%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU759^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n029.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 09:36:58 AM UTC 2026
% Result : Theorem 2.86s 1.84s
% Output : Refutation 2.86s
% Verified :
% SZS Type : Refutation
% Derivation depth : 18
% Number of leaves : 9
% Syntax : Number of formulae : 47 ( 12 unt; 0 typ; 1 def)
% Number of atoms : 4654 (1239 equ; 0 cnn)
% Maximal formula atoms : 280 ( 99 avg)
% Number of connectives : 2461 ( 66 ~; 41 |; 871 &; 351 @)
% ( 5 <=>;1127 =>; 0 <=; 0 <~>)
% Maximal formula depth : 289 ( 46 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 476 ( 473 usr; 287 con; 0-4 aty)
% Number of variables : 94 ( 0 ^; 61 !; 33 ?; 94 :)
% Comments :
%------------------------------------------------------------------------------
thf(func_def_0,type,
in: $i > $i > $o ).
thf(func_def_1,type,
exu: ( $i > $o ) > $o ).
thf(func_def_6,type,
setadjoin: $i > $i > $i ).
thf(func_def_8,type,
powerset: $i > $i ).
thf(func_def_10,type,
setunion: $i > $i ).
thf(func_def_19,type,
descr: ( $i > $o ) > $i ).
thf(func_def_21,type,
dsetconstr: $i > ( $i > $o ) > $i ).
thf(func_def_26,type,
prop2set: $o > $i ).
thf(func_def_36,type,
nonempty: $i > $o ).
thf(func_def_69,type,
set2prop: $i > $o ).
thf(func_def_88,type,
subset: $i > $i > $o ).
thf(func_def_89,type,
disjoint: $i > $i > $o ).
thf(func_def_90,type,
setsmeet: $i > $i > $o ).
thf(func_def_114,type,
binunion: $i > $i > $i ).
thf(func_def_122,type,
binintersect: $i > $i > $i ).
thf(func_def_135,type,
regular: $i > $o ).
thf(func_def_136,type,
setminus: $i > $i > $i ).
thf(func_def_147,type,
symdiff: $i > $i > $i ).
thf(func_def_153,type,
iskpair: $i > $o ).
thf(func_def_158,type,
kpair: $i > $i > $i ).
thf(func_def_160,type,
cartprod: $i > $i > $i ).
thf(func_def_177,type,
singleton: $i > $o ).
thf(func_def_179,type,
ex1: $i > ( $i > $o ) > $o ).
thf(func_def_184,type,
atmost1p: $i > $o ).
thf(func_def_185,type,
atleast2p: $i > $o ).
thf(func_def_186,type,
atmost2p: $i > $o ).
thf(func_def_187,type,
upairsetp: $i > $o ).
thf(func_def_191,type,
kfst: $i > $i ).
thf(func_def_203,type,
ksnd: $i > $i ).
thf(func_def_213,type,
breln: $i > $i > $i > $o ).
thf(func_def_214,type,
dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_222,type,
func: $i > $i > $i > $o ).
thf(func_def_223,type,
funcSet: $i > $i > $i ).
thf(func_def_226,type,
ap: $i > $i > $i > $i > $i ).
thf(func_def_232,type,
lam: $i > $i > ( $i > $i ) > $i ).
thf(func_def_259,type,
if: $i > $o > $i > $i > $i ).
thf(func_def_328,type,
sP1: $o > $i > $i > $i > $o ).
thf(func_def_329,type,
sP2: $i > $i > $i > $i > $o ).
thf(func_def_332,type,
sP5: $i > $i > $o ).
thf(func_def_333,type,
sP6: $i > $o ).
thf(func_def_334,type,
sP7: $i > $i > $o ).
thf(func_def_335,type,
sP8: $i > $i > $o ).
thf(func_def_337,type,
sP10: $i > $i > $i > $o ).
thf(func_def_354,type,
sK27: $i > $o ).
thf(func_def_361,type,
sK34: $i > $o ).
thf(func_def_379,type,
sK52: $i > $i > $o ).
thf(func_def_414,type,
sK87: $o > $i > $i > $i ).
thf(func_def_417,type,
sK90: $i > $i ).
thf(func_def_421,type,
sK94: $i > $o ).
thf(func_def_429,type,
sK102: $i > $i ).
thf(func_def_431,type,
sK104: ( $i > $i ) > $i > $i > $i ).
thf(func_def_445,type,
sK118: $i > $o ).
thf(func_def_458,type,
sK131: $i > $i > $i > $i > $i ).
thf(func_def_459,type,
sK132: $i > $i > $i > $i > $i ).
thf(func_def_460,type,
sK133: $i > $i > $i > $i > $i ).
thf(func_def_461,type,
sK134: $i > $i > $i > $i > $i ).
thf(func_def_486,type,
sK159: $i > $o ).
thf(func_def_490,type,
sK163: $i > $i > ( $i > $i ) > $i ).
thf(func_def_491,type,
sK164: $i > $i ).
thf(func_def_494,type,
sK167: $i > $o ).
thf(func_def_511,type,
sK184: ( $i > $o ) > $i ).
thf(func_def_512,type,
sK185: $i > $o ).
thf(func_def_517,type,
sK190: $i > $i > $i > $i ).
thf(func_def_568,type,
sK241: $i > $i > $i > $i > $i ).
thf(func_def_581,type,
sK254: $i > $i > $i > $i ).
thf(func_def_582,type,
sK255: $i > $i > $i > $i ).
thf(func_def_597,type,
sK270: $i > $i > $i > $i ).
thf(func_def_629,type,
sK302: $i > ( $i > $o ) > $i ).
thf(func_def_630,type,
sK303: $i > $o ).
thf(func_def_638,type,
sK311: $i > $i > $i ).
thf(func_def_639,type,
sK312: $i > $i > $i ).
thf(func_def_640,type,
sK313: $i > $i > $i ).
thf(func_def_641,type,
sK314: $i > $i > $i ).
thf(func_def_642,type,
sK315: $i > $i > $i ).
thf(func_def_643,type,
sK316: $i > $i > $i > $i ).
thf(func_def_644,type,
sK317: $i > $i ).
thf(func_def_645,type,
sK318: $i > $i ).
thf(func_def_646,type,
sK319: $i > $i ).
thf(func_def_647,type,
sK320: $i > $i ).
thf(func_def_648,type,
sK321: $i > $i > $i > $i ).
thf(func_def_649,type,
sK322: $i > $i > $i > $i ).
thf(func_def_650,type,
sK323: $i > $i > $i ).
thf(func_def_651,type,
sK324: $i > $i > $i ).
thf(func_def_652,type,
sK325: $i > $i > $i ).
thf(func_def_653,type,
sK326: $i > $i ).
thf(func_def_655,type,
sK328: $i > $i ).
thf(func_def_656,type,
sK329: $i > $i ).
thf(func_def_660,type,
sK333: $i > $i > $i ).
thf(func_def_681,type,
sK354: $i > $o ).
thf(func_def_682,type,
sK355: $i > $i ).
thf(func_def_683,type,
sK356: ( $i > $o ) > $i ).
thf(func_def_686,type,
sK359: $i > $i > $i ).
thf(func_def_693,type,
sK366: $i > $i > ( $i > $i ) > $i ).
thf(func_def_694,type,
sK367: $i > $i ).
thf(func_def_705,type,
sK378: $i > $i > $i ).
thf(func_def_707,type,
sK380: $i > $o ).
thf(func_def_709,type,
sK382: ( $i > $o ) > $i > $i ).
thf(func_def_710,type,
sK383: $i > $i > $i > $i > $i ).
thf(func_def_724,type,
sK397: $i > $i > $i > $i > $i ).
thf(func_def_748,type,
sK421: $i > $o ).
thf(func_def_750,type,
sK423: ( $i > $o ) > $i > $i ).
thf(func_def_753,type,
sK426: $i > $i ).
thf(func_def_771,type,
sK444: $i > $i > $i ).
thf(func_def_772,type,
sK445: $i > $i > $i ).
thf(func_def_780,type,
sK453: $i > ( $i > $o ) > $i ).
thf(func_def_781,type,
sK454: $i > $o ).
thf(func_def_799,type,
sK472: $i > $o ).
thf(func_def_826,type,
sK499: $i > $i ).
thf(func_def_829,type,
sK502: $i > $i > ( $i > $i ) > $i ).
thf(func_def_835,type,
sK508: $i > $i > $o ).
thf(func_def_836,type,
sK509: $i > $i ).
thf(func_def_837,type,
sK510: $i > $i ).
thf(func_def_838,type,
sK511: ( $i > $i > $o ) > $i > $i ).
thf(func_def_839,type,
sK512: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_840,type,
sK513: ( $i > $i > $o ) > $i > $i ).
thf(func_def_841,type,
sK514: ( $i > $o ) > $i ).
thf(func_def_842,type,
sK515: $i > $o ).
thf(func_def_843,type,
sK516: $i > $i ).
thf(func_def_845,type,
sK518: $i > $o ).
thf(func_def_847,type,
sK520: ( $i > $o ) > $i > $i ).
thf(func_def_854,type,
sK527: $i > $o ).
thf(func_def_855,type,
sK528: $i > $o ).
thf(func_def_856,type,
sK529: $i > $o ).
thf(func_def_857,type,
sK530: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_858,type,
sK531: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_861,type,
sK534: $i > $i > $o ).
thf(func_def_864,type,
sK537: $i > $o ).
thf(func_def_867,type,
sK540: ( $i > $o ) > $i > $i > $i ).
thf(func_def_870,type,
sK543: $i > $o ).
thf(func_def_871,type,
sK544: $i > $i > $i ).
thf(func_def_875,type,
sK548: $i > $o ).
thf(func_def_877,type,
sK550: ( $i > $o ) > $i ).
thf(func_def_878,type,
sK551: ( $i > $o ) > $i ).
thf(func_def_879,type,
sK552: $i > ( $i > $o ) > $i ).
thf(func_def_880,type,
sK553: $i > $o ).
thf(func_def_887,type,
sK560: $i > $i ).
thf(func_def_888,type,
sK561: ( $i > $i ) > $i > $i > $i ).
thf(func_def_895,type,
sK568: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_896,type,
sK569: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_900,type,
sK573: $i > $o ).
thf(func_def_911,type,
sK584: $i > $i ).
thf(func_def_922,type,
sK595: $i > $o ).
thf(func_def_926,type,
sK599: $i > $o ).
thf(func_def_931,type,
sK604: $i > $i ).
thf(func_def_933,type,
sK606: $i > $i ).
thf(func_def_936,type,
sK609: $i > $i > $i > $i ).
thf(func_def_937,type,
sK610: $i > $i > $i > $i ).
thf(func_def_941,type,
sK614: $i > $o ).
thf(func_def_943,type,
sK616: $i > ( $i > $o ) > $i ).
thf(func_def_973,type,
sK646: $i > $i > $o ).
thf(func_def_978,type,
sK651: $i > $i > $o ).
thf(func_def_984,type,
sK657: $i > $i > $i ).
thf(func_def_999,type,
sK672: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1001,type,
sK674: $i > $i ).
thf(func_def_1014,type,
sK687: $i > $o ).
thf(func_def_1017,type,
sK690: $i > $i ).
thf(func_def_1019,type,
sK692: $i > $o ).
thf(func_def_1020,type,
sK693: $i > $o ).
thf(func_def_1021,type,
sK694: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1022,type,
sK695: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1027,type,
sK700: $i > $i > $o ).
thf(func_def_1035,type,
sK708: $i > $o ).
thf(func_def_1037,type,
sK710: ( $i > $o ) > $i > $i ).
thf(func_def_1055,type,
sK728: $i > $i > $o ).
thf(func_def_1068,type,
sK741: $i > $o ).
thf(func_def_1076,type,
sK749: $i > $o ).
thf(func_def_1078,type,
sK751: $i > ( $i > $o ) > $i ).
thf(func_def_1083,type,
sK756: $i > $i > $i > $i > $i ).
thf(func_def_1084,type,
sK757: $i > $i > $i > $i > $i ).
thf(func_def_1095,type,
sK768: $i > $i > $i ).
thf(func_def_1123,type,
sK796: $i > $o ).
thf(func_def_1126,type,
sK799: $i > ( $i > $o ) > $i ).
thf(func_def_1127,type,
sK800: $i > ( $i > $o ) > $i ).
thf(func_def_1128,type,
sK801: $i > $i > $i ).
thf(func_def_1152,type,
sK825: $i > $i > $o ).
thf(func_def_1170,type,
sK843: $i > $i > $i ).
thf(func_def_1174,type,
sK847: $i > $o ).
thf(func_def_1176,type,
sK849: ( $i > $o ) > $i > $i ).
thf(func_def_1177,type,
sK850: ( $i > $o ) > $i > $i ).
thf(func_def_1193,type,
sK866: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_1194,type,
sK867: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_1197,type,
sK870: $i > $o ).
thf(func_def_1198,type,
sK871: $i > $o ).
thf(func_def_1209,type,
sK882: $i > $i ).
thf(func_def_1212,type,
sK885: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1213,type,
sK886: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1217,type,
sK890: $i > $o ).
thf(func_def_1220,type,
ph893:
!>[X0: $tType] : X0 ).
thf(f5305,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f4473,f5282]) ).
thf(f5282,plain,
~ spl892_10,
inference(avatar_split_clause,[status(thm)],[f5281,f4390]) ).
thf(f4390,definition,
( spl892_10
<=> ( ( subset @ sK875 @ sK877 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl892_10])],[avatar_definition]) ).
thf(f5281,plain,
( ( subset @ sK875 @ sK877 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f5280]) ).
thf(f5280,plain,
( ( ( subset @ sK875 @ sK877 )
!= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f5145,f4038]) ).
thf(f4038,plain,
binintersectLsub = $true,
inference(cnf_transformation,[status(thm)],[f2572]) ).
thf(f2572,plain,
( ( powersetT_lem = $true )
& ( iftrueorfalse = $true )
& ( dsetconstr__Cong = $true )
& ( ifSingleton = $true )
& ( setukpairinjR1 = $true )
& ( binintersectTELcontra = $true )
& ( quantDeMorgan3 = $true )
& ( emptyI = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( binunionLsub = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( binintersectLsub = $true )
& ( eta1 = $true )
& ( descrp = $true )
& ( woz13rule0 = $true )
& ( upairequniteq = $true )
& ( symdiffI1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( cartprodsndpairEq = $true )
& ( powersetI1 = $true )
& ( powersetI = $true )
& ( descr__Cong = $true )
& ( ex1I = $true )
& ( sepSubset = $true )
& ( exuEu = $true )
& ( setminusSubset2 = $true )
& ( setukpairinjL2 = $true )
& ( demorgan2a1 = $true )
& ( ( in @ sK876 @ ( powerset @ sK874 ) )
= $true )
& ( ( subset @ ( binintersect @ sK875 @ sK876 ) @ sK877 )
!= $true )
& ( ( in @ sK877 @ ( powerset @ sK874 ) )
= $true )
& ( ( subset @ sK875 @ sK877 )
= $true )
& ( ( in @ sK875 @ ( powerset @ sK874 ) )
= $true )
& ( setextsub = $true )
& ( setukpairinjR = $true )
& ( singletoninpowunion = $true )
& ( subsetE = $true )
& ( powersetsubset = $true )
& ( emptyinunitempty = $true )
& ( binintersectTERcontra = $true )
& ( ksndpairEq = $true )
& ( dsetconstrI = $true )
& ( powersetE = $true )
& ( dpsetconstrER = $true )
& ( setbeta = $true )
& ( symdiffIneg1 = $true )
& ( subbreln = $true )
& ( funcGraphProp2 = $true )
& ( setukpairIL = $true )
& ( app = $true )
& ( notequalI2 = $true )
& ( sepInPowerset = $true )
& ( omega0Ax = $true )
& ( setextAx = $true )
& ( cartprodmempaircEq = $true )
& ( funcext2 = $true )
& ( dsetconstrEL = $true )
& ( singletonsswitch = $true )
& ( complementTE1 = $true )
& ( emptysetE = $true )
& ( notdallE = $true )
& ( singletoninpowerset = $true )
& ( complementTnotintersectT = $true )
& ( secondinupair = $true )
& ( emptyinPowerset = $true )
& ( emptysetsubset = $true )
& ( eqinunit = $true )
& ( setminusER = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( uniqinunit = $true )
& ( complementTcontraSubset = $true )
& ( setunionAx = $true )
& ( ex1E2 = $true )
& ( funcext = $true )
& ( beta1 = $true )
& ( inIntersectImpInUnion2 = $true )
& ( nonemptyI = $true )
& ( symdiffE = $true )
& ( inCongP = $true )
& ( cartprodfstin = $true )
& ( emptysetimpfalse = $true )
& ( dpsetconstrSub = $true )
& ( funcGraphProp3 = $true )
& ( symdiffI2 = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndin = $true )
& ( prop2setI = $true )
& ( iftrueProp1 = $true )
& ( singletonsubset = $true )
& ( prop2set2propI = $true )
& ( complementSubsetComplementIntersect = $true )
& ( demorgan2a2 = $true )
& ( apProp = $true )
& ( notsubsetI = $true )
& ( exuE3e = $true )
& ( binunionEcases = $true )
& ( setunionI = $true )
& ( binunionTIRcontra = $true )
& ( demorgan2b = $true )
& ( quantDeMorgan2 = $true )
& ( powerset__Cong = $true )
& ( iffalseProp1 = $true )
& ( cartprodmempair1 = $true )
& ( ubforcartprodlem2 = $true )
& ( iftrue = $true )
& ( ubforcartprodlem3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectEL = $true )
& ( setminusIRneg = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( setunionE2 = $true )
& ( omega__Cong = $true )
& ( setminusERneg = $true )
& ( binintersectI = $true )
& ( subsetE2 = $true )
& ( exuI2 = $true )
& ( kfstsingleton = $true )
& ( notequalI1 = $true )
& ( setminusSubset1 = $true )
& ( cartprodpairmemEL = $true )
& ( lam2lamEq = $true )
& ( notinsingleton = $true )
& ( contraSubsetComplement = $true )
& ( cartprodpairmemER = $true )
& ( ex1E1 = $true )
& ( upairsetIL = $true )
& ( contrasubsetT1 = $true )
& ( quantDeMorgan1 = $true )
& ( inIntersectImpInUnion = $true )
& ( brelnall1 = $true )
& ( wellorderingAx = $true )
& ( theeq = $true )
& ( notinemptyset = $true )
& ( powersetTI1 = $true )
& ( setminusT_lem = $true )
& ( binunionTILcontra = $true )
& ( setadjoin__Cong = $true )
& ( upairsetE = $true )
& ( upairset2IR = $true )
& ( brelnall2 = $true )
& ( doubleComplementI1 = $true )
& ( binintersectSubset4 = $true )
& ( setadjoinIR = $true )
& ( setukpairinjL = $true )
& ( kpairsurjEq = $true )
& ( setminusEL = $true )
& ( lamp = $true )
& ( setminusLsub = $true )
& ( exu__Cong = $true )
& ( binunionRsub = $true )
& ( setunionsingleton = $true )
& ( bs114d = $true )
& ( contrasubsetT2 = $true )
& ( singletonsuniq = $true )
& ( ap2apEq2 = $true )
& ( funcGraphProp1 = $true )
& ( ifp = $true )
& ( complementImpComplementIntersect = $true )
& ( subsetTI = $true )
& ( vacuousDall = $true )
& ( subsetRefl = $true )
& ( exuE3u = $true )
& ( ap2apEq1 = $true )
& ( subsetemptysetimpeq = $true )
& ( ubforcartprodlem1 = $true )
& ( setunionE = $true )
& ( demorgan1 = $true )
& ( exuI1 = $true )
& ( binintersectSubset5 = $true )
& ( dpsetconstrEL1 = $true )
& ( setadjoinIL = $true )
& ( setadjoinE = $true )
& ( funcGraphProp4 = $true )
& ( noeltsimpempty = $true )
& ( eqimpsubset2 = $true )
& ( setminusI = $true )
& ( kfstpairEq = $true )
& ( iffalseProp2 = $true )
& ( setukpairIR = $true )
& ( symdiffIneg2 = $true )
& ( setukpairinjR11 = $true )
& ( beta2 = $true )
& ( kpairp = $true )
& ( setukpairinjL1 = $true )
& ( binunionTE = $true )
& ( subsetI1 = $true )
& ( binintersectT_lem = $true )
& ( setukpairinjR12 = $true )
& ( binintersectSubset3 = $true )
& ( subsetI2 = $true )
& ( emptysetAx = $true )
& ( setunion__Cong = $true )
& ( demorgan2a = $true )
& ( setadjoinSub2 = $true )
& ( eta2 = $true )
& ( cartprodmempair = $true )
& ( eqimpsubset1 = $true )
& ( setextT = $true )
& ( replAx = $true )
& ( powersetAx = $true )
& ( setadjoinOr = $true )
& ( binunionE = $true )
& ( setadjoinSub = $true )
& ( cartprodpairin = $true )
& ( binintersectSubset1 = $true )
& ( quantDeMorgan4 = $true )
& ( exuE1 = $true )
& ( subsetTrans = $true )
& ( emptyE1 = $true )
& ( binunionIL = $true )
& ( dsetconstrER = $true )
& ( iffalse = $true )
& ( ex1I2 = $true )
& ( demorgan2 = $true )
& ( binintersectER = $true )
& ( powersetTE1 = $true )
& ( subPowSU = $true )
& ( omegaSAx = $true )
& ( complementT_lem = $true )
& ( dpsetconstrERa = $true )
& ( funcextLem = $true )
& ( foundationAx = $true )
& ( setminusELneg = $true )
& ( disjointsetsI1 = $true )
& ( in__Cong = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( dpsetconstrEL2 = $true )
& ( binunionIR = $true )
& ( doubleComplementEq = $true )
& ( upairsubunion = $true )
& ( upairsetIR = $true )
& ( emptyInPowerset = $true )
& ( exuI3 = $true )
& ( doubleComplementSub1 = $true )
& ( omegaIndAx = $true )
& ( ksndsingleton = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan1a = $true )
& ( setext = $true )
& ( dpsetconstrI = $true )
& ( contrasubsetT = $true )
& ( binunionT_lem = $true )
& ( theprop = $true )
& ( upairset2E = $true )
& ( ap2p = $true )
& ( singletonprop = $true )
& ( setunionsingleton2 = $true )
& ( exuE2 = $true )
& ( nonemptyImpWitness = $true )
& ( cartprodfstpairEq = $true )
& ( nonemptyE1 = $true )
& ( doubleComplementSub2 = $true )
& ( funcinfuncset = $true )
& ( setunionsingleton1 = $true )
& ( binunionTEcontra = $true )
& ( subset2powerset = $true )
& ( nonemptyI1 = $true )
& ( eqbreln = $true )
& ( infuncsetfunc = $true )
& ( upairinpowunion = $true )
& ( emptyset__Cong = $true )
& ( setoftrueEq = $true )
& ( doubleComplementE1 = $true )
& ( setminusILneg = $true )
& ( binintersectRsub = $true )
& ( prop2setE = $true )
& ( funcImageSingleton = $true )
& ( demorgan2b2 = $true )
& ( iftrueProp2 = $true )
& ( lam2p = $true )
& ( setukpairinjR2 = $true )
& ( kpairiskpair = $true )
& ( demorgan1b = $true )
& ( contrasubsetT3 = $true )
& ( complementTI1 = $true )
& ( notdexE = $true )
& ( setOfPairsIsBReln = $true )
& ( lamProp = $true )
& ( setadjoinAx = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK874,sK875,sK876,sK877])],[f1061,f2571,f2570,f2569]) ).
thf(f2569,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( ( in @ X2 @ ( powerset @ X0 ) )
= $true )
& ? [X3: $i] :
( ( ( subset @ ( binintersect @ X1 @ X2 ) @ X3 )
!= $true )
& ( ( in @ X3 @ ( powerset @ X0 ) )
= $true )
& ( ( subset @ X1 @ X3 )
= $true ) ) )
& ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) )
=> ( ? [X2: $i] :
( ( ( in @ X2 @ ( powerset @ sK874 ) )
= $true )
& ? [X3: $i] :
( ( ( subset @ ( binintersect @ sK875 @ X2 ) @ X3 )
!= $true )
& ( ( in @ X3 @ ( powerset @ sK874 ) )
= $true )
& ( ( subset @ sK875 @ X3 )
= $true ) ) )
& ( ( in @ sK875 @ ( powerset @ sK874 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2570,plain,
( ? [X2: $i] :
( ( ( in @ X2 @ ( powerset @ sK874 ) )
= $true )
& ? [X3: $i] :
( ( ( subset @ ( binintersect @ sK875 @ X2 ) @ X3 )
!= $true )
& ( ( in @ X3 @ ( powerset @ sK874 ) )
= $true )
& ( ( subset @ sK875 @ X3 )
= $true ) ) )
=> ( ( ( in @ sK876 @ ( powerset @ sK874 ) )
= $true )
& ? [X3: $i] :
( ( ( subset @ ( binintersect @ sK875 @ sK876 ) @ X3 )
!= $true )
& ( ( in @ X3 @ ( powerset @ sK874 ) )
= $true )
& ( ( subset @ sK875 @ X3 )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2571,plain,
( ? [X3: $i] :
( ( ( subset @ ( binintersect @ sK875 @ sK876 ) @ X3 )
!= $true )
& ( ( in @ X3 @ ( powerset @ sK874 ) )
= $true )
& ( ( subset @ sK875 @ X3 )
= $true ) )
=> ( ( ( subset @ ( binintersect @ sK875 @ sK876 ) @ sK877 )
!= $true )
& ( ( in @ sK877 @ ( powerset @ sK874 ) )
= $true )
& ( ( subset @ sK875 @ sK877 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1061,plain,
( ( powersetT_lem = $true )
& ( iftrueorfalse = $true )
& ( dsetconstr__Cong = $true )
& ( ifSingleton = $true )
& ( setukpairinjR1 = $true )
& ( binintersectTELcontra = $true )
& ( quantDeMorgan3 = $true )
& ( emptyI = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( binunionLsub = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( binintersectLsub = $true )
& ( eta1 = $true )
& ( descrp = $true )
& ( woz13rule0 = $true )
& ( upairequniteq = $true )
& ( symdiffI1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( cartprodsndpairEq = $true )
& ( powersetI1 = $true )
& ( powersetI = $true )
& ( descr__Cong = $true )
& ( ex1I = $true )
& ( sepSubset = $true )
& ( exuEu = $true )
& ( setminusSubset2 = $true )
& ( setukpairinjL2 = $true )
& ( demorgan2a1 = $true )
& ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( ( in @ X2 @ ( powerset @ X0 ) )
= $true )
& ? [X3: $i] :
( ( ( subset @ ( binintersect @ X1 @ X2 ) @ X3 )
!= $true )
& ( ( in @ X3 @ ( powerset @ X0 ) )
= $true )
& ( ( subset @ X1 @ X3 )
= $true ) ) )
& ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) )
& ( setextsub = $true )
& ( setukpairinjR = $true )
& ( singletoninpowunion = $true )
& ( subsetE = $true )
& ( powersetsubset = $true )
& ( emptyinunitempty = $true )
& ( binintersectTERcontra = $true )
& ( ksndpairEq = $true )
& ( dsetconstrI = $true )
& ( powersetE = $true )
& ( dpsetconstrER = $true )
& ( setbeta = $true )
& ( symdiffIneg1 = $true )
& ( subbreln = $true )
& ( funcGraphProp2 = $true )
& ( setukpairIL = $true )
& ( app = $true )
& ( notequalI2 = $true )
& ( sepInPowerset = $true )
& ( omega0Ax = $true )
& ( setextAx = $true )
& ( cartprodmempaircEq = $true )
& ( funcext2 = $true )
& ( dsetconstrEL = $true )
& ( singletonsswitch = $true )
& ( complementTE1 = $true )
& ( emptysetE = $true )
& ( notdallE = $true )
& ( singletoninpowerset = $true )
& ( complementTnotintersectT = $true )
& ( secondinupair = $true )
& ( emptyinPowerset = $true )
& ( emptysetsubset = $true )
& ( eqinunit = $true )
& ( setminusER = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( uniqinunit = $true )
& ( complementTcontraSubset = $true )
& ( setunionAx = $true )
& ( ex1E2 = $true )
& ( funcext = $true )
& ( beta1 = $true )
& ( inIntersectImpInUnion2 = $true )
& ( nonemptyI = $true )
& ( symdiffE = $true )
& ( inCongP = $true )
& ( cartprodfstin = $true )
& ( emptysetimpfalse = $true )
& ( dpsetconstrSub = $true )
& ( funcGraphProp3 = $true )
& ( symdiffI2 = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndin = $true )
& ( prop2setI = $true )
& ( iftrueProp1 = $true )
& ( singletonsubset = $true )
& ( prop2set2propI = $true )
& ( complementSubsetComplementIntersect = $true )
& ( demorgan2a2 = $true )
& ( apProp = $true )
& ( notsubsetI = $true )
& ( exuE3e = $true )
& ( binunionEcases = $true )
& ( setunionI = $true )
& ( binunionTIRcontra = $true )
& ( demorgan2b = $true )
& ( quantDeMorgan2 = $true )
& ( powerset__Cong = $true )
& ( iffalseProp1 = $true )
& ( cartprodmempair1 = $true )
& ( ubforcartprodlem2 = $true )
& ( iftrue = $true )
& ( ubforcartprodlem3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectEL = $true )
& ( setminusIRneg = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( setunionE2 = $true )
& ( omega__Cong = $true )
& ( setminusERneg = $true )
& ( binintersectI = $true )
& ( subsetE2 = $true )
& ( exuI2 = $true )
& ( kfstsingleton = $true )
& ( notequalI1 = $true )
& ( setminusSubset1 = $true )
& ( cartprodpairmemEL = $true )
& ( lam2lamEq = $true )
& ( notinsingleton = $true )
& ( contraSubsetComplement = $true )
& ( cartprodpairmemER = $true )
& ( ex1E1 = $true )
& ( upairsetIL = $true )
& ( contrasubsetT1 = $true )
& ( quantDeMorgan1 = $true )
& ( inIntersectImpInUnion = $true )
& ( brelnall1 = $true )
& ( wellorderingAx = $true )
& ( theeq = $true )
& ( notinemptyset = $true )
& ( powersetTI1 = $true )
& ( setminusT_lem = $true )
& ( binunionTILcontra = $true )
& ( setadjoin__Cong = $true )
& ( upairsetE = $true )
& ( upairset2IR = $true )
& ( brelnall2 = $true )
& ( doubleComplementI1 = $true )
& ( binintersectSubset4 = $true )
& ( setadjoinIR = $true )
& ( setukpairinjL = $true )
& ( kpairsurjEq = $true )
& ( setminusEL = $true )
& ( lamp = $true )
& ( setminusLsub = $true )
& ( exu__Cong = $true )
& ( binunionRsub = $true )
& ( setunionsingleton = $true )
& ( bs114d = $true )
& ( contrasubsetT2 = $true )
& ( singletonsuniq = $true )
& ( ap2apEq2 = $true )
& ( funcGraphProp1 = $true )
& ( ifp = $true )
& ( complementImpComplementIntersect = $true )
& ( subsetTI = $true )
& ( vacuousDall = $true )
& ( subsetRefl = $true )
& ( exuE3u = $true )
& ( ap2apEq1 = $true )
& ( subsetemptysetimpeq = $true )
& ( ubforcartprodlem1 = $true )
& ( setunionE = $true )
& ( demorgan1 = $true )
& ( exuI1 = $true )
& ( binintersectSubset5 = $true )
& ( dpsetconstrEL1 = $true )
& ( setadjoinIL = $true )
& ( setadjoinE = $true )
& ( funcGraphProp4 = $true )
& ( noeltsimpempty = $true )
& ( eqimpsubset2 = $true )
& ( setminusI = $true )
& ( kfstpairEq = $true )
& ( iffalseProp2 = $true )
& ( setukpairIR = $true )
& ( symdiffIneg2 = $true )
& ( setukpairinjR11 = $true )
& ( beta2 = $true )
& ( kpairp = $true )
& ( setukpairinjL1 = $true )
& ( binunionTE = $true )
& ( subsetI1 = $true )
& ( binintersectT_lem = $true )
& ( setukpairinjR12 = $true )
& ( binintersectSubset3 = $true )
& ( subsetI2 = $true )
& ( emptysetAx = $true )
& ( setunion__Cong = $true )
& ( demorgan2a = $true )
& ( setadjoinSub2 = $true )
& ( eta2 = $true )
& ( cartprodmempair = $true )
& ( eqimpsubset1 = $true )
& ( setextT = $true )
& ( replAx = $true )
& ( powersetAx = $true )
& ( setadjoinOr = $true )
& ( binunionE = $true )
& ( setadjoinSub = $true )
& ( cartprodpairin = $true )
& ( binintersectSubset1 = $true )
& ( quantDeMorgan4 = $true )
& ( exuE1 = $true )
& ( subsetTrans = $true )
& ( emptyE1 = $true )
& ( binunionIL = $true )
& ( dsetconstrER = $true )
& ( iffalse = $true )
& ( ex1I2 = $true )
& ( demorgan2 = $true )
& ( binintersectER = $true )
& ( powersetTE1 = $true )
& ( subPowSU = $true )
& ( omegaSAx = $true )
& ( complementT_lem = $true )
& ( dpsetconstrERa = $true )
& ( funcextLem = $true )
& ( foundationAx = $true )
& ( setminusELneg = $true )
& ( disjointsetsI1 = $true )
& ( in__Cong = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( dpsetconstrEL2 = $true )
& ( binunionIR = $true )
& ( doubleComplementEq = $true )
& ( upairsubunion = $true )
& ( upairsetIR = $true )
& ( emptyInPowerset = $true )
& ( exuI3 = $true )
& ( doubleComplementSub1 = $true )
& ( omegaIndAx = $true )
& ( ksndsingleton = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan1a = $true )
& ( setext = $true )
& ( dpsetconstrI = $true )
& ( contrasubsetT = $true )
& ( binunionT_lem = $true )
& ( theprop = $true )
& ( upairset2E = $true )
& ( ap2p = $true )
& ( singletonprop = $true )
& ( setunionsingleton2 = $true )
& ( exuE2 = $true )
& ( nonemptyImpWitness = $true )
& ( cartprodfstpairEq = $true )
& ( nonemptyE1 = $true )
& ( doubleComplementSub2 = $true )
& ( funcinfuncset = $true )
& ( setunionsingleton1 = $true )
& ( binunionTEcontra = $true )
& ( subset2powerset = $true )
& ( nonemptyI1 = $true )
& ( eqbreln = $true )
& ( infuncsetfunc = $true )
& ( upairinpowunion = $true )
& ( emptyset__Cong = $true )
& ( setoftrueEq = $true )
& ( doubleComplementE1 = $true )
& ( setminusILneg = $true )
& ( binintersectRsub = $true )
& ( prop2setE = $true )
& ( funcImageSingleton = $true )
& ( demorgan2b2 = $true )
& ( iftrueProp2 = $true )
& ( lam2p = $true )
& ( setukpairinjR2 = $true )
& ( kpairiskpair = $true )
& ( demorgan1b = $true )
& ( contrasubsetT3 = $true )
& ( complementTI1 = $true )
& ( notdexE = $true )
& ( setOfPairsIsBReln = $true )
& ( lamProp = $true )
& ( setadjoinAx = $true ) ),
inference(flattening,[status(thm)],[f1060]) ).
thf(f1060,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( subset @ ( binintersect @ X1 @ X2 ) @ X3 )
!= $true )
& ( ( subset @ X1 @ X3 )
= $true )
& ( ( in @ X3 @ ( powerset @ X0 ) )
= $true ) )
& ( ( in @ X2 @ ( powerset @ X0 ) )
= $true ) )
& ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) )
& ( woz13rule0 = $true )
& ( demorgan2 = $true )
& ( demorgan2b = $true )
& ( demorgan2b2 = $true )
& ( demorgan2a = $true )
& ( demorgan1 = $true )
& ( demorgan1b = $true )
& ( demorgan1a = $true )
& ( demorgan2a2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( demorgan2a1 = $true )
& ( binunionTEcontra = $true )
& ( binunionTE = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( inIntersectImpInUnion2 = $true )
& ( inIntersectImpInUnion = $true )
& ( binunionTIRcontra = $true )
& ( binunionTILcontra = $true )
& ( complementTcontraSubset = $true )
& ( contraSubsetComplement = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( complementSubsetComplementIntersect = $true )
& ( complementImpComplementIntersect = $true )
& ( complementTnotintersectT = $true )
& ( doubleComplementEq = $true )
& ( doubleComplementSub2 = $true )
& ( doubleComplementSub1 = $true )
& ( doubleComplementE1 = $true )
& ( doubleComplementI1 = $true )
& ( contrasubsetT3 = $true )
& ( contrasubsetT2 = $true )
& ( contrasubsetT1 = $true )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f306]) ).
thf(f306,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ( ( complementSubsetComplementIntersect = $true )
=> ( ( complementInPowersetComplementIntersect = $true )
=> ( ( contraSubsetComplement = $true )
=> ( ( complementTcontraSubset = $true )
=> ( ( binunionTILcontra = $true )
=> ( ( binunionTIRcontra = $true )
=> ( ( inIntersectImpInUnion = $true )
=> ( ( inIntersectImpInUnion2 = $true )
=> ( ( inIntersectImpInIntersectUnions = $true )
=> ( ( intersectInPowersetIntersectUnions = $true )
=> ( ( inComplementUnionImpNotIn1 = $true )
=> ( ( inComplementUnionImpInComplement1 = $true )
=> ( ( binunionTE = $true )
=> ( ( binunionTEcontra = $true )
=> ( ( demorgan2a1 = $true )
=> ( ( complementUnionInPowersetComplement = $true )
=> ( ( demorgan2a2 = $true )
=> ( ( demorgan1a = $true )
=> ( ( demorgan1b = $true )
=> ( ( demorgan1 = $true )
=> ( ( demorgan2a = $true )
=> ( ( demorgan2b2 = $true )
=> ( ( demorgan2b = $true )
=> ( ( demorgan2 = $true )
=> ( ( woz13rule0 = $true )
=> ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
=> ! [X2: $i] :
( ( ( in @ X2 @ ( powerset @ X0 ) )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ ( powerset @ X0 ) )
= $true )
=> ( ( ( subset @ X1 @ X3 )
= $true )
=> ( ( subset @ ( binintersect @ X1 @ X2 ) @ X3 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f305]) ).
thf(f305,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X0 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X0 ) )
=> ( ( subset @ X1 @ X3 )
=> ( subset @ ( binintersect @ X1 @ X2 ) @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f281]) ).
thf(f281,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X17: $i] :
( ( in @ X17 @ ( powerset @ X3 ) )
=> ( ( subset @ X11 @ X17 )
=> ( subset @ ( binintersect @ X11 @ X16 ) @ X17 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f280]) ).
thf(f280,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ( inIntersectImpInUnion2
=> ( inIntersectImpInIntersectUnions
=> ( intersectInPowersetIntersectUnions
=> ( inComplementUnionImpNotIn1
=> ( inComplementUnionImpInComplement1
=> ( binunionTE
=> ( binunionTEcontra
=> ( demorgan2a1
=> ( complementUnionInPowersetComplement
=> ( demorgan2a2
=> ( demorgan1a
=> ( demorgan1b
=> ( demorgan1
=> ( demorgan2a
=> ( demorgan2b2
=> ( demorgan2b
=> ( demorgan2
=> ( woz13rule0
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X17: $i] :
( ( in @ X17 @ ( powerset @ X3 ) )
=> ( ( subset @ X11 @ X17 )
=> ( subset @ ( binintersect @ X11 @ X16 ) @ X17 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.iP8AXdSty9/DTF2THF_22688.p',woz13rule1) ).
thf(f5145,plain,
( ( ( subset @ sK875 @ sK877 )
!= $true )
| ( binintersectLsub != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f5107]) ).
thf(f5107,plain,
( ( binintersectLsub != $true )
| ( ( subset @ sK875 @ sK877 )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f4369,f3677]) ).
thf(f3677,plain,
! [X0: $i,X1: $i] :
( ( ( subset @ ( binintersect @ X1 @ X0 ) @ X1 )
= $true )
| ( binintersectLsub != $true ) ),
inference(cnf_transformation,[status(thm)],[f2467]) ).
thf(f2467,plain,
( ( ! [X0: $i,X1: $i] :
( ( subset @ ( binintersect @ X1 @ X0 ) @ X1 )
= $true )
| ( binintersectLsub != $true ) )
& ( ( binintersectLsub = $true )
| ( ( subset @ ( binintersect @ sK810 @ sK809 ) @ sK810 )
!= $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK809,sK810])],[f2465,f2466]) ).
thf(f2466,plain,
( ? [X2: $i,X3: $i] :
( ( subset @ ( binintersect @ X3 @ X2 ) @ X3 )
!= $true )
=> ( ( subset @ ( binintersect @ sK810 @ sK809 ) @ sK810 )
!= $true ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2465,plain,
( ( ! [X0: $i,X1: $i] :
( ( subset @ ( binintersect @ X1 @ X0 ) @ X1 )
= $true )
| ( binintersectLsub != $true ) )
& ( ( binintersectLsub = $true )
| ? [X2: $i,X3: $i] :
( ( subset @ ( binintersect @ X3 @ X2 ) @ X3 )
!= $true ) ) ),
inference(rectify,[status(thm)],[f2464]) ).
thf(f2464,plain,
( ( ! [X0: $i,X1: $i] :
( ( subset @ ( binintersect @ X1 @ X0 ) @ X1 )
= $true )
| ( binintersectLsub != $true ) )
& ( ( binintersectLsub = $true )
| ? [X0: $i,X1: $i] :
( ( subset @ ( binintersect @ X1 @ X0 ) @ X1 )
!= $true ) ) ),
inference(nnf_transformation,[status(thm)],[f812]) ).
thf(f812,plain,
( ! [X0: $i,X1: $i] :
( ( subset @ ( binintersect @ X1 @ X0 ) @ X1 )
= $true )
<=> ( binintersectLsub = $true ) ),
inference(fool_elimination,[status(thm)],[f811]) ).
thf(f811,plain,
! [X0: $i,X1: $i] :
( ( subset @ ( binintersect @ X1 @ X0 ) @ X1 )
= binintersectLsub ),
inference(rectify,[status(thm)],[f111]) ).
thf(f111,axiom,
! [X4: $i,X3: $i] :
( ( subset @ ( binintersect @ X3 @ X4 ) @ X3 )
= binintersectLsub ),
file('/export/starexec/sandbox/tmp/tmp.iP8AXdSty9/DTF2THF_22688.p',binintersectLsub) ).
thf(f4369,plain,
! [X0: $i] :
( ( ( subset @ ( binintersect @ sK875 @ sK876 ) @ X0 )
!= $true )
| ( ( subset @ X0 @ sK877 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4368]) ).
thf(f4368,plain,
! [X0: $i] :
( ( $true != $true )
| ( ( subset @ ( binintersect @ sK875 @ sK876 ) @ X0 )
!= $true )
| ( ( subset @ X0 @ sK877 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f4361,f3841]) ).
thf(f3841,plain,
subsetTrans = $true,
inference(cnf_transformation,[status(thm)],[f2572]) ).
thf(f4361,plain,
! [X0: $i] :
( ( ( subset @ ( binintersect @ sK875 @ sK876 ) @ X0 )
!= $true )
| ( subsetTrans != $true )
| ( ( subset @ X0 @ sK877 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4356]) ).
thf(f4356,plain,
! [X0: $i] :
( ( $true != $true )
| ( ( subset @ X0 @ sK877 )
!= $true )
| ( ( subset @ ( binintersect @ sK875 @ sK876 ) @ X0 )
!= $true )
| ( subsetTrans != $true ) ),
inference(superposition,[status(thm)],[f4020,f3362]) ).
thf(f3362,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( subset @ X2 @ X0 )
= $true )
| ( ( subset @ X1 @ X0 )
!= $true )
| ( subsetTrans != $true )
| ( ( subset @ X2 @ X1 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f2092]) ).
thf(f2092,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X0 )
= $true )
| ( ( subset @ X1 @ X0 )
!= $true )
| ( ( subset @ X2 @ X1 )
!= $true ) )
| ( subsetTrans != $true ) )
& ( ( subsetTrans = $true )
| ( ( ( subset @ sK564 @ sK562 )
!= $true )
& ( ( subset @ sK563 @ sK562 )
= $true )
& ( ( subset @ sK564 @ sK563 )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK562,sK563,sK564])],[f2090,f2091]) ).
thf(f2091,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X5 @ X3 )
!= $true )
& ( ( subset @ X4 @ X3 )
= $true )
& ( ( subset @ X5 @ X4 )
= $true ) )
=> ( ( ( subset @ sK564 @ sK562 )
!= $true )
& ( ( subset @ sK563 @ sK562 )
= $true )
& ( ( subset @ sK564 @ sK563 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2090,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X0 )
= $true )
| ( ( subset @ X1 @ X0 )
!= $true )
| ( ( subset @ X2 @ X1 )
!= $true ) )
| ( subsetTrans != $true ) )
& ( ( subsetTrans = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X5 @ X3 )
!= $true )
& ( ( subset @ X4 @ X3 )
= $true )
& ( ( subset @ X5 @ X4 )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f2089]) ).
thf(f2089,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X0 )
= $true )
| ( ( subset @ X1 @ X0 )
!= $true )
| ( ( subset @ X2 @ X1 )
!= $true ) )
| ( subsetTrans != $true ) )
& ( ( subsetTrans = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X0 )
!= $true )
& ( ( subset @ X1 @ X0 )
= $true )
& ( ( subset @ X2 @ X1 )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1211]) ).
thf(f1211,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X0 )
= $true )
| ( ( subset @ X1 @ X0 )
!= $true )
| ( ( subset @ X2 @ X1 )
!= $true ) )
<=> ( subsetTrans = $true ) ),
inference(flattening,[status(thm)],[f1210]) ).
thf(f1210,plain,
( ( subsetTrans = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X0 )
= $true )
| ( ( subset @ X1 @ X0 )
!= $true )
| ( ( subset @ X2 @ X1 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f514]) ).
thf(f514,plain,
( ( subsetTrans = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X2 @ X1 )
= $true )
=> ( ( ( subset @ X1 @ X0 )
= $true )
=> ( ( subset @ X2 @ X0 )
= $true ) ) ) ),
inference(fool_elimination,[status(thm)],[f513]) ).
thf(f513,plain,
( subsetTrans
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( subset @ X2 @ X1 )
=> ( ( subset @ X1 @ X0 )
=> ( subset @ X2 @ X0 ) ) ) ) ),
inference(rectify,[status(thm)],[f89]) ).
thf(f89,axiom,
( subsetTrans
= ( ! [X5: $i,X4: $i,X3: $i] :
( ( subset @ X3 @ X4 )
=> ( ( subset @ X4 @ X5 )
=> ( subset @ X3 @ X5 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.iP8AXdSty9/DTF2THF_22688.p',subsetTrans) ).
thf(f4020,plain,
( ( subset @ ( binintersect @ sK875 @ sK876 ) @ sK877 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f2572]) ).
thf(f4473,plain,
spl892_10,
inference(avatar_contradiction_clause,[status(thm)],[f4472]) ).
thf(f4472,plain,
( $false
| spl892_10 ),
inference(trivial_inequality_removal,[status(thm)],[f4465]) ).
thf(f4465,plain,
( ( $true != $true )
| spl892_10 ),
inference(superposition,[status(thm)],[f4392,f4018]) ).
thf(f4018,plain,
( ( subset @ sK875 @ sK877 )
= $true ),
inference(cnf_transformation,[status(thm)],[f2572]) ).
thf(f4392,plain,
( ( ( subset @ sK875 @ sK877 )
!= $true )
| spl892_10 ),
inference(avatar_component_clause,[status(thm)],[f4390]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU759^1 : TPTP v9.2.1. Released v3.7.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n029.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.33 % CPULimit : 300
% 0.16/0.33 % WCLimit : 300
% 0.16/0.33 % DateTime : Tue Feb 24 12:33:14 EST 2026
% 0.16/0.33 % CPUTime :
% 0.16/0.33 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.45 ---- Original DTF file ---
% 0.20/0.45 thf(spec,logic,$$dhol).
% 0.20/0.45 %------------------------------------------------------------------------------
% 0.20/0.45 % File : SEU759^1 : TPTP v9.2.1. Released v3.7.0.
% 0.20/0.45 % Domain : Set Theory
% 0.20/0.45 % Problem : Typed Set Theory - First Wizard of Oz Examples - WoZ1 Lemmas
% 0.20/0.45 % Version : Especial.
% 0.20/0.45 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.20/0.45 % (! Z:i.in Z (powerset A) -> subset X Z ->
% 0.20/0.45 % subset (binintersect X Y) Z)))
% 0.20/0.45
% 0.20/0.45 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.20/0.45 % Source : [Bro08]
% 0.20/0.45 % Names : ZFC261g [Bro08]
% 0.20/0.45
% 0.20/0.45 % Status : Theorem
% 0.20/0.45 % Rating : 0.22 v9.1.0, 0.38 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v6.1.0, 0.71 v6.0.0, 0.43 v5.5.0, 0.67 v5.4.0, 0.80 v5.2.0, 1.00 v3.7.0
% 0.20/0.45 % Syntax : Number of formulae : 593 ( 279 unt; 313 typ; 279 def)
% 0.20/0.45 % Number of atoms : 1657 ( 407 equ; 0 cnn)
% 0.20/0.45 % Maximal formula atoms : 280 ( 5 avg)
% 0.20/0.45 % Number of connectives : 3486 ( 73 ~; 12 |; 44 &;2511 @)
% 0.20/0.45 % ( 14 <=>; 832 =>; 0 <=; 0 <~>)
% 0.20/0.45 % Maximal formula depth : 288 ( 2 avg)
% 0.20/0.45 % Number of types : 2 ( 0 usr)
% 0.20/0.45 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.20/0.45 % Number of symbols : 316 ( 313 usr; 279 con; 0-4 aty)
% 0.20/0.45 % Number of variables : 952 ( 74 ^; 843 !; 35 ?; 952 :)
% 0.20/0.45 % SPC : TH0_THM_EQU_NAR
% 0.20/0.45
% 0.20/0.45 % Comments : http://mathgate.info/detsetitem.php?id=320
% 0.20/0.45 %------------------------------------------------------------------------------
% 0.20/0.45 thf(in_type,type,
% 0.20/0.45 in: $i > $i > $o ).
% 0.20/0.45
% 0.20/0.45 thf(exu_type,type,
% 0.20/0.45 exu: ( $i > $o ) > $o ).
% 0.20/0.45
% 0.20/0.45 thf(exu,definition,
% 0.20/0.45 ( exu
% 0.20/0.45 = ( ^ [Xphi: $i > $o] :
% 0.20/0.45 ? [Xx: $i] :
% 0.20/0.45 ( ( Xphi @ Xx )
% 0.20/0.45 & ! [Xy: $i] :
% 0.20/0.45 ( ( Xphi @ Xy )
% 0.20/0.45 => ( Xx = Xy ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(setextAx_type,type,
% 0.20/0.45 setextAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(setextAx,definition,
% 0.20/0.45 ( setextAx
% 0.20/0.45 = ( ! [A: $i,B: $i] :
% 0.20/0.45 ( ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 <=> ( in @ Xx @ B ) )
% 0.20/0.45 => ( A = B ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(emptyset_type,type,
% 0.20/0.45 emptyset: $i ).
% 0.20/0.45
% 0.20/0.45 thf(emptysetAx_type,type,
% 0.20/0.45 emptysetAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(emptysetAx,definition,
% 0.20/0.45 ( emptysetAx
% 0.20/0.45 = ( ! [Xx: $i] :
% 0.20/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(setadjoin_type,type,
% 0.20/0.45 setadjoin: $i > $i > $i ).
% 0.20/0.45
% 0.20/0.45 thf(setadjoinAx_type,type,
% 0.20/0.45 setadjoinAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(setadjoinAx,definition,
% 0.20/0.45 ( setadjoinAx
% 0.20/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.20/0.45 <=> ( ( Xy = Xx )
% 0.20/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(powerset_type,type,
% 0.20/0.45 powerset: $i > $i ).
% 0.20/0.45
% 0.20/0.45 thf(powersetAx_type,type,
% 0.20/0.45 powersetAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(powersetAx,definition,
% 0.20/0.45 ( powersetAx
% 0.20/0.45 = ( ! [A: $i,B: $i] :
% 0.20/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.20/0.45 <=> ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ B )
% 0.20/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(setunion_type,type,
% 0.20/0.45 setunion: $i > $i ).
% 0.20/0.45
% 0.20/0.45 thf(setunionAx_type,type,
% 0.20/0.45 setunionAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(setunionAx,definition,
% 0.20/0.45 ( setunionAx
% 0.20/0.45 = ( ! [A: $i,Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.20/0.45 <=> ? [B: $i] :
% 0.20/0.45 ( ( in @ Xx @ B )
% 0.20/0.45 & ( in @ B @ A ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(omega_type,type,
% 0.20/0.45 omega: $i ).
% 0.20/0.45
% 0.20/0.45 thf(omega0Ax_type,type,
% 0.20/0.45 omega0Ax: $o ).
% 0.20/0.45
% 0.20/0.45 thf(omega0Ax,definition,
% 0.20/0.45 ( omega0Ax
% 0.20/0.45 = ( in @ emptyset @ omega ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(omegaSAx_type,type,
% 0.20/0.45 omegaSAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(omegaSAx,definition,
% 0.20/0.45 ( omegaSAx
% 0.20/0.45 = ( ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ omega )
% 0.20/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(omegaIndAx_type,type,
% 0.20/0.45 omegaIndAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(omegaIndAx,definition,
% 0.20/0.45 ( omegaIndAx
% 0.20/0.45 = ( ! [A: $i] :
% 0.20/0.45 ( ( ( in @ emptyset @ A )
% 0.20/0.45 & ! [Xx: $i] :
% 0.20/0.45 ( ( ( in @ Xx @ omega )
% 0.20/0.45 & ( in @ Xx @ A ) )
% 0.20/0.45 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.20/0.45 => ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ omega )
% 0.20/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(replAx_type,type,
% 0.20/0.45 replAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(replAx,definition,
% 0.20/0.45 ( replAx
% 0.20/0.45 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.20/0.45 ( ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 => ( exu
% 0.20/0.45 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.20/0.45 => ? [B: $i] :
% 0.20/0.45 ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ B )
% 0.20/0.45 <=> ? [Xy: $i] :
% 0.20/0.45 ( ( in @ Xy @ A )
% 0.20/0.45 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(foundationAx_type,type,
% 0.20/0.45 foundationAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(foundationAx,definition,
% 0.20/0.45 ( foundationAx
% 0.20/0.45 = ( ! [A: $i] :
% 0.20/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.20/0.45 => ? [B: $i] :
% 0.20/0.45 ( ( in @ B @ A )
% 0.20/0.45 & ~ ? [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ B )
% 0.20/0.45 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(wellorderingAx_type,type,
% 0.20/0.45 wellorderingAx: $o ).
% 0.20/0.45
% 0.20/0.45 thf(wellorderingAx,definition,
% 0.20/0.45 ( wellorderingAx
% 0.20/0.45 = ( ! [A: $i] :
% 0.20/0.45 ? [B: $i] :
% 0.20/0.45 ( ! [C: $i] :
% 0.20/0.45 ( ( in @ C @ B )
% 0.20/0.45 => ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ C )
% 0.20/0.45 => ( in @ Xx @ A ) ) )
% 0.20/0.45 & ! [Xx: $i,Xy: $i] :
% 0.20/0.45 ( ( ( in @ Xx @ A )
% 0.20/0.45 & ( in @ Xy @ A ) )
% 0.20/0.45 => ( ! [C: $i] :
% 0.20/0.45 ( ( in @ C @ B )
% 0.20/0.45 => ( ( in @ Xx @ C )
% 0.20/0.45 <=> ( in @ Xy @ C ) ) )
% 0.20/0.45 => ( Xx = Xy ) ) )
% 0.20/0.45 & ! [C: $i,D: $i] :
% 0.20/0.45 ( ( ( in @ C @ B )
% 0.20/0.45 & ( in @ D @ B ) )
% 0.20/0.45 => ( ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ C )
% 0.20/0.45 => ( in @ Xx @ D ) )
% 0.20/0.45 | ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ D )
% 0.20/0.45 => ( in @ Xx @ C ) ) ) )
% 0.20/0.45 & ! [C: $i] :
% 0.20/0.45 ( ( ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ C )
% 0.20/0.45 => ( in @ Xx @ A ) )
% 0.20/0.45 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.20/0.45 => ? [D: $i,Xx: $i] :
% 0.20/0.45 ( ( in @ D @ B )
% 0.20/0.45 & ( in @ Xx @ C )
% 0.20/0.45 & ~ ? [Xy: $i] :
% 0.20/0.45 ( ( in @ Xy @ D )
% 0.20/0.45 & ( in @ Xy @ C ) )
% 0.20/0.45 & ! [E: $i] :
% 0.20/0.45 ( ( in @ E @ B )
% 0.20/0.45 => ( ! [Xy: $i] :
% 0.20/0.45 ( ( in @ Xy @ E )
% 0.20/0.45 => ( in @ Xy @ D ) )
% 0.20/0.45 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(descr_type,type,
% 0.20/0.45 descr: ( $i > $o ) > $i ).
% 0.20/0.45
% 0.20/0.45 thf(descrp_type,type,
% 0.20/0.45 descrp: $o ).
% 0.20/0.45
% 0.20/0.45 thf(descrp,definition,
% 0.20/0.45 ( descrp
% 0.20/0.45 = ( ! [Xphi: $i > $o] :
% 0.20/0.45 ( ( exu
% 0.20/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.45 => ( Xphi
% 0.20/0.45 @ ( descr
% 0.20/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(dsetconstr_type,type,
% 0.20/0.45 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.20/0.45
% 0.20/0.45 thf(dsetconstrI_type,type,
% 0.20/0.45 dsetconstrI: $o ).
% 0.20/0.45
% 0.20/0.45 thf(dsetconstrI,definition,
% 0.20/0.45 ( dsetconstrI
% 0.20/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 => ( ( Xphi @ Xx )
% 0.20/0.45 => ( in @ Xx
% 0.20/0.45 @ ( dsetconstr @ A
% 0.20/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(dsetconstrEL_type,type,
% 0.20/0.45 dsetconstrEL: $o ).
% 0.20/0.45
% 0.20/0.45 thf(dsetconstrEL,definition,
% 0.20/0.45 ( dsetconstrEL
% 0.20/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.45 ( ( in @ Xx
% 0.20/0.45 @ ( dsetconstr @ A
% 0.20/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.20/0.45 => ( in @ Xx @ A ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(dsetconstrER_type,type,
% 0.20/0.45 dsetconstrER: $o ).
% 0.20/0.45
% 0.20/0.45 thf(dsetconstrER,definition,
% 0.20/0.45 ( dsetconstrER
% 0.20/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.45 ( ( in @ Xx
% 0.20/0.45 @ ( dsetconstr @ A
% 0.20/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.20/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(exuE1_type,type,
% 0.20/0.45 exuE1: $o ).
% 0.20/0.45
% 0.20/0.45 thf(exuE1,definition,
% 0.20/0.45 ( exuE1
% 0.20/0.45 = ( ! [Xphi: $i > $o] :
% 0.20/0.45 ( ( exu
% 0.20/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.45 => ? [Xx: $i] :
% 0.20/0.45 ( ( Xphi @ Xx )
% 0.20/0.45 & ! [Xy: $i] :
% 0.20/0.45 ( ( Xphi @ Xy )
% 0.20/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(prop2set_type,type,
% 0.20/0.45 prop2set: $o > $i ).
% 0.20/0.45
% 0.20/0.45 thf(prop2setE_type,type,
% 0.20/0.45 prop2setE: $o ).
% 0.20/0.45
% 0.20/0.45 thf(prop2setE,definition,
% 0.20/0.45 ( prop2setE
% 0.20/0.45 = ( ! [Xphi: $o,Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.20/0.45 => Xphi ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(emptysetE_type,type,
% 0.20/0.45 emptysetE: $o ).
% 0.20/0.45
% 0.20/0.45 thf(emptysetE,definition,
% 0.20/0.45 ( emptysetE
% 0.20/0.45 = ( ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ emptyset )
% 0.20/0.45 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(emptysetimpfalse_type,type,
% 0.20/0.45 emptysetimpfalse: $o ).
% 0.20/0.45
% 0.20/0.45 thf(emptysetimpfalse,definition,
% 0.20/0.45 ( emptysetimpfalse
% 0.20/0.45 = ( ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ emptyset )
% 0.20/0.45 => $false ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(notinemptyset_type,type,
% 0.20/0.45 notinemptyset: $o ).
% 0.20/0.45
% 0.20/0.45 thf(notinemptyset,definition,
% 0.20/0.45 ( notinemptyset
% 0.20/0.45 = ( ! [Xx: $i] :
% 0.20/0.45 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(exuE3e_type,type,
% 0.20/0.45 exuE3e: $o ).
% 0.20/0.45
% 0.20/0.45 thf(exuE3e,definition,
% 0.20/0.45 ( exuE3e
% 0.20/0.45 = ( ! [Xphi: $i > $o] :
% 0.20/0.45 ( ( exu
% 0.20/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.45 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(setext_type,type,
% 0.20/0.45 setext: $o ).
% 0.20/0.45
% 0.20/0.45 thf(setext,definition,
% 0.20/0.45 ( setext
% 0.20/0.45 = ( ! [A: $i,B: $i] :
% 0.20/0.45 ( ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 => ( in @ Xx @ B ) )
% 0.20/0.45 => ( ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ B )
% 0.20/0.45 => ( in @ Xx @ A ) )
% 0.20/0.45 => ( A = B ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(emptyI_type,type,
% 0.20/0.45 emptyI: $o ).
% 0.20/0.45
% 0.20/0.45 thf(emptyI,definition,
% 0.20/0.45 ( emptyI
% 0.20/0.45 = ( ! [A: $i] :
% 0.20/0.45 ( ! [Xx: $i] :
% 0.20/0.45 ~ ( in @ Xx @ A )
% 0.20/0.45 => ( A = emptyset ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(noeltsimpempty_type,type,
% 0.20/0.45 noeltsimpempty: $o ).
% 0.20/0.45
% 0.20/0.45 thf(noeltsimpempty,definition,
% 0.20/0.45 ( noeltsimpempty
% 0.20/0.45 = ( ! [A: $i] :
% 0.20/0.45 ( ! [Xx: $i] :
% 0.20/0.45 ~ ( in @ Xx @ A )
% 0.20/0.45 => ( A = emptyset ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(setbeta_type,type,
% 0.20/0.45 setbeta: $o ).
% 0.20/0.45
% 0.20/0.45 thf(setbeta,definition,
% 0.20/0.45 ( setbeta
% 0.20/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 => ( ( in @ Xx
% 0.20/0.45 @ ( dsetconstr @ A
% 0.20/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.20/0.45 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(nonempty_type,type,
% 0.20/0.45 nonempty: $i > $o ).
% 0.20/0.45
% 0.20/0.45 thf(nonempty,definition,
% 0.20/0.45 ( nonempty
% 0.20/0.45 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(nonemptyE1_type,type,
% 0.20/0.45 nonemptyE1: $o ).
% 0.20/0.45
% 0.20/0.45 thf(nonemptyE1,definition,
% 0.20/0.45 ( nonemptyE1
% 0.20/0.45 = ( ! [A: $i] :
% 0.20/0.45 ( ( nonempty @ A )
% 0.20/0.45 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(nonemptyI_type,type,
% 0.20/0.45 nonemptyI: $o ).
% 0.20/0.45
% 0.20/0.45 thf(nonemptyI,definition,
% 0.20/0.45 ( nonemptyI
% 0.20/0.45 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 => ( ( Xphi @ Xx )
% 0.20/0.45 => ( nonempty
% 0.20/0.45 @ ( dsetconstr @ A
% 0.20/0.45 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(nonemptyI1_type,type,
% 0.20/0.45 nonemptyI1: $o ).
% 0.20/0.45
% 0.20/0.45 thf(nonemptyI1,definition,
% 0.20/0.45 ( nonemptyI1
% 0.20/0.45 = ( ! [A: $i] :
% 0.20/0.45 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.20/0.45 => ( nonempty @ A ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(setadjoinIL_type,type,
% 0.20/0.45 setadjoinIL: $o ).
% 0.20/0.45
% 0.20/0.45 thf(setadjoinIL,definition,
% 0.20/0.45 ( setadjoinIL
% 0.20/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(emptyinunitempty_type,type,
% 0.20/0.45 emptyinunitempty: $o ).
% 0.20/0.45
% 0.20/0.45 thf(emptyinunitempty,definition,
% 0.20/0.45 ( emptyinunitempty
% 0.20/0.45 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(setadjoinIR_type,type,
% 0.20/0.45 setadjoinIR: $o ).
% 0.20/0.45
% 0.20/0.45 thf(setadjoinIR,definition,
% 0.20/0.45 ( setadjoinIR
% 0.20/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.45 ( ( in @ Xy @ A )
% 0.20/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(setadjoinE_type,type,
% 0.20/0.45 setadjoinE: $o ).
% 0.20/0.45
% 0.20/0.45 thf(setadjoinE,definition,
% 0.20/0.45 ( setadjoinE
% 0.20/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.20/0.45 => ! [Xphi: $o] :
% 0.20/0.45 ( ( ( Xy = Xx )
% 0.20/0.45 => Xphi )
% 0.20/0.45 => ( ( ( in @ Xy @ A )
% 0.20/0.45 => Xphi )
% 0.20/0.45 => Xphi ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(setadjoinOr_type,type,
% 0.20/0.45 setadjoinOr: $o ).
% 0.20/0.45
% 0.20/0.45 thf(setadjoinOr,definition,
% 0.20/0.45 ( setadjoinOr
% 0.20/0.45 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.20/0.45 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.20/0.45 => ( ( Xy = Xx )
% 0.20/0.45 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(setoftrueEq_type,type,
% 0.20/0.45 setoftrueEq: $o ).
% 0.20/0.45
% 0.20/0.45 thf(setoftrueEq,definition,
% 0.20/0.45 ( setoftrueEq
% 0.20/0.45 = ( ! [A: $i] :
% 0.20/0.45 ( ( dsetconstr @ A
% 0.20/0.45 @ ^ [Xx: $i] : $true )
% 0.20/0.45 = A ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(powersetI_type,type,
% 0.20/0.45 powersetI: $o ).
% 0.20/0.45
% 0.20/0.45 thf(powersetI,definition,
% 0.20/0.45 ( powersetI
% 0.20/0.45 = ( ! [A: $i,B: $i] :
% 0.20/0.45 ( ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ B )
% 0.20/0.45 => ( in @ Xx @ A ) )
% 0.20/0.45 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(emptyinPowerset_type,type,
% 0.20/0.45 emptyinPowerset: $o ).
% 0.20/0.45
% 0.20/0.45 thf(emptyinPowerset,definition,
% 0.20/0.45 ( emptyinPowerset
% 0.20/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(emptyInPowerset_type,type,
% 0.20/0.45 emptyInPowerset: $o ).
% 0.20/0.45
% 0.20/0.45 thf(emptyInPowerset,definition,
% 0.20/0.45 ( emptyInPowerset
% 0.20/0.45 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(powersetE_type,type,
% 0.20/0.45 powersetE: $o ).
% 0.20/0.45
% 0.20/0.45 thf(powersetE,definition,
% 0.20/0.45 ( powersetE
% 0.20/0.45 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.45 ( ( in @ B @ ( powerset @ A ) )
% 0.20/0.45 => ( ( in @ Xx @ B )
% 0.20/0.45 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(setunionI_type,type,
% 0.20/0.45 setunionI: $o ).
% 0.20/0.45
% 0.20/0.45 thf(setunionI,definition,
% 0.20/0.45 ( setunionI
% 0.20/0.45 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.20/0.45 ( ( in @ Xx @ B )
% 0.20/0.45 => ( ( in @ B @ A )
% 0.20/0.45 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(setunionE_type,type,
% 0.20/0.45 setunionE: $o ).
% 0.20/0.45
% 0.20/0.45 thf(setunionE,definition,
% 0.20/0.45 ( setunionE
% 0.20/0.45 = ( ! [A: $i,Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ ( setunion @ A ) )
% 0.20/0.45 => ! [Xphi: $o] :
% 0.20/0.45 ( ! [B: $i] :
% 0.20/0.45 ( ( in @ Xx @ B )
% 0.20/0.45 => ( ( in @ B @ A )
% 0.20/0.45 => Xphi ) )
% 0.20/0.45 => Xphi ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(subPowSU_type,type,
% 0.20/0.45 subPowSU: $o ).
% 0.20/0.45
% 0.20/0.45 thf(subPowSU,definition,
% 0.20/0.45 ( subPowSU
% 0.20/0.45 = ( ! [A: $i,Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(exuE2_type,type,
% 0.20/0.45 exuE2: $o ).
% 0.20/0.45
% 0.20/0.45 thf(exuE2,definition,
% 0.20/0.45 ( exuE2
% 0.20/0.45 = ( ! [Xphi: $i > $o] :
% 0.20/0.45 ( ( exu
% 0.20/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.45 => ? [Xx: $i] :
% 0.20/0.45 ! [Xy: $i] :
% 0.20/0.45 ( ( Xphi @ Xy )
% 0.20/0.45 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(nonemptyImpWitness_type,type,
% 0.20/0.45 nonemptyImpWitness: $o ).
% 0.20/0.45
% 0.20/0.45 thf(nonemptyImpWitness,definition,
% 0.20/0.45 ( nonemptyImpWitness
% 0.20/0.45 = ( ! [A: $i] :
% 0.20/0.45 ( ( nonempty @ A )
% 0.20/0.45 => ? [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 & $true ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(uniqinunit_type,type,
% 0.20/0.45 uniqinunit: $o ).
% 0.20/0.45
% 0.20/0.45 thf(uniqinunit,definition,
% 0.20/0.45 ( uniqinunit
% 0.20/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.20/0.45 => ( Xx = Xy ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(notinsingleton_type,type,
% 0.20/0.45 notinsingleton: $o ).
% 0.20/0.45
% 0.20/0.45 thf(notinsingleton,definition,
% 0.20/0.45 ( notinsingleton
% 0.20/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.45 ( ( Xx != Xy )
% 0.20/0.45 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(eqinunit_type,type,
% 0.20/0.45 eqinunit: $o ).
% 0.20/0.45
% 0.20/0.45 thf(eqinunit,definition,
% 0.20/0.45 ( eqinunit
% 0.20/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.45 ( ( Xx = Xy )
% 0.20/0.45 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(singletonsswitch_type,type,
% 0.20/0.45 singletonsswitch: $o ).
% 0.20/0.45
% 0.20/0.45 thf(singletonsswitch,definition,
% 0.20/0.45 ( singletonsswitch
% 0.20/0.45 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.45 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.20/0.45 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(upairsetE_type,type,
% 0.20/0.45 upairsetE: $o ).
% 0.20/0.45
% 0.20/0.45 thf(upairsetE,definition,
% 0.20/0.45 ( upairsetE
% 0.20/0.45 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.20/0.45 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.20/0.45 => ( ( Xz = Xx )
% 0.20/0.45 | ( Xz = Xy ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(upairsetIL_type,type,
% 0.20/0.45 upairsetIL: $o ).
% 0.20/0.45
% 0.20/0.45 thf(upairsetIL,definition,
% 0.20/0.45 ( upairsetIL
% 0.20/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(upairsetIR_type,type,
% 0.20/0.45 upairsetIR: $o ).
% 0.20/0.45
% 0.20/0.45 thf(upairsetIR,definition,
% 0.20/0.45 ( upairsetIR
% 0.20/0.45 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(emptyE1_type,type,
% 0.20/0.45 emptyE1: $o ).
% 0.20/0.45
% 0.20/0.45 thf(emptyE1,definition,
% 0.20/0.45 ( emptyE1
% 0.20/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.45 ( ? [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 & ( Xphi @ Xx ) )
% 0.20/0.45 => ( ( ( dsetconstr @ A
% 0.20/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.45 = emptyset )
% 0.20/0.45 => $false ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(vacuousDall_type,type,
% 0.20/0.45 vacuousDall: $o ).
% 0.20/0.45
% 0.20/0.45 thf(vacuousDall,definition,
% 0.20/0.45 ( vacuousDall
% 0.20/0.45 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ emptyset )
% 0.20/0.45 => ( Xphi @ Xx ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(quantDeMorgan1_type,type,
% 0.20/0.45 quantDeMorgan1: $o ).
% 0.20/0.45
% 0.20/0.45 thf(quantDeMorgan1,definition,
% 0.20/0.45 ( quantDeMorgan1
% 0.20/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.45 ( ~ ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 => ( Xphi @ Xx ) )
% 0.20/0.45 => ? [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(quantDeMorgan2_type,type,
% 0.20/0.45 quantDeMorgan2: $o ).
% 0.20/0.45
% 0.20/0.45 thf(quantDeMorgan2,definition,
% 0.20/0.45 ( quantDeMorgan2
% 0.20/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.45 ( ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 => ~ ( Xphi @ Xx ) )
% 0.20/0.45 => ~ ? [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(quantDeMorgan3_type,type,
% 0.20/0.45 quantDeMorgan3: $o ).
% 0.20/0.45
% 0.20/0.45 thf(quantDeMorgan3,definition,
% 0.20/0.45 ( quantDeMorgan3
% 0.20/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.45 ( ~ ? [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 & ( Xphi @ Xx ) )
% 0.20/0.45 => ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(quantDeMorgan4_type,type,
% 0.20/0.45 quantDeMorgan4: $o ).
% 0.20/0.45
% 0.20/0.45 thf(quantDeMorgan4,definition,
% 0.20/0.45 ( quantDeMorgan4
% 0.20/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.45 ( ? [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 & ~ ( Xphi @ Xx ) )
% 0.20/0.45 => ~ ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(prop2setI_type,type,
% 0.20/0.45 prop2setI: $o ).
% 0.20/0.45
% 0.20/0.45 thf(prop2setI,definition,
% 0.20/0.45 ( prop2setI
% 0.20/0.45 = ( ! [Xphi: $o] :
% 0.20/0.45 ( Xphi
% 0.20/0.45 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(set2prop_type,type,
% 0.20/0.45 set2prop: $i > $o ).
% 0.20/0.45
% 0.20/0.45 thf(prop2set2propI_type,type,
% 0.20/0.45 prop2set2propI: $o ).
% 0.20/0.45
% 0.20/0.45 thf(prop2set2propI,definition,
% 0.20/0.45 ( prop2set2propI
% 0.20/0.45 = ( ! [Xphi: $o] :
% 0.20/0.45 ( Xphi
% 0.20/0.45 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(notdexE_type,type,
% 0.20/0.45 notdexE: $o ).
% 0.20/0.45
% 0.20/0.45 thf(notdexE,definition,
% 0.20/0.45 ( notdexE
% 0.20/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.45 ( ~ ? [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 & ( Xphi @ Xx ) )
% 0.20/0.45 => ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(notdallE_type,type,
% 0.20/0.45 notdallE: $o ).
% 0.20/0.45
% 0.20/0.45 thf(notdallE,definition,
% 0.20/0.45 ( notdallE
% 0.20/0.45 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.45 ( ~ ! [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 => ( Xphi @ Xx ) )
% 0.20/0.45 => ? [Xx: $i] :
% 0.20/0.45 ( ( in @ Xx @ A )
% 0.20/0.45 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(exuI1_type,type,
% 0.20/0.45 exuI1: $o ).
% 0.20/0.45
% 0.20/0.45 thf(exuI1,definition,
% 0.20/0.45 ( exuI1
% 0.20/0.45 = ( ! [Xphi: $i > $o] :
% 0.20/0.45 ( ? [Xx: $i] :
% 0.20/0.45 ( ( Xphi @ Xx )
% 0.20/0.45 & ! [Xy: $i] :
% 0.20/0.45 ( ( Xphi @ Xy )
% 0.20/0.45 => ( Xx = Xy ) ) )
% 0.20/0.45 => ( exu
% 0.20/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(exuI3_type,type,
% 0.20/0.45 exuI3: $o ).
% 0.20/0.45
% 0.20/0.45 thf(exuI3,definition,
% 0.20/0.45 ( exuI3
% 0.20/0.45 = ( ! [Xphi: $i > $o] :
% 0.20/0.45 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.20/0.45 => ( ! [Xx: $i,Xy: $i] :
% 0.20/0.45 ( ( Xphi @ Xx )
% 0.20/0.45 => ( ( Xphi @ Xy )
% 0.20/0.45 => ( Xx = Xy ) ) )
% 0.20/0.45 => ( exu
% 0.20/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(exuI2_type,type,
% 0.20/0.45 exuI2: $o ).
% 0.20/0.45
% 0.20/0.45 thf(exuI2,definition,
% 0.20/0.45 ( exuI2
% 0.20/0.45 = ( ! [Xphi: $i > $o] :
% 0.20/0.45 ( ? [Xx: $i] :
% 0.20/0.45 ! [Xy: $i] :
% 0.20/0.45 ( ( Xphi @ Xy )
% 0.20/0.45 <=> ( Xy = Xx ) )
% 0.20/0.45 => ( exu
% 0.20/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(inCongP_type,type,
% 0.20/0.45 inCongP: $o ).
% 0.20/0.45
% 0.20/0.45 thf(inCongP,definition,
% 0.20/0.45 ( inCongP
% 0.20/0.45 = ( ! [A: $i,B: $i] :
% 0.20/0.45 ( ( A = B )
% 0.20/0.45 => ! [Xx: $i,Xy: $i] :
% 0.20/0.45 ( ( Xx = Xy )
% 0.20/0.45 => ( ( in @ Xx @ A )
% 0.20/0.45 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(in__Cong_type,type,
% 0.20/0.45 in__Cong: $o ).
% 0.20/0.45
% 0.20/0.45 thf(in__Cong,definition,
% 0.20/0.45 ( in__Cong
% 0.20/0.45 = ( ! [A: $i,B: $i] :
% 0.20/0.45 ( ( A = B )
% 0.20/0.45 => ! [Xx: $i,Xy: $i] :
% 0.20/0.45 ( ( Xx = Xy )
% 0.20/0.45 => ( ( in @ Xx @ A )
% 0.20/0.45 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(exuE3u_type,type,
% 0.20/0.45 exuE3u: $o ).
% 0.20/0.45
% 0.20/0.45 thf(exuE3u,definition,
% 0.20/0.45 ( exuE3u
% 0.20/0.45 = ( ! [Xphi: $i > $o] :
% 0.20/0.45 ( ( exu
% 0.20/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.45 => ! [Xx: $i,Xy: $i] :
% 0.20/0.45 ( ( Xphi @ Xx )
% 0.20/0.45 => ( ( Xphi @ Xy )
% 0.20/0.45 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(exu__Cong_type,type,
% 0.20/0.45 exu__Cong: $o ).
% 0.20/0.45
% 0.20/0.45 thf(exu__Cong,definition,
% 0.20/0.45 ( exu__Cong
% 0.20/0.45 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.20/0.45 ( ! [Xx: $i,Xy: $i] :
% 0.20/0.45 ( ( Xx = Xy )
% 0.20/0.45 => ( ( Xphi @ Xx )
% 0.20/0.45 <=> ( Xpsi @ Xy ) ) )
% 0.20/0.45 => ( ( exu
% 0.20/0.45 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.45 <=> ( exu
% 0.20/0.45 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(emptyset__Cong_type,type,
% 0.20/0.45 emptyset__Cong: $o ).
% 0.20/0.45
% 0.20/0.45 thf(emptyset__Cong,definition,
% 0.20/0.45 ( emptyset__Cong
% 0.20/0.45 = ( emptyset = emptyset ) ) ).
% 0.20/0.45
% 0.20/0.45 thf(setadjoin__Cong_type,type,
% 0.20/0.45 setadjoin__Cong: $o ).
% 0.20/0.45
% 0.20/0.45 thf(setadjoin__Cong,definition,
% 0.20/0.46 ( setadjoin__Cong
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( Xx = Xy )
% 0.20/0.46 => ! [Xz: $i,Xu: $i] :
% 0.20/0.46 ( ( Xz = Xu )
% 0.20/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.20/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(powerset__Cong_type,type,
% 0.20/0.46 powerset__Cong: $o ).
% 0.20/0.46
% 0.20/0.46 thf(powerset__Cong,definition,
% 0.20/0.46 ( powerset__Cong
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( A = B )
% 0.20/0.46 => ( ( powerset @ A )
% 0.20/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setunion__Cong_type,type,
% 0.20/0.46 setunion__Cong: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setunion__Cong,definition,
% 0.20/0.46 ( setunion__Cong
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( A = B )
% 0.20/0.46 => ( ( setunion @ A )
% 0.20/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(omega__Cong_type,type,
% 0.20/0.46 omega__Cong: $o ).
% 0.20/0.46
% 0.20/0.46 thf(omega__Cong,definition,
% 0.20/0.46 ( omega__Cong
% 0.20/0.46 = ( omega = omega ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(exuEu_type,type,
% 0.20/0.46 exuEu: $o ).
% 0.20/0.46
% 0.20/0.46 thf(exuEu,definition,
% 0.20/0.46 ( exuEu
% 0.20/0.46 = ( ! [Xphi: $i > $o] :
% 0.20/0.46 ( ( exu
% 0.20/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.46 => ! [Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( Xphi @ Xx )
% 0.20/0.46 => ( ( Xphi @ Xy )
% 0.20/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(descr__Cong_type,type,
% 0.20/0.46 descr__Cong: $o ).
% 0.20/0.46
% 0.20/0.46 thf(descr__Cong,definition,
% 0.20/0.46 ( descr__Cong
% 0.20/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.20/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( Xx = Xy )
% 0.20/0.46 => ( ( Xphi @ Xx )
% 0.20/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.20/0.46 => ( ( exu
% 0.20/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.46 => ( ( exu
% 0.20/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.20/0.46 => ( ( descr
% 0.20/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.46 = ( descr
% 0.20/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(dsetconstr__Cong_type,type,
% 0.20/0.46 dsetconstr__Cong: $o ).
% 0.20/0.46
% 0.20/0.46 thf(dsetconstr__Cong,definition,
% 0.20/0.46 ( dsetconstr__Cong
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( A = B )
% 0.20/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( ( Xx = Xy )
% 0.20/0.46 => ( ( Xphi @ Xx )
% 0.20/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.20/0.46 => ( ( dsetconstr @ A
% 0.20/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.46 = ( dsetconstr @ B
% 0.20/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(subset_type,type,
% 0.20/0.46 subset: $i > $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(disjoint_type,type,
% 0.20/0.46 disjoint: $i > $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(setsmeet_type,type,
% 0.20/0.46 setsmeet: $i > $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(subsetI1_type,type,
% 0.20/0.46 subsetI1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(subsetI1,definition,
% 0.20/0.46 ( subsetI1
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ Xx @ B ) )
% 0.20/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(eqimpsubset2_type,type,
% 0.20/0.46 eqimpsubset2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(eqimpsubset2,definition,
% 0.20/0.46 ( eqimpsubset2
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( A = B )
% 0.20/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(eqimpsubset1_type,type,
% 0.20/0.46 eqimpsubset1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(eqimpsubset1,definition,
% 0.20/0.46 ( eqimpsubset1
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( A = B )
% 0.20/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(subsetI2_type,type,
% 0.20/0.46 subsetI2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(subsetI2,definition,
% 0.20/0.46 ( subsetI2
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ Xx @ B ) )
% 0.20/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(emptysetsubset_type,type,
% 0.20/0.46 emptysetsubset: $o ).
% 0.20/0.46
% 0.20/0.46 thf(emptysetsubset,definition,
% 0.20/0.46 ( emptysetsubset
% 0.20/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(subsetE_type,type,
% 0.20/0.46 subsetE: $o ).
% 0.20/0.46
% 0.20/0.46 thf(subsetE,definition,
% 0.20/0.46 ( subsetE
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( subset @ A @ B )
% 0.20/0.46 => ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(subsetE2_type,type,
% 0.20/0.46 subsetE2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(subsetE2,definition,
% 0.20/0.46 ( subsetE2
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( subset @ A @ B )
% 0.20/0.46 => ( ~ ( in @ Xx @ B )
% 0.20/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(notsubsetI_type,type,
% 0.20/0.46 notsubsetI: $o ).
% 0.20/0.46
% 0.20/0.46 thf(notsubsetI,definition,
% 0.20/0.46 ( notsubsetI
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ~ ( in @ Xx @ B )
% 0.20/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(notequalI1_type,type,
% 0.20/0.46 notequalI1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(notequalI1,definition,
% 0.20/0.46 ( notequalI1
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ~ ( subset @ A @ B )
% 0.20/0.46 => ( A != B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(notequalI2_type,type,
% 0.20/0.46 notequalI2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(notequalI2,definition,
% 0.20/0.46 ( notequalI2
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ~ ( in @ Xx @ B )
% 0.20/0.46 => ( A != B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(subsetRefl_type,type,
% 0.20/0.46 subsetRefl: $o ).
% 0.20/0.46
% 0.20/0.46 thf(subsetRefl,definition,
% 0.20/0.46 ( subsetRefl
% 0.20/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(subsetTrans_type,type,
% 0.20/0.46 subsetTrans: $o ).
% 0.20/0.46
% 0.20/0.46 thf(subsetTrans,definition,
% 0.20/0.46 ( subsetTrans
% 0.20/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.20/0.46 ( ( subset @ A @ B )
% 0.20/0.46 => ( ( subset @ B @ C )
% 0.20/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setadjoinSub_type,type,
% 0.20/0.46 setadjoinSub: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setadjoinSub,definition,
% 0.20/0.46 ( setadjoinSub
% 0.20/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setadjoinSub2_type,type,
% 0.20/0.46 setadjoinSub2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setadjoinSub2,definition,
% 0.20/0.46 ( setadjoinSub2
% 0.20/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.20/0.46 ( ( subset @ A @ B )
% 0.20/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(subset2powerset_type,type,
% 0.20/0.46 subset2powerset: $o ).
% 0.20/0.46
% 0.20/0.46 thf(subset2powerset,definition,
% 0.20/0.46 ( subset2powerset
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( subset @ A @ B )
% 0.20/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setextsub_type,type,
% 0.20/0.46 setextsub: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setextsub,definition,
% 0.20/0.46 ( setextsub
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( subset @ A @ B )
% 0.20/0.46 => ( ( subset @ B @ A )
% 0.20/0.46 => ( A = B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(subsetemptysetimpeq_type,type,
% 0.20/0.46 subsetemptysetimpeq: $o ).
% 0.20/0.46
% 0.20/0.46 thf(subsetemptysetimpeq,definition,
% 0.20/0.46 ( subsetemptysetimpeq
% 0.20/0.46 = ( ! [A: $i] :
% 0.20/0.46 ( ( subset @ A @ emptyset )
% 0.20/0.46 => ( A = emptyset ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(powersetI1_type,type,
% 0.20/0.46 powersetI1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(powersetI1,definition,
% 0.20/0.46 ( powersetI1
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( subset @ B @ A )
% 0.20/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(powersetE1_type,type,
% 0.20/0.46 powersetE1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(powersetE1,definition,
% 0.20/0.46 ( powersetE1
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.20/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(inPowerset_type,type,
% 0.20/0.46 inPowerset: $o ).
% 0.20/0.46
% 0.20/0.46 thf(inPowerset,definition,
% 0.20/0.46 ( inPowerset
% 0.20/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(powersetsubset_type,type,
% 0.20/0.46 powersetsubset: $o ).
% 0.20/0.46
% 0.20/0.46 thf(powersetsubset,definition,
% 0.20/0.46 ( powersetsubset
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( subset @ A @ B )
% 0.20/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sepInPowerset_type,type,
% 0.20/0.46 sepInPowerset: $o ).
% 0.20/0.46
% 0.20/0.46 thf(sepInPowerset,definition,
% 0.20/0.46 ( sepInPowerset
% 0.20/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.46 ( in
% 0.20/0.46 @ ( dsetconstr @ A
% 0.20/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(sepSubset_type,type,
% 0.20/0.46 sepSubset: $o ).
% 0.20/0.46
% 0.20/0.46 thf(sepSubset,definition,
% 0.20/0.46 ( sepSubset
% 0.20/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.46 ( subset
% 0.20/0.46 @ ( dsetconstr @ A
% 0.20/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.46 @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binunion_type,type,
% 0.20/0.46 binunion: $i > $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(binunionIL_type,type,
% 0.20/0.46 binunionIL: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binunionIL,definition,
% 0.20/0.46 ( binunionIL
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(upairset2IR_type,type,
% 0.20/0.46 upairset2IR: $o ).
% 0.20/0.46
% 0.20/0.46 thf(upairset2IR,definition,
% 0.20/0.46 ( upairset2IR
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binunionIR_type,type,
% 0.20/0.46 binunionIR: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binunionIR,definition,
% 0.20/0.46 ( binunionIR
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ B )
% 0.20/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binunionEcases_type,type,
% 0.20/0.46 binunionEcases: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binunionEcases,definition,
% 0.20/0.46 ( binunionEcases
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.20/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.20/0.46 => ( ( ( in @ Xx @ A )
% 0.20/0.46 => Xphi )
% 0.20/0.46 => ( ( ( in @ Xx @ B )
% 0.20/0.46 => Xphi )
% 0.20/0.46 => Xphi ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binunionE_type,type,
% 0.20/0.46 binunionE: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binunionE,definition,
% 0.20/0.46 ( binunionE
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.20/0.46 => ( ( in @ Xx @ A )
% 0.20/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binunionLsub_type,type,
% 0.20/0.46 binunionLsub: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binunionLsub,definition,
% 0.20/0.46 ( binunionLsub
% 0.20/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binunionRsub_type,type,
% 0.20/0.46 binunionRsub: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binunionRsub,definition,
% 0.20/0.46 ( binunionRsub
% 0.20/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binintersect_type,type,
% 0.20/0.46 binintersect: $i > $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectI_type,type,
% 0.20/0.46 binintersectI: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectI,definition,
% 0.20/0.46 ( binintersectI
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ B )
% 0.20/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectSubset5_type,type,
% 0.20/0.46 binintersectSubset5: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectSubset5,definition,
% 0.20/0.46 ( binintersectSubset5
% 0.20/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.20/0.46 ( ( subset @ C @ A )
% 0.20/0.46 => ( ( subset @ C @ B )
% 0.20/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectEL_type,type,
% 0.20/0.46 binintersectEL: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectEL,definition,
% 0.20/0.46 ( binintersectEL
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.20/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectLsub_type,type,
% 0.20/0.46 binintersectLsub: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectLsub,definition,
% 0.20/0.46 ( binintersectLsub
% 0.20/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectSubset2_type,type,
% 0.20/0.46 binintersectSubset2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectSubset2,definition,
% 0.20/0.46 ( binintersectSubset2
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( subset @ A @ B )
% 0.20/0.46 => ( ( binintersect @ A @ B )
% 0.20/0.46 = A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectSubset3_type,type,
% 0.20/0.46 binintersectSubset3: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectSubset3,definition,
% 0.20/0.46 ( binintersectSubset3
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( ( binintersect @ A @ B )
% 0.20/0.46 = B )
% 0.20/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectER_type,type,
% 0.20/0.46 binintersectER: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectER,definition,
% 0.20/0.46 ( binintersectER
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.20/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(disjointsetsI1_type,type,
% 0.20/0.46 disjointsetsI1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(disjointsetsI1,definition,
% 0.20/0.46 ( disjointsetsI1
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ~ ? [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 & ( in @ Xx @ B ) )
% 0.20/0.46 => ( ( binintersect @ A @ B )
% 0.20/0.46 = emptyset ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectRsub_type,type,
% 0.20/0.46 binintersectRsub: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectRsub,definition,
% 0.20/0.46 ( binintersectRsub
% 0.20/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectSubset4_type,type,
% 0.20/0.46 binintersectSubset4: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectSubset4,definition,
% 0.20/0.46 ( binintersectSubset4
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( subset @ B @ A )
% 0.20/0.46 => ( ( binintersect @ A @ B )
% 0.20/0.46 = B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectSubset1_type,type,
% 0.20/0.46 binintersectSubset1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectSubset1,definition,
% 0.20/0.46 ( binintersectSubset1
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( ( binintersect @ A @ B )
% 0.20/0.46 = A )
% 0.20/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(bs114d_type,type,
% 0.20/0.46 bs114d: $o ).
% 0.20/0.46
% 0.20/0.46 thf(bs114d,definition,
% 0.20/0.46 ( bs114d
% 0.20/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.20/0.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.20/0.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(regular_type,type,
% 0.20/0.46 regular: $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(setminus_type,type,
% 0.20/0.46 setminus: $i > $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(setminusI_type,type,
% 0.20/0.46 setminusI: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setminusI,definition,
% 0.20/0.46 ( setminusI
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ~ ( in @ Xx @ B )
% 0.20/0.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setminusEL_type,type,
% 0.20/0.46 setminusEL: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setminusEL,definition,
% 0.20/0.46 ( setminusEL
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.20/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setminusER_type,type,
% 0.20/0.46 setminusER: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setminusER,definition,
% 0.20/0.46 ( setminusER
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.20/0.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setminusSubset2_type,type,
% 0.20/0.46 setminusSubset2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setminusSubset2,definition,
% 0.20/0.46 ( setminusSubset2
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( subset @ A @ B )
% 0.20/0.46 => ( ( setminus @ A @ B )
% 0.20/0.46 = emptyset ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setminusERneg_type,type,
% 0.20/0.46 setminusERneg: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setminusERneg,definition,
% 0.20/0.46 ( setminusERneg
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.20/0.46 => ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setminusELneg_type,type,
% 0.20/0.46 setminusELneg: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setminusELneg,definition,
% 0.20/0.46 ( setminusELneg
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.20/0.46 => ( ~ ( in @ Xx @ B )
% 0.20/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setminusILneg_type,type,
% 0.20/0.46 setminusILneg: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setminusILneg,definition,
% 0.20/0.46 ( setminusILneg
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ~ ( in @ Xx @ A )
% 0.20/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setminusIRneg_type,type,
% 0.20/0.46 setminusIRneg: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setminusIRneg,definition,
% 0.20/0.46 ( setminusIRneg
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ B )
% 0.20/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setminusLsub_type,type,
% 0.20/0.46 setminusLsub: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setminusLsub,definition,
% 0.20/0.46 ( setminusLsub
% 0.20/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setminusSubset1_type,type,
% 0.20/0.46 setminusSubset1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setminusSubset1,definition,
% 0.20/0.46 ( setminusSubset1
% 0.20/0.46 = ( ! [A: $i,B: $i] :
% 0.20/0.46 ( ( ( setminus @ A @ B )
% 0.20/0.46 = emptyset )
% 0.20/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(symdiff_type,type,
% 0.20/0.46 symdiff: $i > $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(symdiffE_type,type,
% 0.20/0.46 symdiffE: $o ).
% 0.20/0.46
% 0.20/0.46 thf(symdiffE,definition,
% 0.20/0.46 ( symdiffE
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.20/0.46 => ! [Xphi: $o] :
% 0.20/0.46 ( ( ( in @ Xx @ A )
% 0.20/0.46 => ( ~ ( in @ Xx @ B )
% 0.20/0.46 => Xphi ) )
% 0.20/0.46 => ( ( ~ ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ B )
% 0.20/0.46 => Xphi ) )
% 0.20/0.46 => Xphi ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(symdiffI1_type,type,
% 0.20/0.46 symdiffI1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(symdiffI1,definition,
% 0.20/0.46 ( symdiffI1
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ~ ( in @ Xx @ B )
% 0.20/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(symdiffI2_type,type,
% 0.20/0.46 symdiffI2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(symdiffI2,definition,
% 0.20/0.46 ( symdiffI2
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ~ ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ B )
% 0.20/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(symdiffIneg1_type,type,
% 0.20/0.46 symdiffIneg1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(symdiffIneg1,definition,
% 0.20/0.46 ( symdiffIneg1
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ B )
% 0.20/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(symdiffIneg2_type,type,
% 0.20/0.46 symdiffIneg2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(symdiffIneg2,definition,
% 0.20/0.46 ( symdiffIneg2
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ~ ( in @ Xx @ A )
% 0.20/0.46 => ( ~ ( in @ Xx @ B )
% 0.20/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(iskpair_type,type,
% 0.20/0.46 iskpair: $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(secondinupair_type,type,
% 0.20/0.46 secondinupair: $o ).
% 0.20/0.46
% 0.20/0.46 thf(secondinupair,definition,
% 0.20/0.46 ( secondinupair
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairIL_type,type,
% 0.20/0.46 setukpairIL: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairIL,definition,
% 0.20/0.46 ( setukpairIL
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairIR_type,type,
% 0.20/0.46 setukpairIR: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairIR,definition,
% 0.20/0.46 ( setukpairIR
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(kpairiskpair_type,type,
% 0.20/0.46 kpairiskpair: $o ).
% 0.20/0.46
% 0.20/0.46 thf(kpairiskpair,definition,
% 0.20/0.46 ( kpairiskpair
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(kpair_type,type,
% 0.20/0.46 kpair: $i > $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(kpairp_type,type,
% 0.20/0.46 kpairp: $o ).
% 0.20/0.46
% 0.20/0.46 thf(kpairp,definition,
% 0.20/0.46 ( kpairp
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(cartprod_type,type,
% 0.20/0.46 cartprod: $i > $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(singletonsubset_type,type,
% 0.20/0.46 singletonsubset: $o ).
% 0.20/0.46
% 0.20/0.46 thf(singletonsubset,definition,
% 0.20/0.46 ( singletonsubset
% 0.20/0.46 = ( ! [A: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(singletoninpowerset_type,type,
% 0.20/0.46 singletoninpowerset: $o ).
% 0.20/0.46
% 0.20/0.46 thf(singletoninpowerset,definition,
% 0.20/0.46 ( singletoninpowerset
% 0.20/0.46 = ( ! [A: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(singletoninpowunion_type,type,
% 0.20/0.46 singletoninpowunion: $o ).
% 0.20/0.46
% 0.20/0.46 thf(singletoninpowunion,definition,
% 0.20/0.46 ( singletoninpowunion
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(upairset2E_type,type,
% 0.20/0.46 upairset2E: $o ).
% 0.20/0.46
% 0.20/0.46 thf(upairset2E,definition,
% 0.20/0.46 ( upairset2E
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.20/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.20/0.46 => ( ( Xz = Xx )
% 0.20/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(upairsubunion_type,type,
% 0.20/0.46 upairsubunion: $o ).
% 0.20/0.46
% 0.20/0.46 thf(upairsubunion,definition,
% 0.20/0.46 ( upairsubunion
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(upairinpowunion_type,type,
% 0.20/0.46 upairinpowunion: $o ).
% 0.20/0.46
% 0.20/0.46 thf(upairinpowunion,definition,
% 0.20/0.46 ( upairinpowunion
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(ubforcartprodlem1_type,type,
% 0.20/0.46 ubforcartprodlem1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ubforcartprodlem1,definition,
% 0.20/0.46 ( ubforcartprodlem1
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(ubforcartprodlem2_type,type,
% 0.20/0.46 ubforcartprodlem2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ubforcartprodlem2,definition,
% 0.20/0.46 ( ubforcartprodlem2
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(ubforcartprodlem3_type,type,
% 0.20/0.46 ubforcartprodlem3: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ubforcartprodlem3,definition,
% 0.20/0.46 ( ubforcartprodlem3
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodpairin_type,type,
% 0.20/0.46 cartprodpairin: $o ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodpairin,definition,
% 0.20/0.46 ( cartprodpairin
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodmempair1_type,type,
% 0.20/0.46 cartprodmempair1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodmempair1,definition,
% 0.20/0.46 ( cartprodmempair1
% 0.20/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.20/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.20/0.46 => ? [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 & ? [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 & ( Xu
% 0.20/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodmempair_type,type,
% 0.20/0.46 cartprodmempair: $o ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodmempair,definition,
% 0.20/0.46 ( cartprodmempair
% 0.20/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.20/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.20/0.46 => ( iskpair @ Xu ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setunionE2_type,type,
% 0.20/0.46 setunionE2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setunionE2,definition,
% 0.20/0.46 ( setunionE2
% 0.20/0.46 = ( ! [A: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.20/0.46 => ? [X: $i] :
% 0.20/0.46 ( ( in @ X @ A )
% 0.20/0.46 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setunionsingleton1_type,type,
% 0.20/0.46 setunionsingleton1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setunionsingleton1,definition,
% 0.20/0.46 ( setunionsingleton1
% 0.20/0.46 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setunionsingleton2_type,type,
% 0.20/0.46 setunionsingleton2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setunionsingleton2,definition,
% 0.20/0.46 ( setunionsingleton2
% 0.20/0.46 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setunionsingleton_type,type,
% 0.20/0.46 setunionsingleton: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setunionsingleton,definition,
% 0.20/0.46 ( setunionsingleton
% 0.20/0.46 = ( ! [Xx: $i] :
% 0.20/0.46 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.20/0.46 = Xx ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(singleton_type,type,
% 0.20/0.46 singleton: $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(singletonprop_type,type,
% 0.20/0.46 singletonprop: $o ).
% 0.20/0.46
% 0.20/0.46 thf(singletonprop,definition,
% 0.20/0.46 ( singletonprop
% 0.20/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ A )
% 0.20/0.46 => ( ( Xphi @ Xx )
% 0.20/0.46 => ( ( Xphi @ Xy )
% 0.20/0.46 => ( Xx = Xy ) ) ) ) )
% 0.20/0.46 => ( ? [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 & ( Xphi @ Xx ) )
% 0.20/0.46 => ( singleton
% 0.20/0.46 @ ( dsetconstr @ A
% 0.20/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(ex1_type,type,
% 0.20/0.46 ex1: $i > ( $i > $o ) > $o ).
% 0.20/0.46
% 0.20/0.46 thf(ex1,definition,
% 0.20/0.46 ( ex1
% 0.20/0.46 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.20/0.46 ( singleton
% 0.20/0.46 @ ( dsetconstr @ A
% 0.20/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(ex1E1_type,type,
% 0.20/0.46 ex1E1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ex1E1,definition,
% 0.20/0.46 ( ex1E1
% 0.20/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.46 ( ( ex1 @ A
% 0.20/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.46 => ? [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(ex1I_type,type,
% 0.20/0.46 ex1I: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ex1I,definition,
% 0.20/0.46 ( ex1I
% 0.20/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( Xphi @ Xx )
% 0.20/0.46 => ( ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ A )
% 0.20/0.46 => ( ( Xphi @ Xy )
% 0.20/0.46 => ( Xy = Xx ) ) )
% 0.20/0.46 => ( ex1 @ A
% 0.20/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(ex1I2_type,type,
% 0.20/0.46 ex1I2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ex1I2,definition,
% 0.20/0.46 ( ex1I2
% 0.20/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ A )
% 0.20/0.46 => ( ( Xphi @ Xx )
% 0.20/0.46 => ( ( Xphi @ Xy )
% 0.20/0.46 => ( Xx = Xy ) ) ) ) )
% 0.20/0.46 => ( ? [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 & ( Xphi @ Xx ) )
% 0.20/0.46 => ( ex1 @ A
% 0.20/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(singletonsuniq_type,type,
% 0.20/0.46 singletonsuniq: $o ).
% 0.20/0.46
% 0.20/0.46 thf(singletonsuniq,definition,
% 0.20/0.46 ( singletonsuniq
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( ( setadjoin @ Xx @ emptyset )
% 0.20/0.46 = ( setadjoin @ Xy @ emptyset ) )
% 0.20/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(atmost1p_type,type,
% 0.20/0.46 atmost1p: $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(atleast2p_type,type,
% 0.20/0.46 atleast2p: $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(atmost2p_type,type,
% 0.20/0.46 atmost2p: $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(upairsetp_type,type,
% 0.20/0.46 upairsetp: $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjL1_type,type,
% 0.20/0.46 setukpairinjL1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjL1,definition,
% 0.20/0.46 ( setukpairinjL1
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.20/0.46 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.20/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(kfstsingleton_type,type,
% 0.20/0.46 kfstsingleton: $o ).
% 0.20/0.46
% 0.20/0.46 thf(kfstsingleton,definition,
% 0.20/0.46 ( kfstsingleton
% 0.20/0.46 = ( ! [Xu: $i] :
% 0.20/0.46 ( ( iskpair @ Xu )
% 0.20/0.46 => ( singleton
% 0.20/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.20/0.46 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(theprop_type,type,
% 0.20/0.46 theprop: $o ).
% 0.20/0.46
% 0.20/0.46 thf(theprop,definition,
% 0.20/0.46 ( theprop
% 0.20/0.46 = ( ! [X: $i] :
% 0.20/0.46 ( ( singleton @ X )
% 0.20/0.46 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(kfst_type,type,
% 0.20/0.46 kfst: $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(kfstpairEq_type,type,
% 0.20/0.46 kfstpairEq: $o ).
% 0.20/0.46
% 0.20/0.46 thf(kfstpairEq,definition,
% 0.20/0.46 ( kfstpairEq
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.20/0.46 = Xx ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodfstin_type,type,
% 0.20/0.46 cartprodfstin: $o ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodfstin,definition,
% 0.20/0.46 ( cartprodfstin
% 0.20/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.20/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.20/0.46 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjL2_type,type,
% 0.20/0.46 setukpairinjL2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjL2,definition,
% 0.20/0.46 ( setukpairinjL2
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.20/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.20/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.20/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjL_type,type,
% 0.20/0.46 setukpairinjL: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjL,definition,
% 0.20/0.46 ( setukpairinjL
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.20/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.20/0.46 = ( kpair @ Xz @ Xu ) )
% 0.20/0.46 => ( Xx = Xz ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjR11_type,type,
% 0.20/0.46 setukpairinjR11: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjR11,definition,
% 0.20/0.46 ( setukpairinjR11
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( Xx = Xy )
% 0.20/0.46 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.20/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjR12_type,type,
% 0.20/0.46 setukpairinjR12: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjR12,definition,
% 0.20/0.46 ( setukpairinjR12
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( Xx = Xy )
% 0.20/0.46 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.20/0.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjR1_type,type,
% 0.20/0.46 setukpairinjR1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjR1,definition,
% 0.20/0.46 ( setukpairinjR1
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.20/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.20/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.20/0.46 => ( ( Xz = Xu )
% 0.20/0.46 => ( Xy = Xu ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(upairequniteq_type,type,
% 0.20/0.46 upairequniteq: $o ).
% 0.20/0.46
% 0.20/0.46 thf(upairequniteq,definition,
% 0.20/0.46 ( upairequniteq
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.20/0.46 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.20/0.46 = ( setadjoin @ Xz @ emptyset ) )
% 0.20/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjR2_type,type,
% 0.20/0.46 setukpairinjR2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjR2,definition,
% 0.20/0.46 ( setukpairinjR2
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.20/0.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.20/0.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.20/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjR_type,type,
% 0.20/0.46 setukpairinjR: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setukpairinjR,definition,
% 0.20/0.46 ( setukpairinjR
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.20/0.46 ( ( ( kpair @ Xx @ Xy )
% 0.20/0.46 = ( kpair @ Xz @ Xu ) )
% 0.20/0.46 => ( Xy = Xu ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(ksndsingleton_type,type,
% 0.20/0.46 ksndsingleton: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ksndsingleton,definition,
% 0.20/0.46 ( ksndsingleton
% 0.20/0.46 = ( ! [Xu: $i] :
% 0.20/0.46 ( ( iskpair @ Xu )
% 0.20/0.46 => ( singleton
% 0.20/0.46 @ ( dsetconstr @ ( setunion @ Xu )
% 0.20/0.46 @ ^ [Xx: $i] :
% 0.20/0.46 ( Xu
% 0.20/0.46 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(ksnd_type,type,
% 0.20/0.46 ksnd: $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(ksndpairEq_type,type,
% 0.20/0.46 ksndpairEq: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ksndpairEq,definition,
% 0.20/0.46 ( ksndpairEq
% 0.20/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.20/0.46 = Xy ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(kpairsurjEq_type,type,
% 0.20/0.46 kpairsurjEq: $o ).
% 0.20/0.46
% 0.20/0.46 thf(kpairsurjEq,definition,
% 0.20/0.46 ( kpairsurjEq
% 0.20/0.46 = ( ! [Xu: $i] :
% 0.20/0.46 ( ( iskpair @ Xu )
% 0.20/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.20/0.46 = Xu ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodsndin_type,type,
% 0.20/0.46 cartprodsndin: $o ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodsndin,definition,
% 0.20/0.46 ( cartprodsndin
% 0.20/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.20/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.20/0.46 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodpairmemEL_type,type,
% 0.20/0.46 cartprodpairmemEL: $o ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodpairmemEL,definition,
% 0.20/0.46 ( cartprodpairmemEL
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.20/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodpairmemER_type,type,
% 0.20/0.46 cartprodpairmemER: $o ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodpairmemER,definition,
% 0.20/0.46 ( cartprodpairmemER
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.20/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodmempaircEq_type,type,
% 0.20/0.46 cartprodmempaircEq: $o ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodmempaircEq,definition,
% 0.20/0.46 ( cartprodmempaircEq
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( ( kpair @ Xx @ Xy )
% 0.20/0.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodfstpairEq_type,type,
% 0.20/0.46 cartprodfstpairEq: $o ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodfstpairEq,definition,
% 0.20/0.46 ( cartprodfstpairEq
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.20/0.46 = Xx ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodsndpairEq_type,type,
% 0.20/0.46 cartprodsndpairEq: $o ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodsndpairEq,definition,
% 0.20/0.46 ( cartprodsndpairEq
% 0.20/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.20/0.46 = Xy ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodpairsurjEq_type,type,
% 0.20/0.46 cartprodpairsurjEq: $o ).
% 0.20/0.46
% 0.20/0.46 thf(cartprodpairsurjEq,definition,
% 0.20/0.46 ( cartprodpairsurjEq
% 0.20/0.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.20/0.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.20/0.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.20/0.46 = Xu ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(breln_type,type,
% 0.20/0.46 breln: $i > $i > $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(breln,definition,
% 0.20/0.46 ( breln
% 0.20/0.46 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(dpsetconstr_type,type,
% 0.20/0.46 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.20/0.46
% 0.20/0.46 thf(dpsetconstrI_type,type,
% 0.20/0.46 dpsetconstrI: $o ).
% 0.20/0.46
% 0.20/0.46 thf(dpsetconstrI,definition,
% 0.20/0.46 ( dpsetconstrI
% 0.20/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( ( Xphi @ Xx @ Xy )
% 0.20/0.46 => ( in @ ( kpair @ Xx @ Xy )
% 0.20/0.46 @ ( dpsetconstr @ A @ B
% 0.20/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(dpsetconstrSub_type,type,
% 0.20/0.46 dpsetconstrSub: $o ).
% 0.20/0.46
% 0.20/0.46 thf(dpsetconstrSub,definition,
% 0.20/0.46 ( dpsetconstrSub
% 0.20/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.20/0.46 ( subset
% 0.20/0.46 @ ( dpsetconstr @ A @ B
% 0.20/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.20/0.46 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setOfPairsIsBReln_type,type,
% 0.20/0.46 setOfPairsIsBReln: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setOfPairsIsBReln,definition,
% 0.20/0.46 ( setOfPairsIsBReln
% 0.20/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.20/0.46 ( breln @ A @ B
% 0.20/0.46 @ ( dpsetconstr @ A @ B
% 0.20/0.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(dpsetconstrERa_type,type,
% 0.20/0.46 dpsetconstrERa: $o ).
% 0.20/0.46
% 0.20/0.46 thf(dpsetconstrERa,definition,
% 0.20/0.46 ( dpsetconstrERa
% 0.20/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.20/0.46 @ ( dpsetconstr @ A @ B
% 0.20/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.20/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(dpsetconstrEL1_type,type,
% 0.20/0.46 dpsetconstrEL1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(dpsetconstrEL1,definition,
% 0.20/0.46 ( dpsetconstrEL1
% 0.20/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.20/0.46 @ ( dpsetconstr @ A @ B
% 0.20/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.20/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(dpsetconstrEL2_type,type,
% 0.20/0.46 dpsetconstrEL2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(dpsetconstrEL2,definition,
% 0.20/0.46 ( dpsetconstrEL2
% 0.20/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.20/0.46 @ ( dpsetconstr @ A @ B
% 0.20/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.20/0.46 => ( in @ Xy @ B ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(dpsetconstrER_type,type,
% 0.20/0.46 dpsetconstrER: $o ).
% 0.20/0.46
% 0.20/0.46 thf(dpsetconstrER,definition,
% 0.20/0.46 ( dpsetconstrER
% 0.20/0.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( in @ ( kpair @ Xx @ Xy )
% 0.20/0.46 @ ( dpsetconstr @ A @ B
% 0.20/0.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.20/0.46 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(func_type,type,
% 0.20/0.46 func: $i > $i > $i > $o ).
% 0.20/0.46
% 0.20/0.46 thf(funcSet_type,type,
% 0.20/0.46 funcSet: $i > $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(funcImageSingleton_type,type,
% 0.20/0.46 funcImageSingleton: $o ).
% 0.20/0.46
% 0.20/0.46 thf(funcImageSingleton,definition,
% 0.20/0.46 ( funcImageSingleton
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( func @ A @ B @ Xf )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( singleton
% 0.20/0.46 @ ( dsetconstr @ B
% 0.20/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(apProp_type,type,
% 0.20/0.46 apProp: $o ).
% 0.20/0.46
% 0.20/0.46 thf(apProp,definition,
% 0.20/0.46 ( apProp
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( func @ A @ B @ Xf )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in
% 0.20/0.46 @ ( setunion
% 0.20/0.46 @ ( dsetconstr @ B
% 0.20/0.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.20/0.46 @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(ap_type,type,
% 0.20/0.46 ap: $i > $i > $i > $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(app_type,type,
% 0.20/0.46 app: $o ).
% 0.20/0.46
% 0.20/0.46 thf(app,definition,
% 0.20/0.46 ( app
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( func @ A @ B @ Xf )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(infuncsetfunc_type,type,
% 0.20/0.46 infuncsetfunc: $o ).
% 0.20/0.46
% 0.20/0.46 thf(infuncsetfunc,definition,
% 0.20/0.46 ( infuncsetfunc
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.20/0.46 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(ap2p_type,type,
% 0.20/0.46 ap2p: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ap2p,definition,
% 0.20/0.46 ( ap2p
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(funcinfuncset_type,type,
% 0.20/0.46 funcinfuncset: $o ).
% 0.20/0.46
% 0.20/0.46 thf(funcinfuncset,definition,
% 0.20/0.46 ( funcinfuncset
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( func @ A @ B @ Xf )
% 0.20/0.46 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(lamProp_type,type,
% 0.20/0.46 lamProp: $o ).
% 0.20/0.46
% 0.20/0.46 thf(lamProp,definition,
% 0.20/0.46 ( lamProp
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.20/0.46 => ( func @ A @ B
% 0.20/0.46 @ ( dpsetconstr @ A @ B
% 0.20/0.46 @ ^ [Xx: $i,Xy: $i] :
% 0.20/0.46 ( ( Xf @ Xx )
% 0.20/0.46 = Xy ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(lam_type,type,
% 0.20/0.46 lam: $i > $i > ( $i > $i ) > $i ).
% 0.20/0.46
% 0.20/0.46 thf(lamp_type,type,
% 0.20/0.46 lamp: $o ).
% 0.20/0.46
% 0.20/0.46 thf(lamp,definition,
% 0.20/0.46 ( lamp
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.20/0.46 => ( func @ A @ B
% 0.20/0.46 @ ( lam @ A @ B
% 0.20/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(lam2p_type,type,
% 0.20/0.46 lam2p: $o ).
% 0.20/0.46
% 0.20/0.46 thf(lam2p,definition,
% 0.20/0.46 ( lam2p
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.20/0.46 => ( in
% 0.20/0.46 @ ( lam @ A @ B
% 0.20/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.20/0.46 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(brelnall1_type,type,
% 0.20/0.46 brelnall1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(brelnall1,definition,
% 0.20/0.46 ( brelnall1
% 0.20/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.20/0.46 ( ( breln @ A @ B @ R )
% 0.20/0.46 => ! [Xphi: $i > $o] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.20/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ R )
% 0.20/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(brelnall2_type,type,
% 0.20/0.46 brelnall2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(brelnall2,definition,
% 0.20/0.46 ( brelnall2
% 0.20/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.20/0.46 ( ( breln @ A @ B @ R )
% 0.20/0.46 => ! [Xphi: $i > $o] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.20/0.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ R )
% 0.20/0.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(ex1E2_type,type,
% 0.20/0.46 ex1E2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ex1E2,definition,
% 0.20/0.46 ( ex1E2
% 0.20/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.20/0.46 ( ( ex1 @ A
% 0.20/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ A )
% 0.20/0.46 => ( ( Xphi @ Xx )
% 0.20/0.46 => ( ( Xphi @ Xy )
% 0.20/0.46 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(funcGraphProp1_type,type,
% 0.20/0.46 funcGraphProp1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(funcGraphProp1,definition,
% 0.20/0.46 ( funcGraphProp1
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( func @ A @ B @ Xf )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(funcGraphProp3_type,type,
% 0.20/0.46 funcGraphProp3: $o ).
% 0.20/0.46
% 0.20/0.46 thf(funcGraphProp3,definition,
% 0.20/0.46 ( funcGraphProp3
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(funcGraphProp2_type,type,
% 0.20/0.46 funcGraphProp2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(funcGraphProp2,definition,
% 0.20/0.46 ( funcGraphProp2
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( func @ A @ B @ Xf )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.20/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.20/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(funcextLem_type,type,
% 0.20/0.46 funcextLem: $o ).
% 0.20/0.46
% 0.20/0.46 thf(funcextLem,definition,
% 0.20/0.46 ( funcextLem
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( func @ A @ B @ Xf )
% 0.20/0.46 => ! [Xg: $i] :
% 0.20/0.46 ( ( func @ A @ B @ Xg )
% 0.20/0.46 => ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.20/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.20/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(funcGraphProp4_type,type,
% 0.20/0.46 funcGraphProp4: $o ).
% 0.20/0.46
% 0.20/0.46 thf(funcGraphProp4,definition,
% 0.20/0.46 ( funcGraphProp4
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.20/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.20/0.46 = Xy ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(subbreln_type,type,
% 0.20/0.46 subbreln: $o ).
% 0.20/0.46
% 0.20/0.46 thf(subbreln,definition,
% 0.20/0.46 ( subbreln
% 0.20/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.20/0.46 ( ( breln @ A @ B @ R )
% 0.20/0.46 => ! [S: $i] :
% 0.20/0.46 ( ( breln @ A @ B @ S )
% 0.20/0.46 => ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.20/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.20/0.46 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(eqbreln_type,type,
% 0.20/0.46 eqbreln: $o ).
% 0.20/0.46
% 0.20/0.46 thf(eqbreln,definition,
% 0.20/0.46 ( eqbreln
% 0.20/0.46 = ( ! [A: $i,B: $i,R: $i] :
% 0.20/0.46 ( ( breln @ A @ B @ R )
% 0.20/0.46 => ! [S: $i] :
% 0.20/0.46 ( ( breln @ A @ B @ S )
% 0.20/0.46 => ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.20/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.20/0.46 => ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ B )
% 0.20/0.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.20/0.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.20/0.46 => ( R = S ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(funcext_type,type,
% 0.20/0.46 funcext: $o ).
% 0.20/0.46
% 0.20/0.46 thf(funcext,definition,
% 0.20/0.46 ( funcext
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( func @ A @ B @ Xf )
% 0.20/0.46 => ! [Xg: $i] :
% 0.20/0.46 ( ( func @ A @ B @ Xg )
% 0.20/0.46 => ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.20/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.20/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(funcext2_type,type,
% 0.20/0.46 funcext2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(funcext2,definition,
% 0.20/0.46 ( funcext2
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.20/0.46 => ! [Xg: $i] :
% 0.20/0.46 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.20/0.46 => ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.20/0.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.20/0.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(ap2apEq1_type,type,
% 0.20/0.46 ap2apEq1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ap2apEq1,definition,
% 0.20/0.46 ( ap2apEq1
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.20/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(ap2apEq2_type,type,
% 0.20/0.46 ap2apEq2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ap2apEq2,definition,
% 0.20/0.46 ( ap2apEq2
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( func @ A @ B @ Xf )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.20/0.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(beta1_type,type,
% 0.20/0.46 beta1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(beta1,definition,
% 0.20/0.46 ( beta1
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( ap @ A @ B
% 0.20/0.46 @ ( lam @ A @ B
% 0.20/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.20/0.46 @ Xx )
% 0.20/0.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(eta1_type,type,
% 0.20/0.46 eta1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(eta1,definition,
% 0.20/0.46 ( eta1
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( func @ A @ B @ Xf )
% 0.20/0.46 => ( ( lam @ A @ B
% 0.20/0.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.20/0.46 = Xf ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(lam2lamEq_type,type,
% 0.20/0.46 lam2lamEq: $o ).
% 0.20/0.46
% 0.20/0.46 thf(lam2lamEq,definition,
% 0.20/0.46 ( lam2lamEq
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.20/0.46 => ( ( lam @ A @ B
% 0.20/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.20/0.46 = ( lam @ A @ B
% 0.20/0.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(beta2_type,type,
% 0.20/0.46 beta2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(beta2,definition,
% 0.20/0.46 ( beta2
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.20/0.46 ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( ap @ A @ B
% 0.20/0.46 @ ( lam @ A @ B
% 0.20/0.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.20/0.46 @ Xx )
% 0.20/0.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(eta2_type,type,
% 0.20/0.46 eta2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(eta2,definition,
% 0.20/0.46 ( eta2
% 0.20/0.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.20/0.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.20/0.46 => ( ( lam @ A @ B
% 0.20/0.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.20/0.46 = Xf ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(iffalseProp1_type,type,
% 0.20/0.46 iffalseProp1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(iffalseProp1,definition,
% 0.20/0.46 ( iffalseProp1
% 0.20/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ A )
% 0.20/0.46 => ( ~ Xphi
% 0.20/0.46 => ( in @ Xy
% 0.20/0.46 @ ( dsetconstr @ A
% 0.20/0.46 @ ^ [Xz: $i] :
% 0.20/0.46 ( ( Xphi
% 0.20/0.46 & ( Xz = Xx ) )
% 0.20/0.46 | ( ~ Xphi
% 0.20/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(iffalseProp2_type,type,
% 0.20/0.46 iffalseProp2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(iffalseProp2,definition,
% 0.20/0.46 ( iffalseProp2
% 0.20/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ A )
% 0.20/0.46 => ( ~ Xphi
% 0.20/0.46 => ( ( dsetconstr @ A
% 0.20/0.46 @ ^ [Xz: $i] :
% 0.20/0.46 ( ( Xphi
% 0.20/0.46 & ( Xz = Xx ) )
% 0.20/0.46 | ( ~ Xphi
% 0.20/0.46 & ( Xz = Xy ) ) ) )
% 0.20/0.46 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(iftrueProp1_type,type,
% 0.20/0.46 iftrueProp1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(iftrueProp1,definition,
% 0.20/0.46 ( iftrueProp1
% 0.20/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ A )
% 0.20/0.46 => ( Xphi
% 0.20/0.46 => ( in @ Xx
% 0.20/0.46 @ ( dsetconstr @ A
% 0.20/0.46 @ ^ [Xz: $i] :
% 0.20/0.46 ( ( Xphi
% 0.20/0.46 & ( Xz = Xx ) )
% 0.20/0.46 | ( ~ Xphi
% 0.20/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(iftrueProp2_type,type,
% 0.20/0.46 iftrueProp2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(iftrueProp2,definition,
% 0.20/0.46 ( iftrueProp2
% 0.20/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ A )
% 0.20/0.46 => ( Xphi
% 0.20/0.46 => ( ( dsetconstr @ A
% 0.20/0.46 @ ^ [Xz: $i] :
% 0.20/0.46 ( ( Xphi
% 0.20/0.46 & ( Xz = Xx ) )
% 0.20/0.46 | ( ~ Xphi
% 0.20/0.46 & ( Xz = Xy ) ) ) )
% 0.20/0.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(ifSingleton_type,type,
% 0.20/0.46 ifSingleton: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ifSingleton,definition,
% 0.20/0.46 ( ifSingleton
% 0.20/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ A )
% 0.20/0.46 => ( singleton
% 0.20/0.46 @ ( dsetconstr @ A
% 0.20/0.46 @ ^ [Xz: $i] :
% 0.20/0.46 ( ( Xphi
% 0.20/0.46 & ( Xz = Xx ) )
% 0.20/0.46 | ( ~ Xphi
% 0.20/0.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(if_type,type,
% 0.20/0.46 if: $i > $o > $i > $i > $i ).
% 0.20/0.46
% 0.20/0.46 thf(ifp_type,type,
% 0.20/0.46 ifp: $o ).
% 0.20/0.46
% 0.20/0.46 thf(ifp,definition,
% 0.20/0.46 ( ifp
% 0.20/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ A )
% 0.20/0.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(theeq_type,type,
% 0.20/0.46 theeq: $o ).
% 0.20/0.46
% 0.20/0.46 thf(theeq,definition,
% 0.20/0.46 ( theeq
% 0.20/0.46 = ( ! [X: $i] :
% 0.20/0.46 ( ( singleton @ X )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ X )
% 0.20/0.46 => ( ( setunion @ X )
% 0.20/0.46 = Xx ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(iftrue_type,type,
% 0.20/0.46 iftrue: $o ).
% 0.20/0.46
% 0.20/0.46 thf(iftrue,definition,
% 0.20/0.46 ( iftrue
% 0.20/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ A )
% 0.20/0.46 => ( Xphi
% 0.20/0.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.20/0.46 = Xx ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(iffalse_type,type,
% 0.20/0.46 iffalse: $o ).
% 0.20/0.46
% 0.20/0.46 thf(iffalse,definition,
% 0.20/0.46 ( iffalse
% 0.20/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ A )
% 0.20/0.46 => ( ~ Xphi
% 0.20/0.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.20/0.46 = Xy ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(iftrueorfalse_type,type,
% 0.20/0.46 iftrueorfalse: $o ).
% 0.20/0.46
% 0.20/0.46 thf(iftrueorfalse,definition,
% 0.20/0.46 ( iftrueorfalse
% 0.20/0.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ! [Xy: $i] :
% 0.20/0.46 ( ( in @ Xy @ A )
% 0.20/0.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectT_lem_type,type,
% 0.20/0.46 binintersectT_lem: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectT_lem,definition,
% 0.20/0.46 ( binintersectT_lem
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binunionT_lem_type,type,
% 0.20/0.46 binunionT_lem: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binunionT_lem,definition,
% 0.20/0.46 ( binunionT_lem
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(powersetT_lem_type,type,
% 0.20/0.46 powersetT_lem: $o ).
% 0.20/0.46
% 0.20/0.46 thf(powersetT_lem,definition,
% 0.20/0.46 ( powersetT_lem
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setminusT_lem_type,type,
% 0.20/0.46 setminusT_lem: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setminusT_lem,definition,
% 0.20/0.46 ( setminusT_lem
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(complementT_lem_type,type,
% 0.20/0.46 complementT_lem: $o ).
% 0.20/0.46
% 0.20/0.46 thf(complementT_lem,definition,
% 0.20/0.46 ( complementT_lem
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(setextT_type,type,
% 0.20/0.46 setextT: $o ).
% 0.20/0.46
% 0.20/0.46 thf(setextT,definition,
% 0.20/0.46 ( setextT
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ X )
% 0.20/0.46 => ( in @ Xx @ Y ) ) )
% 0.20/0.46 => ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ Y )
% 0.20/0.46 => ( in @ Xx @ X ) ) )
% 0.20/0.46 => ( X = Y ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(subsetTI_type,type,
% 0.20/0.46 subsetTI: $o ).
% 0.20/0.46
% 0.20/0.46 thf(subsetTI,definition,
% 0.20/0.46 ( subsetTI
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ X )
% 0.20/0.46 => ( in @ Xx @ Y ) ) )
% 0.20/0.46 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(powersetTI1_type,type,
% 0.20/0.46 powersetTI1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(powersetTI1,definition,
% 0.20/0.46 ( powersetTI1
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ( ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ X )
% 0.20/0.46 => ( in @ Xx @ Y ) ) )
% 0.20/0.46 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(powersetTE1_type,type,
% 0.20/0.46 powersetTE1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(powersetTE1,definition,
% 0.20/0.46 ( powersetTE1
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ X @ ( powerset @ Y ) )
% 0.20/0.46 => ( ( in @ Xx @ X )
% 0.20/0.46 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(complementTI1_type,type,
% 0.20/0.46 complementTI1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(complementTI1,definition,
% 0.20/0.46 ( complementTI1
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ X )
% 0.20/0.46 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(complementTE1_type,type,
% 0.20/0.46 complementTE1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(complementTE1,definition,
% 0.20/0.46 ( complementTE1
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.20/0.46 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectTELcontra_type,type,
% 0.20/0.46 binintersectTELcontra: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectTELcontra,definition,
% 0.20/0.46 ( binintersectTELcontra
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ~ ( in @ Xx @ X )
% 0.20/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectTERcontra_type,type,
% 0.20/0.46 binintersectTERcontra: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binintersectTERcontra,definition,
% 0.20/0.46 ( binintersectTERcontra
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ~ ( in @ Xx @ Y )
% 0.20/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(contrasubsetT_type,type,
% 0.20/0.46 contrasubsetT: $o ).
% 0.20/0.46
% 0.20/0.46 thf(contrasubsetT,definition,
% 0.20/0.46 ( contrasubsetT
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.20/0.46 => ( ( in @ Xx @ Y )
% 0.20/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(contrasubsetT1_type,type,
% 0.20/0.46 contrasubsetT1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(contrasubsetT1,definition,
% 0.20/0.46 ( contrasubsetT1
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( subset @ X @ Y )
% 0.20/0.46 => ( ~ ( in @ Xx @ Y )
% 0.20/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(contrasubsetT2_type,type,
% 0.20/0.46 contrasubsetT2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(contrasubsetT2,definition,
% 0.20/0.46 ( contrasubsetT2
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ( ( subset @ X @ Y )
% 0.20/0.46 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(contrasubsetT3_type,type,
% 0.20/0.46 contrasubsetT3: $o ).
% 0.20/0.46
% 0.20/0.46 thf(contrasubsetT3,definition,
% 0.20/0.46 ( contrasubsetT3
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.20/0.46 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(doubleComplementI1_type,type,
% 0.20/0.46 doubleComplementI1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(doubleComplementI1,definition,
% 0.20/0.46 ( doubleComplementI1
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ X )
% 0.20/0.46 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(doubleComplementE1_type,type,
% 0.20/0.46 doubleComplementE1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(doubleComplementE1,definition,
% 0.20/0.46 ( doubleComplementE1
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.20/0.46 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(doubleComplementSub1_type,type,
% 0.20/0.46 doubleComplementSub1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(doubleComplementSub1,definition,
% 0.20/0.46 ( doubleComplementSub1
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(doubleComplementSub2_type,type,
% 0.20/0.46 doubleComplementSub2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(doubleComplementSub2,definition,
% 0.20/0.46 ( doubleComplementSub2
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(doubleComplementEq_type,type,
% 0.20/0.46 doubleComplementEq: $o ).
% 0.20/0.46
% 0.20/0.46 thf(doubleComplementEq,definition,
% 0.20/0.46 ( doubleComplementEq
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ( X
% 0.20/0.46 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(complementTnotintersectT_type,type,
% 0.20/0.46 complementTnotintersectT: $o ).
% 0.20/0.46
% 0.20/0.46 thf(complementTnotintersectT,definition,
% 0.20/0.46 ( complementTnotintersectT
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.20/0.46 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(complementImpComplementIntersect_type,type,
% 0.20/0.46 complementImpComplementIntersect: $o ).
% 0.20/0.46
% 0.20/0.46 thf(complementImpComplementIntersect,definition,
% 0.20/0.46 ( complementImpComplementIntersect
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.20/0.46 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(complementSubsetComplementIntersect_type,type,
% 0.20/0.46 complementSubsetComplementIntersect: $o ).
% 0.20/0.46
% 0.20/0.46 thf(complementSubsetComplementIntersect,definition,
% 0.20/0.46 ( complementSubsetComplementIntersect
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(complementInPowersetComplementIntersect_type,type,
% 0.20/0.46 complementInPowersetComplementIntersect: $o ).
% 0.20/0.46
% 0.20/0.46 thf(complementInPowersetComplementIntersect,definition,
% 0.20/0.46 ( complementInPowersetComplementIntersect
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(contraSubsetComplement_type,type,
% 0.20/0.46 contraSubsetComplement: $o ).
% 0.20/0.46
% 0.20/0.46 thf(contraSubsetComplement,definition,
% 0.20/0.46 ( contraSubsetComplement
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ Y )
% 0.20/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(complementTcontraSubset_type,type,
% 0.20/0.46 complementTcontraSubset: $o ).
% 0.20/0.46
% 0.20/0.46 thf(complementTcontraSubset,definition,
% 0.20/0.46 ( complementTcontraSubset
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.20/0.46 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binunionTILcontra_type,type,
% 0.20/0.46 binunionTILcontra: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binunionTILcontra,definition,
% 0.20/0.46 ( binunionTILcontra
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.20/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binunionTIRcontra_type,type,
% 0.20/0.46 binunionTIRcontra: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binunionTIRcontra,definition,
% 0.20/0.46 ( binunionTIRcontra
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.20/0.46 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(inIntersectImpInUnion_type,type,
% 0.20/0.46 inIntersectImpInUnion: $o ).
% 0.20/0.46
% 0.20/0.46 thf(inIntersectImpInUnion,definition,
% 0.20/0.46 ( inIntersectImpInUnion
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Z: $i] :
% 0.20/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.20/0.46 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(inIntersectImpInUnion2_type,type,
% 0.20/0.46 inIntersectImpInUnion2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(inIntersectImpInUnion2,definition,
% 0.20/0.46 ( inIntersectImpInUnion2
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Z: $i] :
% 0.20/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.20/0.46 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(inIntersectImpInIntersectUnions_type,type,
% 0.20/0.46 inIntersectImpInIntersectUnions: $o ).
% 0.20/0.46
% 0.20/0.46 thf(inIntersectImpInIntersectUnions,definition,
% 0.20/0.46 ( inIntersectImpInIntersectUnions
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Z: $i] :
% 0.20/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.20/0.46 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(intersectInPowersetIntersectUnions_type,type,
% 0.20/0.46 intersectInPowersetIntersectUnions: $o ).
% 0.20/0.46
% 0.20/0.46 thf(intersectInPowersetIntersectUnions,definition,
% 0.20/0.46 ( intersectInPowersetIntersectUnions
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Z: $i] :
% 0.20/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.20/0.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(inComplementUnionImpNotIn1_type,type,
% 0.20/0.46 inComplementUnionImpNotIn1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(inComplementUnionImpNotIn1,definition,
% 0.20/0.46 ( inComplementUnionImpNotIn1
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.20/0.46 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(inComplementUnionImpInComplement1_type,type,
% 0.20/0.46 inComplementUnionImpInComplement1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(inComplementUnionImpInComplement1,definition,
% 0.20/0.46 ( inComplementUnionImpInComplement1
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.20/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binunionTE_type,type,
% 0.20/0.46 binunionTE: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binunionTE,definition,
% 0.20/0.46 ( binunionTE
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xphi: $o,Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.20/0.46 => ( ( ( in @ Xx @ X )
% 0.20/0.46 => Xphi )
% 0.20/0.46 => ( ( ( in @ Xx @ Y )
% 0.20/0.46 => Xphi )
% 0.20/0.46 => Xphi ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(binunionTEcontra_type,type,
% 0.20/0.46 binunionTEcontra: $o ).
% 0.20/0.46
% 0.20/0.46 thf(binunionTEcontra,definition,
% 0.20/0.46 ( binunionTEcontra
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ~ ( in @ Xx @ X )
% 0.20/0.46 => ( ~ ( in @ Xx @ Y )
% 0.20/0.46 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan2a1_type,type,
% 0.20/0.46 demorgan2a1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan2a1,definition,
% 0.20/0.46 ( demorgan2a1
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.20/0.46 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(complementUnionInPowersetComplement_type,type,
% 0.20/0.46 complementUnionInPowersetComplement: $o ).
% 0.20/0.46
% 0.20/0.46 thf(complementUnionInPowersetComplement,definition,
% 0.20/0.46 ( complementUnionInPowersetComplement
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan2a2_type,type,
% 0.20/0.46 demorgan2a2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan2a2,definition,
% 0.20/0.46 ( demorgan2a2
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.20/0.46 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan1a_type,type,
% 0.20/0.46 demorgan1a: $o ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan1a,definition,
% 0.20/0.46 ( demorgan1a
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.20/0.46 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan1b_type,type,
% 0.20/0.46 demorgan1b: $o ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan1b,definition,
% 0.20/0.46 ( demorgan1b
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.20/0.46 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan1_type,type,
% 0.20/0.46 demorgan1: $o ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan1,definition,
% 0.20/0.46 ( demorgan1
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.20/0.46 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan2a_type,type,
% 0.20/0.46 demorgan2a: $o ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan2a,definition,
% 0.20/0.46 ( demorgan2a
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.20/0.46 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan2b2_type,type,
% 0.20/0.46 demorgan2b2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan2b2,definition,
% 0.20/0.46 ( demorgan2b2
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.20/0.46 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.20/0.46 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan2b_type,type,
% 0.20/0.46 demorgan2b: $o ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan2b,definition,
% 0.20/0.46 ( demorgan2b
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ A )
% 0.20/0.46 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.20/0.46 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan2_type,type,
% 0.20/0.46 demorgan2: $o ).
% 0.20/0.46
% 0.20/0.46 thf(demorgan2,definition,
% 0.20/0.46 ( demorgan2
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.20/0.46 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(woz13rule0_type,type,
% 0.20/0.46 woz13rule0: $o ).
% 0.20/0.46
% 0.20/0.46 thf(woz13rule0,definition,
% 0.20/0.46 ( woz13rule0
% 0.20/0.46 = ( ! [A: $i,X: $i] :
% 0.20/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.20/0.46 => ! [Y: $i] :
% 0.20/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.20/0.46 => ! [Xx: $i] :
% 0.20/0.46 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.20/0.46 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 thf(woz13rule1,conjecture,
% 0.20/0.46 ( setextAx
% 0.20/0.46 => ( emptysetAx
% 0.20/0.46 => ( setadjoinAx
% 0.20/0.46 => ( powersetAx
% 0.20/0.46 => ( setunionAx
% 0.20/0.46 => ( omega0Ax
% 0.20/0.46 => ( omegaSAx
% 0.20/0.46 => ( omegaIndAx
% 0.20/0.46 => ( replAx
% 0.20/0.46 => ( foundationAx
% 0.20/0.46 => ( wellorderingAx
% 0.20/0.46 => ( descrp
% 0.20/0.46 => ( dsetconstrI
% 0.20/0.46 => ( dsetconstrEL
% 0.20/0.46 => ( dsetconstrER
% 0.20/0.46 => ( exuE1
% 0.20/0.46 => ( prop2setE
% 0.20/0.46 => ( emptysetE
% 0.20/0.46 => ( emptysetimpfalse
% 0.20/0.46 => ( notinemptyset
% 0.20/0.46 => ( exuE3e
% 0.20/0.46 => ( setext
% 0.20/0.46 => ( emptyI
% 0.20/0.46 => ( noeltsimpempty
% 0.20/0.46 => ( setbeta
% 0.20/0.46 => ( nonemptyE1
% 0.20/0.46 => ( nonemptyI
% 0.20/0.46 => ( nonemptyI1
% 0.20/0.46 => ( setadjoinIL
% 0.20/0.46 => ( emptyinunitempty
% 0.20/0.46 => ( setadjoinIR
% 0.20/0.46 => ( setadjoinE
% 0.20/0.46 => ( setadjoinOr
% 0.20/0.46 => ( setoftrueEq
% 0.20/0.46 => ( powersetI
% 0.20/0.46 => ( emptyinPowerset
% 0.20/0.46 => ( emptyInPowerset
% 0.20/0.46 => ( powersetE
% 0.20/0.46 => ( setunionI
% 0.20/0.46 => ( setunionE
% 0.20/0.46 => ( subPowSU
% 0.20/0.46 => ( exuE2
% 0.20/0.46 => ( nonemptyImpWitness
% 0.20/0.46 => ( uniqinunit
% 0.20/0.46 => ( notinsingleton
% 0.20/0.46 => ( eqinunit
% 0.20/0.46 => ( singletonsswitch
% 0.20/0.46 => ( upairsetE
% 0.20/0.46 => ( upairsetIL
% 0.20/0.46 => ( upairsetIR
% 0.20/0.46 => ( emptyE1
% 0.20/0.46 => ( vacuousDall
% 0.20/0.46 => ( quantDeMorgan1
% 0.20/0.46 => ( quantDeMorgan2
% 0.20/0.46 => ( quantDeMorgan3
% 0.20/0.46 => ( quantDeMorgan4
% 0.20/0.46 => ( prop2setI
% 0.20/0.46 => ( prop2set2propI
% 0.20/0.46 => ( notdexE
% 0.20/0.46 => ( notdallE
% 0.20/0.46 => ( exuI1
% 0.20/0.46 => ( exuI3
% 0.20/0.46 => ( exuI2
% 0.20/0.46 => ( inCongP
% 0.20/0.46 => ( in__Cong
% 0.20/0.46 => ( exuE3u
% 0.20/0.46 => ( exu__Cong
% 0.20/0.46 => ( emptyset__Cong
% 0.20/0.46 => ( setadjoin__Cong
% 0.20/0.46 => ( powerset__Cong
% 0.20/0.46 => ( setunion__Cong
% 0.20/0.46 => ( omega__Cong
% 0.20/0.46 => ( exuEu
% 0.20/0.46 => ( descr__Cong
% 0.20/0.46 => ( dsetconstr__Cong
% 0.20/0.46 => ( subsetI1
% 0.20/0.46 => ( eqimpsubset2
% 0.20/0.46 => ( eqimpsubset1
% 0.20/0.46 => ( subsetI2
% 0.20/0.46 => ( emptysetsubset
% 0.20/0.46 => ( subsetE
% 0.20/0.46 => ( subsetE2
% 0.20/0.46 => ( notsubsetI
% 0.20/0.46 => ( notequalI1
% 0.20/0.46 => ( notequalI2
% 0.20/0.46 => ( subsetRefl
% 0.20/0.46 => ( subsetTrans
% 0.20/0.46 => ( setadjoinSub
% 0.20/0.46 => ( setadjoinSub2
% 0.20/0.46 => ( subset2powerset
% 0.20/0.46 => ( setextsub
% 0.20/0.46 => ( subsetemptysetimpeq
% 0.20/0.46 => ( powersetI1
% 0.20/0.46 => ( powersetE1
% 0.20/0.46 => ( inPowerset
% 0.20/0.46 => ( powersetsubset
% 0.20/0.46 => ( sepInPowerset
% 0.20/0.46 => ( sepSubset
% 0.20/0.46 => ( binunionIL
% 0.20/0.46 => ( upairset2IR
% 0.20/0.46 => ( binunionIR
% 0.20/0.46 => ( binunionEcases
% 0.20/0.46 => ( binunionE
% 0.20/0.46 => ( binunionLsub
% 0.20/0.46 => ( binunionRsub
% 0.20/0.46 => ( binintersectI
% 0.20/0.46 => ( binintersectSubset5
% 0.20/0.46 => ( binintersectEL
% 0.20/0.46 => ( binintersectLsub
% 0.20/0.46 => ( binintersectSubset2
% 0.20/0.46 => ( binintersectSubset3
% 0.20/0.46 => ( binintersectER
% 0.20/0.46 => ( disjointsetsI1
% 0.20/0.46 => ( binintersectRsub
% 0.20/0.46 => ( binintersectSubset4
% 0.20/0.46 => ( binintersectSubset1
% 0.20/0.46 => ( bs114d
% 0.20/0.46 => ( setminusI
% 0.20/0.46 => ( setminusEL
% 0.20/0.46 => ( setminusER
% 0.20/0.46 => ( setminusSubset2
% 0.20/0.46 => ( setminusERneg
% 0.20/0.46 => ( setminusELneg
% 0.20/0.46 => ( setminusILneg
% 0.20/0.46 => ( setminusIRneg
% 0.20/0.46 => ( setminusLsub
% 0.20/0.46 => ( setminusSubset1
% 0.20/0.46 => ( symdiffE
% 0.20/0.46 => ( symdiffI1
% 0.20/0.46 => ( symdiffI2
% 0.20/0.46 => ( symdiffIneg1
% 0.20/0.46 => ( symdiffIneg2
% 0.20/0.46 => ( secondinupair
% 0.20/0.46 => ( setukpairIL
% 0.20/0.46 => ( setukpairIR
% 0.20/0.46 => ( kpairiskpair
% 0.20/0.46 => ( kpairp
% 0.20/0.46 => ( singletonsubset
% 0.20/0.46 => ( singletoninpowerset
% 0.20/0.46 => ( singletoninpowunion
% 0.20/0.46 => ( upairset2E
% 0.20/0.46 => ( upairsubunion
% 0.20/0.46 => ( upairinpowunion
% 0.20/0.46 => ( ubforcartprodlem1
% 0.20/0.46 => ( ubforcartprodlem2
% 0.20/0.46 => ( ubforcartprodlem3
% 0.20/0.46 => ( cartprodpairin
% 0.20/0.46 => ( cartprodmempair1
% 0.20/0.46 => ( cartprodmempair
% 0.20/0.46 => ( setunionE2
% 0.20/0.46 => ( setunionsingleton1
% 0.20/0.46 => ( setunionsingleton2
% 0.20/0.46 => ( setunionsingleton
% 0.20/0.46 => ( singletonprop
% 0.20/0.46 => ( ex1E1
% 0.20/0.46 => ( ex1I
% 0.20/0.46 => ( ex1I2
% 0.20/0.46 => ( singletonsuniq
% 0.20/0.46 => ( setukpairinjL1
% 0.20/0.46 => ( kfstsingleton
% 0.20/0.46 => ( theprop
% 0.20/0.46 => ( kfstpairEq
% 0.20/0.46 => ( cartprodfstin
% 0.20/0.46 => ( setukpairinjL2
% 0.20/0.46 => ( setukpairinjL
% 0.20/0.46 => ( setukpairinjR11
% 0.20/0.46 => ( setukpairinjR12
% 0.20/0.46 => ( setukpairinjR1
% 0.20/0.46 => ( upairequniteq
% 0.20/0.46 => ( setukpairinjR2
% 0.20/0.46 => ( setukpairinjR
% 0.20/0.46 => ( ksndsingleton
% 0.20/0.46 => ( ksndpairEq
% 0.20/0.46 => ( kpairsurjEq
% 0.20/0.46 => ( cartprodsndin
% 0.20/0.46 => ( cartprodpairmemEL
% 0.20/0.46 => ( cartprodpairmemER
% 0.20/0.46 => ( cartprodmempaircEq
% 0.20/0.46 => ( cartprodfstpairEq
% 0.20/0.46 => ( cartprodsndpairEq
% 0.20/0.46 => ( cartprodpairsurjEq
% 0.20/0.46 => ( dpsetconstrI
% 0.20/0.46 => ( dpsetconstrSub
% 0.20/0.46 => ( setOfPairsIsBReln
% 0.20/0.46 => ( dpsetconstrERa
% 0.20/0.46 => ( dpsetconstrEL1
% 0.20/0.46 => ( dpsetconstrEL2
% 0.20/0.46 => ( dpsetconstrER
% 0.20/0.46 => ( funcImageSingleton
% 0.20/0.46 => ( apProp
% 0.20/0.46 => ( app
% 0.20/0.46 => ( infuncsetfunc
% 0.20/0.46 => ( ap2p
% 0.20/0.46 => ( funcinfuncset
% 0.20/0.46 => ( lamProp
% 0.20/0.46 => ( lamp
% 0.20/0.46 => ( lam2p
% 0.20/0.46 => ( brelnall1
% 0.20/0.46 => ( brelnall2
% 0.20/0.46 => ( ex1E2
% 0.20/0.46 => ( funcGraphProp1
% 0.20/0.46 => ( funcGraphProp3
% 0.20/0.46 => ( funcGraphProp2
% 0.20/0.46 => ( funcextLem
% 0.20/0.46 => ( funcGraphProp4
% 0.20/0.46 => ( subbreln
% 0.20/0.46 => ( eqbreln
% 0.20/0.46 => ( funcext
% 0.20/0.46 => ( funcext2
% 0.20/0.46 => ( ap2apEq1
% 0.20/0.46 => ( ap2apEq2
% 0.20/0.46 => ( beta1
% 0.20/0.46 => ( eta1
% 0.20/0.46 => ( lam2lamEq
% 0.20/0.46 => ( beta2
% 0.20/0.46 => ( eta2
% 0.20/0.46 => ( iffalseProp1
% 0.20/0.46 => ( iffalseProp2
% 0.20/0.46 => ( iftrueProp1
% 0.20/0.46 => ( iftrueProp2
% 0.20/0.46 => ( ifSingleton
% 0.20/0.46 => ( ifp
% 0.20/0.46 => ( theeq
% 0.20/0.46 => ( iftrue
% 0.20/0.46 => ( iffalse
% 0.20/0.46 => ( iftrueorfalse
% 0.20/0.46 => ( binintersectT_lem
% 0.20/0.46 => ( binunionT_lem
% 0.20/0.46 => ( powersetT_lem
% 0.20/0.46 => ( setminusT_lem
% 0.20/0.46 => ( complementT_lem
% 0.20/0.46 => ( setextT
% 0.20/0.46 => ( subsetTI
% 0.20/0.46 => ( powersetTI1
% 0.20/0.46 => ( powersetTE1
% 0.20/0.46 => ( complementTI1
% 0.20/0.46 => ( complementTE1
% 0.20/0.46 => ( binintersectTELcontra
% 0.20/0.46 => ( binintersectTERcontra
% 0.20/0.46 => ( contrasubsetT
% 0.20/0.46 => ( contrasubsetT1
% 0.20/0.46 => ( contrasubsetT2
% 0.20/0.46 => ( contrasubsetT3
% 0.20/0.46 => ( doubleComplementI1
% 0.20/0.46 => ( doubleComplementE1
% 0.20/0.46 => ( doubleComplementSub1
% 0.20/0.46 => ( doubleComplementSub2
% 0.20/0.46 => ( doubleComplementEq
% 0.20/0.46 => ( complementTnotintersectT
% 0.20/0.46 => ( complementImpComplementIntersect
% 0.20/0.46 => ( complementSubsetComplementIntersect
% 0.20/0.46 => ( complementInPowersetComplementIntersect
% 0.20/0.46 => ( contraSubsetComplement
% 0.20/0.46 => ( complementTcontraSubset
% 0.20/0.46 => ( binunionTILcontra
% 0.20/0.46 => ( binunionTIRcontra
% 0.20/0.46 => ( inIntersectImpInUnion
% 0.20/0.46 => ( inIntersectImpInUnion2
% 0.20/0.46 => ( inIntersectImpInIntersectUnions
% 0.20/0.46 => ( intersectInPowersetIntersectUnions
% 0.20/0.46 => ( inComplementUnionImpNotIn1
% 0.20/0.46 => ( inComplementUnionImpInComplement1
% 0.20/0.46 => ( binunionTE
% 0.20/0.46 => ( binunionTEcontra
% 0.20/0.46 => ( demorgan2a1
% 0.20/0.46 => ( complementUnionInPowersetComplement
% 0.20/0.46 => ( demorgan2a2
% 0.20/0.46 => ( demorgan1a
% 0.20/0.46 => ( demorgan1b
% 0.20/0.46 => ( demorgan1
% 0.20/0.46 => ( demorgan2a
% 0.20/0.46 => ( demorgan2b2
% 0.20/0.46 => ( demorgan2b
% 0.32/0.46 => ( demorgan2
% 0.32/0.46 => ( woz13rule0
% 0.32/0.46 => ! [A: $i,X: $i] :
% 0.32/0.46 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.46 => ! [Y: $i] :
% 0.32/0.46 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.46 => ! [Z: $i] :
% 0.32/0.46 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.46 => ( ( subset @ X @ Z )
% 0.32/0.46 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.46
% 0.32/0.46 %------------------------------------------------------------------------------
% 0.32/0.46 ------------------------
% 2.77/1.52 ---- Embedded in THF ---
% 2.77/1.53 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.77/1.53 %%% Generated on Tue Feb 24 12:33:15 EST 2026
% 2.77/1.53 %%% using '$$dhol' embedding, version 1.3.0.
% 2.77/1.53 %%% Logic specification used:
% 2.77/1.53 %%% thf(spec, logic, $$dhol).
% 2.77/1.53
% 2.77/1.53 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.iP8AXdSty9/DTF2DTF22688.p
% See solution above
% 2.77/1.53 ------------------------
% 2.77/1.56 ---- Cleaned THF ---
% 2.77/1.56 thf(in_type,type,
% 2.77/1.56 in: $i > $i > $o ).
% 2.77/1.56
% 2.77/1.56 thf(exu_type,type,
% 2.77/1.56 exu: ( $i > $o ) > $o ).
% 2.77/1.56
% 2.77/1.56 thf(exu,definition,
% 2.77/1.56 ( exu
% 2.77/1.56 = ( ^ [Xphi: $i > $o] :
% 2.77/1.56 ? [Xx: $i] :
% 2.77/1.56 ( ( Xphi @ Xx )
% 2.77/1.56 & ! [Xy: $i] :
% 2.77/1.56 ( ( Xphi @ Xy )
% 2.77/1.56 => ( Xx = Xy ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setextAx_type,type,
% 2.77/1.56 setextAx: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setextAx,definition,
% 2.77/1.56 ( setextAx
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 <=> ( in @ Xx @ B ) )
% 2.77/1.56 => ( A = B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(emptyset_type,type,
% 2.77/1.56 emptyset: $i ).
% 2.77/1.56
% 2.77/1.56 thf(emptysetAx_type,type,
% 2.77/1.56 emptysetAx: $o ).
% 2.77/1.56
% 2.77/1.56 thf(emptysetAx,definition,
% 2.77/1.56 ( emptysetAx
% 2.77/1.56 = ( ! [Xx: $i] :
% 2.77/1.56 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoin_type,type,
% 2.77/1.56 setadjoin: $i > $i > $i ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoinAx_type,type,
% 2.77/1.56 setadjoinAx: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoinAx,definition,
% 2.77/1.56 ( setadjoinAx
% 2.77/1.56 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.77/1.56 <=> ( ( Xy = Xx )
% 2.77/1.56 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(powerset_type,type,
% 2.77/1.56 powerset: $i > $i ).
% 2.77/1.56
% 2.77/1.56 thf(powersetAx_type,type,
% 2.77/1.56 powersetAx: $o ).
% 2.77/1.56
% 2.77/1.56 thf(powersetAx,definition,
% 2.77/1.56 ( powersetAx
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( in @ B @ ( powerset @ A ) )
% 2.77/1.56 <=> ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ B )
% 2.77/1.56 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setunion_type,type,
% 2.77/1.56 setunion: $i > $i ).
% 2.77/1.56
% 2.77/1.56 thf(setunionAx_type,type,
% 2.77/1.56 setunionAx: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setunionAx,definition,
% 2.77/1.56 ( setunionAx
% 2.77/1.56 = ( ! [A: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ ( setunion @ A ) )
% 2.77/1.56 <=> ? [B: $i] :
% 2.77/1.56 ( ( in @ Xx @ B )
% 2.77/1.56 & ( in @ B @ A ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(omega_type,type,
% 2.77/1.56 omega: $i ).
% 2.77/1.56
% 2.77/1.56 thf(omega0Ax_type,type,
% 2.77/1.56 omega0Ax: $o ).
% 2.77/1.56
% 2.77/1.56 thf(omega0Ax,definition,
% 2.77/1.56 ( omega0Ax
% 2.77/1.56 = ( in @ emptyset @ omega ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(omegaSAx_type,type,
% 2.77/1.56 omegaSAx: $o ).
% 2.77/1.56
% 2.77/1.56 thf(omegaSAx,definition,
% 2.77/1.56 ( omegaSAx
% 2.77/1.56 = ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ omega )
% 2.77/1.56 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(omegaIndAx_type,type,
% 2.77/1.56 omegaIndAx: $o ).
% 2.77/1.56
% 2.77/1.56 thf(omegaIndAx,definition,
% 2.77/1.56 ( omegaIndAx
% 2.77/1.56 = ( ! [A: $i] :
% 2.77/1.56 ( ( ( in @ emptyset @ A )
% 2.77/1.56 & ! [Xx: $i] :
% 2.77/1.56 ( ( ( in @ Xx @ omega )
% 2.77/1.56 & ( in @ Xx @ A ) )
% 2.77/1.56 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ omega )
% 2.77/1.56 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(replAx_type,type,
% 2.77/1.56 replAx: $o ).
% 2.77/1.56
% 2.77/1.56 thf(replAx,definition,
% 2.77/1.56 ( replAx
% 2.77/1.56 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( exu
% 2.77/1.56 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.77/1.56 => ? [B: $i] :
% 2.77/1.56 ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ B )
% 2.77/1.56 <=> ? [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ A )
% 2.77/1.56 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(foundationAx_type,type,
% 2.77/1.56 foundationAx: $o ).
% 2.77/1.56
% 2.77/1.56 thf(foundationAx,definition,
% 2.77/1.56 ( foundationAx
% 2.77/1.56 = ( ! [A: $i] :
% 2.77/1.56 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.77/1.56 => ? [B: $i] :
% 2.77/1.56 ( ( in @ B @ A )
% 2.77/1.56 & ~ ? [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ B )
% 2.77/1.56 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(wellorderingAx_type,type,
% 2.77/1.56 wellorderingAx: $o ).
% 2.77/1.56
% 2.77/1.56 thf(wellorderingAx,definition,
% 2.77/1.56 ( wellorderingAx
% 2.77/1.56 = ( ! [A: $i] :
% 2.77/1.56 ? [B: $i] :
% 2.77/1.56 ( ! [C: $i] :
% 2.77/1.56 ( ( in @ C @ B )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ C )
% 2.77/1.56 => ( in @ Xx @ A ) ) )
% 2.77/1.56 & ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( ( in @ Xx @ A )
% 2.77/1.56 & ( in @ Xy @ A ) )
% 2.77/1.56 => ( ! [C: $i] :
% 2.77/1.56 ( ( in @ C @ B )
% 2.77/1.56 => ( ( in @ Xx @ C )
% 2.77/1.56 <=> ( in @ Xy @ C ) ) )
% 2.77/1.56 => ( Xx = Xy ) ) )
% 2.77/1.56 & ! [C: $i,D: $i] :
% 2.77/1.56 ( ( ( in @ C @ B )
% 2.77/1.56 & ( in @ D @ B ) )
% 2.77/1.56 => ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ C )
% 2.77/1.56 => ( in @ Xx @ D ) )
% 2.77/1.56 | ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ D )
% 2.77/1.56 => ( in @ Xx @ C ) ) ) )
% 2.77/1.56 & ! [C: $i] :
% 2.77/1.56 ( ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ C )
% 2.77/1.56 => ( in @ Xx @ A ) )
% 2.77/1.56 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.77/1.56 => ? [D: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ D @ B )
% 2.77/1.56 & ( in @ Xx @ C )
% 2.77/1.56 & ~ ? [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ D )
% 2.77/1.56 & ( in @ Xy @ C ) )
% 2.77/1.56 & ! [E: $i] :
% 2.77/1.56 ( ( in @ E @ B )
% 2.77/1.56 => ( ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ E )
% 2.77/1.56 => ( in @ Xy @ D ) )
% 2.77/1.56 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(descr_type,type,
% 2.77/1.56 descr: ( $i > $o ) > $i ).
% 2.77/1.56
% 2.77/1.56 thf(descrp_type,type,
% 2.77/1.56 descrp: $o ).
% 2.77/1.56
% 2.77/1.56 thf(descrp,definition,
% 2.77/1.56 ( descrp
% 2.77/1.56 = ( ! [Xphi: $i > $o] :
% 2.77/1.56 ( ( exu
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.56 => ( Xphi
% 2.77/1.56 @ ( descr
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(dsetconstr_type,type,
% 2.77/1.56 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.77/1.56
% 2.77/1.56 thf(dsetconstrI_type,type,
% 2.77/1.56 dsetconstrI: $o ).
% 2.77/1.56
% 2.77/1.56 thf(dsetconstrI,definition,
% 2.77/1.56 ( dsetconstrI
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( Xphi @ Xx )
% 2.77/1.56 => ( in @ Xx
% 2.77/1.56 @ ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(dsetconstrEL_type,type,
% 2.77/1.56 dsetconstrEL: $o ).
% 2.77/1.56
% 2.77/1.56 thf(dsetconstrEL,definition,
% 2.77/1.56 ( dsetconstrEL
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx
% 2.77/1.56 @ ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.77/1.56 => ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(dsetconstrER_type,type,
% 2.77/1.56 dsetconstrER: $o ).
% 2.77/1.56
% 2.77/1.56 thf(dsetconstrER,definition,
% 2.77/1.56 ( dsetconstrER
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx
% 2.77/1.56 @ ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.77/1.56 => ( Xphi @ Xx ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(exuE1_type,type,
% 2.77/1.56 exuE1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(exuE1,definition,
% 2.77/1.56 ( exuE1
% 2.77/1.56 = ( ! [Xphi: $i > $o] :
% 2.77/1.56 ( ( exu
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.56 => ? [Xx: $i] :
% 2.77/1.56 ( ( Xphi @ Xx )
% 2.77/1.56 & ! [Xy: $i] :
% 2.77/1.56 ( ( Xphi @ Xy )
% 2.77/1.56 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(prop2set_type,type,
% 2.77/1.56 prop2set: $o > $i ).
% 2.77/1.56
% 2.77/1.56 thf(prop2setE_type,type,
% 2.77/1.56 prop2setE: $o ).
% 2.77/1.56
% 2.77/1.56 thf(prop2setE,definition,
% 2.77/1.56 ( prop2setE
% 2.77/1.56 = ( ! [Xphi: $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.77/1.56 => Xphi ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(emptysetE_type,type,
% 2.77/1.56 emptysetE: $o ).
% 2.77/1.56
% 2.77/1.56 thf(emptysetE,definition,
% 2.77/1.56 ( emptysetE
% 2.77/1.56 = ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ emptyset )
% 2.77/1.56 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(emptysetimpfalse_type,type,
% 2.77/1.56 emptysetimpfalse: $o ).
% 2.77/1.56
% 2.77/1.56 thf(emptysetimpfalse,definition,
% 2.77/1.56 ( emptysetimpfalse
% 2.77/1.56 = ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ emptyset )
% 2.77/1.56 => $false ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(notinemptyset_type,type,
% 2.77/1.56 notinemptyset: $o ).
% 2.77/1.56
% 2.77/1.56 thf(notinemptyset,definition,
% 2.77/1.56 ( notinemptyset
% 2.77/1.56 = ( ! [Xx: $i] :
% 2.77/1.56 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(exuE3e_type,type,
% 2.77/1.56 exuE3e: $o ).
% 2.77/1.56
% 2.77/1.56 thf(exuE3e,definition,
% 2.77/1.56 ( exuE3e
% 2.77/1.56 = ( ! [Xphi: $i > $o] :
% 2.77/1.56 ( ( exu
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.56 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setext_type,type,
% 2.77/1.56 setext: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setext,definition,
% 2.77/1.56 ( setext
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ Xx @ B ) )
% 2.77/1.56 => ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ B )
% 2.77/1.56 => ( in @ Xx @ A ) )
% 2.77/1.56 => ( A = B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(emptyI_type,type,
% 2.77/1.56 emptyI: $o ).
% 2.77/1.56
% 2.77/1.56 thf(emptyI,definition,
% 2.77/1.56 ( emptyI
% 2.77/1.56 = ( ! [A: $i] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ~ ( in @ Xx @ A )
% 2.77/1.56 => ( A = emptyset ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(noeltsimpempty_type,type,
% 2.77/1.56 noeltsimpempty: $o ).
% 2.77/1.56
% 2.77/1.56 thf(noeltsimpempty,definition,
% 2.77/1.56 ( noeltsimpempty
% 2.77/1.56 = ( ! [A: $i] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ~ ( in @ Xx @ A )
% 2.77/1.56 => ( A = emptyset ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setbeta_type,type,
% 2.77/1.56 setbeta: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setbeta,definition,
% 2.77/1.56 ( setbeta
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx
% 2.77/1.56 @ ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.77/1.56 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(nonempty_type,type,
% 2.77/1.56 nonempty: $i > $o ).
% 2.77/1.56
% 2.77/1.56 thf(nonempty,definition,
% 2.77/1.56 ( nonempty
% 2.77/1.56 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(nonemptyE1_type,type,
% 2.77/1.56 nonemptyE1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(nonemptyE1,definition,
% 2.77/1.56 ( nonemptyE1
% 2.77/1.56 = ( ! [A: $i] :
% 2.77/1.56 ( ( nonempty @ A )
% 2.77/1.56 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(nonemptyI_type,type,
% 2.77/1.56 nonemptyI: $o ).
% 2.77/1.56
% 2.77/1.56 thf(nonemptyI,definition,
% 2.77/1.56 ( nonemptyI
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( Xphi @ Xx )
% 2.77/1.56 => ( nonempty
% 2.77/1.56 @ ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(nonemptyI1_type,type,
% 2.77/1.56 nonemptyI1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(nonemptyI1,definition,
% 2.77/1.56 ( nonemptyI1
% 2.77/1.56 = ( ! [A: $i] :
% 2.77/1.56 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.77/1.56 => ( nonempty @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoinIL_type,type,
% 2.77/1.56 setadjoinIL: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoinIL,definition,
% 2.77/1.56 ( setadjoinIL
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(emptyinunitempty_type,type,
% 2.77/1.56 emptyinunitempty: $o ).
% 2.77/1.56
% 2.77/1.56 thf(emptyinunitempty,definition,
% 2.77/1.56 ( emptyinunitempty
% 2.77/1.56 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoinIR_type,type,
% 2.77/1.56 setadjoinIR: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoinIR,definition,
% 2.77/1.56 ( setadjoinIR
% 2.77/1.56 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ A )
% 2.77/1.56 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoinE_type,type,
% 2.77/1.56 setadjoinE: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoinE,definition,
% 2.77/1.56 ( setadjoinE
% 2.77/1.56 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.77/1.56 => ! [Xphi: $o] :
% 2.77/1.56 ( ( ( Xy = Xx )
% 2.77/1.56 => Xphi )
% 2.77/1.56 => ( ( ( in @ Xy @ A )
% 2.77/1.56 => Xphi )
% 2.77/1.56 => Xphi ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoinOr_type,type,
% 2.77/1.56 setadjoinOr: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoinOr,definition,
% 2.77/1.56 ( setadjoinOr
% 2.77/1.56 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.77/1.56 => ( ( Xy = Xx )
% 2.77/1.56 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setoftrueEq_type,type,
% 2.77/1.56 setoftrueEq: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setoftrueEq,definition,
% 2.77/1.56 ( setoftrueEq
% 2.77/1.56 = ( ! [A: $i] :
% 2.77/1.56 ( ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xx: $i] : $true )
% 2.77/1.56 = A ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(powersetI_type,type,
% 2.77/1.56 powersetI: $o ).
% 2.77/1.56
% 2.77/1.56 thf(powersetI,definition,
% 2.77/1.56 ( powersetI
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ B )
% 2.77/1.56 => ( in @ Xx @ A ) )
% 2.77/1.56 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(emptyinPowerset_type,type,
% 2.77/1.56 emptyinPowerset: $o ).
% 2.77/1.56
% 2.77/1.56 thf(emptyinPowerset,definition,
% 2.77/1.56 ( emptyinPowerset
% 2.77/1.56 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(emptyInPowerset_type,type,
% 2.77/1.56 emptyInPowerset: $o ).
% 2.77/1.56
% 2.77/1.56 thf(emptyInPowerset,definition,
% 2.77/1.56 ( emptyInPowerset
% 2.77/1.56 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(powersetE_type,type,
% 2.77/1.56 powersetE: $o ).
% 2.77/1.56
% 2.77/1.56 thf(powersetE,definition,
% 2.77/1.56 ( powersetE
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ B @ ( powerset @ A ) )
% 2.77/1.56 => ( ( in @ Xx @ B )
% 2.77/1.56 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setunionI_type,type,
% 2.77/1.56 setunionI: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setunionI,definition,
% 2.77/1.56 ( setunionI
% 2.77/1.56 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.77/1.56 ( ( in @ Xx @ B )
% 2.77/1.56 => ( ( in @ B @ A )
% 2.77/1.56 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setunionE_type,type,
% 2.77/1.56 setunionE: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setunionE,definition,
% 2.77/1.56 ( setunionE
% 2.77/1.56 = ( ! [A: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ ( setunion @ A ) )
% 2.77/1.56 => ! [Xphi: $o] :
% 2.77/1.56 ( ! [B: $i] :
% 2.77/1.56 ( ( in @ Xx @ B )
% 2.77/1.56 => ( ( in @ B @ A )
% 2.77/1.56 => Xphi ) )
% 2.77/1.56 => Xphi ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(subPowSU_type,type,
% 2.77/1.56 subPowSU: $o ).
% 2.77/1.56
% 2.77/1.56 thf(subPowSU,definition,
% 2.77/1.56 ( subPowSU
% 2.77/1.56 = ( ! [A: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(exuE2_type,type,
% 2.77/1.56 exuE2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(exuE2,definition,
% 2.77/1.56 ( exuE2
% 2.77/1.56 = ( ! [Xphi: $i > $o] :
% 2.77/1.56 ( ( exu
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.56 => ? [Xx: $i] :
% 2.77/1.56 ! [Xy: $i] :
% 2.77/1.56 ( ( Xphi @ Xy )
% 2.77/1.56 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(nonemptyImpWitness_type,type,
% 2.77/1.56 nonemptyImpWitness: $o ).
% 2.77/1.56
% 2.77/1.56 thf(nonemptyImpWitness,definition,
% 2.77/1.56 ( nonemptyImpWitness
% 2.77/1.56 = ( ! [A: $i] :
% 2.77/1.56 ( ( nonempty @ A )
% 2.77/1.56 => ? [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 & $true ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(uniqinunit_type,type,
% 2.77/1.56 uniqinunit: $o ).
% 2.77/1.56
% 2.77/1.56 thf(uniqinunit,definition,
% 2.77/1.56 ( uniqinunit
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.77/1.56 => ( Xx = Xy ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(notinsingleton_type,type,
% 2.77/1.56 notinsingleton: $o ).
% 2.77/1.56
% 2.77/1.56 thf(notinsingleton,definition,
% 2.77/1.56 ( notinsingleton
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( Xx != Xy )
% 2.77/1.56 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(eqinunit_type,type,
% 2.77/1.56 eqinunit: $o ).
% 2.77/1.56
% 2.77/1.56 thf(eqinunit,definition,
% 2.77/1.56 ( eqinunit
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( Xx = Xy )
% 2.77/1.56 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(singletonsswitch_type,type,
% 2.77/1.56 singletonsswitch: $o ).
% 2.77/1.56
% 2.77/1.56 thf(singletonsswitch,definition,
% 2.77/1.56 ( singletonsswitch
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.77/1.56 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(upairsetE_type,type,
% 2.77/1.56 upairsetE: $o ).
% 2.77/1.56
% 2.77/1.56 thf(upairsetE,definition,
% 2.77/1.56 ( upairsetE
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.77/1.56 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.77/1.56 => ( ( Xz = Xx )
% 2.77/1.56 | ( Xz = Xy ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(upairsetIL_type,type,
% 2.77/1.56 upairsetIL: $o ).
% 2.77/1.56
% 2.77/1.56 thf(upairsetIL,definition,
% 2.77/1.56 ( upairsetIL
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(upairsetIR_type,type,
% 2.77/1.56 upairsetIR: $o ).
% 2.77/1.56
% 2.77/1.56 thf(upairsetIR,definition,
% 2.77/1.56 ( upairsetIR
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(emptyE1_type,type,
% 2.77/1.56 emptyE1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(emptyE1,definition,
% 2.77/1.56 ( emptyE1
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.56 ( ? [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 & ( Xphi @ Xx ) )
% 2.77/1.56 => ( ( ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.56 = emptyset )
% 2.77/1.56 => $false ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(vacuousDall_type,type,
% 2.77/1.56 vacuousDall: $o ).
% 2.77/1.56
% 2.77/1.56 thf(vacuousDall,definition,
% 2.77/1.56 ( vacuousDall
% 2.77/1.56 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ emptyset )
% 2.77/1.56 => ( Xphi @ Xx ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(quantDeMorgan1_type,type,
% 2.77/1.56 quantDeMorgan1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(quantDeMorgan1,definition,
% 2.77/1.56 ( quantDeMorgan1
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.56 ( ~ ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( Xphi @ Xx ) )
% 2.77/1.56 => ? [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(quantDeMorgan2_type,type,
% 2.77/1.56 quantDeMorgan2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(quantDeMorgan2,definition,
% 2.77/1.56 ( quantDeMorgan2
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ~ ( Xphi @ Xx ) )
% 2.77/1.56 => ~ ? [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(quantDeMorgan3_type,type,
% 2.77/1.56 quantDeMorgan3: $o ).
% 2.77/1.56
% 2.77/1.56 thf(quantDeMorgan3,definition,
% 2.77/1.56 ( quantDeMorgan3
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.56 ( ~ ? [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 & ( Xphi @ Xx ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(quantDeMorgan4_type,type,
% 2.77/1.56 quantDeMorgan4: $o ).
% 2.77/1.56
% 2.77/1.56 thf(quantDeMorgan4,definition,
% 2.77/1.56 ( quantDeMorgan4
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.56 ( ? [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 & ~ ( Xphi @ Xx ) )
% 2.77/1.56 => ~ ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(prop2setI_type,type,
% 2.77/1.56 prop2setI: $o ).
% 2.77/1.56
% 2.77/1.56 thf(prop2setI,definition,
% 2.77/1.56 ( prop2setI
% 2.77/1.56 = ( ! [Xphi: $o] :
% 2.77/1.56 ( Xphi
% 2.77/1.56 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(set2prop_type,type,
% 2.77/1.56 set2prop: $i > $o ).
% 2.77/1.56
% 2.77/1.56 thf(prop2set2propI_type,type,
% 2.77/1.56 prop2set2propI: $o ).
% 2.77/1.56
% 2.77/1.56 thf(prop2set2propI,definition,
% 2.77/1.56 ( prop2set2propI
% 2.77/1.56 = ( ! [Xphi: $o] :
% 2.77/1.56 ( Xphi
% 2.77/1.56 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(notdexE_type,type,
% 2.77/1.56 notdexE: $o ).
% 2.77/1.56
% 2.77/1.56 thf(notdexE,definition,
% 2.77/1.56 ( notdexE
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.56 ( ~ ? [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 & ( Xphi @ Xx ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(notdallE_type,type,
% 2.77/1.56 notdallE: $o ).
% 2.77/1.56
% 2.77/1.56 thf(notdallE,definition,
% 2.77/1.56 ( notdallE
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.56 ( ~ ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( Xphi @ Xx ) )
% 2.77/1.56 => ? [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(exuI1_type,type,
% 2.77/1.56 exuI1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(exuI1,definition,
% 2.77/1.56 ( exuI1
% 2.77/1.56 = ( ! [Xphi: $i > $o] :
% 2.77/1.56 ( ? [Xx: $i] :
% 2.77/1.56 ( ( Xphi @ Xx )
% 2.77/1.56 & ! [Xy: $i] :
% 2.77/1.56 ( ( Xphi @ Xy )
% 2.77/1.56 => ( Xx = Xy ) ) )
% 2.77/1.56 => ( exu
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(exuI3_type,type,
% 2.77/1.56 exuI3: $o ).
% 2.77/1.56
% 2.77/1.56 thf(exuI3,definition,
% 2.77/1.56 ( exuI3
% 2.77/1.56 = ( ! [Xphi: $i > $o] :
% 2.77/1.56 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.77/1.56 => ( ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( Xphi @ Xx )
% 2.77/1.56 => ( ( Xphi @ Xy )
% 2.77/1.56 => ( Xx = Xy ) ) )
% 2.77/1.56 => ( exu
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(exuI2_type,type,
% 2.77/1.56 exuI2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(exuI2,definition,
% 2.77/1.56 ( exuI2
% 2.77/1.56 = ( ! [Xphi: $i > $o] :
% 2.77/1.56 ( ? [Xx: $i] :
% 2.77/1.56 ! [Xy: $i] :
% 2.77/1.56 ( ( Xphi @ Xy )
% 2.77/1.56 <=> ( Xy = Xx ) )
% 2.77/1.56 => ( exu
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(inCongP_type,type,
% 2.77/1.56 inCongP: $o ).
% 2.77/1.56
% 2.77/1.56 thf(inCongP,definition,
% 2.77/1.56 ( inCongP
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( A = B )
% 2.77/1.56 => ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( Xx = Xy )
% 2.77/1.56 => ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(in__Cong_type,type,
% 2.77/1.56 in__Cong: $o ).
% 2.77/1.56
% 2.77/1.56 thf(in__Cong,definition,
% 2.77/1.56 ( in__Cong
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( A = B )
% 2.77/1.56 => ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( Xx = Xy )
% 2.77/1.56 => ( ( in @ Xx @ A )
% 2.77/1.56 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(exuE3u_type,type,
% 2.77/1.56 exuE3u: $o ).
% 2.77/1.56
% 2.77/1.56 thf(exuE3u,definition,
% 2.77/1.56 ( exuE3u
% 2.77/1.56 = ( ! [Xphi: $i > $o] :
% 2.77/1.56 ( ( exu
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.56 => ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( Xphi @ Xx )
% 2.77/1.56 => ( ( Xphi @ Xy )
% 2.77/1.56 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(exu__Cong_type,type,
% 2.77/1.56 exu__Cong: $o ).
% 2.77/1.56
% 2.77/1.56 thf(exu__Cong,definition,
% 2.77/1.56 ( exu__Cong
% 2.77/1.56 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.77/1.56 ( ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( Xx = Xy )
% 2.77/1.56 => ( ( Xphi @ Xx )
% 2.77/1.56 <=> ( Xpsi @ Xy ) ) )
% 2.77/1.56 => ( ( exu
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.56 <=> ( exu
% 2.77/1.56 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(emptyset__Cong_type,type,
% 2.77/1.56 emptyset__Cong: $o ).
% 2.77/1.56
% 2.77/1.56 thf(emptyset__Cong,definition,
% 2.77/1.56 ( emptyset__Cong
% 2.77/1.56 = ( emptyset = emptyset ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoin__Cong_type,type,
% 2.77/1.56 setadjoin__Cong: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoin__Cong,definition,
% 2.77/1.56 ( setadjoin__Cong
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( Xx = Xy )
% 2.77/1.56 => ! [Xz: $i,Xu: $i] :
% 2.77/1.56 ( ( Xz = Xu )
% 2.77/1.56 => ( ( setadjoin @ Xx @ Xz )
% 2.77/1.56 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(powerset__Cong_type,type,
% 2.77/1.56 powerset__Cong: $o ).
% 2.77/1.56
% 2.77/1.56 thf(powerset__Cong,definition,
% 2.77/1.56 ( powerset__Cong
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( A = B )
% 2.77/1.56 => ( ( powerset @ A )
% 2.77/1.56 = ( powerset @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setunion__Cong_type,type,
% 2.77/1.56 setunion__Cong: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setunion__Cong,definition,
% 2.77/1.56 ( setunion__Cong
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( A = B )
% 2.77/1.56 => ( ( setunion @ A )
% 2.77/1.56 = ( setunion @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(omega__Cong_type,type,
% 2.77/1.56 omega__Cong: $o ).
% 2.77/1.56
% 2.77/1.56 thf(omega__Cong,definition,
% 2.77/1.56 ( omega__Cong
% 2.77/1.56 = ( omega = omega ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(exuEu_type,type,
% 2.77/1.56 exuEu: $o ).
% 2.77/1.56
% 2.77/1.56 thf(exuEu,definition,
% 2.77/1.56 ( exuEu
% 2.77/1.56 = ( ! [Xphi: $i > $o] :
% 2.77/1.56 ( ( exu
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.56 => ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( Xphi @ Xx )
% 2.77/1.56 => ( ( Xphi @ Xy )
% 2.77/1.56 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(descr__Cong_type,type,
% 2.77/1.56 descr__Cong: $o ).
% 2.77/1.56
% 2.77/1.56 thf(descr__Cong,definition,
% 2.77/1.56 ( descr__Cong
% 2.77/1.56 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.77/1.56 ( ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( Xx = Xy )
% 2.77/1.56 => ( ( Xphi @ Xx )
% 2.77/1.56 <=> ( Xpsi @ Xy ) ) )
% 2.77/1.56 => ( ( exu
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.56 => ( ( exu
% 2.77/1.56 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.77/1.56 => ( ( descr
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.56 = ( descr
% 2.77/1.56 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(dsetconstr__Cong_type,type,
% 2.77/1.56 dsetconstr__Cong: $o ).
% 2.77/1.56
% 2.77/1.56 thf(dsetconstr__Cong,definition,
% 2.77/1.56 ( dsetconstr__Cong
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( A = B )
% 2.77/1.56 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( ( Xx = Xy )
% 2.77/1.56 => ( ( Xphi @ Xx )
% 2.77/1.56 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.77/1.56 => ( ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.56 = ( dsetconstr @ B
% 2.77/1.56 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(subset_type,type,
% 2.77/1.56 subset: $i > $i > $o ).
% 2.77/1.56
% 2.77/1.56 thf(disjoint_type,type,
% 2.77/1.56 disjoint: $i > $i > $o ).
% 2.77/1.56
% 2.77/1.56 thf(setsmeet_type,type,
% 2.77/1.56 setsmeet: $i > $i > $o ).
% 2.77/1.56
% 2.77/1.56 thf(subsetI1_type,type,
% 2.77/1.56 subsetI1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(subsetI1,definition,
% 2.77/1.56 ( subsetI1
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ Xx @ B ) )
% 2.77/1.56 => ( subset @ A @ B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(eqimpsubset2_type,type,
% 2.77/1.56 eqimpsubset2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(eqimpsubset2,definition,
% 2.77/1.56 ( eqimpsubset2
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( A = B )
% 2.77/1.56 => ( subset @ B @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(eqimpsubset1_type,type,
% 2.77/1.56 eqimpsubset1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(eqimpsubset1,definition,
% 2.77/1.56 ( eqimpsubset1
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( A = B )
% 2.77/1.56 => ( subset @ A @ B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(subsetI2_type,type,
% 2.77/1.56 subsetI2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(subsetI2,definition,
% 2.77/1.56 ( subsetI2
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ Xx @ B ) )
% 2.77/1.56 => ( subset @ A @ B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(emptysetsubset_type,type,
% 2.77/1.56 emptysetsubset: $o ).
% 2.77/1.56
% 2.77/1.56 thf(emptysetsubset,definition,
% 2.77/1.56 ( emptysetsubset
% 2.77/1.56 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(subsetE_type,type,
% 2.77/1.56 subsetE: $o ).
% 2.77/1.56
% 2.77/1.56 thf(subsetE,definition,
% 2.77/1.56 ( subsetE
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( subset @ A @ B )
% 2.77/1.56 => ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(subsetE2_type,type,
% 2.77/1.56 subsetE2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(subsetE2,definition,
% 2.77/1.56 ( subsetE2
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( subset @ A @ B )
% 2.77/1.56 => ( ~ ( in @ Xx @ B )
% 2.77/1.56 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(notsubsetI_type,type,
% 2.77/1.56 notsubsetI: $o ).
% 2.77/1.56
% 2.77/1.56 thf(notsubsetI,definition,
% 2.77/1.56 ( notsubsetI
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ~ ( in @ Xx @ B )
% 2.77/1.56 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(notequalI1_type,type,
% 2.77/1.56 notequalI1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(notequalI1,definition,
% 2.77/1.56 ( notequalI1
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ~ ( subset @ A @ B )
% 2.77/1.56 => ( A != B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(notequalI2_type,type,
% 2.77/1.56 notequalI2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(notequalI2,definition,
% 2.77/1.56 ( notequalI2
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ~ ( in @ Xx @ B )
% 2.77/1.56 => ( A != B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(subsetRefl_type,type,
% 2.77/1.56 subsetRefl: $o ).
% 2.77/1.56
% 2.77/1.56 thf(subsetRefl,definition,
% 2.77/1.56 ( subsetRefl
% 2.77/1.56 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(subsetTrans_type,type,
% 2.77/1.56 subsetTrans: $o ).
% 2.77/1.56
% 2.77/1.56 thf(subsetTrans,definition,
% 2.77/1.56 ( subsetTrans
% 2.77/1.56 = ( ! [A: $i,B: $i,C: $i] :
% 2.77/1.56 ( ( subset @ A @ B )
% 2.77/1.56 => ( ( subset @ B @ C )
% 2.77/1.56 => ( subset @ A @ C ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoinSub_type,type,
% 2.77/1.56 setadjoinSub: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoinSub,definition,
% 2.77/1.56 ( setadjoinSub
% 2.77/1.56 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoinSub2_type,type,
% 2.77/1.56 setadjoinSub2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setadjoinSub2,definition,
% 2.77/1.56 ( setadjoinSub2
% 2.77/1.56 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.77/1.56 ( ( subset @ A @ B )
% 2.77/1.56 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(subset2powerset_type,type,
% 2.77/1.56 subset2powerset: $o ).
% 2.77/1.56
% 2.77/1.56 thf(subset2powerset,definition,
% 2.77/1.56 ( subset2powerset
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( subset @ A @ B )
% 2.77/1.56 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setextsub_type,type,
% 2.77/1.56 setextsub: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setextsub,definition,
% 2.77/1.56 ( setextsub
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( subset @ A @ B )
% 2.77/1.56 => ( ( subset @ B @ A )
% 2.77/1.56 => ( A = B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(subsetemptysetimpeq_type,type,
% 2.77/1.56 subsetemptysetimpeq: $o ).
% 2.77/1.56
% 2.77/1.56 thf(subsetemptysetimpeq,definition,
% 2.77/1.56 ( subsetemptysetimpeq
% 2.77/1.56 = ( ! [A: $i] :
% 2.77/1.56 ( ( subset @ A @ emptyset )
% 2.77/1.56 => ( A = emptyset ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(powersetI1_type,type,
% 2.77/1.56 powersetI1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(powersetI1,definition,
% 2.77/1.56 ( powersetI1
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( subset @ B @ A )
% 2.77/1.56 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(powersetE1_type,type,
% 2.77/1.56 powersetE1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(powersetE1,definition,
% 2.77/1.56 ( powersetE1
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( in @ B @ ( powerset @ A ) )
% 2.77/1.56 => ( subset @ B @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(inPowerset_type,type,
% 2.77/1.56 inPowerset: $o ).
% 2.77/1.56
% 2.77/1.56 thf(inPowerset,definition,
% 2.77/1.56 ( inPowerset
% 2.77/1.56 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(powersetsubset_type,type,
% 2.77/1.56 powersetsubset: $o ).
% 2.77/1.56
% 2.77/1.56 thf(powersetsubset,definition,
% 2.77/1.56 ( powersetsubset
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( subset @ A @ B )
% 2.77/1.56 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(sepInPowerset_type,type,
% 2.77/1.56 sepInPowerset: $o ).
% 2.77/1.56
% 2.77/1.56 thf(sepInPowerset,definition,
% 2.77/1.56 ( sepInPowerset
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.56 ( in
% 2.77/1.56 @ ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.56 @ ( powerset @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(sepSubset_type,type,
% 2.77/1.56 sepSubset: $o ).
% 2.77/1.56
% 2.77/1.56 thf(sepSubset,definition,
% 2.77/1.56 ( sepSubset
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.56 ( subset
% 2.77/1.56 @ ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.56 @ A ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binunion_type,type,
% 2.77/1.56 binunion: $i > $i > $i ).
% 2.77/1.56
% 2.77/1.56 thf(binunionIL_type,type,
% 2.77/1.56 binunionIL: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binunionIL,definition,
% 2.77/1.56 ( binunionIL
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(upairset2IR_type,type,
% 2.77/1.56 upairset2IR: $o ).
% 2.77/1.56
% 2.77/1.56 thf(upairset2IR,definition,
% 2.77/1.56 ( upairset2IR
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binunionIR_type,type,
% 2.77/1.56 binunionIR: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binunionIR,definition,
% 2.77/1.56 ( binunionIR
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ B )
% 2.77/1.56 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binunionEcases_type,type,
% 2.77/1.56 binunionEcases: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binunionEcases,definition,
% 2.77/1.56 ( binunionEcases
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.77/1.56 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.77/1.56 => ( ( ( in @ Xx @ A )
% 2.77/1.56 => Xphi )
% 2.77/1.56 => ( ( ( in @ Xx @ B )
% 2.77/1.56 => Xphi )
% 2.77/1.56 => Xphi ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binunionE_type,type,
% 2.77/1.56 binunionE: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binunionE,definition,
% 2.77/1.56 ( binunionE
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.77/1.56 => ( ( in @ Xx @ A )
% 2.77/1.56 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binunionLsub_type,type,
% 2.77/1.56 binunionLsub: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binunionLsub,definition,
% 2.77/1.56 ( binunionLsub
% 2.77/1.56 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binunionRsub_type,type,
% 2.77/1.56 binunionRsub: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binunionRsub,definition,
% 2.77/1.56 ( binunionRsub
% 2.77/1.56 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binintersect_type,type,
% 2.77/1.56 binintersect: $i > $i > $i ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectI_type,type,
% 2.77/1.56 binintersectI: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectI,definition,
% 2.77/1.56 ( binintersectI
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ B )
% 2.77/1.56 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectSubset5_type,type,
% 2.77/1.56 binintersectSubset5: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectSubset5,definition,
% 2.77/1.56 ( binintersectSubset5
% 2.77/1.56 = ( ! [A: $i,B: $i,C: $i] :
% 2.77/1.56 ( ( subset @ C @ A )
% 2.77/1.56 => ( ( subset @ C @ B )
% 2.77/1.56 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectEL_type,type,
% 2.77/1.56 binintersectEL: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectEL,definition,
% 2.77/1.56 ( binintersectEL
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.77/1.56 => ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectLsub_type,type,
% 2.77/1.56 binintersectLsub: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectLsub,definition,
% 2.77/1.56 ( binintersectLsub
% 2.77/1.56 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectSubset2_type,type,
% 2.77/1.56 binintersectSubset2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectSubset2,definition,
% 2.77/1.56 ( binintersectSubset2
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( subset @ A @ B )
% 2.77/1.56 => ( ( binintersect @ A @ B )
% 2.77/1.56 = A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectSubset3_type,type,
% 2.77/1.56 binintersectSubset3: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectSubset3,definition,
% 2.77/1.56 ( binintersectSubset3
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( ( binintersect @ A @ B )
% 2.77/1.56 = B )
% 2.77/1.56 => ( subset @ B @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectER_type,type,
% 2.77/1.56 binintersectER: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectER,definition,
% 2.77/1.56 ( binintersectER
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.77/1.56 => ( in @ Xx @ B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(disjointsetsI1_type,type,
% 2.77/1.56 disjointsetsI1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(disjointsetsI1,definition,
% 2.77/1.56 ( disjointsetsI1
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ~ ? [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 & ( in @ Xx @ B ) )
% 2.77/1.56 => ( ( binintersect @ A @ B )
% 2.77/1.56 = emptyset ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectRsub_type,type,
% 2.77/1.56 binintersectRsub: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectRsub,definition,
% 2.77/1.56 ( binintersectRsub
% 2.77/1.56 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectSubset4_type,type,
% 2.77/1.56 binintersectSubset4: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectSubset4,definition,
% 2.77/1.56 ( binintersectSubset4
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( subset @ B @ A )
% 2.77/1.56 => ( ( binintersect @ A @ B )
% 2.77/1.56 = B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectSubset1_type,type,
% 2.77/1.56 binintersectSubset1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectSubset1,definition,
% 2.77/1.56 ( binintersectSubset1
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( ( binintersect @ A @ B )
% 2.77/1.56 = A )
% 2.77/1.56 => ( subset @ A @ B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(bs114d_type,type,
% 2.77/1.56 bs114d: $o ).
% 2.77/1.56
% 2.77/1.56 thf(bs114d,definition,
% 2.77/1.56 ( bs114d
% 2.77/1.56 = ( ! [A: $i,B: $i,C: $i] :
% 2.77/1.56 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.77/1.56 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(regular_type,type,
% 2.77/1.56 regular: $i > $o ).
% 2.77/1.56
% 2.77/1.56 thf(setminus_type,type,
% 2.77/1.56 setminus: $i > $i > $i ).
% 2.77/1.56
% 2.77/1.56 thf(setminusI_type,type,
% 2.77/1.56 setminusI: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setminusI,definition,
% 2.77/1.56 ( setminusI
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ~ ( in @ Xx @ B )
% 2.77/1.56 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setminusEL_type,type,
% 2.77/1.56 setminusEL: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setminusEL,definition,
% 2.77/1.56 ( setminusEL
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.77/1.56 => ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setminusER_type,type,
% 2.77/1.56 setminusER: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setminusER,definition,
% 2.77/1.56 ( setminusER
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.77/1.56 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setminusSubset2_type,type,
% 2.77/1.56 setminusSubset2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setminusSubset2,definition,
% 2.77/1.56 ( setminusSubset2
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( subset @ A @ B )
% 2.77/1.56 => ( ( setminus @ A @ B )
% 2.77/1.56 = emptyset ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setminusERneg_type,type,
% 2.77/1.56 setminusERneg: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setminusERneg,definition,
% 2.77/1.56 ( setminusERneg
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.77/1.56 => ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setminusELneg_type,type,
% 2.77/1.56 setminusELneg: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setminusELneg,definition,
% 2.77/1.56 ( setminusELneg
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.77/1.56 => ( ~ ( in @ Xx @ B )
% 2.77/1.56 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setminusILneg_type,type,
% 2.77/1.56 setminusILneg: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setminusILneg,definition,
% 2.77/1.56 ( setminusILneg
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ~ ( in @ Xx @ A )
% 2.77/1.56 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setminusIRneg_type,type,
% 2.77/1.56 setminusIRneg: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setminusIRneg,definition,
% 2.77/1.56 ( setminusIRneg
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ B )
% 2.77/1.56 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setminusLsub_type,type,
% 2.77/1.56 setminusLsub: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setminusLsub,definition,
% 2.77/1.56 ( setminusLsub
% 2.77/1.56 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setminusSubset1_type,type,
% 2.77/1.56 setminusSubset1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setminusSubset1,definition,
% 2.77/1.56 ( setminusSubset1
% 2.77/1.56 = ( ! [A: $i,B: $i] :
% 2.77/1.56 ( ( ( setminus @ A @ B )
% 2.77/1.56 = emptyset )
% 2.77/1.56 => ( subset @ A @ B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(symdiff_type,type,
% 2.77/1.56 symdiff: $i > $i > $i ).
% 2.77/1.56
% 2.77/1.56 thf(symdiffE_type,type,
% 2.77/1.56 symdiffE: $o ).
% 2.77/1.56
% 2.77/1.56 thf(symdiffE,definition,
% 2.77/1.56 ( symdiffE
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.77/1.56 => ! [Xphi: $o] :
% 2.77/1.56 ( ( ( in @ Xx @ A )
% 2.77/1.56 => ( ~ ( in @ Xx @ B )
% 2.77/1.56 => Xphi ) )
% 2.77/1.56 => ( ( ~ ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ B )
% 2.77/1.56 => Xphi ) )
% 2.77/1.56 => Xphi ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(symdiffI1_type,type,
% 2.77/1.56 symdiffI1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(symdiffI1,definition,
% 2.77/1.56 ( symdiffI1
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ~ ( in @ Xx @ B )
% 2.77/1.56 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(symdiffI2_type,type,
% 2.77/1.56 symdiffI2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(symdiffI2,definition,
% 2.77/1.56 ( symdiffI2
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ~ ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ B )
% 2.77/1.56 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(symdiffIneg1_type,type,
% 2.77/1.56 symdiffIneg1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(symdiffIneg1,definition,
% 2.77/1.56 ( symdiffIneg1
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ B )
% 2.77/1.56 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(symdiffIneg2_type,type,
% 2.77/1.56 symdiffIneg2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(symdiffIneg2,definition,
% 2.77/1.56 ( symdiffIneg2
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ~ ( in @ Xx @ A )
% 2.77/1.56 => ( ~ ( in @ Xx @ B )
% 2.77/1.56 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(iskpair_type,type,
% 2.77/1.56 iskpair: $i > $o ).
% 2.77/1.56
% 2.77/1.56 thf(secondinupair_type,type,
% 2.77/1.56 secondinupair: $o ).
% 2.77/1.56
% 2.77/1.56 thf(secondinupair,definition,
% 2.77/1.56 ( secondinupair
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairIL_type,type,
% 2.77/1.56 setukpairIL: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairIL,definition,
% 2.77/1.56 ( setukpairIL
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairIR_type,type,
% 2.77/1.56 setukpairIR: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairIR,definition,
% 2.77/1.56 ( setukpairIR
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(kpairiskpair_type,type,
% 2.77/1.56 kpairiskpair: $o ).
% 2.77/1.56
% 2.77/1.56 thf(kpairiskpair,definition,
% 2.77/1.56 ( kpairiskpair
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(kpair_type,type,
% 2.77/1.56 kpair: $i > $i > $i ).
% 2.77/1.56
% 2.77/1.56 thf(kpairp_type,type,
% 2.77/1.56 kpairp: $o ).
% 2.77/1.56
% 2.77/1.56 thf(kpairp,definition,
% 2.77/1.56 ( kpairp
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(cartprod_type,type,
% 2.77/1.56 cartprod: $i > $i > $i ).
% 2.77/1.56
% 2.77/1.56 thf(singletonsubset_type,type,
% 2.77/1.56 singletonsubset: $o ).
% 2.77/1.56
% 2.77/1.56 thf(singletonsubset,definition,
% 2.77/1.56 ( singletonsubset
% 2.77/1.56 = ( ! [A: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(singletoninpowerset_type,type,
% 2.77/1.56 singletoninpowerset: $o ).
% 2.77/1.56
% 2.77/1.56 thf(singletoninpowerset,definition,
% 2.77/1.56 ( singletoninpowerset
% 2.77/1.56 = ( ! [A: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(singletoninpowunion_type,type,
% 2.77/1.56 singletoninpowunion: $o ).
% 2.77/1.56
% 2.77/1.56 thf(singletoninpowunion,definition,
% 2.77/1.56 ( singletoninpowunion
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(upairset2E_type,type,
% 2.77/1.56 upairset2E: $o ).
% 2.77/1.56
% 2.77/1.56 thf(upairset2E,definition,
% 2.77/1.56 ( upairset2E
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.77/1.56 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.77/1.56 => ( ( Xz = Xx )
% 2.77/1.56 | ( Xz = Xy ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(upairsubunion_type,type,
% 2.77/1.56 upairsubunion: $o ).
% 2.77/1.56
% 2.77/1.56 thf(upairsubunion,definition,
% 2.77/1.56 ( upairsubunion
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(upairinpowunion_type,type,
% 2.77/1.56 upairinpowunion: $o ).
% 2.77/1.56
% 2.77/1.56 thf(upairinpowunion,definition,
% 2.77/1.56 ( upairinpowunion
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(ubforcartprodlem1_type,type,
% 2.77/1.56 ubforcartprodlem1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(ubforcartprodlem1,definition,
% 2.77/1.56 ( ubforcartprodlem1
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(ubforcartprodlem2_type,type,
% 2.77/1.56 ubforcartprodlem2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(ubforcartprodlem2,definition,
% 2.77/1.56 ( ubforcartprodlem2
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(ubforcartprodlem3_type,type,
% 2.77/1.56 ubforcartprodlem3: $o ).
% 2.77/1.56
% 2.77/1.56 thf(ubforcartprodlem3,definition,
% 2.77/1.56 ( ubforcartprodlem3
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodpairin_type,type,
% 2.77/1.56 cartprodpairin: $o ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodpairin,definition,
% 2.77/1.56 ( cartprodpairin
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodmempair1_type,type,
% 2.77/1.56 cartprodmempair1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodmempair1,definition,
% 2.77/1.56 ( cartprodmempair1
% 2.77/1.56 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.77/1.56 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.77/1.56 => ? [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 & ? [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 & ( Xu
% 2.77/1.56 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodmempair_type,type,
% 2.77/1.56 cartprodmempair: $o ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodmempair,definition,
% 2.77/1.56 ( cartprodmempair
% 2.77/1.56 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.77/1.56 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.77/1.56 => ( iskpair @ Xu ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setunionE2_type,type,
% 2.77/1.56 setunionE2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setunionE2,definition,
% 2.77/1.56 ( setunionE2
% 2.77/1.56 = ( ! [A: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ ( setunion @ A ) )
% 2.77/1.56 => ? [X: $i] :
% 2.77/1.56 ( ( in @ X @ A )
% 2.77/1.56 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setunionsingleton1_type,type,
% 2.77/1.56 setunionsingleton1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setunionsingleton1,definition,
% 2.77/1.56 ( setunionsingleton1
% 2.77/1.56 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setunionsingleton2_type,type,
% 2.77/1.56 setunionsingleton2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setunionsingleton2,definition,
% 2.77/1.56 ( setunionsingleton2
% 2.77/1.56 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setunionsingleton_type,type,
% 2.77/1.56 setunionsingleton: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setunionsingleton,definition,
% 2.77/1.56 ( setunionsingleton
% 2.77/1.56 = ( ! [Xx: $i] :
% 2.77/1.56 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.77/1.56 = Xx ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(singleton_type,type,
% 2.77/1.56 singleton: $i > $o ).
% 2.77/1.56
% 2.77/1.56 thf(singletonprop_type,type,
% 2.77/1.56 singletonprop: $o ).
% 2.77/1.56
% 2.77/1.56 thf(singletonprop,definition,
% 2.77/1.56 ( singletonprop
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ A )
% 2.77/1.56 => ( ( Xphi @ Xx )
% 2.77/1.56 => ( ( Xphi @ Xy )
% 2.77/1.56 => ( Xx = Xy ) ) ) ) )
% 2.77/1.56 => ( ? [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 & ( Xphi @ Xx ) )
% 2.77/1.56 => ( singleton
% 2.77/1.56 @ ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(ex1_type,type,
% 2.77/1.56 ex1: $i > ( $i > $o ) > $o ).
% 2.77/1.56
% 2.77/1.56 thf(ex1,definition,
% 2.77/1.56 ( ex1
% 2.77/1.56 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.77/1.56 ( singleton
% 2.77/1.56 @ ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(ex1E1_type,type,
% 2.77/1.56 ex1E1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(ex1E1,definition,
% 2.77/1.56 ( ex1E1
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.56 ( ( ex1 @ A
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.56 => ? [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(ex1I_type,type,
% 2.77/1.56 ex1I: $o ).
% 2.77/1.56
% 2.77/1.56 thf(ex1I,definition,
% 2.77/1.56 ( ex1I
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( Xphi @ Xx )
% 2.77/1.56 => ( ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ A )
% 2.77/1.56 => ( ( Xphi @ Xy )
% 2.77/1.56 => ( Xy = Xx ) ) )
% 2.77/1.56 => ( ex1 @ A
% 2.77/1.56 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(ex1I2_type,type,
% 2.77/1.56 ex1I2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(ex1I2,definition,
% 2.77/1.56 ( ex1I2
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ A )
% 2.77/1.56 => ( ( Xphi @ Xx )
% 2.77/1.56 => ( ( Xphi @ Xy )
% 2.77/1.56 => ( Xx = Xy ) ) ) ) )
% 2.77/1.56 => ( ? [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 & ( Xphi @ Xx ) )
% 2.77/1.56 => ( ex1 @ A
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(singletonsuniq_type,type,
% 2.77/1.56 singletonsuniq: $o ).
% 2.77/1.56
% 2.77/1.56 thf(singletonsuniq,definition,
% 2.77/1.56 ( singletonsuniq
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( ( setadjoin @ Xx @ emptyset )
% 2.77/1.56 = ( setadjoin @ Xy @ emptyset ) )
% 2.77/1.56 => ( Xx = Xy ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(atmost1p_type,type,
% 2.77/1.56 atmost1p: $i > $o ).
% 2.77/1.56
% 2.77/1.56 thf(atleast2p_type,type,
% 2.77/1.56 atleast2p: $i > $o ).
% 2.77/1.56
% 2.77/1.56 thf(atmost2p_type,type,
% 2.77/1.56 atmost2p: $i > $o ).
% 2.77/1.56
% 2.77/1.56 thf(upairsetp_type,type,
% 2.77/1.56 upairsetp: $i > $o ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairinjL1_type,type,
% 2.77/1.56 setukpairinjL1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairinjL1,definition,
% 2.77/1.56 ( setukpairinjL1
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.77/1.56 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.77/1.56 => ( Xx = Xz ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(kfstsingleton_type,type,
% 2.77/1.56 kfstsingleton: $o ).
% 2.77/1.56
% 2.77/1.56 thf(kfstsingleton,definition,
% 2.77/1.56 ( kfstsingleton
% 2.77/1.56 = ( ! [Xu: $i] :
% 2.77/1.56 ( ( iskpair @ Xu )
% 2.77/1.56 => ( singleton
% 2.77/1.56 @ ( dsetconstr @ ( setunion @ Xu )
% 2.77/1.56 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(theprop_type,type,
% 2.77/1.56 theprop: $o ).
% 2.77/1.56
% 2.77/1.56 thf(theprop,definition,
% 2.77/1.56 ( theprop
% 2.77/1.56 = ( ! [X: $i] :
% 2.77/1.56 ( ( singleton @ X )
% 2.77/1.56 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(kfst_type,type,
% 2.77/1.56 kfst: $i > $i ).
% 2.77/1.56
% 2.77/1.56 thf(kfstpairEq_type,type,
% 2.77/1.56 kfstpairEq: $o ).
% 2.77/1.56
% 2.77/1.56 thf(kfstpairEq,definition,
% 2.77/1.56 ( kfstpairEq
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.77/1.56 = Xx ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodfstin_type,type,
% 2.77/1.56 cartprodfstin: $o ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodfstin,definition,
% 2.77/1.56 ( cartprodfstin
% 2.77/1.56 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.77/1.56 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.77/1.56 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairinjL2_type,type,
% 2.77/1.56 setukpairinjL2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairinjL2,definition,
% 2.77/1.56 ( setukpairinjL2
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.77/1.56 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.77/1.56 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.77/1.56 => ( Xx = Xz ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairinjL_type,type,
% 2.77/1.56 setukpairinjL: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairinjL,definition,
% 2.77/1.56 ( setukpairinjL
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.77/1.56 ( ( ( kpair @ Xx @ Xy )
% 2.77/1.56 = ( kpair @ Xz @ Xu ) )
% 2.77/1.56 => ( Xx = Xz ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairinjR11_type,type,
% 2.77/1.56 setukpairinjR11: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairinjR11,definition,
% 2.77/1.56 ( setukpairinjR11
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( Xx = Xy )
% 2.77/1.56 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.77/1.56 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairinjR12_type,type,
% 2.77/1.56 setukpairinjR12: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairinjR12,definition,
% 2.77/1.56 ( setukpairinjR12
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( Xx = Xy )
% 2.77/1.56 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.77/1.56 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairinjR1_type,type,
% 2.77/1.56 setukpairinjR1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairinjR1,definition,
% 2.77/1.56 ( setukpairinjR1
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.77/1.56 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.77/1.56 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.77/1.56 => ( ( Xz = Xu )
% 2.77/1.56 => ( Xy = Xu ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(upairequniteq_type,type,
% 2.77/1.56 upairequniteq: $o ).
% 2.77/1.56
% 2.77/1.56 thf(upairequniteq,definition,
% 2.77/1.56 ( upairequniteq
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.77/1.56 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.77/1.56 = ( setadjoin @ Xz @ emptyset ) )
% 2.77/1.56 => ( Xx = Xy ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairinjR2_type,type,
% 2.77/1.56 setukpairinjR2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairinjR2,definition,
% 2.77/1.56 ( setukpairinjR2
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.77/1.56 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.77/1.56 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.77/1.56 => ( Xy = Xu ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairinjR_type,type,
% 2.77/1.56 setukpairinjR: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setukpairinjR,definition,
% 2.77/1.56 ( setukpairinjR
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.77/1.56 ( ( ( kpair @ Xx @ Xy )
% 2.77/1.56 = ( kpair @ Xz @ Xu ) )
% 2.77/1.56 => ( Xy = Xu ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(ksndsingleton_type,type,
% 2.77/1.56 ksndsingleton: $o ).
% 2.77/1.56
% 2.77/1.56 thf(ksndsingleton,definition,
% 2.77/1.56 ( ksndsingleton
% 2.77/1.56 = ( ! [Xu: $i] :
% 2.77/1.56 ( ( iskpair @ Xu )
% 2.77/1.56 => ( singleton
% 2.77/1.56 @ ( dsetconstr @ ( setunion @ Xu )
% 2.77/1.56 @ ^ [Xx: $i] :
% 2.77/1.56 ( Xu
% 2.77/1.56 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(ksnd_type,type,
% 2.77/1.56 ksnd: $i > $i ).
% 2.77/1.56
% 2.77/1.56 thf(ksndpairEq_type,type,
% 2.77/1.56 ksndpairEq: $o ).
% 2.77/1.56
% 2.77/1.56 thf(ksndpairEq,definition,
% 2.77/1.56 ( ksndpairEq
% 2.77/1.56 = ( ! [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.77/1.56 = Xy ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(kpairsurjEq_type,type,
% 2.77/1.56 kpairsurjEq: $o ).
% 2.77/1.56
% 2.77/1.56 thf(kpairsurjEq,definition,
% 2.77/1.56 ( kpairsurjEq
% 2.77/1.56 = ( ! [Xu: $i] :
% 2.77/1.56 ( ( iskpair @ Xu )
% 2.77/1.56 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.77/1.56 = Xu ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodsndin_type,type,
% 2.77/1.56 cartprodsndin: $o ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodsndin,definition,
% 2.77/1.56 ( cartprodsndin
% 2.77/1.56 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.77/1.56 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.77/1.56 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodpairmemEL_type,type,
% 2.77/1.56 cartprodpairmemEL: $o ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodpairmemEL,definition,
% 2.77/1.56 ( cartprodpairmemEL
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.77/1.56 => ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodpairmemER_type,type,
% 2.77/1.56 cartprodpairmemER: $o ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodpairmemER,definition,
% 2.77/1.56 ( cartprodpairmemER
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.77/1.56 => ( in @ Xy @ B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodmempaircEq_type,type,
% 2.77/1.56 cartprodmempaircEq: $o ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodmempaircEq,definition,
% 2.77/1.56 ( cartprodmempaircEq
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( ( kpair @ Xx @ Xy )
% 2.77/1.56 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodfstpairEq_type,type,
% 2.77/1.56 cartprodfstpairEq: $o ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodfstpairEq,definition,
% 2.77/1.56 ( cartprodfstpairEq
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.77/1.56 = Xx ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodsndpairEq_type,type,
% 2.77/1.56 cartprodsndpairEq: $o ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodsndpairEq,definition,
% 2.77/1.56 ( cartprodsndpairEq
% 2.77/1.56 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.77/1.56 = Xy ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodpairsurjEq_type,type,
% 2.77/1.56 cartprodpairsurjEq: $o ).
% 2.77/1.56
% 2.77/1.56 thf(cartprodpairsurjEq,definition,
% 2.77/1.56 ( cartprodpairsurjEq
% 2.77/1.56 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.77/1.56 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.77/1.56 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.77/1.56 = Xu ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(breln_type,type,
% 2.77/1.56 breln: $i > $i > $i > $o ).
% 2.77/1.56
% 2.77/1.56 thf(breln,definition,
% 2.77/1.56 ( breln
% 2.77/1.56 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(dpsetconstr_type,type,
% 2.77/1.56 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.77/1.56
% 2.77/1.56 thf(dpsetconstrI_type,type,
% 2.77/1.56 dpsetconstrI: $o ).
% 2.77/1.56
% 2.77/1.56 thf(dpsetconstrI,definition,
% 2.77/1.56 ( dpsetconstrI
% 2.77/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( ( Xphi @ Xx @ Xy )
% 2.77/1.56 => ( in @ ( kpair @ Xx @ Xy )
% 2.77/1.56 @ ( dpsetconstr @ A @ B
% 2.77/1.56 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(dpsetconstrSub_type,type,
% 2.77/1.56 dpsetconstrSub: $o ).
% 2.77/1.56
% 2.77/1.56 thf(dpsetconstrSub,definition,
% 2.77/1.56 ( dpsetconstrSub
% 2.77/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.77/1.56 ( subset
% 2.77/1.56 @ ( dpsetconstr @ A @ B
% 2.77/1.56 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.77/1.56 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setOfPairsIsBReln_type,type,
% 2.77/1.56 setOfPairsIsBReln: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setOfPairsIsBReln,definition,
% 2.77/1.56 ( setOfPairsIsBReln
% 2.77/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.77/1.56 ( breln @ A @ B
% 2.77/1.56 @ ( dpsetconstr @ A @ B
% 2.77/1.56 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(dpsetconstrERa_type,type,
% 2.77/1.56 dpsetconstrERa: $o ).
% 2.77/1.56
% 2.77/1.56 thf(dpsetconstrERa,definition,
% 2.77/1.56 ( dpsetconstrERa
% 2.77/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.77/1.56 @ ( dpsetconstr @ A @ B
% 2.77/1.56 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.77/1.56 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(dpsetconstrEL1_type,type,
% 2.77/1.56 dpsetconstrEL1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(dpsetconstrEL1,definition,
% 2.77/1.56 ( dpsetconstrEL1
% 2.77/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( in @ ( kpair @ Xx @ Xy )
% 2.77/1.56 @ ( dpsetconstr @ A @ B
% 2.77/1.56 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.77/1.56 => ( in @ Xx @ A ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(dpsetconstrEL2_type,type,
% 2.77/1.56 dpsetconstrEL2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(dpsetconstrEL2,definition,
% 2.77/1.56 ( dpsetconstrEL2
% 2.77/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( in @ ( kpair @ Xx @ Xy )
% 2.77/1.56 @ ( dpsetconstr @ A @ B
% 2.77/1.56 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.77/1.56 => ( in @ Xy @ B ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(dpsetconstrER_type,type,
% 2.77/1.56 dpsetconstrER: $o ).
% 2.77/1.56
% 2.77/1.56 thf(dpsetconstrER,definition,
% 2.77/1.56 ( dpsetconstrER
% 2.77/1.56 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( in @ ( kpair @ Xx @ Xy )
% 2.77/1.56 @ ( dpsetconstr @ A @ B
% 2.77/1.56 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.77/1.56 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(func_type,type,
% 2.77/1.56 func: $i > $i > $i > $o ).
% 2.77/1.56
% 2.77/1.56 thf(funcSet_type,type,
% 2.77/1.56 funcSet: $i > $i > $i ).
% 2.77/1.56
% 2.77/1.56 thf(funcImageSingleton_type,type,
% 2.77/1.56 funcImageSingleton: $o ).
% 2.77/1.56
% 2.77/1.56 thf(funcImageSingleton,definition,
% 2.77/1.56 ( funcImageSingleton
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( func @ A @ B @ Xf )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( singleton
% 2.77/1.56 @ ( dsetconstr @ B
% 2.77/1.56 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(apProp_type,type,
% 2.77/1.56 apProp: $o ).
% 2.77/1.56
% 2.77/1.56 thf(apProp,definition,
% 2.77/1.56 ( apProp
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( func @ A @ B @ Xf )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in
% 2.77/1.56 @ ( setunion
% 2.77/1.56 @ ( dsetconstr @ B
% 2.77/1.56 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.77/1.56 @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(ap_type,type,
% 2.77/1.56 ap: $i > $i > $i > $i > $i ).
% 2.77/1.56
% 2.77/1.56 thf(app_type,type,
% 2.77/1.56 app: $o ).
% 2.77/1.56
% 2.77/1.56 thf(app,definition,
% 2.77/1.56 ( app
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( func @ A @ B @ Xf )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(infuncsetfunc_type,type,
% 2.77/1.56 infuncsetfunc: $o ).
% 2.77/1.56
% 2.77/1.56 thf(infuncsetfunc,definition,
% 2.77/1.56 ( infuncsetfunc
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.56 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(ap2p_type,type,
% 2.77/1.56 ap2p: $o ).
% 2.77/1.56
% 2.77/1.56 thf(ap2p,definition,
% 2.77/1.56 ( ap2p
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(funcinfuncset_type,type,
% 2.77/1.56 funcinfuncset: $o ).
% 2.77/1.56
% 2.77/1.56 thf(funcinfuncset,definition,
% 2.77/1.56 ( funcinfuncset
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( func @ A @ B @ Xf )
% 2.77/1.56 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(lamProp_type,type,
% 2.77/1.56 lamProp: $o ).
% 2.77/1.56
% 2.77/1.56 thf(lamProp,definition,
% 2.77/1.56 ( lamProp
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.56 => ( func @ A @ B
% 2.77/1.56 @ ( dpsetconstr @ A @ B
% 2.77/1.56 @ ^ [Xx: $i,Xy: $i] :
% 2.77/1.56 ( ( Xf @ Xx )
% 2.77/1.56 = Xy ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(lam_type,type,
% 2.77/1.56 lam: $i > $i > ( $i > $i ) > $i ).
% 2.77/1.56
% 2.77/1.56 thf(lamp_type,type,
% 2.77/1.56 lamp: $o ).
% 2.77/1.56
% 2.77/1.56 thf(lamp,definition,
% 2.77/1.56 ( lamp
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.56 => ( func @ A @ B
% 2.77/1.56 @ ( lam @ A @ B
% 2.77/1.56 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(lam2p_type,type,
% 2.77/1.56 lam2p: $o ).
% 2.77/1.56
% 2.77/1.56 thf(lam2p,definition,
% 2.77/1.56 ( lam2p
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.56 => ( in
% 2.77/1.56 @ ( lam @ A @ B
% 2.77/1.56 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.77/1.56 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(brelnall1_type,type,
% 2.77/1.56 brelnall1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(brelnall1,definition,
% 2.77/1.56 ( brelnall1
% 2.77/1.56 = ( ! [A: $i,B: $i,R: $i] :
% 2.77/1.56 ( ( breln @ A @ B @ R )
% 2.77/1.56 => ! [Xphi: $i > $o] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.77/1.56 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ R )
% 2.77/1.56 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(brelnall2_type,type,
% 2.77/1.56 brelnall2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(brelnall2,definition,
% 2.77/1.56 ( brelnall2
% 2.77/1.56 = ( ! [A: $i,B: $i,R: $i] :
% 2.77/1.56 ( ( breln @ A @ B @ R )
% 2.77/1.56 => ! [Xphi: $i > $o] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.77/1.56 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ R )
% 2.77/1.56 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(ex1E2_type,type,
% 2.77/1.56 ex1E2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(ex1E2,definition,
% 2.77/1.56 ( ex1E2
% 2.77/1.56 = ( ! [A: $i,Xphi: $i > $o] :
% 2.77/1.56 ( ( ex1 @ A
% 2.77/1.56 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ A )
% 2.77/1.56 => ( ( Xphi @ Xx )
% 2.77/1.56 => ( ( Xphi @ Xy )
% 2.77/1.56 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(funcGraphProp1_type,type,
% 2.77/1.56 funcGraphProp1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(funcGraphProp1,definition,
% 2.77/1.56 ( funcGraphProp1
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( func @ A @ B @ Xf )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(funcGraphProp3_type,type,
% 2.77/1.56 funcGraphProp3: $o ).
% 2.77/1.56
% 2.77/1.56 thf(funcGraphProp3,definition,
% 2.77/1.56 ( funcGraphProp3
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(funcGraphProp2_type,type,
% 2.77/1.56 funcGraphProp2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(funcGraphProp2,definition,
% 2.77/1.56 ( funcGraphProp2
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( func @ A @ B @ Xf )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.77/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.56 = Xy ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(funcextLem_type,type,
% 2.77/1.56 funcextLem: $o ).
% 2.77/1.56
% 2.77/1.56 thf(funcextLem,definition,
% 2.77/1.56 ( funcextLem
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( func @ A @ B @ Xf )
% 2.77/1.56 => ! [Xg: $i] :
% 2.77/1.56 ( ( func @ A @ B @ Xg )
% 2.77/1.56 => ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.56 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.77/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(funcGraphProp4_type,type,
% 2.77/1.56 funcGraphProp4: $o ).
% 2.77/1.56
% 2.77/1.56 thf(funcGraphProp4,definition,
% 2.77/1.56 ( funcGraphProp4
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.77/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.56 = Xy ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(subbreln_type,type,
% 2.77/1.56 subbreln: $o ).
% 2.77/1.56
% 2.77/1.56 thf(subbreln,definition,
% 2.77/1.56 ( subbreln
% 2.77/1.56 = ( ! [A: $i,B: $i,R: $i] :
% 2.77/1.56 ( ( breln @ A @ B @ R )
% 2.77/1.56 => ! [S: $i] :
% 2.77/1.56 ( ( breln @ A @ B @ S )
% 2.77/1.56 => ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.77/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.77/1.56 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(eqbreln_type,type,
% 2.77/1.56 eqbreln: $o ).
% 2.77/1.56
% 2.77/1.56 thf(eqbreln,definition,
% 2.77/1.56 ( eqbreln
% 2.77/1.56 = ( ! [A: $i,B: $i,R: $i] :
% 2.77/1.56 ( ( breln @ A @ B @ R )
% 2.77/1.56 => ! [S: $i] :
% 2.77/1.56 ( ( breln @ A @ B @ S )
% 2.77/1.56 => ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.77/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.77/1.56 => ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ B )
% 2.77/1.56 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.77/1.56 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.77/1.56 => ( R = S ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(funcext_type,type,
% 2.77/1.56 funcext: $o ).
% 2.77/1.56
% 2.77/1.56 thf(funcext,definition,
% 2.77/1.56 ( funcext
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( func @ A @ B @ Xf )
% 2.77/1.56 => ! [Xg: $i] :
% 2.77/1.56 ( ( func @ A @ B @ Xg )
% 2.77/1.56 => ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.56 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.77/1.56 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(funcext2_type,type,
% 2.77/1.56 funcext2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(funcext2,definition,
% 2.77/1.56 ( funcext2
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.56 => ! [Xg: $i] :
% 2.77/1.56 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.77/1.56 => ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.56 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.77/1.56 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(ap2apEq1_type,type,
% 2.77/1.56 ap2apEq1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(ap2apEq1,definition,
% 2.77/1.56 ( ap2apEq1
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.56 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(ap2apEq2_type,type,
% 2.77/1.56 ap2apEq2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(ap2apEq2,definition,
% 2.77/1.56 ( ap2apEq2
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( func @ A @ B @ Xf )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.77/1.56 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(beta1_type,type,
% 2.77/1.56 beta1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(beta1,definition,
% 2.77/1.56 ( beta1
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( ap @ A @ B
% 2.77/1.56 @ ( lam @ A @ B
% 2.77/1.56 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.77/1.56 @ Xx )
% 2.77/1.56 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(eta1_type,type,
% 2.77/1.56 eta1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(eta1,definition,
% 2.77/1.56 ( eta1
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( func @ A @ B @ Xf )
% 2.77/1.56 => ( ( lam @ A @ B
% 2.77/1.56 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.77/1.56 = Xf ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(lam2lamEq_type,type,
% 2.77/1.56 lam2lamEq: $o ).
% 2.77/1.56
% 2.77/1.56 thf(lam2lamEq,definition,
% 2.77/1.56 ( lam2lamEq
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.56 => ( ( lam @ A @ B
% 2.77/1.56 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.77/1.56 = ( lam @ A @ B
% 2.77/1.56 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(beta2_type,type,
% 2.77/1.56 beta2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(beta2,definition,
% 2.77/1.56 ( beta2
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.77/1.56 ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( ap @ A @ B
% 2.77/1.56 @ ( lam @ A @ B
% 2.77/1.56 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.77/1.56 @ Xx )
% 2.77/1.56 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(eta2_type,type,
% 2.77/1.56 eta2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(eta2,definition,
% 2.77/1.56 ( eta2
% 2.77/1.56 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.77/1.56 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.77/1.56 => ( ( lam @ A @ B
% 2.77/1.56 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.77/1.56 = Xf ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(iffalseProp1_type,type,
% 2.77/1.56 iffalseProp1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(iffalseProp1,definition,
% 2.77/1.56 ( iffalseProp1
% 2.77/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ A )
% 2.77/1.56 => ( ~ Xphi
% 2.77/1.56 => ( in @ Xy
% 2.77/1.56 @ ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xz: $i] :
% 2.77/1.56 ( ( Xphi
% 2.77/1.56 & ( Xz = Xx ) )
% 2.77/1.56 | ( ~ Xphi
% 2.77/1.56 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(iffalseProp2_type,type,
% 2.77/1.56 iffalseProp2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(iffalseProp2,definition,
% 2.77/1.56 ( iffalseProp2
% 2.77/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ A )
% 2.77/1.56 => ( ~ Xphi
% 2.77/1.56 => ( ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xz: $i] :
% 2.77/1.56 ( ( Xphi
% 2.77/1.56 & ( Xz = Xx ) )
% 2.77/1.56 | ( ~ Xphi
% 2.77/1.56 & ( Xz = Xy ) ) ) )
% 2.77/1.56 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(iftrueProp1_type,type,
% 2.77/1.56 iftrueProp1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(iftrueProp1,definition,
% 2.77/1.56 ( iftrueProp1
% 2.77/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ A )
% 2.77/1.56 => ( Xphi
% 2.77/1.56 => ( in @ Xx
% 2.77/1.56 @ ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xz: $i] :
% 2.77/1.56 ( ( Xphi
% 2.77/1.56 & ( Xz = Xx ) )
% 2.77/1.56 | ( ~ Xphi
% 2.77/1.56 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(iftrueProp2_type,type,
% 2.77/1.56 iftrueProp2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(iftrueProp2,definition,
% 2.77/1.56 ( iftrueProp2
% 2.77/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ A )
% 2.77/1.56 => ( Xphi
% 2.77/1.56 => ( ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xz: $i] :
% 2.77/1.56 ( ( Xphi
% 2.77/1.56 & ( Xz = Xx ) )
% 2.77/1.56 | ( ~ Xphi
% 2.77/1.56 & ( Xz = Xy ) ) ) )
% 2.77/1.56 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(ifSingleton_type,type,
% 2.77/1.56 ifSingleton: $o ).
% 2.77/1.56
% 2.77/1.56 thf(ifSingleton,definition,
% 2.77/1.56 ( ifSingleton
% 2.77/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ A )
% 2.77/1.56 => ( singleton
% 2.77/1.56 @ ( dsetconstr @ A
% 2.77/1.56 @ ^ [Xz: $i] :
% 2.77/1.56 ( ( Xphi
% 2.77/1.56 & ( Xz = Xx ) )
% 2.77/1.56 | ( ~ Xphi
% 2.77/1.56 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(if_type,type,
% 2.77/1.56 if: $i > $o > $i > $i > $i ).
% 2.77/1.56
% 2.77/1.56 thf(ifp_type,type,
% 2.77/1.56 ifp: $o ).
% 2.77/1.56
% 2.77/1.56 thf(ifp,definition,
% 2.77/1.56 ( ifp
% 2.77/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ A )
% 2.77/1.56 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(theeq_type,type,
% 2.77/1.56 theeq: $o ).
% 2.77/1.56
% 2.77/1.56 thf(theeq,definition,
% 2.77/1.56 ( theeq
% 2.77/1.56 = ( ! [X: $i] :
% 2.77/1.56 ( ( singleton @ X )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ X )
% 2.77/1.56 => ( ( setunion @ X )
% 2.77/1.56 = Xx ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(iftrue_type,type,
% 2.77/1.56 iftrue: $o ).
% 2.77/1.56
% 2.77/1.56 thf(iftrue,definition,
% 2.77/1.56 ( iftrue
% 2.77/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ A )
% 2.77/1.56 => ( Xphi
% 2.77/1.56 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.77/1.56 = Xx ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(iffalse_type,type,
% 2.77/1.56 iffalse: $o ).
% 2.77/1.56
% 2.77/1.56 thf(iffalse,definition,
% 2.77/1.56 ( iffalse
% 2.77/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ A )
% 2.77/1.56 => ( ~ Xphi
% 2.77/1.56 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.77/1.56 = Xy ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(iftrueorfalse_type,type,
% 2.77/1.56 iftrueorfalse: $o ).
% 2.77/1.56
% 2.77/1.56 thf(iftrueorfalse,definition,
% 2.77/1.56 ( iftrueorfalse
% 2.77/1.56 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ! [Xy: $i] :
% 2.77/1.56 ( ( in @ Xy @ A )
% 2.77/1.56 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectT_lem_type,type,
% 2.77/1.56 binintersectT_lem: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectT_lem,definition,
% 2.77/1.56 ( binintersectT_lem
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binunionT_lem_type,type,
% 2.77/1.56 binunionT_lem: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binunionT_lem,definition,
% 2.77/1.56 ( binunionT_lem
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(powersetT_lem_type,type,
% 2.77/1.56 powersetT_lem: $o ).
% 2.77/1.56
% 2.77/1.56 thf(powersetT_lem,definition,
% 2.77/1.56 ( powersetT_lem
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setminusT_lem_type,type,
% 2.77/1.56 setminusT_lem: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setminusT_lem,definition,
% 2.77/1.56 ( setminusT_lem
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(complementT_lem_type,type,
% 2.77/1.56 complementT_lem: $o ).
% 2.77/1.56
% 2.77/1.56 thf(complementT_lem,definition,
% 2.77/1.56 ( complementT_lem
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(setextT_type,type,
% 2.77/1.56 setextT: $o ).
% 2.77/1.56
% 2.77/1.56 thf(setextT,definition,
% 2.77/1.56 ( setextT
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ X )
% 2.77/1.56 => ( in @ Xx @ Y ) ) )
% 2.77/1.56 => ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ Y )
% 2.77/1.56 => ( in @ Xx @ X ) ) )
% 2.77/1.56 => ( X = Y ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(subsetTI_type,type,
% 2.77/1.56 subsetTI: $o ).
% 2.77/1.56
% 2.77/1.56 thf(subsetTI,definition,
% 2.77/1.56 ( subsetTI
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ X )
% 2.77/1.56 => ( in @ Xx @ Y ) ) )
% 2.77/1.56 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(powersetTI1_type,type,
% 2.77/1.56 powersetTI1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(powersetTI1,definition,
% 2.77/1.56 ( powersetTI1
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ( ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ X )
% 2.77/1.56 => ( in @ Xx @ Y ) ) )
% 2.77/1.56 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(powersetTE1_type,type,
% 2.77/1.56 powersetTE1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(powersetTE1,definition,
% 2.77/1.56 ( powersetTE1
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ X @ ( powerset @ Y ) )
% 2.77/1.56 => ( ( in @ Xx @ X )
% 2.77/1.56 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(complementTI1_type,type,
% 2.77/1.56 complementTI1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(complementTI1,definition,
% 2.77/1.56 ( complementTI1
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ X )
% 2.77/1.56 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(complementTE1_type,type,
% 2.77/1.56 complementTE1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(complementTE1,definition,
% 2.77/1.56 ( complementTE1
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.77/1.56 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectTELcontra_type,type,
% 2.77/1.56 binintersectTELcontra: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectTELcontra,definition,
% 2.77/1.56 ( binintersectTELcontra
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ~ ( in @ Xx @ X )
% 2.77/1.56 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectTERcontra_type,type,
% 2.77/1.56 binintersectTERcontra: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binintersectTERcontra,definition,
% 2.77/1.56 ( binintersectTERcontra
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ~ ( in @ Xx @ Y )
% 2.77/1.56 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(contrasubsetT_type,type,
% 2.77/1.56 contrasubsetT: $o ).
% 2.77/1.56
% 2.77/1.56 thf(contrasubsetT,definition,
% 2.77/1.56 ( contrasubsetT
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.77/1.56 => ( ( in @ Xx @ Y )
% 2.77/1.56 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(contrasubsetT1_type,type,
% 2.77/1.56 contrasubsetT1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(contrasubsetT1,definition,
% 2.77/1.56 ( contrasubsetT1
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( subset @ X @ Y )
% 2.77/1.56 => ( ~ ( in @ Xx @ Y )
% 2.77/1.56 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(contrasubsetT2_type,type,
% 2.77/1.56 contrasubsetT2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(contrasubsetT2,definition,
% 2.77/1.56 ( contrasubsetT2
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ( ( subset @ X @ Y )
% 2.77/1.56 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(contrasubsetT3_type,type,
% 2.77/1.56 contrasubsetT3: $o ).
% 2.77/1.56
% 2.77/1.56 thf(contrasubsetT3,definition,
% 2.77/1.56 ( contrasubsetT3
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.77/1.56 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(doubleComplementI1_type,type,
% 2.77/1.56 doubleComplementI1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(doubleComplementI1,definition,
% 2.77/1.56 ( doubleComplementI1
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ X )
% 2.77/1.56 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(doubleComplementE1_type,type,
% 2.77/1.56 doubleComplementE1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(doubleComplementE1,definition,
% 2.77/1.56 ( doubleComplementE1
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 2.77/1.56 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(doubleComplementSub1_type,type,
% 2.77/1.56 doubleComplementSub1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(doubleComplementSub1,definition,
% 2.77/1.56 ( doubleComplementSub1
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(doubleComplementSub2_type,type,
% 2.77/1.56 doubleComplementSub2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(doubleComplementSub2,definition,
% 2.77/1.56 ( doubleComplementSub2
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(doubleComplementEq_type,type,
% 2.77/1.56 doubleComplementEq: $o ).
% 2.77/1.56
% 2.77/1.56 thf(doubleComplementEq,definition,
% 2.77/1.56 ( doubleComplementEq
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ( X
% 2.77/1.56 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(complementTnotintersectT_type,type,
% 2.77/1.56 complementTnotintersectT: $o ).
% 2.77/1.56
% 2.77/1.56 thf(complementTnotintersectT,definition,
% 2.77/1.56 ( complementTnotintersectT
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.77/1.56 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(complementImpComplementIntersect_type,type,
% 2.77/1.56 complementImpComplementIntersect: $o ).
% 2.77/1.56
% 2.77/1.56 thf(complementImpComplementIntersect,definition,
% 2.77/1.56 ( complementImpComplementIntersect
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.77/1.56 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(complementSubsetComplementIntersect_type,type,
% 2.77/1.56 complementSubsetComplementIntersect: $o ).
% 2.77/1.56
% 2.77/1.56 thf(complementSubsetComplementIntersect,definition,
% 2.77/1.56 ( complementSubsetComplementIntersect
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(complementInPowersetComplementIntersect_type,type,
% 2.77/1.56 complementInPowersetComplementIntersect: $o ).
% 2.77/1.56
% 2.77/1.56 thf(complementInPowersetComplementIntersect,definition,
% 2.77/1.56 ( complementInPowersetComplementIntersect
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(contraSubsetComplement_type,type,
% 2.77/1.56 contraSubsetComplement: $o ).
% 2.77/1.56
% 2.77/1.56 thf(contraSubsetComplement,definition,
% 2.77/1.56 ( contraSubsetComplement
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ Y )
% 2.77/1.56 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(complementTcontraSubset_type,type,
% 2.77/1.56 complementTcontraSubset: $o ).
% 2.77/1.56
% 2.77/1.56 thf(complementTcontraSubset,definition,
% 2.77/1.56 ( complementTcontraSubset
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.77/1.56 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binunionTILcontra_type,type,
% 2.77/1.56 binunionTILcontra: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binunionTILcontra,definition,
% 2.77/1.56 ( binunionTILcontra
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.77/1.56 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binunionTIRcontra_type,type,
% 2.77/1.56 binunionTIRcontra: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binunionTIRcontra,definition,
% 2.77/1.56 ( binunionTIRcontra
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.77/1.56 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(inIntersectImpInUnion_type,type,
% 2.77/1.56 inIntersectImpInUnion: $o ).
% 2.77/1.56
% 2.77/1.56 thf(inIntersectImpInUnion,definition,
% 2.77/1.56 ( inIntersectImpInUnion
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Z: $i] :
% 2.77/1.56 ( ( in @ Z @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.77/1.56 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(inIntersectImpInUnion2_type,type,
% 2.77/1.56 inIntersectImpInUnion2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(inIntersectImpInUnion2,definition,
% 2.77/1.56 ( inIntersectImpInUnion2
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Z: $i] :
% 2.77/1.56 ( ( in @ Z @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.77/1.56 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(inIntersectImpInIntersectUnions_type,type,
% 2.77/1.56 inIntersectImpInIntersectUnions: $o ).
% 2.77/1.56
% 2.77/1.56 thf(inIntersectImpInIntersectUnions,definition,
% 2.77/1.56 ( inIntersectImpInIntersectUnions
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Z: $i] :
% 2.77/1.56 ( ( in @ Z @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.77/1.56 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(intersectInPowersetIntersectUnions_type,type,
% 2.77/1.56 intersectInPowersetIntersectUnions: $o ).
% 2.77/1.56
% 2.77/1.56 thf(intersectInPowersetIntersectUnions,definition,
% 2.77/1.56 ( intersectInPowersetIntersectUnions
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Z: $i] :
% 2.77/1.56 ( ( in @ Z @ ( powerset @ A ) )
% 2.77/1.56 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(inComplementUnionImpNotIn1_type,type,
% 2.77/1.56 inComplementUnionImpNotIn1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(inComplementUnionImpNotIn1,definition,
% 2.77/1.56 ( inComplementUnionImpNotIn1
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.77/1.56 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(inComplementUnionImpInComplement1_type,type,
% 2.77/1.56 inComplementUnionImpInComplement1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(inComplementUnionImpInComplement1,definition,
% 2.77/1.56 ( inComplementUnionImpInComplement1
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.77/1.56 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binunionTE_type,type,
% 2.77/1.56 binunionTE: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binunionTE,definition,
% 2.77/1.56 ( binunionTE
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xphi: $o,Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.77/1.56 => ( ( ( in @ Xx @ X )
% 2.77/1.56 => Xphi )
% 2.77/1.56 => ( ( ( in @ Xx @ Y )
% 2.77/1.56 => Xphi )
% 2.77/1.56 => Xphi ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(binunionTEcontra_type,type,
% 2.77/1.56 binunionTEcontra: $o ).
% 2.77/1.56
% 2.77/1.56 thf(binunionTEcontra,definition,
% 2.77/1.56 ( binunionTEcontra
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ~ ( in @ Xx @ X )
% 2.77/1.56 => ( ~ ( in @ Xx @ Y )
% 2.77/1.56 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan2a1_type,type,
% 2.77/1.56 demorgan2a1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan2a1,definition,
% 2.77/1.56 ( demorgan2a1
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.77/1.56 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(complementUnionInPowersetComplement_type,type,
% 2.77/1.56 complementUnionInPowersetComplement: $o ).
% 2.77/1.56
% 2.77/1.56 thf(complementUnionInPowersetComplement,definition,
% 2.77/1.56 ( complementUnionInPowersetComplement
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan2a2_type,type,
% 2.77/1.56 demorgan2a2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan2a2,definition,
% 2.77/1.56 ( demorgan2a2
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.77/1.56 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan1a_type,type,
% 2.77/1.56 demorgan1a: $o ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan1a,definition,
% 2.77/1.56 ( demorgan1a
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 2.77/1.56 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan1b_type,type,
% 2.77/1.56 demorgan1b: $o ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan1b,definition,
% 2.77/1.56 ( demorgan1b
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.77/1.56 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan1_type,type,
% 2.77/1.56 demorgan1: $o ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan1,definition,
% 2.77/1.56 ( demorgan1
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 2.77/1.56 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan2a_type,type,
% 2.77/1.56 demorgan2a: $o ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan2a,definition,
% 2.77/1.56 ( demorgan2a
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.77/1.56 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan2b2_type,type,
% 2.77/1.56 demorgan2b2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan2b2,definition,
% 2.77/1.56 ( demorgan2b2
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.77/1.56 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 2.77/1.56 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan2b_type,type,
% 2.77/1.56 demorgan2b: $o ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan2b,definition,
% 2.77/1.56 ( demorgan2b
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ A )
% 2.77/1.56 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.77/1.56 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan2_type,type,
% 2.77/1.56 demorgan2: $o ).
% 2.77/1.56
% 2.77/1.56 thf(demorgan2,definition,
% 2.77/1.56 ( demorgan2
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 2.77/1.56 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(woz13rule0_type,type,
% 2.77/1.56 woz13rule0: $o ).
% 2.77/1.56
% 2.77/1.56 thf(woz13rule0,definition,
% 2.77/1.56 ( woz13rule0
% 2.77/1.56 = ( ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Xx: $i] :
% 2.77/1.56 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.77/1.56 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.77/1.56
% 2.77/1.56 thf(woz13rule1,conjecture,
% 2.77/1.56 ( setextAx
% 2.77/1.56 => ( emptysetAx
% 2.77/1.56 => ( setadjoinAx
% 2.77/1.56 => ( powersetAx
% 2.77/1.56 => ( setunionAx
% 2.77/1.56 => ( omega0Ax
% 2.77/1.56 => ( omegaSAx
% 2.77/1.56 => ( omegaIndAx
% 2.77/1.56 => ( replAx
% 2.77/1.56 => ( foundationAx
% 2.77/1.56 => ( wellorderingAx
% 2.77/1.56 => ( descrp
% 2.77/1.56 => ( dsetconstrI
% 2.77/1.56 => ( dsetconstrEL
% 2.77/1.56 => ( dsetconstrER
% 2.77/1.56 => ( exuE1
% 2.77/1.56 => ( prop2setE
% 2.77/1.56 => ( emptysetE
% 2.77/1.56 => ( emptysetimpfalse
% 2.77/1.56 => ( notinemptyset
% 2.77/1.56 => ( exuE3e
% 2.77/1.56 => ( setext
% 2.77/1.56 => ( emptyI
% 2.77/1.56 => ( noeltsimpempty
% 2.77/1.56 => ( setbeta
% 2.77/1.56 => ( nonemptyE1
% 2.77/1.56 => ( nonemptyI
% 2.77/1.56 => ( nonemptyI1
% 2.77/1.56 => ( setadjoinIL
% 2.77/1.56 => ( emptyinunitempty
% 2.77/1.56 => ( setadjoinIR
% 2.77/1.56 => ( setadjoinE
% 2.77/1.56 => ( setadjoinOr
% 2.77/1.56 => ( setoftrueEq
% 2.77/1.56 => ( powersetI
% 2.77/1.56 => ( emptyinPowerset
% 2.77/1.56 => ( emptyInPowerset
% 2.77/1.56 => ( powersetE
% 2.77/1.56 => ( setunionI
% 2.77/1.56 => ( setunionE
% 2.77/1.56 => ( subPowSU
% 2.77/1.56 => ( exuE2
% 2.77/1.56 => ( nonemptyImpWitness
% 2.77/1.56 => ( uniqinunit
% 2.77/1.56 => ( notinsingleton
% 2.77/1.56 => ( eqinunit
% 2.77/1.56 => ( singletonsswitch
% 2.77/1.56 => ( upairsetE
% 2.77/1.56 => ( upairsetIL
% 2.77/1.56 => ( upairsetIR
% 2.77/1.56 => ( emptyE1
% 2.77/1.56 => ( vacuousDall
% 2.77/1.56 => ( quantDeMorgan1
% 2.77/1.56 => ( quantDeMorgan2
% 2.77/1.56 => ( quantDeMorgan3
% 2.77/1.56 => ( quantDeMorgan4
% 2.77/1.56 => ( prop2setI
% 2.77/1.56 => ( prop2set2propI
% 2.77/1.56 => ( notdexE
% 2.77/1.56 => ( notdallE
% 2.77/1.56 => ( exuI1
% 2.77/1.56 => ( exuI3
% 2.77/1.56 => ( exuI2
% 2.77/1.56 => ( inCongP
% 2.77/1.56 => ( in__Cong
% 2.77/1.56 => ( exuE3u
% 2.77/1.56 => ( exu__Cong
% 2.77/1.56 => ( emptyset__Cong
% 2.77/1.56 => ( setadjoin__Cong
% 2.77/1.56 => ( powerset__Cong
% 2.77/1.56 => ( setunion__Cong
% 2.77/1.56 => ( omega__Cong
% 2.77/1.56 => ( exuEu
% 2.77/1.56 => ( descr__Cong
% 2.77/1.56 => ( dsetconstr__Cong
% 2.77/1.56 => ( subsetI1
% 2.77/1.56 => ( eqimpsubset2
% 2.77/1.56 => ( eqimpsubset1
% 2.77/1.56 => ( subsetI2
% 2.77/1.56 => ( emptysetsubset
% 2.77/1.56 => ( subsetE
% 2.77/1.56 => ( subsetE2
% 2.77/1.56 => ( notsubsetI
% 2.77/1.56 => ( notequalI1
% 2.77/1.56 => ( notequalI2
% 2.77/1.56 => ( subsetRefl
% 2.77/1.56 => ( subsetTrans
% 2.77/1.56 => ( setadjoinSub
% 2.77/1.56 => ( setadjoinSub2
% 2.77/1.56 => ( subset2powerset
% 2.77/1.56 => ( setextsub
% 2.77/1.56 => ( subsetemptysetimpeq
% 2.77/1.56 => ( powersetI1
% 2.77/1.56 => ( powersetE1
% 2.77/1.56 => ( inPowerset
% 2.77/1.56 => ( powersetsubset
% 2.77/1.56 => ( sepInPowerset
% 2.77/1.56 => ( sepSubset
% 2.77/1.56 => ( binunionIL
% 2.77/1.56 => ( upairset2IR
% 2.77/1.56 => ( binunionIR
% 2.77/1.56 => ( binunionEcases
% 2.77/1.56 => ( binunionE
% 2.77/1.56 => ( binunionLsub
% 2.77/1.56 => ( binunionRsub
% 2.77/1.56 => ( binintersectI
% 2.77/1.56 => ( binintersectSubset5
% 2.77/1.56 => ( binintersectEL
% 2.77/1.56 => ( binintersectLsub
% 2.77/1.56 => ( binintersectSubset2
% 2.77/1.56 => ( binintersectSubset3
% 2.77/1.56 => ( binintersectER
% 2.77/1.56 => ( disjointsetsI1
% 2.77/1.56 => ( binintersectRsub
% 2.77/1.56 => ( binintersectSubset4
% 2.77/1.56 => ( binintersectSubset1
% 2.77/1.56 => ( bs114d
% 2.77/1.56 => ( setminusI
% 2.77/1.56 => ( setminusEL
% 2.77/1.56 => ( setminusER
% 2.77/1.56 => ( setminusSubset2
% 2.77/1.56 => ( setminusERneg
% 2.77/1.56 => ( setminusELneg
% 2.77/1.56 => ( setminusILneg
% 2.77/1.56 => ( setminusIRneg
% 2.77/1.56 => ( setminusLsub
% 2.77/1.56 => ( setminusSubset1
% 2.77/1.56 => ( symdiffE
% 2.77/1.56 => ( symdiffI1
% 2.77/1.56 => ( symdiffI2
% 2.77/1.56 => ( symdiffIneg1
% 2.77/1.56 => ( symdiffIneg2
% 2.77/1.56 => ( secondinupair
% 2.77/1.56 => ( setukpairIL
% 2.77/1.56 => ( setukpairIR
% 2.77/1.56 => ( kpairiskpair
% 2.77/1.56 => ( kpairp
% 2.77/1.56 => ( singletonsubset
% 2.77/1.56 => ( singletoninpowerset
% 2.77/1.56 => ( singletoninpowunion
% 2.77/1.56 => ( upairset2E
% 2.77/1.56 => ( upairsubunion
% 2.77/1.56 => ( upairinpowunion
% 2.77/1.56 => ( ubforcartprodlem1
% 2.77/1.56 => ( ubforcartprodlem2
% 2.77/1.56 => ( ubforcartprodlem3
% 2.77/1.56 => ( cartprodpairin
% 2.77/1.56 => ( cartprodmempair1
% 2.77/1.56 => ( cartprodmempair
% 2.77/1.56 => ( setunionE2
% 2.77/1.56 => ( setunionsingleton1
% 2.77/1.56 => ( setunionsingleton2
% 2.77/1.56 => ( setunionsingleton
% 2.77/1.56 => ( singletonprop
% 2.77/1.56 => ( ex1E1
% 2.77/1.56 => ( ex1I
% 2.77/1.56 => ( ex1I2
% 2.77/1.56 => ( singletonsuniq
% 2.77/1.56 => ( setukpairinjL1
% 2.77/1.56 => ( kfstsingleton
% 2.77/1.56 => ( theprop
% 2.77/1.56 => ( kfstpairEq
% 2.77/1.56 => ( cartprodfstin
% 2.77/1.56 => ( setukpairinjL2
% 2.77/1.56 => ( setukpairinjL
% 2.77/1.56 => ( setukpairinjR11
% 2.77/1.56 => ( setukpairinjR12
% 2.77/1.56 => ( setukpairinjR1
% 2.77/1.56 => ( upairequniteq
% 2.77/1.56 => ( setukpairinjR2
% 2.77/1.56 => ( setukpairinjR
% 2.77/1.56 => ( ksndsingleton
% 2.77/1.56 => ( ksndpairEq
% 2.77/1.56 => ( kpairsurjEq
% 2.77/1.56 => ( cartprodsndin
% 2.77/1.56 => ( cartprodpairmemEL
% 2.77/1.56 => ( cartprodpairmemER
% 2.77/1.56 => ( cartprodmempaircEq
% 2.77/1.56 => ( cartprodfstpairEq
% 2.77/1.56 => ( cartprodsndpairEq
% 2.77/1.56 => ( cartprodpairsurjEq
% 2.77/1.56 => ( dpsetconstrI
% 2.77/1.56 => ( dpsetconstrSub
% 2.77/1.56 => ( setOfPairsIsBReln
% 2.77/1.56 => ( dpsetconstrERa
% 2.77/1.56 => ( dpsetconstrEL1
% 2.77/1.56 => ( dpsetconstrEL2
% 2.77/1.56 => ( dpsetconstrER
% 2.77/1.56 => ( funcImageSingleton
% 2.77/1.56 => ( apProp
% 2.77/1.56 => ( app
% 2.77/1.56 => ( infuncsetfunc
% 2.77/1.56 => ( ap2p
% 2.77/1.56 => ( funcinfuncset
% 2.77/1.56 => ( lamProp
% 2.77/1.56 => ( lamp
% 2.77/1.56 => ( lam2p
% 2.77/1.56 => ( brelnall1
% 2.77/1.56 => ( brelnall2
% 2.77/1.56 => ( ex1E2
% 2.77/1.56 => ( funcGraphProp1
% 2.77/1.56 => ( funcGraphProp3
% 2.77/1.56 => ( funcGraphProp2
% 2.77/1.56 => ( funcextLem
% 2.77/1.56 => ( funcGraphProp4
% 2.77/1.56 => ( subbreln
% 2.77/1.56 => ( eqbreln
% 2.77/1.56 => ( funcext
% 2.77/1.56 => ( funcext2
% 2.77/1.56 => ( ap2apEq1
% 2.77/1.56 => ( ap2apEq2
% 2.77/1.56 => ( beta1
% 2.77/1.56 => ( eta1
% 2.77/1.56 => ( lam2lamEq
% 2.77/1.56 => ( beta2
% 2.77/1.56 => ( eta2
% 2.77/1.56 => ( iffalseProp1
% 2.77/1.56 => ( iffalseProp2
% 2.77/1.56 => ( iftrueProp1
% 2.77/1.56 => ( iftrueProp2
% 2.77/1.56 => ( ifSingleton
% 2.77/1.56 => ( ifp
% 2.77/1.56 => ( theeq
% 2.77/1.56 => ( iftrue
% 2.77/1.56 => ( iffalse
% 2.77/1.56 => ( iftrueorfalse
% 2.77/1.56 => ( binintersectT_lem
% 2.77/1.56 => ( binunionT_lem
% 2.77/1.56 => ( powersetT_lem
% 2.77/1.56 => ( setminusT_lem
% 2.77/1.56 => ( complementT_lem
% 2.77/1.56 => ( setextT
% 2.77/1.56 => ( subsetTI
% 2.77/1.56 => ( powersetTI1
% 2.77/1.56 => ( powersetTE1
% 2.77/1.56 => ( complementTI1
% 2.77/1.56 => ( complementTE1
% 2.77/1.56 => ( binintersectTELcontra
% 2.77/1.56 => ( binintersectTERcontra
% 2.77/1.56 => ( contrasubsetT
% 2.77/1.56 => ( contrasubsetT1
% 2.77/1.56 => ( contrasubsetT2
% 2.77/1.56 => ( contrasubsetT3
% 2.77/1.56 => ( doubleComplementI1
% 2.77/1.56 => ( doubleComplementE1
% 2.77/1.56 => ( doubleComplementSub1
% 2.77/1.56 => ( doubleComplementSub2
% 2.77/1.56 => ( doubleComplementEq
% 2.77/1.56 => ( complementTnotintersectT
% 2.77/1.56 => ( complementImpComplementIntersect
% 2.77/1.56 => ( complementSubsetComplementIntersect
% 2.77/1.56 => ( complementInPowersetComplementIntersect
% 2.77/1.56 => ( contraSubsetComplement
% 2.77/1.56 => ( complementTcontraSubset
% 2.77/1.56 => ( binunionTILcontra
% 2.77/1.56 => ( binunionTIRcontra
% 2.77/1.56 => ( inIntersectImpInUnion
% 2.77/1.56 => ( inIntersectImpInUnion2
% 2.77/1.56 => ( inIntersectImpInIntersectUnions
% 2.77/1.56 => ( intersectInPowersetIntersectUnions
% 2.77/1.56 => ( inComplementUnionImpNotIn1
% 2.77/1.56 => ( inComplementUnionImpInComplement1
% 2.77/1.56 => ( binunionTE
% 2.77/1.56 => ( binunionTEcontra
% 2.77/1.56 => ( demorgan2a1
% 2.77/1.56 => ( complementUnionInPowersetComplement
% 2.77/1.56 => ( demorgan2a2
% 2.77/1.56 => ( demorgan1a
% 2.77/1.56 => ( demorgan1b
% 2.77/1.56 => ( demorgan1
% 2.77/1.56 => ( demorgan2a
% 2.77/1.56 => ( demorgan2b2
% 2.77/1.56 => ( demorgan2b
% 2.77/1.56 => ( demorgan2
% 2.77/1.56 => ( woz13rule0
% 2.77/1.56 => ! [A: $i,X: $i] :
% 2.77/1.56 ( ( in @ X @ ( powerset @ A ) )
% 2.77/1.56 => ! [Y: $i] :
% 2.77/1.56 ( ( in @ Y @ ( powerset @ A ) )
% 2.77/1.56 => ! [Z: $i] :
% 2.77/1.56 ( ( in @ Z @ ( powerset @ A ) )
% 2.77/1.56 => ( ( subset @ X @ Z )
% 2.77/1.56 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.77/1.56 ------------------------
% 2.86/1.60 % (23017)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22688 for (2999ds/183Mi)
% 2.86/1.60 % (23018)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_22688 for (2999ds/4Mi)
% 2.86/1.60 % (23018)Instruction limit reached!
% 2.86/1.60 % (23018)------------------------------
% 2.86/1.60 % (23018)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.60 % (23018)Termination reason: Unknown
% 2.86/1.60 % (23018)Termination phase: shuffling
% 2.86/1.60
% 2.86/1.60 % (23018)Memory used [KB]: 1535
% 2.86/1.60 % (23018)Time elapsed: 0.003 s
% 2.86/1.60 % (23018)Instructions burned: 6 (million)
% 2.86/1.60 % (23018)------------------------------
% 2.86/1.60 % (23018)------------------------------
% 2.86/1.61 % (23019)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_22688 for (2999ds/27Mi)
% 2.86/1.61 % (23020)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22688 for (2999ds/2Mi)
% 2.86/1.61 % (23021)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_22688 for (2999ds/2Mi)
% 2.86/1.61 % (23022)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_22688 for (2999ds/275Mi)
% 2.86/1.61 % (23023)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22688 for (2999ds/18Mi)
% 2.86/1.61 % (23020)Instruction limit reached!
% 2.86/1.61 % (23020)------------------------------
% 2.86/1.61 % (23020)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.61 % (23020)Termination reason: Unknown
% 2.86/1.61 % (23020)Termination phase: shuffling
% 2.86/1.61
% 2.86/1.61 % (23020)Memory used [KB]: 1535
% 2.86/1.61 % (23020)Time elapsed: 0.003 s
% 2.86/1.61 % (23020)Instructions burned: 3 (million)
% 2.86/1.61 % (23020)------------------------------
% 2.86/1.61 % (23020)------------------------------
% 2.86/1.61 % (23021)Instruction limit reached!
% 2.86/1.61 % (23021)------------------------------
% 2.86/1.61 % (23021)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.61 % (23021)Termination reason: Unknown
% 2.86/1.61 % (23021)Termination phase: shuffling
% 2.86/1.61
% 2.86/1.61 % (23021)Memory used [KB]: 1535
% 2.86/1.61 % (23021)Time elapsed: 0.003 s
% 2.86/1.61 % (23021)Instructions burned: 3 (million)
% 2.86/1.61 % (23021)------------------------------
% 2.86/1.61 % (23021)------------------------------
% 2.86/1.61 % (23024)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22688 for (2999ds/3Mi)
% 2.86/1.61 % (23024)Instruction limit reached!
% 2.86/1.61 % (23024)------------------------------
% 2.86/1.61 % (23024)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.61 % (23024)Termination reason: Unknown
% 2.86/1.61 % (23024)Termination phase: shuffling
% 2.86/1.61
% 2.86/1.61 % (23024)Memory used [KB]: 1535
% 2.86/1.61 % (23024)Time elapsed: 0.002 s
% 2.86/1.61 % (23024)Instructions burned: 3 (million)
% 2.86/1.61 % (23024)------------------------------
% 2.86/1.61 % (23024)------------------------------
% 2.86/1.62 % (23023)Instruction limit reached!
% 2.86/1.62 % (23023)------------------------------
% 2.86/1.62 % (23023)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.62 % (23023)Termination reason: Unknown
% 2.86/1.62 % (23023)Termination phase: shuffling
% 2.86/1.62
% 2.86/1.62 % (23023)Memory used [KB]: 1791
% 2.86/1.62 % (23023)Time elapsed: 0.013 s
% 2.86/1.62 % (23023)Instructions burned: 18 (million)
% 2.86/1.62 % (23023)------------------------------
% 2.86/1.62 % (23023)------------------------------
% 2.86/1.62 % (23027)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_22688 for (2999ds/3Mi)
% 2.86/1.62 % (23019)Instruction limit reached!
% 2.86/1.62 % (23019)------------------------------
% 2.86/1.62 % (23019)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.62 % (23019)Termination reason: Unknown
% 2.86/1.62 % (23019)Termination phase: shuffling
% 2.86/1.62
% 2.86/1.62 % (23019)Memory used [KB]: 1918
% 2.86/1.62 % (23019)Time elapsed: 0.016 s
% 2.86/1.62 % (23019)Instructions burned: 28 (million)
% 2.86/1.62 % (23019)------------------------------
% 2.86/1.62 % (23019)------------------------------
% 2.86/1.62 % (23027)Instruction limit reached!
% 2.86/1.62 % (23027)------------------------------
% 2.86/1.62 % (23027)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.62 % (23027)Termination reason: Unknown
% 2.86/1.62 % (23027)Termination phase: shuffling
% 2.86/1.62
% 2.86/1.62 % (23027)Memory used [KB]: 1535
% 2.86/1.62 % (23027)Time elapsed: 0.003 s
% 2.86/1.62 % (23027)Instructions burned: 6 (million)
% 2.86/1.62 % (23027)------------------------------
% 2.86/1.62 % (23027)------------------------------
% 2.86/1.63 % (23025)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_22688 for (2999ds/37Mi)
% 2.86/1.63 % (23026)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_22688 for (2999ds/15Mi)
% 2.86/1.63 % (23030)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_22688 for (2999ds/16Mi)
% 2.86/1.64 % (23026)Instruction limit reached!
% 2.86/1.64 % (23026)------------------------------
% 2.86/1.64 % (23026)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.64 % (23026)Termination reason: Unknown
% 2.86/1.64 % (23026)Termination phase: shuffling
% 2.86/1.64
% 2.86/1.64 % (23026)Memory used [KB]: 1791
% 2.86/1.64 % (23026)Time elapsed: 0.011 s
% 2.86/1.64 % (23026)Instructions burned: 15 (million)
% 2.86/1.64 % (23026)------------------------------
% 2.86/1.64 % (23026)------------------------------
% 2.86/1.64 % (23028)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_22688 for (2999ds/1041Mi)
% 2.86/1.64 % (23030)Instruction limit reached!
% 2.86/1.64 % (23030)------------------------------
% 2.86/1.64 % (23030)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.64 % (23030)Termination reason: Unknown
% 2.86/1.64 % (23030)Termination phase: shuffling
% 2.86/1.64
% 2.86/1.64 % (23030)Memory used [KB]: 1791
% 2.86/1.64 % (23030)Time elapsed: 0.006 s
% 2.86/1.64 % (23030)Instructions burned: 18 (million)
% 2.86/1.64 % (23030)------------------------------
% 2.86/1.64 % (23030)------------------------------
% 2.86/1.64 % (23029)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_22688 for (2999ds/7Mi)
% 2.86/1.65 % (23029)Instruction limit reached!
% 2.86/1.65 % (23029)------------------------------
% 2.86/1.65 % (23029)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.65 % (23029)Termination reason: Unknown
% 2.86/1.65 % (23029)Termination phase: shuffling
% 2.86/1.65
% 2.86/1.65 % (23029)Memory used [KB]: 1663
% 2.86/1.65 % (23029)Time elapsed: 0.006 s
% 2.86/1.65 % (23029)Instructions burned: 7 (million)
% 2.86/1.65 % (23029)------------------------------
% 2.86/1.65 % (23029)------------------------------
% 2.86/1.65 % (23025)Instruction limit reached!
% 2.86/1.65 % (23025)------------------------------
% 2.86/1.65 % (23025)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.65 % (23025)Termination reason: Unknown
% 2.86/1.65 % (23025)Termination phase: shuffling
% 2.86/1.65
% 2.86/1.65 % (23025)Memory used [KB]: 2302
% 2.86/1.65 % (23025)Time elapsed: 0.021 s
% 2.86/1.65 % (23025)Instructions burned: 38 (million)
% 2.86/1.65 % (23025)------------------------------
% 2.86/1.65 % (23025)------------------------------
% 2.86/1.65 % (23017)Instruction limit reached!
% 2.86/1.65 % (23017)------------------------------
% 2.86/1.65 % (23017)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.65 % (23017)Termination reason: Unknown
% 2.86/1.65 % (23017)Termination phase: Saturation
% 2.86/1.65
% 2.86/1.65 % (23017)Memory used [KB]: 7419
% 2.86/1.65 % (23017)Time elapsed: 0.048 s
% 2.86/1.65 % (23017)Instructions burned: 183 (million)
% 2.86/1.65 % (23017)------------------------------
% 2.86/1.65 % (23017)------------------------------
% 2.86/1.65 % (23032)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_22688 for (2999ds/3Mi)
% 2.86/1.65 % (23032)Instruction limit reached!
% 2.86/1.65 % (23032)------------------------------
% 2.86/1.65 % (23032)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.65 % (23032)Termination reason: Unknown
% 2.86/1.65 % (23032)Termination phase: shuffling
% 2.86/1.65
% 2.86/1.65 % (23032)Memory used [KB]: 1535
% 2.86/1.65 % (23032)Time elapsed: 0.002 s
% 2.86/1.65 % (23032)Instructions burned: 3 (million)
% 2.86/1.65 % (23032)------------------------------
% 2.86/1.65 % (23032)------------------------------
% 2.86/1.66 % (23031)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_22688 for (2999ds/3Mi)
% 2.86/1.66 % (23031)Instruction limit reached!
% 2.86/1.66 % (23031)------------------------------
% 2.86/1.66 % (23031)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.66 % (23031)Termination reason: Unknown
% 2.86/1.66 % (23031)Termination phase: shuffling
% 2.86/1.66
% 2.86/1.66 % (23031)Memory used [KB]: 1535
% 2.86/1.66 % (23031)Time elapsed: 0.004 s
% 2.86/1.66 % (23031)Instructions burned: 3 (million)
% 2.86/1.66 % (23031)------------------------------
% 2.86/1.66 % (23031)------------------------------
% 2.86/1.66 % (23034)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_22688 for (2999ds/3Mi)
% 2.86/1.66 % (23034)Instruction limit reached!
% 2.86/1.66 % (23034)------------------------------
% 2.86/1.66 % (23034)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.66 % (23034)Termination reason: Unknown
% 2.86/1.66 % (23034)Termination phase: shuffling
% 2.86/1.66
% 2.86/1.66 % (23034)Memory used [KB]: 1535
% 2.86/1.66 % (23034)Time elapsed: 0.002 s
% 2.86/1.66 % (23034)Instructions burned: 3 (million)
% 2.86/1.66 % (23034)------------------------------
% 2.86/1.66 % (23034)------------------------------
% 2.86/1.66 % (23036)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_22688 for (2999ds/18Mi)
% 2.86/1.66 % (23033)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_22688 for (2999ds/7Mi)
% 2.86/1.66 % (23036)Instruction limit reached!
% 2.86/1.66 % (23036)------------------------------
% 2.86/1.66 % (23036)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.66 % (23036)Termination reason: Unknown
% 2.86/1.66 % (23036)Termination phase: shuffling
% 2.86/1.66
% 2.86/1.66 % (23036)Memory used [KB]: 1918
% 2.86/1.66 % (23036)Time elapsed: 0.007 s
% 2.86/1.66 % (23036)Instructions burned: 21 (million)
% 2.86/1.66 % (23036)------------------------------
% 2.86/1.66 % (23036)------------------------------
% 2.86/1.67 % (23035)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_22688 for (2999ds/4Mi)
% 2.86/1.67 % (23033)Instruction limit reached!
% 2.86/1.67 % (23033)------------------------------
% 2.86/1.67 % (23033)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.67 % (23033)Termination reason: Unknown
% 2.86/1.67 % (23033)Termination phase: shuffling
% 2.86/1.67
% 2.86/1.67 % (23033)Memory used [KB]: 1535
% 2.86/1.67 % (23033)Time elapsed: 0.006 s
% 2.86/1.67 % (23033)Instructions burned: 7 (million)
% 2.86/1.67 % (23033)------------------------------
% 2.86/1.67 % (23033)------------------------------
% 2.86/1.67 % (23035)Instruction limit reached!
% 2.86/1.67 % (23035)------------------------------
% 2.86/1.67 % (23035)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.67 % (23035)Termination reason: Unknown
% 2.86/1.67 % (23035)Termination phase: shuffling
% 2.86/1.67
% 2.86/1.67 % (23035)Memory used [KB]: 1535
% 2.86/1.67 % (23035)Time elapsed: 0.004 s
% 2.86/1.67 % (23035)Instructions burned: 4 (million)
% 2.86/1.67 % (23035)------------------------------
% 2.86/1.67 % (23035)------------------------------
% 2.86/1.67 % (23038)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_22688 for (2999ds/6Mi)
% 2.86/1.67 % (23038)Instruction limit reached!
% 2.86/1.67 % (23038)------------------------------
% 2.86/1.67 % (23038)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.67 % (23038)Termination reason: Unknown
% 2.86/1.67 % (23038)Termination phase: shuffling
% 2.86/1.67
% 2.86/1.67 % (23038)Memory used [KB]: 1535
% 2.86/1.67 % (23038)Time elapsed: 0.003 s
% 2.86/1.67 % (23038)Instructions burned: 7 (million)
% 2.86/1.67 % (23038)------------------------------
% 2.86/1.67 % (23038)------------------------------
% 2.86/1.67 % (23039)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_22688 for (2999ds/902Mi)
% 2.86/1.67 % (23037)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_22688 for (2999ds/710Mi)
% 2.86/1.68 % (23042)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_22688 for (2999ds/6Mi)
% 2.86/1.68 % (23042)Instruction limit reached!
% 2.86/1.68 % (23042)------------------------------
% 2.86/1.68 % (23042)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.68 % (23042)Termination reason: Unknown
% 2.86/1.68 % (23042)Termination phase: shuffling
% 2.86/1.68
% 2.86/1.68 % (23042)Memory used [KB]: 1535
% 2.86/1.68 % (23042)Time elapsed: 0.003 s
% 2.86/1.68 % (23042)Instructions burned: 7 (million)
% 2.86/1.68 % (23042)------------------------------
% 2.86/1.68 % (23042)------------------------------
% 2.86/1.68 % (23041)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_22688 for (2999ds/5Mi)
% 2.86/1.68 % (23040)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_22688 for (2999ds/21Mi)
% 2.86/1.69 % (23041)Instruction limit reached!
% 2.86/1.69 % (23041)------------------------------
% 2.86/1.69 % (23041)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.69 % (23041)Termination reason: Unknown
% 2.86/1.69 % (23041)Termination phase: shuffling
% 2.86/1.69
% 2.86/1.69 % (23041)Memory used [KB]: 1535
% 2.86/1.69 % (23041)Time elapsed: 0.004 s
% 2.86/1.69 % (23041)Instructions burned: 7 (million)
% 2.86/1.69 % (23041)------------------------------
% 2.86/1.69 % (23041)------------------------------
% 2.86/1.69 % (23043)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_22688 for (2998ds/377Mi)
% 2.86/1.70 % (23040)Instruction limit reached!
% 2.86/1.70 % (23040)------------------------------
% 2.86/1.70 % (23040)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.70 % (23040)Termination reason: Unknown
% 2.86/1.70 % (23040)Termination phase: shuffling
% 2.86/1.70
% 2.86/1.70 % (23040)Memory used [KB]: 1918
% 2.86/1.70 % (23040)Time elapsed: 0.013 s
% 2.86/1.70 % (23040)Instructions burned: 22 (million)
% 2.86/1.70 % (23040)------------------------------
% 2.86/1.70 % (23040)------------------------------
% 2.86/1.70 % (23044)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_22688 for (2998ds/779Mi)
% 2.86/1.71 % (23022)Instruction limit reached!
% 2.86/1.71 % (23022)------------------------------
% 2.86/1.71 % (23022)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.71 % (23022)Termination reason: Unknown
% 2.86/1.71 % (23022)Termination phase: Saturation
% 2.86/1.71
% 2.86/1.71 % (23022)Memory used [KB]: 9978
% 2.86/1.71 % (23022)Time elapsed: 0.104 s
% 2.86/1.71 % (23022)Instructions burned: 277 (million)
% 2.86/1.71 % (23022)------------------------------
% 2.86/1.71 % (23022)------------------------------
% 2.86/1.71 % (23045)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_22688 for (2998ds/19Mi)
% 2.86/1.72 % (23046)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_22688 for (2998ds/879Mi)
% 2.86/1.72 % (23045)Instruction limit reached!
% 2.86/1.72 % (23045)------------------------------
% 2.86/1.72 % (23045)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.72 % (23045)Termination reason: Unknown
% 2.86/1.72 % (23045)Termination phase: shuffling
% 2.86/1.72
% 2.86/1.72 % (23045)Memory used [KB]: 1791
% 2.86/1.72 % (23045)Time elapsed: 0.012 s
% 2.86/1.72 % (23045)Instructions burned: 19 (million)
% 2.86/1.72 % (23045)------------------------------
% 2.86/1.72 % (23045)------------------------------
% 2.86/1.74 % (23047)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_22688 for (2998ds/17Mi)
% 2.86/1.75 % (23047)Instruction limit reached!
% 2.86/1.75 % (23047)------------------------------
% 2.86/1.75 % (23047)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.75 % (23047)Termination reason: Unknown
% 2.86/1.75 % (23047)Termination phase: shuffling
% 2.86/1.75
% 2.86/1.75 % (23047)Memory used [KB]: 1791
% 2.86/1.75 % (23047)Time elapsed: 0.011 s
% 2.86/1.75 % (23047)Instructions burned: 17 (million)
% 2.86/1.75 % (23047)------------------------------
% 2.86/1.75 % (23047)------------------------------
% 2.86/1.77 % (23048)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_22688 for (2998ds/3Mi)
% 2.86/1.77 % (23048)Instruction limit reached!
% 2.86/1.77 % (23048)------------------------------
% 2.86/1.77 % (23048)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.77 % (23048)Termination reason: Unknown
% 2.86/1.77 % (23048)Termination phase: shuffling
% 2.86/1.77
% 2.86/1.77 % (23048)Memory used [KB]: 1535
% 2.86/1.77 % (23048)Time elapsed: 0.004 s
% 2.86/1.77 % (23048)Instructions burned: 4 (million)
% 2.86/1.77 % (23048)------------------------------
% 2.86/1.77 % (23048)------------------------------
% 2.86/1.78 % (23049)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_22688 for (2998ds/30Mi)
% 2.86/1.79 % (23043)Instruction limit reached!
% 2.86/1.79 % (23043)------------------------------
% 2.86/1.79 % (23043)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.79 % (23043)Termination reason: Unknown
% 2.86/1.79 % (23043)Termination phase: Saturation
% 2.86/1.79
% 2.86/1.79 % (23043)Memory used [KB]: 9466
% 2.86/1.79 % (23043)Time elapsed: 0.103 s
% 2.86/1.79 % (23043)Instructions burned: 379 (million)
% 2.86/1.79 % (23043)------------------------------
% 2.86/1.79 % (23043)------------------------------
% 2.86/1.79 % (23049)Instruction limit reached!
% 2.86/1.79 % (23049)------------------------------
% 2.86/1.79 % (23049)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.79 % (23049)Termination reason: Unknown
% 2.86/1.79 % (23049)Termination phase: shuffling
% 2.86/1.79
% 2.86/1.79 % (23049)Memory used [KB]: 2046
% 2.86/1.79 % (23049)Time elapsed: 0.011 s
% 2.86/1.79 % (23049)Instructions burned: 33 (million)
% 2.86/1.79 % (23049)------------------------------
% 2.86/1.79 % (23049)------------------------------
% 2.86/1.80 % (23050)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_22688 for (2997ds/127Mi)
% 2.86/1.80 % (23051)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_22688 for (2997ds/100Mi)
% 2.86/1.83 % (23039)First to succeed.
% 2.86/1.83 % (23051)Instruction limit reached!
% 2.86/1.83 % (23051)------------------------------
% 2.86/1.83 % (23051)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.83 % (23051)Termination reason: Unknown
% 2.86/1.83 % (23051)Termination phase: Property scanning
% 2.86/1.83
% 2.86/1.83 % (23051)Memory used [KB]: 2686
% 2.86/1.83 % (23051)Time elapsed: 0.029 s
% 2.86/1.83 % (23051)Instructions burned: 102 (million)
% 2.86/1.83 % (23051)------------------------------
% 2.86/1.83 % (23051)------------------------------
% 2.86/1.83 % (23050)Instruction limit reached!
% 2.86/1.83 % (23050)------------------------------
% 2.86/1.83 % (23050)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.83 % (23050)Termination reason: Unknown
% 2.86/1.83 % (23050)Termination phase: Property scanning
% 2.86/1.83
% 2.86/1.83 % (23050)Memory used [KB]: 2686
% 2.86/1.83 % (23050)Time elapsed: 0.033 s
% 2.86/1.83 % (23050)Instructions burned: 127 (million)
% 2.86/1.83 % (23050)------------------------------
% 2.86/1.83 % (23050)------------------------------
% 2.86/1.84 % (23039)Refutation found. Thanks to Tanya!
% 2.86/1.84 % SZS status Theorem for DTF2THF_22688
% 2.86/1.84 % SZS output start Proof for DTF2THF_22688
% 2.86/1.84 thf(func_def_0, type, in: $i > $i > $o).
% 2.86/1.84 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.86/1.84 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.86/1.84 thf(func_def_8, type, powerset: $i > $i).
% 2.86/1.84 thf(func_def_10, type, setunion: $i > $i).
% 2.86/1.84 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.86/1.84 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.86/1.84 thf(func_def_26, type, prop2set: $o > $i).
% 2.86/1.84 thf(func_def_36, type, nonempty: $i > $o).
% 2.86/1.84 thf(func_def_69, type, set2prop: $i > $o).
% 2.86/1.84 thf(func_def_88, type, subset: $i > $i > $o).
% 2.86/1.84 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.86/1.84 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.86/1.84 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.86/1.84 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.86/1.84 thf(func_def_135, type, regular: $i > $o).
% 2.86/1.84 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.86/1.84 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.86/1.84 thf(func_def_153, type, iskpair: $i > $o).
% 2.86/1.84 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.86/1.84 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.86/1.84 thf(func_def_177, type, singleton: $i > $o).
% 2.86/1.84 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.86/1.84 thf(func_def_184, type, atmost1p: $i > $o).
% 2.86/1.84 thf(func_def_185, type, atleast2p: $i > $o).
% 2.86/1.84 thf(func_def_186, type, atmost2p: $i > $o).
% 2.86/1.84 thf(func_def_187, type, upairsetp: $i > $o).
% 2.86/1.84 thf(func_def_191, type, kfst: $i > $i).
% 2.86/1.84 thf(func_def_203, type, ksnd: $i > $i).
% 2.86/1.84 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.86/1.84 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.86/1.84 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.86/1.84 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.86/1.84 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.86/1.84 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.86/1.84 thf(func_def_328, type, sP1: $o > $i > $i > $i > $o).
% 2.86/1.84 thf(func_def_329, type, sP2: $i > $i > $i > $i > $o).
% 2.86/1.84 thf(func_def_332, type, sP5: $i > $i > $o).
% 2.86/1.84 thf(func_def_333, type, sP6: $i > $o).
% 2.86/1.84 thf(func_def_334, type, sP7: $i > $i > $o).
% 2.86/1.84 thf(func_def_335, type, sP8: $i > $i > $o).
% 2.86/1.84 thf(func_def_337, type, sP10: $i > $i > $i > $o).
% 2.86/1.84 thf(func_def_354, type, sK27: $i > $o).
% 2.86/1.84 thf(func_def_361, type, sK34: $i > $o).
% 2.86/1.84 thf(func_def_379, type, sK52: $i > $i > $o).
% 2.86/1.84 thf(func_def_414, type, sK87: $o > $i > $i > $i).
% 2.86/1.84 thf(func_def_417, type, sK90: $i > $i).
% 2.86/1.84 thf(func_def_421, type, sK94: $i > $o).
% 2.86/1.84 thf(func_def_429, type, sK102: $i > $i).
% 2.86/1.84 thf(func_def_431, type, sK104: ($i > $i) > $i > $i > $i).
% 2.86/1.84 thf(func_def_445, type, sK118: $i > $o).
% 2.86/1.84 thf(func_def_458, type, sK131: $i > $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_459, type, sK132: $i > $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_460, type, sK133: $i > $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_461, type, sK134: $i > $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_486, type, sK159: $i > $o).
% 2.86/1.84 thf(func_def_490, type, sK163: $i > $i > ($i > $i) > $i).
% 2.86/1.84 thf(func_def_491, type, sK164: $i > $i).
% 2.86/1.84 thf(func_def_494, type, sK167: $i > $o).
% 2.86/1.84 thf(func_def_511, type, sK184: ($i > $o) > $i).
% 2.86/1.84 thf(func_def_512, type, sK185: $i > $o).
% 2.86/1.84 thf(func_def_517, type, sK190: $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_568, type, sK241: $i > $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_581, type, sK254: $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_582, type, sK255: $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_597, type, sK270: $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_629, type, sK302: $i > ($i > $o) > $i).
% 2.86/1.84 thf(func_def_630, type, sK303: $i > $o).
% 2.86/1.84 thf(func_def_638, type, sK311: $i > $i > $i).
% 2.86/1.84 thf(func_def_639, type, sK312: $i > $i > $i).
% 2.86/1.84 thf(func_def_640, type, sK313: $i > $i > $i).
% 2.86/1.84 thf(func_def_641, type, sK314: $i > $i > $i).
% 2.86/1.84 thf(func_def_642, type, sK315: $i > $i > $i).
% 2.86/1.84 thf(func_def_643, type, sK316: $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_644, type, sK317: $i > $i).
% 2.86/1.84 thf(func_def_645, type, sK318: $i > $i).
% 2.86/1.84 thf(func_def_646, type, sK319: $i > $i).
% 2.86/1.84 thf(func_def_647, type, sK320: $i > $i).
% 2.86/1.84 thf(func_def_648, type, sK321: $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_649, type, sK322: $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_650, type, sK323: $i > $i > $i).
% 2.86/1.84 thf(func_def_651, type, sK324: $i > $i > $i).
% 2.86/1.84 thf(func_def_652, type, sK325: $i > $i > $i).
% 2.86/1.84 thf(func_def_653, type, sK326: $i > $i).
% 2.86/1.84 thf(func_def_655, type, sK328: $i > $i).
% 2.86/1.84 thf(func_def_656, type, sK329: $i > $i).
% 2.86/1.84 thf(func_def_660, type, sK333: $i > $i > $i).
% 2.86/1.84 thf(func_def_681, type, sK354: $i > $o).
% 2.86/1.84 thf(func_def_682, type, sK355: $i > $i).
% 2.86/1.84 thf(func_def_683, type, sK356: ($i > $o) > $i).
% 2.86/1.84 thf(func_def_686, type, sK359: $i > $i > $i).
% 2.86/1.84 thf(func_def_693, type, sK366: $i > $i > ($i > $i) > $i).
% 2.86/1.84 thf(func_def_694, type, sK367: $i > $i).
% 2.86/1.84 thf(func_def_705, type, sK378: $i > $i > $i).
% 2.86/1.84 thf(func_def_707, type, sK380: $i > $o).
% 2.86/1.84 thf(func_def_709, type, sK382: ($i > $o) > $i > $i).
% 2.86/1.84 thf(func_def_710, type, sK383: $i > $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_724, type, sK397: $i > $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_748, type, sK421: $i > $o).
% 2.86/1.84 thf(func_def_750, type, sK423: ($i > $o) > $i > $i).
% 2.86/1.84 thf(func_def_753, type, sK426: $i > $i).
% 2.86/1.84 thf(func_def_771, type, sK444: $i > $i > $i).
% 2.86/1.84 thf(func_def_772, type, sK445: $i > $i > $i).
% 2.86/1.84 thf(func_def_780, type, sK453: $i > ($i > $o) > $i).
% 2.86/1.84 thf(func_def_781, type, sK454: $i > $o).
% 2.86/1.84 thf(func_def_799, type, sK472: $i > $o).
% 2.86/1.84 thf(func_def_826, type, sK499: $i > $i).
% 2.86/1.84 thf(func_def_829, type, sK502: $i > $i > ($i > $i) > $i).
% 2.86/1.84 thf(func_def_835, type, sK508: $i > $i > $o).
% 2.86/1.84 thf(func_def_836, type, sK509: $i > $i).
% 2.86/1.84 thf(func_def_837, type, sK510: $i > $i).
% 2.86/1.84 thf(func_def_838, type, sK511: ($i > $i > $o) > $i > $i).
% 2.86/1.84 thf(func_def_839, type, sK512: $i > ($i > $i > $o) > $i > $i).
% 2.86/1.84 thf(func_def_840, type, sK513: ($i > $i > $o) > $i > $i).
% 2.86/1.84 thf(func_def_841, type, sK514: ($i > $o) > $i).
% 2.86/1.84 thf(func_def_842, type, sK515: $i > $o).
% 2.86/1.84 thf(func_def_843, type, sK516: $i > $i).
% 2.86/1.84 thf(func_def_845, type, sK518: $i > $o).
% 2.86/1.84 thf(func_def_847, type, sK520: ($i > $o) > $i > $i).
% 2.86/1.84 thf(func_def_854, type, sK527: $i > $o).
% 2.86/1.84 thf(func_def_855, type, sK528: $i > $o).
% 2.86/1.84 thf(func_def_856, type, sK529: $i > $o).
% 2.86/1.84 thf(func_def_857, type, sK530: ($i > $o) > ($i > $o) > $i).
% 2.86/1.84 thf(func_def_858, type, sK531: ($i > $o) > ($i > $o) > $i).
% 2.86/1.84 thf(func_def_861, type, sK534: $i > $i > $o).
% 2.86/1.84 thf(func_def_864, type, sK537: $i > $o).
% 2.86/1.84 thf(func_def_867, type, sK540: ($i > $o) > $i > $i > $i).
% 2.86/1.84 thf(func_def_870, type, sK543: $i > $o).
% 2.86/1.84 thf(func_def_871, type, sK544: $i > $i > $i).
% 2.86/1.84 thf(func_def_875, type, sK548: $i > $o).
% 2.86/1.84 thf(func_def_877, type, sK550: ($i > $o) > $i).
% 2.86/1.84 thf(func_def_878, type, sK551: ($i > $o) > $i).
% 2.86/1.84 thf(func_def_879, type, sK552: $i > ($i > $o) > $i).
% 2.86/1.84 thf(func_def_880, type, sK553: $i > $o).
% 2.86/1.84 thf(func_def_887, type, sK560: $i > $i).
% 2.86/1.84 thf(func_def_888, type, sK561: ($i > $i) > $i > $i > $i).
% 2.86/1.84 thf(func_def_895, type, sK568: ($i > $o) > $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_896, type, sK569: ($i > $o) > $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_900, type, sK573: $i > $o).
% 2.86/1.84 thf(func_def_911, type, sK584: $i > $i).
% 2.86/1.84 thf(func_def_922, type, sK595: $i > $o).
% 2.86/1.84 thf(func_def_926, type, sK599: $i > $o).
% 2.86/1.84 thf(func_def_931, type, sK604: $i > $i).
% 2.86/1.84 thf(func_def_933, type, sK606: $i > $i).
% 2.86/1.84 thf(func_def_936, type, sK609: $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_937, type, sK610: $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_941, type, sK614: $i > $o).
% 2.86/1.84 thf(func_def_943, type, sK616: $i > ($i > $o) > $i).
% 2.86/1.84 thf(func_def_973, type, sK646: $i > $i > $o).
% 2.86/1.84 thf(func_def_978, type, sK651: $i > $i > $o).
% 2.86/1.84 thf(func_def_984, type, sK657: $i > $i > $i).
% 2.86/1.84 thf(func_def_999, type, sK672: $i > ($i > $i) > $i > $i).
% 2.86/1.84 thf(func_def_1001, type, sK674: $i > $i).
% 2.86/1.84 thf(func_def_1014, type, sK687: $i > $o).
% 2.86/1.84 thf(func_def_1017, type, sK690: $i > $i).
% 2.86/1.84 thf(func_def_1019, type, sK692: $i > $o).
% 2.86/1.84 thf(func_def_1020, type, sK693: $i > $o).
% 2.86/1.84 thf(func_def_1021, type, sK694: ($i > $o) > ($i > $o) > $i).
% 2.86/1.84 thf(func_def_1022, type, sK695: ($i > $o) > ($i > $o) > $i).
% 2.86/1.84 thf(func_def_1027, type, sK700: $i > $i > $o).
% 2.86/1.84 thf(func_def_1035, type, sK708: $i > $o).
% 2.86/1.84 thf(func_def_1037, type, sK710: ($i > $o) > $i > $i).
% 2.86/1.84 thf(func_def_1055, type, sK728: $i > $i > $o).
% 2.86/1.84 thf(func_def_1068, type, sK741: $i > $o).
% 2.86/1.84 thf(func_def_1076, type, sK749: $i > $o).
% 2.86/1.84 thf(func_def_1078, type, sK751: $i > ($i > $o) > $i).
% 2.86/1.84 thf(func_def_1083, type, sK756: $i > $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_1084, type, sK757: $i > $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_1095, type, sK768: $i > $i > $i).
% 2.86/1.84 thf(func_def_1123, type, sK796: $i > $o).
% 2.86/1.84 thf(func_def_1126, type, sK799: $i > ($i > $o) > $i).
% 2.86/1.84 thf(func_def_1127, type, sK800: $i > ($i > $o) > $i).
% 2.86/1.84 thf(func_def_1128, type, sK801: $i > $i > $i).
% 2.86/1.84 thf(func_def_1152, type, sK825: $i > $i > $o).
% 2.86/1.84 thf(func_def_1170, type, sK843: $i > $i > $i).
% 2.86/1.84 thf(func_def_1174, type, sK847: $i > $o).
% 2.86/1.84 thf(func_def_1176, type, sK849: ($i > $o) > $i > $i).
% 2.86/1.84 thf(func_def_1177, type, sK850: ($i > $o) > $i > $i).
% 2.86/1.84 thf(func_def_1193, type, sK866: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.86/1.84 thf(func_def_1194, type, sK867: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.86/1.84 thf(func_def_1197, type, sK870: $i > $o).
% 2.86/1.84 thf(func_def_1198, type, sK871: $i > $o).
% 2.86/1.84 thf(func_def_1209, type, sK882: $i > $i).
% 2.86/1.84 thf(func_def_1212, type, sK885: ($i > $o) > $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_1213, type, sK886: ($i > $o) > $i > $i > $i > $i).
% 2.86/1.84 thf(func_def_1217, type, sK890: $i > $o).
% 2.86/1.84 thf(func_def_1220, type, ph893: !>[X0: $tType]:(X0)).
% 2.86/1.84 thf(f5305,plain,(
% 2.86/1.84 $false),
% 2.86/1.84 inference(avatar_sat_refutation,[status(thm)],[f4473,f5282])).
% 2.86/1.84 thf(f5282,plain,(
% 2.86/1.84 ~spl892_10),
% 2.86/1.84 inference(avatar_split_clause,[status(thm)],[f5281,f4390])).
% 2.86/1.84 thf(f4390,definition,(
% 2.86/1.84 spl892_10 <=> ((subset @ sK875 @ sK877) = $true)),
% 2.86/1.84 introduced(definition,[new_symbols(naming,[spl892_10])],[avatar_definition])).
% 2.86/1.84 thf(f5281,plain,(
% 2.86/1.84 ((subset @ sK875 @ sK877) != $true)),
% 2.86/1.84 inference(trivial_inequality_removal,[status(thm)],[f5280])).
% 2.86/1.84 thf(f5280,plain,(
% 2.86/1.84 ((subset @ sK875 @ sK877) != $true) | ($true != $true)),
% 2.86/1.84 inference(forward_demodulation,[status(thm)],[f5145,f4038])).
% 2.86/1.84 thf(f4038,plain,(
% 2.86/1.84 (binintersectLsub = $true)),
% 2.86/1.84 inference(cnf_transformation,[status(thm)],[f2572])).
% 2.86/1.84 thf(f2572,plain,(
% 2.86/1.84 (powersetT_lem = $true) & (iftrueorfalse = $true) & (dsetconstr__Cong = $true) & (ifSingleton = $true) & (setukpairinjR1 = $true) & (binintersectTELcontra = $true) & (quantDeMorgan3 = $true) & (emptyI = $true) & (complementInPowersetComplementIntersect = $true) & (binunionLsub = $true) & (inComplementUnionImpNotIn1 = $true) & (binintersectLsub = $true) & (eta1 = $true) & (descrp = $true) & (woz13rule0 = $true) & (upairequniteq = $true) & (symdiffI1 = $true) & (intersectInPowersetIntersectUnions = $true) & (cartprodsndpairEq = $true) & (powersetI1 = $true) & (powersetI = $true) & (descr__Cong = $true) & (ex1I = $true) & (sepSubset = $true) & (exuEu = $true) & (setminusSubset2 = $true) & (setukpairinjL2 = $true) & (demorgan2a1 = $true) & ((((in @ sK876 @ (powerset @ sK874)) = $true) & (((subset @ (binintersect @ sK875 @ sK876) @ sK877) != $true) & ((in @ sK877 @ (powerset @ sK874)) = $true) & ((subset @ sK875 @ sK877) = $true))) & ((in @ sK875 @ (powerset @ sK874)) = $true)) & (setextsub = $true) & (setukpairinjR = $true) & (singletoninpowunion = $true) & (subsetE = $true) & (powersetsubset = $true) & (emptyinunitempty = $true) & (binintersectTERcontra = $true) & (ksndpairEq = $true) & (dsetconstrI = $true) & (powersetE = $true) & (dpsetconstrER = $true) & (setbeta = $true) & (symdiffIneg1 = $true) & (subbreln = $true) & (funcGraphProp2 = $true) & (setukpairIL = $true) & (app = $true) & (notequalI2 = $true) & (sepInPowerset = $true) & (omega0Ax = $true) & (setextAx = $true) & (cartprodmempaircEq = $true) & (funcext2 = $true) & (dsetconstrEL = $true) & (singletonsswitch = $true) & (complementTE1 = $true) & (emptysetE = $true) & (notdallE = $true) & (singletoninpowerset = $true) & (complementTnotintersectT = $true) & (secondinupair = $true) & (emptyinPowerset = $true) & (emptysetsubset = $true) & (eqinunit = $true) & (setminusER = $true) & (inIntersectImpInIntersectUnions = $true) & (uniqinunit = $true) & (complementTcontraSubset = $true) & (setunionAx = $true) & (ex1E2 = $true) & (funcext = $true) & (beta1 = $true) & (inIntersectImpInUnion2 = $true) & (nonemptyI = $true) & (symdiffE = $true) & (inCongP = $true) & (cartprodfstin = $true) & (emptysetimpfalse = $true) & (dpsetconstrSub = $true) & (funcGraphProp3 = $true) & (symdiffI2 = $true) & (cartprodpairsurjEq = $true) & (cartprodsndin = $true) & (prop2setI = $true) & (iftrueProp1 = $true) & (singletonsubset = $true) & (prop2set2propI = $true) & (complementSubsetComplementIntersect = $true) & (demorgan2a2 = $true) & (apProp = $true) & (notsubsetI = $true) & (exuE3e = $true) & (binunionEcases = $true) & (setunionI = $true) & (binunionTIRcontra = $true) & (demorgan2b = $true) & (quantDeMorgan2 = $true) & (powerset__Cong = $true) & (iffalseProp1 = $true) & (cartprodmempair1 = $true) & (ubforcartprodlem2 = $true) & (iftrue = $true) & (ubforcartprodlem3 = $true) & (binintersectSubset2 = $true) & (binintersectEL = $true) & (setminusIRneg = $true) & (inPowerset = $true) & (powersetE1 = $true) & (setunionE2 = $true) & (omega__Cong = $true) & (setminusERneg = $true) & (binintersectI = $true) & (subsetE2 = $true) & (exuI2 = $true) & (kfstsingleton = $true) & (notequalI1 = $true) & (setminusSubset1 = $true) & (cartprodpairmemEL = $true) & (lam2lamEq = $true) & (notinsingleton = $true) & (contraSubsetComplement = $true) & (cartprodpairmemER = $true) & (ex1E1 = $true) & (upairsetIL = $true) & (contrasubsetT1 = $true) & (quantDeMorgan1 = $true) & (inIntersectImpInUnion = $true) & (brelnall1 = $true) & (wellorderingAx = $true) & (theeq = $true) & (notinemptyset = $true) & (powersetTI1 = $true) & (setminusT_lem = $true) & (binunionTILcontra = $true) & (setadjoin__Cong = $true) & (upairsetE = $true) & (upairset2IR = $true) & (brelnall2 = $true) & (doubleComplementI1 = $true) & (binintersectSubset4 = $true) & (setadjoinIR = $true) & (setukpairinjL = $true) & (kpairsurjEq = $true) & (setminusEL = $true) & (lamp = $true) & (setminusLsub = $true) & (exu__Cong = $true) & (binunionRsub = $true) & (setunionsingleton = $true) & (bs114d = $true) & (contrasubsetT2 = $true) & (singletonsuniq = $true) & (ap2apEq2 = $true) & (funcGraphProp1 = $true) & (ifp = $true) & (complementImpComplementIntersect = $true) & (subsetTI = $true) & (vacuousDall = $true) & (subsetRefl = $true) & (exuE3u = $true) & (ap2apEq1 = $true) & (subsetemptysetimpeq = $true) & (ubforcartprodlem1 = $true) & (setunionE = $true) & (demorgan1 = $true) & (exuI1 = $true) & (binintersectSubset5 = $true) & (dpsetconstrEL1 = $true) & (setadjoinIL = $true) & (setadjoinE = $true) & (funcGraphProp4 = $true) & (noeltsimpempty = $true) & (eqimpsubset2 = $true) & (setminusI = $true) & (kfstpairEq = $true) & (iffalseProp2 = $true) & (setukpairIR = $true) & (symdiffIneg2 = $true) & (setukpairinjR11 = $true) & (beta2 = $true) & (kpairp = $true) & (setukpairinjL1 = $true) & (binunionTE = $true) & (subsetI1 = $true) & (binintersectT_lem = $true) & (setukpairinjR12 = $true) & (binintersectSubset3 = $true) & (subsetI2 = $true) & (emptysetAx = $true) & (setunion__Cong = $true) & (demorgan2a = $true) & (setadjoinSub2 = $true) & (eta2 = $true) & (cartprodmempair = $true) & (eqimpsubset1 = $true) & (setextT = $true) & (replAx = $true) & (powersetAx = $true) & (setadjoinOr = $true) & (binunionE = $true) & (setadjoinSub = $true) & (cartprodpairin = $true) & (binintersectSubset1 = $true) & (quantDeMorgan4 = $true) & (exuE1 = $true) & (subsetTrans = $true) & (emptyE1 = $true) & (binunionIL = $true) & (dsetconstrER = $true) & (iffalse = $true) & (ex1I2 = $true) & (demorgan2 = $true) & (binintersectER = $true) & (powersetTE1 = $true) & (subPowSU = $true) & (omegaSAx = $true) & (complementT_lem = $true) & (dpsetconstrERa = $true) & (funcextLem = $true) & (foundationAx = $true) & (setminusELneg = $true) & (disjointsetsI1 = $true) & (in__Cong = $true) & (inComplementUnionImpInComplement1 = $true) & (dpsetconstrEL2 = $true) & (binunionIR = $true) & (doubleComplementEq = $true) & (upairsubunion = $true) & (upairsetIR = $true) & (emptyInPowerset = $true) & (exuI3 = $true) & (doubleComplementSub1 = $true) & (omegaIndAx = $true) & (ksndsingleton = $true) & (complementUnionInPowersetComplement = $true) & (demorgan1a = $true) & (setext = $true) & (dpsetconstrI = $true) & (contrasubsetT = $true) & (binunionT_lem = $true) & (theprop = $true) & (upairset2E = $true) & (ap2p = $true) & (singletonprop = $true) & (setunionsingleton2 = $true) & (exuE2 = $true) & (nonemptyImpWitness = $true) & (cartprodfstpairEq = $true) & (nonemptyE1 = $true) & (doubleComplementSub2 = $true) & (funcinfuncset = $true) & (setunionsingleton1 = $true) & (binunionTEcontra = $true) & (subset2powerset = $true) & (nonemptyI1 = $true) & (eqbreln = $true) & (infuncsetfunc = $true) & (upairinpowunion = $true) & (emptyset__Cong = $true) & (setoftrueEq = $true) & (doubleComplementE1 = $true) & (setminusILneg = $true) & (binintersectRsub = $true) & (prop2setE = $true) & (funcImageSingleton = $true) & (demorgan2b2 = $true) & (iftrueProp2 = $true) & (lam2p = $true) & (setukpairinjR2 = $true) & (kpairiskpair = $true) & (demorgan1b = $true) & (contrasubsetT3 = $true) & (complementTI1 = $true) & (notdexE = $true) & (setOfPairsIsBReln = $true) & (lamProp = $true) & (setadjoinAx = $true)),
% 2.86/1.84 inference(skolemisation,[status(esa),new_symbols(skolem,[sK874,sK875,sK876,sK877])],[f1061,f2571,f2570,f2569])).
% 2.86/1.84 thf(f2569,plain,(
% 2.86/1.84 ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (((in @ X2 @ (powerset @ X0)) = $true) & ? [X3 : $i] : (((subset @ (binintersect @ X1 @ X2) @ X3) != $true) & ((in @ X3 @ (powerset @ X0)) = $true) & ((subset @ X1 @ X3) = $true))) & ((in @ X1 @ (powerset @ X0)) = $true)) => (? [X2 : $i] : (((in @ X2 @ (powerset @ sK874)) = $true) & ? [X3 : $i] : (((subset @ (binintersect @ sK875 @ X2) @ X3) != $true) & ((in @ X3 @ (powerset @ sK874)) = $true) & ((subset @ sK875 @ X3) = $true))) & ((in @ sK875 @ (powerset @ sK874)) = $true))),
% 2.86/1.84 introduced(definition,[],[choice_axiom])).
% 2.86/1.84 thf(f2570,plain,(
% 2.86/1.84 ? [X2 : $i] : (((in @ X2 @ (powerset @ sK874)) = $true) & ? [X3 : $i] : (((subset @ (binintersect @ sK875 @ X2) @ X3) != $true) & ((in @ X3 @ (powerset @ sK874)) = $true) & ((subset @ sK875 @ X3) = $true))) => (((in @ sK876 @ (powerset @ sK874)) = $true) & ? [X3 : $i] : (((subset @ (binintersect @ sK875 @ sK876) @ X3) != $true) & ((in @ X3 @ (powerset @ sK874)) = $true) & ((subset @ sK875 @ X3) = $true)))),
% 2.86/1.84 introduced(definition,[],[choice_axiom])).
% 2.86/1.84 thf(f2571,plain,(
% 2.86/1.84 ? [X3 : $i] : (((subset @ (binintersect @ sK875 @ sK876) @ X3) != $true) & ((in @ X3 @ (powerset @ sK874)) = $true) & ((subset @ sK875 @ X3) = $true)) => (((subset @ (binintersect @ sK875 @ sK876) @ sK877) != $true) & ((in @ sK877 @ (powerset @ sK874)) = $true) & ((subset @ sK875 @ sK877) = $true))),
% 2.86/1.84 introduced(definition,[],[choice_axiom])).
% 2.86/1.84 thf(f1061,plain,(
% 2.86/1.84 (powersetT_lem = $true) & (iftrueorfalse = $true) & (dsetconstr__Cong = $true) & (ifSingleton = $true) & (setukpairinjR1 = $true) & (binintersectTELcontra = $true) & (quantDeMorgan3 = $true) & (emptyI = $true) & (complementInPowersetComplementIntersect = $true) & (binunionLsub = $true) & (inComplementUnionImpNotIn1 = $true) & (binintersectLsub = $true) & (eta1 = $true) & (descrp = $true) & (woz13rule0 = $true) & (upairequniteq = $true) & (symdiffI1 = $true) & (intersectInPowersetIntersectUnions = $true) & (cartprodsndpairEq = $true) & (powersetI1 = $true) & (powersetI = $true) & (descr__Cong = $true) & (ex1I = $true) & (sepSubset = $true) & (exuEu = $true) & (setminusSubset2 = $true) & (setukpairinjL2 = $true) & (demorgan2a1 = $true) & ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (((in @ X2 @ (powerset @ X0)) = $true) & ? [X3 : $i] : (((subset @ (binintersect @ X1 @ X2) @ X3) != $true) & ((in @ X3 @ (powerset @ X0)) = $true) & ((subset @ X1 @ X3) = $true))) & ((in @ X1 @ (powerset @ X0)) = $true)) & (setextsub = $true) & (setukpairinjR = $true) & (singletoninpowunion = $true) & (subsetE = $true) & (powersetsubset = $true) & (emptyinunitempty = $true) & (binintersectTERcontra = $true) & (ksndpairEq = $true) & (dsetconstrI = $true) & (powersetE = $true) & (dpsetconstrER = $true) & (setbeta = $true) & (symdiffIneg1 = $true) & (subbreln = $true) & (funcGraphProp2 = $true) & (setukpairIL = $true) & (app = $true) & (notequalI2 = $true) & (sepInPowerset = $true) & (omega0Ax = $true) & (setextAx = $true) & (cartprodmempaircEq = $true) & (funcext2 = $true) & (dsetconstrEL = $true) & (singletonsswitch = $true) & (complementTE1 = $true) & (emptysetE = $true) & (notdallE = $true) & (singletoninpowerset = $true) & (complementTnotintersectT = $true) & (secondinupair = $true) & (emptyinPowerset = $true) & (emptysetsubset = $true) & (eqinunit = $true) & (setminusER = $true) & (inIntersectImpInIntersectUnions = $true) & (uniqinunit = $true) & (complementTcontraSubset = $true) & (setunionAx = $true) & (ex1E2 = $true) & (funcext = $true) & (beta1 = $true) & (inIntersectImpInUnion2 = $true) & (nonemptyI = $true) & (symdiffE = $true) & (inCongP = $true) & (cartprodfstin = $true) & (emptysetimpfalse = $true) & (dpsetconstrSub = $true) & (funcGraphProp3 = $true) & (symdiffI2 = $true) & (cartprodpairsurjEq = $true) & (cartprodsndin = $true) & (prop2setI = $true) & (iftrueProp1 = $true) & (singletonsubset = $true) & (prop2set2propI = $true) & (complementSubsetComplementIntersect = $true) & (demorgan2a2 = $true) & (apProp = $true) & (notsubsetI = $true) & (exuE3e = $true) & (binunionEcases = $true) & (setunionI = $true) & (binunionTIRcontra = $true) & (demorgan2b = $true) & (quantDeMorgan2 = $true) & (powerset__Cong = $true) & (iffalseProp1 = $true) & (cartprodmempair1 = $true) & (ubforcartprodlem2 = $true) & (iftrue = $true) & (ubforcartprodlem3 = $true) & (binintersectSubset2 = $true) & (binintersectEL = $true) & (setminusIRneg = $true) & (inPowerset = $true) & (powersetE1 = $true) & (setunionE2 = $true) & (omega__Cong = $true) & (setminusERneg = $true) & (binintersectI = $true) & (subsetE2 = $true) & (exuI2 = $true) & (kfstsingleton = $true) & (notequalI1 = $true) & (setminusSubset1 = $true) & (cartprodpairmemEL = $true) & (lam2lamEq = $true) & (notinsingleton = $true) & (contraSubsetComplement = $true) & (cartprodpairmemER = $true) & (ex1E1 = $true) & (upairsetIL = $true) & (contrasubsetT1 = $true) & (quantDeMorgan1 = $true) & (inIntersectImpInUnion = $true) & (brelnall1 = $true) & (wellorderingAx = $true) & (theeq = $true) & (notinemptyset = $true) & (powersetTI1 = $true) & (setminusT_lem = $true) & (binunionTILcontra = $true) & (setadjoin__Cong = $true) & (upairsetE = $true) & (upairset2IR = $true) & (brelnall2 = $true) & (doubleComplementI1 = $true) & (binintersectSubset4 = $true) & (setadjoinIR = $true) & (setukpairinjL = $true) & (kpairsurjEq = $true) & (setminusEL = $true) & (lamp = $true) & (setminusLsub = $true) & (exu__Cong = $true) & (binunionRsub = $true) & (setunionsingleton = $true) & (bs114d = $true) & (contrasubsetT2 = $true) & (singletonsuniq = $true) & (ap2apEq2 = $true) & (funcGraphProp1 = $true) & (ifp = $true) & (complementImpComplementIntersect = $true) & (subsetTI = $true) & (vacuousDall = $true) & (subsetRefl = $true) & (exuE3u = $true) & (ap2apEq1 = $true) & (subsetemptysetimpeq = $true) & (ubforcartprodlem1 = $true) & (setunionE = $true) & (demorgan1 = $true) & (exuI1 = $true) & (binintersectSubset5 = $true) & (dpsetconstrEL1 = $true) & (setadjoinIL = $true) & (setadjoinE = $true) & (funcGraphProp4 = $true) & (noeltsimpempty = $true) & (eqimpsubset2 = $true) & (setminusI = $true) & (kfstpairEq = $true) & (iffalseProp2 = $true) & (setukpairIR = $true) & (symdiffIneg2 = $true) & (setukpairinjR11 = $true) & (beta2 = $true) & (kpairp = $true) & (setukpairinjL1 = $true) & (binunionTE = $true) & (subsetI1 = $true) & (binintersectT_lem = $true) & (setukpairinjR12 = $true) & (binintersectSubset3 = $true) & (subsetI2 = $true) & (emptysetAx = $true) & (setunion__Cong = $true) & (demorgan2a = $true) & (setadjoinSub2 = $true) & (eta2 = $true) & (cartprodmempair = $true) & (eqimpsubset1 = $true) & (setextT = $true) & (replAx = $true) & (powersetAx = $true) & (setadjoinOr = $true) & (binunionE = $true) & (setadjoinSub = $true) & (cartprodpairin = $true) & (binintersectSubset1 = $true) & (quantDeMorgan4 = $true) & (exuE1 = $true) & (subsetTrans = $true) & (emptyE1 = $true) & (binunionIL = $true) & (dsetconstrER = $true) & (iffalse = $true) & (ex1I2 = $true) & (demorgan2 = $true) & (binintersectER = $true) & (powersetTE1 = $true) & (subPowSU = $true) & (omegaSAx = $true) & (complementT_lem = $true) & (dpsetconstrERa = $true) & (funcextLem = $true) & (foundationAx = $true) & (setminusELneg = $true) & (disjointsetsI1 = $true) & (in__Cong = $true) & (inComplementUnionImpInComplement1 = $true) & (dpsetconstrEL2 = $true) & (binunionIR = $true) & (doubleComplementEq = $true) & (upairsubunion = $true) & (upairsetIR = $true) & (emptyInPowerset = $true) & (exuI3 = $true) & (doubleComplementSub1 = $true) & (omegaIndAx = $true) & (ksndsingleton = $true) & (complementUnionInPowersetComplement = $true) & (demorgan1a = $true) & (setext = $true) & (dpsetconstrI = $true) & (contrasubsetT = $true) & (binunionT_lem = $true) & (theprop = $true) & (upairset2E = $true) & (ap2p = $true) & (singletonprop = $true) & (setunionsingleton2 = $true) & (exuE2 = $true) & (nonemptyImpWitness = $true) & (cartprodfstpairEq = $true) & (nonemptyE1 = $true) & (doubleComplementSub2 = $true) & (funcinfuncset = $true) & (setunionsingleton1 = $true) & (binunionTEcontra = $true) & (subset2powerset = $true) & (nonemptyI1 = $true) & (eqbreln = $true) & (infuncsetfunc = $true) & (upairinpowunion = $true) & (emptyset__Cong = $true) & (setoftrueEq = $true) & (doubleComplementE1 = $true) & (setminusILneg = $true) & (binintersectRsub = $true) & (prop2setE = $true) & (funcImageSingleton = $true) & (demorgan2b2 = $true) & (iftrueProp2 = $true) & (lam2p = $true) & (setukpairinjR2 = $true) & (kpairiskpair = $true) & (demorgan1b = $true) & (contrasubsetT3 = $true) & (complementTI1 = $true) & (notdexE = $true) & (setOfPairsIsBReln = $true) & (lamProp = $true) & (setadjoinAx = $true)),
% 2.86/1.84 inference(flattening,[status(thm)],[f1060])).
% 2.86/1.84 thf(f1060,plain,(
% 2.86/1.84 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : ((((subset @ (binintersect @ X1 @ X2) @ X3) != $true) & ((subset @ X1 @ X3) = $true)) & ((in @ X3 @ (powerset @ X0)) = $true)) & ((in @ X2 @ (powerset @ X0)) = $true)) & ((in @ X1 @ (powerset @ X0)) = $true)) & (woz13rule0 = $true)) & (demorgan2 = $true)) & (demorgan2b = $true)) & (demorgan2b2 = $true)) & (demorgan2a = $true)) & (demorgan1 = $true)) & (demorgan1b = $true)) & (demorgan1a = $true)) & (demorgan2a2 = $true)) & (complementUnionInPowersetComplement = $true)) & (demorgan2a1 = $true)) & (binunionTEcontra = $true)) & (binunionTE = $true)) & (inComplementUnionImpInComplement1 = $true)) & (inComplementUnionImpNotIn1 = $true)) & (intersectInPowersetIntersectUnions = $true)) & (inIntersectImpInIntersectUnions = $true)) & (inIntersectImpInUnion2 = $true)) & (inIntersectImpInUnion = $true)) & (binunionTIRcontra = $true)) & (binunionTILcontra = $true)) & (complementTcontraSubset = $true)) & (contraSubsetComplement = $true)) & (complementInPowersetComplementIntersect = $true)) & (complementSubsetComplementIntersect = $true)) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.86/1.84 inference(ennf_transformation,[status(thm)],[f306])).
% 2.86/1.84 thf(f306,plain,(
% 2.86/1.84 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ((complementSubsetComplementIntersect = $true) => ((complementInPowersetComplementIntersect = $true) => ((contraSubsetComplement = $true) => ((complementTcontraSubset = $true) => ((binunionTILcontra = $true) => ((binunionTIRcontra = $true) => ((inIntersectImpInUnion = $true) => ((inIntersectImpInUnion2 = $true) => ((inIntersectImpInIntersectUnions = $true) => ((intersectInPowersetIntersectUnions = $true) => ((inComplementUnionImpNotIn1 = $true) => ((inComplementUnionImpInComplement1 = $true) => ((binunionTE = $true) => ((binunionTEcontra = $true) => ((demorgan2a1 = $true) => ((complementUnionInPowersetComplement = $true) => ((demorgan2a2 = $true) => ((demorgan1a = $true) => ((demorgan1b = $true) => ((demorgan1 = $true) => ((demorgan2a = $true) => ((demorgan2b2 = $true) => ((demorgan2b = $true) => ((demorgan2 = $true) => ((woz13rule0 = $true) => ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) => ! [X2 : $i] : (((in @ X2 @ (powerset @ X0)) = $true) => ! [X3 : $i] : (((in @ X3 @ (powerset @ X0)) = $true) => (((subset @ X1 @ X3) = $true) => ((subset @ (binintersect @ X1 @ X2) @ X3) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.86/1.84 inference(fool_elimination,[status(thm)],[f305])).
% 2.86/1.84 thf(f305,plain,(
% 2.86/1.84 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => ! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X0)) => ! [X3 : $i] : ((in @ X3 @ (powerset @ X0)) => ((subset @ X1 @ X3) => (subset @ (binintersect @ X1 @ X2) @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.86/1.84 inference(rectify,[status(thm)],[f281])).
% 2.86/1.84 thf(f281,negated_conjecture,(
% 2.86/1.84 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X17 : $i] : ((in @ X17 @ (powerset @ X3)) => ((subset @ X11 @ X17) => (subset @ (binintersect @ X11 @ X16) @ X17))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.86/1.84 inference(negated_conjecture,[status(cth)],[f280])).
% 2.86/1.84 thf(f280,conjecture,(
% 2.86/1.84 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => (inIntersectImpInUnion2 => (inIntersectImpInIntersectUnions => (intersectInPowersetIntersectUnions => (inComplementUnionImpNotIn1 => (inComplementUnionImpInComplement1 => (binunionTE => (binunionTEcontra => (demorgan2a1 => (complementUnionInPowersetComplement => (demorgan2a2 => (demorgan1a => (demorgan1b => (demorgan1 => (demorgan2a => (demorgan2b2 => (demorgan2b => (demorgan2 => (woz13rule0 => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X17 : $i] : ((in @ X17 @ (powerset @ X3)) => ((subset @ X11 @ X17) => (subset @ (binintersect @ X11 @ X16) @ X17)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.86/1.84 file('/export/starexec/sandbox/tmp/tmp.iP8AXdSty9/DTF2THF_22688.p',woz13rule1)).
% 2.86/1.84 thf(f5145,plain,(
% 2.86/1.84 ((subset @ sK875 @ sK877) != $true) | (binintersectLsub != $true)),
% 2.86/1.84 inference(trivial_inequality_removal,[status(thm)],[f5107])).
% 2.86/1.84 thf(f5107,plain,(
% 2.86/1.84 (binintersectLsub != $true) | ((subset @ sK875 @ sK877) != $true) | ($true != $true)),
% 2.86/1.84 inference(superposition,[status(thm)],[f4369,f3677])).
% 2.86/1.84 thf(f3677,plain,(
% 2.86/1.84 ( ! [X0 : $i,X1 : $i] : (((subset @ (binintersect @ X1 @ X0) @ X1) = $true) | (binintersectLsub != $true)) )),
% 2.86/1.84 inference(cnf_transformation,[status(thm)],[f2467])).
% 2.86/1.84 thf(f2467,plain,(
% 2.86/1.84 (! [X0 : $i,X1 : $i] : ((subset @ (binintersect @ X1 @ X0) @ X1) = $true) | (binintersectLsub != $true)) & ((binintersectLsub = $true) | ((subset @ (binintersect @ sK810 @ sK809) @ sK810) != $true))),
% 2.86/1.84 inference(skolemisation,[status(esa),new_symbols(skolem,[sK809,sK810])],[f2465,f2466])).
% 2.86/1.84 thf(f2466,plain,(
% 2.86/1.84 ? [X2 : $i,X3 : $i] : ((subset @ (binintersect @ X3 @ X2) @ X3) != $true) => ((subset @ (binintersect @ sK810 @ sK809) @ sK810) != $true)),
% 2.86/1.84 introduced(definition,[],[choice_axiom])).
% 2.86/1.84 thf(f2465,plain,(
% 2.86/1.84 (! [X0 : $i,X1 : $i] : ((subset @ (binintersect @ X1 @ X0) @ X1) = $true) | (binintersectLsub != $true)) & ((binintersectLsub = $true) | ? [X2 : $i,X3 : $i] : ((subset @ (binintersect @ X3 @ X2) @ X3) != $true))),
% 2.86/1.84 inference(rectify,[status(thm)],[f2464])).
% 2.86/1.84 thf(f2464,plain,(
% 2.86/1.84 (! [X0 : $i,X1 : $i] : ((subset @ (binintersect @ X1 @ X0) @ X1) = $true) | (binintersectLsub != $true)) & ((binintersectLsub = $true) | ? [X0 : $i,X1 : $i] : ((subset @ (binintersect @ X1 @ X0) @ X1) != $true))),
% 2.86/1.84 inference(nnf_transformation,[status(thm)],[f812])).
% 2.86/1.84 thf(f812,plain,(
% 2.86/1.84 ! [X0 : $i,X1 : $i] : ((subset @ (binintersect @ X1 @ X0) @ X1) = $true) <=> (binintersectLsub = $true)),
% 2.86/1.84 inference(fool_elimination,[status(thm)],[f811])).
% 2.86/1.84 thf(f811,plain,(
% 2.86/1.84 (! [X0 : $i,X1 : $i] : (subset @ (binintersect @ X1 @ X0) @ X1) = binintersectLsub)),
% 2.86/1.84 inference(rectify,[status(thm)],[f111])).
% 2.86/1.84 thf(f111,axiom,(
% 2.86/1.84 (! [X4 : $i,X3 : $i] : (subset @ (binintersect @ X3 @ X4) @ X3) = binintersectLsub)),
% 2.86/1.84 file('/export/starexec/sandbox/tmp/tmp.iP8AXdSty9/DTF2THF_22688.p',binintersectLsub)).
% 2.86/1.84 thf(f4369,plain,(
% 2.86/1.84 ( ! [X0 : $i] : (((subset @ (binintersect @ sK875 @ sK876) @ X0) != $true) | ((subset @ X0 @ sK877) != $true)) )),
% 2.86/1.84 inference(trivial_inequality_removal,[status(thm)],[f4368])).
% 2.86/1.84 thf(f4368,plain,(
% 2.86/1.84 ( ! [X0 : $i] : (($true != $true) | ((subset @ (binintersect @ sK875 @ sK876) @ X0) != $true) | ((subset @ X0 @ sK877) != $true)) )),
% 2.86/1.84 inference(forward_demodulation,[status(thm)],[f4361,f3841])).
% 2.86/1.84 thf(f3841,plain,(
% 2.86/1.84 (subsetTrans = $true)),
% 2.86/1.84 inference(cnf_transformation,[status(thm)],[f2572])).
% 2.86/1.84 thf(f4361,plain,(
% 2.86/1.84 ( ! [X0 : $i] : (((subset @ (binintersect @ sK875 @ sK876) @ X0) != $true) | (subsetTrans != $true) | ((subset @ X0 @ sK877) != $true)) )),
% 2.86/1.84 inference(trivial_inequality_removal,[status(thm)],[f4356])).
% 2.86/1.84 thf(f4356,plain,(
% 2.86/1.84 ( ! [X0 : $i] : (($true != $true) | ((subset @ X0 @ sK877) != $true) | ((subset @ (binintersect @ sK875 @ sK876) @ X0) != $true) | (subsetTrans != $true)) )),
% 2.86/1.84 inference(superposition,[status(thm)],[f4020,f3362])).
% 2.86/1.84 thf(f3362,plain,(
% 2.86/1.84 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((subset @ X2 @ X0) = $true) | ((subset @ X1 @ X0) != $true) | (subsetTrans != $true) | ((subset @ X2 @ X1) != $true)) )),
% 2.86/1.84 inference(cnf_transformation,[status(thm)],[f2092])).
% 2.86/1.84 thf(f2092,plain,(
% 2.86/1.84 (! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X0) = $true) | ((subset @ X1 @ X0) != $true) | ((subset @ X2 @ X1) != $true)) | (subsetTrans != $true)) & ((subsetTrans = $true) | (((subset @ sK564 @ sK562) != $true) & ((subset @ sK563 @ sK562) = $true) & ((subset @ sK564 @ sK563) = $true)))),
% 2.86/1.84 inference(skolemisation,[status(esa),new_symbols(skolem,[sK562,sK563,sK564])],[f2090,f2091])).
% 2.86/1.84 thf(f2091,plain,(
% 2.86/1.84 ? [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X5 @ X3) != $true) & ((subset @ X4 @ X3) = $true) & ((subset @ X5 @ X4) = $true)) => (((subset @ sK564 @ sK562) != $true) & ((subset @ sK563 @ sK562) = $true) & ((subset @ sK564 @ sK563) = $true))),
% 2.86/1.84 introduced(definition,[],[choice_axiom])).
% 2.86/1.84 thf(f2090,plain,(
% 2.86/1.84 (! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X0) = $true) | ((subset @ X1 @ X0) != $true) | ((subset @ X2 @ X1) != $true)) | (subsetTrans != $true)) & ((subsetTrans = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X5 @ X3) != $true) & ((subset @ X4 @ X3) = $true) & ((subset @ X5 @ X4) = $true)))),
% 2.86/1.84 inference(rectify,[status(thm)],[f2089])).
% 2.86/1.84 thf(f2089,plain,(
% 2.86/1.84 (! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X0) = $true) | ((subset @ X1 @ X0) != $true) | ((subset @ X2 @ X1) != $true)) | (subsetTrans != $true)) & ((subsetTrans = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X0) != $true) & ((subset @ X1 @ X0) = $true) & ((subset @ X2 @ X1) = $true)))),
% 2.86/1.84 inference(nnf_transformation,[status(thm)],[f1211])).
% 2.86/1.84 thf(f1211,plain,(
% 2.86/1.84 ! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X0) = $true) | ((subset @ X1 @ X0) != $true) | ((subset @ X2 @ X1) != $true)) <=> (subsetTrans = $true)),
% 2.86/1.84 inference(flattening,[status(thm)],[f1210])).
% 2.86/1.84 thf(f1210,plain,(
% 2.86/1.84 (subsetTrans = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : ((((subset @ X2 @ X0) = $true) | ((subset @ X1 @ X0) != $true)) | ((subset @ X2 @ X1) != $true))),
% 2.86/1.84 inference(ennf_transformation,[status(thm)],[f514])).
% 2.86/1.84 thf(f514,plain,(
% 2.86/1.84 (subsetTrans = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((subset @ X2 @ X1) = $true) => (((subset @ X1 @ X0) = $true) => ((subset @ X2 @ X0) = $true)))),
% 2.86/1.84 inference(fool_elimination,[status(thm)],[f513])).
% 2.86/1.84 thf(f513,plain,(
% 2.86/1.84 (subsetTrans = ! [X0 : $i,X1 : $i,X2 : $i] : ((subset @ X2 @ X1) => ((subset @ X1 @ X0) => (subset @ X2 @ X0))))),
% 2.86/1.84 inference(rectify,[status(thm)],[f89])).
% 2.86/1.84 thf(f89,axiom,(
% 2.86/1.84 (subsetTrans = ! [X5 : $i,X4 : $i,X3 : $i] : ((subset @ X3 @ X4) => ((subset @ X4 @ X5) => (subset @ X3 @ X5))))),
% 2.86/1.84 file('/export/starexec/sandbox/tmp/tmp.iP8AXdSty9/DTF2THF_22688.p',subsetTrans)).
% 2.86/1.84 thf(f4020,plain,(
% 2.86/1.84 ((subset @ (binintersect @ sK875 @ sK876) @ sK877) != $true)),
% 2.86/1.84 inference(cnf_transformation,[status(thm)],[f2572])).
% 2.86/1.84 thf(f4473,plain,(
% 2.86/1.84 spl892_10),
% 2.86/1.84 inference(avatar_contradiction_clause,[status(thm)],[f4472])).
% 2.86/1.84 thf(f4472,plain,(
% 2.86/1.84 $false | spl892_10),
% 2.86/1.84 inference(trivial_inequality_removal,[status(thm)],[f4465])).
% 2.86/1.84 thf(f4465,plain,(
% 2.86/1.84 ($true != $true) | spl892_10),
% 2.86/1.84 inference(superposition,[status(thm)],[f4392,f4018])).
% 2.86/1.84 thf(f4018,plain,(
% 2.86/1.84 ((subset @ sK875 @ sK877) = $true)),
% 2.86/1.84 inference(cnf_transformation,[status(thm)],[f2572])).
% 2.86/1.84 thf(f4392,plain,(
% 2.86/1.84 ((subset @ sK875 @ sK877) != $true) | spl892_10),
% 2.86/1.84 inference(avatar_component_clause,[status(thm)],[f4390])).
% 2.86/1.84 % SZS output end Proof for DTF2THF_22688
% 2.86/1.84 % (23039)------------------------------
% 2.86/1.84 % (23039)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.86/1.84 % (23039)Termination reason: Refutation
% 2.86/1.84
% 2.86/1.84 % (23039)Memory used [KB]: 11129
% 2.86/1.84 % (23039)Time elapsed: 0.162 s
% 2.86/1.84 % (23039)Instructions burned: 537 (million)
% 2.86/1.84 % (23039)------------------------------
% 2.86/1.84 % (23039)------------------------------
% 2.86/1.84 % (23016)Success in time 0.253 s
%------------------------------------------------------------------------------