%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU693^1 : 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 : n009.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 09:36:27 AM UTC 2026
% Result : Theorem 2.62s 1.63s
% Output : Refutation 2.62s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 3
% Syntax : Number of formulae : 11 ( 2 unt; 0 typ; 0 def)
% Number of atoms : 3244 ( 862 equ; 0 cnn)
% Maximal formula atoms : 212 ( 294 avg)
% Number of connectives : 1650 ( 11 ~; 0 |; 639 &; 154 @)
% ( 0 <=>; 846 =>; 0 <=; 0 <~>)
% Maximal formula depth : 219 ( 138 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 392 ( 389 usr; 215 con; 0-4 aty)
% Number of variables : 30 ( 0 ^; 16 !; 14 ?; 30 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_259,type,
sP0: $i > $i > $o ).
thf(func_def_260,type,
sP1: $i > $o ).
thf(func_def_261,type,
sP2: $i > $i > $o ).
thf(func_def_262,type,
sP3: $i > $i > $o ).
thf(func_def_266,type,
sP7: $i > $i > $i > $o > $o ).
thf(func_def_267,type,
sP8: $i > $i > $i > $i > $o ).
thf(func_def_285,type,
sK26: $i > $i > $i ).
thf(func_def_291,type,
sK32: $o > $i > $i > $i ).
thf(func_def_297,type,
sK38: $i > $i ).
thf(func_def_298,type,
sK39: $i > $i ).
thf(func_def_308,type,
sK49: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_309,type,
sK50: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_313,type,
sK54: $i > $o ).
thf(func_def_315,type,
sK56: $i > $i > $i ).
thf(func_def_316,type,
sK57: $i > $i > $i ).
thf(func_def_317,type,
sK58: $i > $i > $i ).
thf(func_def_318,type,
sK59: $i > $i > $i ).
thf(func_def_319,type,
sK60: $i > $i > $i ).
thf(func_def_320,type,
sK61: $i > $i > $i > $i ).
thf(func_def_321,type,
sK62: $i > $i ).
thf(func_def_322,type,
sK63: $i > $i ).
thf(func_def_323,type,
sK64: $i > $i ).
thf(func_def_324,type,
sK65: $i > $i ).
thf(func_def_325,type,
sK66: $i > $i > $i > $i ).
thf(func_def_326,type,
sK67: $i > $i > $i > $i ).
thf(func_def_327,type,
sK68: $i > $i > $i ).
thf(func_def_328,type,
sK69: $i > $i > $i ).
thf(func_def_329,type,
sK70: $i > $i > $i ).
thf(func_def_330,type,
sK71: $i > $i ).
thf(func_def_332,type,
sK73: $i > $i ).
thf(func_def_333,type,
sK74: $i > $i ).
thf(func_def_343,type,
sK84: $i > $o ).
thf(func_def_357,type,
sK98: $i > $i > $i ).
thf(func_def_382,type,
sK123: $i > $i > $i > $i ).
thf(func_def_383,type,
sK124: $i > $i > $i > $i ).
thf(func_def_386,type,
sK127: $i > $o ).
thf(func_def_398,type,
sK139: $i > $i > $o ).
thf(func_def_400,type,
sK141: $i > $i ).
thf(func_def_401,type,
sK142: $i > $i ).
thf(func_def_402,type,
sK143: $i > ( $i > $i > $o ) > $i ).
thf(func_def_403,type,
sK144: $i > ( $i > $i > $o ) > $i ).
thf(func_def_404,type,
sK145: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_405,type,
sK146: $i > $i > $o ).
thf(func_def_410,type,
sK151: $i > $o ).
thf(func_def_414,type,
sK155: ( $i > $i ) > $i > $i > $i ).
thf(func_def_417,type,
sK158: $i > $i ).
thf(func_def_419,type,
sK160: $i > $i > $o ).
thf(func_def_431,type,
sK172: $i > $o ).
thf(func_def_433,type,
sK174: $i > ( $i > $o ) > $i ).
thf(func_def_446,type,
sK187: $i > $i ).
thf(func_def_448,type,
sK189: $i > ( $i > $i ) > $i > $i ).
thf(func_def_449,type,
sK190: $i > $o ).
thf(func_def_450,type,
sK191: $i > $o ).
thf(func_def_451,type,
sK192: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_452,type,
sK193: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_463,type,
sK204: $i > $i > $i ).
thf(func_def_464,type,
sK205: $i > $i > $i ).
thf(func_def_482,type,
sK223: $i > $i > $i ).
thf(func_def_485,type,
sK226: $i > $i > $o ).
thf(func_def_490,type,
sK231: $i > ( $i > $o ) > $i ).
thf(func_def_491,type,
sK232: $i > $o ).
thf(func_def_495,type,
sK236: $i > $i > $i > $i > $i ).
thf(func_def_500,type,
sK241: ( $i > $o ) > $i ).
thf(func_def_501,type,
sK242: $i > $o ).
thf(func_def_502,type,
sK243: $i > $i ).
thf(func_def_515,type,
sK256: $i > $i > $i > $i > $i ).
thf(func_def_516,type,
sK257: $i > $i > $i > $i > $i ).
thf(func_def_530,type,
sK271: $i > $i > $i ).
thf(func_def_532,type,
sK273: $i > $i ).
thf(func_def_534,type,
sK275: $i > $o ).
thf(func_def_543,type,
sK284: $i > $o ).
thf(func_def_545,type,
sK286: ( $i > $o ) > $i > $i ).
thf(func_def_546,type,
sK287: ( $i > $o ) > $i > $i ).
thf(func_def_551,type,
sK292: $i > $i ).
thf(func_def_567,type,
sK308: ( $i > $o ) > $i ).
thf(func_def_568,type,
sK309: $i > $o ).
thf(func_def_569,type,
sK310: $i > $i ).
thf(func_def_573,type,
sK314: $i > $i > $i ).
thf(func_def_576,type,
sK317: $i > $o ).
thf(func_def_583,type,
sK324: ( $i > $o ) > $i > $i > $i ).
thf(func_def_586,type,
sK327: $i > $o ).
thf(func_def_587,type,
sK328: $i > $o ).
thf(func_def_590,type,
sK331: $i > ( $i > $o ) > $i ).
thf(func_def_594,type,
sK335: $i > $i > $o ).
thf(func_def_605,type,
sK346: $i > $i > $i > $i > $i ).
thf(func_def_610,type,
sK351: $i > $o ).
thf(func_def_611,type,
sK352: ( $i > $o ) > $i ).
thf(func_def_617,type,
sK358: $i > ( $i > $o ) > $i ).
thf(func_def_618,type,
sK359: $i > $o ).
thf(func_def_628,type,
sK369: $i > $i > $i ).
thf(func_def_667,type,
sK408: $i > $o ).
thf(func_def_670,type,
sK411: $i > $i ).
thf(func_def_676,type,
sK417: $i > $i > $i > $i > $i ).
thf(func_def_677,type,
sK418: $i > $i > $i > $i > $i ).
thf(func_def_678,type,
sK419: $i > $i > $i > $i > $i ).
thf(func_def_679,type,
sK420: $i > $i > $i > $i > $i ).
thf(func_def_682,type,
sK423: $i > $i > $o ).
thf(func_def_685,type,
sK426: $i > $o ).
thf(func_def_692,type,
sK433: $i > $i > $o ).
thf(func_def_695,type,
sK436: $i > $o ).
thf(func_def_697,type,
sK438: $i > ( $i > $o ) > $i ).
thf(func_def_718,type,
sK459: $i > $o ).
thf(func_def_719,type,
sK460: $i > $o ).
thf(func_def_720,type,
sK461: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_721,type,
sK462: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_726,type,
sK467: $i > $i ).
thf(func_def_728,type,
sK469: $i > ( $i > $o ) > $i ).
thf(func_def_729,type,
sK470: $i > $o ).
thf(func_def_737,type,
sK478: $i > $o ).
thf(func_def_741,type,
sK482: ( $i > $o ) > $i > $i ).
thf(func_def_743,type,
sK484: $i > $o ).
thf(func_def_745,type,
sK486: $i > ( $i > $i ) > $i > $i ).
thf(func_def_747,type,
sK488: $i > $i ).
thf(func_def_749,type,
sK490: $i > $o ).
thf(func_def_752,type,
sK493: $i > ( $i > $o ) > $i ).
thf(func_def_753,type,
sK494: $i > ( $i > $o ) > $i ).
thf(func_def_776,type,
sK517: $i > $i > $i ).
thf(func_def_778,type,
sK519: $i > $o ).
thf(func_def_780,type,
sK521: $i > $i > $o ).
thf(func_def_788,type,
sK529: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_789,type,
sK530: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_790,type,
sK531: $i > $o ).
thf(func_def_791,type,
sK532: $i > $o ).
thf(func_def_806,type,
sK547: $i > $o ).
thf(func_def_813,type,
sK554: ( $i > $o ) > $i > $i ).
thf(func_def_815,type,
sK556: $i > $o ).
thf(func_def_829,type,
sK570: $i > $o ).
thf(func_def_844,type,
sK585: $i > $i ).
thf(func_def_847,type,
sK588: $i > $i > $i ).
thf(func_def_852,type,
sK593: $i > $i > $i > $i > $i ).
thf(func_def_857,type,
sK598: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_858,type,
sK599: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_862,type,
sK603: $i > $o ).
thf(func_def_868,type,
sK609: $i > $o ).
thf(func_def_891,type,
sK632: ( $i > $o ) > $i ).
thf(func_def_892,type,
sK633: ( $i > $o ) > $i ).
thf(func_def_893,type,
sK634: $i > $o ).
thf(func_def_896,type,
sK637: $i > $o ).
thf(func_def_898,type,
sK639: $i > $o ).
thf(func_def_900,type,
sK641: ( $i > $o ) > $i > $i ).
thf(func_def_907,type,
ph648:
!>[X0: $tType] : X0 ).
thf(f4063,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f2769]) ).
thf(f2769,plain,
( ( ap @ sK544 @ sK543 @ sK545 @ sK546 )
!= ( ap @ sK544 @ sK543 @ sK545 @ sK546 ) ),
inference(cnf_transformation,[status(thm)],[f1745]) ).
thf(f1745,plain,
( ( exuI3 = $true )
& ( singletonprop = $true )
& ( kpairsurjEq = $true )
& ( cartprodpairin = $true )
& ( powersetI1 = $true )
& ( kfstsingleton = $true )
& ( dsetconstrI = $true )
& ( setadjoinSub2 = $true )
& ( emptysetsubset = $true )
& ( eqimpsubset1 = $true )
& ( exu__Cong = $true )
& ( setminusERneg = $true )
& ( dpsetconstrEL1 = $true )
& ( setukpairinjR = $true )
& ( powersetE1 = $true )
& ( funcGraphProp1 = $true )
& ( exuE3u = $true )
& ( powersetE = $true )
& ( setukpairinjR11 = $true )
& ( ex1E2 = $true )
& ( upairsubunion = $true )
& ( app = $true )
& ( setadjoinIR = $true )
& ( ubforcartprodlem1 = $true )
& ( binintersectI = $true )
& ( subset2powerset = $true )
& ( emptyI = $true )
& ( setunion__Cong = $true )
& ( ubforcartprodlem2 = $true )
& ( theprop = $true )
& ( binunionEcases = $true )
& ( setminusSubset2 = $true )
& ( infuncsetfunc = $true )
& ( disjointsetsI1 = $true )
& ( noeltsimpempty = $true )
& ( binintersectSubset5 = $true )
& ( setminusI = $true )
& ( symdiffIneg1 = $true )
& ( quantDeMorgan4 = $true )
& ( eqbreln = $true )
& ( subsetE2 = $true )
& ( subsetRefl = $true )
& ( dpsetconstrI = $true )
& ( notdexE = $true )
& ( setadjoinAx = $true )
& ( dsetconstrEL = $true )
& ( inPowerset = $true )
& ( powersetI = $true )
& ( binintersectSubset2 = $true )
& ( cartprodpairsurjEq = $true )
& ( dsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( funcGraphProp4 = $true )
& ( cartprodmempair1 = $true )
& ( binintersectEL = $true )
& ( lamp = $true )
& ( cartprodsndpairEq = $true )
& ( foundationAx = $true )
& ( $true
= ( in @ sK545 @ ( funcSet @ sK544 @ sK543 ) ) )
& ( ( ap @ sK544 @ sK543 @ sK545 @ sK546 )
!= ( ap @ sK544 @ sK543 @ sK545 @ sK546 ) )
& ( $true
= ( in @ sK546 @ sK544 ) )
& ( setminusELneg = $true )
& ( nonemptyE1 = $true )
& ( nonemptyImpWitness = $true )
& ( vacuousDall = $true )
& ( symdiffI1 = $true )
& ( notinsingleton = $true )
& ( setukpairinjR2 = $true )
& ( binintersectSubset1 = $true )
& ( dsetconstr__Cong = $true )
& ( kpairp = $true )
& ( prop2setI = $true )
& ( brelnall1 = $true )
& ( brelnall2 = $true )
& ( funcImageSingleton = $true )
& ( notinemptyset = $true )
& ( powersetAx = $true )
& ( sepSubset = $true )
& ( bs114d = $true )
& ( setminusIRneg = $true )
& ( upairset2E = $true )
& ( cartprodfstin = $true )
& ( setukpairinjL2 = $true )
& ( exuI2 = $true )
& ( funcext = $true )
& ( binintersectLsub = $true )
& ( subbreln = $true )
& ( subsetE = $true )
& ( ex1I = $true )
& ( notequalI1 = $true )
& ( cartprodmempaircEq = $true )
& ( setminusILneg = $true )
& ( cartprodpairmemER = $true )
& ( lamProp = $true )
& ( singletonsubset = $true )
& ( upairsetE = $true )
& ( emptyinunitempty = $true )
& ( omega0Ax = $true )
& ( setunionI = $true )
& ( binintersectSubset4 = $true )
& ( omegaIndAx = $true )
& ( setukpairinjL = $true )
& ( omega__Cong = $true )
& ( uniqinunit = $true )
& ( subPowSU = $true )
& ( funcext2 = $true )
& ( singletoninpowerset = $true )
& ( binintersectSubset3 = $true )
& ( setextsub = $true )
& ( setunionE = $true )
& ( prop2set2propI = $true )
& ( cartprodfstpairEq = $true )
& ( sepInPowerset = $true )
& ( funcinfuncset = $true )
& ( singletonsuniq = $true )
& ( inCongP = $true )
& ( dpsetconstrERa = $true )
& ( setunionsingleton2 = $true )
& ( setukpairIR = $true )
& ( notequalI2 = $true )
& ( setunionsingleton1 = $true )
& ( upairset2IR = $true )
& ( ksndsingleton = $true )
& ( subsetI2 = $true )
& ( binunionRsub = $true )
& ( powersetsubset = $true )
& ( eqimpsubset2 = $true )
& ( upairequniteq = $true )
& ( setoftrueEq = $true )
& ( emptysetAx = $true )
& ( setukpairIL = $true )
& ( setadjoinE = $true )
& ( ksndpairEq = $true )
& ( setunionAx = $true )
& ( kpairiskpair = $true )
& ( setextAx = $true )
& ( funcextLem = $true )
& ( exuE3e = $true )
& ( ubforcartprodlem3 = $true )
& ( setukpairinjR12 = $true )
& ( setadjoinIL = $true )
& ( nonemptyI = $true )
& ( replAx = $true )
& ( setminusLsub = $true )
& ( binintersectRsub = $true )
& ( symdiffI2 = $true )
& ( setadjoin__Cong = $true )
& ( ap2p = $true )
& ( emptyE1 = $true )
& ( notdallE = $true )
& ( dpsetconstrSub = $true )
& ( in__Cong = $true )
& ( upairsetIR = $true )
& ( subsetemptysetimpeq = $true )
& ( binunionIR = $true )
& ( symdiffE = $true )
& ( subsetTrans = $true )
& ( setminusSubset1 = $true )
& ( cartprodsndin = $true )
& ( setext = $true )
& ( emptysetE = $true )
& ( setunionE2 = $true )
& ( upairsetIL = $true )
& ( lam2p = $true )
& ( funcGraphProp2 = $true )
& ( apProp = $true )
& ( emptyset__Cong = $true )
& ( setadjoinSub = $true )
& ( setunionsingleton = $true )
& ( setadjoinOr = $true )
& ( secondinupair = $true )
& ( binintersectER = $true )
& ( upairinpowunion = $true )
& ( setOfPairsIsBReln = $true )
& ( singletonsswitch = $true )
& ( ex1I2 = $true )
& ( kfstpairEq = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( quantDeMorgan1 = $true )
& ( ex1E1 = $true )
& ( emptyinPowerset = $true )
& ( singletoninpowunion = $true )
& ( prop2setE = $true )
& ( omegaSAx = $true )
& ( cartprodmempair = $true )
& ( quantDeMorgan2 = $true )
& ( binunionIL = $true )
& ( setukpairinjL1 = $true )
& ( exuE1 = $true )
& ( nonemptyI1 = $true )
& ( exuE2 = $true )
& ( emptysetimpfalse = $true )
& ( symdiffIneg2 = $true )
& ( setminusER = $true )
& ( setbeta = $true )
& ( cartprodpairmemEL = $true )
& ( quantDeMorgan3 = $true )
& ( dpsetconstrER = $true )
& ( exuI1 = $true )
& ( subsetI1 = $true )
& ( emptyInPowerset = $true )
& ( notsubsetI = $true )
& ( powerset__Cong = $true )
& ( eqinunit = $true )
& ( setukpairinjR1 = $true )
& ( exuEu = $true )
& ( setminusEL = $true )
& ( descr__Cong = $true )
& ( funcGraphProp3 = $true )
& ( wellorderingAx = $true )
& ( descrp = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK543,sK544,sK545,sK546])],[f792,f1744,f1743]) ).
thf(f1743,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( funcSet @ X1 @ X0 ) ) )
& ? [X3: $i] :
( ( ( ap @ X1 @ X0 @ X2 @ X3 )
!= ( ap @ X1 @ X0 @ X2 @ X3 ) )
& ( $true
= ( in @ X3 @ X1 ) ) ) )
=> ( ( $true
= ( in @ sK545 @ ( funcSet @ sK544 @ sK543 ) ) )
& ? [X3: $i] :
( ( ( ap @ sK544 @ sK543 @ sK545 @ X3 )
!= ( ap @ sK544 @ sK543 @ sK545 @ X3 ) )
& ( $true
= ( in @ X3 @ sK544 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1744,plain,
( ? [X3: $i] :
( ( ( ap @ sK544 @ sK543 @ sK545 @ X3 )
!= ( ap @ sK544 @ sK543 @ sK545 @ X3 ) )
& ( $true
= ( in @ X3 @ sK544 ) ) )
=> ( ( ( ap @ sK544 @ sK543 @ sK545 @ sK546 )
!= ( ap @ sK544 @ sK543 @ sK545 @ sK546 ) )
& ( $true
= ( in @ sK546 @ sK544 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f792,plain,
( ( exuI3 = $true )
& ( singletonprop = $true )
& ( kpairsurjEq = $true )
& ( cartprodpairin = $true )
& ( powersetI1 = $true )
& ( kfstsingleton = $true )
& ( dsetconstrI = $true )
& ( setadjoinSub2 = $true )
& ( emptysetsubset = $true )
& ( eqimpsubset1 = $true )
& ( exu__Cong = $true )
& ( setminusERneg = $true )
& ( dpsetconstrEL1 = $true )
& ( setukpairinjR = $true )
& ( powersetE1 = $true )
& ( funcGraphProp1 = $true )
& ( exuE3u = $true )
& ( powersetE = $true )
& ( setukpairinjR11 = $true )
& ( ex1E2 = $true )
& ( upairsubunion = $true )
& ( app = $true )
& ( setadjoinIR = $true )
& ( ubforcartprodlem1 = $true )
& ( binintersectI = $true )
& ( subset2powerset = $true )
& ( emptyI = $true )
& ( setunion__Cong = $true )
& ( ubforcartprodlem2 = $true )
& ( theprop = $true )
& ( binunionEcases = $true )
& ( setminusSubset2 = $true )
& ( infuncsetfunc = $true )
& ( disjointsetsI1 = $true )
& ( noeltsimpempty = $true )
& ( binintersectSubset5 = $true )
& ( setminusI = $true )
& ( symdiffIneg1 = $true )
& ( quantDeMorgan4 = $true )
& ( eqbreln = $true )
& ( subsetE2 = $true )
& ( subsetRefl = $true )
& ( dpsetconstrI = $true )
& ( notdexE = $true )
& ( setadjoinAx = $true )
& ( dsetconstrEL = $true )
& ( inPowerset = $true )
& ( powersetI = $true )
& ( binintersectSubset2 = $true )
& ( cartprodpairsurjEq = $true )
& ( dsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( funcGraphProp4 = $true )
& ( cartprodmempair1 = $true )
& ( binintersectEL = $true )
& ( lamp = $true )
& ( cartprodsndpairEq = $true )
& ( foundationAx = $true )
& ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( funcSet @ X1 @ X0 ) ) )
& ? [X3: $i] :
( ( ( ap @ X1 @ X0 @ X2 @ X3 )
!= ( ap @ X1 @ X0 @ X2 @ X3 ) )
& ( $true
= ( in @ X3 @ X1 ) ) ) )
& ( setminusELneg = $true )
& ( nonemptyE1 = $true )
& ( nonemptyImpWitness = $true )
& ( vacuousDall = $true )
& ( symdiffI1 = $true )
& ( notinsingleton = $true )
& ( setukpairinjR2 = $true )
& ( binintersectSubset1 = $true )
& ( dsetconstr__Cong = $true )
& ( kpairp = $true )
& ( prop2setI = $true )
& ( brelnall1 = $true )
& ( brelnall2 = $true )
& ( funcImageSingleton = $true )
& ( notinemptyset = $true )
& ( powersetAx = $true )
& ( sepSubset = $true )
& ( bs114d = $true )
& ( setminusIRneg = $true )
& ( upairset2E = $true )
& ( cartprodfstin = $true )
& ( setukpairinjL2 = $true )
& ( exuI2 = $true )
& ( funcext = $true )
& ( binintersectLsub = $true )
& ( subbreln = $true )
& ( subsetE = $true )
& ( ex1I = $true )
& ( notequalI1 = $true )
& ( cartprodmempaircEq = $true )
& ( setminusILneg = $true )
& ( cartprodpairmemER = $true )
& ( lamProp = $true )
& ( singletonsubset = $true )
& ( upairsetE = $true )
& ( emptyinunitempty = $true )
& ( omega0Ax = $true )
& ( setunionI = $true )
& ( binintersectSubset4 = $true )
& ( omegaIndAx = $true )
& ( setukpairinjL = $true )
& ( omega__Cong = $true )
& ( uniqinunit = $true )
& ( subPowSU = $true )
& ( funcext2 = $true )
& ( singletoninpowerset = $true )
& ( binintersectSubset3 = $true )
& ( setextsub = $true )
& ( setunionE = $true )
& ( prop2set2propI = $true )
& ( cartprodfstpairEq = $true )
& ( sepInPowerset = $true )
& ( funcinfuncset = $true )
& ( singletonsuniq = $true )
& ( inCongP = $true )
& ( dpsetconstrERa = $true )
& ( setunionsingleton2 = $true )
& ( setukpairIR = $true )
& ( notequalI2 = $true )
& ( setunionsingleton1 = $true )
& ( upairset2IR = $true )
& ( ksndsingleton = $true )
& ( subsetI2 = $true )
& ( binunionRsub = $true )
& ( powersetsubset = $true )
& ( eqimpsubset2 = $true )
& ( upairequniteq = $true )
& ( setoftrueEq = $true )
& ( emptysetAx = $true )
& ( setukpairIL = $true )
& ( setadjoinE = $true )
& ( ksndpairEq = $true )
& ( setunionAx = $true )
& ( kpairiskpair = $true )
& ( setextAx = $true )
& ( funcextLem = $true )
& ( exuE3e = $true )
& ( ubforcartprodlem3 = $true )
& ( setukpairinjR12 = $true )
& ( setadjoinIL = $true )
& ( nonemptyI = $true )
& ( replAx = $true )
& ( setminusLsub = $true )
& ( binintersectRsub = $true )
& ( symdiffI2 = $true )
& ( setadjoin__Cong = $true )
& ( ap2p = $true )
& ( emptyE1 = $true )
& ( notdallE = $true )
& ( dpsetconstrSub = $true )
& ( in__Cong = $true )
& ( upairsetIR = $true )
& ( subsetemptysetimpeq = $true )
& ( binunionIR = $true )
& ( symdiffE = $true )
& ( subsetTrans = $true )
& ( setminusSubset1 = $true )
& ( cartprodsndin = $true )
& ( setext = $true )
& ( emptysetE = $true )
& ( setunionE2 = $true )
& ( upairsetIL = $true )
& ( lam2p = $true )
& ( funcGraphProp2 = $true )
& ( apProp = $true )
& ( emptyset__Cong = $true )
& ( setadjoinSub = $true )
& ( setunionsingleton = $true )
& ( setadjoinOr = $true )
& ( secondinupair = $true )
& ( binintersectER = $true )
& ( upairinpowunion = $true )
& ( setOfPairsIsBReln = $true )
& ( singletonsswitch = $true )
& ( ex1I2 = $true )
& ( kfstpairEq = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( quantDeMorgan1 = $true )
& ( ex1E1 = $true )
& ( emptyinPowerset = $true )
& ( singletoninpowunion = $true )
& ( prop2setE = $true )
& ( omegaSAx = $true )
& ( cartprodmempair = $true )
& ( quantDeMorgan2 = $true )
& ( binunionIL = $true )
& ( setukpairinjL1 = $true )
& ( exuE1 = $true )
& ( nonemptyI1 = $true )
& ( exuE2 = $true )
& ( emptysetimpfalse = $true )
& ( symdiffIneg2 = $true )
& ( setminusER = $true )
& ( setbeta = $true )
& ( cartprodpairmemEL = $true )
& ( quantDeMorgan3 = $true )
& ( dpsetconstrER = $true )
& ( exuI1 = $true )
& ( subsetI1 = $true )
& ( emptyInPowerset = $true )
& ( notsubsetI = $true )
& ( powerset__Cong = $true )
& ( eqinunit = $true )
& ( setukpairinjR1 = $true )
& ( exuEu = $true )
& ( setminusEL = $true )
& ( descr__Cong = $true )
& ( funcGraphProp3 = $true )
& ( wellorderingAx = $true )
& ( descrp = $true ) ),
inference(flattening,[status(thm)],[f791]) ).
thf(f791,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( funcSet @ X1 @ X0 ) ) )
& ? [X3: $i] :
( ( ( ap @ X1 @ X0 @ X2 @ X3 )
!= ( ap @ X1 @ X0 @ X2 @ X3 ) )
& ( $true
= ( in @ X3 @ X1 ) ) ) )
& ( 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)],[f429]) ).
thf(f429,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 )
=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ ( funcSet @ X1 @ X0 ) ) )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ X1 ) )
=> ( ( ap @ X1 @ X0 @ X2 @ X3 )
= ( ap @ X1 @ X0 @ X2 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f428]) ).
thf(f428,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
=> ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X2 @ ( funcSet @ X1 @ X0 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( ap @ X1 @ X0 @ X2 @ X3 )
= ( ap @ X1 @ X0 @ X2 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f217]) ).
thf(f217,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
=> ! [X4: $i,X3: $i,X12: $i] :
( ( in @ X12 @ ( funcSet @ X3 @ X4 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( ap @ X3 @ X4 @ X12 @ X1 )
= ( ap @ X3 @ X4 @ X12 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f216]) ).
thf(f216,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
=> ! [X4: $i,X3: $i,X12: $i] :
( ( in @ X12 @ ( funcSet @ X3 @ X4 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( ap @ X3 @ X4 @ X12 @ X1 )
= ( ap @ X3 @ X4 @ X12 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.7DY03sHzZu/DTF2THF_23302.p',ap2apEq1) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU693^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n009.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:30:55 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.31/0.46 ---- Original DTF file ---
% 0.31/0.46 thf(spec,logic,$$dhol).
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 % File : SEU693^1 : TPTP v9.2.1. Released v3.7.0.
% 0.31/0.46 % Domain : Set Theory
% 0.31/0.46 % Problem : Functions - Extensionality and Beta Reduction
% 0.31/0.46 % Version : Especial.
% 0.31/0.46 % English : (! A:i.! B:i.! f:i.in f (funcSet A B) -> (! x:i.in x A ->
% 0.31/0.46 % ap A B f x = ap A B f x))
% 0.31/0.46
% 0.31/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.31/0.46 % Source : [Bro08]
% 0.31/0.46 % Names : ZFC195g [Bro08]
% 0.31/0.46
% 0.31/0.46 % Status : Theorem
% 0.31/0.46 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.15 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.12 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.1.0, 0.60 v5.0.0, 0.40 v4.1.0, 0.67 v4.0.0, 0.33 v3.7.0
% 0.31/0.46 % Syntax : Number of formulae : 462 ( 215 unt; 246 typ; 215 def)
% 0.31/0.46 % Number of atoms : 1181 ( 320 equ; 0 cnn)
% 0.31/0.46 % Maximal formula atoms : 212 ( 5 avg)
% 0.31/0.46 % Number of connectives : 2249 ( 47 ~; 7 |; 35 &;1597 @)
% 0.31/0.46 % ( 14 <=>; 549 =>; 0 <=; 0 <~>)
% 0.31/0.46 % Maximal formula depth : 218 ( 2 avg)
% 0.31/0.46 % Number of types : 2 ( 0 usr)
% 0.31/0.46 % Number of type conns : 126 ( 126 >; 0 *; 0 +; 0 <<)
% 0.31/0.46 % Number of symbols : 249 ( 246 usr; 213 con; 0-4 aty)
% 0.31/0.46 % Number of variables : 706 ( 69 ^; 601 !; 36 ?; 706 :)
% 0.31/0.46 % SPC : TH0_THM_EQU_NAR
% 0.31/0.46
% 0.31/0.46 % Comments : http://mathgate.info/detsetitem.php?id=247
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 thf(in_type,type,
% 0.31/0.46 in: $i > $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(exu_type,type,
% 0.31/0.46 exu: ( $i > $o ) > $o ).
% 0.31/0.46
% 0.31/0.46 thf(exu,definition,
% 0.31/0.46 ( exu
% 0.31/0.46 = ( ^ [Xphi: $i > $o] :
% 0.31/0.46 ? [Xx: $i] :
% 0.31/0.46 ( ( Xphi @ Xx )
% 0.31/0.46 & ! [Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setextAx_type,type,
% 0.31/0.46 setextAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setextAx,definition,
% 0.31/0.46 ( setextAx
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 <=> ( in @ Xx @ B ) )
% 0.31/0.46 => ( A = B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptyset_type,type,
% 0.31/0.46 emptyset: $i ).
% 0.31/0.46
% 0.31/0.46 thf(emptysetAx_type,type,
% 0.31/0.46 emptysetAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptysetAx,definition,
% 0.31/0.46 ( emptysetAx
% 0.31/0.46 = ( ! [Xx: $i] :
% 0.31/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoin_type,type,
% 0.31/0.46 setadjoin: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinAx_type,type,
% 0.31/0.46 setadjoinAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinAx,definition,
% 0.31/0.46 ( setadjoinAx
% 0.31/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.46 <=> ( ( Xy = Xx )
% 0.31/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(powerset_type,type,
% 0.31/0.46 powerset: $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(powersetAx_type,type,
% 0.31/0.46 powersetAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(powersetAx,definition,
% 0.31/0.46 ( powersetAx
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.46 <=> ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunion_type,type,
% 0.31/0.46 setunion: $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(setunionAx_type,type,
% 0.31/0.46 setunionAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionAx,definition,
% 0.31/0.46 ( setunionAx
% 0.31/0.46 = ( ! [A: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.46 <=> ? [B: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(omega_type,type,
% 0.31/0.46 omega: $i ).
% 0.31/0.46
% 0.31/0.46 thf(omega0Ax_type,type,
% 0.31/0.46 omega0Ax: $o ).
% 0.31/0.46
% 0.31/0.46 thf(omega0Ax,definition,
% 0.31/0.46 ( omega0Ax
% 0.31/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(omegaSAx_type,type,
% 0.31/0.46 omegaSAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(omegaSAx,definition,
% 0.31/0.46 ( omegaSAx
% 0.31/0.46 = ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ omega )
% 0.31/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(omegaIndAx_type,type,
% 0.31/0.46 omegaIndAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(omegaIndAx,definition,
% 0.31/0.46 ( omegaIndAx
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ( ( in @ emptyset @ A )
% 0.31/0.46 & ! [Xx: $i] :
% 0.31/0.46 ( ( ( in @ Xx @ omega )
% 0.31/0.46 & ( in @ Xx @ A ) )
% 0.31/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ omega )
% 0.31/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(replAx_type,type,
% 0.31/0.46 replAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(replAx,definition,
% 0.31/0.46 ( replAx
% 0.31/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( exu
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.31/0.46 => ? [B: $i] :
% 0.31/0.46 ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 <=> ? [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ A )
% 0.31/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(foundationAx_type,type,
% 0.31/0.46 foundationAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(foundationAx,definition,
% 0.31/0.46 ( foundationAx
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/0.46 => ? [B: $i] :
% 0.31/0.46 ( ( in @ B @ A )
% 0.31/0.46 & ~ ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(wellorderingAx_type,type,
% 0.31/0.46 wellorderingAx: $o ).
% 0.31/0.46
% 0.31/0.46 thf(wellorderingAx,definition,
% 0.31/0.46 ( wellorderingAx
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ? [B: $i] :
% 0.31/0.46 ( ! [C: $i] :
% 0.31/0.46 ( ( in @ C @ B )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ C )
% 0.31/0.46 => ( in @ Xx @ A ) ) )
% 0.31/0.46 & ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( ( in @ Xx @ A )
% 0.31/0.46 & ( in @ Xy @ A ) )
% 0.31/0.46 => ( ! [C: $i] :
% 0.31/0.46 ( ( in @ C @ B )
% 0.31/0.46 => ( ( in @ Xx @ C )
% 0.31/0.46 <=> ( in @ Xy @ C ) ) )
% 0.31/0.46 => ( Xx = Xy ) ) )
% 0.31/0.46 & ! [C: $i,D: $i] :
% 0.31/0.46 ( ( ( in @ C @ B )
% 0.31/0.46 & ( in @ D @ B ) )
% 0.31/0.46 => ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ C )
% 0.31/0.46 => ( in @ Xx @ D ) )
% 0.31/0.46 | ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ D )
% 0.31/0.46 => ( in @ Xx @ C ) ) ) )
% 0.31/0.46 & ! [C: $i] :
% 0.31/0.46 ( ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ C )
% 0.31/0.46 => ( in @ Xx @ A ) )
% 0.31/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.31/0.46 => ? [D: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ D @ B )
% 0.31/0.46 & ( in @ Xx @ C )
% 0.31/0.46 & ~ ? [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ D )
% 0.31/0.46 & ( in @ Xy @ C ) )
% 0.31/0.46 & ! [E: $i] :
% 0.31/0.46 ( ( in @ E @ B )
% 0.31/0.46 => ( ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ E )
% 0.31/0.46 => ( in @ Xy @ D ) )
% 0.31/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(descr_type,type,
% 0.31/0.46 descr: ( $i > $o ) > $i ).
% 0.31/0.46
% 0.31/0.46 thf(descrp_type,type,
% 0.31/0.46 descrp: $o ).
% 0.31/0.46
% 0.31/0.46 thf(descrp,definition,
% 0.31/0.46 ( descrp
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ( Xphi
% 0.31/0.46 @ ( descr
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstr_type,type,
% 0.31/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrI_type,type,
% 0.31/0.46 dsetconstrI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrI,definition,
% 0.31/0.46 ( dsetconstrI
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 => ( in @ Xx
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrEL_type,type,
% 0.31/0.46 dsetconstrEL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrEL,definition,
% 0.31/0.46 ( dsetconstrEL
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrER_type,type,
% 0.31/0.46 dsetconstrER: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstrER,definition,
% 0.31/0.46 ( dsetconstrER
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuE1_type,type,
% 0.31/0.46 exuE1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuE1,definition,
% 0.31/0.46 ( exuE1
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ? [Xx: $i] :
% 0.31/0.46 ( ( Xphi @ Xx )
% 0.31/0.46 & ! [Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(prop2set_type,type,
% 0.31/0.46 prop2set: $o > $i ).
% 0.31/0.46
% 0.31/0.46 thf(prop2setE_type,type,
% 0.31/0.46 prop2setE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(prop2setE,definition,
% 0.31/0.46 ( prop2setE
% 0.31/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.31/0.46 => Xphi ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptysetE_type,type,
% 0.31/0.46 emptysetE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptysetE,definition,
% 0.31/0.46 ( emptysetE
% 0.31/0.46 = ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ emptyset )
% 0.31/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptysetimpfalse_type,type,
% 0.31/0.46 emptysetimpfalse: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptysetimpfalse,definition,
% 0.31/0.46 ( emptysetimpfalse
% 0.31/0.46 = ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ emptyset )
% 0.31/0.46 => $false ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(notinemptyset_type,type,
% 0.31/0.46 notinemptyset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(notinemptyset,definition,
% 0.31/0.46 ( notinemptyset
% 0.31/0.46 = ( ! [Xx: $i] :
% 0.31/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuE3e_type,type,
% 0.31/0.46 exuE3e: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuE3e,definition,
% 0.31/0.46 ( exuE3e
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setext_type,type,
% 0.31/0.46 setext: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setext,definition,
% 0.31/0.46 ( setext
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ Xx @ B ) )
% 0.31/0.46 => ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ A ) )
% 0.31/0.46 => ( A = B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptyI_type,type,
% 0.31/0.46 emptyI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptyI,definition,
% 0.31/0.46 ( emptyI
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ~ ( in @ Xx @ A )
% 0.31/0.46 => ( A = emptyset ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(noeltsimpempty_type,type,
% 0.31/0.46 noeltsimpempty: $o ).
% 0.31/0.46
% 0.31/0.46 thf(noeltsimpempty,definition,
% 0.31/0.46 ( noeltsimpempty
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ~ ( in @ Xx @ A )
% 0.31/0.46 => ( A = emptyset ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setbeta_type,type,
% 0.31/0.46 setbeta: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setbeta,definition,
% 0.31/0.46 ( setbeta
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( in @ Xx
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(nonempty_type,type,
% 0.31/0.46 nonempty: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(nonempty,definition,
% 0.31/0.46 ( nonempty
% 0.31/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(nonemptyE1_type,type,
% 0.31/0.46 nonemptyE1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(nonemptyE1,definition,
% 0.31/0.46 ( nonemptyE1
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ( nonempty @ A )
% 0.31/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(nonemptyI_type,type,
% 0.31/0.46 nonemptyI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(nonemptyI,definition,
% 0.31/0.46 ( nonemptyI
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 => ( nonempty
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(nonemptyI1_type,type,
% 0.31/0.46 nonemptyI1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(nonemptyI1,definition,
% 0.31/0.46 ( nonemptyI1
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinIL_type,type,
% 0.31/0.46 setadjoinIL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinIL,definition,
% 0.31/0.46 ( setadjoinIL
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptyinunitempty_type,type,
% 0.31/0.46 emptyinunitempty: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptyinunitempty,definition,
% 0.31/0.46 ( emptyinunitempty
% 0.31/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinIR_type,type,
% 0.31/0.46 setadjoinIR: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinIR,definition,
% 0.31/0.46 ( setadjoinIR
% 0.31/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ A )
% 0.31/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinE_type,type,
% 0.31/0.46 setadjoinE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinE,definition,
% 0.31/0.46 ( setadjoinE
% 0.31/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.46 => ! [Xphi: $o] :
% 0.31/0.46 ( ( ( Xy = Xx )
% 0.31/0.46 => Xphi )
% 0.31/0.46 => ( ( ( in @ Xy @ A )
% 0.31/0.46 => Xphi )
% 0.31/0.46 => Xphi ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinOr_type,type,
% 0.31/0.46 setadjoinOr: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinOr,definition,
% 0.31/0.46 ( setadjoinOr
% 0.31/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/0.46 => ( ( Xy = Xx )
% 0.31/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setoftrueEq_type,type,
% 0.31/0.46 setoftrueEq: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setoftrueEq,definition,
% 0.31/0.46 ( setoftrueEq
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xx: $i] : $true )
% 0.31/0.46 = A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(powersetI_type,type,
% 0.31/0.46 powersetI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(powersetI,definition,
% 0.31/0.46 ( powersetI
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ A ) )
% 0.31/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptyinPowerset_type,type,
% 0.31/0.46 emptyinPowerset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptyinPowerset,definition,
% 0.31/0.46 ( emptyinPowerset
% 0.31/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptyInPowerset_type,type,
% 0.31/0.46 emptyInPowerset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptyInPowerset,definition,
% 0.31/0.46 ( emptyInPowerset
% 0.31/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(powersetE_type,type,
% 0.31/0.46 powersetE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(powersetE,definition,
% 0.31/0.46 ( powersetE
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.46 => ( ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunionI_type,type,
% 0.31/0.46 setunionI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionI,definition,
% 0.31/0.46 ( setunionI
% 0.31/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 => ( ( in @ B @ A )
% 0.31/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunionE_type,type,
% 0.31/0.46 setunionE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionE,definition,
% 0.31/0.46 ( setunionE
% 0.31/0.46 = ( ! [A: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.46 => ! [Xphi: $o] :
% 0.31/0.46 ( ! [B: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 => ( ( in @ B @ A )
% 0.31/0.46 => Xphi ) )
% 0.31/0.46 => Xphi ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subPowSU_type,type,
% 0.31/0.46 subPowSU: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subPowSU,definition,
% 0.31/0.46 ( subPowSU
% 0.31/0.46 = ( ! [A: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuE2_type,type,
% 0.31/0.46 exuE2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuE2,definition,
% 0.31/0.46 ( exuE2
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ? [Xx: $i] :
% 0.31/0.46 ! [Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xy )
% 0.31/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(nonemptyImpWitness_type,type,
% 0.31/0.46 nonemptyImpWitness: $o ).
% 0.31/0.46
% 0.31/0.46 thf(nonemptyImpWitness,definition,
% 0.31/0.46 ( nonemptyImpWitness
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ( nonempty @ A )
% 0.31/0.46 => ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & $true ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(uniqinunit_type,type,
% 0.31/0.46 uniqinunit: $o ).
% 0.31/0.46
% 0.31/0.46 thf(uniqinunit,definition,
% 0.31/0.46 ( uniqinunit
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(notinsingleton_type,type,
% 0.31/0.46 notinsingleton: $o ).
% 0.31/0.46
% 0.31/0.46 thf(notinsingleton,definition,
% 0.31/0.46 ( notinsingleton
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xx != Xy )
% 0.31/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(eqinunit_type,type,
% 0.31/0.46 eqinunit: $o ).
% 0.31/0.46
% 0.31/0.46 thf(eqinunit,definition,
% 0.31/0.46 ( eqinunit
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xx = Xy )
% 0.31/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(singletonsswitch_type,type,
% 0.31/0.46 singletonsswitch: $o ).
% 0.31/0.46
% 0.31/0.46 thf(singletonsswitch,definition,
% 0.31/0.46 ( singletonsswitch
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairsetE_type,type,
% 0.31/0.46 upairsetE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairsetE,definition,
% 0.31/0.46 ( upairsetE
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/0.46 => ( ( Xz = Xx )
% 0.31/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairsetIL_type,type,
% 0.31/0.46 upairsetIL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairsetIL,definition,
% 0.31/0.46 ( upairsetIL
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairsetIR_type,type,
% 0.31/0.46 upairsetIR: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairsetIR,definition,
% 0.31/0.46 ( upairsetIR
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptyE1_type,type,
% 0.31/0.46 emptyE1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptyE1,definition,
% 0.31/0.46 ( emptyE1
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( Xphi @ Xx ) )
% 0.31/0.46 => ( ( ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 = emptyset )
% 0.31/0.46 => $false ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(vacuousDall_type,type,
% 0.31/0.46 vacuousDall: $o ).
% 0.31/0.46
% 0.31/0.46 thf(vacuousDall,definition,
% 0.31/0.46 ( vacuousDall
% 0.31/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ emptyset )
% 0.31/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(quantDeMorgan1_type,type,
% 0.31/0.46 quantDeMorgan1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(quantDeMorgan1,definition,
% 0.31/0.46 ( quantDeMorgan1
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ~ ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( Xphi @ Xx ) )
% 0.31/0.46 => ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(quantDeMorgan2_type,type,
% 0.31/0.46 quantDeMorgan2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(quantDeMorgan2,definition,
% 0.31/0.46 ( quantDeMorgan2
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ~ ( Xphi @ Xx ) )
% 0.31/0.46 => ~ ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(quantDeMorgan3_type,type,
% 0.31/0.46 quantDeMorgan3: $o ).
% 0.31/0.46
% 0.31/0.46 thf(quantDeMorgan3,definition,
% 0.31/0.46 ( quantDeMorgan3
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ~ ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( Xphi @ Xx ) )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(quantDeMorgan4_type,type,
% 0.31/0.46 quantDeMorgan4: $o ).
% 0.31/0.46
% 0.31/0.46 thf(quantDeMorgan4,definition,
% 0.31/0.46 ( quantDeMorgan4
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ~ ( Xphi @ Xx ) )
% 0.31/0.46 => ~ ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(prop2setI_type,type,
% 0.31/0.46 prop2setI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(prop2setI,definition,
% 0.31/0.46 ( prop2setI
% 0.31/0.46 = ( ! [Xphi: $o] :
% 0.31/0.46 ( Xphi
% 0.31/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(set2prop_type,type,
% 0.31/0.46 set2prop: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(prop2set2propI_type,type,
% 0.31/0.46 prop2set2propI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(prop2set2propI,definition,
% 0.31/0.46 ( prop2set2propI
% 0.31/0.46 = ( ! [Xphi: $o] :
% 0.31/0.46 ( Xphi
% 0.31/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(notdexE_type,type,
% 0.31/0.46 notdexE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(notdexE,definition,
% 0.31/0.46 ( notdexE
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ~ ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( Xphi @ Xx ) )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(notdallE_type,type,
% 0.31/0.46 notdallE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(notdallE,definition,
% 0.31/0.46 ( notdallE
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ~ ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( Xphi @ Xx ) )
% 0.31/0.46 => ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuI1_type,type,
% 0.31/0.46 exuI1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuI1,definition,
% 0.31/0.46 ( exuI1
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ? [Xx: $i] :
% 0.31/0.46 ( ( Xphi @ Xx )
% 0.31/0.46 & ! [Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) )
% 0.31/0.46 => ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuI3_type,type,
% 0.31/0.46 exuI3: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuI3,definition,
% 0.31/0.46 ( exuI3
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.31/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xx )
% 0.31/0.46 => ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) )
% 0.31/0.46 => ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuI2_type,type,
% 0.31/0.46 exuI2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuI2,definition,
% 0.31/0.46 ( exuI2
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ? [Xx: $i] :
% 0.31/0.46 ! [Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xy )
% 0.31/0.46 <=> ( Xy = Xx ) )
% 0.31/0.46 => ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(inCongP_type,type,
% 0.31/0.46 inCongP: $o ).
% 0.31/0.46
% 0.31/0.46 thf(inCongP,definition,
% 0.31/0.46 ( inCongP
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xx = Xy )
% 0.31/0.46 => ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(in__Cong_type,type,
% 0.31/0.46 in__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(in__Cong,definition,
% 0.31/0.46 ( in__Cong
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xx = Xy )
% 0.31/0.46 => ( ( in @ Xx @ A )
% 0.31/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuE3u_type,type,
% 0.31/0.46 exuE3u: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuE3u,definition,
% 0.31/0.46 ( exuE3u
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xx )
% 0.31/0.46 => ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exu__Cong_type,type,
% 0.31/0.46 exu__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exu__Cong,definition,
% 0.31/0.46 ( exu__Cong
% 0.31/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xx = Xy )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.31/0.46 => ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 <=> ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptyset__Cong_type,type,
% 0.31/0.46 emptyset__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptyset__Cong,definition,
% 0.31/0.46 ( emptyset__Cong
% 0.31/0.46 = ( emptyset = emptyset ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoin__Cong_type,type,
% 0.31/0.46 setadjoin__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoin__Cong,definition,
% 0.31/0.46 ( setadjoin__Cong
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xx = Xy )
% 0.31/0.46 => ! [Xz: $i,Xu: $i] :
% 0.31/0.46 ( ( Xz = Xu )
% 0.31/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.31/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(powerset__Cong_type,type,
% 0.31/0.46 powerset__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(powerset__Cong,definition,
% 0.31/0.46 ( powerset__Cong
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ( ( powerset @ A )
% 0.31/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunion__Cong_type,type,
% 0.31/0.46 setunion__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunion__Cong,definition,
% 0.31/0.46 ( setunion__Cong
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ( ( setunion @ A )
% 0.31/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(omega__Cong_type,type,
% 0.31/0.46 omega__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(omega__Cong,definition,
% 0.31/0.46 ( omega__Cong
% 0.31/0.46 = ( omega = omega ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(exuEu_type,type,
% 0.31/0.46 exuEu: $o ).
% 0.31/0.46
% 0.31/0.46 thf(exuEu,definition,
% 0.31/0.46 ( exuEu
% 0.31/0.46 = ( ! [Xphi: $i > $o] :
% 0.31/0.46 ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xphi @ Xx )
% 0.31/0.46 => ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(descr__Cong_type,type,
% 0.31/0.46 descr__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(descr__Cong,definition,
% 0.31/0.46 ( descr__Cong
% 0.31/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xx = Xy )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.31/0.46 => ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ( ( exu
% 0.31/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.31/0.46 => ( ( descr
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 = ( descr
% 0.31/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstr__Cong_type,type,
% 0.31/0.46 dsetconstr__Cong: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dsetconstr__Cong,definition,
% 0.31/0.46 ( dsetconstr__Cong
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( ( Xx = Xy )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.31/0.46 => ( ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 = ( dsetconstr @ B
% 0.31/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subset_type,type,
% 0.31/0.46 subset: $i > $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(disjoint_type,type,
% 0.31/0.46 disjoint: $i > $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(setsmeet_type,type,
% 0.31/0.46 setsmeet: $i > $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(subsetI1_type,type,
% 0.31/0.46 subsetI1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subsetI1,definition,
% 0.31/0.46 ( subsetI1
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ Xx @ B ) )
% 0.31/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(eqimpsubset2_type,type,
% 0.31/0.46 eqimpsubset2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(eqimpsubset2,definition,
% 0.31/0.46 ( eqimpsubset2
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(eqimpsubset1_type,type,
% 0.31/0.46 eqimpsubset1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(eqimpsubset1,definition,
% 0.31/0.46 ( eqimpsubset1
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( A = B )
% 0.31/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subsetI2_type,type,
% 0.31/0.46 subsetI2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subsetI2,definition,
% 0.31/0.46 ( subsetI2
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ Xx @ B ) )
% 0.31/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(emptysetsubset_type,type,
% 0.31/0.46 emptysetsubset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(emptysetsubset,definition,
% 0.31/0.46 ( emptysetsubset
% 0.31/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subsetE_type,type,
% 0.31/0.46 subsetE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subsetE,definition,
% 0.31/0.46 ( subsetE
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subsetE2_type,type,
% 0.31/0.46 subsetE2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subsetE2,definition,
% 0.31/0.46 ( subsetE2
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(notsubsetI_type,type,
% 0.31/0.46 notsubsetI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(notsubsetI,definition,
% 0.31/0.46 ( notsubsetI
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(notequalI1_type,type,
% 0.31/0.46 notequalI1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(notequalI1,definition,
% 0.31/0.46 ( notequalI1
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ~ ( subset @ A @ B )
% 0.31/0.46 => ( A != B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(notequalI2_type,type,
% 0.31/0.46 notequalI2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(notequalI2,definition,
% 0.31/0.46 ( notequalI2
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => ( A != B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subsetRefl_type,type,
% 0.31/0.46 subsetRefl: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subsetRefl,definition,
% 0.31/0.46 ( subsetRefl
% 0.31/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subsetTrans_type,type,
% 0.31/0.46 subsetTrans: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subsetTrans,definition,
% 0.31/0.46 ( subsetTrans
% 0.31/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( ( subset @ B @ C )
% 0.31/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinSub_type,type,
% 0.31/0.46 setadjoinSub: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinSub,definition,
% 0.31/0.46 ( setadjoinSub
% 0.31/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinSub2_type,type,
% 0.31/0.46 setadjoinSub2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setadjoinSub2,definition,
% 0.31/0.46 ( setadjoinSub2
% 0.31/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subset2powerset_type,type,
% 0.31/0.46 subset2powerset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subset2powerset,definition,
% 0.31/0.46 ( subset2powerset
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setextsub_type,type,
% 0.31/0.46 setextsub: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setextsub,definition,
% 0.31/0.46 ( setextsub
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( ( subset @ B @ A )
% 0.31/0.46 => ( A = B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subsetemptysetimpeq_type,type,
% 0.31/0.46 subsetemptysetimpeq: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subsetemptysetimpeq,definition,
% 0.31/0.46 ( subsetemptysetimpeq
% 0.31/0.46 = ( ! [A: $i] :
% 0.31/0.46 ( ( subset @ A @ emptyset )
% 0.31/0.46 => ( A = emptyset ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(powersetI1_type,type,
% 0.31/0.46 powersetI1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(powersetI1,definition,
% 0.31/0.46 ( powersetI1
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( subset @ B @ A )
% 0.31/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(powersetE1_type,type,
% 0.31/0.46 powersetE1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(powersetE1,definition,
% 0.31/0.46 ( powersetE1
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(inPowerset_type,type,
% 0.31/0.46 inPowerset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(inPowerset,definition,
% 0.31/0.46 ( inPowerset
% 0.31/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(powersetsubset_type,type,
% 0.31/0.46 powersetsubset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(powersetsubset,definition,
% 0.31/0.46 ( powersetsubset
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sepInPowerset_type,type,
% 0.31/0.46 sepInPowerset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(sepInPowerset,definition,
% 0.31/0.46 ( sepInPowerset
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( in
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(sepSubset_type,type,
% 0.31/0.46 sepSubset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(sepSubset,definition,
% 0.31/0.46 ( sepSubset
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( subset
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binunion_type,type,
% 0.31/0.46 binunion: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(binunionIL_type,type,
% 0.31/0.46 binunionIL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binunionIL,definition,
% 0.31/0.46 ( binunionIL
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairset2IR_type,type,
% 0.31/0.46 upairset2IR: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairset2IR,definition,
% 0.31/0.46 ( upairset2IR
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binunionIR_type,type,
% 0.31/0.46 binunionIR: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binunionIR,definition,
% 0.31/0.46 ( binunionIR
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binunionEcases_type,type,
% 0.31/0.46 binunionEcases: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binunionEcases,definition,
% 0.31/0.46 ( binunionEcases
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.31/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/0.46 => ( ( ( in @ Xx @ A )
% 0.31/0.46 => Xphi )
% 0.31/0.46 => ( ( ( in @ Xx @ B )
% 0.31/0.46 => Xphi )
% 0.31/0.46 => Xphi ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binunionE_type,type,
% 0.31/0.46 binunionE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binunionE,definition,
% 0.31/0.46 ( binunionE
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/0.46 => ( ( in @ Xx @ A )
% 0.31/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binunionLsub_type,type,
% 0.31/0.46 binunionLsub: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binunionLsub,definition,
% 0.31/0.46 ( binunionLsub
% 0.31/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binunionRsub_type,type,
% 0.31/0.46 binunionRsub: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binunionRsub,definition,
% 0.31/0.46 ( binunionRsub
% 0.31/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersect_type,type,
% 0.31/0.46 binintersect: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectI_type,type,
% 0.31/0.46 binintersectI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectI,definition,
% 0.31/0.46 ( binintersectI
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset5_type,type,
% 0.31/0.46 binintersectSubset5: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset5,definition,
% 0.31/0.46 ( binintersectSubset5
% 0.31/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.46 ( ( subset @ C @ A )
% 0.31/0.46 => ( ( subset @ C @ B )
% 0.31/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectEL_type,type,
% 0.31/0.46 binintersectEL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectEL,definition,
% 0.31/0.46 ( binintersectEL
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectLsub_type,type,
% 0.31/0.46 binintersectLsub: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectLsub,definition,
% 0.31/0.46 ( binintersectLsub
% 0.31/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset2_type,type,
% 0.31/0.46 binintersectSubset2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset2,definition,
% 0.31/0.46 ( binintersectSubset2
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( ( binintersect @ A @ B )
% 0.31/0.46 = A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset3_type,type,
% 0.31/0.46 binintersectSubset3: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset3,definition,
% 0.31/0.46 ( binintersectSubset3
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( ( binintersect @ A @ B )
% 0.31/0.46 = B )
% 0.31/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectER_type,type,
% 0.31/0.46 binintersectER: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectER,definition,
% 0.31/0.46 ( binintersectER
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(disjointsetsI1_type,type,
% 0.31/0.46 disjointsetsI1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(disjointsetsI1,definition,
% 0.31/0.46 ( disjointsetsI1
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ~ ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( in @ Xx @ B ) )
% 0.31/0.46 => ( ( binintersect @ A @ B )
% 0.31/0.46 = emptyset ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectRsub_type,type,
% 0.31/0.46 binintersectRsub: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectRsub,definition,
% 0.31/0.46 ( binintersectRsub
% 0.31/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset4_type,type,
% 0.31/0.46 binintersectSubset4: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset4,definition,
% 0.31/0.46 ( binintersectSubset4
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( subset @ B @ A )
% 0.31/0.46 => ( ( binintersect @ A @ B )
% 0.31/0.46 = B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset1_type,type,
% 0.31/0.46 binintersectSubset1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(binintersectSubset1,definition,
% 0.31/0.46 ( binintersectSubset1
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( ( binintersect @ A @ B )
% 0.31/0.46 = A )
% 0.31/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(bs114d_type,type,
% 0.31/0.46 bs114d: $o ).
% 0.31/0.46
% 0.31/0.46 thf(bs114d,definition,
% 0.31/0.46 ( bs114d
% 0.31/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.31/0.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(regular_type,type,
% 0.31/0.46 regular: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminus_type,type,
% 0.31/0.46 setminus: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(setminusI_type,type,
% 0.31/0.46 setminusI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusI,definition,
% 0.31/0.46 ( setminusI
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusEL_type,type,
% 0.31/0.46 setminusEL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusEL,definition,
% 0.31/0.46 ( setminusEL
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusER_type,type,
% 0.31/0.46 setminusER: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusER,definition,
% 0.31/0.46 ( setminusER
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusSubset2_type,type,
% 0.31/0.46 setminusSubset2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusSubset2,definition,
% 0.31/0.46 ( setminusSubset2
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( subset @ A @ B )
% 0.31/0.46 => ( ( setminus @ A @ B )
% 0.31/0.46 = emptyset ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusERneg_type,type,
% 0.31/0.46 setminusERneg: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusERneg,definition,
% 0.31/0.46 ( setminusERneg
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.46 => ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusELneg_type,type,
% 0.31/0.46 setminusELneg: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusELneg,definition,
% 0.31/0.46 ( setminusELneg
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusILneg_type,type,
% 0.31/0.46 setminusILneg: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusILneg,definition,
% 0.31/0.46 ( setminusILneg
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ~ ( in @ Xx @ A )
% 0.31/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusIRneg_type,type,
% 0.31/0.46 setminusIRneg: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusIRneg,definition,
% 0.31/0.46 ( setminusIRneg
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ B )
% 0.31/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusLsub_type,type,
% 0.31/0.46 setminusLsub: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusLsub,definition,
% 0.31/0.46 ( setminusLsub
% 0.31/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setminusSubset1_type,type,
% 0.31/0.46 setminusSubset1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setminusSubset1,definition,
% 0.31/0.46 ( setminusSubset1
% 0.31/0.46 = ( ! [A: $i,B: $i] :
% 0.31/0.46 ( ( ( setminus @ A @ B )
% 0.31/0.46 = emptyset )
% 0.31/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(symdiff_type,type,
% 0.31/0.46 symdiff: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffE_type,type,
% 0.31/0.46 symdiffE: $o ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffE,definition,
% 0.31/0.46 ( symdiffE
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.31/0.46 => ! [Xphi: $o] :
% 0.31/0.46 ( ( ( in @ Xx @ A )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => Xphi ) )
% 0.31/0.46 => ( ( ~ ( in @ Xx @ A )
% 0.31/0.46 => ( ( in @ Xx @ B )
% 0.31/0.46 => Xphi ) )
% 0.31/0.46 => Xphi ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffI1_type,type,
% 0.31/0.46 symdiffI1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffI1,definition,
% 0.31/0.46 ( symdiffI1
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffI2_type,type,
% 0.31/0.46 symdiffI2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffI2,definition,
% 0.31/0.46 ( symdiffI2
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ~ ( in @ Xx @ A )
% 0.31/0.46 => ( ( in @ Xx @ B )
% 0.31/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffIneg1_type,type,
% 0.31/0.46 symdiffIneg1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffIneg1,definition,
% 0.31/0.46 ( symdiffIneg1
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( in @ Xx @ B )
% 0.31/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffIneg2_type,type,
% 0.31/0.46 symdiffIneg2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(symdiffIneg2,definition,
% 0.31/0.46 ( symdiffIneg2
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ~ ( in @ Xx @ A )
% 0.31/0.46 => ( ~ ( in @ Xx @ B )
% 0.31/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(iskpair_type,type,
% 0.31/0.46 iskpair: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(secondinupair_type,type,
% 0.31/0.46 secondinupair: $o ).
% 0.31/0.46
% 0.31/0.46 thf(secondinupair,definition,
% 0.31/0.46 ( secondinupair
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairIL_type,type,
% 0.31/0.46 setukpairIL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairIL,definition,
% 0.31/0.46 ( setukpairIL
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairIR_type,type,
% 0.31/0.46 setukpairIR: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairIR,definition,
% 0.31/0.46 ( setukpairIR
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(kpairiskpair_type,type,
% 0.31/0.46 kpairiskpair: $o ).
% 0.31/0.46
% 0.31/0.46 thf(kpairiskpair,definition,
% 0.31/0.46 ( kpairiskpair
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(kpair_type,type,
% 0.31/0.46 kpair: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(kpairp_type,type,
% 0.31/0.46 kpairp: $o ).
% 0.31/0.46
% 0.31/0.46 thf(kpairp,definition,
% 0.31/0.46 ( kpairp
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprod_type,type,
% 0.31/0.46 cartprod: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(singletonsubset_type,type,
% 0.31/0.46 singletonsubset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(singletonsubset,definition,
% 0.31/0.46 ( singletonsubset
% 0.31/0.46 = ( ! [A: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(singletoninpowerset_type,type,
% 0.31/0.46 singletoninpowerset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(singletoninpowerset,definition,
% 0.31/0.46 ( singletoninpowerset
% 0.31/0.46 = ( ! [A: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(singletoninpowunion_type,type,
% 0.31/0.46 singletoninpowunion: $o ).
% 0.31/0.46
% 0.31/0.46 thf(singletoninpowunion,definition,
% 0.31/0.46 ( singletoninpowunion
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairset2E_type,type,
% 0.31/0.46 upairset2E: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairset2E,definition,
% 0.31/0.46 ( upairset2E
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/0.46 => ( ( Xz = Xx )
% 0.31/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairsubunion_type,type,
% 0.31/0.46 upairsubunion: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairsubunion,definition,
% 0.31/0.46 ( upairsubunion
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairinpowunion_type,type,
% 0.31/0.46 upairinpowunion: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairinpowunion,definition,
% 0.31/0.46 ( upairinpowunion
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem1_type,type,
% 0.31/0.46 ubforcartprodlem1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem1,definition,
% 0.31/0.46 ( ubforcartprodlem1
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem2_type,type,
% 0.31/0.46 ubforcartprodlem2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem2,definition,
% 0.31/0.46 ( ubforcartprodlem2
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem3_type,type,
% 0.31/0.46 ubforcartprodlem3: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ubforcartprodlem3,definition,
% 0.31/0.46 ( ubforcartprodlem3
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodpairin_type,type,
% 0.31/0.46 cartprodpairin: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodpairin,definition,
% 0.31/0.46 ( cartprodpairin
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodmempair1_type,type,
% 0.31/0.46 cartprodmempair1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodmempair1,definition,
% 0.31/0.46 ( cartprodmempair1
% 0.31/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.46 => ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ? [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 & ( Xu
% 0.31/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodmempair_type,type,
% 0.31/0.46 cartprodmempair: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodmempair,definition,
% 0.31/0.46 ( cartprodmempair
% 0.31/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.46 => ( iskpair @ Xu ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunionE2_type,type,
% 0.31/0.46 setunionE2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionE2,definition,
% 0.31/0.46 ( setunionE2
% 0.31/0.46 = ( ! [A: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.46 => ? [X: $i] :
% 0.31/0.46 ( ( in @ X @ A )
% 0.31/0.46 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton1_type,type,
% 0.31/0.46 setunionsingleton1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton1,definition,
% 0.31/0.46 ( setunionsingleton1
% 0.31/0.46 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton2_type,type,
% 0.31/0.46 setunionsingleton2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton2,definition,
% 0.31/0.46 ( setunionsingleton2
% 0.31/0.46 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton_type,type,
% 0.31/0.46 setunionsingleton: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setunionsingleton,definition,
% 0.31/0.46 ( setunionsingleton
% 0.31/0.46 = ( ! [Xx: $i] :
% 0.31/0.46 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.31/0.46 = Xx ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(singleton_type,type,
% 0.31/0.46 singleton: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(singleton,definition,
% 0.31/0.46 ( singleton
% 0.31/0.46 = ( ^ [A: $i] :
% 0.31/0.46 ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( A
% 0.31/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(singletonprop_type,type,
% 0.31/0.46 singletonprop: $o ).
% 0.31/0.46
% 0.31/0.46 thf(singletonprop,definition,
% 0.31/0.46 ( singletonprop
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ A )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 => ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) )
% 0.31/0.46 => ( ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( Xphi @ Xx ) )
% 0.31/0.46 => ( singleton
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ex1_type,type,
% 0.31/0.46 ex1: $i > ( $i > $o ) > $o ).
% 0.31/0.46
% 0.31/0.46 thf(ex1,definition,
% 0.31/0.46 ( ex1
% 0.31/0.46 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( singleton
% 0.31/0.46 @ ( dsetconstr @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ex1E1_type,type,
% 0.31/0.46 ex1E1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ex1E1,definition,
% 0.31/0.46 ( ex1E1
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ( ex1 @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ex1I_type,type,
% 0.31/0.46 ex1I: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ex1I,definition,
% 0.31/0.46 ( ex1I
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 => ( ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ A )
% 0.31/0.46 => ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xy = Xx ) ) )
% 0.31/0.46 => ( ex1 @ A
% 0.31/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ex1I2_type,type,
% 0.31/0.46 ex1I2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ex1I2,definition,
% 0.31/0.46 ( ex1I2
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ A )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 => ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) )
% 0.31/0.46 => ( ? [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 & ( Xphi @ Xx ) )
% 0.31/0.46 => ( ex1 @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(singletonsuniq_type,type,
% 0.31/0.46 singletonsuniq: $o ).
% 0.31/0.46
% 0.31/0.46 thf(singletonsuniq,definition,
% 0.31/0.46 ( singletonsuniq
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( ( setadjoin @ Xx @ emptyset )
% 0.31/0.46 = ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(atmost1p_type,type,
% 0.31/0.46 atmost1p: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(atleast2p_type,type,
% 0.31/0.46 atleast2p: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(atmost2p_type,type,
% 0.31/0.46 atmost2p: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairsetp_type,type,
% 0.31/0.46 upairsetp: $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjL1_type,type,
% 0.31/0.46 setukpairinjL1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjL1,definition,
% 0.31/0.46 ( setukpairinjL1
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.46 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(kfstsingleton_type,type,
% 0.31/0.46 kfstsingleton: $o ).
% 0.31/0.46
% 0.31/0.46 thf(kfstsingleton,definition,
% 0.31/0.46 ( kfstsingleton
% 0.31/0.46 = ( ! [Xu: $i] :
% 0.31/0.46 ( ( iskpair @ Xu )
% 0.31/0.46 => ( singleton
% 0.31/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.31/0.46 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(theprop_type,type,
% 0.31/0.46 theprop: $o ).
% 0.31/0.46
% 0.31/0.46 thf(theprop,definition,
% 0.31/0.46 ( theprop
% 0.31/0.46 = ( ! [X: $i] :
% 0.31/0.46 ( ( singleton @ X )
% 0.31/0.46 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(kfst_type,type,
% 0.31/0.46 kfst: $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(kfstpairEq_type,type,
% 0.31/0.46 kfstpairEq: $o ).
% 0.31/0.46
% 0.31/0.46 thf(kfstpairEq,definition,
% 0.31/0.46 ( kfstpairEq
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.31/0.46 = Xx ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodfstin_type,type,
% 0.31/0.46 cartprodfstin: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodfstin,definition,
% 0.31/0.46 ( cartprodfstin
% 0.31/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.46 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjL2_type,type,
% 0.31/0.46 setukpairinjL2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjL2,definition,
% 0.31/0.46 ( setukpairinjL2
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjL_type,type,
% 0.31/0.46 setukpairinjL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjL,definition,
% 0.31/0.46 ( setukpairinjL
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.31/0.46 = ( kpair @ Xz @ Xu ) )
% 0.31/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjR11_type,type,
% 0.31/0.46 setukpairinjR11: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjR11,definition,
% 0.31/0.46 ( setukpairinjR11
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xx = Xy )
% 0.31/0.46 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjR12_type,type,
% 0.31/0.46 setukpairinjR12: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjR12,definition,
% 0.31/0.46 ( setukpairinjR12
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xx = Xy )
% 0.31/0.46 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjR1_type,type,
% 0.31/0.46 setukpairinjR1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjR1,definition,
% 0.31/0.46 ( setukpairinjR1
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.46 => ( ( Xz = Xu )
% 0.31/0.46 => ( Xy = Xu ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(upairequniteq_type,type,
% 0.31/0.46 upairequniteq: $o ).
% 0.31/0.46
% 0.31/0.46 thf(upairequniteq,definition,
% 0.31/0.46 ( upairequniteq
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.46 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.46 = ( setadjoin @ Xz @ emptyset ) )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjR2_type,type,
% 0.31/0.46 setukpairinjR2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjR2,definition,
% 0.31/0.46 ( setukpairinjR2
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjR_type,type,
% 0.31/0.46 setukpairinjR: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setukpairinjR,definition,
% 0.31/0.46 ( setukpairinjR
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.31/0.46 = ( kpair @ Xz @ Xu ) )
% 0.31/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ksndsingleton_type,type,
% 0.31/0.46 ksndsingleton: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ksndsingleton,definition,
% 0.31/0.46 ( ksndsingleton
% 0.31/0.46 = ( ! [Xu: $i] :
% 0.31/0.46 ( ( iskpair @ Xu )
% 0.31/0.46 => ( singleton
% 0.31/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.31/0.46 @ ^ [Xx: $i] :
% 0.31/0.46 ( Xu
% 0.31/0.46 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ksnd_type,type,
% 0.31/0.46 ksnd: $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(ksndpairEq_type,type,
% 0.31/0.46 ksndpairEq: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ksndpairEq,definition,
% 0.31/0.46 ( ksndpairEq
% 0.31/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.31/0.46 = Xy ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(kpairsurjEq_type,type,
% 0.31/0.46 kpairsurjEq: $o ).
% 0.31/0.46
% 0.31/0.46 thf(kpairsurjEq,definition,
% 0.31/0.46 ( kpairsurjEq
% 0.31/0.46 = ( ! [Xu: $i] :
% 0.31/0.46 ( ( iskpair @ Xu )
% 0.31/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.31/0.46 = Xu ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodsndin_type,type,
% 0.31/0.46 cartprodsndin: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodsndin,definition,
% 0.31/0.46 ( cartprodsndin
% 0.31/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.46 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodpairmemEL_type,type,
% 0.31/0.46 cartprodpairmemEL: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodpairmemEL,definition,
% 0.31/0.46 ( cartprodpairmemEL
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.31/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodpairmemER_type,type,
% 0.31/0.46 cartprodpairmemER: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodpairmemER,definition,
% 0.31/0.46 ( cartprodpairmemER
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.31/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodmempaircEq_type,type,
% 0.31/0.46 cartprodmempaircEq: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodmempaircEq,definition,
% 0.31/0.46 ( cartprodmempaircEq
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( ( kpair @ Xx @ Xy )
% 0.31/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodfstpairEq_type,type,
% 0.31/0.46 cartprodfstpairEq: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodfstpairEq,definition,
% 0.31/0.46 ( cartprodfstpairEq
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.31/0.46 = Xx ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodsndpairEq_type,type,
% 0.31/0.46 cartprodsndpairEq: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodsndpairEq,definition,
% 0.31/0.46 ( cartprodsndpairEq
% 0.31/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.31/0.46 = Xy ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodpairsurjEq_type,type,
% 0.31/0.46 cartprodpairsurjEq: $o ).
% 0.31/0.46
% 0.31/0.46 thf(cartprodpairsurjEq,definition,
% 0.31/0.46 ( cartprodpairsurjEq
% 0.31/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.31/0.46 = Xu ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(breln_type,type,
% 0.31/0.46 breln: $i > $i > $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(breln,definition,
% 0.31/0.46 ( breln
% 0.31/0.46 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dpsetconstr_type,type,
% 0.31/0.46 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.31/0.46
% 0.31/0.46 thf(dpsetconstrI_type,type,
% 0.31/0.46 dpsetconstrI: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dpsetconstrI,definition,
% 0.31/0.46 ( dpsetconstrI
% 0.31/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( ( Xphi @ Xx @ Xy )
% 0.31/0.46 => ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.46 @ ( dpsetconstr @ A @ B
% 0.31/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dpsetconstrSub_type,type,
% 0.31/0.46 dpsetconstrSub: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dpsetconstrSub,definition,
% 0.31/0.46 ( dpsetconstrSub
% 0.31/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/0.46 ( subset
% 0.31/0.46 @ ( dpsetconstr @ A @ B
% 0.31/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.31/0.46 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(setOfPairsIsBReln_type,type,
% 0.31/0.46 setOfPairsIsBReln: $o ).
% 0.31/0.46
% 0.31/0.46 thf(setOfPairsIsBReln,definition,
% 0.31/0.46 ( setOfPairsIsBReln
% 0.31/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/0.46 ( breln @ A @ B
% 0.31/0.46 @ ( dpsetconstr @ A @ B
% 0.31/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dpsetconstrERa_type,type,
% 0.31/0.46 dpsetconstrERa: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dpsetconstrERa,definition,
% 0.31/0.46 ( dpsetconstrERa
% 0.31/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.46 @ ( dpsetconstr @ A @ B
% 0.31/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dpsetconstrEL1_type,type,
% 0.31/0.46 dpsetconstrEL1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dpsetconstrEL1,definition,
% 0.31/0.46 ( dpsetconstrEL1
% 0.31/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.46 @ ( dpsetconstr @ A @ B
% 0.31/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dpsetconstrEL2_type,type,
% 0.31/0.46 dpsetconstrEL2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dpsetconstrEL2,definition,
% 0.31/0.46 ( dpsetconstrEL2
% 0.31/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.46 @ ( dpsetconstr @ A @ B
% 0.31/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(dpsetconstrER_type,type,
% 0.31/0.46 dpsetconstrER: $o ).
% 0.31/0.46
% 0.31/0.46 thf(dpsetconstrER,definition,
% 0.31/0.46 ( dpsetconstrER
% 0.31/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.46 @ ( dpsetconstr @ A @ B
% 0.31/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(func_type,type,
% 0.31/0.46 func: $i > $i > $i > $o ).
% 0.31/0.46
% 0.31/0.46 thf(funcSet_type,type,
% 0.31/0.46 funcSet: $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(funcImageSingleton_type,type,
% 0.31/0.46 funcImageSingleton: $o ).
% 0.31/0.46
% 0.31/0.46 thf(funcImageSingleton,definition,
% 0.31/0.46 ( funcImageSingleton
% 0.31/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.46 ( ( func @ A @ B @ Xf )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( singleton
% 0.31/0.46 @ ( dsetconstr @ B
% 0.31/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(apProp_type,type,
% 0.31/0.46 apProp: $o ).
% 0.31/0.46
% 0.31/0.46 thf(apProp,definition,
% 0.31/0.46 ( apProp
% 0.31/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.46 ( ( func @ A @ B @ Xf )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in
% 0.31/0.46 @ ( setunion
% 0.31/0.46 @ ( dsetconstr @ B
% 0.31/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.31/0.46 @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ap_type,type,
% 0.31/0.46 ap: $i > $i > $i > $i > $i ).
% 0.31/0.46
% 0.31/0.46 thf(app_type,type,
% 0.31/0.46 app: $o ).
% 0.31/0.46
% 0.31/0.46 thf(app,definition,
% 0.31/0.46 ( app
% 0.31/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.46 ( ( func @ A @ B @ Xf )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(infuncsetfunc_type,type,
% 0.31/0.46 infuncsetfunc: $o ).
% 0.31/0.46
% 0.31/0.46 thf(infuncsetfunc,definition,
% 0.31/0.46 ( infuncsetfunc
% 0.31/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.46 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ap2p_type,type,
% 0.31/0.46 ap2p: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ap2p,definition,
% 0.31/0.46 ( ap2p
% 0.31/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(funcinfuncset_type,type,
% 0.31/0.46 funcinfuncset: $o ).
% 0.31/0.46
% 0.31/0.46 thf(funcinfuncset,definition,
% 0.31/0.46 ( funcinfuncset
% 0.31/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.46 ( ( func @ A @ B @ Xf )
% 0.31/0.46 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(lamProp_type,type,
% 0.31/0.46 lamProp: $o ).
% 0.31/0.46
% 0.31/0.46 thf(lamProp,definition,
% 0.31/0.46 ( lamProp
% 0.31/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.46 => ( func @ A @ B
% 0.31/0.46 @ ( dpsetconstr @ A @ B
% 0.31/0.46 @ ^ [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xf @ Xx )
% 0.31/0.46 = Xy ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(lam_type,type,
% 0.31/0.46 lam: $i > $i > ( $i > $i ) > $i ).
% 0.31/0.46
% 0.31/0.46 thf(lam,definition,
% 0.31/0.46 ( lam
% 0.31/0.46 = ( ^ [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.46 ( dpsetconstr @ A @ B
% 0.31/0.46 @ ^ [Xx: $i,Xy: $i] :
% 0.31/0.46 ( ( Xf @ Xx )
% 0.31/0.46 = Xy ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(lamp_type,type,
% 0.31/0.46 lamp: $o ).
% 0.31/0.46
% 0.31/0.46 thf(lamp,definition,
% 0.31/0.46 ( lamp
% 0.31/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.46 => ( func @ A @ B
% 0.31/0.46 @ ( lam @ A @ B
% 0.31/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(lam2p_type,type,
% 0.31/0.46 lam2p: $o ).
% 0.31/0.46
% 0.31/0.46 thf(lam2p,definition,
% 0.31/0.46 ( lam2p
% 0.31/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.46 => ( in
% 0.31/0.46 @ ( lam @ A @ B
% 0.31/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.31/0.46 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(brelnall1_type,type,
% 0.31/0.46 brelnall1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(brelnall1,definition,
% 0.31/0.46 ( brelnall1
% 0.31/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/0.46 ( ( breln @ A @ B @ R )
% 0.31/0.46 => ! [Xphi: $i > $o] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ R )
% 0.31/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(brelnall2_type,type,
% 0.31/0.46 brelnall2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(brelnall2,definition,
% 0.31/0.46 ( brelnall2
% 0.31/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/0.46 ( ( breln @ A @ B @ R )
% 0.31/0.46 => ! [Xphi: $i > $o] :
% 0.31/0.46 ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ R )
% 0.31/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ex1E2_type,type,
% 0.31/0.46 ex1E2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(ex1E2,definition,
% 0.31/0.46 ( ex1E2
% 0.31/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.46 ( ( ex1 @ A
% 0.31/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ A )
% 0.31/0.46 => ( ( Xphi @ Xx )
% 0.31/0.46 => ( ( Xphi @ Xy )
% 0.31/0.46 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(funcGraphProp1_type,type,
% 0.31/0.46 funcGraphProp1: $o ).
% 0.31/0.46
% 0.31/0.46 thf(funcGraphProp1,definition,
% 0.31/0.46 ( funcGraphProp1
% 0.31/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.46 ( ( func @ A @ B @ Xf )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(funcGraphProp3_type,type,
% 0.31/0.46 funcGraphProp3: $o ).
% 0.31/0.46
% 0.31/0.46 thf(funcGraphProp3,definition,
% 0.31/0.46 ( funcGraphProp3
% 0.31/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(funcGraphProp2_type,type,
% 0.31/0.46 funcGraphProp2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(funcGraphProp2,definition,
% 0.31/0.46 ( funcGraphProp2
% 0.31/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.46 ( ( func @ A @ B @ Xf )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.31/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(funcextLem_type,type,
% 0.31/0.46 funcextLem: $o ).
% 0.31/0.46
% 0.31/0.46 thf(funcextLem,definition,
% 0.31/0.46 ( funcextLem
% 0.31/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.46 ( ( func @ A @ B @ Xf )
% 0.31/0.46 => ! [Xg: $i] :
% 0.31/0.46 ( ( func @ A @ B @ Xg )
% 0.31/0.46 => ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.31/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(funcGraphProp4_type,type,
% 0.31/0.46 funcGraphProp4: $o ).
% 0.31/0.46
% 0.31/0.46 thf(funcGraphProp4,definition,
% 0.31/0.46 ( funcGraphProp4
% 0.31/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.31/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(subbreln_type,type,
% 0.31/0.46 subbreln: $o ).
% 0.31/0.46
% 0.31/0.46 thf(subbreln,definition,
% 0.31/0.46 ( subbreln
% 0.31/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/0.46 ( ( breln @ A @ B @ R )
% 0.31/0.46 => ! [S: $i] :
% 0.31/0.46 ( ( breln @ A @ B @ S )
% 0.31/0.46 => ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.31/0.46 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(eqbreln_type,type,
% 0.31/0.46 eqbreln: $o ).
% 0.31/0.46
% 0.31/0.46 thf(eqbreln,definition,
% 0.31/0.46 ( eqbreln
% 0.31/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/0.46 ( ( breln @ A @ B @ R )
% 0.31/0.46 => ! [S: $i] :
% 0.31/0.46 ( ( breln @ A @ B @ S )
% 0.31/0.46 => ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.31/0.46 => ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ! [Xy: $i] :
% 0.31/0.46 ( ( in @ Xy @ B )
% 0.31/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.31/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.31/0.46 => ( R = S ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(funcext_type,type,
% 0.31/0.46 funcext: $o ).
% 0.31/0.46
% 0.31/0.46 thf(funcext,definition,
% 0.31/0.46 ( funcext
% 0.31/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.46 ( ( func @ A @ B @ Xf )
% 0.31/0.46 => ! [Xg: $i] :
% 0.31/0.46 ( ( func @ A @ B @ Xg )
% 0.31/0.46 => ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(funcext2_type,type,
% 0.31/0.46 funcext2: $o ).
% 0.31/0.46
% 0.31/0.46 thf(funcext2,definition,
% 0.31/0.46 ( funcext2
% 0.31/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.46 => ! [Xg: $i] :
% 0.31/0.46 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.31/0.46 => ( ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 thf(ap2apEq1,conjecture,
% 0.31/0.46 ( setextAx
% 0.31/0.46 => ( emptysetAx
% 0.31/0.46 => ( setadjoinAx
% 0.31/0.46 => ( powersetAx
% 0.31/0.46 => ( setunionAx
% 0.31/0.46 => ( omega0Ax
% 0.31/0.46 => ( omegaSAx
% 0.31/0.46 => ( omegaIndAx
% 0.31/0.46 => ( replAx
% 0.31/0.46 => ( foundationAx
% 0.31/0.46 => ( wellorderingAx
% 0.31/0.46 => ( descrp
% 0.31/0.46 => ( dsetconstrI
% 0.31/0.46 => ( dsetconstrEL
% 0.31/0.46 => ( dsetconstrER
% 0.31/0.46 => ( exuE1
% 0.31/0.46 => ( prop2setE
% 0.31/0.46 => ( emptysetE
% 0.31/0.46 => ( emptysetimpfalse
% 0.31/0.46 => ( notinemptyset
% 0.31/0.46 => ( exuE3e
% 0.31/0.46 => ( setext
% 0.31/0.46 => ( emptyI
% 0.31/0.46 => ( noeltsimpempty
% 0.31/0.46 => ( setbeta
% 0.31/0.46 => ( nonemptyE1
% 0.31/0.46 => ( nonemptyI
% 0.31/0.46 => ( nonemptyI1
% 0.31/0.46 => ( setadjoinIL
% 0.31/0.46 => ( emptyinunitempty
% 0.31/0.46 => ( setadjoinIR
% 0.31/0.46 => ( setadjoinE
% 0.31/0.46 => ( setadjoinOr
% 0.31/0.46 => ( setoftrueEq
% 0.31/0.46 => ( powersetI
% 0.31/0.46 => ( emptyinPowerset
% 0.31/0.46 => ( emptyInPowerset
% 0.31/0.46 => ( powersetE
% 0.31/0.46 => ( setunionI
% 0.31/0.46 => ( setunionE
% 0.31/0.46 => ( subPowSU
% 0.31/0.46 => ( exuE2
% 0.31/0.46 => ( nonemptyImpWitness
% 0.31/0.46 => ( uniqinunit
% 0.31/0.46 => ( notinsingleton
% 0.31/0.46 => ( eqinunit
% 0.31/0.46 => ( singletonsswitch
% 0.31/0.46 => ( upairsetE
% 0.31/0.46 => ( upairsetIL
% 0.31/0.46 => ( upairsetIR
% 0.31/0.46 => ( emptyE1
% 0.31/0.46 => ( vacuousDall
% 0.31/0.46 => ( quantDeMorgan1
% 0.31/0.46 => ( quantDeMorgan2
% 0.31/0.46 => ( quantDeMorgan3
% 0.31/0.46 => ( quantDeMorgan4
% 0.31/0.46 => ( prop2setI
% 0.31/0.46 => ( prop2set2propI
% 0.31/0.46 => ( notdexE
% 0.31/0.46 => ( notdallE
% 0.31/0.46 => ( exuI1
% 0.31/0.46 => ( exuI3
% 0.31/0.46 => ( exuI2
% 0.31/0.46 => ( inCongP
% 0.31/0.46 => ( in__Cong
% 0.31/0.46 => ( exuE3u
% 0.31/0.46 => ( exu__Cong
% 0.31/0.46 => ( emptyset__Cong
% 0.31/0.46 => ( setadjoin__Cong
% 0.31/0.46 => ( powerset__Cong
% 0.31/0.46 => ( setunion__Cong
% 0.31/0.46 => ( omega__Cong
% 0.31/0.46 => ( exuEu
% 0.31/0.46 => ( descr__Cong
% 0.31/0.46 => ( dsetconstr__Cong
% 0.31/0.46 => ( subsetI1
% 0.31/0.46 => ( eqimpsubset2
% 0.31/0.46 => ( eqimpsubset1
% 0.31/0.46 => ( subsetI2
% 0.31/0.46 => ( emptysetsubset
% 0.31/0.46 => ( subsetE
% 0.31/0.46 => ( subsetE2
% 0.31/0.46 => ( notsubsetI
% 0.31/0.46 => ( notequalI1
% 0.31/0.46 => ( notequalI2
% 0.31/0.46 => ( subsetRefl
% 0.31/0.46 => ( subsetTrans
% 0.31/0.46 => ( setadjoinSub
% 0.31/0.46 => ( setadjoinSub2
% 0.31/0.46 => ( subset2powerset
% 0.31/0.46 => ( setextsub
% 0.31/0.46 => ( subsetemptysetimpeq
% 0.31/0.46 => ( powersetI1
% 0.31/0.46 => ( powersetE1
% 0.31/0.46 => ( inPowerset
% 0.31/0.46 => ( powersetsubset
% 0.31/0.46 => ( sepInPowerset
% 0.31/0.46 => ( sepSubset
% 0.31/0.46 => ( binunionIL
% 0.31/0.46 => ( upairset2IR
% 0.31/0.46 => ( binunionIR
% 0.31/0.46 => ( binunionEcases
% 0.31/0.46 => ( binunionE
% 0.31/0.46 => ( binunionLsub
% 0.31/0.46 => ( binunionRsub
% 0.31/0.46 => ( binintersectI
% 0.31/0.46 => ( binintersectSubset5
% 0.31/0.46 => ( binintersectEL
% 0.31/0.46 => ( binintersectLsub
% 0.31/0.46 => ( binintersectSubset2
% 0.31/0.46 => ( binintersectSubset3
% 0.31/0.46 => ( binintersectER
% 0.31/0.46 => ( disjointsetsI1
% 0.31/0.46 => ( binintersectRsub
% 0.31/0.46 => ( binintersectSubset4
% 0.31/0.46 => ( binintersectSubset1
% 0.31/0.46 => ( bs114d
% 0.31/0.46 => ( setminusI
% 0.31/0.46 => ( setminusEL
% 0.31/0.46 => ( setminusER
% 0.31/0.46 => ( setminusSubset2
% 0.31/0.46 => ( setminusERneg
% 0.31/0.46 => ( setminusELneg
% 0.31/0.46 => ( setminusILneg
% 0.31/0.46 => ( setminusIRneg
% 0.31/0.46 => ( setminusLsub
% 0.31/0.46 => ( setminusSubset1
% 0.31/0.46 => ( symdiffE
% 0.31/0.46 => ( symdiffI1
% 0.31/0.46 => ( symdiffI2
% 0.31/0.46 => ( symdiffIneg1
% 0.31/0.46 => ( symdiffIneg2
% 0.31/0.46 => ( secondinupair
% 0.31/0.46 => ( setukpairIL
% 0.31/0.46 => ( setukpairIR
% 0.31/0.46 => ( kpairiskpair
% 0.31/0.46 => ( kpairp
% 0.31/0.46 => ( singletonsubset
% 0.31/0.46 => ( singletoninpowerset
% 0.31/0.46 => ( singletoninpowunion
% 0.31/0.46 => ( upairset2E
% 0.31/0.46 => ( upairsubunion
% 0.31/0.46 => ( upairinpowunion
% 0.31/0.46 => ( ubforcartprodlem1
% 0.31/0.46 => ( ubforcartprodlem2
% 0.31/0.46 => ( ubforcartprodlem3
% 0.31/0.46 => ( cartprodpairin
% 0.31/0.46 => ( cartprodmempair1
% 0.31/0.46 => ( cartprodmempair
% 0.31/0.46 => ( setunionE2
% 0.31/0.46 => ( setunionsingleton1
% 0.31/0.46 => ( setunionsingleton2
% 0.31/0.46 => ( setunionsingleton
% 0.31/0.46 => ( singletonprop
% 0.31/0.46 => ( ex1E1
% 0.31/0.46 => ( ex1I
% 0.31/0.46 => ( ex1I2
% 0.31/0.46 => ( singletonsuniq
% 0.31/0.46 => ( setukpairinjL1
% 0.31/0.46 => ( kfstsingleton
% 0.31/0.46 => ( theprop
% 0.31/0.46 => ( kfstpairEq
% 0.31/0.46 => ( cartprodfstin
% 0.31/0.46 => ( setukpairinjL2
% 0.31/0.46 => ( setukpairinjL
% 0.31/0.46 => ( setukpairinjR11
% 0.31/0.46 => ( setukpairinjR12
% 0.31/0.46 => ( setukpairinjR1
% 0.31/0.46 => ( upairequniteq
% 0.31/0.46 => ( setukpairinjR2
% 0.31/0.46 => ( setukpairinjR
% 0.31/0.46 => ( ksndsingleton
% 0.31/0.46 => ( ksndpairEq
% 0.31/0.46 => ( kpairsurjEq
% 0.31/0.46 => ( cartprodsndin
% 0.31/0.46 => ( cartprodpairmemEL
% 0.31/0.46 => ( cartprodpairmemER
% 0.31/0.46 => ( cartprodmempaircEq
% 0.31/0.46 => ( cartprodfstpairEq
% 0.31/0.46 => ( cartprodsndpairEq
% 0.31/0.46 => ( cartprodpairsurjEq
% 0.31/0.46 => ( dpsetconstrI
% 0.31/0.46 => ( dpsetconstrSub
% 0.31/0.46 => ( setOfPairsIsBReln
% 0.31/0.46 => ( dpsetconstrERa
% 0.31/0.46 => ( dpsetconstrEL1
% 0.31/0.46 => ( dpsetconstrEL2
% 0.31/0.46 => ( dpsetconstrER
% 0.31/0.46 => ( funcImageSingleton
% 0.31/0.46 => ( apProp
% 0.31/0.46 => ( app
% 0.31/0.46 => ( infuncsetfunc
% 0.31/0.46 => ( ap2p
% 0.31/0.46 => ( funcinfuncset
% 0.31/0.46 => ( lamProp
% 0.31/0.46 => ( lamp
% 0.31/0.46 => ( lam2p
% 0.31/0.46 => ( brelnall1
% 0.31/0.46 => ( brelnall2
% 0.31/0.46 => ( ex1E2
% 0.31/0.46 => ( funcGraphProp1
% 0.31/0.46 => ( funcGraphProp3
% 0.31/0.46 => ( funcGraphProp2
% 0.31/0.46 => ( funcextLem
% 0.31/0.46 => ( funcGraphProp4
% 0.31/0.46 => ( subbreln
% 0.31/0.46 => ( eqbreln
% 0.31/0.46 => ( funcext
% 0.31/0.46 => ( funcext2
% 0.31/0.46 => ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.46 => ! [Xx: $i] :
% 0.31/0.46 ( ( in @ Xx @ A )
% 0.31/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.46
% 0.31/0.46 %------------------------------------------------------------------------------
% 0.31/0.46 ------------------------
% 2.38/1.46 ---- Embedded in THF ---
% 2.38/1.46 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.38/1.46 %%% Generated on Tue Feb 24 12:30:56 EST 2026
% 2.38/1.46 %%% using '$$dhol' embedding, version 1.3.0.
% 2.38/1.46 %%% Logic specification used:
% 2.38/1.46 %%% thf(spec, logic, $$dhol).
% 2.38/1.46
% 2.38/1.46 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.7DY03sHzZu/DTF2DTF23302.p
% See solution above
% 2.38/1.46 ------------------------
% 2.57/1.51 ---- Cleaned THF ---
% 2.57/1.51 thf(in_type,type,
% 2.57/1.51 in: $i > $i > $o ).
% 2.57/1.51
% 2.57/1.51 thf(exu_type,type,
% 2.57/1.51 exu: ( $i > $o ) > $o ).
% 2.57/1.51
% 2.57/1.51 thf(exu,definition,
% 2.57/1.51 ( exu
% 2.57/1.51 = ( ^ [Xphi: $i > $o] :
% 2.57/1.51 ? [Xx: $i] :
% 2.57/1.51 ( ( Xphi @ Xx )
% 2.57/1.51 & ! [Xy: $i] :
% 2.57/1.51 ( ( Xphi @ Xy )
% 2.57/1.51 => ( Xx = Xy ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setextAx_type,type,
% 2.57/1.51 setextAx: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setextAx,definition,
% 2.57/1.51 ( setextAx
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 <=> ( in @ Xx @ B ) )
% 2.57/1.51 => ( A = B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(emptyset_type,type,
% 2.57/1.51 emptyset: $i ).
% 2.57/1.51
% 2.57/1.51 thf(emptysetAx_type,type,
% 2.57/1.51 emptysetAx: $o ).
% 2.57/1.51
% 2.57/1.51 thf(emptysetAx,definition,
% 2.57/1.51 ( emptysetAx
% 2.57/1.51 = ( ! [Xx: $i] :
% 2.57/1.51 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoin_type,type,
% 2.57/1.51 setadjoin: $i > $i > $i ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoinAx_type,type,
% 2.57/1.51 setadjoinAx: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoinAx,definition,
% 2.57/1.51 ( setadjoinAx
% 2.57/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.57/1.51 <=> ( ( Xy = Xx )
% 2.57/1.51 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(powerset_type,type,
% 2.57/1.51 powerset: $i > $i ).
% 2.57/1.51
% 2.57/1.51 thf(powersetAx_type,type,
% 2.57/1.51 powersetAx: $o ).
% 2.57/1.51
% 2.57/1.51 thf(powersetAx,definition,
% 2.57/1.51 ( powersetAx
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.57/1.51 <=> ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ B )
% 2.57/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setunion_type,type,
% 2.57/1.51 setunion: $i > $i ).
% 2.57/1.51
% 2.57/1.51 thf(setunionAx_type,type,
% 2.57/1.51 setunionAx: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setunionAx,definition,
% 2.57/1.51 ( setunionAx
% 2.57/1.51 = ( ! [A: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.57/1.51 <=> ? [B: $i] :
% 2.57/1.51 ( ( in @ Xx @ B )
% 2.57/1.51 & ( in @ B @ A ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(omega_type,type,
% 2.57/1.51 omega: $i ).
% 2.57/1.51
% 2.57/1.51 thf(omega0Ax_type,type,
% 2.57/1.51 omega0Ax: $o ).
% 2.57/1.51
% 2.57/1.51 thf(omega0Ax,definition,
% 2.57/1.51 ( omega0Ax
% 2.57/1.51 = ( in @ emptyset @ omega ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(omegaSAx_type,type,
% 2.57/1.51 omegaSAx: $o ).
% 2.57/1.51
% 2.57/1.51 thf(omegaSAx,definition,
% 2.57/1.51 ( omegaSAx
% 2.57/1.51 = ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ omega )
% 2.57/1.51 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(omegaIndAx_type,type,
% 2.57/1.51 omegaIndAx: $o ).
% 2.57/1.51
% 2.57/1.51 thf(omegaIndAx,definition,
% 2.57/1.51 ( omegaIndAx
% 2.57/1.51 = ( ! [A: $i] :
% 2.57/1.51 ( ( ( in @ emptyset @ A )
% 2.57/1.51 & ! [Xx: $i] :
% 2.57/1.51 ( ( ( in @ Xx @ omega )
% 2.57/1.51 & ( in @ Xx @ A ) )
% 2.57/1.51 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.57/1.51 => ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ omega )
% 2.57/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(replAx_type,type,
% 2.57/1.51 replAx: $o ).
% 2.57/1.51
% 2.57/1.51 thf(replAx,definition,
% 2.57/1.51 ( replAx
% 2.57/1.51 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( exu
% 2.57/1.51 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.57/1.51 => ? [B: $i] :
% 2.57/1.51 ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ B )
% 2.57/1.51 <=> ? [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ A )
% 2.57/1.51 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(foundationAx_type,type,
% 2.57/1.51 foundationAx: $o ).
% 2.57/1.51
% 2.57/1.51 thf(foundationAx,definition,
% 2.57/1.51 ( foundationAx
% 2.57/1.51 = ( ! [A: $i] :
% 2.57/1.51 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.57/1.51 => ? [B: $i] :
% 2.57/1.51 ( ( in @ B @ A )
% 2.57/1.51 & ~ ? [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ B )
% 2.57/1.51 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(wellorderingAx_type,type,
% 2.57/1.51 wellorderingAx: $o ).
% 2.57/1.51
% 2.57/1.51 thf(wellorderingAx,definition,
% 2.57/1.51 ( wellorderingAx
% 2.57/1.51 = ( ! [A: $i] :
% 2.57/1.51 ? [B: $i] :
% 2.57/1.51 ( ! [C: $i] :
% 2.57/1.51 ( ( in @ C @ B )
% 2.57/1.51 => ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ C )
% 2.57/1.51 => ( in @ Xx @ A ) ) )
% 2.57/1.51 & ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( ( in @ Xx @ A )
% 2.57/1.51 & ( in @ Xy @ A ) )
% 2.57/1.51 => ( ! [C: $i] :
% 2.57/1.51 ( ( in @ C @ B )
% 2.57/1.51 => ( ( in @ Xx @ C )
% 2.57/1.51 <=> ( in @ Xy @ C ) ) )
% 2.57/1.51 => ( Xx = Xy ) ) )
% 2.57/1.51 & ! [C: $i,D: $i] :
% 2.57/1.51 ( ( ( in @ C @ B )
% 2.57/1.51 & ( in @ D @ B ) )
% 2.57/1.51 => ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ C )
% 2.57/1.51 => ( in @ Xx @ D ) )
% 2.57/1.51 | ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ D )
% 2.57/1.51 => ( in @ Xx @ C ) ) ) )
% 2.57/1.51 & ! [C: $i] :
% 2.57/1.51 ( ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ C )
% 2.57/1.51 => ( in @ Xx @ A ) )
% 2.57/1.51 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.57/1.51 => ? [D: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ D @ B )
% 2.57/1.51 & ( in @ Xx @ C )
% 2.57/1.51 & ~ ? [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ D )
% 2.57/1.51 & ( in @ Xy @ C ) )
% 2.57/1.51 & ! [E: $i] :
% 2.57/1.51 ( ( in @ E @ B )
% 2.57/1.51 => ( ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ E )
% 2.57/1.51 => ( in @ Xy @ D ) )
% 2.57/1.51 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(descr_type,type,
% 2.57/1.51 descr: ( $i > $o ) > $i ).
% 2.57/1.51
% 2.57/1.51 thf(descrp_type,type,
% 2.57/1.51 descrp: $o ).
% 2.57/1.51
% 2.57/1.51 thf(descrp,definition,
% 2.57/1.51 ( descrp
% 2.57/1.51 = ( ! [Xphi: $i > $o] :
% 2.57/1.51 ( ( exu
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.57/1.51 => ( Xphi
% 2.57/1.51 @ ( descr
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(dsetconstr_type,type,
% 2.57/1.51 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.57/1.51
% 2.57/1.51 thf(dsetconstrI_type,type,
% 2.57/1.51 dsetconstrI: $o ).
% 2.57/1.51
% 2.57/1.51 thf(dsetconstrI,definition,
% 2.57/1.51 ( dsetconstrI
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( ( Xphi @ Xx )
% 2.57/1.51 => ( in @ Xx
% 2.57/1.51 @ ( dsetconstr @ A
% 2.57/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(dsetconstrEL_type,type,
% 2.57/1.51 dsetconstrEL: $o ).
% 2.57/1.51
% 2.57/1.51 thf(dsetconstrEL,definition,
% 2.57/1.51 ( dsetconstrEL
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx
% 2.57/1.51 @ ( dsetconstr @ A
% 2.57/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.57/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(dsetconstrER_type,type,
% 2.57/1.51 dsetconstrER: $o ).
% 2.57/1.51
% 2.57/1.51 thf(dsetconstrER,definition,
% 2.57/1.51 ( dsetconstrER
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx
% 2.57/1.51 @ ( dsetconstr @ A
% 2.57/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.57/1.51 => ( Xphi @ Xx ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(exuE1_type,type,
% 2.57/1.51 exuE1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(exuE1,definition,
% 2.57/1.51 ( exuE1
% 2.57/1.51 = ( ! [Xphi: $i > $o] :
% 2.57/1.51 ( ( exu
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.57/1.51 => ? [Xx: $i] :
% 2.57/1.51 ( ( Xphi @ Xx )
% 2.57/1.51 & ! [Xy: $i] :
% 2.57/1.51 ( ( Xphi @ Xy )
% 2.57/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(prop2set_type,type,
% 2.57/1.51 prop2set: $o > $i ).
% 2.57/1.51
% 2.57/1.51 thf(prop2setE_type,type,
% 2.57/1.51 prop2setE: $o ).
% 2.57/1.51
% 2.57/1.51 thf(prop2setE,definition,
% 2.57/1.51 ( prop2setE
% 2.57/1.51 = ( ! [Xphi: $o,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.57/1.51 => Xphi ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(emptysetE_type,type,
% 2.57/1.51 emptysetE: $o ).
% 2.57/1.51
% 2.57/1.51 thf(emptysetE,definition,
% 2.57/1.51 ( emptysetE
% 2.57/1.51 = ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ emptyset )
% 2.57/1.51 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(emptysetimpfalse_type,type,
% 2.57/1.51 emptysetimpfalse: $o ).
% 2.57/1.51
% 2.57/1.51 thf(emptysetimpfalse,definition,
% 2.57/1.51 ( emptysetimpfalse
% 2.57/1.51 = ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ emptyset )
% 2.57/1.51 => $false ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(notinemptyset_type,type,
% 2.57/1.51 notinemptyset: $o ).
% 2.57/1.51
% 2.57/1.51 thf(notinemptyset,definition,
% 2.57/1.51 ( notinemptyset
% 2.57/1.51 = ( ! [Xx: $i] :
% 2.57/1.51 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(exuE3e_type,type,
% 2.57/1.51 exuE3e: $o ).
% 2.57/1.51
% 2.57/1.51 thf(exuE3e,definition,
% 2.57/1.51 ( exuE3e
% 2.57/1.51 = ( ! [Xphi: $i > $o] :
% 2.57/1.51 ( ( exu
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.57/1.51 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setext_type,type,
% 2.57/1.51 setext: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setext,definition,
% 2.57/1.51 ( setext
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ Xx @ B ) )
% 2.57/1.51 => ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ B )
% 2.57/1.51 => ( in @ Xx @ A ) )
% 2.57/1.51 => ( A = B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(emptyI_type,type,
% 2.57/1.51 emptyI: $o ).
% 2.57/1.51
% 2.57/1.51 thf(emptyI,definition,
% 2.57/1.51 ( emptyI
% 2.57/1.51 = ( ! [A: $i] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ~ ( in @ Xx @ A )
% 2.57/1.51 => ( A = emptyset ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(noeltsimpempty_type,type,
% 2.57/1.51 noeltsimpempty: $o ).
% 2.57/1.51
% 2.57/1.51 thf(noeltsimpempty,definition,
% 2.57/1.51 ( noeltsimpempty
% 2.57/1.51 = ( ! [A: $i] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ~ ( in @ Xx @ A )
% 2.57/1.51 => ( A = emptyset ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setbeta_type,type,
% 2.57/1.51 setbeta: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setbeta,definition,
% 2.57/1.51 ( setbeta
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( ( in @ Xx
% 2.57/1.51 @ ( dsetconstr @ A
% 2.57/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.57/1.51 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(nonempty_type,type,
% 2.57/1.51 nonempty: $i > $o ).
% 2.57/1.51
% 2.57/1.51 thf(nonempty,definition,
% 2.57/1.51 ( nonempty
% 2.57/1.51 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(nonemptyE1_type,type,
% 2.57/1.51 nonemptyE1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(nonemptyE1,definition,
% 2.57/1.51 ( nonemptyE1
% 2.57/1.51 = ( ! [A: $i] :
% 2.57/1.51 ( ( nonempty @ A )
% 2.57/1.51 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(nonemptyI_type,type,
% 2.57/1.51 nonemptyI: $o ).
% 2.57/1.51
% 2.57/1.51 thf(nonemptyI,definition,
% 2.57/1.51 ( nonemptyI
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( ( Xphi @ Xx )
% 2.57/1.51 => ( nonempty
% 2.57/1.51 @ ( dsetconstr @ A
% 2.57/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(nonemptyI1_type,type,
% 2.57/1.51 nonemptyI1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(nonemptyI1,definition,
% 2.57/1.51 ( nonemptyI1
% 2.57/1.51 = ( ! [A: $i] :
% 2.57/1.51 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.57/1.51 => ( nonempty @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoinIL_type,type,
% 2.57/1.51 setadjoinIL: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoinIL,definition,
% 2.57/1.51 ( setadjoinIL
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(emptyinunitempty_type,type,
% 2.57/1.51 emptyinunitempty: $o ).
% 2.57/1.51
% 2.57/1.51 thf(emptyinunitempty,definition,
% 2.57/1.51 ( emptyinunitempty
% 2.57/1.51 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoinIR_type,type,
% 2.57/1.51 setadjoinIR: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoinIR,definition,
% 2.57/1.51 ( setadjoinIR
% 2.57/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ A )
% 2.57/1.51 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoinE_type,type,
% 2.57/1.51 setadjoinE: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoinE,definition,
% 2.57/1.51 ( setadjoinE
% 2.57/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.57/1.51 => ! [Xphi: $o] :
% 2.57/1.51 ( ( ( Xy = Xx )
% 2.57/1.51 => Xphi )
% 2.57/1.51 => ( ( ( in @ Xy @ A )
% 2.57/1.51 => Xphi )
% 2.57/1.51 => Xphi ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoinOr_type,type,
% 2.57/1.51 setadjoinOr: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoinOr,definition,
% 2.57/1.51 ( setadjoinOr
% 2.57/1.51 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.57/1.51 => ( ( Xy = Xx )
% 2.57/1.51 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setoftrueEq_type,type,
% 2.57/1.51 setoftrueEq: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setoftrueEq,definition,
% 2.57/1.51 ( setoftrueEq
% 2.57/1.51 = ( ! [A: $i] :
% 2.57/1.51 ( ( dsetconstr @ A
% 2.57/1.51 @ ^ [Xx: $i] : $true )
% 2.57/1.51 = A ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(powersetI_type,type,
% 2.57/1.51 powersetI: $o ).
% 2.57/1.51
% 2.57/1.51 thf(powersetI,definition,
% 2.57/1.51 ( powersetI
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ B )
% 2.57/1.51 => ( in @ Xx @ A ) )
% 2.57/1.51 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(emptyinPowerset_type,type,
% 2.57/1.51 emptyinPowerset: $o ).
% 2.57/1.51
% 2.57/1.51 thf(emptyinPowerset,definition,
% 2.57/1.51 ( emptyinPowerset
% 2.57/1.51 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(emptyInPowerset_type,type,
% 2.57/1.51 emptyInPowerset: $o ).
% 2.57/1.51
% 2.57/1.51 thf(emptyInPowerset,definition,
% 2.57/1.51 ( emptyInPowerset
% 2.57/1.51 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(powersetE_type,type,
% 2.57/1.51 powersetE: $o ).
% 2.57/1.51
% 2.57/1.51 thf(powersetE,definition,
% 2.57/1.51 ( powersetE
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.57/1.51 => ( ( in @ Xx @ B )
% 2.57/1.51 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setunionI_type,type,
% 2.57/1.51 setunionI: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setunionI,definition,
% 2.57/1.51 ( setunionI
% 2.57/1.51 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.57/1.51 ( ( in @ Xx @ B )
% 2.57/1.51 => ( ( in @ B @ A )
% 2.57/1.51 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setunionE_type,type,
% 2.57/1.51 setunionE: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setunionE,definition,
% 2.57/1.51 ( setunionE
% 2.57/1.51 = ( ! [A: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.57/1.51 => ! [Xphi: $o] :
% 2.57/1.51 ( ! [B: $i] :
% 2.57/1.51 ( ( in @ Xx @ B )
% 2.57/1.51 => ( ( in @ B @ A )
% 2.57/1.51 => Xphi ) )
% 2.57/1.51 => Xphi ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(subPowSU_type,type,
% 2.57/1.51 subPowSU: $o ).
% 2.57/1.51
% 2.57/1.51 thf(subPowSU,definition,
% 2.57/1.51 ( subPowSU
% 2.57/1.51 = ( ! [A: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(exuE2_type,type,
% 2.57/1.51 exuE2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(exuE2,definition,
% 2.57/1.51 ( exuE2
% 2.57/1.51 = ( ! [Xphi: $i > $o] :
% 2.57/1.51 ( ( exu
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.57/1.51 => ? [Xx: $i] :
% 2.57/1.51 ! [Xy: $i] :
% 2.57/1.51 ( ( Xphi @ Xy )
% 2.57/1.51 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(nonemptyImpWitness_type,type,
% 2.57/1.51 nonemptyImpWitness: $o ).
% 2.57/1.51
% 2.57/1.51 thf(nonemptyImpWitness,definition,
% 2.57/1.51 ( nonemptyImpWitness
% 2.57/1.51 = ( ! [A: $i] :
% 2.57/1.51 ( ( nonempty @ A )
% 2.57/1.51 => ? [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 & $true ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(uniqinunit_type,type,
% 2.57/1.51 uniqinunit: $o ).
% 2.57/1.51
% 2.57/1.51 thf(uniqinunit,definition,
% 2.57/1.51 ( uniqinunit
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.57/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(notinsingleton_type,type,
% 2.57/1.51 notinsingleton: $o ).
% 2.57/1.51
% 2.57/1.51 thf(notinsingleton,definition,
% 2.57/1.51 ( notinsingleton
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( Xx != Xy )
% 2.57/1.51 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(eqinunit_type,type,
% 2.57/1.51 eqinunit: $o ).
% 2.57/1.51
% 2.57/1.51 thf(eqinunit,definition,
% 2.57/1.51 ( eqinunit
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( Xx = Xy )
% 2.57/1.51 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(singletonsswitch_type,type,
% 2.57/1.51 singletonsswitch: $o ).
% 2.57/1.51
% 2.57/1.51 thf(singletonsswitch,definition,
% 2.57/1.51 ( singletonsswitch
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.57/1.51 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(upairsetE_type,type,
% 2.57/1.51 upairsetE: $o ).
% 2.57/1.51
% 2.57/1.51 thf(upairsetE,definition,
% 2.57/1.51 ( upairsetE
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.57/1.51 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.57/1.51 => ( ( Xz = Xx )
% 2.57/1.51 | ( Xz = Xy ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(upairsetIL_type,type,
% 2.57/1.51 upairsetIL: $o ).
% 2.57/1.51
% 2.57/1.51 thf(upairsetIL,definition,
% 2.57/1.51 ( upairsetIL
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(upairsetIR_type,type,
% 2.57/1.51 upairsetIR: $o ).
% 2.57/1.51
% 2.57/1.51 thf(upairsetIR,definition,
% 2.57/1.51 ( upairsetIR
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(emptyE1_type,type,
% 2.57/1.51 emptyE1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(emptyE1,definition,
% 2.57/1.51 ( emptyE1
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.57/1.51 ( ? [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 & ( Xphi @ Xx ) )
% 2.57/1.51 => ( ( ( dsetconstr @ A
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.57/1.51 = emptyset )
% 2.57/1.51 => $false ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(vacuousDall_type,type,
% 2.57/1.51 vacuousDall: $o ).
% 2.57/1.51
% 2.57/1.51 thf(vacuousDall,definition,
% 2.57/1.51 ( vacuousDall
% 2.57/1.51 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ emptyset )
% 2.57/1.51 => ( Xphi @ Xx ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(quantDeMorgan1_type,type,
% 2.57/1.51 quantDeMorgan1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(quantDeMorgan1,definition,
% 2.57/1.51 ( quantDeMorgan1
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.57/1.51 ( ~ ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( Xphi @ Xx ) )
% 2.57/1.51 => ? [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(quantDeMorgan2_type,type,
% 2.57/1.51 quantDeMorgan2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(quantDeMorgan2,definition,
% 2.57/1.51 ( quantDeMorgan2
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ~ ( Xphi @ Xx ) )
% 2.57/1.51 => ~ ? [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(quantDeMorgan3_type,type,
% 2.57/1.51 quantDeMorgan3: $o ).
% 2.57/1.51
% 2.57/1.51 thf(quantDeMorgan3,definition,
% 2.57/1.51 ( quantDeMorgan3
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.57/1.51 ( ~ ? [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 & ( Xphi @ Xx ) )
% 2.57/1.51 => ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(quantDeMorgan4_type,type,
% 2.57/1.51 quantDeMorgan4: $o ).
% 2.57/1.51
% 2.57/1.51 thf(quantDeMorgan4,definition,
% 2.57/1.51 ( quantDeMorgan4
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.57/1.51 ( ? [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 & ~ ( Xphi @ Xx ) )
% 2.57/1.51 => ~ ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(prop2setI_type,type,
% 2.57/1.51 prop2setI: $o ).
% 2.57/1.51
% 2.57/1.51 thf(prop2setI,definition,
% 2.57/1.51 ( prop2setI
% 2.57/1.51 = ( ! [Xphi: $o] :
% 2.57/1.51 ( Xphi
% 2.57/1.51 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(set2prop_type,type,
% 2.57/1.51 set2prop: $i > $o ).
% 2.57/1.51
% 2.57/1.51 thf(prop2set2propI_type,type,
% 2.57/1.51 prop2set2propI: $o ).
% 2.57/1.51
% 2.57/1.51 thf(prop2set2propI,definition,
% 2.57/1.51 ( prop2set2propI
% 2.57/1.51 = ( ! [Xphi: $o] :
% 2.57/1.51 ( Xphi
% 2.57/1.51 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(notdexE_type,type,
% 2.57/1.51 notdexE: $o ).
% 2.57/1.51
% 2.57/1.51 thf(notdexE,definition,
% 2.57/1.51 ( notdexE
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.57/1.51 ( ~ ? [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 & ( Xphi @ Xx ) )
% 2.57/1.51 => ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(notdallE_type,type,
% 2.57/1.51 notdallE: $o ).
% 2.57/1.51
% 2.57/1.51 thf(notdallE,definition,
% 2.57/1.51 ( notdallE
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.57/1.51 ( ~ ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( Xphi @ Xx ) )
% 2.57/1.51 => ? [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(exuI1_type,type,
% 2.57/1.51 exuI1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(exuI1,definition,
% 2.57/1.51 ( exuI1
% 2.57/1.51 = ( ! [Xphi: $i > $o] :
% 2.57/1.51 ( ? [Xx: $i] :
% 2.57/1.51 ( ( Xphi @ Xx )
% 2.57/1.51 & ! [Xy: $i] :
% 2.57/1.51 ( ( Xphi @ Xy )
% 2.57/1.51 => ( Xx = Xy ) ) )
% 2.57/1.51 => ( exu
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(exuI3_type,type,
% 2.57/1.51 exuI3: $o ).
% 2.57/1.51
% 2.57/1.51 thf(exuI3,definition,
% 2.57/1.51 ( exuI3
% 2.57/1.51 = ( ! [Xphi: $i > $o] :
% 2.57/1.51 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.57/1.51 => ( ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( Xphi @ Xx )
% 2.57/1.51 => ( ( Xphi @ Xy )
% 2.57/1.51 => ( Xx = Xy ) ) )
% 2.57/1.51 => ( exu
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(exuI2_type,type,
% 2.57/1.51 exuI2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(exuI2,definition,
% 2.57/1.51 ( exuI2
% 2.57/1.51 = ( ! [Xphi: $i > $o] :
% 2.57/1.51 ( ? [Xx: $i] :
% 2.57/1.51 ! [Xy: $i] :
% 2.57/1.51 ( ( Xphi @ Xy )
% 2.57/1.51 <=> ( Xy = Xx ) )
% 2.57/1.51 => ( exu
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(inCongP_type,type,
% 2.57/1.51 inCongP: $o ).
% 2.57/1.51
% 2.57/1.51 thf(inCongP,definition,
% 2.57/1.51 ( inCongP
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( A = B )
% 2.57/1.51 => ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( Xx = Xy )
% 2.57/1.51 => ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(in__Cong_type,type,
% 2.57/1.51 in__Cong: $o ).
% 2.57/1.51
% 2.57/1.51 thf(in__Cong,definition,
% 2.57/1.51 ( in__Cong
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( A = B )
% 2.57/1.51 => ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( Xx = Xy )
% 2.57/1.51 => ( ( in @ Xx @ A )
% 2.57/1.51 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(exuE3u_type,type,
% 2.57/1.51 exuE3u: $o ).
% 2.57/1.51
% 2.57/1.51 thf(exuE3u,definition,
% 2.57/1.51 ( exuE3u
% 2.57/1.51 = ( ! [Xphi: $i > $o] :
% 2.57/1.51 ( ( exu
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.57/1.51 => ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( Xphi @ Xx )
% 2.57/1.51 => ( ( Xphi @ Xy )
% 2.57/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(exu__Cong_type,type,
% 2.57/1.51 exu__Cong: $o ).
% 2.57/1.51
% 2.57/1.51 thf(exu__Cong,definition,
% 2.57/1.51 ( exu__Cong
% 2.57/1.51 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.57/1.51 ( ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( Xx = Xy )
% 2.57/1.51 => ( ( Xphi @ Xx )
% 2.57/1.51 <=> ( Xpsi @ Xy ) ) )
% 2.57/1.51 => ( ( exu
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.57/1.51 <=> ( exu
% 2.57/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(emptyset__Cong_type,type,
% 2.57/1.51 emptyset__Cong: $o ).
% 2.57/1.51
% 2.57/1.51 thf(emptyset__Cong,definition,
% 2.57/1.51 ( emptyset__Cong
% 2.57/1.51 = ( emptyset = emptyset ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoin__Cong_type,type,
% 2.57/1.51 setadjoin__Cong: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoin__Cong,definition,
% 2.57/1.51 ( setadjoin__Cong
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( Xx = Xy )
% 2.57/1.51 => ! [Xz: $i,Xu: $i] :
% 2.57/1.51 ( ( Xz = Xu )
% 2.57/1.51 => ( ( setadjoin @ Xx @ Xz )
% 2.57/1.51 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(powerset__Cong_type,type,
% 2.57/1.51 powerset__Cong: $o ).
% 2.57/1.51
% 2.57/1.51 thf(powerset__Cong,definition,
% 2.57/1.51 ( powerset__Cong
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( A = B )
% 2.57/1.51 => ( ( powerset @ A )
% 2.57/1.51 = ( powerset @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setunion__Cong_type,type,
% 2.57/1.51 setunion__Cong: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setunion__Cong,definition,
% 2.57/1.51 ( setunion__Cong
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( A = B )
% 2.57/1.51 => ( ( setunion @ A )
% 2.57/1.51 = ( setunion @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(omega__Cong_type,type,
% 2.57/1.51 omega__Cong: $o ).
% 2.57/1.51
% 2.57/1.51 thf(omega__Cong,definition,
% 2.57/1.51 ( omega__Cong
% 2.57/1.51 = ( omega = omega ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(exuEu_type,type,
% 2.57/1.51 exuEu: $o ).
% 2.57/1.51
% 2.57/1.51 thf(exuEu,definition,
% 2.57/1.51 ( exuEu
% 2.57/1.51 = ( ! [Xphi: $i > $o] :
% 2.57/1.51 ( ( exu
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.57/1.51 => ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( Xphi @ Xx )
% 2.57/1.51 => ( ( Xphi @ Xy )
% 2.57/1.51 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(descr__Cong_type,type,
% 2.57/1.51 descr__Cong: $o ).
% 2.57/1.51
% 2.57/1.51 thf(descr__Cong,definition,
% 2.57/1.51 ( descr__Cong
% 2.57/1.51 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.57/1.51 ( ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( Xx = Xy )
% 2.57/1.51 => ( ( Xphi @ Xx )
% 2.57/1.51 <=> ( Xpsi @ Xy ) ) )
% 2.57/1.51 => ( ( exu
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.57/1.51 => ( ( exu
% 2.57/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.57/1.51 => ( ( descr
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.57/1.51 = ( descr
% 2.57/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(dsetconstr__Cong_type,type,
% 2.57/1.51 dsetconstr__Cong: $o ).
% 2.57/1.51
% 2.57/1.51 thf(dsetconstr__Cong,definition,
% 2.57/1.51 ( dsetconstr__Cong
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( A = B )
% 2.57/1.51 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( ( Xx = Xy )
% 2.57/1.51 => ( ( Xphi @ Xx )
% 2.57/1.51 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.57/1.51 => ( ( dsetconstr @ A
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.57/1.51 = ( dsetconstr @ B
% 2.57/1.51 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(subset_type,type,
% 2.57/1.51 subset: $i > $i > $o ).
% 2.57/1.51
% 2.57/1.51 thf(disjoint_type,type,
% 2.57/1.51 disjoint: $i > $i > $o ).
% 2.57/1.51
% 2.57/1.51 thf(setsmeet_type,type,
% 2.57/1.51 setsmeet: $i > $i > $o ).
% 2.57/1.51
% 2.57/1.51 thf(subsetI1_type,type,
% 2.57/1.51 subsetI1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(subsetI1,definition,
% 2.57/1.51 ( subsetI1
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ Xx @ B ) )
% 2.57/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(eqimpsubset2_type,type,
% 2.57/1.51 eqimpsubset2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(eqimpsubset2,definition,
% 2.57/1.51 ( eqimpsubset2
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( A = B )
% 2.57/1.51 => ( subset @ B @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(eqimpsubset1_type,type,
% 2.57/1.51 eqimpsubset1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(eqimpsubset1,definition,
% 2.57/1.51 ( eqimpsubset1
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( A = B )
% 2.57/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(subsetI2_type,type,
% 2.57/1.51 subsetI2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(subsetI2,definition,
% 2.57/1.51 ( subsetI2
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ Xx @ B ) )
% 2.57/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(emptysetsubset_type,type,
% 2.57/1.51 emptysetsubset: $o ).
% 2.57/1.51
% 2.57/1.51 thf(emptysetsubset,definition,
% 2.57/1.51 ( emptysetsubset
% 2.57/1.51 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(subsetE_type,type,
% 2.57/1.51 subsetE: $o ).
% 2.57/1.51
% 2.57/1.51 thf(subsetE,definition,
% 2.57/1.51 ( subsetE
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( subset @ A @ B )
% 2.57/1.51 => ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(subsetE2_type,type,
% 2.57/1.51 subsetE2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(subsetE2,definition,
% 2.57/1.51 ( subsetE2
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( subset @ A @ B )
% 2.57/1.51 => ( ~ ( in @ Xx @ B )
% 2.57/1.51 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(notsubsetI_type,type,
% 2.57/1.51 notsubsetI: $o ).
% 2.57/1.51
% 2.57/1.51 thf(notsubsetI,definition,
% 2.57/1.51 ( notsubsetI
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( ~ ( in @ Xx @ B )
% 2.57/1.51 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(notequalI1_type,type,
% 2.57/1.51 notequalI1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(notequalI1,definition,
% 2.57/1.51 ( notequalI1
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ~ ( subset @ A @ B )
% 2.57/1.51 => ( A != B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(notequalI2_type,type,
% 2.57/1.51 notequalI2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(notequalI2,definition,
% 2.57/1.51 ( notequalI2
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( ~ ( in @ Xx @ B )
% 2.57/1.51 => ( A != B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(subsetRefl_type,type,
% 2.57/1.51 subsetRefl: $o ).
% 2.57/1.51
% 2.57/1.51 thf(subsetRefl,definition,
% 2.57/1.51 ( subsetRefl
% 2.57/1.51 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(subsetTrans_type,type,
% 2.57/1.51 subsetTrans: $o ).
% 2.57/1.51
% 2.57/1.51 thf(subsetTrans,definition,
% 2.57/1.51 ( subsetTrans
% 2.57/1.51 = ( ! [A: $i,B: $i,C: $i] :
% 2.57/1.51 ( ( subset @ A @ B )
% 2.57/1.51 => ( ( subset @ B @ C )
% 2.57/1.51 => ( subset @ A @ C ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoinSub_type,type,
% 2.57/1.51 setadjoinSub: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoinSub,definition,
% 2.57/1.51 ( setadjoinSub
% 2.57/1.51 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoinSub2_type,type,
% 2.57/1.51 setadjoinSub2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setadjoinSub2,definition,
% 2.57/1.51 ( setadjoinSub2
% 2.57/1.51 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.57/1.51 ( ( subset @ A @ B )
% 2.57/1.51 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(subset2powerset_type,type,
% 2.57/1.51 subset2powerset: $o ).
% 2.57/1.51
% 2.57/1.51 thf(subset2powerset,definition,
% 2.57/1.51 ( subset2powerset
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( subset @ A @ B )
% 2.57/1.51 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setextsub_type,type,
% 2.57/1.51 setextsub: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setextsub,definition,
% 2.57/1.51 ( setextsub
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( subset @ A @ B )
% 2.57/1.51 => ( ( subset @ B @ A )
% 2.57/1.51 => ( A = B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(subsetemptysetimpeq_type,type,
% 2.57/1.51 subsetemptysetimpeq: $o ).
% 2.57/1.51
% 2.57/1.51 thf(subsetemptysetimpeq,definition,
% 2.57/1.51 ( subsetemptysetimpeq
% 2.57/1.51 = ( ! [A: $i] :
% 2.57/1.51 ( ( subset @ A @ emptyset )
% 2.57/1.51 => ( A = emptyset ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(powersetI1_type,type,
% 2.57/1.51 powersetI1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(powersetI1,definition,
% 2.57/1.51 ( powersetI1
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( subset @ B @ A )
% 2.57/1.51 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(powersetE1_type,type,
% 2.57/1.51 powersetE1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(powersetE1,definition,
% 2.57/1.51 ( powersetE1
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( in @ B @ ( powerset @ A ) )
% 2.57/1.51 => ( subset @ B @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(inPowerset_type,type,
% 2.57/1.51 inPowerset: $o ).
% 2.57/1.51
% 2.57/1.51 thf(inPowerset,definition,
% 2.57/1.51 ( inPowerset
% 2.57/1.51 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(powersetsubset_type,type,
% 2.57/1.51 powersetsubset: $o ).
% 2.57/1.51
% 2.57/1.51 thf(powersetsubset,definition,
% 2.57/1.51 ( powersetsubset
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( subset @ A @ B )
% 2.57/1.51 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(sepInPowerset_type,type,
% 2.57/1.51 sepInPowerset: $o ).
% 2.57/1.51
% 2.57/1.51 thf(sepInPowerset,definition,
% 2.57/1.51 ( sepInPowerset
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.57/1.51 ( in
% 2.57/1.51 @ ( dsetconstr @ A
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.57/1.51 @ ( powerset @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(sepSubset_type,type,
% 2.57/1.51 sepSubset: $o ).
% 2.57/1.51
% 2.57/1.51 thf(sepSubset,definition,
% 2.57/1.51 ( sepSubset
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.57/1.51 ( subset
% 2.57/1.51 @ ( dsetconstr @ A
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.57/1.51 @ A ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(binunion_type,type,
% 2.57/1.51 binunion: $i > $i > $i ).
% 2.57/1.51
% 2.57/1.51 thf(binunionIL_type,type,
% 2.57/1.51 binunionIL: $o ).
% 2.57/1.51
% 2.57/1.51 thf(binunionIL,definition,
% 2.57/1.51 ( binunionIL
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(upairset2IR_type,type,
% 2.57/1.51 upairset2IR: $o ).
% 2.57/1.51
% 2.57/1.51 thf(upairset2IR,definition,
% 2.57/1.51 ( upairset2IR
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(binunionIR_type,type,
% 2.57/1.51 binunionIR: $o ).
% 2.57/1.51
% 2.57/1.51 thf(binunionIR,definition,
% 2.57/1.51 ( binunionIR
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ B )
% 2.57/1.51 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(binunionEcases_type,type,
% 2.57/1.51 binunionEcases: $o ).
% 2.57/1.51
% 2.57/1.51 thf(binunionEcases,definition,
% 2.57/1.51 ( binunionEcases
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.57/1.51 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.57/1.51 => ( ( ( in @ Xx @ A )
% 2.57/1.51 => Xphi )
% 2.57/1.51 => ( ( ( in @ Xx @ B )
% 2.57/1.51 => Xphi )
% 2.57/1.51 => Xphi ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(binunionE_type,type,
% 2.57/1.51 binunionE: $o ).
% 2.57/1.51
% 2.57/1.51 thf(binunionE,definition,
% 2.57/1.51 ( binunionE
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.57/1.51 => ( ( in @ Xx @ A )
% 2.57/1.51 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(binunionLsub_type,type,
% 2.57/1.51 binunionLsub: $o ).
% 2.57/1.51
% 2.57/1.51 thf(binunionLsub,definition,
% 2.57/1.51 ( binunionLsub
% 2.57/1.51 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(binunionRsub_type,type,
% 2.57/1.51 binunionRsub: $o ).
% 2.57/1.51
% 2.57/1.51 thf(binunionRsub,definition,
% 2.57/1.51 ( binunionRsub
% 2.57/1.51 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(binintersect_type,type,
% 2.57/1.51 binintersect: $i > $i > $i ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectI_type,type,
% 2.57/1.51 binintersectI: $o ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectI,definition,
% 2.57/1.51 ( binintersectI
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( ( in @ Xx @ B )
% 2.57/1.51 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectSubset5_type,type,
% 2.57/1.51 binintersectSubset5: $o ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectSubset5,definition,
% 2.57/1.51 ( binintersectSubset5
% 2.57/1.51 = ( ! [A: $i,B: $i,C: $i] :
% 2.57/1.51 ( ( subset @ C @ A )
% 2.57/1.51 => ( ( subset @ C @ B )
% 2.57/1.51 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectEL_type,type,
% 2.57/1.51 binintersectEL: $o ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectEL,definition,
% 2.57/1.51 ( binintersectEL
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.57/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectLsub_type,type,
% 2.57/1.51 binintersectLsub: $o ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectLsub,definition,
% 2.57/1.51 ( binintersectLsub
% 2.57/1.51 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectSubset2_type,type,
% 2.57/1.51 binintersectSubset2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectSubset2,definition,
% 2.57/1.51 ( binintersectSubset2
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( subset @ A @ B )
% 2.57/1.51 => ( ( binintersect @ A @ B )
% 2.57/1.51 = A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectSubset3_type,type,
% 2.57/1.51 binintersectSubset3: $o ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectSubset3,definition,
% 2.57/1.51 ( binintersectSubset3
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( ( binintersect @ A @ B )
% 2.57/1.51 = B )
% 2.57/1.51 => ( subset @ B @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectER_type,type,
% 2.57/1.51 binintersectER: $o ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectER,definition,
% 2.57/1.51 ( binintersectER
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.57/1.51 => ( in @ Xx @ B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(disjointsetsI1_type,type,
% 2.57/1.51 disjointsetsI1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(disjointsetsI1,definition,
% 2.57/1.51 ( disjointsetsI1
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ~ ? [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 & ( in @ Xx @ B ) )
% 2.57/1.51 => ( ( binintersect @ A @ B )
% 2.57/1.51 = emptyset ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectRsub_type,type,
% 2.57/1.51 binintersectRsub: $o ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectRsub,definition,
% 2.57/1.51 ( binintersectRsub
% 2.57/1.51 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectSubset4_type,type,
% 2.57/1.51 binintersectSubset4: $o ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectSubset4,definition,
% 2.57/1.51 ( binintersectSubset4
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( subset @ B @ A )
% 2.57/1.51 => ( ( binintersect @ A @ B )
% 2.57/1.51 = B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectSubset1_type,type,
% 2.57/1.51 binintersectSubset1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(binintersectSubset1,definition,
% 2.57/1.51 ( binintersectSubset1
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( ( binintersect @ A @ B )
% 2.57/1.51 = A )
% 2.57/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(bs114d_type,type,
% 2.57/1.51 bs114d: $o ).
% 2.57/1.51
% 2.57/1.51 thf(bs114d,definition,
% 2.57/1.51 ( bs114d
% 2.57/1.51 = ( ! [A: $i,B: $i,C: $i] :
% 2.57/1.51 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.57/1.51 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(regular_type,type,
% 2.57/1.51 regular: $i > $o ).
% 2.57/1.51
% 2.57/1.51 thf(setminus_type,type,
% 2.57/1.51 setminus: $i > $i > $i ).
% 2.57/1.51
% 2.57/1.51 thf(setminusI_type,type,
% 2.57/1.51 setminusI: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setminusI,definition,
% 2.57/1.51 ( setminusI
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( ~ ( in @ Xx @ B )
% 2.57/1.51 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setminusEL_type,type,
% 2.57/1.51 setminusEL: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setminusEL,definition,
% 2.57/1.51 ( setminusEL
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.57/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setminusER_type,type,
% 2.57/1.51 setminusER: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setminusER,definition,
% 2.57/1.51 ( setminusER
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.57/1.51 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setminusSubset2_type,type,
% 2.57/1.51 setminusSubset2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setminusSubset2,definition,
% 2.57/1.51 ( setminusSubset2
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( subset @ A @ B )
% 2.57/1.51 => ( ( setminus @ A @ B )
% 2.57/1.51 = emptyset ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setminusERneg_type,type,
% 2.57/1.51 setminusERneg: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setminusERneg,definition,
% 2.57/1.51 ( setminusERneg
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.57/1.51 => ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setminusELneg_type,type,
% 2.57/1.51 setminusELneg: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setminusELneg,definition,
% 2.57/1.51 ( setminusELneg
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.57/1.51 => ( ~ ( in @ Xx @ B )
% 2.57/1.51 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setminusILneg_type,type,
% 2.57/1.51 setminusILneg: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setminusILneg,definition,
% 2.57/1.51 ( setminusILneg
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ~ ( in @ Xx @ A )
% 2.57/1.51 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setminusIRneg_type,type,
% 2.57/1.51 setminusIRneg: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setminusIRneg,definition,
% 2.57/1.51 ( setminusIRneg
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ B )
% 2.57/1.51 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setminusLsub_type,type,
% 2.57/1.51 setminusLsub: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setminusLsub,definition,
% 2.57/1.51 ( setminusLsub
% 2.57/1.51 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setminusSubset1_type,type,
% 2.57/1.51 setminusSubset1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setminusSubset1,definition,
% 2.57/1.51 ( setminusSubset1
% 2.57/1.51 = ( ! [A: $i,B: $i] :
% 2.57/1.51 ( ( ( setminus @ A @ B )
% 2.57/1.51 = emptyset )
% 2.57/1.51 => ( subset @ A @ B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(symdiff_type,type,
% 2.57/1.51 symdiff: $i > $i > $i ).
% 2.57/1.51
% 2.57/1.51 thf(symdiffE_type,type,
% 2.57/1.51 symdiffE: $o ).
% 2.57/1.51
% 2.57/1.51 thf(symdiffE,definition,
% 2.57/1.51 ( symdiffE
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.57/1.51 => ! [Xphi: $o] :
% 2.57/1.51 ( ( ( in @ Xx @ A )
% 2.57/1.51 => ( ~ ( in @ Xx @ B )
% 2.57/1.51 => Xphi ) )
% 2.57/1.51 => ( ( ~ ( in @ Xx @ A )
% 2.57/1.51 => ( ( in @ Xx @ B )
% 2.57/1.51 => Xphi ) )
% 2.57/1.51 => Xphi ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(symdiffI1_type,type,
% 2.57/1.51 symdiffI1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(symdiffI1,definition,
% 2.57/1.51 ( symdiffI1
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( ~ ( in @ Xx @ B )
% 2.57/1.51 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(symdiffI2_type,type,
% 2.57/1.51 symdiffI2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(symdiffI2,definition,
% 2.57/1.51 ( symdiffI2
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ~ ( in @ Xx @ A )
% 2.57/1.51 => ( ( in @ Xx @ B )
% 2.57/1.51 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(symdiffIneg1_type,type,
% 2.57/1.51 symdiffIneg1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(symdiffIneg1,definition,
% 2.57/1.51 ( symdiffIneg1
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( ( in @ Xx @ B )
% 2.57/1.51 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(symdiffIneg2_type,type,
% 2.57/1.51 symdiffIneg2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(symdiffIneg2,definition,
% 2.57/1.51 ( symdiffIneg2
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ~ ( in @ Xx @ A )
% 2.57/1.51 => ( ~ ( in @ Xx @ B )
% 2.57/1.51 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(iskpair_type,type,
% 2.57/1.51 iskpair: $i > $o ).
% 2.57/1.51
% 2.57/1.51 thf(secondinupair_type,type,
% 2.57/1.51 secondinupair: $o ).
% 2.57/1.51
% 2.57/1.51 thf(secondinupair,definition,
% 2.57/1.51 ( secondinupair
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairIL_type,type,
% 2.57/1.51 setukpairIL: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairIL,definition,
% 2.57/1.51 ( setukpairIL
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairIR_type,type,
% 2.57/1.51 setukpairIR: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairIR,definition,
% 2.57/1.51 ( setukpairIR
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(kpairiskpair_type,type,
% 2.57/1.51 kpairiskpair: $o ).
% 2.57/1.51
% 2.57/1.51 thf(kpairiskpair,definition,
% 2.57/1.51 ( kpairiskpair
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(kpair_type,type,
% 2.57/1.51 kpair: $i > $i > $i ).
% 2.57/1.51
% 2.57/1.51 thf(kpairp_type,type,
% 2.57/1.51 kpairp: $o ).
% 2.57/1.51
% 2.57/1.51 thf(kpairp,definition,
% 2.57/1.51 ( kpairp
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(cartprod_type,type,
% 2.57/1.51 cartprod: $i > $i > $i ).
% 2.57/1.51
% 2.57/1.51 thf(singletonsubset_type,type,
% 2.57/1.51 singletonsubset: $o ).
% 2.57/1.51
% 2.57/1.51 thf(singletonsubset,definition,
% 2.57/1.51 ( singletonsubset
% 2.57/1.51 = ( ! [A: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(singletoninpowerset_type,type,
% 2.57/1.51 singletoninpowerset: $o ).
% 2.57/1.51
% 2.57/1.51 thf(singletoninpowerset,definition,
% 2.57/1.51 ( singletoninpowerset
% 2.57/1.51 = ( ! [A: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(singletoninpowunion_type,type,
% 2.57/1.51 singletoninpowunion: $o ).
% 2.57/1.51
% 2.57/1.51 thf(singletoninpowunion,definition,
% 2.57/1.51 ( singletoninpowunion
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(upairset2E_type,type,
% 2.57/1.51 upairset2E: $o ).
% 2.57/1.51
% 2.57/1.51 thf(upairset2E,definition,
% 2.57/1.51 ( upairset2E
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.57/1.51 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.57/1.51 => ( ( Xz = Xx )
% 2.57/1.51 | ( Xz = Xy ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(upairsubunion_type,type,
% 2.57/1.51 upairsubunion: $o ).
% 2.57/1.51
% 2.57/1.51 thf(upairsubunion,definition,
% 2.57/1.51 ( upairsubunion
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(upairinpowunion_type,type,
% 2.57/1.51 upairinpowunion: $o ).
% 2.57/1.51
% 2.57/1.51 thf(upairinpowunion,definition,
% 2.57/1.51 ( upairinpowunion
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(ubforcartprodlem1_type,type,
% 2.57/1.51 ubforcartprodlem1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(ubforcartprodlem1,definition,
% 2.57/1.51 ( ubforcartprodlem1
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(ubforcartprodlem2_type,type,
% 2.57/1.51 ubforcartprodlem2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(ubforcartprodlem2,definition,
% 2.57/1.51 ( ubforcartprodlem2
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(ubforcartprodlem3_type,type,
% 2.57/1.51 ubforcartprodlem3: $o ).
% 2.57/1.51
% 2.57/1.51 thf(ubforcartprodlem3,definition,
% 2.57/1.51 ( ubforcartprodlem3
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodpairin_type,type,
% 2.57/1.51 cartprodpairin: $o ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodpairin,definition,
% 2.57/1.51 ( cartprodpairin
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodmempair1_type,type,
% 2.57/1.51 cartprodmempair1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodmempair1,definition,
% 2.57/1.51 ( cartprodmempair1
% 2.57/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.57/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.57/1.51 => ? [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 & ? [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 & ( Xu
% 2.57/1.51 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodmempair_type,type,
% 2.57/1.51 cartprodmempair: $o ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodmempair,definition,
% 2.57/1.51 ( cartprodmempair
% 2.57/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.57/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.57/1.51 => ( iskpair @ Xu ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setunionE2_type,type,
% 2.57/1.51 setunionE2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setunionE2,definition,
% 2.57/1.51 ( setunionE2
% 2.57/1.51 = ( ! [A: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ ( setunion @ A ) )
% 2.57/1.51 => ? [X: $i] :
% 2.57/1.51 ( ( in @ X @ A )
% 2.57/1.51 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setunionsingleton1_type,type,
% 2.57/1.51 setunionsingleton1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setunionsingleton1,definition,
% 2.57/1.51 ( setunionsingleton1
% 2.57/1.51 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setunionsingleton2_type,type,
% 2.57/1.51 setunionsingleton2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setunionsingleton2,definition,
% 2.57/1.51 ( setunionsingleton2
% 2.57/1.51 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setunionsingleton_type,type,
% 2.57/1.51 setunionsingleton: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setunionsingleton,definition,
% 2.57/1.51 ( setunionsingleton
% 2.57/1.51 = ( ! [Xx: $i] :
% 2.57/1.51 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.57/1.51 = Xx ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(singleton_type,type,
% 2.57/1.51 singleton: $i > $o ).
% 2.57/1.51
% 2.57/1.51 thf(singleton,definition,
% 2.57/1.51 ( singleton
% 2.57/1.51 = ( ^ [A: $i] :
% 2.57/1.51 ? [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 & ( A
% 2.57/1.51 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(singletonprop_type,type,
% 2.57/1.51 singletonprop: $o ).
% 2.57/1.51
% 2.57/1.51 thf(singletonprop,definition,
% 2.57/1.51 ( singletonprop
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ A )
% 2.57/1.51 => ( ( Xphi @ Xx )
% 2.57/1.51 => ( ( Xphi @ Xy )
% 2.57/1.51 => ( Xx = Xy ) ) ) ) )
% 2.57/1.51 => ( ? [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 & ( Xphi @ Xx ) )
% 2.57/1.51 => ( singleton
% 2.57/1.51 @ ( dsetconstr @ A
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(ex1_type,type,
% 2.57/1.51 ex1: $i > ( $i > $o ) > $o ).
% 2.57/1.51
% 2.57/1.51 thf(ex1,definition,
% 2.57/1.51 ( ex1
% 2.57/1.51 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.57/1.51 ( singleton
% 2.57/1.51 @ ( dsetconstr @ A
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(ex1E1_type,type,
% 2.57/1.51 ex1E1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(ex1E1,definition,
% 2.57/1.51 ( ex1E1
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.57/1.51 ( ( ex1 @ A
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.57/1.51 => ? [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(ex1I_type,type,
% 2.57/1.51 ex1I: $o ).
% 2.57/1.51
% 2.57/1.51 thf(ex1I,definition,
% 2.57/1.51 ( ex1I
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( ( Xphi @ Xx )
% 2.57/1.51 => ( ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ A )
% 2.57/1.51 => ( ( Xphi @ Xy )
% 2.57/1.51 => ( Xy = Xx ) ) )
% 2.57/1.51 => ( ex1 @ A
% 2.57/1.51 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(ex1I2_type,type,
% 2.57/1.51 ex1I2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(ex1I2,definition,
% 2.57/1.51 ( ex1I2
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ A )
% 2.57/1.51 => ( ( Xphi @ Xx )
% 2.57/1.51 => ( ( Xphi @ Xy )
% 2.57/1.51 => ( Xx = Xy ) ) ) ) )
% 2.57/1.51 => ( ? [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 & ( Xphi @ Xx ) )
% 2.57/1.51 => ( ex1 @ A
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(singletonsuniq_type,type,
% 2.57/1.51 singletonsuniq: $o ).
% 2.57/1.51
% 2.57/1.51 thf(singletonsuniq,definition,
% 2.57/1.51 ( singletonsuniq
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( ( setadjoin @ Xx @ emptyset )
% 2.57/1.51 = ( setadjoin @ Xy @ emptyset ) )
% 2.57/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(atmost1p_type,type,
% 2.57/1.51 atmost1p: $i > $o ).
% 2.57/1.51
% 2.57/1.51 thf(atleast2p_type,type,
% 2.57/1.51 atleast2p: $i > $o ).
% 2.57/1.51
% 2.57/1.51 thf(atmost2p_type,type,
% 2.57/1.51 atmost2p: $i > $o ).
% 2.57/1.51
% 2.57/1.51 thf(upairsetp_type,type,
% 2.57/1.51 upairsetp: $i > $o ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairinjL1_type,type,
% 2.57/1.51 setukpairinjL1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairinjL1,definition,
% 2.57/1.51 ( setukpairinjL1
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.57/1.51 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.57/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(kfstsingleton_type,type,
% 2.57/1.51 kfstsingleton: $o ).
% 2.57/1.51
% 2.57/1.51 thf(kfstsingleton,definition,
% 2.57/1.51 ( kfstsingleton
% 2.57/1.51 = ( ! [Xu: $i] :
% 2.57/1.51 ( ( iskpair @ Xu )
% 2.57/1.51 => ( singleton
% 2.57/1.51 @ ( dsetconstr @ ( setunion @ Xu )
% 2.57/1.51 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(theprop_type,type,
% 2.57/1.51 theprop: $o ).
% 2.57/1.51
% 2.57/1.51 thf(theprop,definition,
% 2.57/1.51 ( theprop
% 2.57/1.51 = ( ! [X: $i] :
% 2.57/1.51 ( ( singleton @ X )
% 2.57/1.51 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(kfst_type,type,
% 2.57/1.51 kfst: $i > $i ).
% 2.57/1.51
% 2.57/1.51 thf(kfstpairEq_type,type,
% 2.57/1.51 kfstpairEq: $o ).
% 2.57/1.51
% 2.57/1.51 thf(kfstpairEq,definition,
% 2.57/1.51 ( kfstpairEq
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.57/1.51 = Xx ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodfstin_type,type,
% 2.57/1.51 cartprodfstin: $o ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodfstin,definition,
% 2.57/1.51 ( cartprodfstin
% 2.57/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.57/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.57/1.51 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairinjL2_type,type,
% 2.57/1.51 setukpairinjL2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairinjL2,definition,
% 2.57/1.51 ( setukpairinjL2
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.57/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.57/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.57/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairinjL_type,type,
% 2.57/1.51 setukpairinjL: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairinjL,definition,
% 2.57/1.51 ( setukpairinjL
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.57/1.51 ( ( ( kpair @ Xx @ Xy )
% 2.57/1.51 = ( kpair @ Xz @ Xu ) )
% 2.57/1.51 => ( Xx = Xz ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairinjR11_type,type,
% 2.57/1.51 setukpairinjR11: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairinjR11,definition,
% 2.57/1.51 ( setukpairinjR11
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( Xx = Xy )
% 2.57/1.51 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.57/1.51 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairinjR12_type,type,
% 2.57/1.51 setukpairinjR12: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairinjR12,definition,
% 2.57/1.51 ( setukpairinjR12
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( Xx = Xy )
% 2.57/1.51 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.57/1.51 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairinjR1_type,type,
% 2.57/1.51 setukpairinjR1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairinjR1,definition,
% 2.57/1.51 ( setukpairinjR1
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.57/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.57/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.57/1.51 => ( ( Xz = Xu )
% 2.57/1.51 => ( Xy = Xu ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(upairequniteq_type,type,
% 2.57/1.51 upairequniteq: $o ).
% 2.57/1.51
% 2.57/1.51 thf(upairequniteq,definition,
% 2.57/1.51 ( upairequniteq
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.57/1.51 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.57/1.51 = ( setadjoin @ Xz @ emptyset ) )
% 2.57/1.51 => ( Xx = Xy ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairinjR2_type,type,
% 2.57/1.51 setukpairinjR2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairinjR2,definition,
% 2.57/1.51 ( setukpairinjR2
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.57/1.51 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.57/1.51 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.57/1.51 => ( Xy = Xu ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairinjR_type,type,
% 2.57/1.51 setukpairinjR: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setukpairinjR,definition,
% 2.57/1.51 ( setukpairinjR
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.57/1.51 ( ( ( kpair @ Xx @ Xy )
% 2.57/1.51 = ( kpair @ Xz @ Xu ) )
% 2.57/1.51 => ( Xy = Xu ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(ksndsingleton_type,type,
% 2.57/1.51 ksndsingleton: $o ).
% 2.57/1.51
% 2.57/1.51 thf(ksndsingleton,definition,
% 2.57/1.51 ( ksndsingleton
% 2.57/1.51 = ( ! [Xu: $i] :
% 2.57/1.51 ( ( iskpair @ Xu )
% 2.57/1.51 => ( singleton
% 2.57/1.51 @ ( dsetconstr @ ( setunion @ Xu )
% 2.57/1.51 @ ^ [Xx: $i] :
% 2.57/1.51 ( Xu
% 2.57/1.51 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(ksnd_type,type,
% 2.57/1.51 ksnd: $i > $i ).
% 2.57/1.51
% 2.57/1.51 thf(ksndpairEq_type,type,
% 2.57/1.51 ksndpairEq: $o ).
% 2.57/1.51
% 2.57/1.51 thf(ksndpairEq,definition,
% 2.57/1.51 ( ksndpairEq
% 2.57/1.51 = ( ! [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.57/1.51 = Xy ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(kpairsurjEq_type,type,
% 2.57/1.51 kpairsurjEq: $o ).
% 2.57/1.51
% 2.57/1.51 thf(kpairsurjEq,definition,
% 2.57/1.51 ( kpairsurjEq
% 2.57/1.51 = ( ! [Xu: $i] :
% 2.57/1.51 ( ( iskpair @ Xu )
% 2.57/1.51 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.57/1.51 = Xu ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodsndin_type,type,
% 2.57/1.51 cartprodsndin: $o ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodsndin,definition,
% 2.57/1.51 ( cartprodsndin
% 2.57/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.57/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.57/1.51 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodpairmemEL_type,type,
% 2.57/1.51 cartprodpairmemEL: $o ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodpairmemEL,definition,
% 2.57/1.51 ( cartprodpairmemEL
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.57/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodpairmemER_type,type,
% 2.57/1.51 cartprodpairmemER: $o ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodpairmemER,definition,
% 2.57/1.51 ( cartprodpairmemER
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.57/1.51 => ( in @ Xy @ B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodmempaircEq_type,type,
% 2.57/1.51 cartprodmempaircEq: $o ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodmempaircEq,definition,
% 2.57/1.51 ( cartprodmempaircEq
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( ( kpair @ Xx @ Xy )
% 2.57/1.51 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodfstpairEq_type,type,
% 2.57/1.51 cartprodfstpairEq: $o ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodfstpairEq,definition,
% 2.57/1.51 ( cartprodfstpairEq
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.57/1.51 = Xx ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodsndpairEq_type,type,
% 2.57/1.51 cartprodsndpairEq: $o ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodsndpairEq,definition,
% 2.57/1.51 ( cartprodsndpairEq
% 2.57/1.51 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.57/1.51 = Xy ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodpairsurjEq_type,type,
% 2.57/1.51 cartprodpairsurjEq: $o ).
% 2.57/1.51
% 2.57/1.51 thf(cartprodpairsurjEq,definition,
% 2.57/1.51 ( cartprodpairsurjEq
% 2.57/1.51 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.57/1.51 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.57/1.51 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.57/1.51 = Xu ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(breln_type,type,
% 2.57/1.51 breln: $i > $i > $i > $o ).
% 2.57/1.51
% 2.57/1.51 thf(breln,definition,
% 2.57/1.51 ( breln
% 2.57/1.51 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(dpsetconstr_type,type,
% 2.57/1.51 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.57/1.51
% 2.57/1.51 thf(dpsetconstrI_type,type,
% 2.57/1.51 dpsetconstrI: $o ).
% 2.57/1.51
% 2.57/1.51 thf(dpsetconstrI,definition,
% 2.57/1.51 ( dpsetconstrI
% 2.57/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( ( Xphi @ Xx @ Xy )
% 2.57/1.51 => ( in @ ( kpair @ Xx @ Xy )
% 2.57/1.51 @ ( dpsetconstr @ A @ B
% 2.57/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(dpsetconstrSub_type,type,
% 2.57/1.51 dpsetconstrSub: $o ).
% 2.57/1.51
% 2.57/1.51 thf(dpsetconstrSub,definition,
% 2.57/1.51 ( dpsetconstrSub
% 2.57/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.57/1.51 ( subset
% 2.57/1.51 @ ( dpsetconstr @ A @ B
% 2.57/1.51 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.57/1.51 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(setOfPairsIsBReln_type,type,
% 2.57/1.51 setOfPairsIsBReln: $o ).
% 2.57/1.51
% 2.57/1.51 thf(setOfPairsIsBReln,definition,
% 2.57/1.51 ( setOfPairsIsBReln
% 2.57/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.57/1.51 ( breln @ A @ B
% 2.57/1.51 @ ( dpsetconstr @ A @ B
% 2.57/1.51 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(dpsetconstrERa_type,type,
% 2.57/1.51 dpsetconstrERa: $o ).
% 2.57/1.51
% 2.57/1.51 thf(dpsetconstrERa,definition,
% 2.57/1.51 ( dpsetconstrERa
% 2.57/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.57/1.51 @ ( dpsetconstr @ A @ B
% 2.57/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.57/1.51 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(dpsetconstrEL1_type,type,
% 2.57/1.51 dpsetconstrEL1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(dpsetconstrEL1,definition,
% 2.57/1.51 ( dpsetconstrEL1
% 2.57/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 2.57/1.51 @ ( dpsetconstr @ A @ B
% 2.57/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.57/1.51 => ( in @ Xx @ A ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(dpsetconstrEL2_type,type,
% 2.57/1.51 dpsetconstrEL2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(dpsetconstrEL2,definition,
% 2.57/1.51 ( dpsetconstrEL2
% 2.57/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 2.57/1.51 @ ( dpsetconstr @ A @ B
% 2.57/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.57/1.51 => ( in @ Xy @ B ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(dpsetconstrER_type,type,
% 2.57/1.51 dpsetconstrER: $o ).
% 2.57/1.51
% 2.57/1.51 thf(dpsetconstrER,definition,
% 2.57/1.51 ( dpsetconstrER
% 2.57/1.51 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( in @ ( kpair @ Xx @ Xy )
% 2.57/1.51 @ ( dpsetconstr @ A @ B
% 2.57/1.51 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.57/1.51 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(func_type,type,
% 2.57/1.51 func: $i > $i > $i > $o ).
% 2.57/1.51
% 2.57/1.51 thf(funcSet_type,type,
% 2.57/1.51 funcSet: $i > $i > $i ).
% 2.57/1.51
% 2.57/1.51 thf(funcImageSingleton_type,type,
% 2.57/1.51 funcImageSingleton: $o ).
% 2.57/1.51
% 2.57/1.51 thf(funcImageSingleton,definition,
% 2.57/1.51 ( funcImageSingleton
% 2.57/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.57/1.51 ( ( func @ A @ B @ Xf )
% 2.57/1.51 => ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( singleton
% 2.57/1.51 @ ( dsetconstr @ B
% 2.57/1.51 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(apProp_type,type,
% 2.57/1.51 apProp: $o ).
% 2.57/1.51
% 2.57/1.51 thf(apProp,definition,
% 2.57/1.51 ( apProp
% 2.57/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.57/1.51 ( ( func @ A @ B @ Xf )
% 2.57/1.51 => ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( in
% 2.57/1.51 @ ( setunion
% 2.57/1.51 @ ( dsetconstr @ B
% 2.57/1.51 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.57/1.51 @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(ap_type,type,
% 2.57/1.51 ap: $i > $i > $i > $i > $i ).
% 2.57/1.51
% 2.57/1.51 thf(app_type,type,
% 2.57/1.51 app: $o ).
% 2.57/1.51
% 2.57/1.51 thf(app,definition,
% 2.57/1.51 ( app
% 2.57/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.57/1.51 ( ( func @ A @ B @ Xf )
% 2.57/1.51 => ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(infuncsetfunc_type,type,
% 2.57/1.51 infuncsetfunc: $o ).
% 2.57/1.51
% 2.57/1.51 thf(infuncsetfunc,definition,
% 2.57/1.51 ( infuncsetfunc
% 2.57/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.57/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.57/1.51 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(ap2p_type,type,
% 2.57/1.51 ap2p: $o ).
% 2.57/1.51
% 2.57/1.51 thf(ap2p,definition,
% 2.57/1.51 ( ap2p
% 2.57/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.57/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.57/1.51 => ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(funcinfuncset_type,type,
% 2.57/1.51 funcinfuncset: $o ).
% 2.57/1.51
% 2.57/1.51 thf(funcinfuncset,definition,
% 2.57/1.51 ( funcinfuncset
% 2.57/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.57/1.51 ( ( func @ A @ B @ Xf )
% 2.57/1.51 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(lamProp_type,type,
% 2.57/1.51 lamProp: $o ).
% 2.57/1.51
% 2.57/1.51 thf(lamProp,definition,
% 2.57/1.51 ( lamProp
% 2.57/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.57/1.51 => ( func @ A @ B
% 2.57/1.51 @ ( dpsetconstr @ A @ B
% 2.57/1.51 @ ^ [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( Xf @ Xx )
% 2.57/1.51 = Xy ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(lam_type,type,
% 2.57/1.51 lam: $i > $i > ( $i > $i ) > $i ).
% 2.57/1.51
% 2.57/1.51 thf(lam,definition,
% 2.57/1.51 ( lam
% 2.57/1.51 = ( ^ [A: $i,B: $i,Xf: $i > $i] :
% 2.57/1.51 ( dpsetconstr @ A @ B
% 2.57/1.51 @ ^ [Xx: $i,Xy: $i] :
% 2.57/1.51 ( ( Xf @ Xx )
% 2.57/1.51 = Xy ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(lamp_type,type,
% 2.57/1.51 lamp: $o ).
% 2.57/1.51
% 2.57/1.51 thf(lamp,definition,
% 2.57/1.51 ( lamp
% 2.57/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.57/1.51 => ( func @ A @ B
% 2.57/1.51 @ ( lam @ A @ B
% 2.57/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(lam2p_type,type,
% 2.57/1.51 lam2p: $o ).
% 2.57/1.51
% 2.57/1.51 thf(lam2p,definition,
% 2.57/1.51 ( lam2p
% 2.57/1.51 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.57/1.51 => ( in
% 2.57/1.51 @ ( lam @ A @ B
% 2.57/1.51 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.57/1.51 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(brelnall1_type,type,
% 2.57/1.51 brelnall1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(brelnall1,definition,
% 2.57/1.51 ( brelnall1
% 2.57/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.57/1.51 ( ( breln @ A @ B @ R )
% 2.57/1.51 => ! [Xphi: $i > $o] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.57/1.51 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.57/1.51 => ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ R )
% 2.57/1.51 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(brelnall2_type,type,
% 2.57/1.51 brelnall2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(brelnall2,definition,
% 2.57/1.51 ( brelnall2
% 2.57/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.57/1.51 ( ( breln @ A @ B @ R )
% 2.57/1.51 => ! [Xphi: $i > $o] :
% 2.57/1.51 ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.57/1.51 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.57/1.51 => ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ R )
% 2.57/1.51 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(ex1E2_type,type,
% 2.57/1.51 ex1E2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(ex1E2,definition,
% 2.57/1.51 ( ex1E2
% 2.57/1.51 = ( ! [A: $i,Xphi: $i > $o] :
% 2.57/1.51 ( ( ex1 @ A
% 2.57/1.51 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.57/1.51 => ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ A )
% 2.57/1.51 => ( ( Xphi @ Xx )
% 2.57/1.51 => ( ( Xphi @ Xy )
% 2.57/1.51 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(funcGraphProp1_type,type,
% 2.57/1.51 funcGraphProp1: $o ).
% 2.57/1.51
% 2.57/1.51 thf(funcGraphProp1,definition,
% 2.57/1.51 ( funcGraphProp1
% 2.57/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.57/1.51 ( ( func @ A @ B @ Xf )
% 2.57/1.51 => ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(funcGraphProp3_type,type,
% 2.57/1.51 funcGraphProp3: $o ).
% 2.57/1.51
% 2.57/1.51 thf(funcGraphProp3,definition,
% 2.57/1.51 ( funcGraphProp3
% 2.57/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.57/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.57/1.51 => ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(funcGraphProp2_type,type,
% 2.57/1.51 funcGraphProp2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(funcGraphProp2,definition,
% 2.57/1.51 ( funcGraphProp2
% 2.57/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.57/1.51 ( ( func @ A @ B @ Xf )
% 2.57/1.51 => ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.57/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.57/1.51 = Xy ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(funcextLem_type,type,
% 2.57/1.51 funcextLem: $o ).
% 2.57/1.51
% 2.57/1.51 thf(funcextLem,definition,
% 2.57/1.51 ( funcextLem
% 2.57/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.57/1.51 ( ( func @ A @ B @ Xf )
% 2.57/1.51 => ! [Xg: $i] :
% 2.57/1.51 ( ( func @ A @ B @ Xg )
% 2.57/1.51 => ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.57/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.57/1.51 => ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.57/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(funcGraphProp4_type,type,
% 2.57/1.51 funcGraphProp4: $o ).
% 2.57/1.51
% 2.57/1.51 thf(funcGraphProp4,definition,
% 2.57/1.51 ( funcGraphProp4
% 2.57/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.57/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.57/1.51 => ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.57/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.57/1.51 = Xy ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(subbreln_type,type,
% 2.57/1.51 subbreln: $o ).
% 2.57/1.51
% 2.57/1.51 thf(subbreln,definition,
% 2.57/1.51 ( subbreln
% 2.57/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.57/1.51 ( ( breln @ A @ B @ R )
% 2.57/1.51 => ! [S: $i] :
% 2.57/1.51 ( ( breln @ A @ B @ S )
% 2.57/1.51 => ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.57/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.57/1.51 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(eqbreln_type,type,
% 2.57/1.51 eqbreln: $o ).
% 2.57/1.51
% 2.57/1.51 thf(eqbreln,definition,
% 2.57/1.51 ( eqbreln
% 2.57/1.51 = ( ! [A: $i,B: $i,R: $i] :
% 2.57/1.51 ( ( breln @ A @ B @ R )
% 2.57/1.51 => ! [S: $i] :
% 2.57/1.51 ( ( breln @ A @ B @ S )
% 2.57/1.51 => ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.57/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.57/1.51 => ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ! [Xy: $i] :
% 2.57/1.51 ( ( in @ Xy @ B )
% 2.57/1.51 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.57/1.51 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.57/1.51 => ( R = S ) ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(funcext_type,type,
% 2.57/1.51 funcext: $o ).
% 2.57/1.51
% 2.57/1.51 thf(funcext,definition,
% 2.57/1.51 ( funcext
% 2.57/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.57/1.51 ( ( func @ A @ B @ Xf )
% 2.57/1.51 => ! [Xg: $i] :
% 2.57/1.51 ( ( func @ A @ B @ Xg )
% 2.57/1.51 => ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.57/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.57/1.51 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(funcext2_type,type,
% 2.57/1.51 funcext2: $o ).
% 2.57/1.51
% 2.57/1.51 thf(funcext2,definition,
% 2.57/1.51 ( funcext2
% 2.57/1.51 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.57/1.51 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.57/1.51 => ! [Xg: $i] :
% 2.57/1.51 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.57/1.51 => ( ! [Xx: $i] :
% 2.57/1.51 ( ( in @ Xx @ A )
% 2.57/1.51 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.57/1.51 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.57/1.51 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.57/1.51
% 2.57/1.51 thf(ap2apEq1,conjecture,
% 2.57/1.51 ( setextAx
% 2.57/1.51 => ( emptysetAx
% 2.57/1.51 => ( setadjoinAx
% 2.57/1.51 => ( powersetAx
% 2.57/1.51 => ( setunionAx
% 2.57/1.51 => ( omega0Ax
% 2.57/1.51 => ( omegaSAx
% 2.57/1.51 => ( omegaIndAx
% 2.57/1.51 => ( replAx
% 2.57/1.51 => ( foundationAx
% 2.57/1.51 => ( wellorderingAx
% 2.57/1.51 => ( descrp
% 2.57/1.51 => ( dsetconstrI
% 2.57/1.51 => ( dsetconstrEL
% 2.57/1.51 => ( dsetconstrER
% 2.57/1.51 => ( exuE1
% 2.57/1.51 => ( prop2setE
% 2.57/1.51 => ( emptysetE
% 2.57/1.51 => ( emptysetimpfalse
% 2.57/1.51 => ( notinemptyset
% 2.57/1.51 => ( exuE3e
% 2.57/1.51 => ( setext
% 2.57/1.51 => ( emptyI
% 2.57/1.51 => ( noeltsimpempty
% 2.57/1.51 => ( setbeta
% 2.57/1.51 => ( nonemptyE1
% 2.57/1.51 => ( nonemptyI
% 2.57/1.51 => ( nonemptyI1
% 2.57/1.51 => ( setadjoinIL
% 2.57/1.51 => ( emptyinunitempty
% 2.57/1.51 => ( setadjoinIR
% 2.57/1.51 => ( setadjoinE
% 2.57/1.51 => ( setadjoinOr
% 2.57/1.51 => ( setoftrueEq
% 2.57/1.51 => ( powersetI
% 2.57/1.51 => ( emptyinPowerset
% 2.57/1.51 => ( emptyInPowerset
% 2.57/1.51 => ( powersetE
% 2.57/1.51 => ( setunionI
% 2.57/1.51 => ( setunionE
% 2.57/1.51 => ( subPowSU
% 2.57/1.51 => ( exuE2
% 2.57/1.51 => ( nonemptyImpWitness
% 2.57/1.51 => ( uniqinunit
% 2.57/1.51 => ( notinsingleton
% 2.57/1.51 => ( eqinunit
% 2.57/1.51 => ( singletonsswitch
% 2.57/1.51 => ( upairsetE
% 2.57/1.51 => ( upairsetIL
% 2.57/1.51 => ( upairsetIR
% 2.57/1.51 => ( emptyE1
% 2.57/1.51 => ( vacuousDall
% 2.57/1.51 => ( quantDeMorgan1
% 2.57/1.51 => ( quantDeMorgan2
% 2.57/1.51 => ( quantDeMorgan3
% 2.57/1.51 => ( quantDeMorgan4
% 2.57/1.51 => ( prop2setI
% 2.57/1.51 => ( prop2set2propI
% 2.57/1.51 => ( notdexE
% 2.57/1.51 => ( notdallE
% 2.57/1.51 => ( exuI1
% 2.57/1.51 => ( exuI3
% 2.57/1.51 => ( exuI2
% 2.57/1.51 => ( inCongP
% 2.57/1.51 => ( in__Cong
% 2.57/1.51 => ( exuE3u
% 2.57/1.51 => ( exu__Cong
% 2.57/1.51 => ( emptyset__Cong
% 2.57/1.51 => ( setadjoin__Cong
% 2.57/1.51 => ( powerset__Cong
% 2.57/1.51 => ( setunion__Cong
% 2.57/1.51 => ( omega__Cong
% 2.57/1.51 => ( exuEu
% 2.57/1.51 => ( descr__Cong
% 2.57/1.51 => ( dsetconstr__Cong
% 2.57/1.51 => ( subsetI1
% 2.57/1.51 => ( eqimpsubset2
% 2.57/1.51 => ( eqimpsubset1
% 2.57/1.51 => ( subsetI2
% 2.57/1.51 => ( emptysetsubset
% 2.57/1.51 => ( subsetE
% 2.57/1.51 => ( subsetE2
% 2.57/1.51 => ( notsubsetI
% 2.57/1.51 => ( notequalI1
% 2.57/1.51 => ( notequalI2
% 2.57/1.51 => ( subsetRefl
% 2.57/1.51 => ( subsetTrans
% 2.57/1.51 => ( setadjoinSub
% 2.57/1.51 => ( setadjoinSub2
% 2.57/1.51 => ( subset2powerset
% 2.57/1.51 => ( setextsub
% 2.57/1.51 => ( subsetemptysetimpeq
% 2.57/1.51 => ( powersetI1
% 2.57/1.51 => ( powersetE1
% 2.57/1.51 => ( inPowerset
% 2.57/1.51 => ( powersetsubset
% 2.57/1.51 => ( sepInPowerset
% 2.57/1.51 => ( sepSubset
% 2.57/1.51 => ( binunionIL
% 2.57/1.51 => ( upairset2IR
% 2.57/1.51 => ( binunionIR
% 2.57/1.51 => ( binunionEcases
% 2.57/1.51 => ( binunionE
% 2.57/1.51 => ( binunionLsub
% 2.57/1.51 => ( binunionRsub
% 2.57/1.51 => ( binintersectI
% 2.57/1.51 => ( binintersectSubset5
% 2.57/1.51 => ( binintersectEL
% 2.57/1.51 => ( binintersectLsub
% 2.57/1.51 => ( binintersectSubset2
% 2.57/1.51 => ( binintersectSubset3
% 2.57/1.51 => ( binintersectER
% 2.57/1.51 => ( disjointsetsI1
% 2.57/1.51 => ( binintersectRsub
% 2.57/1.51 => ( binintersectSubset4
% 2.57/1.51 => ( binintersectSubset1
% 2.57/1.51 => ( bs114d
% 2.57/1.51 => ( setminusI
% 2.57/1.51 => ( setminusEL
% 2.57/1.51 => ( setminusER
% 2.57/1.51 => ( setminusSubset2
% 2.57/1.51 => ( setminusERneg
% 2.57/1.51 => ( setminusELneg
% 2.57/1.51 => ( setminusILneg
% 2.57/1.51 => ( setminusIRneg
% 2.57/1.51 => ( setminusLsub
% 2.57/1.51 => ( setminusSubset1
% 2.57/1.51 => ( symdiffE
% 2.57/1.51 => ( symdiffI1
% 2.57/1.51 => ( symdiffI2
% 2.57/1.51 => ( symdiffIneg1
% 2.57/1.51 => ( symdiffIneg2
% 2.57/1.51 => ( secondinupair
% 2.57/1.51 => ( setukpairIL
% 2.57/1.51 => ( setukpairIR
% 2.57/1.51 => ( kpairiskpair
% 2.57/1.51 => ( kpairp
% 2.57/1.51 => ( singletonsubset
% 2.57/1.51 => ( singletoninpowerset
% 2.57/1.51 => ( singletoninpowunion
% 2.57/1.51 => ( upairset2E
% 2.57/1.51 => ( upairsubunion
% 2.57/1.51 => ( upairinpowunion
% 2.57/1.51 => ( ubforcartprodlem1
% 2.57/1.51 => ( ubforcartprodlem2
% 2.57/1.51 => ( ubforcartprodlem3
% 2.57/1.51 => ( cartprodpairin
% 2.57/1.51 => ( cartprodmempair1
% 2.57/1.51 => ( cartprodmempair
% 2.57/1.51 => ( setunionE2
% 2.57/1.51 => ( setunionsingleton1
% 2.57/1.51 => ( setunionsingleton2
% 2.57/1.51 => ( setunionsingleton
% 2.57/1.51 => ( singletonprop
% 2.57/1.51 => ( ex1E1
% 2.57/1.51 => ( ex1I
% 2.57/1.51 => ( ex1I2
% 2.57/1.51 => ( singletonsuniq
% 2.57/1.51 => ( setukpairinjL1
% 2.57/1.51 => ( kfstsingleton
% 2.57/1.51 => ( theprop
% 2.57/1.51 => ( kfstpairEq
% 2.57/1.51 => ( cartprodfstin
% 2.57/1.51 => ( setukpairinjL2
% 2.57/1.52 => ( setukpairinjL
% 2.57/1.52 => ( setukpairinjR11
% 2.57/1.52 => ( setukpairinjR12
% 2.57/1.52 => ( setukpairinjR1
% 2.57/1.52 => ( upairequniteq
% 2.57/1.52 => ( setukpairinjR2
% 2.57/1.52 => ( setukpairinjR
% 2.57/1.52 => ( ksndsingleton
% 2.57/1.52 => ( ksndpairEq
% 2.57/1.52 => ( kpairsurjEq
% 2.57/1.52 => ( cartprodsndin
% 2.57/1.52 => ( cartprodpairmemEL
% 2.57/1.52 => ( cartprodpairmemER
% 2.57/1.52 => ( cartprodmempaircEq
% 2.57/1.52 => ( cartprodfstpairEq
% 2.57/1.52 => ( cartprodsndpairEq
% 2.57/1.52 => ( cartprodpairsurjEq
% 2.57/1.52 => ( dpsetconstrI
% 2.57/1.52 => ( dpsetconstrSub
% 2.57/1.52 => ( setOfPairsIsBReln
% 2.57/1.52 => ( dpsetconstrERa
% 2.57/1.52 => ( dpsetconstrEL1
% 2.57/1.52 => ( dpsetconstrEL2
% 2.57/1.52 => ( dpsetconstrER
% 2.57/1.52 => ( funcImageSingleton
% 2.57/1.52 => ( apProp
% 2.57/1.52 => ( app
% 2.57/1.52 => ( infuncsetfunc
% 2.57/1.52 => ( ap2p
% 2.57/1.52 => ( funcinfuncset
% 2.57/1.52 => ( lamProp
% 2.57/1.52 => ( lamp
% 2.57/1.52 => ( lam2p
% 2.57/1.52 => ( brelnall1
% 2.57/1.52 => ( brelnall2
% 2.57/1.52 => ( ex1E2
% 2.57/1.52 => ( funcGraphProp1
% 2.57/1.52 => ( funcGraphProp3
% 2.57/1.52 => ( funcGraphProp2
% 2.57/1.52 => ( funcextLem
% 2.57/1.52 => ( funcGraphProp4
% 2.57/1.52 => ( subbreln
% 2.57/1.52 => ( eqbreln
% 2.57/1.52 => ( funcext
% 2.57/1.52 => ( funcext2
% 2.57/1.52 => ! [A: $i,B: $i,Xf: $i] :
% 2.57/1.52 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.57/1.52 => ! [Xx: $i] :
% 2.57/1.52 ( ( in @ Xx @ A )
% 2.57/1.52 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.57/1.52 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.57/1.52 ------------------------
% 2.57/1.54 % (23576)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_23302 for (2999ds/27Mi)
% 2.57/1.54 % (23575)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_23302 for (2999ds/4Mi)
% 2.57/1.54 % (23575)Instruction limit reached!
% 2.57/1.54 % (23575)------------------------------
% 2.57/1.54 % (23575)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.54 % (23575)Termination reason: Unknown
% 2.57/1.54 % (23575)Termination phase: shuffling
% 2.57/1.54
% 2.57/1.54 % (23575)Memory used [KB]: 1407
% 2.57/1.54 % (23575)Time elapsed: 0.003 s
% 2.57/1.54 % (23575)Instructions burned: 5 (million)
% 2.57/1.54 % (23575)------------------------------
% 2.57/1.54 % (23575)------------------------------
% 2.57/1.54 % (23574)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_23302 for (2999ds/183Mi)
% 2.57/1.54 % (23577)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_23302 for (2999ds/2Mi)
% 2.57/1.54 % (23579)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_23302 for (2999ds/275Mi)
% 2.57/1.54 % (23580)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_23302 for (2999ds/18Mi)
% 2.57/1.54 % (23578)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_23302 for (2999ds/2Mi)
% 2.57/1.54 % (23576)Instruction limit reached!
% 2.57/1.54 % (23576)------------------------------
% 2.57/1.54 % (23576)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.54 % (23576)Termination reason: Unknown
% 2.57/1.54 % (23576)Termination phase: shuffling
% 2.57/1.54
% 2.57/1.54 % (23576)Memory used [KB]: 1918
% 2.57/1.54 % (23576)Time elapsed: 0.010 s
% 2.57/1.54 % (23576)Instructions burned: 29 (million)
% 2.57/1.54 % (23576)------------------------------
% 2.57/1.54 % (23576)------------------------------
% 2.57/1.54 % (23577)Instruction limit reached!
% 2.57/1.54 % (23577)------------------------------
% 2.57/1.54 % (23577)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.54 % (23577)Termination reason: Unknown
% 2.57/1.54 % (23577)Termination phase: shuffling
% 2.57/1.54
% 2.57/1.54 % (23577)Memory used [KB]: 1407
% 2.57/1.54 % (23577)Time elapsed: 0.004 s
% 2.57/1.54 % (23577)Instructions burned: 3 (million)
% 2.57/1.54 % (23577)------------------------------
% 2.57/1.54 % (23577)------------------------------
% 2.57/1.54 % (23578)Instruction limit reached!
% 2.57/1.54 % (23578)------------------------------
% 2.57/1.54 % (23578)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.54 % (23578)Termination reason: Unknown
% 2.57/1.54 % (23578)Termination phase: shuffling
% 2.57/1.54
% 2.57/1.54 % (23578)Memory used [KB]: 1407
% 2.57/1.54 % (23578)Time elapsed: 0.003 s
% 2.57/1.54 % (23578)Instructions burned: 3 (million)
% 2.57/1.54 % (23578)------------------------------
% 2.57/1.54 % (23578)------------------------------
% 2.57/1.55 % (23581)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_23302 for (2999ds/3Mi)
% 2.57/1.55 % (23581)Instruction limit reached!
% 2.57/1.55 % (23581)------------------------------
% 2.57/1.55 % (23581)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.55 % (23581)Termination reason: Unknown
% 2.57/1.55 % (23581)Termination phase: shuffling
% 2.57/1.55
% 2.57/1.55 % (23581)Memory used [KB]: 1407
% 2.57/1.55 % (23581)Time elapsed: 0.003 s
% 2.57/1.55 % (23581)Instructions burned: 5 (million)
% 2.57/1.55 % (23581)------------------------------
% 2.57/1.55 % (23581)------------------------------
% 2.57/1.55 % (23580)Instruction limit reached!
% 2.57/1.55 % (23580)------------------------------
% 2.57/1.55 % (23580)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.55 % (23580)Termination reason: Unknown
% 2.57/1.55 % (23580)Termination phase: shuffling
% 2.57/1.55
% 2.57/1.55 % (23580)Memory used [KB]: 1663
% 2.57/1.55 % (23580)Time elapsed: 0.013 s
% 2.57/1.55 % (23580)Instructions burned: 18 (million)
% 2.57/1.55 % (23580)------------------------------
% 2.57/1.55 % (23580)------------------------------
% 2.57/1.55 % (23582)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_23302 for (2999ds/37Mi)
% 2.57/1.56 % (23583)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_23302 for (2999ds/15Mi)
% 2.57/1.56 % (23584)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_23302 for (2999ds/3Mi)
% 2.57/1.56 % (23585)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_23302 for (2999ds/1041Mi)
% 2.57/1.56 % (23584)Instruction limit reached!
% 2.57/1.56 % (23584)------------------------------
% 2.57/1.56 % (23584)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.56 % (23584)Termination reason: Unknown
% 2.57/1.56 % (23584)Termination phase: shuffling
% 2.57/1.56
% 2.57/1.56 % (23584)Memory used [KB]: 1407
% 2.57/1.56 % (23584)Time elapsed: 0.004 s
% 2.57/1.56 % (23584)Instructions burned: 3 (million)
% 2.57/1.56 % (23584)------------------------------
% 2.57/1.56 % (23584)------------------------------
% 2.57/1.57 % (23582)Instruction limit reached!
% 2.57/1.57 % (23582)------------------------------
% 2.57/1.57 % (23582)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.57 % (23582)Termination reason: Unknown
% 2.57/1.57 % (23582)Termination phase: Property scanning
% 2.57/1.57
% 2.57/1.57 % (23582)Memory used [KB]: 2174
% 2.57/1.57 % (23582)Time elapsed: 0.012 s
% 2.57/1.57 % (23582)Instructions burned: 39 (million)
% 2.57/1.57 % (23582)------------------------------
% 2.57/1.57 % (23582)------------------------------
% 2.57/1.57 % (23586)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_23302 for (2999ds/7Mi)
% 2.57/1.57 % (23583)Instruction limit reached!
% 2.57/1.57 % (23583)------------------------------
% 2.57/1.57 % (23583)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.57 % (23583)Termination reason: Unknown
% 2.57/1.57 % (23583)Termination phase: shuffling
% 2.57/1.57
% 2.57/1.57 % (23583)Memory used [KB]: 1663
% 2.57/1.57 % (23583)Time elapsed: 0.011 s
% 2.57/1.57 % (23583)Instructions burned: 15 (million)
% 2.57/1.57 % (23583)------------------------------
% 2.57/1.57 % (23583)------------------------------
% 2.57/1.57 % (23586)Instruction limit reached!
% 2.57/1.57 % (23586)------------------------------
% 2.57/1.57 % (23586)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.57 % (23586)Termination reason: Unknown
% 2.57/1.57 % (23586)Termination phase: shuffling
% 2.57/1.57
% 2.57/1.57 % (23586)Memory used [KB]: 1535
% 2.57/1.57 % (23586)Time elapsed: 0.005 s
% 2.57/1.57 % (23586)Instructions burned: 7 (million)
% 2.57/1.57 % (23586)------------------------------
% 2.57/1.57 % (23586)------------------------------
% 2.57/1.58 % (23588)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_23302 for (2999ds/3Mi)
% 2.57/1.58 % (23588)Instruction limit reached!
% 2.57/1.58 % (23588)------------------------------
% 2.57/1.58 % (23588)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.58 % (23588)Termination reason: Unknown
% 2.57/1.58 % (23588)Termination phase: shuffling
% 2.57/1.58
% 2.57/1.58 % (23588)Memory used [KB]: 1407
% 2.57/1.58 % (23588)Time elapsed: 0.002 s
% 2.57/1.58 % (23588)Instructions burned: 3 (million)
% 2.57/1.58 % (23588)------------------------------
% 2.57/1.58 % (23588)------------------------------
% 2.57/1.58 % (23587)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_23302 for (2999ds/16Mi)
% 2.57/1.59 % (23590)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_23302 for (2999ds/7Mi)
% 2.57/1.59 % (23591)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_23302 for (2999ds/3Mi)
% 2.57/1.59 % (23591)Instruction limit reached!
% 2.57/1.59 % (23591)------------------------------
% 2.57/1.59 % (23591)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.59 % (23591)Termination reason: Unknown
% 2.57/1.59 % (23591)Termination phase: shuffling
% 2.57/1.59
% 2.57/1.59 % (23591)Memory used [KB]: 1407
% 2.57/1.59 % (23591)Time elapsed: 0.002 s
% 2.57/1.59 % (23591)Instructions burned: 3 (million)
% 2.57/1.59 % (23591)------------------------------
% 2.57/1.59 % (23591)------------------------------
% 2.57/1.59 % (23590)Instruction limit reached!
% 2.57/1.59 % (23590)------------------------------
% 2.57/1.59 % (23590)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.59 % (23590)Termination reason: Unknown
% 2.57/1.59 % (23590)Termination phase: shuffling
% 2.57/1.59
% 2.57/1.59 % (23590)Memory used [KB]: 1535
% 2.57/1.59 % (23590)Time elapsed: 0.003 s
% 2.57/1.59 % (23590)Instructions burned: 8 (million)
% 2.57/1.59 % (23590)------------------------------
% 2.57/1.59 % (23590)------------------------------
% 2.57/1.59 % (23589)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_23302 for (2999ds/3Mi)
% 2.57/1.59 % (23587)Instruction limit reached!
% 2.57/1.59 % (23587)------------------------------
% 2.57/1.59 % (23587)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.59 % (23587)Termination reason: Unknown
% 2.57/1.59 % (23587)Termination phase: shuffling
% 2.57/1.59
% 2.57/1.59 % (23589)Instruction limit reached!
% 2.57/1.59 % (23589)------------------------------
% 2.57/1.59 % (23589)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.57/1.59 % (23587)Memory used [KB]: 1663
% 2.57/1.59 % (23589)Termination reason: Unknown
% 2.57/1.59 % (23587)Time elapsed: 0.011 s
% 2.57/1.59 % (23587)Instructions burned: 17 (million)
% 2.57/1.59 % (23589)Termination phase: shuffling
% 2.57/1.59
% 2.57/1.59 % (23587)------------------------------
% 2.57/1.59 % (23587)------------------------------
% 2.57/1.59 % (23589)Memory used [KB]: 1407
% 2.57/1.59 % (23589)Time elapsed: 0.004 s
% 2.57/1.59 % (23589)Instructions burned: 4 (million)
% 2.57/1.59 % (23589)------------------------------
% 2.57/1.59 % (23589)------------------------------
% 2.57/1.60 % (23592)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_23302 for (2999ds/4Mi)
% 2.62/1.60 % (23593)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_23302 for (2999ds/18Mi)
% 2.62/1.60 % (23592)Instruction limit reached!
% 2.62/1.60 % (23592)------------------------------
% 2.62/1.60 % (23592)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.60 % (23592)Termination reason: Unknown
% 2.62/1.60 % (23592)Termination phase: shuffling
% 2.62/1.60
% 2.62/1.60 % (23592)Memory used [KB]: 1407
% 2.62/1.60 % (23592)Time elapsed: 0.003 s
% 2.62/1.60 % (23592)Instructions burned: 6 (million)
% 2.62/1.60 % (23592)------------------------------
% 2.62/1.60 % (23592)------------------------------
% 2.62/1.60 % (23593)Instruction limit reached!
% 2.62/1.60 % (23593)------------------------------
% 2.62/1.60 % (23593)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.60 % (23593)Termination reason: Unknown
% 2.62/1.60 % (23593)Termination phase: shuffling
% 2.62/1.60
% 2.62/1.60 % (23593)Memory used [KB]: 1791
% 2.62/1.60 % (23593)Time elapsed: 0.007 s
% 2.62/1.60 % (23593)Instructions burned: 21 (million)
% 2.62/1.60 % (23593)------------------------------
% 2.62/1.60 % (23593)------------------------------
% 2.62/1.61 % (23596)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_23302 for (2999ds/902Mi)
% 2.62/1.61 % (23574)Instruction limit reached!
% 2.62/1.61 % (23574)------------------------------
% 2.62/1.61 % (23574)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.61 % (23574)Termination reason: Unknown
% 2.62/1.61 % (23574)Termination phase: Saturation
% 2.62/1.61
% 2.62/1.61 % (23574)Memory used [KB]: 7547
% 2.62/1.61 % (23574)Time elapsed: 0.068 s
% 2.62/1.61 % (23574)Instructions burned: 184 (million)
% 2.62/1.61 % (23574)------------------------------
% 2.62/1.61 % (23574)------------------------------
% 2.62/1.61 % (23595)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_23302 for (2999ds/6Mi)
% 2.62/1.61 % (23594)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_23302 for (2999ds/710Mi)
% 2.62/1.61 % (23595)Instruction limit reached!
% 2.62/1.61 % (23595)------------------------------
% 2.62/1.61 % (23595)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.61 % (23595)Termination reason: Unknown
% 2.62/1.61 % (23595)Termination phase: shuffling
% 2.62/1.61
% 2.62/1.61 % (23595)Memory used [KB]: 1535
% 2.62/1.61 % (23595)Time elapsed: 0.006 s
% 2.62/1.61 % (23595)Instructions burned: 7 (million)
% 2.62/1.61 % (23595)------------------------------
% 2.62/1.61 % (23595)------------------------------
% 2.62/1.61 % (23597)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_23302 for (2999ds/21Mi)
% 2.62/1.62 % (23597)Instruction limit reached!
% 2.62/1.62 % (23597)------------------------------
% 2.62/1.62 % (23597)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.62 % (23597)Termination reason: Unknown
% 2.62/1.62 % (23597)Termination phase: shuffling
% 2.62/1.62
% 2.62/1.62 % (23597)Memory used [KB]: 1791
% 2.62/1.62 % (23597)Time elapsed: 0.007 s
% 2.62/1.62 % (23597)Instructions burned: 21 (million)
% 2.62/1.62 % (23597)------------------------------
% 2.62/1.62 % (23597)------------------------------
% 2.62/1.62 % (23598)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_23302 for (2999ds/5Mi)
% 2.62/1.62 % (23598)Instruction limit reached!
% 2.62/1.62 % (23598)------------------------------
% 2.62/1.62 % (23598)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.62 % (23598)Termination reason: Unknown
% 2.62/1.62 % (23598)Termination phase: shuffling
% 2.62/1.62
% 2.62/1.62 % (23598)Memory used [KB]: 1535
% 2.62/1.62 % (23598)Time elapsed: 0.003 s
% 2.62/1.62 % (23598)Instructions burned: 8 (million)
% 2.62/1.62 % (23598)------------------------------
% 2.62/1.62 % (23598)------------------------------
% 2.62/1.62 % (23599)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_23302 for (2999ds/6Mi)
% 2.62/1.62 % (23599)Instruction limit reached!
% 2.62/1.62 % (23599)------------------------------
% 2.62/1.62 % (23599)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.62 % (23599)Termination reason: Unknown
% 2.62/1.62 % (23599)Termination phase: shuffling
% 2.62/1.62
% 2.62/1.62 % (23599)Memory used [KB]: 1535
% 2.62/1.62 % (23599)Time elapsed: 0.004 s
% 2.62/1.62 % (23599)Instructions burned: 8 (million)
% 2.62/1.62 % (23599)------------------------------
% 2.62/1.62 % (23599)------------------------------
% 2.62/1.63 % (23600)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_23302 for (2999ds/377Mi)
% 2.62/1.63 % (23579)First to succeed.
% 2.62/1.63 % (23601)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_23302 for (2999ds/779Mi)
% 2.62/1.63 % (23579)Refutation found. Thanks to Tanya!
% 2.62/1.63 % SZS status Theorem for DTF2THF_23302
% 2.62/1.63 % SZS output start Proof for DTF2THF_23302
% 2.62/1.63 thf(func_def_0, type, in: $i > $i > $o).
% 2.62/1.63 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.62/1.63 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.62/1.63 thf(func_def_8, type, powerset: $i > $i).
% 2.62/1.63 thf(func_def_10, type, setunion: $i > $i).
% 2.62/1.63 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.62/1.63 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.62/1.63 thf(func_def_26, type, prop2set: $o > $i).
% 2.62/1.63 thf(func_def_36, type, nonempty: $i > $o).
% 2.62/1.63 thf(func_def_69, type, set2prop: $i > $o).
% 2.62/1.63 thf(func_def_88, type, subset: $i > $i > $o).
% 2.62/1.63 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.62/1.63 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.62/1.63 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.62/1.63 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.62/1.63 thf(func_def_135, type, regular: $i > $o).
% 2.62/1.63 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.62/1.63 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.62/1.63 thf(func_def_153, type, iskpair: $i > $o).
% 2.62/1.63 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.62/1.63 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.62/1.63 thf(func_def_177, type, singleton: $i > $o).
% 2.62/1.63 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.62/1.63 thf(func_def_184, type, atmost1p: $i > $o).
% 2.62/1.63 thf(func_def_185, type, atleast2p: $i > $o).
% 2.62/1.63 thf(func_def_186, type, atmost2p: $i > $o).
% 2.62/1.63 thf(func_def_187, type, upairsetp: $i > $o).
% 2.62/1.63 thf(func_def_191, type, kfst: $i > $i).
% 2.62/1.63 thf(func_def_203, type, ksnd: $i > $i).
% 2.62/1.63 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.62/1.63 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.62/1.63 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.62/1.63 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.62/1.63 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.62/1.63 thf(func_def_259, type, sP0: $i > $i > $o).
% 2.62/1.63 thf(func_def_260, type, sP1: $i > $o).
% 2.62/1.63 thf(func_def_261, type, sP2: $i > $i > $o).
% 2.62/1.63 thf(func_def_262, type, sP3: $i > $i > $o).
% 2.62/1.63 thf(func_def_266, type, sP7: $i > $i > $i > $o > $o).
% 2.62/1.63 thf(func_def_267, type, sP8: $i > $i > $i > $i > $o).
% 2.62/1.63 thf(func_def_285, type, sK26: $i > $i > $i).
% 2.62/1.63 thf(func_def_291, type, sK32: $o > $i > $i > $i).
% 2.62/1.63 thf(func_def_297, type, sK38: $i > $i).
% 2.62/1.63 thf(func_def_298, type, sK39: $i > $i).
% 2.62/1.63 thf(func_def_308, type, sK49: ($i > $o) > $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_309, type, sK50: ($i > $o) > $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_313, type, sK54: $i > $o).
% 2.62/1.63 thf(func_def_315, type, sK56: $i > $i > $i).
% 2.62/1.63 thf(func_def_316, type, sK57: $i > $i > $i).
% 2.62/1.63 thf(func_def_317, type, sK58: $i > $i > $i).
% 2.62/1.63 thf(func_def_318, type, sK59: $i > $i > $i).
% 2.62/1.63 thf(func_def_319, type, sK60: $i > $i > $i).
% 2.62/1.63 thf(func_def_320, type, sK61: $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_321, type, sK62: $i > $i).
% 2.62/1.63 thf(func_def_322, type, sK63: $i > $i).
% 2.62/1.63 thf(func_def_323, type, sK64: $i > $i).
% 2.62/1.63 thf(func_def_324, type, sK65: $i > $i).
% 2.62/1.63 thf(func_def_325, type, sK66: $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_326, type, sK67: $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_327, type, sK68: $i > $i > $i).
% 2.62/1.63 thf(func_def_328, type, sK69: $i > $i > $i).
% 2.62/1.63 thf(func_def_329, type, sK70: $i > $i > $i).
% 2.62/1.63 thf(func_def_330, type, sK71: $i > $i).
% 2.62/1.63 thf(func_def_332, type, sK73: $i > $i).
% 2.62/1.63 thf(func_def_333, type, sK74: $i > $i).
% 2.62/1.63 thf(func_def_343, type, sK84: $i > $o).
% 2.62/1.63 thf(func_def_357, type, sK98: $i > $i > $i).
% 2.62/1.63 thf(func_def_382, type, sK123: $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_383, type, sK124: $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_386, type, sK127: $i > $o).
% 2.62/1.63 thf(func_def_398, type, sK139: $i > $i > $o).
% 2.62/1.63 thf(func_def_400, type, sK141: $i > $i).
% 2.62/1.63 thf(func_def_401, type, sK142: $i > $i).
% 2.62/1.63 thf(func_def_402, type, sK143: $i > ($i > $i > $o) > $i).
% 2.62/1.63 thf(func_def_403, type, sK144: $i > ($i > $i > $o) > $i).
% 2.62/1.63 thf(func_def_404, type, sK145: $i > $i > ($i > $i > $o) > $i).
% 2.62/1.63 thf(func_def_405, type, sK146: $i > $i > $o).
% 2.62/1.63 thf(func_def_410, type, sK151: $i > $o).
% 2.62/1.63 thf(func_def_414, type, sK155: ($i > $i) > $i > $i > $i).
% 2.62/1.63 thf(func_def_417, type, sK158: $i > $i).
% 2.62/1.63 thf(func_def_419, type, sK160: $i > $i > $o).
% 2.62/1.63 thf(func_def_431, type, sK172: $i > $o).
% 2.62/1.63 thf(func_def_433, type, sK174: $i > ($i > $o) > $i).
% 2.62/1.63 thf(func_def_446, type, sK187: $i > $i).
% 2.62/1.63 thf(func_def_448, type, sK189: $i > ($i > $i) > $i > $i).
% 2.62/1.63 thf(func_def_449, type, sK190: $i > $o).
% 2.62/1.63 thf(func_def_450, type, sK191: $i > $o).
% 2.62/1.63 thf(func_def_451, type, sK192: ($i > $o) > ($i > $o) > $i).
% 2.62/1.63 thf(func_def_452, type, sK193: ($i > $o) > ($i > $o) > $i).
% 2.62/1.63 thf(func_def_463, type, sK204: $i > $i > $i).
% 2.62/1.63 thf(func_def_464, type, sK205: $i > $i > $i).
% 2.62/1.63 thf(func_def_482, type, sK223: $i > $i > $i).
% 2.62/1.63 thf(func_def_485, type, sK226: $i > $i > $o).
% 2.62/1.63 thf(func_def_490, type, sK231: $i > ($i > $o) > $i).
% 2.62/1.63 thf(func_def_491, type, sK232: $i > $o).
% 2.62/1.63 thf(func_def_495, type, sK236: $i > $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_500, type, sK241: ($i > $o) > $i).
% 2.62/1.63 thf(func_def_501, type, sK242: $i > $o).
% 2.62/1.63 thf(func_def_502, type, sK243: $i > $i).
% 2.62/1.63 thf(func_def_515, type, sK256: $i > $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_516, type, sK257: $i > $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_530, type, sK271: $i > $i > $i).
% 2.62/1.63 thf(func_def_532, type, sK273: $i > $i).
% 2.62/1.63 thf(func_def_534, type, sK275: $i > $o).
% 2.62/1.63 thf(func_def_543, type, sK284: $i > $o).
% 2.62/1.63 thf(func_def_545, type, sK286: ($i > $o) > $i > $i).
% 2.62/1.63 thf(func_def_546, type, sK287: ($i > $o) > $i > $i).
% 2.62/1.63 thf(func_def_551, type, sK292: $i > $i).
% 2.62/1.63 thf(func_def_567, type, sK308: ($i > $o) > $i).
% 2.62/1.63 thf(func_def_568, type, sK309: $i > $o).
% 2.62/1.63 thf(func_def_569, type, sK310: $i > $i).
% 2.62/1.63 thf(func_def_573, type, sK314: $i > $i > $i).
% 2.62/1.63 thf(func_def_576, type, sK317: $i > $o).
% 2.62/1.63 thf(func_def_583, type, sK324: ($i > $o) > $i > $i > $i).
% 2.62/1.63 thf(func_def_586, type, sK327: $i > $o).
% 2.62/1.63 thf(func_def_587, type, sK328: $i > $o).
% 2.62/1.63 thf(func_def_590, type, sK331: $i > ($i > $o) > $i).
% 2.62/1.63 thf(func_def_594, type, sK335: $i > $i > $o).
% 2.62/1.63 thf(func_def_605, type, sK346: $i > $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_610, type, sK351: $i > $o).
% 2.62/1.63 thf(func_def_611, type, sK352: ($i > $o) > $i).
% 2.62/1.63 thf(func_def_617, type, sK358: $i > ($i > $o) > $i).
% 2.62/1.63 thf(func_def_618, type, sK359: $i > $o).
% 2.62/1.63 thf(func_def_628, type, sK369: $i > $i > $i).
% 2.62/1.63 thf(func_def_667, type, sK408: $i > $o).
% 2.62/1.63 thf(func_def_670, type, sK411: $i > $i).
% 2.62/1.63 thf(func_def_676, type, sK417: $i > $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_677, type, sK418: $i > $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_678, type, sK419: $i > $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_679, type, sK420: $i > $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_682, type, sK423: $i > $i > $o).
% 2.62/1.63 thf(func_def_685, type, sK426: $i > $o).
% 2.62/1.63 thf(func_def_692, type, sK433: $i > $i > $o).
% 2.62/1.63 thf(func_def_695, type, sK436: $i > $o).
% 2.62/1.63 thf(func_def_697, type, sK438: $i > ($i > $o) > $i).
% 2.62/1.63 thf(func_def_718, type, sK459: $i > $o).
% 2.62/1.63 thf(func_def_719, type, sK460: $i > $o).
% 2.62/1.63 thf(func_def_720, type, sK461: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.62/1.63 thf(func_def_721, type, sK462: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.62/1.63 thf(func_def_726, type, sK467: $i > $i).
% 2.62/1.63 thf(func_def_728, type, sK469: $i > ($i > $o) > $i).
% 2.62/1.63 thf(func_def_729, type, sK470: $i > $o).
% 2.62/1.63 thf(func_def_737, type, sK478: $i > $o).
% 2.62/1.63 thf(func_def_741, type, sK482: ($i > $o) > $i > $i).
% 2.62/1.63 thf(func_def_743, type, sK484: $i > $o).
% 2.62/1.63 thf(func_def_745, type, sK486: $i > ($i > $i) > $i > $i).
% 2.62/1.63 thf(func_def_747, type, sK488: $i > $i).
% 2.62/1.63 thf(func_def_749, type, sK490: $i > $o).
% 2.62/1.63 thf(func_def_752, type, sK493: $i > ($i > $o) > $i).
% 2.62/1.63 thf(func_def_753, type, sK494: $i > ($i > $o) > $i).
% 2.62/1.63 thf(func_def_776, type, sK517: $i > $i > $i).
% 2.62/1.63 thf(func_def_778, type, sK519: $i > $o).
% 2.62/1.63 thf(func_def_780, type, sK521: $i > $i > $o).
% 2.62/1.63 thf(func_def_788, type, sK529: ($i > $o) > ($i > $o) > $i).
% 2.62/1.63 thf(func_def_789, type, sK530: ($i > $o) > ($i > $o) > $i).
% 2.62/1.63 thf(func_def_790, type, sK531: $i > $o).
% 2.62/1.63 thf(func_def_791, type, sK532: $i > $o).
% 2.62/1.63 thf(func_def_806, type, sK547: $i > $o).
% 2.62/1.63 thf(func_def_813, type, sK554: ($i > $o) > $i > $i).
% 2.62/1.63 thf(func_def_815, type, sK556: $i > $o).
% 2.62/1.63 thf(func_def_829, type, sK570: $i > $o).
% 2.62/1.63 thf(func_def_844, type, sK585: $i > $i).
% 2.62/1.63 thf(func_def_847, type, sK588: $i > $i > $i).
% 2.62/1.63 thf(func_def_852, type, sK593: $i > $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_857, type, sK598: ($i > $o) > $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_858, type, sK599: ($i > $o) > $i > $i > $i > $i).
% 2.62/1.63 thf(func_def_862, type, sK603: $i > $o).
% 2.62/1.63 thf(func_def_868, type, sK609: $i > $o).
% 2.62/1.63 thf(func_def_891, type, sK632: ($i > $o) > $i).
% 2.62/1.63 thf(func_def_892, type, sK633: ($i > $o) > $i).
% 2.62/1.63 thf(func_def_893, type, sK634: $i > $o).
% 2.62/1.63 thf(func_def_896, type, sK637: $i > $o).
% 2.62/1.63 thf(func_def_898, type, sK639: $i > $o).
% 2.62/1.63 thf(func_def_900, type, sK641: ($i > $o) > $i > $i).
% 2.62/1.63 thf(func_def_907, type, ph648: !>[X0: $tType]:(X0)).
% 2.62/1.63 thf(f4063,plain,(
% 2.62/1.63 $false),
% 2.62/1.63 inference(trivial_inequality_removal,[status(thm)],[f2769])).
% 2.62/1.63 thf(f2769,plain,(
% 2.62/1.63 ((ap @ sK544 @ sK543 @ sK545 @ sK546) != (ap @ sK544 @ sK543 @ sK545 @ sK546))),
% 2.62/1.63 inference(cnf_transformation,[status(thm)],[f1745])).
% 2.62/1.63 thf(f1745,plain,(
% 2.62/1.63 (exuI3 = $true) & (singletonprop = $true) & (kpairsurjEq = $true) & (cartprodpairin = $true) & (powersetI1 = $true) & (kfstsingleton = $true) & (dsetconstrI = $true) & (setadjoinSub2 = $true) & (emptysetsubset = $true) & (eqimpsubset1 = $true) & (exu__Cong = $true) & (setminusERneg = $true) & (dpsetconstrEL1 = $true) & (setukpairinjR = $true) & (powersetE1 = $true) & (funcGraphProp1 = $true) & (exuE3u = $true) & (powersetE = $true) & (setukpairinjR11 = $true) & (ex1E2 = $true) & (upairsubunion = $true) & (app = $true) & (setadjoinIR = $true) & (ubforcartprodlem1 = $true) & (binintersectI = $true) & (subset2powerset = $true) & (emptyI = $true) & (setunion__Cong = $true) & (ubforcartprodlem2 = $true) & (theprop = $true) & (binunionEcases = $true) & (setminusSubset2 = $true) & (infuncsetfunc = $true) & (disjointsetsI1 = $true) & (noeltsimpempty = $true) & (binintersectSubset5 = $true) & (setminusI = $true) & (symdiffIneg1 = $true) & (quantDeMorgan4 = $true) & (eqbreln = $true) & (subsetE2 = $true) & (subsetRefl = $true) & (dpsetconstrI = $true) & (notdexE = $true) & (setadjoinAx = $true) & (dsetconstrEL = $true) & (inPowerset = $true) & (powersetI = $true) & (binintersectSubset2 = $true) & (cartprodpairsurjEq = $true) & (dsetconstrER = $true) & (dpsetconstrEL2 = $true) & (funcGraphProp4 = $true) & (cartprodmempair1 = $true) & (binintersectEL = $true) & (lamp = $true) & (cartprodsndpairEq = $true) & (foundationAx = $true) & (($true = (in @ sK545 @ (funcSet @ sK544 @ sK543))) & (((ap @ sK544 @ sK543 @ sK545 @ sK546) != (ap @ sK544 @ sK543 @ sK545 @ sK546)) & ($true = (in @ sK546 @ sK544)))) & (setminusELneg = $true) & (nonemptyE1 = $true) & (nonemptyImpWitness = $true) & (vacuousDall = $true) & (symdiffI1 = $true) & (notinsingleton = $true) & (setukpairinjR2 = $true) & (binintersectSubset1 = $true) & (dsetconstr__Cong = $true) & (kpairp = $true) & (prop2setI = $true) & (brelnall1 = $true) & (brelnall2 = $true) & (funcImageSingleton = $true) & (notinemptyset = $true) & (powersetAx = $true) & (sepSubset = $true) & (bs114d = $true) & (setminusIRneg = $true) & (upairset2E = $true) & (cartprodfstin = $true) & (setukpairinjL2 = $true) & (exuI2 = $true) & (funcext = $true) & (binintersectLsub = $true) & (subbreln = $true) & (subsetE = $true) & (ex1I = $true) & (notequalI1 = $true) & (cartprodmempaircEq = $true) & (setminusILneg = $true) & (cartprodpairmemER = $true) & (lamProp = $true) & (singletonsubset = $true) & (upairsetE = $true) & (emptyinunitempty = $true) & (omega0Ax = $true) & (setunionI = $true) & (binintersectSubset4 = $true) & (omegaIndAx = $true) & (setukpairinjL = $true) & (omega__Cong = $true) & (uniqinunit = $true) & (subPowSU = $true) & (funcext2 = $true) & (singletoninpowerset = $true) & (binintersectSubset3 = $true) & (setextsub = $true) & (setunionE = $true) & (prop2set2propI = $true) & (cartprodfstpairEq = $true) & (sepInPowerset = $true) & (funcinfuncset = $true) & (singletonsuniq = $true) & (inCongP = $true) & (dpsetconstrERa = $true) & (setunionsingleton2 = $true) & (setukpairIR = $true) & (notequalI2 = $true) & (setunionsingleton1 = $true) & (upairset2IR = $true) & (ksndsingleton = $true) & (subsetI2 = $true) & (binunionRsub = $true) & (powersetsubset = $true) & (eqimpsubset2 = $true) & (upairequniteq = $true) & (setoftrueEq = $true) & (emptysetAx = $true) & (setukpairIL = $true) & (setadjoinE = $true) & (ksndpairEq = $true) & (setunionAx = $true) & (kpairiskpair = $true) & (setextAx = $true) & (funcextLem = $true) & (exuE3e = $true) & (ubforcartprodlem3 = $true) & (setukpairinjR12 = $true) & (setadjoinIL = $true) & (nonemptyI = $true) & (replAx = $true) & (setminusLsub = $true) & (binintersectRsub = $true) & (symdiffI2 = $true) & (setadjoin__Cong = $true) & (ap2p = $true) & (emptyE1 = $true) & (notdallE = $true) & (dpsetconstrSub = $true) & (in__Cong = $true) & (upairsetIR = $true) & (subsetemptysetimpeq = $true) & (binunionIR = $true) & (symdiffE = $true) & (subsetTrans = $true) & (setminusSubset1 = $true) & (cartprodsndin = $true) & (setext = $true) & (emptysetE = $true) & (setunionE2 = $true) & (upairsetIL = $true) & (lam2p = $true) & (funcGraphProp2 = $true) & (apProp = $true) & (emptyset__Cong = $true) & (setadjoinSub = $true) & (setunionsingleton = $true) & (setadjoinOr = $true) & (secondinupair = $true) & (binintersectER = $true) & (upairinpowunion = $true) & (setOfPairsIsBReln = $true) & (singletonsswitch = $true) & (ex1I2 = $true) & (kfstpairEq = $true) & (binunionLsub = $true) & (binunionE = $true) & (quantDeMorgan1 = $true) & (ex1E1 = $true) & (emptyinPowerset = $true) & (singletoninpowunion = $true) & (prop2setE = $true) & (omegaSAx = $true) & (cartprodmempair = $true) & (quantDeMorgan2 = $true) & (binunionIL = $true) & (setukpairinjL1 = $true) & (exuE1 = $true) & (nonemptyI1 = $true) & (exuE2 = $true) & (emptysetimpfalse = $true) & (symdiffIneg2 = $true) & (setminusER = $true) & (setbeta = $true) & (cartprodpairmemEL = $true) & (quantDeMorgan3 = $true) & (dpsetconstrER = $true) & (exuI1 = $true) & (subsetI1 = $true) & (emptyInPowerset = $true) & (notsubsetI = $true) & (powerset__Cong = $true) & (eqinunit = $true) & (setukpairinjR1 = $true) & (exuEu = $true) & (setminusEL = $true) & (descr__Cong = $true) & (funcGraphProp3 = $true) & (wellorderingAx = $true) & (descrp = $true)),
% 2.62/1.63 inference(skolemisation,[status(esa),new_symbols(skolem,[sK543,sK544,sK545,sK546])],[f792,f1744,f1743])).
% 2.62/1.63 thf(f1743,plain,(
% 2.62/1.63 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (funcSet @ X1 @ X0))) & ? [X3 : $i] : (((ap @ X1 @ X0 @ X2 @ X3) != (ap @ X1 @ X0 @ X2 @ X3)) & ($true = (in @ X3 @ X1)))) => (($true = (in @ sK545 @ (funcSet @ sK544 @ sK543))) & ? [X3 : $i] : (((ap @ sK544 @ sK543 @ sK545 @ X3) != (ap @ sK544 @ sK543 @ sK545 @ X3)) & ($true = (in @ X3 @ sK544))))),
% 2.62/1.63 introduced(definition,[],[choice_axiom])).
% 2.62/1.63 thf(f1744,plain,(
% 2.62/1.63 ? [X3 : $i] : (((ap @ sK544 @ sK543 @ sK545 @ X3) != (ap @ sK544 @ sK543 @ sK545 @ X3)) & ($true = (in @ X3 @ sK544))) => (((ap @ sK544 @ sK543 @ sK545 @ sK546) != (ap @ sK544 @ sK543 @ sK545 @ sK546)) & ($true = (in @ sK546 @ sK544)))),
% 2.62/1.63 introduced(definition,[],[choice_axiom])).
% 2.62/1.63 thf(f792,plain,(
% 2.62/1.63 (exuI3 = $true) & (singletonprop = $true) & (kpairsurjEq = $true) & (cartprodpairin = $true) & (powersetI1 = $true) & (kfstsingleton = $true) & (dsetconstrI = $true) & (setadjoinSub2 = $true) & (emptysetsubset = $true) & (eqimpsubset1 = $true) & (exu__Cong = $true) & (setminusERneg = $true) & (dpsetconstrEL1 = $true) & (setukpairinjR = $true) & (powersetE1 = $true) & (funcGraphProp1 = $true) & (exuE3u = $true) & (powersetE = $true) & (setukpairinjR11 = $true) & (ex1E2 = $true) & (upairsubunion = $true) & (app = $true) & (setadjoinIR = $true) & (ubforcartprodlem1 = $true) & (binintersectI = $true) & (subset2powerset = $true) & (emptyI = $true) & (setunion__Cong = $true) & (ubforcartprodlem2 = $true) & (theprop = $true) & (binunionEcases = $true) & (setminusSubset2 = $true) & (infuncsetfunc = $true) & (disjointsetsI1 = $true) & (noeltsimpempty = $true) & (binintersectSubset5 = $true) & (setminusI = $true) & (symdiffIneg1 = $true) & (quantDeMorgan4 = $true) & (eqbreln = $true) & (subsetE2 = $true) & (subsetRefl = $true) & (dpsetconstrI = $true) & (notdexE = $true) & (setadjoinAx = $true) & (dsetconstrEL = $true) & (inPowerset = $true) & (powersetI = $true) & (binintersectSubset2 = $true) & (cartprodpairsurjEq = $true) & (dsetconstrER = $true) & (dpsetconstrEL2 = $true) & (funcGraphProp4 = $true) & (cartprodmempair1 = $true) & (binintersectEL = $true) & (lamp = $true) & (cartprodsndpairEq = $true) & (foundationAx = $true) & ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (funcSet @ X1 @ X0))) & ? [X3 : $i] : (((ap @ X1 @ X0 @ X2 @ X3) != (ap @ X1 @ X0 @ X2 @ X3)) & ($true = (in @ X3 @ X1)))) & (setminusELneg = $true) & (nonemptyE1 = $true) & (nonemptyImpWitness = $true) & (vacuousDall = $true) & (symdiffI1 = $true) & (notinsingleton = $true) & (setukpairinjR2 = $true) & (binintersectSubset1 = $true) & (dsetconstr__Cong = $true) & (kpairp = $true) & (prop2setI = $true) & (brelnall1 = $true) & (brelnall2 = $true) & (funcImageSingleton = $true) & (notinemptyset = $true) & (powersetAx = $true) & (sepSubset = $true) & (bs114d = $true) & (setminusIRneg = $true) & (upairset2E = $true) & (cartprodfstin = $true) & (setukpairinjL2 = $true) & (exuI2 = $true) & (funcext = $true) & (binintersectLsub = $true) & (subbreln = $true) & (subsetE = $true) & (ex1I = $true) & (notequalI1 = $true) & (cartprodmempaircEq = $true) & (setminusILneg = $true) & (cartprodpairmemER = $true) & (lamProp = $true) & (singletonsubset = $true) & (upairsetE = $true) & (emptyinunitempty = $true) & (omega0Ax = $true) & (setunionI = $true) & (binintersectSubset4 = $true) & (omegaIndAx = $true) & (setukpairinjL = $true) & (omega__Cong = $true) & (uniqinunit = $true) & (subPowSU = $true) & (funcext2 = $true) & (singletoninpowerset = $true) & (binintersectSubset3 = $true) & (setextsub = $true) & (setunionE = $true) & (prop2set2propI = $true) & (cartprodfstpairEq = $true) & (sepInPowerset = $true) & (funcinfuncset = $true) & (singletonsuniq = $true) & (inCongP = $true) & (dpsetconstrERa = $true) & (setunionsingleton2 = $true) & (setukpairIR = $true) & (notequalI2 = $true) & (setunionsingleton1 = $true) & (upairset2IR = $true) & (ksndsingleton = $true) & (subsetI2 = $true) & (binunionRsub = $true) & (powersetsubset = $true) & (eqimpsubset2 = $true) & (upairequniteq = $true) & (setoftrueEq = $true) & (emptysetAx = $true) & (setukpairIL = $true) & (setadjoinE = $true) & (ksndpairEq = $true) & (setunionAx = $true) & (kpairiskpair = $true) & (setextAx = $true) & (funcextLem = $true) & (exuE3e = $true) & (ubforcartprodlem3 = $true) & (setukpairinjR12 = $true) & (setadjoinIL = $true) & (nonemptyI = $true) & (replAx = $true) & (setminusLsub = $true) & (binintersectRsub = $true) & (symdiffI2 = $true) & (setadjoin__Cong = $true) & (ap2p = $true) & (emptyE1 = $true) & (notdallE = $true) & (dpsetconstrSub = $true) & (in__Cong = $true) & (upairsetIR = $true) & (subsetemptysetimpeq = $true) & (binunionIR = $true) & (symdiffE = $true) & (subsetTrans = $true) & (setminusSubset1 = $true) & (cartprodsndin = $true) & (setext = $true) & (emptysetE = $true) & (setunionE2 = $true) & (upairsetIL = $true) & (lam2p = $true) & (funcGraphProp2 = $true) & (apProp = $true) & (emptyset__Cong = $true) & (setadjoinSub = $true) & (setunionsingleton = $true) & (setadjoinOr = $true) & (secondinupair = $true) & (binintersectER = $true) & (upairinpowunion = $true) & (setOfPairsIsBReln = $true) & (singletonsswitch = $true) & (ex1I2 = $true) & (kfstpairEq = $true) & (binunionLsub = $true) & (binunionE = $true) & (quantDeMorgan1 = $true) & (ex1E1 = $true) & (emptyinPowerset = $true) & (singletoninpowunion = $true) & (prop2setE = $true) & (omegaSAx = $true) & (cartprodmempair = $true) & (quantDeMorgan2 = $true) & (binunionIL = $true) & (setukpairinjL1 = $true) & (exuE1 = $true) & (nonemptyI1 = $true) & (exuE2 = $true) & (emptysetimpfalse = $true) & (symdiffIneg2 = $true) & (setminusER = $true) & (setbeta = $true) & (cartprodpairmemEL = $true) & (quantDeMorgan3 = $true) & (dpsetconstrER = $true) & (exuI1 = $true) & (subsetI1 = $true) & (emptyInPowerset = $true) & (notsubsetI = $true) & (powerset__Cong = $true) & (eqinunit = $true) & (setukpairinjR1 = $true) & (exuEu = $true) & (setminusEL = $true) & (descr__Cong = $true) & (funcGraphProp3 = $true) & (wellorderingAx = $true) & (descrp = $true)),
% 2.62/1.63 inference(flattening,[status(thm)],[f791])).
% 2.62/1.63 thf(f791,plain,(
% 2.62/1.63 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (funcSet @ X1 @ X0))) & ? [X3 : $i] : (((ap @ X1 @ X0 @ X2 @ X3) != (ap @ X1 @ X0 @ X2 @ X3)) & ($true = (in @ X3 @ X1)))) & (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)),
% 2.62/1.63 inference(ennf_transformation,[status(thm)],[f429])).
% 2.62/1.63 thf(f429,plain,(
% 2.62/1.63 ~((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) => ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ (funcSet @ X1 @ X0))) => ! [X3 : $i] : (($true = (in @ X3 @ X1)) => ((ap @ X1 @ X0 @ X2 @ X3) = (ap @ X1 @ X0 @ X2 @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.62/1.63 inference(fool_elimination,[status(thm)],[f428])).
% 2.62/1.63 thf(f428,plain,(
% 2.62/1.63 ~(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 => ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X2 @ (funcSet @ X1 @ X0)) => ! [X3 : $i] : ((in @ X3 @ X1) => ((ap @ X1 @ X0 @ X2 @ X3) = (ap @ X1 @ X0 @ X2 @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.62/1.63 inference(rectify,[status(thm)],[f217])).
% 2.62/1.63 thf(f217,negated_conjecture,(
% 2.62/1.63 ~(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 => ! [X4 : $i,X3 : $i,X12 : $i] : ((in @ X12 @ (funcSet @ X3 @ X4)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((ap @ X3 @ X4 @ X12 @ X1) = (ap @ X3 @ X4 @ X12 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.62/1.63 inference(negated_conjecture,[status(cth)],[f216])).
% 2.62/1.63 thf(f216,conjecture,(
% 2.62/1.63 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 => ! [X4 : $i,X3 : $i,X12 : $i] : ((in @ X12 @ (funcSet @ X3 @ X4)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((ap @ X3 @ X4 @ X12 @ X1) = (ap @ X3 @ X4 @ X12 @ X1))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.62/1.63 file('/export/starexec/sandbox2/tmp/tmp.7DY03sHzZu/DTF2THF_23302.p',ap2apEq1)).
% 2.62/1.63 % SZS output end Proof for DTF2THF_23302
% 2.62/1.63 % (23579)------------------------------
% 2.62/1.63 % (23579)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.62/1.63 % (23579)Termination reason: Refutation
% 2.62/1.63
% 2.62/1.63 % (23579)Memory used [KB]: 8699
% 2.62/1.63 % (23579)Time elapsed: 0.091 s
% 2.62/1.63 % (23579)Instructions burned: 162 (million)
% 2.62/1.63 % (23579)------------------------------
% 2.62/1.63 % (23579)------------------------------
% 2.62/1.63 % (23573)Success in time 0.107 s
%------------------------------------------------------------------------------