprodpairmemER = $true) & (notequalI2 = $true) & (binintersectLsub = $true) & (emptysetsubset = $true) & (bs114d = $true) & (powersetsubset = $true) & (setminusSubset2 = $true) & (omega__Cong = $true) & (quantDeMorgan3 = $true) & (upairsetIR = $true) & (setukpairIL = $true) & (binunionLsub = $true) & (binintersectSubset1 = $true) & (eqimpsubset2 = $true) & (dsetconstrER = $true) & (setunionsingleton1 = $true) & (setadjoin__Cong = $true) & (cartprodfstin = $true) & (setukpairinjR12 = $true) & (kpairiskpair = $true) & (quantDeMorgan1 = $true) & (binunionRsub = $true) & (funcImageSingleton = $true) & (beta2 = $true) & (dsetconstr__Cong = $true) & (setextsub = $true) & (subsetTrans = $true) & (eqimpsubset1 = $true) & (disjointsetsI1 = $true) & (ubforcartprodlem2 = $true) & (emptysetE = $true) & (ubforcartprodlem1 = $true) & (emptysetimpfalse = $true) & (dsetconstrI = $true) & (binintersectSubset5 = $true) & (powersetAx = $true) & (binunionIL = $true) & (cartprodsndpairEq = $true) & (setunionAx = $true) & (kfstpairEq = $true) & (emptyE1 = $true) & (subbreln = $true) & (ex1E1 = $true) & (inCongP = $true) & (setunionE2 = $true) & (dpsetconstrEL2 = $true) & (setunionsingleton2 = $true)),
% 2.47/1.59 inference(flattening,[status(thm)],[f753])).
% 2.47/1.59 thf(f753,plain,(
% 2.47/1.59 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (((lam @ X0 @ X2 @ (^[Y0 : $i]: (ap @ X0 @ X2 @ X1 @ Y0))) != X1) & ($true = (in @ X1 @ (funcSet @ X0 @ X2)))) & (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)),
% 2.47/1.59 inference(ennf_transformation,[status(thm)],[f313])).
% 2.47/1.59 thf(f313,plain,(
% 2.47/1.59 ~((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) => ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (funcSet @ X0 @ X2))) => ((lam @ X0 @ X2 @ (^[Y0 : $i]: (ap @ X0 @ X2 @ X1 @ Y0))) = X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.47/1.59 inference(fool_elimination,[status(thm)],[f312])).
% 2.47/1.59 thf(f312,plain,(
% 2.47/1.59 ~(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 => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ (funcSet @ X0 @ X2)) => (X1 = (lam @ X0 @ X2 @ (^[X3 : $i] : (ap @ X0 @ X2 @ X1 @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.47/1.59 inference(rectify,[status(thm)],[f222])).
% 2.47/1.59 thf(f222,negated_conjecture,(
% 2.47/1.59 ~(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 => ! [X3 : $i,X12 : $i,X4 : $i] : ((in @ X12 @ (funcSet @ X3 @ X4)) => (X12 = (lam @ X3 @ X4 @ (^[X1 : $i] : (ap @ X3 @ X4 @ X12 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.47/1.59 inference(negated_conjecture,[status(cth)],[f221])).
% 2.47/1.59 thf(f221,conjecture,(
% 2.47/1.59 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 => ! [X3 : $i,X12 : $i,X4 : $i] : ((in @ X12 @ (funcSet @ X3 @ X4)) => (X12 = (lam @ X3 @ X4 @ (^[X1 : $i] : (ap @ X3 @ X4 @ X12 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.47/1.59 file('/export/starexec/sandbox2/tmp/tmp.GXJcYPDlCU/DTF2THF_6599.p',eta2)).
% 2.47/1.59 thf(f3207,plain,(
% 2.47/1.59 (infuncsetfunc != $true) | ($true = (func @ sK341 @ sK343 @ sK342))),
% 2.47/1.59 inference(trivial_inequality_removal,[status(thm)],[f3204])).
% 2.47/1.59 thf(f3204,plain,(
% 2.47/1.59 ($true != $true) | ($true = (func @ sK341 @ sK343 @ sK342)) | (infuncsetfunc != $true)),
% 2.47/1.59 inference(superposition,[status(thm)],[f2707,f2600])).
% 2.47/1.59 thf(f2600,plain,(
% 2.47/1.59 ($true = (in @ sK342 @ (funcSet @ sK341 @ sK343)))),
% 2.47/1.59 inference(cnf_transformation,[status(thm)],[f1474])).
% 2.47/1.59 thf(f2707,plain,(
% 2.47/1.59 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true != (in @ X0 @ (funcSet @ X2 @ X1))) | (infuncsetfunc != $true) | ($true = (func @ X2 @ X1 @ X0))) )),
% 2.47/1.59 inference(cnf_transformation,[status(thm)],[f1591])).
% 2.47/1.59 thf(f1591,plain,(
% 2.47/1.59 (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X2 @ X1 @ X0)) | ($true != (in @ X0 @ (funcSet @ X2 @ X1)))) | (infuncsetfunc != $true)) & ((infuncsetfunc = $true) | (($true != (func @ sK425 @ sK424 @ sK423)) & ($true = (in @ sK423 @ (funcSet @ sK425 @ sK424)))))),
% 2.47/1.59 inference(skolemisation,[status(esa),new_symbols(skolem,[sK423,sK424,sK425])],[f1589,f1590])).
% 2.47/1.59 thf(f1590,plain,(
% 2.47/1.59 ? [X3 : $i,X4 : $i,X5 : $i] : (($true != (func @ X5 @ X4 @ X3)) & ($true = (in @ X3 @ (funcSet @ X5 @ X4)))) => (($true != (func @ sK425 @ sK424 @ sK423)) & ($true = (in @ sK423 @ (funcSet @ sK425 @ sK424))))),
% 2.47/1.59 introduced(definition,[],[choice_axiom])).
% 2.47/1.59 thf(f1589,plain,(
% 2.47/1.59 (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X2 @ X1 @ X0)) | ($true != (in @ X0 @ (funcSet @ X2 @ X1)))) | (infuncsetfunc != $true)) & ((infuncsetfunc = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (($true != (func @ X5 @ X4 @ X3)) & ($true = (in @ X3 @ (funcSet @ X5 @ X4)))))),
% 2.47/1.59 inference(rectify,[status(thm)],[f1588])).
% 2.47/1.59 thf(f1588,plain,(
% 2.47/1.59 (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X2 @ X1 @ X0)) | ($true != (in @ X0 @ (funcSet @ X2 @ X1)))) | (infuncsetfunc != $true)) & ((infuncsetfunc = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (func @ X2 @ X1 @ X0)) & ($true = (in @ X0 @ (funcSet @ X2 @ X1)))))),
% 2.47/1.59 inference(nnf_transformation,[status(thm)],[f915])).
% 2.47/1.59 thf(f915,plain,(
% 2.47/1.59 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X2 @ X1 @ X0)) | ($true != (in @ X0 @ (funcSet @ X2 @ X1)))) <=> (infuncsetfunc = $true)),
% 2.47/1.59 inference(ennf_transformation,[status(thm)],[f548])).
% 2.47/1.59 thf(f548,plain,(
% 2.47/1.59 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (funcSet @ X2 @ X1))) => ($true = (func @ X2 @ X1 @ X0))) <=> (infuncsetfunc = $true)),
% 2.47/1.59 inference(fool_elimination,[status(thm)],[f547])).
% 2.47/1.59 thf(f547,plain,(
% 2.47/1.59 (infuncsetfunc = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ (funcSet @ X2 @ X1)) => (func @ X2 @ X1 @ X0)))),
% 2.47/1.59 inference(rectify,[status(thm)],[f197])).
% 2.47/1.59 thf(f197,axiom,(
% 2.47/1.59 (infuncsetfunc = ! [X12 : $i,X4 : $i,X3 : $i] : ((in @ X12 @ (funcSet @ X3 @ X4)) => (func @ X3 @ X4 @ X12)))),
% 2.47/1.59 file('/export/starexec/sandbox2/tmp/tmp.GXJcYPDlCU/DTF2THF_6599.p',infuncsetfunc)).
% 2.47/1.59 thf(f3215,plain,(
% 2.47/1.59 ($true != (func @ sK341 @ sK343 @ sK342))),
% 2.47/1.59 inference(trivial_inequality_removal,[status(thm)],[f3214])).
% 2.47/1.59 thf(f3214,plain,(
% 2.47/1.59 ($true != $true) | ($true != (func @ sK341 @ sK343 @ sK342))),
% 2.47/1.59 inference(forward_demodulation,[status(thm)],[f3213,f2484])).
% 2.47/1.59 thf(f2484,plain,(
% 2.47/1.59 (eta1 = $true)),
% 2.47/1.59 inference(cnf_transformation,[status(thm)],[f1474])).
% 2.47/1.59 thf(f3213,plain,(
% 2.47/1.59 ($true != (func @ sK341 @ sK343 @ sK342)) | (eta1 != $true)),
% 2.47/1.59 inference(trivial_inequality_removal,[status(thm)],[f3212])).
% 2.47/1.59 thf(f3212,plain,(
% 2.47/1.59 (eta1 != $true) | (sK342 != sK342) | ($true != (func @ sK341 @ sK343 @ sK342))),
% 2.47/1.59 inference(superposition,[status(thm)],[f3203,f3189])).
% 2.47/1.59 thf(f3189,plain,(
% 2.47/1.59 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((lam @ X4 @ X3 @ (ap @ X4 @ X3 @ X5)) = X5) | (eta1 != $true) | ($true != (func @ X4 @ X3 @ X5))) )),
% 2.47/1.59 inference(beta_eta_normalization,[status(thm)],[f2747])).
% 2.47/1.59 thf(f2747,plain,(
% 2.47/1.59 ( ! [X3 : $i,X4 : $i,X5 : $i] : (($true != (func @ X4 @ X3 @ X5)) | (eta1 != $true) | ((lam @ X4 @ X3 @ (^[Y0 : $i]: (ap @ X4 @ X3 @ X5 @ Y0))) = X5)) )),
% 2.47/1.59 inference(cnf_transformation,[status(thm)],[f1635])).
% 2.47/1.59 thf(f1635,plain,(
% 2.47/1.59 ((eta1 = $true) | ((sK450 != (lam @ sK449 @ sK448 @ (^[Y0 : $i]: (ap @ sK449 @ sK448 @ sK450 @ Y0)))) & ($true = (func @ sK449 @ sK448 @ sK450)))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((lam @ X4 @ X3 @ (^[Y0 : $i]: (ap @ X4 @ X3 @ X5 @ Y0))) = X5) | ($true != (func @ X4 @ X3 @ X5))) | (eta1 != $true))),
% 2.47/1.59 inference(skolemisation,[status(esa),new_symbols(skolem,[sK448,sK449,sK450])],[f1633,f1634])).
% 2.47/1.59 thf(f1634,plain,(
% 2.47/1.59 ? [X0 : $i,X1 : $i,X2 : $i] : (((lam @ X1 @ X0 @ (^[Y0 : $i]: (ap @ X1 @ X0 @ X2 @ Y0))) != X2) & ($true = (func @ X1 @ X0 @ X2))) => ((sK450 != (lam @ sK449 @ sK448 @ (^[Y0 : $i]: (ap @ sK449 @ sK448 @ sK450 @ Y0)))) & ($true = (func @ sK449 @ sK448 @ sK450)))),
% 2.47/1.59 introduced(definition,[],[choice_axiom])).
% 2.47/1.59 thf(f1633,plain,(
% 2.47/1.59 ((eta1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((lam @ X1 @ X0 @ (^[Y0 : $i]: (ap @ X1 @ X0 @ X2 @ Y0))) != X2) & ($true = (func @ X1 @ X0 @ X2)))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((lam @ X4 @ X3 @ (^[Y0 : $i]: (ap @ X4 @ X3 @ X5 @ Y0))) = X5) | ($true != (func @ X4 @ X3 @ X5))) | (eta1 != $true))),
% 2.47/1.59 inference(rectify,[status(thm)],[f1632])).
% 2.47/1.59 thf(f1632,plain,(
% 2.47/1.59 ((eta1 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((lam @ X1 @ X0 @ (^[Y0 : $i]: (ap @ X1 @ X0 @ X2 @ Y0))) != X2) & ($true = (func @ X1 @ X0 @ X2)))) & (! [X0 : $i,X1 : $i,X2 : $i] : (((lam @ X1 @ X0 @ (^[Y0 : $i]: (ap @ X1 @ X0 @ X2 @ Y0))) = X2) | ($true != (func @ X1 @ X0 @ X2))) | (eta1 != $true))),
% 2.47/1.59 inference(nnf_transformation,[status(thm)],[f802])).
% 2.47/1.59 thf(f802,plain,(
% 2.47/1.59 (eta1 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((lam @ X1 @ X0 @ (^[Y0 : $i]: (ap @ X1 @ X0 @ X2 @ Y0))) = X2) | ($true != (func @ X1 @ X0 @ X2)))),
% 2.47/1.59 inference(ennf_transformation,[status(thm)],[f617])).
% 2.47/1.59 thf(f617,plain,(
% 2.47/1.59 (eta1 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (func @ X1 @ X0 @ X2)) => ((lam @ X1 @ X0 @ (^[Y0 : $i]: (ap @ X1 @ X0 @ X2 @ Y0))) = X2))),
% 2.47/1.59 inference(fool_elimination,[status(thm)],[f616])).
% 2.47/1.59 thf(f616,plain,(
% 2.47/1.59 (! [X0 : $i,X1 : $i,X2 : $i] : ((func @ X1 @ X0 @ X2) => ((lam @ X1 @ X0 @ (^[X3 : $i] : (ap @ X1 @ X0 @ X2 @ X3))) = X2)) = eta1)),
% 2.47/1.59 inference(rectify,[status(thm)],[f218])).
% 2.47/1.59 thf(f218,axiom,(
% 2.47/1.59 (! [X4 : $i,X3 : $i,X12 : $i] : ((func @ X3 @ X4 @ X12) => ((lam @ X3 @ X4 @ (^[X1 : $i] : (ap @ X3 @ X4 @ X12 @ X1))) = X12)) = eta1)),
% 2.47/1.59 file('/export/starexec/sandbox2/tmp/tmp.GXJcYPDlCU/DTF2THF_6599.p',eta1)).
% 2.47/1.59 thf(f3203,plain,(
% 2.47/1.59 ((lam @ sK341 @ sK343 @ (ap @ sK341 @ sK343 @ sK342)) != sK342)),
% 2.47/1.59 inference(beta_eta_normalization,[status(thm)],[f2601])).
% 2.47/1.59 thf(f2601,plain,(
% 2.47/1.59 ((lam @ sK341 @ sK343 @ (^[Y0 : $i]: (ap @ sK341 @ sK343 @ sK342 @ Y0))) != sK342)),
% 2.47/1.59 inference(cnf_transformation,[status(thm)],[f1474])).
% 2.47/1.59 % SZS output end Proof for DTF2THF_6599
% 2.47/1.59 % (6846)------------------------------
% 2.47/1.59 % (6846)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.47/1.59 % (6846)Termination reason: Refutation
% 2.47/1.59
% 2.47/1.59 % (6846)Memory used [KB]: 9210
% 2.47/1.59 % (6846)Time elapsed: 0.112 s
% 2.47/1.59 % (6846)Instructions burned: 272 (million)
% 2.47/1.59 % (6846)------------------------------
% 2.47/1.59 % (6846)------------------------------
% 2.47/1.59 % (6823)Success in time 0.204 s
%------------------------------------------------------------------------------