%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU766^1 : 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 : n019.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 09:37:01 AM UTC 2026
% Result : Theorem 2.95s 1.84s
% Output : Refutation 2.95s
% Verified :
% SZS Type : Refutation
% Derivation depth : 18
% Number of leaves : 6
% Syntax : Number of formulae : 35 ( 10 unt; 0 typ; 0 def)
% Number of atoms : 4737 (1254 equ; 0 cnn)
% Maximal formula atoms : 286 ( 135 avg)
% Number of connectives : 2585 ( 65 ~; 41 |; 887 &; 436 @)
% ( 3 <=>;1153 =>; 0 <=; 0 <~>)
% Maximal formula depth : 293 ( 62 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 479 ( 476 usr; 290 con; 0-4 aty)
% Number of variables : 61 ( 0 ^; 39 !; 22 ?; 61 :)
% 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_334,type,
sP0: $i > $i > $o ).
thf(func_def_335,type,
sP1: $i > $o ).
thf(func_def_336,type,
sP2: $i > $i > $o ).
thf(func_def_337,type,
sP3: $i > $i > $o ).
thf(func_def_338,type,
sP4: $o > $i > $i > $i > $o ).
thf(func_def_340,type,
sP6: $i > $i > $i > $o ).
thf(func_def_345,type,
sP11: $i > $i > $i > $i > $o ).
thf(func_def_350,type,
sK16: ( $i > $o ) > $i > $i > $i ).
thf(func_def_353,type,
sK19: $i > $o ).
thf(func_def_360,type,
sK26: $i > ( $i > $o ) > $i ).
thf(func_def_361,type,
sK27: $i > $o ).
thf(func_def_379,type,
sK45: $i > $i ).
thf(func_def_391,type,
sK57: $i > $i ).
thf(func_def_394,type,
sK60: $i > $i > $i ).
thf(func_def_398,type,
sK64: $i > $i > $i ).
thf(func_def_399,type,
sK65: $i > $i > $i ).
thf(func_def_400,type,
sK66: $i > $i > $i ).
thf(func_def_401,type,
sK67: $i > $i > $i ).
thf(func_def_402,type,
sK68: $i > $i > $i > $i ).
thf(func_def_403,type,
sK69: $i > $i ).
thf(func_def_404,type,
sK70: $i > $i ).
thf(func_def_405,type,
sK71: $i > $i ).
thf(func_def_406,type,
sK72: $i > $i ).
thf(func_def_407,type,
sK73: $i > $i > $i ).
thf(func_def_408,type,
sK74: $i > $i > $i > $i ).
thf(func_def_409,type,
sK75: $i > $i > $i > $i ).
thf(func_def_410,type,
sK76: $i > $i > $i ).
thf(func_def_411,type,
sK77: $i > $i > $i ).
thf(func_def_413,type,
sK79: $i > $i ).
thf(func_def_414,type,
sK80: $i > $i ).
thf(func_def_415,type,
sK81: $i > $i ).
thf(func_def_416,type,
sK82: $i > $i > $i ).
thf(func_def_417,type,
sK83: $i > ( $i > $i ) > $i > $i ).
thf(func_def_419,type,
sK85: $i > $i ).
thf(func_def_423,type,
sK89: $i > $i ).
thf(func_def_424,type,
sK90: ( $i > $i ) > $i > $i > $i ).
thf(func_def_434,type,
sK100: $i > $o ).
thf(func_def_435,type,
sK101: $i > $i ).
thf(func_def_436,type,
sK102: ( $i > $o ) > $i ).
thf(func_def_437,type,
sK103: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_438,type,
sK104: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_439,type,
sK105: $i > $o ).
thf(func_def_440,type,
sK106: $i > $o ).
thf(func_def_445,type,
sK111: $i > $o ).
thf(func_def_448,type,
sK114: $i > ( $i > $o ) > $i ).
thf(func_def_460,type,
sK126: $i > $o ).
thf(func_def_475,type,
sK141: $i > $i > $i > $i ).
thf(func_def_476,type,
sK142: $i > $i > $i > $i ).
thf(func_def_509,type,
sK175: $i > $i > $o ).
thf(func_def_517,type,
sK183: $i > $i > $i ).
thf(func_def_539,type,
sK205: $i > $i > $o ).
thf(func_def_540,type,
sK206: $i > $i ).
thf(func_def_541,type,
sK207: $i > $i ).
thf(func_def_542,type,
sK208: ( $i > $i > $o ) > $i > $i ).
thf(func_def_543,type,
sK209: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_544,type,
sK210: ( $i > $i > $o ) > $i > $i ).
thf(func_def_549,type,
sK215: $i > $o ).
thf(func_def_550,type,
sK216: $i > $o ).
thf(func_def_551,type,
sK217: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_552,type,
sK218: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_553,type,
sK219: $i > $i > $i > $i > $i ).
thf(func_def_563,type,
sK229: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_564,type,
sK230: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_568,type,
sK234: $i > $o ).
thf(func_def_576,type,
sK242: $i > $i > $i ).
thf(func_def_584,type,
sK250: $i > $i > $i > $i ).
thf(func_def_588,type,
sK254: $i > $i > $i > $i ).
thf(func_def_602,type,
sK268: $i > $i > $i > $i > $i ).
thf(func_def_613,type,
sK279: $i > $i > $i > $i > $i ).
thf(func_def_614,type,
sK280: $i > $i > $i > $i > $i ).
thf(func_def_632,type,
sK298: $i > $o ).
thf(func_def_652,type,
sK318: $i > $o ).
thf(func_def_657,type,
sK323: $i > $o ).
thf(func_def_685,type,
sK351: $i > $o ).
thf(func_def_686,type,
sK352: $i > $i > $o ).
thf(func_def_718,type,
sK384: $i > $o ).
thf(func_def_745,type,
sK411: $i > $o ).
thf(func_def_746,type,
sK412: $i > $o ).
thf(func_def_769,type,
sK435: $i > $i ).
thf(func_def_788,type,
sK454: $i > $i > $o ).
thf(func_def_805,type,
sK471: $i > $i > $i ).
thf(func_def_806,type,
sK472: $i > $i > $i ).
thf(func_def_813,type,
sK479: $i > $o ).
thf(func_def_815,type,
sK481: ( $i > $o ) > $i > $i ).
thf(func_def_816,type,
sK482: $i > $i > $i > $i ).
thf(func_def_820,type,
sK486: $i > ( $i > $o ) > $i ).
thf(func_def_821,type,
sK487: $i > $o ).
thf(func_def_832,type,
sK498: $i > $i ).
thf(func_def_834,type,
sK500: $i > ( $i > $i ) > $i > $i ).
thf(func_def_836,type,
sK502: $i > $i ).
thf(func_def_856,type,
sK522: $i > $i > $i ).
thf(func_def_859,type,
sK525: $o > $i > $i > $i ).
thf(func_def_864,type,
sK530: $i > $o ).
thf(func_def_866,type,
sK532: ( $i > $o ) > $i > $i ).
thf(func_def_890,type,
sK556: $i > $i > $o ).
thf(func_def_905,type,
sK571: $i > $i > $o ).
thf(func_def_920,type,
sK586: ( $i > $i ) > $i > $i > $i ).
thf(func_def_923,type,
sK589: $i > $i ).
thf(func_def_926,type,
sK592: $i > $i > $o ).
thf(func_def_928,type,
sK594: $i > $o ).
thf(func_def_929,type,
sK595: $i > $o ).
thf(func_def_930,type,
sK596: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_931,type,
sK597: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_933,type,
sK599: $i > $o ).
thf(func_def_935,type,
sK601: $i > ( $i > $o ) > $i ).
thf(func_def_936,type,
sK602: $i > $i ).
thf(func_def_947,type,
sK613: $i > $o ).
thf(func_def_949,type,
sK615: ( $i > $o ) > $i > $i ).
thf(func_def_950,type,
sK616: $i > $o ).
thf(func_def_954,type,
sK620: $i > $o ).
thf(func_def_956,type,
sK622: ( $i > $o ) > $i > $i ).
thf(func_def_957,type,
sK623: ( $i > $o ) > $i > $i ).
thf(func_def_980,type,
sK646: ( $i > $o ) > $i ).
thf(func_def_981,type,
sK647: ( $i > $o ) > $i ).
thf(func_def_982,type,
sK648: $i > $o ).
thf(func_def_1011,type,
sK677: $i > $i > $i ).
thf(func_def_1028,type,
sK694: $i > $i ).
thf(func_def_1030,type,
sK696: $i > $i ).
thf(func_def_1031,type,
sK697: $i > $o ).
thf(func_def_1032,type,
sK698: ( $i > $o ) > $i ).
thf(func_def_1045,type,
sK711: $i > $i > $i ).
thf(func_def_1053,type,
sK719: $i > $i > $i > $i > $i ).
thf(func_def_1106,type,
sK772: $i > $o ).
thf(func_def_1108,type,
sK774: ( $i > $o ) > $i ).
thf(func_def_1109,type,
sK775: $i > $o ).
thf(func_def_1110,type,
sK776: $i > $i ).
thf(func_def_1111,type,
sK777: $i > $o ).
thf(func_def_1134,type,
sK800: $i > $o ).
thf(func_def_1136,type,
sK802: ( $i > $o ) > $i > $i ).
thf(func_def_1137,type,
sK803: ( $i > $o ) > $i > $i ).
thf(func_def_1141,type,
sK807: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1144,type,
sK810: $i > $i ).
thf(func_def_1162,type,
sK828: $i > ( $i > $o ) > $i ).
thf(func_def_1163,type,
sK829: $i > $o ).
thf(func_def_1169,type,
sK835: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1170,type,
sK836: $i > $i ).
thf(func_def_1175,type,
sK841: $i > $i > $i ).
thf(func_def_1176,type,
sK842: $i > $i > $i ).
thf(func_def_1183,type,
sK849: $i > $i > $i > $i > $i ).
thf(func_def_1184,type,
sK850: $i > $i > $i > $i > $i ).
thf(func_def_1185,type,
sK851: $i > $i > $i > $i > $i ).
thf(func_def_1186,type,
sK852: $i > $i > $i > $i > $i ).
thf(func_def_1190,type,
sK856: $i > $i > $o ).
thf(func_def_1192,type,
sK858: $i > ( $i > $o ) > $i ).
thf(func_def_1193,type,
sK859: $i > $o ).
thf(func_def_1196,type,
sK862: $i > $o ).
thf(func_def_1205,type,
sK871: $i > $o ).
thf(func_def_1207,type,
sK873: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1208,type,
sK874: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1218,type,
sK884: $i > $o ).
thf(func_def_1227,type,
sK893: $i > $i > $i > $i ).
thf(func_def_1255,type,
ph921:
!>[X0: $tType] : X0 ).
thf(f4683,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f4672]) ).
thf(f4672,plain,
$true != $true,
inference(superposition,[status(thm)],[f4491,f3731]) ).
thf(f3731,plain,
( ( subset @ sK740 @ ( setminus @ sK739 @ sK741 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f2394]) ).
thf(f2394,plain,
( ( doubleComplementEq = $true )
& ( dpsetconstrER = $true )
& ( funcextLem = $true )
& ( notdallE = $true )
& ( emptysetE = $true )
& ( woz13rule1 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( brelnall2 = $true )
& ( binunionT_lem = $true )
& ( funcinfuncset = $true )
& ( subsetTI = $true )
& ( notdexE = $true )
& ( prop2setE = $true )
& ( eqbreln = $true )
& ( binunionEcases = $true )
& ( emptyinunitempty = $true )
& ( quantDeMorgan3 = $true )
& ( beta2 = $true )
& ( doubleComplementSub1 = $true )
& ( prop2set2propI = $true )
& ( upairsetIL = $true )
& ( contrasubsetT1 = $true )
& ( powersetT_lem = $true )
& ( binintersectI = $true )
& ( setukpairinjL2 = $true )
& ( funcGraphProp1 = $true )
& ( funcGraphProp3 = $true )
& ( omega0Ax = $true )
& ( binintersectEL = $true )
& ( subsetE = $true )
& ( doubleComplementE1 = $true )
& ( subPowSU = $true )
& ( binunionLsub = $true )
& ( nonemptyE1 = $true )
& ( setminusIRneg = $true )
& ( prop2setI = $true )
& ( woz13rule3 = $true )
& ( dpsetconstrERa = $true )
& ( demorgan1a = $true )
& ( woz1_3 = $true )
& ( ubforcartprodlem2 = $true )
& ( emptyset__Cong = $true )
& ( setminusSubset1 = $true )
& ( sepInPowerset = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( eqimpsubset1 = $true )
& ( ap2apEq1 = $true )
& ( vacuousDall = $true )
& ( iftrue = $true )
& ( setminusT_lem = $true )
& ( upairequniteq = $true )
& ( woz13rule0 = $true )
& ( funcext = $true )
& ( emptysetimpfalse = $true )
& ( setunionsingleton2 = $true )
& ( exuE3e = $true )
& ( funcGraphProp4 = $true )
& ( emptyI = $true )
& ( ex1E2 = $true )
& ( setminusLsub = $true )
& ( notequalI1 = $true )
& ( setadjoinAx = $true )
& ( funcImageSingleton = $true )
& ( notinemptyset = $true )
& ( eqimpsubset2 = $true )
& ( notsubsetI = $true )
& ( binunionTILcontra = $true )
& ( demorgan2a2 = $true )
& ( setunionAx = $true )
& ( setadjoin__Cong = $true )
& ( setminusERneg = $true )
& ( setadjoinSub = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( funcext2 = $true )
& ( setunionE = $true )
& ( noeltsimpempty = $true )
& ( cartprodpairsurjEq = $true )
& ( woz1_2 = $true )
& ( doubleComplementSub2 = $true )
& ( setextT = $true )
& ( subsetRefl = $true )
& ( setunionE2 = $true )
& ( cartprodpairin = $true )
& ( woz13rule2 = $true )
& ( setadjoinIL = $true )
& ( setminusSubset2 = $true )
& ( binunionIL = $true )
& ( omegaIndAx = $true )
& ( lam2lamEq = $true )
& ( theeq = $true )
& ( iffalseProp2 = $true )
& ( setOfPairsIsBReln = $true )
& ( binintersectSubset3 = $true )
& ( iftrueorfalse = $true )
& ( inIntersectImpInUnion = $true )
& ( upairsetIR = $true )
& ( setukpairinjR2 = $true )
& ( eta2 = $true )
& ( brelnall1 = $true )
& ( binunionRsub = $true )
& ( funcGraphProp2 = $true )
& ( subsetemptysetimpeq = $true )
& ( setadjoinOr = $true )
& ( lamp = $true )
& ( cartprodpairmemEL = $true )
& ( upairsetE = $true )
& ( upairinpowunion = $true )
& ( complementT_lem = $true )
& ( singletonprop = $true )
& ( kpairsurjEq = $true )
& ( ap2p = $true )
& ( setunionI = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR = $true )
& ( subbreln = $true )
& ( notinsingleton = $true )
& ( upairset2IR = $true )
& ( powersetTE1 = $true )
& ( emptyinPowerset = $true )
& ( in__Cong = $true )
& ( kfstpairEq = $true )
& ( ubforcartprodlem1 = $true )
& ( exuEu = $true )
& ( setadjoinE = $true )
& ( setunion__Cong = $true )
& ( quantDeMorgan4 = $true )
& ( cartprodmempair1 = $true )
& ( demorgan2a = $true )
& ( cartprodmempair = $true )
& ( singletonsswitch = $true )
& ( cartprodfstpairEq = $true )
& ( ubforcartprodlem3 = $true )
& ( eta1 = $true )
& ( dpsetconstrSub = $true )
& ( binintersectSubset4 = $true )
& ( binintersectSubset1 = $true )
& ( setunionsingleton1 = $true )
& ( descr__Cong = $true )
& ( upairset2E = $true )
& ( demorgan1 = $true )
& ( binintersectRsub = $true )
& ( setminusELneg = $true )
& ( sepSubset = $true )
& ( woz1_1 = $true )
& ( setminusER = $true )
& ( nonemptyI1 = $true )
& ( binintersectSubset2 = $true )
& ( dpsetconstrI = $true )
& ( binintersectLsub = $true )
& ( binintersectT_lem = $true )
& ( doubleComplementI1 = $true )
& ( powersetTI1 = $true )
& ( setextAx = $true )
& ( lamProp = $true )
& ( exuE2 = $true )
& ( powersetAx = $true )
& ( cartprodfstin = $true )
& ( uniqinunit = $true )
& ( cartprodsndin = $true )
& ( contrasubsetT = $true )
& ( powersetI = $true )
& ( powerset__Cong = $true )
& ( symdiffE = $true )
& ( setminusEL = $true )
& ( complementImpComplementIntersect = $true )
& ( setukpairIR = $true )
& ( iffalseProp1 = $true )
& ( dpsetconstrEL1 = $true )
& ( exuI3 = $true )
& ( setukpairinjL = $true )
& ( notequalI2 = $true )
& ( secondinupair = $true )
& ( complementTE1 = $true )
& ( setextsub = $true )
& ( binunionTEcontra = $true )
& ( contraSubsetComplement = $true )
& ( dsetconstrEL = $true )
& ( symdiffIneg1 = $true )
& ( descrp = $true )
& ( iftrueProp2 = $true )
& ( infuncsetfunc = $true )
& ( cartprodmempaircEq = $true )
& ( wellorderingAx = $true )
& ( bs114d = $true )
& ( symdiffI2 = $true )
& ( nonemptyI = $true )
& ( binunionIR = $true )
& ( demorgan1b = $true )
& ( emptysetAx = $true )
& ( singletoninpowunion = $true )
& ( singletonsuniq = $true )
& ( iftrueProp1 = $true )
& ( symdiffIneg2 = $true )
& ( ifSingleton = $true )
& ( inCongP = $true )
& ( emptyE1 = $true )
& ( subset2powerset = $true )
& ( setbeta = $true )
& ( omegaSAx = $true )
& ( eqinunit = $true )
& ( setadjoinSub2 = $true )
& ( ifp = $true )
& ( contrasubsetT3 = $true )
& ( inPowerset = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( cartprodpairmemER = $true )
& ( singletoninpowerset = $true )
& ( upairsubunion = $true )
& ( iffalse = $true )
& ( replAx = $true )
& ( setoftrueEq = $true )
& ( ap2apEq2 = $true )
& ( demorgan2a1 = $true )
& ( theprop = $true )
& ( ( subset @ sK741 @ ( setminus @ sK739 @ sK740 ) )
!= $true )
& ( ( subset @ sK740 @ ( setminus @ sK739 @ sK741 ) )
= $true )
& ( $true
= ( in @ sK741 @ ( powerset @ sK739 ) ) )
& ( ( in @ sK740 @ ( powerset @ sK739 ) )
= $true )
& ( exu__Cong = $true )
& ( dsetconstrER = $true )
& ( ex1I2 = $true )
& ( powersetI1 = $true )
& ( dsetconstrI = $true )
& ( binintersectER = $true )
& ( emptysetsubset = $true )
& ( singletonsubset = $true )
& ( quantDeMorgan1 = $true )
& ( exuI2 = $true )
& ( setukpairinjL1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( setunionsingleton = $true )
& ( setukpairIL = $true )
& ( kpairiskpair = $true )
& ( setminusILneg = $true )
& ( powersetE1 = $true )
& ( setukpairinjR11 = $true )
& ( demorgan2b = $true )
& ( demorgan2b2 = $true )
& ( powersetsubset = $true )
& ( nonemptyImpWitness = $true )
& ( setadjoinIR = $true )
& ( binintersectSubset5 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( ex1I = $true )
& ( quantDeMorgan2 = $true )
& ( kfstsingleton = $true )
& ( dsetconstr__Cong = $true )
& ( contrasubsetT2 = $true )
& ( dpsetconstrEL2 = $true )
& ( cartprodsndpairEq = $true )
& ( binunionTE = $true )
& ( binintersectTELcontra = $true )
& ( omega__Cong = $true )
& ( subsetE2 = $true )
& ( binintersectTERcontra = $true )
& ( ex1E1 = $true )
& ( binunionE = $true )
& ( complementTnotintersectT = $true )
& ( exuE1 = $true )
& ( woz13rule4 = $true )
& ( disjointsetsI1 = $true )
& ( complementTcontraSubset = $true )
& ( exuI1 = $true )
& ( complementTI1 = $true )
& ( subsetI1 = $true )
& ( exuE3u = $true )
& ( ksndsingleton = $true )
& ( lam2p = $true )
& ( app = $true )
& ( symdiffI1 = $true )
& ( kpairp = $true )
& ( subsetTrans = $true )
& ( setminusI = $true )
& ( setukpairinjR12 = $true )
& ( demorgan2 = $true )
& ( apProp = $true )
& ( setext = $true )
& ( inIntersectImpInUnion2 = $true )
& ( foundationAx = $true )
& ( emptyInPowerset = $true )
& ( binunionTIRcontra = $true )
& ( powersetE = $true )
& ( beta1 = $true )
& ( ksndpairEq = $true )
& ( subsetI2 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK739,sK740,sK741])],[f998,f2393,f2392]) ).
thf(f2392,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( ( subset @ X2 @ ( setminus @ X0 @ X1 ) )
!= $true )
& ( ( subset @ X1 @ ( setminus @ X0 @ X2 ) )
= $true )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) )
& ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) )
=> ( ? [X2: $i] :
( ( ( subset @ X2 @ ( setminus @ sK739 @ sK740 ) )
!= $true )
& ( ( subset @ sK740 @ ( setminus @ sK739 @ X2 ) )
= $true )
& ( ( in @ X2 @ ( powerset @ sK739 ) )
= $true ) )
& ( ( in @ sK740 @ ( powerset @ sK739 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2393,plain,
( ? [X2: $i] :
( ( ( subset @ X2 @ ( setminus @ sK739 @ sK740 ) )
!= $true )
& ( ( subset @ sK740 @ ( setminus @ sK739 @ X2 ) )
= $true )
& ( ( in @ X2 @ ( powerset @ sK739 ) )
= $true ) )
=> ( ( ( subset @ sK741 @ ( setminus @ sK739 @ sK740 ) )
!= $true )
& ( ( subset @ sK740 @ ( setminus @ sK739 @ sK741 ) )
= $true )
& ( $true
= ( in @ sK741 @ ( powerset @ sK739 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f998,plain,
( ( doubleComplementEq = $true )
& ( dpsetconstrER = $true )
& ( funcextLem = $true )
& ( notdallE = $true )
& ( emptysetE = $true )
& ( woz13rule1 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( brelnall2 = $true )
& ( binunionT_lem = $true )
& ( funcinfuncset = $true )
& ( subsetTI = $true )
& ( notdexE = $true )
& ( prop2setE = $true )
& ( eqbreln = $true )
& ( binunionEcases = $true )
& ( emptyinunitempty = $true )
& ( quantDeMorgan3 = $true )
& ( beta2 = $true )
& ( doubleComplementSub1 = $true )
& ( prop2set2propI = $true )
& ( upairsetIL = $true )
& ( contrasubsetT1 = $true )
& ( powersetT_lem = $true )
& ( binintersectI = $true )
& ( setukpairinjL2 = $true )
& ( funcGraphProp1 = $true )
& ( funcGraphProp3 = $true )
& ( omega0Ax = $true )
& ( binintersectEL = $true )
& ( subsetE = $true )
& ( doubleComplementE1 = $true )
& ( subPowSU = $true )
& ( binunionLsub = $true )
& ( nonemptyE1 = $true )
& ( setminusIRneg = $true )
& ( prop2setI = $true )
& ( woz13rule3 = $true )
& ( dpsetconstrERa = $true )
& ( demorgan1a = $true )
& ( woz1_3 = $true )
& ( ubforcartprodlem2 = $true )
& ( emptyset__Cong = $true )
& ( setminusSubset1 = $true )
& ( sepInPowerset = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( eqimpsubset1 = $true )
& ( ap2apEq1 = $true )
& ( vacuousDall = $true )
& ( iftrue = $true )
& ( setminusT_lem = $true )
& ( upairequniteq = $true )
& ( woz13rule0 = $true )
& ( funcext = $true )
& ( emptysetimpfalse = $true )
& ( setunionsingleton2 = $true )
& ( exuE3e = $true )
& ( funcGraphProp4 = $true )
& ( emptyI = $true )
& ( ex1E2 = $true )
& ( setminusLsub = $true )
& ( notequalI1 = $true )
& ( setadjoinAx = $true )
& ( funcImageSingleton = $true )
& ( notinemptyset = $true )
& ( eqimpsubset2 = $true )
& ( notsubsetI = $true )
& ( binunionTILcontra = $true )
& ( demorgan2a2 = $true )
& ( setunionAx = $true )
& ( setadjoin__Cong = $true )
& ( setminusERneg = $true )
& ( setadjoinSub = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( funcext2 = $true )
& ( setunionE = $true )
& ( noeltsimpempty = $true )
& ( cartprodpairsurjEq = $true )
& ( woz1_2 = $true )
& ( doubleComplementSub2 = $true )
& ( setextT = $true )
& ( subsetRefl = $true )
& ( setunionE2 = $true )
& ( cartprodpairin = $true )
& ( woz13rule2 = $true )
& ( setadjoinIL = $true )
& ( setminusSubset2 = $true )
& ( binunionIL = $true )
& ( omegaIndAx = $true )
& ( lam2lamEq = $true )
& ( theeq = $true )
& ( iffalseProp2 = $true )
& ( setOfPairsIsBReln = $true )
& ( binintersectSubset3 = $true )
& ( iftrueorfalse = $true )
& ( inIntersectImpInUnion = $true )
& ( upairsetIR = $true )
& ( setukpairinjR2 = $true )
& ( eta2 = $true )
& ( brelnall1 = $true )
& ( binunionRsub = $true )
& ( funcGraphProp2 = $true )
& ( subsetemptysetimpeq = $true )
& ( setadjoinOr = $true )
& ( lamp = $true )
& ( cartprodpairmemEL = $true )
& ( upairsetE = $true )
& ( upairinpowunion = $true )
& ( complementT_lem = $true )
& ( singletonprop = $true )
& ( kpairsurjEq = $true )
& ( ap2p = $true )
& ( setunionI = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR = $true )
& ( subbreln = $true )
& ( notinsingleton = $true )
& ( upairset2IR = $true )
& ( powersetTE1 = $true )
& ( emptyinPowerset = $true )
& ( in__Cong = $true )
& ( kfstpairEq = $true )
& ( ubforcartprodlem1 = $true )
& ( exuEu = $true )
& ( setadjoinE = $true )
& ( setunion__Cong = $true )
& ( quantDeMorgan4 = $true )
& ( cartprodmempair1 = $true )
& ( demorgan2a = $true )
& ( cartprodmempair = $true )
& ( singletonsswitch = $true )
& ( cartprodfstpairEq = $true )
& ( ubforcartprodlem3 = $true )
& ( eta1 = $true )
& ( dpsetconstrSub = $true )
& ( binintersectSubset4 = $true )
& ( binintersectSubset1 = $true )
& ( setunionsingleton1 = $true )
& ( descr__Cong = $true )
& ( upairset2E = $true )
& ( demorgan1 = $true )
& ( binintersectRsub = $true )
& ( setminusELneg = $true )
& ( sepSubset = $true )
& ( woz1_1 = $true )
& ( setminusER = $true )
& ( nonemptyI1 = $true )
& ( binintersectSubset2 = $true )
& ( dpsetconstrI = $true )
& ( binintersectLsub = $true )
& ( binintersectT_lem = $true )
& ( doubleComplementI1 = $true )
& ( powersetTI1 = $true )
& ( setextAx = $true )
& ( lamProp = $true )
& ( exuE2 = $true )
& ( powersetAx = $true )
& ( cartprodfstin = $true )
& ( uniqinunit = $true )
& ( cartprodsndin = $true )
& ( contrasubsetT = $true )
& ( powersetI = $true )
& ( powerset__Cong = $true )
& ( symdiffE = $true )
& ( setminusEL = $true )
& ( complementImpComplementIntersect = $true )
& ( setukpairIR = $true )
& ( iffalseProp1 = $true )
& ( dpsetconstrEL1 = $true )
& ( exuI3 = $true )
& ( setukpairinjL = $true )
& ( notequalI2 = $true )
& ( secondinupair = $true )
& ( complementTE1 = $true )
& ( setextsub = $true )
& ( binunionTEcontra = $true )
& ( contraSubsetComplement = $true )
& ( dsetconstrEL = $true )
& ( symdiffIneg1 = $true )
& ( descrp = $true )
& ( iftrueProp2 = $true )
& ( infuncsetfunc = $true )
& ( cartprodmempaircEq = $true )
& ( wellorderingAx = $true )
& ( bs114d = $true )
& ( symdiffI2 = $true )
& ( nonemptyI = $true )
& ( binunionIR = $true )
& ( demorgan1b = $true )
& ( emptysetAx = $true )
& ( singletoninpowunion = $true )
& ( singletonsuniq = $true )
& ( iftrueProp1 = $true )
& ( symdiffIneg2 = $true )
& ( ifSingleton = $true )
& ( inCongP = $true )
& ( emptyE1 = $true )
& ( subset2powerset = $true )
& ( setbeta = $true )
& ( omegaSAx = $true )
& ( eqinunit = $true )
& ( setadjoinSub2 = $true )
& ( ifp = $true )
& ( contrasubsetT3 = $true )
& ( inPowerset = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( cartprodpairmemER = $true )
& ( singletoninpowerset = $true )
& ( upairsubunion = $true )
& ( iffalse = $true )
& ( replAx = $true )
& ( setoftrueEq = $true )
& ( ap2apEq2 = $true )
& ( demorgan2a1 = $true )
& ( theprop = $true )
& ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( ( subset @ X2 @ ( setminus @ X0 @ X1 ) )
!= $true )
& ( ( subset @ X1 @ ( setminus @ X0 @ X2 ) )
= $true )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) )
& ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) )
& ( exu__Cong = $true )
& ( dsetconstrER = $true )
& ( ex1I2 = $true )
& ( powersetI1 = $true )
& ( dsetconstrI = $true )
& ( binintersectER = $true )
& ( emptysetsubset = $true )
& ( singletonsubset = $true )
& ( quantDeMorgan1 = $true )
& ( exuI2 = $true )
& ( setukpairinjL1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( setunionsingleton = $true )
& ( setukpairIL = $true )
& ( kpairiskpair = $true )
& ( setminusILneg = $true )
& ( powersetE1 = $true )
& ( setukpairinjR11 = $true )
& ( demorgan2b = $true )
& ( demorgan2b2 = $true )
& ( powersetsubset = $true )
& ( nonemptyImpWitness = $true )
& ( setadjoinIR = $true )
& ( binintersectSubset5 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( ex1I = $true )
& ( quantDeMorgan2 = $true )
& ( kfstsingleton = $true )
& ( dsetconstr__Cong = $true )
& ( contrasubsetT2 = $true )
& ( dpsetconstrEL2 = $true )
& ( cartprodsndpairEq = $true )
& ( binunionTE = $true )
& ( binintersectTELcontra = $true )
& ( omega__Cong = $true )
& ( subsetE2 = $true )
& ( binintersectTERcontra = $true )
& ( ex1E1 = $true )
& ( binunionE = $true )
& ( complementTnotintersectT = $true )
& ( exuE1 = $true )
& ( woz13rule4 = $true )
& ( disjointsetsI1 = $true )
& ( complementTcontraSubset = $true )
& ( exuI1 = $true )
& ( complementTI1 = $true )
& ( subsetI1 = $true )
& ( exuE3u = $true )
& ( ksndsingleton = $true )
& ( lam2p = $true )
& ( app = $true )
& ( symdiffI1 = $true )
& ( kpairp = $true )
& ( subsetTrans = $true )
& ( setminusI = $true )
& ( setukpairinjR12 = $true )
& ( demorgan2 = $true )
& ( apProp = $true )
& ( setext = $true )
& ( inIntersectImpInUnion2 = $true )
& ( foundationAx = $true )
& ( emptyInPowerset = $true )
& ( binunionTIRcontra = $true )
& ( powersetE = $true )
& ( beta1 = $true )
& ( ksndpairEq = $true )
& ( subsetI2 = $true ) ),
inference(flattening,[status(thm)],[f997]) ).
thf(f997,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( ( subset @ X2 @ ( setminus @ X0 @ X1 ) )
!= $true )
& ( ( subset @ X1 @ ( setminus @ X0 @ X2 ) )
= $true )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) )
& ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) )
& ( woz1_3 = $true )
& ( woz1_2 = $true )
& ( woz1_1 = $true )
& ( woz13rule4 = $true )
& ( woz13rule3 = $true )
& ( woz13rule2 = $true )
& ( woz13rule1 = $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)],[f626]) ).
thf(f626,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 )
=> ( ( woz13rule1 = $true )
=> ( ( woz13rule2 = $true )
=> ( ( woz13rule3 = $true )
=> ( ( woz13rule4 = $true )
=> ( ( woz1_1 = $true )
=> ( ( woz1_2 = $true )
=> ( ( woz1_3 = $true )
=> ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
=> ( ( ( subset @ X1 @ ( setminus @ X0 @ X2 ) )
= $true )
=> ( ( subset @ X2 @ ( setminus @ X0 @ X1 ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f625]) ).
thf(f625,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
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X0 ) )
=> ( ( subset @ X1 @ ( setminus @ X0 @ X2 ) )
=> ( subset @ X2 @ ( setminus @ X0 @ X1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f288]) ).
thf(f288,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
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( ( subset @ X11 @ ( setminus @ X3 @ X16 ) )
=> ( subset @ X16 @ ( setminus @ X3 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f287]) ).
thf(f287,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
=> ( woz13rule1
=> ( woz13rule2
=> ( woz13rule3
=> ( woz13rule4
=> ( woz1_1
=> ( woz1_2
=> ( woz1_3
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( ( subset @ X11 @ ( setminus @ X3 @ X16 ) )
=> ( subset @ X16 @ ( setminus @ X3 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.WxwUZkQ1Ii/DTF2THF_28543.p',woz1_4) ).
thf(f4491,plain,
( ( subset @ sK740 @ ( setminus @ sK739 @ sK741 ) )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f4490]) ).
thf(f4490,plain,
( ( $true != $true )
| ( ( subset @ sK740 @ ( setminus @ sK739 @ sK741 ) )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f4489,f3729]) ).
thf(f3729,plain,
( ( in @ sK740 @ ( powerset @ sK739 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f2394]) ).
thf(f4489,plain,
( ( ( subset @ sK740 @ ( setminus @ sK739 @ sK741 ) )
!= $true )
| ( ( in @ sK740 @ ( powerset @ sK739 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4488]) ).
thf(f4488,plain,
( ( ( in @ sK740 @ ( powerset @ sK739 ) )
!= $true )
| ( ( subset @ sK740 @ ( setminus @ sK739 @ sK741 ) )
!= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f4487,f3685]) ).
thf(f3685,plain,
complementTcontraSubset = $true,
inference(cnf_transformation,[status(thm)],[f2394]) ).
thf(f4487,plain,
( ( complementTcontraSubset != $true )
| ( ( in @ sK740 @ ( powerset @ sK739 ) )
!= $true )
| ( ( subset @ sK740 @ ( setminus @ sK739 @ sK741 ) )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4486]) ).
thf(f4486,plain,
( ( $true != $true )
| ( complementTcontraSubset != $true )
| ( ( in @ sK740 @ ( powerset @ sK739 ) )
!= $true )
| ( ( subset @ sK740 @ ( setminus @ sK739 @ sK741 ) )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f4476,f3730]) ).
thf(f3730,plain,
( $true
= ( in @ sK741 @ ( powerset @ sK739 ) ) ),
inference(cnf_transformation,[status(thm)],[f2394]) ).
thf(f4476,plain,
( ( $true
!= ( in @ sK741 @ ( powerset @ sK739 ) ) )
| ( ( in @ sK740 @ ( powerset @ sK739 ) )
!= $true )
| ( ( subset @ sK740 @ ( setminus @ sK739 @ sK741 ) )
!= $true )
| ( complementTcontraSubset != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4470]) ).
thf(f4470,plain,
( ( complementTcontraSubset != $true )
| ( $true
!= ( in @ sK741 @ ( powerset @ sK739 ) ) )
| ( ( subset @ sK740 @ ( setminus @ sK739 @ sK741 ) )
!= $true )
| ( ( in @ sK740 @ ( powerset @ sK739 ) )
!= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f3732,f3151]) ).
thf(f3151,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( subset @ X2 @ ( setminus @ X0 @ X1 ) )
= $true )
| ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true )
| ( ( subset @ X1 @ ( setminus @ X0 @ X2 ) )
!= $true )
| ( complementTcontraSubset != $true )
| ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f1810]) ).
thf(f1810,plain,
( ( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) )
| ( ( subset @ X1 @ ( setminus @ X0 @ X2 ) )
!= $true )
| ( ( subset @ X2 @ ( setminus @ X0 @ X1 ) )
= $true ) )
| ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true ) )
| ( complementTcontraSubset != $true ) )
& ( ( complementTcontraSubset = $true )
| ( ( ( in @ sK361 @ ( powerset @ sK359 ) )
= $true )
& ( ( subset @ sK360 @ ( setminus @ sK359 @ sK361 ) )
= $true )
& ( ( subset @ sK361 @ ( setminus @ sK359 @ sK360 ) )
!= $true )
& ( ( in @ sK360 @ ( powerset @ sK359 ) )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK359,sK360,sK361])],[f1807,f1809,f1808]) ).
thf(f1808,plain,
( ? [X3: $i,X4: $i] :
( ? [X5: $i] :
( ( ( in @ X5 @ ( powerset @ X3 ) )
= $true )
& ( ( subset @ X4 @ ( setminus @ X3 @ X5 ) )
= $true )
& ( ( subset @ X5 @ ( setminus @ X3 @ X4 ) )
!= $true ) )
& ( ( in @ X4 @ ( powerset @ X3 ) )
= $true ) )
=> ( ? [X5: $i] :
( ( ( in @ X5 @ ( powerset @ sK359 ) )
= $true )
& ( ( subset @ sK360 @ ( setminus @ sK359 @ X5 ) )
= $true )
& ( ( subset @ X5 @ ( setminus @ sK359 @ sK360 ) )
!= $true ) )
& ( ( in @ sK360 @ ( powerset @ sK359 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1809,plain,
( ? [X5: $i] :
( ( ( in @ X5 @ ( powerset @ sK359 ) )
= $true )
& ( ( subset @ sK360 @ ( setminus @ sK359 @ X5 ) )
= $true )
& ( ( subset @ X5 @ ( setminus @ sK359 @ sK360 ) )
!= $true ) )
=> ( ( ( in @ sK361 @ ( powerset @ sK359 ) )
= $true )
& ( ( subset @ sK360 @ ( setminus @ sK359 @ sK361 ) )
= $true )
& ( ( subset @ sK361 @ ( setminus @ sK359 @ sK360 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1807,plain,
( ( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) )
| ( ( subset @ X1 @ ( setminus @ X0 @ X2 ) )
!= $true )
| ( ( subset @ X2 @ ( setminus @ X0 @ X1 ) )
= $true ) )
| ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true ) )
| ( complementTcontraSubset != $true ) )
& ( ( complementTcontraSubset = $true )
| ? [X3: $i,X4: $i] :
( ? [X5: $i] :
( ( ( in @ X5 @ ( powerset @ X3 ) )
= $true )
& ( ( subset @ X4 @ ( setminus @ X3 @ X5 ) )
= $true )
& ( ( subset @ X5 @ ( setminus @ X3 @ X4 ) )
!= $true ) )
& ( ( in @ X4 @ ( powerset @ X3 ) )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f1806]) ).
thf(f1806,plain,
( ( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) )
| ( ( subset @ X1 @ ( setminus @ X0 @ X2 ) )
!= $true )
| ( ( subset @ X2 @ ( setminus @ X0 @ X1 ) )
= $true ) )
| ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true ) )
| ( complementTcontraSubset != $true ) )
& ( ( complementTcontraSubset = $true )
| ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
& ( ( subset @ X1 @ ( setminus @ X0 @ X2 ) )
= $true )
& ( ( subset @ X2 @ ( setminus @ X0 @ X1 ) )
!= $true ) )
& ( ( in @ X1 @ ( powerset @ X0 ) )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1002]) ).
thf(f1002,plain,
( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) )
| ( ( subset @ X1 @ ( setminus @ X0 @ X2 ) )
!= $true )
| ( ( subset @ X2 @ ( setminus @ X0 @ X1 ) )
= $true ) )
| ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true ) )
<=> ( complementTcontraSubset = $true ) ),
inference(flattening,[status(thm)],[f1001]) ).
thf(f1001,plain,
( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( ( subset @ X2 @ ( setminus @ X0 @ X1 ) )
= $true )
| ( ( subset @ X1 @ ( setminus @ X0 @ X2 ) )
!= $true )
| ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) ) )
| ( ( in @ X1 @ ( powerset @ X0 ) )
!= $true ) )
<=> ( complementTcontraSubset = $true ) ),
inference(ennf_transformation,[status(thm)],[f502]) ).
thf(f502,plain,
( ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
=> ( ( ( subset @ X1 @ ( setminus @ X0 @ X2 ) )
= $true )
=> ( ( subset @ X2 @ ( setminus @ X0 @ X1 ) )
= $true ) ) ) )
<=> ( complementTcontraSubset = $true ) ),
inference(fool_elimination,[status(thm)],[f501]) ).
thf(f501,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X0 ) )
=> ( ( subset @ X1 @ ( setminus @ X0 @ X2 ) )
=> ( subset @ X2 @ ( setminus @ X0 @ X1 ) ) ) ) )
= complementTcontraSubset ),
inference(rectify,[status(thm)],[f258]) ).
thf(f258,axiom,
! [X3: $i,X11: $i] :
( ( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( ( subset @ X11 @ ( setminus @ X3 @ X16 ) )
=> ( subset @ X16 @ ( setminus @ X3 @ X11 ) ) ) ) )
= complementTcontraSubset ),
file('/export/starexec/sandbox/tmp/tmp.WxwUZkQ1Ii/DTF2THF_28543.p',complementTcontraSubset) ).
thf(f3732,plain,
( ( subset @ sK741 @ ( setminus @ sK739 @ sK740 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f2394]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU766^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n019.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:33:24 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.30/0.46 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : SEU766^1 : TPTP v9.2.1. Released v3.7.0.
% 0.30/0.46 % Domain : Set Theory
% 0.30/0.46 % Problem : Typed Set Theory - First Wizard of Oz Examples - WoZ1 Problems
% 0.30/0.46 % Version : Especial.
% 0.30/0.46 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.30/0.46 % subset X (setminus A Y) -> subset Y (setminus A X)))
% 0.30/0.46
% 0.30/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.30/0.46 % Source : [Bro08]
% 0.30/0.46 % Names : ZFC268g [Bro08]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.11 v9.1.0, 0.25 v9.0.0, 0.20 v8.2.0, 0.23 v8.1.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.2.0, 0.60 v5.1.0, 0.80 v5.0.0, 0.60 v4.1.0, 0.67 v3.7.0
% 0.30/0.46 % Syntax : Number of formulae : 607 ( 286 unt; 320 typ; 286 def)
% 0.30/0.46 % Number of atoms : 1712 ( 415 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 286 ( 5 avg)
% 0.30/0.46 % Number of connectives : 3657 ( 73 ~; 12 |; 44 &;2648 @)
% 0.30/0.46 % ( 14 <=>; 866 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 292 ( 2 avg)
% 0.30/0.46 % Number of types : 2 ( 0 usr)
% 0.30/0.46 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 323 ( 320 usr; 286 con; 0-4 aty)
% 0.30/0.46 % Number of variables : 980 ( 74 ^; 871 !; 35 ?; 980 :)
% 0.30/0.46 % SPC : TH0_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : http://mathgate.info/detsetitem.php?id=327
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 thf(in_type,type,
% 0.30/0.46 in: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu_type,type,
% 0.30/0.46 exu: ( $i > $o ) > $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu,definition,
% 0.30/0.46 ( exu
% 0.30/0.46 = ( ^ [Xphi: $i > $o] :
% 0.30/0.46 ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextAx_type,type,
% 0.30/0.46 setextAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextAx,definition,
% 0.30/0.46 ( setextAx
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 <=> ( in @ Xx @ B ) )
% 0.30/0.46 => ( A = B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset_type,type,
% 0.30/0.46 emptyset: $i ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetAx_type,type,
% 0.30/0.46 emptysetAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetAx,definition,
% 0.30/0.46 ( emptysetAx
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin_type,type,
% 0.30/0.46 setadjoin: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinAx_type,type,
% 0.30/0.46 setadjoinAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinAx,definition,
% 0.30/0.46 ( setadjoinAx
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 <=> ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powerset_type,type,
% 0.30/0.46 powerset: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(powersetAx_type,type,
% 0.30/0.46 powersetAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetAx,definition,
% 0.30/0.46 ( powersetAx
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 <=> ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunion_type,type,
% 0.30/0.46 setunion: $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setunionAx_type,type,
% 0.30/0.46 setunionAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionAx,definition,
% 0.30/0.46 ( setunionAx
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 <=> ? [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omega_type,type,
% 0.30/0.46 omega: $i ).
% 0.30/0.46
% 0.30/0.46 thf(omega0Ax_type,type,
% 0.30/0.46 omega0Ax: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omega0Ax,definition,
% 0.30/0.46 ( omega0Ax
% 0.30/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omegaSAx_type,type,
% 0.30/0.46 omegaSAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omegaSAx,definition,
% 0.30/0.46 ( omegaSAx
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ omega )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omegaIndAx_type,type,
% 0.30/0.46 omegaIndAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omegaIndAx,definition,
% 0.30/0.46 ( omegaIndAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( ( in @ emptyset @ A )
% 0.30/0.46 & ! [Xx: $i] :
% 0.30/0.46 ( ( ( in @ Xx @ omega )
% 0.30/0.46 & ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ omega )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(replAx_type,type,
% 0.30/0.46 replAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(replAx,definition,
% 0.30/0.46 ( replAx
% 0.30/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.30/0.46 => ? [B: $i] :
% 0.30/0.46 ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 <=> ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(foundationAx_type,type,
% 0.30/0.46 foundationAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(foundationAx,definition,
% 0.30/0.46 ( foundationAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ? [B: $i] :
% 0.30/0.46 ( ( in @ B @ A )
% 0.30/0.46 & ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(wellorderingAx_type,type,
% 0.30/0.46 wellorderingAx: $o ).
% 0.30/0.46
% 0.30/0.46 thf(wellorderingAx,definition,
% 0.30/0.46 ( wellorderingAx
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ? [B: $i] :
% 0.30/0.46 ( ! [C: $i] :
% 0.30/0.46 ( ( in @ C @ B )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ A ) ) )
% 0.30/0.46 & ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( ( in @ Xx @ A )
% 0.30/0.46 & ( in @ Xy @ A ) )
% 0.30/0.46 => ( ! [C: $i] :
% 0.30/0.46 ( ( in @ C @ B )
% 0.30/0.46 => ( ( in @ Xx @ C )
% 0.30/0.46 <=> ( in @ Xy @ C ) ) )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 & ! [C: $i,D: $i] :
% 0.30/0.46 ( ( ( in @ C @ B )
% 0.30/0.46 & ( in @ D @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ D ) )
% 0.30/0.46 | ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ D )
% 0.30/0.46 => ( in @ Xx @ C ) ) ) )
% 0.30/0.46 & ! [C: $i] :
% 0.30/0.46 ( ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ C )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.30/0.46 => ? [D: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ D @ B )
% 0.30/0.46 & ( in @ Xx @ C )
% 0.30/0.46 & ~ ? [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ D )
% 0.30/0.46 & ( in @ Xy @ C ) )
% 0.30/0.46 & ! [E: $i] :
% 0.30/0.46 ( ( in @ E @ B )
% 0.30/0.46 => ( ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ E )
% 0.30/0.46 => ( in @ Xy @ D ) )
% 0.30/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(descr_type,type,
% 0.30/0.46 descr: ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(descrp_type,type,
% 0.30/0.46 descrp: $o ).
% 0.30/0.46
% 0.30/0.46 thf(descrp,definition,
% 0.30/0.46 ( descrp
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ( Xphi
% 0.30/0.46 @ ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr_type,type,
% 0.30/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrI_type,type,
% 0.30/0.46 dsetconstrI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrI,definition,
% 0.30/0.46 ( dsetconstrI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrEL_type,type,
% 0.30/0.46 dsetconstrEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrEL,definition,
% 0.30/0.46 ( dsetconstrEL
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrER_type,type,
% 0.30/0.46 dsetconstrER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstrER,definition,
% 0.30/0.46 ( dsetconstrER
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE1_type,type,
% 0.30/0.46 exuE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE1,definition,
% 0.30/0.46 ( exuE1
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set_type,type,
% 0.30/0.46 prop2set: $o > $i ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setE_type,type,
% 0.30/0.46 prop2setE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setE,definition,
% 0.30/0.46 ( prop2setE
% 0.30/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetE_type,type,
% 0.30/0.46 emptysetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetE,definition,
% 0.30/0.46 ( emptysetE
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetimpfalse_type,type,
% 0.30/0.46 emptysetimpfalse: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetimpfalse,definition,
% 0.30/0.46 ( emptysetimpfalse
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => $false ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notinemptyset_type,type,
% 0.30/0.46 notinemptyset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notinemptyset,definition,
% 0.30/0.46 ( notinemptyset
% 0.30/0.46 = ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3e_type,type,
% 0.30/0.46 exuE3e: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3e,definition,
% 0.30/0.46 ( exuE3e
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setext_type,type,
% 0.30/0.46 setext: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setext,definition,
% 0.30/0.46 ( setext
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( A = B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyI_type,type,
% 0.30/0.46 emptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyI,definition,
% 0.30/0.46 ( emptyI
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ A )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(noeltsimpempty_type,type,
% 0.30/0.46 noeltsimpempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(noeltsimpempty,definition,
% 0.30/0.46 ( noeltsimpempty
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ~ ( in @ Xx @ A )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setbeta_type,type,
% 0.30/0.46 setbeta: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setbeta,definition,
% 0.30/0.46 ( setbeta
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.30/0.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonempty_type,type,
% 0.30/0.46 nonempty: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonempty,definition,
% 0.30/0.46 ( nonempty
% 0.30/0.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1_type,type,
% 0.30/0.46 nonemptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyE1,definition,
% 0.30/0.46 ( nonemptyE1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI_type,type,
% 0.30/0.46 nonemptyI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI,definition,
% 0.30/0.46 ( nonemptyI
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 => ( nonempty
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1_type,type,
% 0.30/0.46 nonemptyI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyI1,definition,
% 0.30/0.46 ( nonemptyI1
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.30/0.46 => ( nonempty @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL_type,type,
% 0.30/0.46 setadjoinIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIL,definition,
% 0.30/0.46 ( setadjoinIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty_type,type,
% 0.30/0.46 emptyinunitempty: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinunitempty,definition,
% 0.30/0.46 ( emptyinunitempty
% 0.30/0.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR_type,type,
% 0.30/0.46 setadjoinIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinIR,definition,
% 0.30/0.46 ( setadjoinIR
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ A )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE_type,type,
% 0.30/0.46 setadjoinE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinE,definition,
% 0.30/0.46 ( setadjoinE
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ( ( Xy = Xx )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xy @ A )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr_type,type,
% 0.30/0.46 setadjoinOr: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinOr,definition,
% 0.30/0.46 ( setadjoinOr
% 0.30/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.30/0.46 => ( ( Xy = Xx )
% 0.30/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq_type,type,
% 0.30/0.46 setoftrueEq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setoftrueEq,definition,
% 0.30/0.46 ( setoftrueEq
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : $true )
% 0.30/0.46 = A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI_type,type,
% 0.30/0.46 powersetI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI,definition,
% 0.30/0.46 ( powersetI
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) )
% 0.30/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset_type,type,
% 0.30/0.46 emptyinPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyinPowerset,definition,
% 0.30/0.46 ( emptyinPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset_type,type,
% 0.30/0.46 emptyInPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyInPowerset,definition,
% 0.30/0.46 ( emptyInPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE_type,type,
% 0.30/0.46 powersetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE,definition,
% 0.30/0.46 ( powersetE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI_type,type,
% 0.30/0.46 setunionI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionI,definition,
% 0.30/0.46 ( setunionI
% 0.30/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE_type,type,
% 0.30/0.46 setunionE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunionE,definition,
% 0.30/0.46 ( setunionE
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ! [B: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( ( in @ B @ A )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU_type,type,
% 0.30/0.46 subPowSU: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subPowSU,definition,
% 0.30/0.46 ( subPowSU
% 0.30/0.46 = ( ! [A: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2_type,type,
% 0.30/0.46 exuE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE2,definition,
% 0.30/0.46 ( exuE2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness_type,type,
% 0.30/0.46 nonemptyImpWitness: $o ).
% 0.30/0.46
% 0.30/0.46 thf(nonemptyImpWitness,definition,
% 0.30/0.46 ( nonemptyImpWitness
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( nonempty @ A )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & $true ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(uniqinunit_type,type,
% 0.30/0.46 uniqinunit: $o ).
% 0.30/0.46
% 0.30/0.46 thf(uniqinunit,definition,
% 0.30/0.46 ( uniqinunit
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notinsingleton_type,type,
% 0.30/0.46 notinsingleton: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notinsingleton,definition,
% 0.30/0.46 ( notinsingleton
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx != Xy )
% 0.30/0.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqinunit_type,type,
% 0.30/0.46 eqinunit: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqinunit,definition,
% 0.30/0.46 ( eqinunit
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsswitch_type,type,
% 0.30/0.46 singletonsswitch: $o ).
% 0.30/0.46
% 0.30/0.46 thf(singletonsswitch,definition,
% 0.30/0.46 ( singletonsswitch
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetE_type,type,
% 0.30/0.46 upairsetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetE,definition,
% 0.30/0.46 ( upairsetE
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.46 => ( ( Xz = Xx )
% 0.30/0.46 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIL_type,type,
% 0.30/0.46 upairsetIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIL,definition,
% 0.30/0.46 ( upairsetIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIR_type,type,
% 0.30/0.46 upairsetIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairsetIR,definition,
% 0.30/0.46 ( upairsetIR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyE1_type,type,
% 0.30/0.46 emptyE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyE1,definition,
% 0.30/0.46 ( emptyE1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ( ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = emptyset )
% 0.30/0.46 => $false ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(vacuousDall_type,type,
% 0.30/0.46 vacuousDall: $o ).
% 0.30/0.46
% 0.30/0.46 thf(vacuousDall,definition,
% 0.30/0.46 ( vacuousDall
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ emptyset )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan1_type,type,
% 0.30/0.46 quantDeMorgan1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan1,definition,
% 0.30/0.46 ( quantDeMorgan1
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan2_type,type,
% 0.30/0.46 quantDeMorgan2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan2,definition,
% 0.30/0.46 ( quantDeMorgan2
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) )
% 0.30/0.46 => ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan3_type,type,
% 0.30/0.46 quantDeMorgan3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan3,definition,
% 0.30/0.46 ( quantDeMorgan3
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan4_type,type,
% 0.30/0.46 quantDeMorgan4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(quantDeMorgan4,definition,
% 0.30/0.46 ( quantDeMorgan4
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) )
% 0.30/0.46 => ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setI_type,type,
% 0.30/0.46 prop2setI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2setI,definition,
% 0.30/0.46 ( prop2setI
% 0.30/0.46 = ( ! [Xphi: $o] :
% 0.30/0.46 ( Xphi
% 0.30/0.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(set2prop_type,type,
% 0.30/0.46 set2prop: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set2propI_type,type,
% 0.30/0.46 prop2set2propI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(prop2set2propI,definition,
% 0.30/0.46 ( prop2set2propI
% 0.30/0.46 = ( ! [Xphi: $o] :
% 0.30/0.46 ( Xphi
% 0.30/0.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notdexE_type,type,
% 0.30/0.46 notdexE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notdexE,definition,
% 0.30/0.46 ( notdexE
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notdallE_type,type,
% 0.30/0.46 notdallE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notdallE,definition,
% 0.30/0.46 ( notdallE
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( ~ ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( Xphi @ Xx ) )
% 0.30/0.46 => ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI1_type,type,
% 0.30/0.46 exuI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI1,definition,
% 0.30/0.46 ( exuI1
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 & ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI3_type,type,
% 0.30/0.46 exuI3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI3,definition,
% 0.30/0.46 ( exuI3
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.30/0.46 => ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuI2_type,type,
% 0.30/0.46 exuI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuI2,definition,
% 0.30/0.46 ( exuI2
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ? [Xx: $i] :
% 0.30/0.46 ! [Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xy )
% 0.30/0.46 <=> ( Xy = Xx ) )
% 0.30/0.46 => ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inCongP_type,type,
% 0.30/0.46 inCongP: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inCongP,definition,
% 0.30/0.46 ( inCongP
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(in__Cong_type,type,
% 0.30/0.46 in__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(in__Cong,definition,
% 0.30/0.46 ( in__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3u_type,type,
% 0.30/0.46 exuE3u: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuE3u,definition,
% 0.30/0.46 ( exuE3u
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exu__Cong_type,type,
% 0.30/0.46 exu__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exu__Cong,definition,
% 0.30/0.46 ( exu__Cong
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 <=> ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset__Cong_type,type,
% 0.30/0.46 emptyset__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptyset__Cong,definition,
% 0.30/0.46 ( emptyset__Cong
% 0.30/0.46 = ( emptyset = emptyset ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin__Cong_type,type,
% 0.30/0.46 setadjoin__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoin__Cong,definition,
% 0.30/0.46 ( setadjoin__Cong
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ! [Xz: $i,Xu: $i] :
% 0.30/0.46 ( ( Xz = Xu )
% 0.30/0.46 => ( ( setadjoin @ Xx @ Xz )
% 0.30/0.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powerset__Cong_type,type,
% 0.30/0.46 powerset__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powerset__Cong,definition,
% 0.30/0.46 ( powerset__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( powerset @ A )
% 0.30/0.46 = ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setunion__Cong_type,type,
% 0.30/0.46 setunion__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setunion__Cong,definition,
% 0.30/0.46 ( setunion__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( ( setunion @ A )
% 0.30/0.46 = ( setunion @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(omega__Cong_type,type,
% 0.30/0.46 omega__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(omega__Cong,definition,
% 0.30/0.46 ( omega__Cong
% 0.30/0.46 = ( omega = omega ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(exuEu_type,type,
% 0.30/0.46 exuEu: $o ).
% 0.30/0.46
% 0.30/0.46 thf(exuEu,definition,
% 0.30/0.46 ( exuEu
% 0.30/0.46 = ( ! [Xphi: $i > $o] :
% 0.30/0.46 ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xphi @ Xx )
% 0.30/0.46 => ( ( Xphi @ Xy )
% 0.30/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(descr__Cong_type,type,
% 0.30/0.46 descr__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(descr__Cong,definition,
% 0.30/0.46 ( descr__Cong
% 0.30/0.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i,Xy: $i] :
% 0.30/0.46 ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 => ( ( exu
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.30/0.46 => ( ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = ( descr
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr__Cong_type,type,
% 0.30/0.46 dsetconstr__Cong: $o ).
% 0.30/0.46
% 0.30/0.46 thf(dsetconstr__Cong,definition,
% 0.30/0.46 ( dsetconstr__Cong
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ! [Xy: $i] :
% 0.30/0.46 ( ( in @ Xy @ B )
% 0.30/0.46 => ( ( Xx = Xy )
% 0.30/0.46 => ( ( Xphi @ Xx )
% 0.30/0.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.30/0.46 => ( ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 = ( dsetconstr @ B
% 0.30/0.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subset_type,type,
% 0.30/0.46 subset: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(disjoint_type,type,
% 0.30/0.46 disjoint: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(setsmeet_type,type,
% 0.30/0.46 setsmeet: $i > $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI1_type,type,
% 0.30/0.46 subsetI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI1,definition,
% 0.30/0.46 ( subsetI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset2_type,type,
% 0.30/0.46 eqimpsubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset2,definition,
% 0.30/0.46 ( eqimpsubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset1_type,type,
% 0.30/0.46 eqimpsubset1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(eqimpsubset1,definition,
% 0.30/0.46 ( eqimpsubset1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( A = B )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI2_type,type,
% 0.30/0.46 subsetI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetI2,definition,
% 0.30/0.46 ( subsetI2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ! [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetsubset_type,type,
% 0.30/0.46 emptysetsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(emptysetsubset,definition,
% 0.30/0.46 ( emptysetsubset
% 0.30/0.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE_type,type,
% 0.30/0.46 subsetE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE,definition,
% 0.30/0.46 ( subsetE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE2_type,type,
% 0.30/0.46 subsetE2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetE2,definition,
% 0.30/0.46 ( subsetE2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notsubsetI_type,type,
% 0.30/0.46 notsubsetI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notsubsetI,definition,
% 0.30/0.46 ( notsubsetI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI1_type,type,
% 0.30/0.46 notequalI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI1,definition,
% 0.30/0.46 ( notequalI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ~ ( subset @ A @ B )
% 0.30/0.46 => ( A != B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI2_type,type,
% 0.30/0.46 notequalI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(notequalI2,definition,
% 0.30/0.46 ( notequalI2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( A != B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetRefl_type,type,
% 0.30/0.46 subsetRefl: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetRefl,definition,
% 0.30/0.46 ( subsetRefl
% 0.30/0.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTrans_type,type,
% 0.30/0.46 subsetTrans: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetTrans,definition,
% 0.30/0.46 ( subsetTrans
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( subset @ B @ C )
% 0.30/0.46 => ( subset @ A @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub_type,type,
% 0.30/0.46 setadjoinSub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub,definition,
% 0.30/0.46 ( setadjoinSub
% 0.30/0.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub2_type,type,
% 0.30/0.46 setadjoinSub2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setadjoinSub2,definition,
% 0.30/0.46 ( setadjoinSub2
% 0.30/0.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subset2powerset_type,type,
% 0.30/0.46 subset2powerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subset2powerset,definition,
% 0.30/0.46 ( subset2powerset
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setextsub_type,type,
% 0.30/0.46 setextsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setextsub,definition,
% 0.30/0.46 ( setextsub
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( subset @ B @ A )
% 0.30/0.46 => ( A = B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(subsetemptysetimpeq_type,type,
% 0.30/0.46 subsetemptysetimpeq: $o ).
% 0.30/0.46
% 0.30/0.46 thf(subsetemptysetimpeq,definition,
% 0.30/0.46 ( subsetemptysetimpeq
% 0.30/0.46 = ( ! [A: $i] :
% 0.30/0.46 ( ( subset @ A @ emptyset )
% 0.30/0.46 => ( A = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI1_type,type,
% 0.30/0.46 powersetI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetI1,definition,
% 0.30/0.46 ( powersetI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ B @ A )
% 0.30/0.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE1_type,type,
% 0.30/0.46 powersetE1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetE1,definition,
% 0.30/0.46 ( powersetE1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(inPowerset_type,type,
% 0.30/0.46 inPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(inPowerset,definition,
% 0.30/0.46 ( inPowerset
% 0.30/0.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(powersetsubset_type,type,
% 0.30/0.46 powersetsubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(powersetsubset,definition,
% 0.30/0.46 ( powersetsubset
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sepInPowerset_type,type,
% 0.30/0.46 sepInPowerset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(sepInPowerset,definition,
% 0.30/0.46 ( sepInPowerset
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( in
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 @ ( powerset @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sepSubset_type,type,
% 0.30/0.46 sepSubset: $o ).
% 0.30/0.46
% 0.30/0.46 thf(sepSubset,definition,
% 0.30/0.46 ( sepSubset
% 0.30/0.46 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.46 ( subset
% 0.30/0.46 @ ( dsetconstr @ A
% 0.30/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.46 @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunion_type,type,
% 0.30/0.46 binunion: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIL_type,type,
% 0.30/0.46 binunionIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIL,definition,
% 0.30/0.46 ( binunionIL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2IR_type,type,
% 0.30/0.46 upairset2IR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(upairset2IR,definition,
% 0.30/0.46 ( upairset2IR
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIR_type,type,
% 0.30/0.46 binunionIR: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionIR,definition,
% 0.30/0.46 ( binunionIR
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionEcases_type,type,
% 0.30/0.46 binunionEcases: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionEcases,definition,
% 0.30/0.46 ( binunionEcases
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.30/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/0.46 => ( ( ( in @ Xx @ A )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => ( ( ( in @ Xx @ B )
% 0.30/0.46 => Xphi )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionE_type,type,
% 0.30/0.46 binunionE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionE,definition,
% 0.30/0.46 ( binunionE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionLsub_type,type,
% 0.30/0.46 binunionLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionLsub,definition,
% 0.30/0.46 ( binunionLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binunionRsub_type,type,
% 0.30/0.46 binunionRsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binunionRsub,definition,
% 0.30/0.46 ( binunionRsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersect_type,type,
% 0.30/0.46 binintersect: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectI_type,type,
% 0.30/0.46 binintersectI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectI,definition,
% 0.30/0.46 ( binintersectI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset5_type,type,
% 0.30/0.46 binintersectSubset5: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset5,definition,
% 0.30/0.46 ( binintersectSubset5
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( subset @ C @ A )
% 0.30/0.46 => ( ( subset @ C @ B )
% 0.30/0.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectEL_type,type,
% 0.30/0.46 binintersectEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectEL,definition,
% 0.30/0.46 ( binintersectEL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectLsub_type,type,
% 0.30/0.46 binintersectLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectLsub,definition,
% 0.30/0.46 ( binintersectLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset2_type,type,
% 0.30/0.46 binintersectSubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset2,definition,
% 0.30/0.46 ( binintersectSubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset3_type,type,
% 0.30/0.46 binintersectSubset3: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset3,definition,
% 0.30/0.46 ( binintersectSubset3
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( binintersect @ A @ B )
% 0.30/0.46 = B )
% 0.30/0.46 => ( subset @ B @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectER_type,type,
% 0.30/0.46 binintersectER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectER,definition,
% 0.30/0.46 ( binintersectER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(disjointsetsI1_type,type,
% 0.30/0.46 disjointsetsI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(disjointsetsI1,definition,
% 0.30/0.46 ( disjointsetsI1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ~ ? [Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 & ( in @ Xx @ B ) )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectRsub_type,type,
% 0.30/0.46 binintersectRsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectRsub,definition,
% 0.30/0.46 ( binintersectRsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset4_type,type,
% 0.30/0.46 binintersectSubset4: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset4,definition,
% 0.30/0.46 ( binintersectSubset4
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ B @ A )
% 0.30/0.46 => ( ( binintersect @ A @ B )
% 0.30/0.46 = B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset1_type,type,
% 0.30/0.46 binintersectSubset1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(binintersectSubset1,definition,
% 0.30/0.46 ( binintersectSubset1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( binintersect @ A @ B )
% 0.30/0.46 = A )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(bs114d_type,type,
% 0.30/0.46 bs114d: $o ).
% 0.30/0.46
% 0.30/0.46 thf(bs114d,definition,
% 0.30/0.46 ( bs114d
% 0.30/0.46 = ( ! [A: $i,B: $i,C: $i] :
% 0.30/0.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.30/0.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(regular_type,type,
% 0.30/0.46 regular: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminus_type,type,
% 0.30/0.46 setminus: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(setminusI_type,type,
% 0.30/0.46 setminusI: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusI,definition,
% 0.30/0.46 ( setminusI
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusEL_type,type,
% 0.30/0.46 setminusEL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusEL,definition,
% 0.30/0.46 ( setminusEL
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusER_type,type,
% 0.30/0.46 setminusER: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusER,definition,
% 0.30/0.46 ( setminusER
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset2_type,type,
% 0.30/0.46 setminusSubset2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset2,definition,
% 0.30/0.46 ( setminusSubset2
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( subset @ A @ B )
% 0.30/0.46 => ( ( setminus @ A @ B )
% 0.30/0.46 = emptyset ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusERneg_type,type,
% 0.30/0.46 setminusERneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusERneg,definition,
% 0.30/0.46 ( setminusERneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ( ( in @ Xx @ A )
% 0.30/0.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusELneg_type,type,
% 0.30/0.46 setminusELneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusELneg,definition,
% 0.30/0.46 ( setminusELneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusILneg_type,type,
% 0.30/0.46 setminusILneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusILneg,definition,
% 0.30/0.46 ( setminusILneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusIRneg_type,type,
% 0.30/0.46 setminusIRneg: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusIRneg,definition,
% 0.30/0.46 ( setminusIRneg
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusLsub_type,type,
% 0.30/0.46 setminusLsub: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusLsub,definition,
% 0.30/0.46 ( setminusLsub
% 0.30/0.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset1_type,type,
% 0.30/0.46 setminusSubset1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setminusSubset1,definition,
% 0.30/0.46 ( setminusSubset1
% 0.30/0.46 = ( ! [A: $i,B: $i] :
% 0.30/0.46 ( ( ( setminus @ A @ B )
% 0.30/0.46 = emptyset )
% 0.30/0.46 => ( subset @ A @ B ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiff_type,type,
% 0.30/0.46 symdiff: $i > $i > $i ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffE_type,type,
% 0.30/0.46 symdiffE: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffE,definition,
% 0.30/0.46 ( symdiffE
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.30/0.46 => ! [Xphi: $o] :
% 0.30/0.46 ( ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => ( ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => Xphi ) )
% 0.30/0.46 => Xphi ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI1_type,type,
% 0.30/0.46 symdiffI1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI1,definition,
% 0.30/0.46 ( symdiffI1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI2_type,type,
% 0.30/0.46 symdiffI2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffI2,definition,
% 0.30/0.46 ( symdiffI2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg1_type,type,
% 0.30/0.46 symdiffIneg1: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg1,definition,
% 0.30/0.46 ( symdiffIneg1
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ( in @ Xx @ A )
% 0.30/0.46 => ( ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg2_type,type,
% 0.30/0.46 symdiffIneg2: $o ).
% 0.30/0.46
% 0.30/0.46 thf(symdiffIneg2,definition,
% 0.30/0.46 ( symdiffIneg2
% 0.30/0.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.46 ( ~ ( in @ Xx @ A )
% 0.30/0.46 => ( ~ ( in @ Xx @ B )
% 0.30/0.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(iskpair_type,type,
% 0.30/0.46 iskpair: $i > $o ).
% 0.30/0.46
% 0.30/0.46 thf(secondinupair_type,type,
% 0.30/0.46 secondinupair: $o ).
% 0.30/0.46
% 0.30/0.46 thf(secondinupair,definition,
% 0.30/0.46 ( secondinupair
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIL_type,type,
% 0.30/0.46 setukpairIL: $o ).
% 0.30/0.46
% 0.30/0.46 thf(setukpairIL,definition,
% 0.30/0.46 ( setukpairIL
% 0.30/0.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairIR_type,type,
% 0.30/0.47 setukpairIR: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairIR,definition,
% 0.30/0.47 ( setukpairIR
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(kpairiskpair_type,type,
% 0.30/0.47 kpairiskpair: $o ).
% 0.30/0.47
% 0.30/0.47 thf(kpairiskpair,definition,
% 0.30/0.47 ( kpairiskpair
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(kpair_type,type,
% 0.30/0.47 kpair: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(kpairp_type,type,
% 0.30/0.47 kpairp: $o ).
% 0.30/0.47
% 0.30/0.47 thf(kpairp,definition,
% 0.30/0.47 ( kpairp
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprod_type,type,
% 0.30/0.47 cartprod: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(singletonsubset_type,type,
% 0.30/0.47 singletonsubset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(singletonsubset,definition,
% 0.30/0.47 ( singletonsubset
% 0.30/0.47 = ( ! [A: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(singletoninpowerset_type,type,
% 0.30/0.47 singletoninpowerset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(singletoninpowerset,definition,
% 0.30/0.47 ( singletoninpowerset
% 0.30/0.47 = ( ! [A: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(singletoninpowunion_type,type,
% 0.30/0.47 singletoninpowunion: $o ).
% 0.30/0.47
% 0.30/0.47 thf(singletoninpowunion,definition,
% 0.30/0.47 ( singletoninpowunion
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairset2E_type,type,
% 0.30/0.47 upairset2E: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairset2E,definition,
% 0.30/0.47 ( upairset2E
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.30/0.47 => ( ( Xz = Xx )
% 0.30/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairsubunion_type,type,
% 0.30/0.47 upairsubunion: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairsubunion,definition,
% 0.30/0.47 ( upairsubunion
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairinpowunion_type,type,
% 0.30/0.47 upairinpowunion: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairinpowunion,definition,
% 0.30/0.47 ( upairinpowunion
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ubforcartprodlem1_type,type,
% 0.30/0.47 ubforcartprodlem1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ubforcartprodlem1,definition,
% 0.30/0.47 ( ubforcartprodlem1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ubforcartprodlem2_type,type,
% 0.30/0.47 ubforcartprodlem2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ubforcartprodlem2,definition,
% 0.30/0.47 ( ubforcartprodlem2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ubforcartprodlem3_type,type,
% 0.30/0.47 ubforcartprodlem3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ubforcartprodlem3,definition,
% 0.30/0.47 ( ubforcartprodlem3
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairin_type,type,
% 0.30/0.47 cartprodpairin: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairin,definition,
% 0.30/0.47 ( cartprodpairin
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodmempair1_type,type,
% 0.30/0.47 cartprodmempair1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodmempair1,definition,
% 0.30/0.47 ( cartprodmempair1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ? [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 & ( Xu
% 0.30/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodmempair_type,type,
% 0.30/0.47 cartprodmempair: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodmempair,definition,
% 0.30/0.47 ( cartprodmempair
% 0.30/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ( iskpair @ Xu ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setunionE2_type,type,
% 0.30/0.47 setunionE2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setunionE2,definition,
% 0.30/0.47 ( setunionE2
% 0.30/0.47 = ( ! [A: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.30/0.47 => ? [X: $i] :
% 0.30/0.47 ( ( in @ X @ A )
% 0.30/0.47 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setunionsingleton1_type,type,
% 0.30/0.47 setunionsingleton1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setunionsingleton1,definition,
% 0.30/0.47 ( setunionsingleton1
% 0.30/0.47 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setunionsingleton2_type,type,
% 0.30/0.47 setunionsingleton2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setunionsingleton2,definition,
% 0.30/0.47 ( setunionsingleton2
% 0.30/0.47 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setunionsingleton_type,type,
% 0.30/0.47 setunionsingleton: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setunionsingleton,definition,
% 0.30/0.47 ( setunionsingleton
% 0.30/0.47 = ( ! [Xx: $i] :
% 0.30/0.47 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.30/0.47 = Xx ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(singleton_type,type,
% 0.30/0.47 singleton: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(singletonprop_type,type,
% 0.30/0.47 singletonprop: $o ).
% 0.30/0.47
% 0.30/0.47 thf(singletonprop,definition,
% 0.30/0.47 ( singletonprop
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) )
% 0.30/0.47 => ( ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ( Xphi @ Xx ) )
% 0.30/0.47 => ( singleton
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ex1_type,type,
% 0.30/0.47 ex1: $i > ( $i > $o ) > $o ).
% 0.30/0.47
% 0.30/0.47 thf(ex1,definition,
% 0.30/0.47 ( ex1
% 0.30/0.47 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( singleton
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ex1E1_type,type,
% 0.30/0.47 ex1E1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ex1E1,definition,
% 0.30/0.47 ( ex1E1
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ( ex1 @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ex1I_type,type,
% 0.30/0.47 ex1I: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ex1I,definition,
% 0.30/0.47 ( ex1I
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 => ( ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xy = Xx ) ) )
% 0.30/0.47 => ( ex1 @ A
% 0.30/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ex1I2_type,type,
% 0.30/0.47 ex1I2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ex1I2,definition,
% 0.30/0.47 ( ex1I2
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) )
% 0.30/0.47 => ( ? [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 & ( Xphi @ Xx ) )
% 0.30/0.47 => ( ex1 @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(singletonsuniq_type,type,
% 0.30/0.47 singletonsuniq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(singletonsuniq,definition,
% 0.30/0.47 ( singletonsuniq
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( ( setadjoin @ Xx @ emptyset )
% 0.30/0.47 = ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(atmost1p_type,type,
% 0.30/0.47 atmost1p: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(atleast2p_type,type,
% 0.30/0.47 atleast2p: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(atmost2p_type,type,
% 0.30/0.47 atmost2p: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairsetp_type,type,
% 0.30/0.47 upairsetp: $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjL1_type,type,
% 0.30/0.47 setukpairinjL1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjL1,definition,
% 0.30/0.47 ( setukpairinjL1
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.47 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(kfstsingleton_type,type,
% 0.30/0.47 kfstsingleton: $o ).
% 0.30/0.47
% 0.30/0.47 thf(kfstsingleton,definition,
% 0.30/0.47 ( kfstsingleton
% 0.30/0.47 = ( ! [Xu: $i] :
% 0.30/0.47 ( ( iskpair @ Xu )
% 0.30/0.47 => ( singleton
% 0.30/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/0.47 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(theprop_type,type,
% 0.30/0.47 theprop: $o ).
% 0.30/0.47
% 0.30/0.47 thf(theprop,definition,
% 0.30/0.47 ( theprop
% 0.30/0.47 = ( ! [X: $i] :
% 0.30/0.47 ( ( singleton @ X )
% 0.30/0.47 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(kfst_type,type,
% 0.30/0.47 kfst: $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(kfstpairEq_type,type,
% 0.30/0.47 kfstpairEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(kfstpairEq,definition,
% 0.30/0.47 ( kfstpairEq
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/0.47 = Xx ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodfstin_type,type,
% 0.30/0.47 cartprodfstin: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodfstin,definition,
% 0.30/0.47 ( cartprodfstin
% 0.30/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjL2_type,type,
% 0.30/0.47 setukpairinjL2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjL2,definition,
% 0.30/0.47 ( setukpairinjL2
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjL_type,type,
% 0.30/0.47 setukpairinjL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjL,definition,
% 0.30/0.47 ( setukpairinjL
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.30/0.47 = ( kpair @ Xz @ Xu ) )
% 0.30/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR11_type,type,
% 0.30/0.47 setukpairinjR11: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR11,definition,
% 0.30/0.47 ( setukpairinjR11
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR12_type,type,
% 0.30/0.47 setukpairinjR12: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR12,definition,
% 0.30/0.47 ( setukpairinjR12
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xx = Xy )
% 0.30/0.47 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR1_type,type,
% 0.30/0.47 setukpairinjR1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR1,definition,
% 0.30/0.47 ( setukpairinjR1
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.47 => ( ( Xz = Xu )
% 0.30/0.47 => ( Xy = Xu ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(upairequniteq_type,type,
% 0.30/0.47 upairequniteq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(upairequniteq,definition,
% 0.30/0.47 ( upairequniteq
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.30/0.47 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ Xz @ emptyset ) )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR2_type,type,
% 0.30/0.47 setukpairinjR2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR2,definition,
% 0.30/0.47 ( setukpairinjR2
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.30/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.30/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR_type,type,
% 0.30/0.47 setukpairinjR: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setukpairinjR,definition,
% 0.30/0.47 ( setukpairinjR
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.30/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.30/0.47 = ( kpair @ Xz @ Xu ) )
% 0.30/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ksndsingleton_type,type,
% 0.30/0.47 ksndsingleton: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ksndsingleton,definition,
% 0.30/0.47 ( ksndsingleton
% 0.30/0.47 = ( ! [Xu: $i] :
% 0.30/0.47 ( ( iskpair @ Xu )
% 0.30/0.47 => ( singleton
% 0.30/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.30/0.47 @ ^ [Xx: $i] :
% 0.30/0.47 ( Xu
% 0.30/0.47 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ksnd_type,type,
% 0.30/0.47 ksnd: $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(ksndpairEq_type,type,
% 0.30/0.47 ksndpairEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ksndpairEq,definition,
% 0.30/0.47 ( ksndpairEq
% 0.30/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/0.47 = Xy ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(kpairsurjEq_type,type,
% 0.30/0.47 kpairsurjEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(kpairsurjEq,definition,
% 0.30/0.47 ( kpairsurjEq
% 0.30/0.47 = ( ! [Xu: $i] :
% 0.30/0.47 ( ( iskpair @ Xu )
% 0.30/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/0.47 = Xu ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodsndin_type,type,
% 0.30/0.47 cartprodsndin: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodsndin,definition,
% 0.30/0.47 ( cartprodsndin
% 0.30/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairmemEL_type,type,
% 0.30/0.47 cartprodpairmemEL: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairmemEL,definition,
% 0.30/0.47 ( cartprodpairmemEL
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairmemER_type,type,
% 0.30/0.47 cartprodpairmemER: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairmemER,definition,
% 0.30/0.47 ( cartprodpairmemER
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodmempaircEq_type,type,
% 0.30/0.47 cartprodmempaircEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodmempaircEq,definition,
% 0.30/0.47 ( cartprodmempaircEq
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( kpair @ Xx @ Xy )
% 0.30/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodfstpairEq_type,type,
% 0.30/0.47 cartprodfstpairEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodfstpairEq,definition,
% 0.30/0.47 ( cartprodfstpairEq
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.30/0.47 = Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodsndpairEq_type,type,
% 0.30/0.47 cartprodsndpairEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodsndpairEq,definition,
% 0.30/0.47 ( cartprodsndpairEq
% 0.30/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.30/0.47 = Xy ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairsurjEq_type,type,
% 0.30/0.47 cartprodpairsurjEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(cartprodpairsurjEq,definition,
% 0.30/0.47 ( cartprodpairsurjEq
% 0.30/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.30/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.30/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.30/0.47 = Xu ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(breln_type,type,
% 0.30/0.47 breln: $i > $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(breln,definition,
% 0.30/0.47 ( breln
% 0.30/0.47 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstr_type,type,
% 0.30/0.47 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrI_type,type,
% 0.30/0.47 dpsetconstrI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrI,definition,
% 0.30/0.47 ( dpsetconstrI
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( Xphi @ Xx @ Xy )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrSub_type,type,
% 0.30/0.47 dpsetconstrSub: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrSub,definition,
% 0.30/0.47 ( dpsetconstrSub
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.47 ( subset
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.30/0.47 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setOfPairsIsBReln_type,type,
% 0.30/0.47 setOfPairsIsBReln: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setOfPairsIsBReln,definition,
% 0.30/0.47 ( setOfPairsIsBReln
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.30/0.47 ( breln @ A @ B
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrERa_type,type,
% 0.30/0.47 dpsetconstrERa: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrERa,definition,
% 0.30/0.47 ( dpsetconstrERa
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrEL1_type,type,
% 0.30/0.47 dpsetconstrEL1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrEL1,definition,
% 0.30/0.47 ( dpsetconstrEL1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrEL2_type,type,
% 0.30/0.47 dpsetconstrEL2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrEL2,definition,
% 0.30/0.47 ( dpsetconstrEL2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrER_type,type,
% 0.30/0.47 dpsetconstrER: $o ).
% 0.30/0.47
% 0.30/0.47 thf(dpsetconstrER,definition,
% 0.30/0.47 ( dpsetconstrER
% 0.30/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.30/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(func_type,type,
% 0.30/0.47 func: $i > $i > $i > $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcSet_type,type,
% 0.30/0.47 funcSet: $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(funcImageSingleton_type,type,
% 0.30/0.47 funcImageSingleton: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcImageSingleton,definition,
% 0.30/0.47 ( funcImageSingleton
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( singleton
% 0.30/0.47 @ ( dsetconstr @ B
% 0.30/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(apProp_type,type,
% 0.30/0.47 apProp: $o ).
% 0.30/0.47
% 0.30/0.47 thf(apProp,definition,
% 0.30/0.47 ( apProp
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in
% 0.30/0.47 @ ( setunion
% 0.30/0.47 @ ( dsetconstr @ B
% 0.30/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.30/0.47 @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ap_type,type,
% 0.30/0.47 ap: $i > $i > $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(app_type,type,
% 0.30/0.47 app: $o ).
% 0.30/0.47
% 0.30/0.47 thf(app,definition,
% 0.30/0.47 ( app
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(infuncsetfunc_type,type,
% 0.30/0.47 infuncsetfunc: $o ).
% 0.30/0.47
% 0.30/0.47 thf(infuncsetfunc,definition,
% 0.30/0.47 ( infuncsetfunc
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ap2p_type,type,
% 0.30/0.47 ap2p: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ap2p,definition,
% 0.30/0.47 ( ap2p
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcinfuncset_type,type,
% 0.30/0.47 funcinfuncset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcinfuncset,definition,
% 0.30/0.47 ( funcinfuncset
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(lamProp_type,type,
% 0.30/0.47 lamProp: $o ).
% 0.30/0.47
% 0.30/0.47 thf(lamProp,definition,
% 0.30/0.47 ( lamProp
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ( func @ A @ B
% 0.30/0.47 @ ( dpsetconstr @ A @ B
% 0.30/0.47 @ ^ [Xx: $i,Xy: $i] :
% 0.30/0.47 ( ( Xf @ Xx )
% 0.30/0.47 = Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(lam_type,type,
% 0.30/0.47 lam: $i > $i > ( $i > $i ) > $i ).
% 0.30/0.47
% 0.30/0.47 thf(lamp_type,type,
% 0.30/0.47 lamp: $o ).
% 0.30/0.47
% 0.30/0.47 thf(lamp,definition,
% 0.30/0.47 ( lamp
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ( func @ A @ B
% 0.30/0.47 @ ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(lam2p_type,type,
% 0.30/0.47 lam2p: $o ).
% 0.30/0.47
% 0.30/0.47 thf(lam2p,definition,
% 0.30/0.47 ( lam2p
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ( in
% 0.30/0.47 @ ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.30/0.47 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(brelnall1_type,type,
% 0.30/0.47 brelnall1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(brelnall1,definition,
% 0.30/0.47 ( brelnall1
% 0.30/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ R )
% 0.30/0.47 => ! [Xphi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ R )
% 0.30/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(brelnall2_type,type,
% 0.30/0.47 brelnall2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(brelnall2,definition,
% 0.30/0.47 ( brelnall2
% 0.30/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ R )
% 0.30/0.47 => ! [Xphi: $i > $o] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ R )
% 0.30/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ex1E2_type,type,
% 0.30/0.47 ex1E2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ex1E2,definition,
% 0.30/0.47 ( ex1E2
% 0.30/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.30/0.47 ( ( ex1 @ A
% 0.30/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ( Xphi @ Xx )
% 0.30/0.47 => ( ( Xphi @ Xy )
% 0.30/0.47 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp1_type,type,
% 0.30/0.47 funcGraphProp1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp1,definition,
% 0.30/0.47 ( funcGraphProp1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp3_type,type,
% 0.30/0.47 funcGraphProp3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp3,definition,
% 0.30/0.47 ( funcGraphProp3
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp2_type,type,
% 0.30/0.47 funcGraphProp2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp2,definition,
% 0.30/0.47 ( funcGraphProp2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcextLem_type,type,
% 0.30/0.47 funcextLem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcextLem,definition,
% 0.30/0.47 ( funcextLem
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xg: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xg )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp4_type,type,
% 0.30/0.47 funcGraphProp4: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcGraphProp4,definition,
% 0.30/0.47 ( funcGraphProp4
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subbreln_type,type,
% 0.30/0.47 subbreln: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subbreln,definition,
% 0.30/0.47 ( subbreln
% 0.30/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ S )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.47 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eqbreln_type,type,
% 0.30/0.47 eqbreln: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eqbreln,definition,
% 0.30/0.47 ( eqbreln
% 0.30/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ R )
% 0.30/0.47 => ! [S: $i] :
% 0.30/0.47 ( ( breln @ A @ B @ S )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ B )
% 0.30/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.30/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.30/0.47 => ( R = S ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcext_type,type,
% 0.30/0.47 funcext: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcext,definition,
% 0.30/0.47 ( funcext
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xg: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xg )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(funcext2_type,type,
% 0.30/0.47 funcext2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(funcext2,definition,
% 0.30/0.47 ( funcext2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ! [Xg: $i] :
% 0.30/0.47 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.30/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ap2apEq1_type,type,
% 0.30/0.47 ap2apEq1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ap2apEq1,definition,
% 0.30/0.47 ( ap2apEq1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ap2apEq2_type,type,
% 0.30/0.47 ap2apEq2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ap2apEq2,definition,
% 0.30/0.47 ( ap2apEq2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.30/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(beta1_type,type,
% 0.30/0.47 beta1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(beta1,definition,
% 0.30/0.47 ( beta1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B
% 0.30/0.47 @ ( lam @ A @ B
% 0.30/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.30/0.47 @ Xx )
% 0.30/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eta1_type,type,
% 0.30/0.47 eta1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eta1,definition,
% 0.30/0.47 ( eta1
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( func @ A @ B @ Xf )
% 0.30/0.47 => ( ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.30/0.47 = Xf ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(lam2lamEq_type,type,
% 0.30/0.47 lam2lamEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(lam2lamEq,definition,
% 0.30/0.47 ( lam2lamEq
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ( ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.30/0.47 = ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(beta2_type,type,
% 0.30/0.47 beta2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(beta2,definition,
% 0.30/0.47 ( beta2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.30/0.47 ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( ap @ A @ B
% 0.30/0.47 @ ( lam @ A @ B
% 0.30/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.30/0.47 @ Xx )
% 0.30/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(eta2_type,type,
% 0.30/0.47 eta2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(eta2,definition,
% 0.30/0.47 ( eta2
% 0.30/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.30/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.30/0.47 => ( ( lam @ A @ B
% 0.30/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.30/0.47 = Xf ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iffalseProp1_type,type,
% 0.30/0.47 iffalseProp1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iffalseProp1,definition,
% 0.30/0.47 ( iffalseProp1
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ~ Xphi
% 0.30/0.47 => ( in @ Xy
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xz: $i] :
% 0.30/0.47 ( ( Xphi
% 0.30/0.47 & ( Xz = Xx ) )
% 0.30/0.47 | ( ~ Xphi
% 0.30/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iffalseProp2_type,type,
% 0.30/0.47 iffalseProp2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iffalseProp2,definition,
% 0.30/0.47 ( iffalseProp2
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ~ Xphi
% 0.30/0.47 => ( ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xz: $i] :
% 0.30/0.47 ( ( Xphi
% 0.30/0.47 & ( Xz = Xx ) )
% 0.30/0.47 | ( ~ Xphi
% 0.30/0.47 & ( Xz = Xy ) ) ) )
% 0.30/0.47 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueProp1_type,type,
% 0.30/0.47 iftrueProp1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueProp1,definition,
% 0.30/0.47 ( iftrueProp1
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( Xphi
% 0.30/0.47 => ( in @ Xx
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xz: $i] :
% 0.30/0.47 ( ( Xphi
% 0.30/0.47 & ( Xz = Xx ) )
% 0.30/0.47 | ( ~ Xphi
% 0.30/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueProp2_type,type,
% 0.30/0.47 iftrueProp2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueProp2,definition,
% 0.30/0.47 ( iftrueProp2
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( Xphi
% 0.30/0.47 => ( ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xz: $i] :
% 0.30/0.47 ( ( Xphi
% 0.30/0.47 & ( Xz = Xx ) )
% 0.30/0.47 | ( ~ Xphi
% 0.30/0.47 & ( Xz = Xy ) ) ) )
% 0.30/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(ifSingleton_type,type,
% 0.30/0.47 ifSingleton: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ifSingleton,definition,
% 0.30/0.47 ( ifSingleton
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( singleton
% 0.30/0.47 @ ( dsetconstr @ A
% 0.30/0.47 @ ^ [Xz: $i] :
% 0.30/0.47 ( ( Xphi
% 0.30/0.47 & ( Xz = Xx ) )
% 0.30/0.47 | ( ~ Xphi
% 0.30/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(if_type,type,
% 0.30/0.47 if: $i > $o > $i > $i > $i ).
% 0.30/0.47
% 0.30/0.47 thf(ifp_type,type,
% 0.30/0.47 ifp: $o ).
% 0.30/0.47
% 0.30/0.47 thf(ifp,definition,
% 0.30/0.47 ( ifp
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(theeq_type,type,
% 0.30/0.47 theeq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(theeq,definition,
% 0.30/0.47 ( theeq
% 0.30/0.47 = ( ! [X: $i] :
% 0.30/0.47 ( ( singleton @ X )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ X )
% 0.30/0.47 => ( ( setunion @ X )
% 0.30/0.47 = Xx ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iftrue_type,type,
% 0.30/0.47 iftrue: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iftrue,definition,
% 0.30/0.47 ( iftrue
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( Xphi
% 0.30/0.47 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.30/0.47 = Xx ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iffalse_type,type,
% 0.30/0.47 iffalse: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iffalse,definition,
% 0.30/0.47 ( iffalse
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( ~ Xphi
% 0.30/0.47 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.30/0.47 = Xy ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueorfalse_type,type,
% 0.30/0.47 iftrueorfalse: $o ).
% 0.30/0.47
% 0.30/0.47 thf(iftrueorfalse,definition,
% 0.30/0.47 ( iftrueorfalse
% 0.30/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ! [Xy: $i] :
% 0.30/0.47 ( ( in @ Xy @ A )
% 0.30/0.47 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectT_lem_type,type,
% 0.30/0.47 binintersectT_lem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectT_lem,definition,
% 0.30/0.47 ( binintersectT_lem
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionT_lem_type,type,
% 0.30/0.47 binunionT_lem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionT_lem,definition,
% 0.30/0.47 ( binunionT_lem
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetT_lem_type,type,
% 0.30/0.47 powersetT_lem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetT_lem,definition,
% 0.30/0.47 ( powersetT_lem
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setminusT_lem_type,type,
% 0.30/0.47 setminusT_lem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setminusT_lem,definition,
% 0.30/0.47 ( setminusT_lem
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementT_lem_type,type,
% 0.30/0.47 complementT_lem: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementT_lem,definition,
% 0.30/0.47 ( complementT_lem
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(setextT_type,type,
% 0.30/0.47 setextT: $o ).
% 0.30/0.47
% 0.30/0.47 thf(setextT,definition,
% 0.30/0.47 ( setextT
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ( in @ Xx @ Y ) ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ Y )
% 0.30/0.47 => ( in @ Xx @ X ) ) )
% 0.30/0.47 => ( X = Y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(subsetTI_type,type,
% 0.30/0.47 subsetTI: $o ).
% 0.30/0.47
% 0.30/0.47 thf(subsetTI,definition,
% 0.30/0.47 ( subsetTI
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ( in @ Xx @ Y ) ) )
% 0.30/0.47 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetTI1_type,type,
% 0.30/0.47 powersetTI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetTI1,definition,
% 0.30/0.47 ( powersetTI1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ( in @ Xx @ Y ) ) )
% 0.30/0.47 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(powersetTE1_type,type,
% 0.30/0.47 powersetTE1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(powersetTE1,definition,
% 0.30/0.47 ( powersetTE1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ X @ ( powerset @ Y ) )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementTI1_type,type,
% 0.30/0.47 complementTI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementTI1,definition,
% 0.30/0.47 ( complementTI1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementTE1_type,type,
% 0.30/0.47 complementTE1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementTE1,definition,
% 0.30/0.47 ( complementTE1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.47 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectTELcontra_type,type,
% 0.30/0.47 binintersectTELcontra: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectTELcontra,definition,
% 0.30/0.47 ( binintersectTELcontra
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ X )
% 0.30/0.47 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectTERcontra_type,type,
% 0.30/0.47 binintersectTERcontra: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binintersectTERcontra,definition,
% 0.30/0.47 ( binintersectTERcontra
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ Y )
% 0.30/0.47 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT_type,type,
% 0.30/0.47 contrasubsetT: $o ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT,definition,
% 0.30/0.47 ( contrasubsetT
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.47 => ( ( in @ Xx @ Y )
% 0.30/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT1_type,type,
% 0.30/0.47 contrasubsetT1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT1,definition,
% 0.30/0.47 ( contrasubsetT1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( subset @ X @ Y )
% 0.30/0.47 => ( ~ ( in @ Xx @ Y )
% 0.30/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT2_type,type,
% 0.30/0.47 contrasubsetT2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT2,definition,
% 0.30/0.47 ( contrasubsetT2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ Y )
% 0.30/0.47 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT3_type,type,
% 0.30/0.47 contrasubsetT3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(contrasubsetT3,definition,
% 0.30/0.47 ( contrasubsetT3
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.30/0.47 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementI1_type,type,
% 0.30/0.47 doubleComplementI1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementI1,definition,
% 0.30/0.47 ( doubleComplementI1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ X )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementE1_type,type,
% 0.30/0.47 doubleComplementE1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementE1,definition,
% 0.30/0.47 ( doubleComplementE1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.30/0.47 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementSub1_type,type,
% 0.30/0.47 doubleComplementSub1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementSub1,definition,
% 0.30/0.47 ( doubleComplementSub1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementSub2_type,type,
% 0.30/0.47 doubleComplementSub2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementSub2,definition,
% 0.30/0.47 ( doubleComplementSub2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementEq_type,type,
% 0.30/0.47 doubleComplementEq: $o ).
% 0.30/0.47
% 0.30/0.47 thf(doubleComplementEq,definition,
% 0.30/0.47 ( doubleComplementEq
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ( X
% 0.30/0.47 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementTnotintersectT_type,type,
% 0.30/0.47 complementTnotintersectT: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementTnotintersectT,definition,
% 0.30/0.47 ( complementTnotintersectT
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.47 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementImpComplementIntersect_type,type,
% 0.30/0.47 complementImpComplementIntersect: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementImpComplementIntersect,definition,
% 0.30/0.47 ( complementImpComplementIntersect
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementSubsetComplementIntersect_type,type,
% 0.30/0.47 complementSubsetComplementIntersect: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementSubsetComplementIntersect,definition,
% 0.30/0.47 ( complementSubsetComplementIntersect
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementInPowersetComplementIntersect_type,type,
% 0.30/0.47 complementInPowersetComplementIntersect: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementInPowersetComplementIntersect,definition,
% 0.30/0.47 ( complementInPowersetComplementIntersect
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(contraSubsetComplement_type,type,
% 0.30/0.47 contraSubsetComplement: $o ).
% 0.30/0.47
% 0.30/0.47 thf(contraSubsetComplement,definition,
% 0.30/0.47 ( contraSubsetComplement
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ Y )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementTcontraSubset_type,type,
% 0.30/0.47 complementTcontraSubset: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementTcontraSubset,definition,
% 0.30/0.47 ( complementTcontraSubset
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.47 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionTILcontra_type,type,
% 0.30/0.47 binunionTILcontra: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionTILcontra,definition,
% 0.30/0.47 ( binunionTILcontra
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionTIRcontra_type,type,
% 0.30/0.47 binunionTIRcontra: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionTIRcontra,definition,
% 0.30/0.47 ( binunionTIRcontra
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.47 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(inIntersectImpInUnion_type,type,
% 0.30/0.47 inIntersectImpInUnion: $o ).
% 0.30/0.47
% 0.30/0.47 thf(inIntersectImpInUnion,definition,
% 0.30/0.47 ( inIntersectImpInUnion
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.47 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(inIntersectImpInUnion2_type,type,
% 0.30/0.47 inIntersectImpInUnion2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(inIntersectImpInUnion2,definition,
% 0.30/0.47 ( inIntersectImpInUnion2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.47 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(inIntersectImpInIntersectUnions_type,type,
% 0.30/0.47 inIntersectImpInIntersectUnions: $o ).
% 0.30/0.47
% 0.30/0.47 thf(inIntersectImpInIntersectUnions,definition,
% 0.30/0.47 ( inIntersectImpInIntersectUnions
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.47 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(intersectInPowersetIntersectUnions_type,type,
% 0.30/0.47 intersectInPowersetIntersectUnions: $o ).
% 0.30/0.47
% 0.30/0.47 thf(intersectInPowersetIntersectUnions,definition,
% 0.30/0.47 ( intersectInPowersetIntersectUnions
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(inComplementUnionImpNotIn1_type,type,
% 0.30/0.47 inComplementUnionImpNotIn1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(inComplementUnionImpNotIn1,definition,
% 0.30/0.47 ( inComplementUnionImpNotIn1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(inComplementUnionImpInComplement1_type,type,
% 0.30/0.47 inComplementUnionImpInComplement1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(inComplementUnionImpInComplement1,definition,
% 0.30/0.47 ( inComplementUnionImpInComplement1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionTE_type,type,
% 0.30/0.47 binunionTE: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionTE,definition,
% 0.30/0.47 ( binunionTE
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xphi: $o,Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.30/0.47 => ( ( ( in @ Xx @ X )
% 0.30/0.47 => Xphi )
% 0.30/0.47 => ( ( ( in @ Xx @ Y )
% 0.30/0.47 => Xphi )
% 0.30/0.47 => Xphi ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(binunionTEcontra_type,type,
% 0.30/0.47 binunionTEcontra: $o ).
% 0.30/0.47
% 0.30/0.47 thf(binunionTEcontra,definition,
% 0.30/0.47 ( binunionTEcontra
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ~ ( in @ Xx @ X )
% 0.30/0.47 => ( ~ ( in @ Xx @ Y )
% 0.30/0.47 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2a1_type,type,
% 0.30/0.47 demorgan2a1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2a1,definition,
% 0.30/0.47 ( demorgan2a1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(complementUnionInPowersetComplement_type,type,
% 0.30/0.47 complementUnionInPowersetComplement: $o ).
% 0.30/0.47
% 0.30/0.47 thf(complementUnionInPowersetComplement,definition,
% 0.30/0.47 ( complementUnionInPowersetComplement
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2a2_type,type,
% 0.30/0.47 demorgan2a2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2a2,definition,
% 0.30/0.47 ( demorgan2a2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan1a_type,type,
% 0.30/0.47 demorgan1a: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan1a,definition,
% 0.30/0.47 ( demorgan1a
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.30/0.47 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan1b_type,type,
% 0.30/0.47 demorgan1b: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan1b,definition,
% 0.30/0.47 ( demorgan1b
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan1_type,type,
% 0.30/0.47 demorgan1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan1,definition,
% 0.30/0.47 ( demorgan1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.30/0.47 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2a_type,type,
% 0.30/0.47 demorgan2a: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2a,definition,
% 0.30/0.47 ( demorgan2a
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.30/0.47 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2b2_type,type,
% 0.30/0.47 demorgan2b2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2b2,definition,
% 0.30/0.47 ( demorgan2b2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.30/0.47 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2b_type,type,
% 0.30/0.47 demorgan2b: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2b,definition,
% 0.30/0.47 ( demorgan2b
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ A )
% 0.30/0.47 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.30/0.47 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2_type,type,
% 0.30/0.47 demorgan2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(demorgan2,definition,
% 0.30/0.47 ( demorgan2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.30/0.47 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule0_type,type,
% 0.30/0.47 woz13rule0: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule0,definition,
% 0.30/0.47 ( woz13rule0
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Xx: $i] :
% 0.30/0.47 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.30/0.47 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule1_type,type,
% 0.30/0.47 woz13rule1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule1,definition,
% 0.30/0.47 ( woz13rule1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ Z )
% 0.30/0.47 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule2_type,type,
% 0.30/0.47 woz13rule2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule2,definition,
% 0.30/0.47 ( woz13rule2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ Y @ Z )
% 0.30/0.47 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule3_type,type,
% 0.30/0.47 woz13rule3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule3,definition,
% 0.30/0.47 ( woz13rule3
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ Y )
% 0.30/0.47 => ( ( subset @ X @ Z )
% 0.30/0.47 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule4_type,type,
% 0.30/0.47 woz13rule4: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz13rule4,definition,
% 0.30/0.47 ( woz13rule4
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ! [W: $i] :
% 0.30/0.47 ( ( in @ W @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ Z )
% 0.30/0.47 => ( ( subset @ Y @ W )
% 0.30/0.47 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_1_type,type,
% 0.30/0.47 woz1_1: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_1,definition,
% 0.30/0.47 ( woz1_1
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_2_type,type,
% 0.30/0.47 woz1_2: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_2,definition,
% 0.30/0.47 ( woz1_2
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ! [W: $i] :
% 0.30/0.47 ( ( in @ W @ ( powerset @ A ) )
% 0.30/0.47 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.30/0.47 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_3_type,type,
% 0.30/0.47 woz1_3: $o ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_3,definition,
% 0.30/0.47 ( woz1_3
% 0.30/0.47 = ( ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ! [Z: $i] :
% 0.30/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.30/0.47 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 thf(woz1_4,conjecture,
% 0.30/0.47 ( setextAx
% 0.30/0.47 => ( emptysetAx
% 0.30/0.47 => ( setadjoinAx
% 0.30/0.47 => ( powersetAx
% 0.30/0.47 => ( setunionAx
% 0.30/0.47 => ( omega0Ax
% 0.30/0.47 => ( omegaSAx
% 0.30/0.47 => ( omegaIndAx
% 0.30/0.47 => ( replAx
% 0.30/0.47 => ( foundationAx
% 0.30/0.47 => ( wellorderingAx
% 0.30/0.47 => ( descrp
% 0.30/0.47 => ( dsetconstrI
% 0.30/0.47 => ( dsetconstrEL
% 0.30/0.47 => ( dsetconstrER
% 0.30/0.47 => ( exuE1
% 0.30/0.47 => ( prop2setE
% 0.30/0.47 => ( emptysetE
% 0.30/0.47 => ( emptysetimpfalse
% 0.30/0.47 => ( notinemptyset
% 0.30/0.47 => ( exuE3e
% 0.30/0.47 => ( setext
% 0.30/0.47 => ( emptyI
% 0.30/0.47 => ( noeltsimpempty
% 0.30/0.47 => ( setbeta
% 0.30/0.47 => ( nonemptyE1
% 0.30/0.47 => ( nonemptyI
% 0.30/0.47 => ( nonemptyI1
% 0.30/0.47 => ( setadjoinIL
% 0.30/0.47 => ( emptyinunitempty
% 0.30/0.47 => ( setadjoinIR
% 0.30/0.47 => ( setadjoinE
% 0.30/0.47 => ( setadjoinOr
% 0.30/0.47 => ( setoftrueEq
% 0.30/0.47 => ( powersetI
% 0.30/0.47 => ( emptyinPowerset
% 0.30/0.47 => ( emptyInPowerset
% 0.30/0.47 => ( powersetE
% 0.30/0.47 => ( setunionI
% 0.30/0.47 => ( setunionE
% 0.30/0.47 => ( subPowSU
% 0.30/0.47 => ( exuE2
% 0.30/0.47 => ( nonemptyImpWitness
% 0.30/0.47 => ( uniqinunit
% 0.30/0.47 => ( notinsingleton
% 0.30/0.47 => ( eqinunit
% 0.30/0.47 => ( singletonsswitch
% 0.30/0.47 => ( upairsetE
% 0.30/0.47 => ( upairsetIL
% 0.30/0.47 => ( upairsetIR
% 0.30/0.47 => ( emptyE1
% 0.30/0.47 => ( vacuousDall
% 0.30/0.47 => ( quantDeMorgan1
% 0.30/0.47 => ( quantDeMorgan2
% 0.30/0.47 => ( quantDeMorgan3
% 0.30/0.47 => ( quantDeMorgan4
% 0.30/0.47 => ( prop2setI
% 0.30/0.47 => ( prop2set2propI
% 0.30/0.47 => ( notdexE
% 0.30/0.47 => ( notdallE
% 0.30/0.47 => ( exuI1
% 0.30/0.47 => ( exuI3
% 0.30/0.47 => ( exuI2
% 0.30/0.47 => ( inCongP
% 0.30/0.47 => ( in__Cong
% 0.30/0.47 => ( exuE3u
% 0.30/0.47 => ( exu__Cong
% 0.30/0.47 => ( emptyset__Cong
% 0.30/0.47 => ( setadjoin__Cong
% 0.30/0.47 => ( powerset__Cong
% 0.30/0.47 => ( setunion__Cong
% 0.30/0.47 => ( omega__Cong
% 0.30/0.47 => ( exuEu
% 0.30/0.47 => ( descr__Cong
% 0.30/0.47 => ( dsetconstr__Cong
% 0.30/0.47 => ( subsetI1
% 0.30/0.47 => ( eqimpsubset2
% 0.30/0.47 => ( eqimpsubset1
% 0.30/0.47 => ( subsetI2
% 0.30/0.47 => ( emptysetsubset
% 0.30/0.47 => ( subsetE
% 0.30/0.47 => ( subsetE2
% 0.30/0.47 => ( notsubsetI
% 0.30/0.47 => ( notequalI1
% 0.30/0.47 => ( notequalI2
% 0.30/0.47 => ( subsetRefl
% 0.30/0.47 => ( subsetTrans
% 0.30/0.47 => ( setadjoinSub
% 0.30/0.47 => ( setadjoinSub2
% 0.30/0.47 => ( subset2powerset
% 0.30/0.47 => ( setextsub
% 0.30/0.47 => ( subsetemptysetimpeq
% 0.30/0.47 => ( powersetI1
% 0.30/0.47 => ( powersetE1
% 0.30/0.47 => ( inPowerset
% 0.30/0.47 => ( powersetsubset
% 0.30/0.47 => ( sepInPowerset
% 0.30/0.47 => ( sepSubset
% 0.30/0.47 => ( binunionIL
% 0.30/0.47 => ( upairset2IR
% 0.30/0.47 => ( binunionIR
% 0.30/0.47 => ( binunionEcases
% 0.30/0.47 => ( binunionE
% 0.30/0.47 => ( binunionLsub
% 0.30/0.47 => ( binunionRsub
% 0.30/0.47 => ( binintersectI
% 0.30/0.47 => ( binintersectSubset5
% 0.30/0.47 => ( binintersectEL
% 0.30/0.47 => ( binintersectLsub
% 0.30/0.47 => ( binintersectSubset2
% 0.30/0.47 => ( binintersectSubset3
% 0.30/0.47 => ( binintersectER
% 0.30/0.47 => ( disjointsetsI1
% 0.30/0.47 => ( binintersectRsub
% 0.30/0.47 => ( binintersectSubset4
% 0.30/0.47 => ( binintersectSubset1
% 0.30/0.47 => ( bs114d
% 0.30/0.47 => ( setminusI
% 0.30/0.47 => ( setminusEL
% 0.30/0.47 => ( setminusER
% 0.30/0.47 => ( setminusSubset2
% 0.30/0.47 => ( setminusERneg
% 0.30/0.47 => ( setminusELneg
% 0.30/0.47 => ( setminusILneg
% 0.30/0.47 => ( setminusIRneg
% 0.30/0.47 => ( setminusLsub
% 0.30/0.47 => ( setminusSubset1
% 0.30/0.47 => ( symdiffE
% 0.30/0.47 => ( symdiffI1
% 0.30/0.47 => ( symdiffI2
% 0.30/0.47 => ( symdiffIneg1
% 0.30/0.47 => ( symdiffIneg2
% 0.30/0.47 => ( secondinupair
% 0.30/0.47 => ( setukpairIL
% 0.30/0.47 => ( setukpairIR
% 0.30/0.47 => ( kpairiskpair
% 0.30/0.47 => ( kpairp
% 0.30/0.47 => ( singletonsubset
% 0.30/0.47 => ( singletoninpowerset
% 0.30/0.47 => ( singletoninpowunion
% 0.30/0.47 => ( upairset2E
% 0.30/0.47 => ( upairsubunion
% 0.30/0.47 => ( upairinpowunion
% 0.30/0.47 => ( ubforcartprodlem1
% 0.30/0.47 => ( ubforcartprodlem2
% 0.30/0.47 => ( ubforcartprodlem3
% 0.30/0.47 => ( cartprodpairin
% 0.30/0.47 => ( cartprodmempair1
% 0.30/0.47 => ( cartprodmempair
% 0.30/0.47 => ( setunionE2
% 0.30/0.47 => ( setunionsingleton1
% 0.30/0.47 => ( setunionsingleton2
% 0.30/0.47 => ( setunionsingleton
% 0.30/0.47 => ( singletonprop
% 0.30/0.47 => ( ex1E1
% 0.30/0.47 => ( ex1I
% 0.30/0.47 => ( ex1I2
% 0.30/0.47 => ( singletonsuniq
% 0.30/0.47 => ( setukpairinjL1
% 0.30/0.47 => ( kfstsingleton
% 0.30/0.47 => ( theprop
% 0.30/0.47 => ( kfstpairEq
% 0.30/0.47 => ( cartprodfstin
% 0.30/0.47 => ( setukpairinjL2
% 0.30/0.47 => ( setukpairinjL
% 0.30/0.47 => ( setukpairinjR11
% 0.30/0.47 => ( setukpairinjR12
% 0.30/0.47 => ( setukpairinjR1
% 0.30/0.47 => ( upairequniteq
% 0.30/0.47 => ( setukpairinjR2
% 0.30/0.47 => ( setukpairinjR
% 0.30/0.47 => ( ksndsingleton
% 0.30/0.47 => ( ksndpairEq
% 0.30/0.47 => ( kpairsurjEq
% 0.30/0.47 => ( cartprodsndin
% 0.30/0.47 => ( cartprodpairmemEL
% 0.30/0.47 => ( cartprodpairmemER
% 0.30/0.47 => ( cartprodmempaircEq
% 0.30/0.47 => ( cartprodfstpairEq
% 0.30/0.47 => ( cartprodsndpairEq
% 0.30/0.47 => ( cartprodpairsurjEq
% 0.30/0.47 => ( dpsetconstrI
% 0.30/0.47 => ( dpsetconstrSub
% 0.30/0.47 => ( setOfPairsIsBReln
% 0.30/0.47 => ( dpsetconstrERa
% 0.30/0.47 => ( dpsetconstrEL1
% 0.30/0.47 => ( dpsetconstrEL2
% 0.30/0.47 => ( dpsetconstrER
% 0.30/0.47 => ( funcImageSingleton
% 0.30/0.47 => ( apProp
% 0.30/0.47 => ( app
% 0.30/0.47 => ( infuncsetfunc
% 0.30/0.47 => ( ap2p
% 0.30/0.47 => ( funcinfuncset
% 0.30/0.47 => ( lamProp
% 0.30/0.47 => ( lamp
% 0.30/0.47 => ( lam2p
% 0.30/0.47 => ( brelnall1
% 0.30/0.47 => ( brelnall2
% 0.30/0.47 => ( ex1E2
% 0.30/0.47 => ( funcGraphProp1
% 0.30/0.47 => ( funcGraphProp3
% 0.30/0.47 => ( funcGraphProp2
% 0.30/0.47 => ( funcextLem
% 0.30/0.47 => ( funcGraphProp4
% 0.30/0.47 => ( subbreln
% 0.30/0.47 => ( eqbreln
% 0.30/0.47 => ( funcext
% 0.30/0.47 => ( funcext2
% 0.30/0.47 => ( ap2apEq1
% 0.30/0.47 => ( ap2apEq2
% 0.30/0.47 => ( beta1
% 0.30/0.47 => ( eta1
% 0.30/0.47 => ( lam2lamEq
% 0.30/0.47 => ( beta2
% 0.30/0.47 => ( eta2
% 0.30/0.47 => ( iffalseProp1
% 0.30/0.47 => ( iffalseProp2
% 0.30/0.47 => ( iftrueProp1
% 0.30/0.47 => ( iftrueProp2
% 0.30/0.47 => ( ifSingleton
% 0.30/0.47 => ( ifp
% 0.30/0.47 => ( theeq
% 0.30/0.47 => ( iftrue
% 0.30/0.47 => ( iffalse
% 0.30/0.47 => ( iftrueorfalse
% 0.30/0.47 => ( binintersectT_lem
% 0.30/0.47 => ( binunionT_lem
% 0.30/0.47 => ( powersetT_lem
% 0.30/0.47 => ( setminusT_lem
% 0.30/0.47 => ( complementT_lem
% 0.30/0.47 => ( setextT
% 0.30/0.47 => ( subsetTI
% 0.30/0.47 => ( powersetTI1
% 0.30/0.47 => ( powersetTE1
% 0.30/0.47 => ( complementTI1
% 0.30/0.47 => ( complementTE1
% 0.30/0.47 => ( binintersectTELcontra
% 0.30/0.47 => ( binintersectTERcontra
% 0.30/0.47 => ( contrasubsetT
% 0.30/0.47 => ( contrasubsetT1
% 0.30/0.47 => ( contrasubsetT2
% 0.30/0.47 => ( contrasubsetT3
% 0.30/0.47 => ( doubleComplementI1
% 0.30/0.47 => ( doubleComplementE1
% 0.30/0.47 => ( doubleComplementSub1
% 0.30/0.47 => ( doubleComplementSub2
% 0.30/0.47 => ( doubleComplementEq
% 0.30/0.47 => ( complementTnotintersectT
% 0.30/0.47 => ( complementImpComplementIntersect
% 0.30/0.47 => ( complementSubsetComplementIntersect
% 0.30/0.47 => ( complementInPowersetComplementIntersect
% 0.30/0.47 => ( contraSubsetComplement
% 0.30/0.47 => ( complementTcontraSubset
% 0.30/0.47 => ( binunionTILcontra
% 0.30/0.47 => ( binunionTIRcontra
% 0.30/0.47 => ( inIntersectImpInUnion
% 0.30/0.47 => ( inIntersectImpInUnion2
% 0.30/0.47 => ( inIntersectImpInIntersectUnions
% 0.30/0.47 => ( intersectInPowersetIntersectUnions
% 0.30/0.47 => ( inComplementUnionImpNotIn1
% 0.30/0.47 => ( inComplementUnionImpInComplement1
% 0.30/0.47 => ( binunionTE
% 0.30/0.47 => ( binunionTEcontra
% 0.30/0.47 => ( demorgan2a1
% 0.30/0.47 => ( complementUnionInPowersetComplement
% 0.30/0.47 => ( demorgan2a2
% 0.30/0.47 => ( demorgan1a
% 0.30/0.47 => ( demorgan1b
% 0.30/0.47 => ( demorgan1
% 0.30/0.47 => ( demorgan2a
% 0.30/0.47 => ( demorgan2b2
% 0.30/0.47 => ( demorgan2b
% 0.30/0.47 => ( demorgan2
% 0.30/0.47 => ( woz13rule0
% 0.30/0.47 => ( woz13rule1
% 0.30/0.47 => ( woz13rule2
% 0.30/0.47 => ( woz13rule3
% 0.30/0.47 => ( woz13rule4
% 0.30/0.47 => ( woz1_1
% 0.30/0.47 => ( woz1_2
% 0.30/0.47 => ( woz1_3
% 0.30/0.47 => ! [A: $i,X: $i] :
% 0.30/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.30/0.47 => ! [Y: $i] :
% 0.30/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.30/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.30/0.47 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.47
% 0.30/0.47 %------------------------------------------------------------------------------
% 0.30/0.47 ------------------------
% 2.77/1.56 ---- Embedded in THF ---
% 2.77/1.56 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.77/1.56 %%% Generated on Tue Feb 24 12:33:25 EST 2026
% 2.77/1.56 %%% using '$$dhol' embedding, version 1.3.0.
% 2.77/1.56 %%% Logic specification used:
% 2.77/1.56 %%% thf(spec, logic, $$dhol).
% 2.77/1.56
% 2.77/1.56 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.WxwUZkQ1Ii/DTF2DTF28543.p
% See solution above
% 2.77/1.56 ------------------------
% 2.91/1.60 ---- Cleaned THF ---
% 2.91/1.60 thf(in_type,type,
% 2.91/1.60 in: $i > $i > $o ).
% 2.91/1.60
% 2.91/1.60 thf(exu_type,type,
% 2.91/1.60 exu: ( $i > $o ) > $o ).
% 2.91/1.60
% 2.91/1.60 thf(exu,definition,
% 2.91/1.60 ( exu
% 2.91/1.60 = ( ^ [Xphi: $i > $o] :
% 2.91/1.60 ? [Xx: $i] :
% 2.91/1.60 ( ( Xphi @ Xx )
% 2.91/1.60 & ! [Xy: $i] :
% 2.91/1.60 ( ( Xphi @ Xy )
% 2.91/1.60 => ( Xx = Xy ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setextAx_type,type,
% 2.91/1.60 setextAx: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setextAx,definition,
% 2.91/1.60 ( setextAx
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 <=> ( in @ Xx @ B ) )
% 2.91/1.60 => ( A = B ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(emptyset_type,type,
% 2.91/1.60 emptyset: $i ).
% 2.91/1.60
% 2.91/1.60 thf(emptysetAx_type,type,
% 2.91/1.60 emptysetAx: $o ).
% 2.91/1.60
% 2.91/1.60 thf(emptysetAx,definition,
% 2.91/1.60 ( emptysetAx
% 2.91/1.60 = ( ! [Xx: $i] :
% 2.91/1.60 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoin_type,type,
% 2.91/1.60 setadjoin: $i > $i > $i ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoinAx_type,type,
% 2.91/1.60 setadjoinAx: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoinAx,definition,
% 2.91/1.60 ( setadjoinAx
% 2.91/1.60 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.91/1.60 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.91/1.60 <=> ( ( Xy = Xx )
% 2.91/1.60 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(powerset_type,type,
% 2.91/1.60 powerset: $i > $i ).
% 2.91/1.60
% 2.91/1.60 thf(powersetAx_type,type,
% 2.91/1.60 powersetAx: $o ).
% 2.91/1.60
% 2.91/1.60 thf(powersetAx,definition,
% 2.91/1.60 ( powersetAx
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( in @ B @ ( powerset @ A ) )
% 2.91/1.60 <=> ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ B )
% 2.91/1.60 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setunion_type,type,
% 2.91/1.60 setunion: $i > $i ).
% 2.91/1.60
% 2.91/1.60 thf(setunionAx_type,type,
% 2.91/1.60 setunionAx: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setunionAx,definition,
% 2.91/1.60 ( setunionAx
% 2.91/1.60 = ( ! [A: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ ( setunion @ A ) )
% 2.91/1.60 <=> ? [B: $i] :
% 2.91/1.60 ( ( in @ Xx @ B )
% 2.91/1.60 & ( in @ B @ A ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(omega_type,type,
% 2.91/1.60 omega: $i ).
% 2.91/1.60
% 2.91/1.60 thf(omega0Ax_type,type,
% 2.91/1.60 omega0Ax: $o ).
% 2.91/1.60
% 2.91/1.60 thf(omega0Ax,definition,
% 2.91/1.60 ( omega0Ax
% 2.91/1.60 = ( in @ emptyset @ omega ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(omegaSAx_type,type,
% 2.91/1.60 omegaSAx: $o ).
% 2.91/1.60
% 2.91/1.60 thf(omegaSAx,definition,
% 2.91/1.60 ( omegaSAx
% 2.91/1.60 = ( ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ omega )
% 2.91/1.60 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(omegaIndAx_type,type,
% 2.91/1.60 omegaIndAx: $o ).
% 2.91/1.60
% 2.91/1.60 thf(omegaIndAx,definition,
% 2.91/1.60 ( omegaIndAx
% 2.91/1.60 = ( ! [A: $i] :
% 2.91/1.60 ( ( ( in @ emptyset @ A )
% 2.91/1.60 & ! [Xx: $i] :
% 2.91/1.60 ( ( ( in @ Xx @ omega )
% 2.91/1.60 & ( in @ Xx @ A ) )
% 2.91/1.60 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.91/1.60 => ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ omega )
% 2.91/1.60 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(replAx_type,type,
% 2.91/1.60 replAx: $o ).
% 2.91/1.60
% 2.91/1.60 thf(replAx,definition,
% 2.91/1.60 ( replAx
% 2.91/1.60 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.91/1.60 ( ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( exu
% 2.91/1.60 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.91/1.60 => ? [B: $i] :
% 2.91/1.60 ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ B )
% 2.91/1.60 <=> ? [Xy: $i] :
% 2.91/1.60 ( ( in @ Xy @ A )
% 2.91/1.60 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(foundationAx_type,type,
% 2.91/1.60 foundationAx: $o ).
% 2.91/1.60
% 2.91/1.60 thf(foundationAx,definition,
% 2.91/1.60 ( foundationAx
% 2.91/1.60 = ( ! [A: $i] :
% 2.91/1.60 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.91/1.60 => ? [B: $i] :
% 2.91/1.60 ( ( in @ B @ A )
% 2.91/1.60 & ~ ? [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ B )
% 2.91/1.60 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(wellorderingAx_type,type,
% 2.91/1.60 wellorderingAx: $o ).
% 2.91/1.60
% 2.91/1.60 thf(wellorderingAx,definition,
% 2.91/1.60 ( wellorderingAx
% 2.91/1.60 = ( ! [A: $i] :
% 2.91/1.60 ? [B: $i] :
% 2.91/1.60 ( ! [C: $i] :
% 2.91/1.60 ( ( in @ C @ B )
% 2.91/1.60 => ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ C )
% 2.91/1.60 => ( in @ Xx @ A ) ) )
% 2.91/1.60 & ! [Xx: $i,Xy: $i] :
% 2.91/1.60 ( ( ( in @ Xx @ A )
% 2.91/1.60 & ( in @ Xy @ A ) )
% 2.91/1.60 => ( ! [C: $i] :
% 2.91/1.60 ( ( in @ C @ B )
% 2.91/1.60 => ( ( in @ Xx @ C )
% 2.91/1.60 <=> ( in @ Xy @ C ) ) )
% 2.91/1.60 => ( Xx = Xy ) ) )
% 2.91/1.60 & ! [C: $i,D: $i] :
% 2.91/1.60 ( ( ( in @ C @ B )
% 2.91/1.60 & ( in @ D @ B ) )
% 2.91/1.60 => ( ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ C )
% 2.91/1.60 => ( in @ Xx @ D ) )
% 2.91/1.60 | ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ D )
% 2.91/1.60 => ( in @ Xx @ C ) ) ) )
% 2.91/1.60 & ! [C: $i] :
% 2.91/1.60 ( ( ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ C )
% 2.91/1.60 => ( in @ Xx @ A ) )
% 2.91/1.60 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.91/1.60 => ? [D: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ D @ B )
% 2.91/1.60 & ( in @ Xx @ C )
% 2.91/1.60 & ~ ? [Xy: $i] :
% 2.91/1.60 ( ( in @ Xy @ D )
% 2.91/1.60 & ( in @ Xy @ C ) )
% 2.91/1.60 & ! [E: $i] :
% 2.91/1.60 ( ( in @ E @ B )
% 2.91/1.60 => ( ! [Xy: $i] :
% 2.91/1.60 ( ( in @ Xy @ E )
% 2.91/1.60 => ( in @ Xy @ D ) )
% 2.91/1.60 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(descr_type,type,
% 2.91/1.60 descr: ( $i > $o ) > $i ).
% 2.91/1.60
% 2.91/1.60 thf(descrp_type,type,
% 2.91/1.60 descrp: $o ).
% 2.91/1.60
% 2.91/1.60 thf(descrp,definition,
% 2.91/1.60 ( descrp
% 2.91/1.60 = ( ! [Xphi: $i > $o] :
% 2.91/1.60 ( ( exu
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.60 => ( Xphi
% 2.91/1.60 @ ( descr
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(dsetconstr_type,type,
% 2.91/1.60 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.91/1.60
% 2.91/1.60 thf(dsetconstrI_type,type,
% 2.91/1.60 dsetconstrI: $o ).
% 2.91/1.60
% 2.91/1.60 thf(dsetconstrI,definition,
% 2.91/1.60 ( dsetconstrI
% 2.91/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( ( Xphi @ Xx )
% 2.91/1.60 => ( in @ Xx
% 2.91/1.60 @ ( dsetconstr @ A
% 2.91/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(dsetconstrEL_type,type,
% 2.91/1.60 dsetconstrEL: $o ).
% 2.91/1.60
% 2.91/1.60 thf(dsetconstrEL,definition,
% 2.91/1.60 ( dsetconstrEL
% 2.91/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx
% 2.91/1.60 @ ( dsetconstr @ A
% 2.91/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.91/1.60 => ( in @ Xx @ A ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(dsetconstrER_type,type,
% 2.91/1.60 dsetconstrER: $o ).
% 2.91/1.60
% 2.91/1.60 thf(dsetconstrER,definition,
% 2.91/1.60 ( dsetconstrER
% 2.91/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx
% 2.91/1.60 @ ( dsetconstr @ A
% 2.91/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.91/1.60 => ( Xphi @ Xx ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(exuE1_type,type,
% 2.91/1.60 exuE1: $o ).
% 2.91/1.60
% 2.91/1.60 thf(exuE1,definition,
% 2.91/1.60 ( exuE1
% 2.91/1.60 = ( ! [Xphi: $i > $o] :
% 2.91/1.60 ( ( exu
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.60 => ? [Xx: $i] :
% 2.91/1.60 ( ( Xphi @ Xx )
% 2.91/1.60 & ! [Xy: $i] :
% 2.91/1.60 ( ( Xphi @ Xy )
% 2.91/1.60 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(prop2set_type,type,
% 2.91/1.60 prop2set: $o > $i ).
% 2.91/1.60
% 2.91/1.60 thf(prop2setE_type,type,
% 2.91/1.60 prop2setE: $o ).
% 2.91/1.60
% 2.91/1.60 thf(prop2setE,definition,
% 2.91/1.60 ( prop2setE
% 2.91/1.60 = ( ! [Xphi: $o,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.91/1.60 => Xphi ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(emptysetE_type,type,
% 2.91/1.60 emptysetE: $o ).
% 2.91/1.60
% 2.91/1.60 thf(emptysetE,definition,
% 2.91/1.60 ( emptysetE
% 2.91/1.60 = ( ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ emptyset )
% 2.91/1.60 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(emptysetimpfalse_type,type,
% 2.91/1.60 emptysetimpfalse: $o ).
% 2.91/1.60
% 2.91/1.60 thf(emptysetimpfalse,definition,
% 2.91/1.60 ( emptysetimpfalse
% 2.91/1.60 = ( ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ emptyset )
% 2.91/1.60 => $false ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(notinemptyset_type,type,
% 2.91/1.60 notinemptyset: $o ).
% 2.91/1.60
% 2.91/1.60 thf(notinemptyset,definition,
% 2.91/1.60 ( notinemptyset
% 2.91/1.60 = ( ! [Xx: $i] :
% 2.91/1.60 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(exuE3e_type,type,
% 2.91/1.60 exuE3e: $o ).
% 2.91/1.60
% 2.91/1.60 thf(exuE3e,definition,
% 2.91/1.60 ( exuE3e
% 2.91/1.60 = ( ! [Xphi: $i > $o] :
% 2.91/1.60 ( ( exu
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.60 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setext_type,type,
% 2.91/1.60 setext: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setext,definition,
% 2.91/1.60 ( setext
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( in @ Xx @ B ) )
% 2.91/1.60 => ( ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ B )
% 2.91/1.60 => ( in @ Xx @ A ) )
% 2.91/1.60 => ( A = B ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(emptyI_type,type,
% 2.91/1.60 emptyI: $o ).
% 2.91/1.60
% 2.91/1.60 thf(emptyI,definition,
% 2.91/1.60 ( emptyI
% 2.91/1.60 = ( ! [A: $i] :
% 2.91/1.60 ( ! [Xx: $i] :
% 2.91/1.60 ~ ( in @ Xx @ A )
% 2.91/1.60 => ( A = emptyset ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(noeltsimpempty_type,type,
% 2.91/1.60 noeltsimpempty: $o ).
% 2.91/1.60
% 2.91/1.60 thf(noeltsimpempty,definition,
% 2.91/1.60 ( noeltsimpempty
% 2.91/1.60 = ( ! [A: $i] :
% 2.91/1.60 ( ! [Xx: $i] :
% 2.91/1.60 ~ ( in @ Xx @ A )
% 2.91/1.60 => ( A = emptyset ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setbeta_type,type,
% 2.91/1.60 setbeta: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setbeta,definition,
% 2.91/1.60 ( setbeta
% 2.91/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( ( in @ Xx
% 2.91/1.60 @ ( dsetconstr @ A
% 2.91/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.91/1.60 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(nonempty_type,type,
% 2.91/1.60 nonempty: $i > $o ).
% 2.91/1.60
% 2.91/1.60 thf(nonempty,definition,
% 2.91/1.60 ( nonempty
% 2.91/1.60 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(nonemptyE1_type,type,
% 2.91/1.60 nonemptyE1: $o ).
% 2.91/1.60
% 2.91/1.60 thf(nonemptyE1,definition,
% 2.91/1.60 ( nonemptyE1
% 2.91/1.60 = ( ! [A: $i] :
% 2.91/1.60 ( ( nonempty @ A )
% 2.91/1.60 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(nonemptyI_type,type,
% 2.91/1.60 nonemptyI: $o ).
% 2.91/1.60
% 2.91/1.60 thf(nonemptyI,definition,
% 2.91/1.60 ( nonemptyI
% 2.91/1.60 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( ( Xphi @ Xx )
% 2.91/1.60 => ( nonempty
% 2.91/1.60 @ ( dsetconstr @ A
% 2.91/1.60 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(nonemptyI1_type,type,
% 2.91/1.60 nonemptyI1: $o ).
% 2.91/1.60
% 2.91/1.60 thf(nonemptyI1,definition,
% 2.91/1.60 ( nonemptyI1
% 2.91/1.60 = ( ! [A: $i] :
% 2.91/1.60 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.91/1.60 => ( nonempty @ A ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoinIL_type,type,
% 2.91/1.60 setadjoinIL: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoinIL,definition,
% 2.91/1.60 ( setadjoinIL
% 2.91/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(emptyinunitempty_type,type,
% 2.91/1.60 emptyinunitempty: $o ).
% 2.91/1.60
% 2.91/1.60 thf(emptyinunitempty,definition,
% 2.91/1.60 ( emptyinunitempty
% 2.91/1.60 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoinIR_type,type,
% 2.91/1.60 setadjoinIR: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoinIR,definition,
% 2.91/1.60 ( setadjoinIR
% 2.91/1.60 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.91/1.60 ( ( in @ Xy @ A )
% 2.91/1.60 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoinE_type,type,
% 2.91/1.60 setadjoinE: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoinE,definition,
% 2.91/1.60 ( setadjoinE
% 2.91/1.60 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.91/1.60 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.91/1.60 => ! [Xphi: $o] :
% 2.91/1.60 ( ( ( Xy = Xx )
% 2.91/1.60 => Xphi )
% 2.91/1.60 => ( ( ( in @ Xy @ A )
% 2.91/1.60 => Xphi )
% 2.91/1.60 => Xphi ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoinOr_type,type,
% 2.91/1.60 setadjoinOr: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoinOr,definition,
% 2.91/1.60 ( setadjoinOr
% 2.91/1.60 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.91/1.60 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.91/1.60 => ( ( Xy = Xx )
% 2.91/1.60 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setoftrueEq_type,type,
% 2.91/1.60 setoftrueEq: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setoftrueEq,definition,
% 2.91/1.60 ( setoftrueEq
% 2.91/1.60 = ( ! [A: $i] :
% 2.91/1.60 ( ( dsetconstr @ A
% 2.91/1.60 @ ^ [Xx: $i] : $true )
% 2.91/1.60 = A ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(powersetI_type,type,
% 2.91/1.60 powersetI: $o ).
% 2.91/1.60
% 2.91/1.60 thf(powersetI,definition,
% 2.91/1.60 ( powersetI
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ B )
% 2.91/1.60 => ( in @ Xx @ A ) )
% 2.91/1.60 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(emptyinPowerset_type,type,
% 2.91/1.60 emptyinPowerset: $o ).
% 2.91/1.60
% 2.91/1.60 thf(emptyinPowerset,definition,
% 2.91/1.60 ( emptyinPowerset
% 2.91/1.60 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(emptyInPowerset_type,type,
% 2.91/1.60 emptyInPowerset: $o ).
% 2.91/1.60
% 2.91/1.60 thf(emptyInPowerset,definition,
% 2.91/1.60 ( emptyInPowerset
% 2.91/1.60 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(powersetE_type,type,
% 2.91/1.60 powersetE: $o ).
% 2.91/1.60
% 2.91/1.60 thf(powersetE,definition,
% 2.91/1.60 ( powersetE
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ B @ ( powerset @ A ) )
% 2.91/1.60 => ( ( in @ Xx @ B )
% 2.91/1.60 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setunionI_type,type,
% 2.91/1.60 setunionI: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setunionI,definition,
% 2.91/1.60 ( setunionI
% 2.91/1.60 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.91/1.60 ( ( in @ Xx @ B )
% 2.91/1.60 => ( ( in @ B @ A )
% 2.91/1.60 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setunionE_type,type,
% 2.91/1.60 setunionE: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setunionE,definition,
% 2.91/1.60 ( setunionE
% 2.91/1.60 = ( ! [A: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ ( setunion @ A ) )
% 2.91/1.60 => ! [Xphi: $o] :
% 2.91/1.60 ( ! [B: $i] :
% 2.91/1.60 ( ( in @ Xx @ B )
% 2.91/1.60 => ( ( in @ B @ A )
% 2.91/1.60 => Xphi ) )
% 2.91/1.60 => Xphi ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(subPowSU_type,type,
% 2.91/1.60 subPowSU: $o ).
% 2.91/1.60
% 2.91/1.60 thf(subPowSU,definition,
% 2.91/1.60 ( subPowSU
% 2.91/1.60 = ( ! [A: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(exuE2_type,type,
% 2.91/1.60 exuE2: $o ).
% 2.91/1.60
% 2.91/1.60 thf(exuE2,definition,
% 2.91/1.60 ( exuE2
% 2.91/1.60 = ( ! [Xphi: $i > $o] :
% 2.91/1.60 ( ( exu
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.60 => ? [Xx: $i] :
% 2.91/1.60 ! [Xy: $i] :
% 2.91/1.60 ( ( Xphi @ Xy )
% 2.91/1.60 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(nonemptyImpWitness_type,type,
% 2.91/1.60 nonemptyImpWitness: $o ).
% 2.91/1.60
% 2.91/1.60 thf(nonemptyImpWitness,definition,
% 2.91/1.60 ( nonemptyImpWitness
% 2.91/1.60 = ( ! [A: $i] :
% 2.91/1.60 ( ( nonempty @ A )
% 2.91/1.60 => ? [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 & $true ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(uniqinunit_type,type,
% 2.91/1.60 uniqinunit: $o ).
% 2.91/1.60
% 2.91/1.60 thf(uniqinunit,definition,
% 2.91/1.60 ( uniqinunit
% 2.91/1.60 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.60 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.91/1.60 => ( Xx = Xy ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(notinsingleton_type,type,
% 2.91/1.60 notinsingleton: $o ).
% 2.91/1.60
% 2.91/1.60 thf(notinsingleton,definition,
% 2.91/1.60 ( notinsingleton
% 2.91/1.60 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.60 ( ( Xx != Xy )
% 2.91/1.60 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(eqinunit_type,type,
% 2.91/1.60 eqinunit: $o ).
% 2.91/1.60
% 2.91/1.60 thf(eqinunit,definition,
% 2.91/1.60 ( eqinunit
% 2.91/1.60 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.60 ( ( Xx = Xy )
% 2.91/1.60 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(singletonsswitch_type,type,
% 2.91/1.60 singletonsswitch: $o ).
% 2.91/1.60
% 2.91/1.60 thf(singletonsswitch,definition,
% 2.91/1.60 ( singletonsswitch
% 2.91/1.60 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.60 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.91/1.60 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(upairsetE_type,type,
% 2.91/1.60 upairsetE: $o ).
% 2.91/1.60
% 2.91/1.60 thf(upairsetE,definition,
% 2.91/1.60 ( upairsetE
% 2.91/1.60 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.91/1.60 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.91/1.60 => ( ( Xz = Xx )
% 2.91/1.60 | ( Xz = Xy ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(upairsetIL_type,type,
% 2.91/1.60 upairsetIL: $o ).
% 2.91/1.60
% 2.91/1.60 thf(upairsetIL,definition,
% 2.91/1.60 ( upairsetIL
% 2.91/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(upairsetIR_type,type,
% 2.91/1.60 upairsetIR: $o ).
% 2.91/1.60
% 2.91/1.60 thf(upairsetIR,definition,
% 2.91/1.60 ( upairsetIR
% 2.91/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(emptyE1_type,type,
% 2.91/1.60 emptyE1: $o ).
% 2.91/1.60
% 2.91/1.60 thf(emptyE1,definition,
% 2.91/1.60 ( emptyE1
% 2.91/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.60 ( ? [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 & ( Xphi @ Xx ) )
% 2.91/1.60 => ( ( ( dsetconstr @ A
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.60 = emptyset )
% 2.91/1.60 => $false ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(vacuousDall_type,type,
% 2.91/1.60 vacuousDall: $o ).
% 2.91/1.60
% 2.91/1.60 thf(vacuousDall,definition,
% 2.91/1.60 ( vacuousDall
% 2.91/1.60 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ emptyset )
% 2.91/1.60 => ( Xphi @ Xx ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(quantDeMorgan1_type,type,
% 2.91/1.60 quantDeMorgan1: $o ).
% 2.91/1.60
% 2.91/1.60 thf(quantDeMorgan1,definition,
% 2.91/1.60 ( quantDeMorgan1
% 2.91/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.60 ( ~ ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( Xphi @ Xx ) )
% 2.91/1.60 => ? [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(quantDeMorgan2_type,type,
% 2.91/1.60 quantDeMorgan2: $o ).
% 2.91/1.60
% 2.91/1.60 thf(quantDeMorgan2,definition,
% 2.91/1.60 ( quantDeMorgan2
% 2.91/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.60 ( ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ~ ( Xphi @ Xx ) )
% 2.91/1.60 => ~ ? [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(quantDeMorgan3_type,type,
% 2.91/1.60 quantDeMorgan3: $o ).
% 2.91/1.60
% 2.91/1.60 thf(quantDeMorgan3,definition,
% 2.91/1.60 ( quantDeMorgan3
% 2.91/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.60 ( ~ ? [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 & ( Xphi @ Xx ) )
% 2.91/1.60 => ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(quantDeMorgan4_type,type,
% 2.91/1.60 quantDeMorgan4: $o ).
% 2.91/1.60
% 2.91/1.60 thf(quantDeMorgan4,definition,
% 2.91/1.60 ( quantDeMorgan4
% 2.91/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.60 ( ? [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 & ~ ( Xphi @ Xx ) )
% 2.91/1.60 => ~ ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(prop2setI_type,type,
% 2.91/1.60 prop2setI: $o ).
% 2.91/1.60
% 2.91/1.60 thf(prop2setI,definition,
% 2.91/1.60 ( prop2setI
% 2.91/1.60 = ( ! [Xphi: $o] :
% 2.91/1.60 ( Xphi
% 2.91/1.60 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(set2prop_type,type,
% 2.91/1.60 set2prop: $i > $o ).
% 2.91/1.60
% 2.91/1.60 thf(prop2set2propI_type,type,
% 2.91/1.60 prop2set2propI: $o ).
% 2.91/1.60
% 2.91/1.60 thf(prop2set2propI,definition,
% 2.91/1.60 ( prop2set2propI
% 2.91/1.60 = ( ! [Xphi: $o] :
% 2.91/1.60 ( Xphi
% 2.91/1.60 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(notdexE_type,type,
% 2.91/1.60 notdexE: $o ).
% 2.91/1.60
% 2.91/1.60 thf(notdexE,definition,
% 2.91/1.60 ( notdexE
% 2.91/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.60 ( ~ ? [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 & ( Xphi @ Xx ) )
% 2.91/1.60 => ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(notdallE_type,type,
% 2.91/1.60 notdallE: $o ).
% 2.91/1.60
% 2.91/1.60 thf(notdallE,definition,
% 2.91/1.60 ( notdallE
% 2.91/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.60 ( ~ ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( Xphi @ Xx ) )
% 2.91/1.60 => ? [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(exuI1_type,type,
% 2.91/1.60 exuI1: $o ).
% 2.91/1.60
% 2.91/1.60 thf(exuI1,definition,
% 2.91/1.60 ( exuI1
% 2.91/1.60 = ( ! [Xphi: $i > $o] :
% 2.91/1.60 ( ? [Xx: $i] :
% 2.91/1.60 ( ( Xphi @ Xx )
% 2.91/1.60 & ! [Xy: $i] :
% 2.91/1.60 ( ( Xphi @ Xy )
% 2.91/1.60 => ( Xx = Xy ) ) )
% 2.91/1.60 => ( exu
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(exuI3_type,type,
% 2.91/1.60 exuI3: $o ).
% 2.91/1.60
% 2.91/1.60 thf(exuI3,definition,
% 2.91/1.60 ( exuI3
% 2.91/1.60 = ( ! [Xphi: $i > $o] :
% 2.91/1.60 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.91/1.60 => ( ! [Xx: $i,Xy: $i] :
% 2.91/1.60 ( ( Xphi @ Xx )
% 2.91/1.60 => ( ( Xphi @ Xy )
% 2.91/1.60 => ( Xx = Xy ) ) )
% 2.91/1.60 => ( exu
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(exuI2_type,type,
% 2.91/1.60 exuI2: $o ).
% 2.91/1.60
% 2.91/1.60 thf(exuI2,definition,
% 2.91/1.60 ( exuI2
% 2.91/1.60 = ( ! [Xphi: $i > $o] :
% 2.91/1.60 ( ? [Xx: $i] :
% 2.91/1.60 ! [Xy: $i] :
% 2.91/1.60 ( ( Xphi @ Xy )
% 2.91/1.60 <=> ( Xy = Xx ) )
% 2.91/1.60 => ( exu
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(inCongP_type,type,
% 2.91/1.60 inCongP: $o ).
% 2.91/1.60
% 2.91/1.60 thf(inCongP,definition,
% 2.91/1.60 ( inCongP
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( A = B )
% 2.91/1.60 => ! [Xx: $i,Xy: $i] :
% 2.91/1.60 ( ( Xx = Xy )
% 2.91/1.60 => ( ( in @ Xx @ A )
% 2.91/1.60 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(in__Cong_type,type,
% 2.91/1.60 in__Cong: $o ).
% 2.91/1.60
% 2.91/1.60 thf(in__Cong,definition,
% 2.91/1.60 ( in__Cong
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( A = B )
% 2.91/1.60 => ! [Xx: $i,Xy: $i] :
% 2.91/1.60 ( ( Xx = Xy )
% 2.91/1.60 => ( ( in @ Xx @ A )
% 2.91/1.60 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(exuE3u_type,type,
% 2.91/1.60 exuE3u: $o ).
% 2.91/1.60
% 2.91/1.60 thf(exuE3u,definition,
% 2.91/1.60 ( exuE3u
% 2.91/1.60 = ( ! [Xphi: $i > $o] :
% 2.91/1.60 ( ( exu
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.60 => ! [Xx: $i,Xy: $i] :
% 2.91/1.60 ( ( Xphi @ Xx )
% 2.91/1.60 => ( ( Xphi @ Xy )
% 2.91/1.60 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(exu__Cong_type,type,
% 2.91/1.60 exu__Cong: $o ).
% 2.91/1.60
% 2.91/1.60 thf(exu__Cong,definition,
% 2.91/1.60 ( exu__Cong
% 2.91/1.60 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.91/1.60 ( ! [Xx: $i,Xy: $i] :
% 2.91/1.60 ( ( Xx = Xy )
% 2.91/1.60 => ( ( Xphi @ Xx )
% 2.91/1.60 <=> ( Xpsi @ Xy ) ) )
% 2.91/1.60 => ( ( exu
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.60 <=> ( exu
% 2.91/1.60 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(emptyset__Cong_type,type,
% 2.91/1.60 emptyset__Cong: $o ).
% 2.91/1.60
% 2.91/1.60 thf(emptyset__Cong,definition,
% 2.91/1.60 ( emptyset__Cong
% 2.91/1.60 = ( emptyset = emptyset ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoin__Cong_type,type,
% 2.91/1.60 setadjoin__Cong: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoin__Cong,definition,
% 2.91/1.60 ( setadjoin__Cong
% 2.91/1.60 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.60 ( ( Xx = Xy )
% 2.91/1.60 => ! [Xz: $i,Xu: $i] :
% 2.91/1.60 ( ( Xz = Xu )
% 2.91/1.60 => ( ( setadjoin @ Xx @ Xz )
% 2.91/1.60 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(powerset__Cong_type,type,
% 2.91/1.60 powerset__Cong: $o ).
% 2.91/1.60
% 2.91/1.60 thf(powerset__Cong,definition,
% 2.91/1.60 ( powerset__Cong
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( A = B )
% 2.91/1.60 => ( ( powerset @ A )
% 2.91/1.60 = ( powerset @ B ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setunion__Cong_type,type,
% 2.91/1.60 setunion__Cong: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setunion__Cong,definition,
% 2.91/1.60 ( setunion__Cong
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( A = B )
% 2.91/1.60 => ( ( setunion @ A )
% 2.91/1.60 = ( setunion @ B ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(omega__Cong_type,type,
% 2.91/1.60 omega__Cong: $o ).
% 2.91/1.60
% 2.91/1.60 thf(omega__Cong,definition,
% 2.91/1.60 ( omega__Cong
% 2.91/1.60 = ( omega = omega ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(exuEu_type,type,
% 2.91/1.60 exuEu: $o ).
% 2.91/1.60
% 2.91/1.60 thf(exuEu,definition,
% 2.91/1.60 ( exuEu
% 2.91/1.60 = ( ! [Xphi: $i > $o] :
% 2.91/1.60 ( ( exu
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.60 => ! [Xx: $i,Xy: $i] :
% 2.91/1.60 ( ( Xphi @ Xx )
% 2.91/1.60 => ( ( Xphi @ Xy )
% 2.91/1.60 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(descr__Cong_type,type,
% 2.91/1.60 descr__Cong: $o ).
% 2.91/1.60
% 2.91/1.60 thf(descr__Cong,definition,
% 2.91/1.60 ( descr__Cong
% 2.91/1.60 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.91/1.60 ( ! [Xx: $i,Xy: $i] :
% 2.91/1.60 ( ( Xx = Xy )
% 2.91/1.60 => ( ( Xphi @ Xx )
% 2.91/1.60 <=> ( Xpsi @ Xy ) ) )
% 2.91/1.60 => ( ( exu
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.60 => ( ( exu
% 2.91/1.60 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.91/1.60 => ( ( descr
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.60 = ( descr
% 2.91/1.60 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(dsetconstr__Cong_type,type,
% 2.91/1.60 dsetconstr__Cong: $o ).
% 2.91/1.60
% 2.91/1.60 thf(dsetconstr__Cong,definition,
% 2.91/1.60 ( dsetconstr__Cong
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( A = B )
% 2.91/1.60 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.91/1.60 ( ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ! [Xy: $i] :
% 2.91/1.60 ( ( in @ Xy @ B )
% 2.91/1.60 => ( ( Xx = Xy )
% 2.91/1.60 => ( ( Xphi @ Xx )
% 2.91/1.60 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.91/1.60 => ( ( dsetconstr @ A
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.60 = ( dsetconstr @ B
% 2.91/1.60 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(subset_type,type,
% 2.91/1.60 subset: $i > $i > $o ).
% 2.91/1.60
% 2.91/1.60 thf(disjoint_type,type,
% 2.91/1.60 disjoint: $i > $i > $o ).
% 2.91/1.60
% 2.91/1.60 thf(setsmeet_type,type,
% 2.91/1.60 setsmeet: $i > $i > $o ).
% 2.91/1.60
% 2.91/1.60 thf(subsetI1_type,type,
% 2.91/1.60 subsetI1: $o ).
% 2.91/1.60
% 2.91/1.60 thf(subsetI1,definition,
% 2.91/1.60 ( subsetI1
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( in @ Xx @ B ) )
% 2.91/1.60 => ( subset @ A @ B ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(eqimpsubset2_type,type,
% 2.91/1.60 eqimpsubset2: $o ).
% 2.91/1.60
% 2.91/1.60 thf(eqimpsubset2,definition,
% 2.91/1.60 ( eqimpsubset2
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( A = B )
% 2.91/1.60 => ( subset @ B @ A ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(eqimpsubset1_type,type,
% 2.91/1.60 eqimpsubset1: $o ).
% 2.91/1.60
% 2.91/1.60 thf(eqimpsubset1,definition,
% 2.91/1.60 ( eqimpsubset1
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( A = B )
% 2.91/1.60 => ( subset @ A @ B ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(subsetI2_type,type,
% 2.91/1.60 subsetI2: $o ).
% 2.91/1.60
% 2.91/1.60 thf(subsetI2,definition,
% 2.91/1.60 ( subsetI2
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ! [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( in @ Xx @ B ) )
% 2.91/1.60 => ( subset @ A @ B ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(emptysetsubset_type,type,
% 2.91/1.60 emptysetsubset: $o ).
% 2.91/1.60
% 2.91/1.60 thf(emptysetsubset,definition,
% 2.91/1.60 ( emptysetsubset
% 2.91/1.60 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(subsetE_type,type,
% 2.91/1.60 subsetE: $o ).
% 2.91/1.60
% 2.91/1.60 thf(subsetE,definition,
% 2.91/1.60 ( subsetE
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( subset @ A @ B )
% 2.91/1.60 => ( ( in @ Xx @ A )
% 2.91/1.60 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(subsetE2_type,type,
% 2.91/1.60 subsetE2: $o ).
% 2.91/1.60
% 2.91/1.60 thf(subsetE2,definition,
% 2.91/1.60 ( subsetE2
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( subset @ A @ B )
% 2.91/1.60 => ( ~ ( in @ Xx @ B )
% 2.91/1.60 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(notsubsetI_type,type,
% 2.91/1.60 notsubsetI: $o ).
% 2.91/1.60
% 2.91/1.60 thf(notsubsetI,definition,
% 2.91/1.60 ( notsubsetI
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( ~ ( in @ Xx @ B )
% 2.91/1.60 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(notequalI1_type,type,
% 2.91/1.60 notequalI1: $o ).
% 2.91/1.60
% 2.91/1.60 thf(notequalI1,definition,
% 2.91/1.60 ( notequalI1
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ~ ( subset @ A @ B )
% 2.91/1.60 => ( A != B ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(notequalI2_type,type,
% 2.91/1.60 notequalI2: $o ).
% 2.91/1.60
% 2.91/1.60 thf(notequalI2,definition,
% 2.91/1.60 ( notequalI2
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( ~ ( in @ Xx @ B )
% 2.91/1.60 => ( A != B ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(subsetRefl_type,type,
% 2.91/1.60 subsetRefl: $o ).
% 2.91/1.60
% 2.91/1.60 thf(subsetRefl,definition,
% 2.91/1.60 ( subsetRefl
% 2.91/1.60 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(subsetTrans_type,type,
% 2.91/1.60 subsetTrans: $o ).
% 2.91/1.60
% 2.91/1.60 thf(subsetTrans,definition,
% 2.91/1.60 ( subsetTrans
% 2.91/1.60 = ( ! [A: $i,B: $i,C: $i] :
% 2.91/1.60 ( ( subset @ A @ B )
% 2.91/1.60 => ( ( subset @ B @ C )
% 2.91/1.60 => ( subset @ A @ C ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoinSub_type,type,
% 2.91/1.60 setadjoinSub: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoinSub,definition,
% 2.91/1.60 ( setadjoinSub
% 2.91/1.60 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoinSub2_type,type,
% 2.91/1.60 setadjoinSub2: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setadjoinSub2,definition,
% 2.91/1.60 ( setadjoinSub2
% 2.91/1.60 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.91/1.60 ( ( subset @ A @ B )
% 2.91/1.60 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(subset2powerset_type,type,
% 2.91/1.60 subset2powerset: $o ).
% 2.91/1.60
% 2.91/1.60 thf(subset2powerset,definition,
% 2.91/1.60 ( subset2powerset
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( subset @ A @ B )
% 2.91/1.60 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setextsub_type,type,
% 2.91/1.60 setextsub: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setextsub,definition,
% 2.91/1.60 ( setextsub
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( subset @ A @ B )
% 2.91/1.60 => ( ( subset @ B @ A )
% 2.91/1.60 => ( A = B ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(subsetemptysetimpeq_type,type,
% 2.91/1.60 subsetemptysetimpeq: $o ).
% 2.91/1.60
% 2.91/1.60 thf(subsetemptysetimpeq,definition,
% 2.91/1.60 ( subsetemptysetimpeq
% 2.91/1.60 = ( ! [A: $i] :
% 2.91/1.60 ( ( subset @ A @ emptyset )
% 2.91/1.60 => ( A = emptyset ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(powersetI1_type,type,
% 2.91/1.60 powersetI1: $o ).
% 2.91/1.60
% 2.91/1.60 thf(powersetI1,definition,
% 2.91/1.60 ( powersetI1
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( subset @ B @ A )
% 2.91/1.60 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(powersetE1_type,type,
% 2.91/1.60 powersetE1: $o ).
% 2.91/1.60
% 2.91/1.60 thf(powersetE1,definition,
% 2.91/1.60 ( powersetE1
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( in @ B @ ( powerset @ A ) )
% 2.91/1.60 => ( subset @ B @ A ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(inPowerset_type,type,
% 2.91/1.60 inPowerset: $o ).
% 2.91/1.60
% 2.91/1.60 thf(inPowerset,definition,
% 2.91/1.60 ( inPowerset
% 2.91/1.60 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(powersetsubset_type,type,
% 2.91/1.60 powersetsubset: $o ).
% 2.91/1.60
% 2.91/1.60 thf(powersetsubset,definition,
% 2.91/1.60 ( powersetsubset
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( subset @ A @ B )
% 2.91/1.60 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(sepInPowerset_type,type,
% 2.91/1.60 sepInPowerset: $o ).
% 2.91/1.60
% 2.91/1.60 thf(sepInPowerset,definition,
% 2.91/1.60 ( sepInPowerset
% 2.91/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.60 ( in
% 2.91/1.60 @ ( dsetconstr @ A
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.60 @ ( powerset @ A ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(sepSubset_type,type,
% 2.91/1.60 sepSubset: $o ).
% 2.91/1.60
% 2.91/1.60 thf(sepSubset,definition,
% 2.91/1.60 ( sepSubset
% 2.91/1.60 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.60 ( subset
% 2.91/1.60 @ ( dsetconstr @ A
% 2.91/1.60 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.60 @ A ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(binunion_type,type,
% 2.91/1.60 binunion: $i > $i > $i ).
% 2.91/1.60
% 2.91/1.60 thf(binunionIL_type,type,
% 2.91/1.60 binunionIL: $o ).
% 2.91/1.60
% 2.91/1.60 thf(binunionIL,definition,
% 2.91/1.60 ( binunionIL
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(upairset2IR_type,type,
% 2.91/1.60 upairset2IR: $o ).
% 2.91/1.60
% 2.91/1.60 thf(upairset2IR,definition,
% 2.91/1.60 ( upairset2IR
% 2.91/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(binunionIR_type,type,
% 2.91/1.60 binunionIR: $o ).
% 2.91/1.60
% 2.91/1.60 thf(binunionIR,definition,
% 2.91/1.60 ( binunionIR
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ B )
% 2.91/1.60 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(binunionEcases_type,type,
% 2.91/1.60 binunionEcases: $o ).
% 2.91/1.60
% 2.91/1.60 thf(binunionEcases,definition,
% 2.91/1.60 ( binunionEcases
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.91/1.60 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.91/1.60 => ( ( ( in @ Xx @ A )
% 2.91/1.60 => Xphi )
% 2.91/1.60 => ( ( ( in @ Xx @ B )
% 2.91/1.60 => Xphi )
% 2.91/1.60 => Xphi ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(binunionE_type,type,
% 2.91/1.60 binunionE: $o ).
% 2.91/1.60
% 2.91/1.60 thf(binunionE,definition,
% 2.91/1.60 ( binunionE
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.91/1.60 => ( ( in @ Xx @ A )
% 2.91/1.60 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(binunionLsub_type,type,
% 2.91/1.60 binunionLsub: $o ).
% 2.91/1.60
% 2.91/1.60 thf(binunionLsub,definition,
% 2.91/1.60 ( binunionLsub
% 2.91/1.60 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(binunionRsub_type,type,
% 2.91/1.60 binunionRsub: $o ).
% 2.91/1.60
% 2.91/1.60 thf(binunionRsub,definition,
% 2.91/1.60 ( binunionRsub
% 2.91/1.60 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(binintersect_type,type,
% 2.91/1.60 binintersect: $i > $i > $i ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectI_type,type,
% 2.91/1.60 binintersectI: $o ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectI,definition,
% 2.91/1.60 ( binintersectI
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( ( in @ Xx @ B )
% 2.91/1.60 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectSubset5_type,type,
% 2.91/1.60 binintersectSubset5: $o ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectSubset5,definition,
% 2.91/1.60 ( binintersectSubset5
% 2.91/1.60 = ( ! [A: $i,B: $i,C: $i] :
% 2.91/1.60 ( ( subset @ C @ A )
% 2.91/1.60 => ( ( subset @ C @ B )
% 2.91/1.60 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectEL_type,type,
% 2.91/1.60 binintersectEL: $o ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectEL,definition,
% 2.91/1.60 ( binintersectEL
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.91/1.60 => ( in @ Xx @ A ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectLsub_type,type,
% 2.91/1.60 binintersectLsub: $o ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectLsub,definition,
% 2.91/1.60 ( binintersectLsub
% 2.91/1.60 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectSubset2_type,type,
% 2.91/1.60 binintersectSubset2: $o ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectSubset2,definition,
% 2.91/1.60 ( binintersectSubset2
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( subset @ A @ B )
% 2.91/1.60 => ( ( binintersect @ A @ B )
% 2.91/1.60 = A ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectSubset3_type,type,
% 2.91/1.60 binintersectSubset3: $o ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectSubset3,definition,
% 2.91/1.60 ( binintersectSubset3
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( ( binintersect @ A @ B )
% 2.91/1.60 = B )
% 2.91/1.60 => ( subset @ B @ A ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectER_type,type,
% 2.91/1.60 binintersectER: $o ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectER,definition,
% 2.91/1.60 ( binintersectER
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.91/1.60 => ( in @ Xx @ B ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(disjointsetsI1_type,type,
% 2.91/1.60 disjointsetsI1: $o ).
% 2.91/1.60
% 2.91/1.60 thf(disjointsetsI1,definition,
% 2.91/1.60 ( disjointsetsI1
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ~ ? [Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 & ( in @ Xx @ B ) )
% 2.91/1.60 => ( ( binintersect @ A @ B )
% 2.91/1.60 = emptyset ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectRsub_type,type,
% 2.91/1.60 binintersectRsub: $o ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectRsub,definition,
% 2.91/1.60 ( binintersectRsub
% 2.91/1.60 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectSubset4_type,type,
% 2.91/1.60 binintersectSubset4: $o ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectSubset4,definition,
% 2.91/1.60 ( binintersectSubset4
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( subset @ B @ A )
% 2.91/1.60 => ( ( binintersect @ A @ B )
% 2.91/1.60 = B ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectSubset1_type,type,
% 2.91/1.60 binintersectSubset1: $o ).
% 2.91/1.60
% 2.91/1.60 thf(binintersectSubset1,definition,
% 2.91/1.60 ( binintersectSubset1
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( ( binintersect @ A @ B )
% 2.91/1.60 = A )
% 2.91/1.60 => ( subset @ A @ B ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(bs114d_type,type,
% 2.91/1.60 bs114d: $o ).
% 2.91/1.60
% 2.91/1.60 thf(bs114d,definition,
% 2.91/1.60 ( bs114d
% 2.91/1.60 = ( ! [A: $i,B: $i,C: $i] :
% 2.91/1.60 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.91/1.60 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(regular_type,type,
% 2.91/1.60 regular: $i > $o ).
% 2.91/1.60
% 2.91/1.60 thf(setminus_type,type,
% 2.91/1.60 setminus: $i > $i > $i ).
% 2.91/1.60
% 2.91/1.60 thf(setminusI_type,type,
% 2.91/1.60 setminusI: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setminusI,definition,
% 2.91/1.60 ( setminusI
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( ~ ( in @ Xx @ B )
% 2.91/1.60 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setminusEL_type,type,
% 2.91/1.60 setminusEL: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setminusEL,definition,
% 2.91/1.60 ( setminusEL
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.91/1.60 => ( in @ Xx @ A ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setminusER_type,type,
% 2.91/1.60 setminusER: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setminusER,definition,
% 2.91/1.60 ( setminusER
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.91/1.60 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setminusSubset2_type,type,
% 2.91/1.60 setminusSubset2: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setminusSubset2,definition,
% 2.91/1.60 ( setminusSubset2
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( subset @ A @ B )
% 2.91/1.60 => ( ( setminus @ A @ B )
% 2.91/1.60 = emptyset ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setminusERneg_type,type,
% 2.91/1.60 setminusERneg: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setminusERneg,definition,
% 2.91/1.60 ( setminusERneg
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.91/1.60 => ( ( in @ Xx @ A )
% 2.91/1.60 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setminusELneg_type,type,
% 2.91/1.60 setminusELneg: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setminusELneg,definition,
% 2.91/1.60 ( setminusELneg
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.91/1.60 => ( ~ ( in @ Xx @ B )
% 2.91/1.60 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setminusILneg_type,type,
% 2.91/1.60 setminusILneg: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setminusILneg,definition,
% 2.91/1.60 ( setminusILneg
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ~ ( in @ Xx @ A )
% 2.91/1.60 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setminusIRneg_type,type,
% 2.91/1.60 setminusIRneg: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setminusIRneg,definition,
% 2.91/1.60 ( setminusIRneg
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ B )
% 2.91/1.60 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setminusLsub_type,type,
% 2.91/1.60 setminusLsub: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setminusLsub,definition,
% 2.91/1.60 ( setminusLsub
% 2.91/1.60 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setminusSubset1_type,type,
% 2.91/1.60 setminusSubset1: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setminusSubset1,definition,
% 2.91/1.60 ( setminusSubset1
% 2.91/1.60 = ( ! [A: $i,B: $i] :
% 2.91/1.60 ( ( ( setminus @ A @ B )
% 2.91/1.60 = emptyset )
% 2.91/1.60 => ( subset @ A @ B ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(symdiff_type,type,
% 2.91/1.60 symdiff: $i > $i > $i ).
% 2.91/1.60
% 2.91/1.60 thf(symdiffE_type,type,
% 2.91/1.60 symdiffE: $o ).
% 2.91/1.60
% 2.91/1.60 thf(symdiffE,definition,
% 2.91/1.60 ( symdiffE
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.91/1.60 => ! [Xphi: $o] :
% 2.91/1.60 ( ( ( in @ Xx @ A )
% 2.91/1.60 => ( ~ ( in @ Xx @ B )
% 2.91/1.60 => Xphi ) )
% 2.91/1.60 => ( ( ~ ( in @ Xx @ A )
% 2.91/1.60 => ( ( in @ Xx @ B )
% 2.91/1.60 => Xphi ) )
% 2.91/1.60 => Xphi ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(symdiffI1_type,type,
% 2.91/1.60 symdiffI1: $o ).
% 2.91/1.60
% 2.91/1.60 thf(symdiffI1,definition,
% 2.91/1.60 ( symdiffI1
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( ~ ( in @ Xx @ B )
% 2.91/1.60 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(symdiffI2_type,type,
% 2.91/1.60 symdiffI2: $o ).
% 2.91/1.60
% 2.91/1.60 thf(symdiffI2,definition,
% 2.91/1.60 ( symdiffI2
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ~ ( in @ Xx @ A )
% 2.91/1.60 => ( ( in @ Xx @ B )
% 2.91/1.60 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(symdiffIneg1_type,type,
% 2.91/1.60 symdiffIneg1: $o ).
% 2.91/1.60
% 2.91/1.60 thf(symdiffIneg1,definition,
% 2.91/1.60 ( symdiffIneg1
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( ( in @ Xx @ B )
% 2.91/1.60 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(symdiffIneg2_type,type,
% 2.91/1.60 symdiffIneg2: $o ).
% 2.91/1.60
% 2.91/1.60 thf(symdiffIneg2,definition,
% 2.91/1.60 ( symdiffIneg2
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ~ ( in @ Xx @ A )
% 2.91/1.60 => ( ~ ( in @ Xx @ B )
% 2.91/1.60 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(iskpair_type,type,
% 2.91/1.60 iskpair: $i > $o ).
% 2.91/1.60
% 2.91/1.60 thf(secondinupair_type,type,
% 2.91/1.60 secondinupair: $o ).
% 2.91/1.60
% 2.91/1.60 thf(secondinupair,definition,
% 2.91/1.60 ( secondinupair
% 2.91/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setukpairIL_type,type,
% 2.91/1.60 setukpairIL: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setukpairIL,definition,
% 2.91/1.60 ( setukpairIL
% 2.91/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(setukpairIR_type,type,
% 2.91/1.60 setukpairIR: $o ).
% 2.91/1.60
% 2.91/1.60 thf(setukpairIR,definition,
% 2.91/1.60 ( setukpairIR
% 2.91/1.60 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(kpairiskpair_type,type,
% 2.91/1.60 kpairiskpair: $o ).
% 2.91/1.60
% 2.91/1.60 thf(kpairiskpair,definition,
% 2.91/1.60 ( kpairiskpair
% 2.91/1.60 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(kpair_type,type,
% 2.91/1.60 kpair: $i > $i > $i ).
% 2.91/1.60
% 2.91/1.60 thf(kpairp_type,type,
% 2.91/1.60 kpairp: $o ).
% 2.91/1.60
% 2.91/1.60 thf(kpairp,definition,
% 2.91/1.60 ( kpairp
% 2.91/1.60 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(cartprod_type,type,
% 2.91/1.60 cartprod: $i > $i > $i ).
% 2.91/1.60
% 2.91/1.60 thf(singletonsubset_type,type,
% 2.91/1.60 singletonsubset: $o ).
% 2.91/1.60
% 2.91/1.60 thf(singletonsubset,definition,
% 2.91/1.60 ( singletonsubset
% 2.91/1.60 = ( ! [A: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(singletoninpowerset_type,type,
% 2.91/1.60 singletoninpowerset: $o ).
% 2.91/1.60
% 2.91/1.60 thf(singletoninpowerset,definition,
% 2.91/1.60 ( singletoninpowerset
% 2.91/1.60 = ( ! [A: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(singletoninpowunion_type,type,
% 2.91/1.60 singletoninpowunion: $o ).
% 2.91/1.60
% 2.91/1.60 thf(singletoninpowunion,definition,
% 2.91/1.60 ( singletoninpowunion
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(upairset2E_type,type,
% 2.91/1.60 upairset2E: $o ).
% 2.91/1.60
% 2.91/1.60 thf(upairset2E,definition,
% 2.91/1.60 ( upairset2E
% 2.91/1.60 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.91/1.60 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.91/1.60 => ( ( Xz = Xx )
% 2.91/1.60 | ( Xz = Xy ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(upairsubunion_type,type,
% 2.91/1.60 upairsubunion: $o ).
% 2.91/1.60
% 2.91/1.60 thf(upairsubunion,definition,
% 2.91/1.60 ( upairsubunion
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ! [Xy: $i] :
% 2.91/1.60 ( ( in @ Xy @ B )
% 2.91/1.60 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(upairinpowunion_type,type,
% 2.91/1.60 upairinpowunion: $o ).
% 2.91/1.60
% 2.91/1.60 thf(upairinpowunion,definition,
% 2.91/1.60 ( upairinpowunion
% 2.91/1.60 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.60 ( ( in @ Xx @ A )
% 2.91/1.60 => ! [Xy: $i] :
% 2.91/1.60 ( ( in @ Xy @ B )
% 2.91/1.60 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.91/1.60
% 2.91/1.60 thf(ubforcartprodlem1_type,type,
% 2.91/1.60 ubforcartprodlem1: $o ).
% 2.91/1.60
% 2.91/1.61 thf(ubforcartprodlem1,definition,
% 2.91/1.61 ( ubforcartprodlem1
% 2.91/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(ubforcartprodlem2_type,type,
% 2.91/1.61 ubforcartprodlem2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(ubforcartprodlem2,definition,
% 2.91/1.61 ( ubforcartprodlem2
% 2.91/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(ubforcartprodlem3_type,type,
% 2.91/1.61 ubforcartprodlem3: $o ).
% 2.91/1.61
% 2.91/1.61 thf(ubforcartprodlem3,definition,
% 2.91/1.61 ( ubforcartprodlem3
% 2.91/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodpairin_type,type,
% 2.91/1.61 cartprodpairin: $o ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodpairin,definition,
% 2.91/1.61 ( cartprodpairin
% 2.91/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodmempair1_type,type,
% 2.91/1.61 cartprodmempair1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodmempair1,definition,
% 2.91/1.61 ( cartprodmempair1
% 2.91/1.61 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.91/1.61 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.91/1.61 => ? [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 & ? [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 & ( Xu
% 2.91/1.61 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodmempair_type,type,
% 2.91/1.61 cartprodmempair: $o ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodmempair,definition,
% 2.91/1.61 ( cartprodmempair
% 2.91/1.61 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.91/1.61 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.91/1.61 => ( iskpair @ Xu ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(setunionE2_type,type,
% 2.91/1.61 setunionE2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(setunionE2,definition,
% 2.91/1.61 ( setunionE2
% 2.91/1.61 = ( ! [A: $i,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ ( setunion @ A ) )
% 2.91/1.61 => ? [X: $i] :
% 2.91/1.61 ( ( in @ X @ A )
% 2.91/1.61 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(setunionsingleton1_type,type,
% 2.91/1.61 setunionsingleton1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(setunionsingleton1,definition,
% 2.91/1.61 ( setunionsingleton1
% 2.91/1.61 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(setunionsingleton2_type,type,
% 2.91/1.61 setunionsingleton2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(setunionsingleton2,definition,
% 2.91/1.61 ( setunionsingleton2
% 2.91/1.61 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(setunionsingleton_type,type,
% 2.91/1.61 setunionsingleton: $o ).
% 2.91/1.61
% 2.91/1.61 thf(setunionsingleton,definition,
% 2.91/1.61 ( setunionsingleton
% 2.91/1.61 = ( ! [Xx: $i] :
% 2.91/1.61 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.91/1.61 = Xx ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(singleton_type,type,
% 2.91/1.61 singleton: $i > $o ).
% 2.91/1.61
% 2.91/1.61 thf(singletonprop_type,type,
% 2.91/1.61 singletonprop: $o ).
% 2.91/1.61
% 2.91/1.61 thf(singletonprop,definition,
% 2.91/1.61 ( singletonprop
% 2.91/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.61 ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ A )
% 2.91/1.61 => ( ( Xphi @ Xx )
% 2.91/1.61 => ( ( Xphi @ Xy )
% 2.91/1.61 => ( Xx = Xy ) ) ) ) )
% 2.91/1.61 => ( ? [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 & ( Xphi @ Xx ) )
% 2.91/1.61 => ( singleton
% 2.91/1.61 @ ( dsetconstr @ A
% 2.91/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(ex1_type,type,
% 2.91/1.61 ex1: $i > ( $i > $o ) > $o ).
% 2.91/1.61
% 2.91/1.61 thf(ex1,definition,
% 2.91/1.61 ( ex1
% 2.91/1.61 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.91/1.61 ( singleton
% 2.91/1.61 @ ( dsetconstr @ A
% 2.91/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(ex1E1_type,type,
% 2.91/1.61 ex1E1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(ex1E1,definition,
% 2.91/1.61 ( ex1E1
% 2.91/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.61 ( ( ex1 @ A
% 2.91/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.61 => ? [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(ex1I_type,type,
% 2.91/1.61 ex1I: $o ).
% 2.91/1.61
% 2.91/1.61 thf(ex1I,definition,
% 2.91/1.61 ( ex1I
% 2.91/1.61 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( Xphi @ Xx )
% 2.91/1.61 => ( ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ A )
% 2.91/1.61 => ( ( Xphi @ Xy )
% 2.91/1.61 => ( Xy = Xx ) ) )
% 2.91/1.61 => ( ex1 @ A
% 2.91/1.61 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(ex1I2_type,type,
% 2.91/1.61 ex1I2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(ex1I2,definition,
% 2.91/1.61 ( ex1I2
% 2.91/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.61 ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ A )
% 2.91/1.61 => ( ( Xphi @ Xx )
% 2.91/1.61 => ( ( Xphi @ Xy )
% 2.91/1.61 => ( Xx = Xy ) ) ) ) )
% 2.91/1.61 => ( ? [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 & ( Xphi @ Xx ) )
% 2.91/1.61 => ( ex1 @ A
% 2.91/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(singletonsuniq_type,type,
% 2.91/1.61 singletonsuniq: $o ).
% 2.91/1.61
% 2.91/1.61 thf(singletonsuniq,definition,
% 2.91/1.61 ( singletonsuniq
% 2.91/1.61 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.61 ( ( ( setadjoin @ Xx @ emptyset )
% 2.91/1.61 = ( setadjoin @ Xy @ emptyset ) )
% 2.91/1.61 => ( Xx = Xy ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(atmost1p_type,type,
% 2.91/1.61 atmost1p: $i > $o ).
% 2.91/1.61
% 2.91/1.61 thf(atleast2p_type,type,
% 2.91/1.61 atleast2p: $i > $o ).
% 2.91/1.61
% 2.91/1.61 thf(atmost2p_type,type,
% 2.91/1.61 atmost2p: $i > $o ).
% 2.91/1.61
% 2.91/1.61 thf(upairsetp_type,type,
% 2.91/1.61 upairsetp: $i > $o ).
% 2.91/1.61
% 2.91/1.61 thf(setukpairinjL1_type,type,
% 2.91/1.61 setukpairinjL1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(setukpairinjL1,definition,
% 2.91/1.61 ( setukpairinjL1
% 2.91/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.91/1.61 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.91/1.61 => ( Xx = Xz ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(kfstsingleton_type,type,
% 2.91/1.61 kfstsingleton: $o ).
% 2.91/1.61
% 2.91/1.61 thf(kfstsingleton,definition,
% 2.91/1.61 ( kfstsingleton
% 2.91/1.61 = ( ! [Xu: $i] :
% 2.91/1.61 ( ( iskpair @ Xu )
% 2.91/1.61 => ( singleton
% 2.91/1.61 @ ( dsetconstr @ ( setunion @ Xu )
% 2.91/1.61 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(theprop_type,type,
% 2.91/1.61 theprop: $o ).
% 2.91/1.61
% 2.91/1.61 thf(theprop,definition,
% 2.91/1.61 ( theprop
% 2.91/1.61 = ( ! [X: $i] :
% 2.91/1.61 ( ( singleton @ X )
% 2.91/1.61 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(kfst_type,type,
% 2.91/1.61 kfst: $i > $i ).
% 2.91/1.61
% 2.91/1.61 thf(kfstpairEq_type,type,
% 2.91/1.61 kfstpairEq: $o ).
% 2.91/1.61
% 2.91/1.61 thf(kfstpairEq,definition,
% 2.91/1.61 ( kfstpairEq
% 2.91/1.61 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.61 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.91/1.61 = Xx ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodfstin_type,type,
% 2.91/1.61 cartprodfstin: $o ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodfstin,definition,
% 2.91/1.61 ( cartprodfstin
% 2.91/1.61 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.91/1.61 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.91/1.61 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(setukpairinjL2_type,type,
% 2.91/1.61 setukpairinjL2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(setukpairinjL2,definition,
% 2.91/1.61 ( setukpairinjL2
% 2.91/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.91/1.61 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.91/1.61 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.91/1.61 => ( Xx = Xz ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(setukpairinjL_type,type,
% 2.91/1.61 setukpairinjL: $o ).
% 2.91/1.61
% 2.91/1.61 thf(setukpairinjL,definition,
% 2.91/1.61 ( setukpairinjL
% 2.91/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.91/1.61 ( ( ( kpair @ Xx @ Xy )
% 2.91/1.61 = ( kpair @ Xz @ Xu ) )
% 2.91/1.61 => ( Xx = Xz ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(setukpairinjR11_type,type,
% 2.91/1.61 setukpairinjR11: $o ).
% 2.91/1.61
% 2.91/1.61 thf(setukpairinjR11,definition,
% 2.91/1.61 ( setukpairinjR11
% 2.91/1.61 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.61 ( ( Xx = Xy )
% 2.91/1.61 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.91/1.61 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(setukpairinjR12_type,type,
% 2.91/1.61 setukpairinjR12: $o ).
% 2.91/1.61
% 2.91/1.61 thf(setukpairinjR12,definition,
% 2.91/1.61 ( setukpairinjR12
% 2.91/1.61 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.61 ( ( Xx = Xy )
% 2.91/1.61 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.91/1.61 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(setukpairinjR1_type,type,
% 2.91/1.61 setukpairinjR1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(setukpairinjR1,definition,
% 2.91/1.61 ( setukpairinjR1
% 2.91/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.91/1.61 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.91/1.61 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.91/1.61 => ( ( Xz = Xu )
% 2.91/1.61 => ( Xy = Xu ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(upairequniteq_type,type,
% 2.91/1.61 upairequniteq: $o ).
% 2.91/1.61
% 2.91/1.61 thf(upairequniteq,definition,
% 2.91/1.61 ( upairequniteq
% 2.91/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.91/1.61 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.91/1.61 = ( setadjoin @ Xz @ emptyset ) )
% 2.91/1.61 => ( Xx = Xy ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(setukpairinjR2_type,type,
% 2.91/1.61 setukpairinjR2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(setukpairinjR2,definition,
% 2.91/1.61 ( setukpairinjR2
% 2.91/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.91/1.61 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.91/1.61 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.91/1.61 => ( Xy = Xu ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(setukpairinjR_type,type,
% 2.91/1.61 setukpairinjR: $o ).
% 2.91/1.61
% 2.91/1.61 thf(setukpairinjR,definition,
% 2.91/1.61 ( setukpairinjR
% 2.91/1.61 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.91/1.61 ( ( ( kpair @ Xx @ Xy )
% 2.91/1.61 = ( kpair @ Xz @ Xu ) )
% 2.91/1.61 => ( Xy = Xu ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(ksndsingleton_type,type,
% 2.91/1.61 ksndsingleton: $o ).
% 2.91/1.61
% 2.91/1.61 thf(ksndsingleton,definition,
% 2.91/1.61 ( ksndsingleton
% 2.91/1.61 = ( ! [Xu: $i] :
% 2.91/1.61 ( ( iskpair @ Xu )
% 2.91/1.61 => ( singleton
% 2.91/1.61 @ ( dsetconstr @ ( setunion @ Xu )
% 2.91/1.61 @ ^ [Xx: $i] :
% 2.91/1.61 ( Xu
% 2.91/1.61 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(ksnd_type,type,
% 2.91/1.61 ksnd: $i > $i ).
% 2.91/1.61
% 2.91/1.61 thf(ksndpairEq_type,type,
% 2.91/1.61 ksndpairEq: $o ).
% 2.91/1.61
% 2.91/1.61 thf(ksndpairEq,definition,
% 2.91/1.61 ( ksndpairEq
% 2.91/1.61 = ( ! [Xx: $i,Xy: $i] :
% 2.91/1.61 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.91/1.61 = Xy ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(kpairsurjEq_type,type,
% 2.91/1.61 kpairsurjEq: $o ).
% 2.91/1.61
% 2.91/1.61 thf(kpairsurjEq,definition,
% 2.91/1.61 ( kpairsurjEq
% 2.91/1.61 = ( ! [Xu: $i] :
% 2.91/1.61 ( ( iskpair @ Xu )
% 2.91/1.61 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.91/1.61 = Xu ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodsndin_type,type,
% 2.91/1.61 cartprodsndin: $o ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodsndin,definition,
% 2.91/1.61 ( cartprodsndin
% 2.91/1.61 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.91/1.61 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.91/1.61 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodpairmemEL_type,type,
% 2.91/1.61 cartprodpairmemEL: $o ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodpairmemEL,definition,
% 2.91/1.61 ( cartprodpairmemEL
% 2.91/1.61 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.91/1.61 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.91/1.61 => ( in @ Xx @ A ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodpairmemER_type,type,
% 2.91/1.61 cartprodpairmemER: $o ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodpairmemER,definition,
% 2.91/1.61 ( cartprodpairmemER
% 2.91/1.61 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.91/1.61 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.91/1.61 => ( in @ Xy @ B ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodmempaircEq_type,type,
% 2.91/1.61 cartprodmempaircEq: $o ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodmempaircEq,definition,
% 2.91/1.61 ( cartprodmempaircEq
% 2.91/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( ( kpair @ Xx @ Xy )
% 2.91/1.61 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodfstpairEq_type,type,
% 2.91/1.61 cartprodfstpairEq: $o ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodfstpairEq,definition,
% 2.91/1.61 ( cartprodfstpairEq
% 2.91/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.91/1.61 = Xx ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodsndpairEq_type,type,
% 2.91/1.61 cartprodsndpairEq: $o ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodsndpairEq,definition,
% 2.91/1.61 ( cartprodsndpairEq
% 2.91/1.61 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.91/1.61 = Xy ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodpairsurjEq_type,type,
% 2.91/1.61 cartprodpairsurjEq: $o ).
% 2.91/1.61
% 2.91/1.61 thf(cartprodpairsurjEq,definition,
% 2.91/1.61 ( cartprodpairsurjEq
% 2.91/1.61 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.91/1.61 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.91/1.61 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.91/1.61 = Xu ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(breln_type,type,
% 2.91/1.61 breln: $i > $i > $i > $o ).
% 2.91/1.61
% 2.91/1.61 thf(breln,definition,
% 2.91/1.61 ( breln
% 2.91/1.61 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(dpsetconstr_type,type,
% 2.91/1.61 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.91/1.61
% 2.91/1.61 thf(dpsetconstrI_type,type,
% 2.91/1.61 dpsetconstrI: $o ).
% 2.91/1.61
% 2.91/1.61 thf(dpsetconstrI,definition,
% 2.91/1.61 ( dpsetconstrI
% 2.91/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( ( Xphi @ Xx @ Xy )
% 2.91/1.61 => ( in @ ( kpair @ Xx @ Xy )
% 2.91/1.61 @ ( dpsetconstr @ A @ B
% 2.91/1.61 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(dpsetconstrSub_type,type,
% 2.91/1.61 dpsetconstrSub: $o ).
% 2.91/1.61
% 2.91/1.61 thf(dpsetconstrSub,definition,
% 2.91/1.61 ( dpsetconstrSub
% 2.91/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.91/1.61 ( subset
% 2.91/1.61 @ ( dpsetconstr @ A @ B
% 2.91/1.61 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.91/1.61 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(setOfPairsIsBReln_type,type,
% 2.91/1.61 setOfPairsIsBReln: $o ).
% 2.91/1.61
% 2.91/1.61 thf(setOfPairsIsBReln,definition,
% 2.91/1.61 ( setOfPairsIsBReln
% 2.91/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.91/1.61 ( breln @ A @ B
% 2.91/1.61 @ ( dpsetconstr @ A @ B
% 2.91/1.61 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(dpsetconstrERa_type,type,
% 2.91/1.61 dpsetconstrERa: $o ).
% 2.91/1.61
% 2.91/1.61 thf(dpsetconstrERa,definition,
% 2.91/1.61 ( dpsetconstrERa
% 2.91/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.91/1.61 @ ( dpsetconstr @ A @ B
% 2.91/1.61 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.91/1.61 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(dpsetconstrEL1_type,type,
% 2.91/1.61 dpsetconstrEL1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(dpsetconstrEL1,definition,
% 2.91/1.61 ( dpsetconstrEL1
% 2.91/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.91/1.61 ( ( in @ ( kpair @ Xx @ Xy )
% 2.91/1.61 @ ( dpsetconstr @ A @ B
% 2.91/1.61 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.91/1.61 => ( in @ Xx @ A ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(dpsetconstrEL2_type,type,
% 2.91/1.61 dpsetconstrEL2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(dpsetconstrEL2,definition,
% 2.91/1.61 ( dpsetconstrEL2
% 2.91/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.91/1.61 ( ( in @ ( kpair @ Xx @ Xy )
% 2.91/1.61 @ ( dpsetconstr @ A @ B
% 2.91/1.61 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.91/1.61 => ( in @ Xy @ B ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(dpsetconstrER_type,type,
% 2.91/1.61 dpsetconstrER: $o ).
% 2.91/1.61
% 2.91/1.61 thf(dpsetconstrER,definition,
% 2.91/1.61 ( dpsetconstrER
% 2.91/1.61 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.91/1.61 ( ( in @ ( kpair @ Xx @ Xy )
% 2.91/1.61 @ ( dpsetconstr @ A @ B
% 2.91/1.61 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.91/1.61 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(func_type,type,
% 2.91/1.61 func: $i > $i > $i > $o ).
% 2.91/1.61
% 2.91/1.61 thf(funcSet_type,type,
% 2.91/1.61 funcSet: $i > $i > $i ).
% 2.91/1.61
% 2.91/1.61 thf(funcImageSingleton_type,type,
% 2.91/1.61 funcImageSingleton: $o ).
% 2.91/1.61
% 2.91/1.61 thf(funcImageSingleton,definition,
% 2.91/1.61 ( funcImageSingleton
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( func @ A @ B @ Xf )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( singleton
% 2.91/1.61 @ ( dsetconstr @ B
% 2.91/1.61 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(apProp_type,type,
% 2.91/1.61 apProp: $o ).
% 2.91/1.61
% 2.91/1.61 thf(apProp,definition,
% 2.91/1.61 ( apProp
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( func @ A @ B @ Xf )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( in
% 2.91/1.61 @ ( setunion
% 2.91/1.61 @ ( dsetconstr @ B
% 2.91/1.61 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.91/1.61 @ B ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(ap_type,type,
% 2.91/1.61 ap: $i > $i > $i > $i > $i ).
% 2.91/1.61
% 2.91/1.61 thf(app_type,type,
% 2.91/1.61 app: $o ).
% 2.91/1.61
% 2.91/1.61 thf(app,definition,
% 2.91/1.61 ( app
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( func @ A @ B @ Xf )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(infuncsetfunc_type,type,
% 2.91/1.61 infuncsetfunc: $o ).
% 2.91/1.61
% 2.91/1.61 thf(infuncsetfunc,definition,
% 2.91/1.61 ( infuncsetfunc
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.91/1.61 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(ap2p_type,type,
% 2.91/1.61 ap2p: $o ).
% 2.91/1.61
% 2.91/1.61 thf(ap2p,definition,
% 2.91/1.61 ( ap2p
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(funcinfuncset_type,type,
% 2.91/1.61 funcinfuncset: $o ).
% 2.91/1.61
% 2.91/1.61 thf(funcinfuncset,definition,
% 2.91/1.61 ( funcinfuncset
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( func @ A @ B @ Xf )
% 2.91/1.61 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(lamProp_type,type,
% 2.91/1.61 lamProp: $o ).
% 2.91/1.61
% 2.91/1.61 thf(lamProp,definition,
% 2.91/1.61 ( lamProp
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.91/1.61 ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.91/1.61 => ( func @ A @ B
% 2.91/1.61 @ ( dpsetconstr @ A @ B
% 2.91/1.61 @ ^ [Xx: $i,Xy: $i] :
% 2.91/1.61 ( ( Xf @ Xx )
% 2.91/1.61 = Xy ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(lam_type,type,
% 2.91/1.61 lam: $i > $i > ( $i > $i ) > $i ).
% 2.91/1.61
% 2.91/1.61 thf(lamp_type,type,
% 2.91/1.61 lamp: $o ).
% 2.91/1.61
% 2.91/1.61 thf(lamp,definition,
% 2.91/1.61 ( lamp
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.91/1.61 ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.91/1.61 => ( func @ A @ B
% 2.91/1.61 @ ( lam @ A @ B
% 2.91/1.61 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(lam2p_type,type,
% 2.91/1.61 lam2p: $o ).
% 2.91/1.61
% 2.91/1.61 thf(lam2p,definition,
% 2.91/1.61 ( lam2p
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.91/1.61 ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.91/1.61 => ( in
% 2.91/1.61 @ ( lam @ A @ B
% 2.91/1.61 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.91/1.61 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(brelnall1_type,type,
% 2.91/1.61 brelnall1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(brelnall1,definition,
% 2.91/1.61 ( brelnall1
% 2.91/1.61 = ( ! [A: $i,B: $i,R: $i] :
% 2.91/1.61 ( ( breln @ A @ B @ R )
% 2.91/1.61 => ! [Xphi: $i > $o] :
% 2.91/1.61 ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.91/1.61 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ R )
% 2.91/1.61 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(brelnall2_type,type,
% 2.91/1.61 brelnall2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(brelnall2,definition,
% 2.91/1.61 ( brelnall2
% 2.91/1.61 = ( ! [A: $i,B: $i,R: $i] :
% 2.91/1.61 ( ( breln @ A @ B @ R )
% 2.91/1.61 => ! [Xphi: $i > $o] :
% 2.91/1.61 ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.91/1.61 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ R )
% 2.91/1.61 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(ex1E2_type,type,
% 2.91/1.61 ex1E2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(ex1E2,definition,
% 2.91/1.61 ( ex1E2
% 2.91/1.61 = ( ! [A: $i,Xphi: $i > $o] :
% 2.91/1.61 ( ( ex1 @ A
% 2.91/1.61 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ A )
% 2.91/1.61 => ( ( Xphi @ Xx )
% 2.91/1.61 => ( ( Xphi @ Xy )
% 2.91/1.61 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(funcGraphProp1_type,type,
% 2.91/1.61 funcGraphProp1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(funcGraphProp1,definition,
% 2.91/1.61 ( funcGraphProp1
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( func @ A @ B @ Xf )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(funcGraphProp3_type,type,
% 2.91/1.61 funcGraphProp3: $o ).
% 2.91/1.61
% 2.91/1.61 thf(funcGraphProp3,definition,
% 2.91/1.61 ( funcGraphProp3
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(funcGraphProp2_type,type,
% 2.91/1.61 funcGraphProp2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(funcGraphProp2,definition,
% 2.91/1.61 ( funcGraphProp2
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( func @ A @ B @ Xf )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.91/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.91/1.61 = Xy ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(funcextLem_type,type,
% 2.91/1.61 funcextLem: $o ).
% 2.91/1.61
% 2.91/1.61 thf(funcextLem,definition,
% 2.91/1.61 ( funcextLem
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( func @ A @ B @ Xf )
% 2.91/1.61 => ! [Xg: $i] :
% 2.91/1.61 ( ( func @ A @ B @ Xg )
% 2.91/1.61 => ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.91/1.61 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.91/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(funcGraphProp4_type,type,
% 2.91/1.61 funcGraphProp4: $o ).
% 2.91/1.61
% 2.91/1.61 thf(funcGraphProp4,definition,
% 2.91/1.61 ( funcGraphProp4
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.91/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.91/1.61 = Xy ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(subbreln_type,type,
% 2.91/1.61 subbreln: $o ).
% 2.91/1.61
% 2.91/1.61 thf(subbreln,definition,
% 2.91/1.61 ( subbreln
% 2.91/1.61 = ( ! [A: $i,B: $i,R: $i] :
% 2.91/1.61 ( ( breln @ A @ B @ R )
% 2.91/1.61 => ! [S: $i] :
% 2.91/1.61 ( ( breln @ A @ B @ S )
% 2.91/1.61 => ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.91/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.91/1.61 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(eqbreln_type,type,
% 2.91/1.61 eqbreln: $o ).
% 2.91/1.61
% 2.91/1.61 thf(eqbreln,definition,
% 2.91/1.61 ( eqbreln
% 2.91/1.61 = ( ! [A: $i,B: $i,R: $i] :
% 2.91/1.61 ( ( breln @ A @ B @ R )
% 2.91/1.61 => ! [S: $i] :
% 2.91/1.61 ( ( breln @ A @ B @ S )
% 2.91/1.61 => ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.91/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.91/1.61 => ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ B )
% 2.91/1.61 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.91/1.61 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.91/1.61 => ( R = S ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(funcext_type,type,
% 2.91/1.61 funcext: $o ).
% 2.91/1.61
% 2.91/1.61 thf(funcext,definition,
% 2.91/1.61 ( funcext
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( func @ A @ B @ Xf )
% 2.91/1.61 => ! [Xg: $i] :
% 2.91/1.61 ( ( func @ A @ B @ Xg )
% 2.91/1.61 => ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.91/1.61 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.91/1.61 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(funcext2_type,type,
% 2.91/1.61 funcext2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(funcext2,definition,
% 2.91/1.61 ( funcext2
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.91/1.61 => ! [Xg: $i] :
% 2.91/1.61 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.91/1.61 => ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.91/1.61 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.91/1.61 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(ap2apEq1_type,type,
% 2.91/1.61 ap2apEq1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(ap2apEq1,definition,
% 2.91/1.61 ( ap2apEq1
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.91/1.61 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(ap2apEq2_type,type,
% 2.91/1.61 ap2apEq2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(ap2apEq2,definition,
% 2.91/1.61 ( ap2apEq2
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( func @ A @ B @ Xf )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.91/1.61 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(beta1_type,type,
% 2.91/1.61 beta1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(beta1,definition,
% 2.91/1.61 ( beta1
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.91/1.61 ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( ap @ A @ B
% 2.91/1.61 @ ( lam @ A @ B
% 2.91/1.61 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.91/1.61 @ Xx )
% 2.91/1.61 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(eta1_type,type,
% 2.91/1.61 eta1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(eta1,definition,
% 2.91/1.61 ( eta1
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( func @ A @ B @ Xf )
% 2.91/1.61 => ( ( lam @ A @ B
% 2.91/1.61 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.91/1.61 = Xf ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(lam2lamEq_type,type,
% 2.91/1.61 lam2lamEq: $o ).
% 2.91/1.61
% 2.91/1.61 thf(lam2lamEq,definition,
% 2.91/1.61 ( lam2lamEq
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.91/1.61 ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.91/1.61 => ( ( lam @ A @ B
% 2.91/1.61 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.91/1.61 = ( lam @ A @ B
% 2.91/1.61 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(beta2_type,type,
% 2.91/1.61 beta2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(beta2,definition,
% 2.91/1.61 ( beta2
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.91/1.61 ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( ap @ A @ B
% 2.91/1.61 @ ( lam @ A @ B
% 2.91/1.61 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.91/1.61 @ Xx )
% 2.91/1.61 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(eta2_type,type,
% 2.91/1.61 eta2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(eta2,definition,
% 2.91/1.61 ( eta2
% 2.91/1.61 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.91/1.61 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.91/1.61 => ( ( lam @ A @ B
% 2.91/1.61 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.91/1.61 = Xf ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(iffalseProp1_type,type,
% 2.91/1.61 iffalseProp1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(iffalseProp1,definition,
% 2.91/1.61 ( iffalseProp1
% 2.91/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ A )
% 2.91/1.61 => ( ~ Xphi
% 2.91/1.61 => ( in @ Xy
% 2.91/1.61 @ ( dsetconstr @ A
% 2.91/1.61 @ ^ [Xz: $i] :
% 2.91/1.61 ( ( Xphi
% 2.91/1.61 & ( Xz = Xx ) )
% 2.91/1.61 | ( ~ Xphi
% 2.91/1.61 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(iffalseProp2_type,type,
% 2.91/1.61 iffalseProp2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(iffalseProp2,definition,
% 2.91/1.61 ( iffalseProp2
% 2.91/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ A )
% 2.91/1.61 => ( ~ Xphi
% 2.91/1.61 => ( ( dsetconstr @ A
% 2.91/1.61 @ ^ [Xz: $i] :
% 2.91/1.61 ( ( Xphi
% 2.91/1.61 & ( Xz = Xx ) )
% 2.91/1.61 | ( ~ Xphi
% 2.91/1.61 & ( Xz = Xy ) ) ) )
% 2.91/1.61 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(iftrueProp1_type,type,
% 2.91/1.61 iftrueProp1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(iftrueProp1,definition,
% 2.91/1.61 ( iftrueProp1
% 2.91/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ A )
% 2.91/1.61 => ( Xphi
% 2.91/1.61 => ( in @ Xx
% 2.91/1.61 @ ( dsetconstr @ A
% 2.91/1.61 @ ^ [Xz: $i] :
% 2.91/1.61 ( ( Xphi
% 2.91/1.61 & ( Xz = Xx ) )
% 2.91/1.61 | ( ~ Xphi
% 2.91/1.61 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(iftrueProp2_type,type,
% 2.91/1.61 iftrueProp2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(iftrueProp2,definition,
% 2.91/1.61 ( iftrueProp2
% 2.91/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ A )
% 2.91/1.61 => ( Xphi
% 2.91/1.61 => ( ( dsetconstr @ A
% 2.91/1.61 @ ^ [Xz: $i] :
% 2.91/1.61 ( ( Xphi
% 2.91/1.61 & ( Xz = Xx ) )
% 2.91/1.61 | ( ~ Xphi
% 2.91/1.61 & ( Xz = Xy ) ) ) )
% 2.91/1.61 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(ifSingleton_type,type,
% 2.91/1.61 ifSingleton: $o ).
% 2.91/1.61
% 2.91/1.61 thf(ifSingleton,definition,
% 2.91/1.61 ( ifSingleton
% 2.91/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ A )
% 2.91/1.61 => ( singleton
% 2.91/1.61 @ ( dsetconstr @ A
% 2.91/1.61 @ ^ [Xz: $i] :
% 2.91/1.61 ( ( Xphi
% 2.91/1.61 & ( Xz = Xx ) )
% 2.91/1.61 | ( ~ Xphi
% 2.91/1.61 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(if_type,type,
% 2.91/1.61 if: $i > $o > $i > $i > $i ).
% 2.91/1.61
% 2.91/1.61 thf(ifp_type,type,
% 2.91/1.61 ifp: $o ).
% 2.91/1.61
% 2.91/1.61 thf(ifp,definition,
% 2.91/1.61 ( ifp
% 2.91/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ A )
% 2.91/1.61 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(theeq_type,type,
% 2.91/1.61 theeq: $o ).
% 2.91/1.61
% 2.91/1.61 thf(theeq,definition,
% 2.91/1.61 ( theeq
% 2.91/1.61 = ( ! [X: $i] :
% 2.91/1.61 ( ( singleton @ X )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ X )
% 2.91/1.61 => ( ( setunion @ X )
% 2.91/1.61 = Xx ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(iftrue_type,type,
% 2.91/1.61 iftrue: $o ).
% 2.91/1.61
% 2.91/1.61 thf(iftrue,definition,
% 2.91/1.61 ( iftrue
% 2.91/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ A )
% 2.91/1.61 => ( Xphi
% 2.91/1.61 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.91/1.61 = Xx ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(iffalse_type,type,
% 2.91/1.61 iffalse: $o ).
% 2.91/1.61
% 2.91/1.61 thf(iffalse,definition,
% 2.91/1.61 ( iffalse
% 2.91/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ A )
% 2.91/1.61 => ( ~ Xphi
% 2.91/1.61 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.91/1.61 = Xy ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(iftrueorfalse_type,type,
% 2.91/1.61 iftrueorfalse: $o ).
% 2.91/1.61
% 2.91/1.61 thf(iftrueorfalse,definition,
% 2.91/1.61 ( iftrueorfalse
% 2.91/1.61 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ! [Xy: $i] :
% 2.91/1.61 ( ( in @ Xy @ A )
% 2.91/1.61 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(binintersectT_lem_type,type,
% 2.91/1.61 binintersectT_lem: $o ).
% 2.91/1.61
% 2.91/1.61 thf(binintersectT_lem,definition,
% 2.91/1.61 ( binintersectT_lem
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(binunionT_lem_type,type,
% 2.91/1.61 binunionT_lem: $o ).
% 2.91/1.61
% 2.91/1.61 thf(binunionT_lem,definition,
% 2.91/1.61 ( binunionT_lem
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(powersetT_lem_type,type,
% 2.91/1.61 powersetT_lem: $o ).
% 2.91/1.61
% 2.91/1.61 thf(powersetT_lem,definition,
% 2.91/1.61 ( powersetT_lem
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(setminusT_lem_type,type,
% 2.91/1.61 setminusT_lem: $o ).
% 2.91/1.61
% 2.91/1.61 thf(setminusT_lem,definition,
% 2.91/1.61 ( setminusT_lem
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(complementT_lem_type,type,
% 2.91/1.61 complementT_lem: $o ).
% 2.91/1.61
% 2.91/1.61 thf(complementT_lem,definition,
% 2.91/1.61 ( complementT_lem
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(setextT_type,type,
% 2.91/1.61 setextT: $o ).
% 2.91/1.61
% 2.91/1.61 thf(setextT,definition,
% 2.91/1.61 ( setextT
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ X )
% 2.91/1.61 => ( in @ Xx @ Y ) ) )
% 2.91/1.61 => ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ Y )
% 2.91/1.61 => ( in @ Xx @ X ) ) )
% 2.91/1.61 => ( X = Y ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(subsetTI_type,type,
% 2.91/1.61 subsetTI: $o ).
% 2.91/1.61
% 2.91/1.61 thf(subsetTI,definition,
% 2.91/1.61 ( subsetTI
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ X )
% 2.91/1.61 => ( in @ Xx @ Y ) ) )
% 2.91/1.61 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(powersetTI1_type,type,
% 2.91/1.61 powersetTI1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(powersetTI1,definition,
% 2.91/1.61 ( powersetTI1
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ X )
% 2.91/1.61 => ( in @ Xx @ Y ) ) )
% 2.91/1.61 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(powersetTE1_type,type,
% 2.91/1.61 powersetTE1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(powersetTE1,definition,
% 2.91/1.61 ( powersetTE1
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ X @ ( powerset @ Y ) )
% 2.91/1.61 => ( ( in @ Xx @ X )
% 2.91/1.61 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(complementTI1_type,type,
% 2.91/1.61 complementTI1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(complementTI1,definition,
% 2.91/1.61 ( complementTI1
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ X )
% 2.91/1.61 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(complementTE1_type,type,
% 2.91/1.61 complementTE1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(complementTE1,definition,
% 2.91/1.61 ( complementTE1
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.91/1.61 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(binintersectTELcontra_type,type,
% 2.91/1.61 binintersectTELcontra: $o ).
% 2.91/1.61
% 2.91/1.61 thf(binintersectTELcontra,definition,
% 2.91/1.61 ( binintersectTELcontra
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ~ ( in @ Xx @ X )
% 2.91/1.61 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(binintersectTERcontra_type,type,
% 2.91/1.61 binintersectTERcontra: $o ).
% 2.91/1.61
% 2.91/1.61 thf(binintersectTERcontra,definition,
% 2.91/1.61 ( binintersectTERcontra
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ~ ( in @ Xx @ Y )
% 2.91/1.61 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(contrasubsetT_type,type,
% 2.91/1.61 contrasubsetT: $o ).
% 2.91/1.61
% 2.91/1.61 thf(contrasubsetT,definition,
% 2.91/1.61 ( contrasubsetT
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.91/1.61 => ( ( in @ Xx @ Y )
% 2.91/1.61 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(contrasubsetT1_type,type,
% 2.91/1.61 contrasubsetT1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(contrasubsetT1,definition,
% 2.91/1.61 ( contrasubsetT1
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( subset @ X @ Y )
% 2.91/1.61 => ( ~ ( in @ Xx @ Y )
% 2.91/1.61 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(contrasubsetT2_type,type,
% 2.91/1.61 contrasubsetT2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(contrasubsetT2,definition,
% 2.91/1.61 ( contrasubsetT2
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( ( subset @ X @ Y )
% 2.91/1.61 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(contrasubsetT3_type,type,
% 2.91/1.61 contrasubsetT3: $o ).
% 2.91/1.61
% 2.91/1.61 thf(contrasubsetT3,definition,
% 2.91/1.61 ( contrasubsetT3
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.91/1.61 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(doubleComplementI1_type,type,
% 2.91/1.61 doubleComplementI1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(doubleComplementI1,definition,
% 2.91/1.61 ( doubleComplementI1
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ X )
% 2.91/1.61 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(doubleComplementE1_type,type,
% 2.91/1.61 doubleComplementE1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(doubleComplementE1,definition,
% 2.91/1.61 ( doubleComplementE1
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 2.91/1.61 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(doubleComplementSub1_type,type,
% 2.91/1.61 doubleComplementSub1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(doubleComplementSub1,definition,
% 2.91/1.61 ( doubleComplementSub1
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(doubleComplementSub2_type,type,
% 2.91/1.61 doubleComplementSub2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(doubleComplementSub2,definition,
% 2.91/1.61 ( doubleComplementSub2
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(doubleComplementEq_type,type,
% 2.91/1.61 doubleComplementEq: $o ).
% 2.91/1.61
% 2.91/1.61 thf(doubleComplementEq,definition,
% 2.91/1.61 ( doubleComplementEq
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ( X
% 2.91/1.61 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(complementTnotintersectT_type,type,
% 2.91/1.61 complementTnotintersectT: $o ).
% 2.91/1.61
% 2.91/1.61 thf(complementTnotintersectT,definition,
% 2.91/1.61 ( complementTnotintersectT
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.91/1.61 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(complementImpComplementIntersect_type,type,
% 2.91/1.61 complementImpComplementIntersect: $o ).
% 2.91/1.61
% 2.91/1.61 thf(complementImpComplementIntersect,definition,
% 2.91/1.61 ( complementImpComplementIntersect
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.91/1.61 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(complementSubsetComplementIntersect_type,type,
% 2.91/1.61 complementSubsetComplementIntersect: $o ).
% 2.91/1.61
% 2.91/1.61 thf(complementSubsetComplementIntersect,definition,
% 2.91/1.61 ( complementSubsetComplementIntersect
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(complementInPowersetComplementIntersect_type,type,
% 2.91/1.61 complementInPowersetComplementIntersect: $o ).
% 2.91/1.61
% 2.91/1.61 thf(complementInPowersetComplementIntersect,definition,
% 2.91/1.61 ( complementInPowersetComplementIntersect
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(contraSubsetComplement_type,type,
% 2.91/1.61 contraSubsetComplement: $o ).
% 2.91/1.61
% 2.91/1.61 thf(contraSubsetComplement,definition,
% 2.91/1.61 ( contraSubsetComplement
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ Y )
% 2.91/1.61 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(complementTcontraSubset_type,type,
% 2.91/1.61 complementTcontraSubset: $o ).
% 2.91/1.61
% 2.91/1.61 thf(complementTcontraSubset,definition,
% 2.91/1.61 ( complementTcontraSubset
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.91/1.61 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(binunionTILcontra_type,type,
% 2.91/1.61 binunionTILcontra: $o ).
% 2.91/1.61
% 2.91/1.61 thf(binunionTILcontra,definition,
% 2.91/1.61 ( binunionTILcontra
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.91/1.61 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(binunionTIRcontra_type,type,
% 2.91/1.61 binunionTIRcontra: $o ).
% 2.91/1.61
% 2.91/1.61 thf(binunionTIRcontra,definition,
% 2.91/1.61 ( binunionTIRcontra
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.91/1.61 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(inIntersectImpInUnion_type,type,
% 2.91/1.61 inIntersectImpInUnion: $o ).
% 2.91/1.61
% 2.91/1.61 thf(inIntersectImpInUnion,definition,
% 2.91/1.61 ( inIntersectImpInUnion
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Z: $i] :
% 2.91/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.91/1.61 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(inIntersectImpInUnion2_type,type,
% 2.91/1.61 inIntersectImpInUnion2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(inIntersectImpInUnion2,definition,
% 2.91/1.61 ( inIntersectImpInUnion2
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Z: $i] :
% 2.91/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.91/1.61 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(inIntersectImpInIntersectUnions_type,type,
% 2.91/1.61 inIntersectImpInIntersectUnions: $o ).
% 2.91/1.61
% 2.91/1.61 thf(inIntersectImpInIntersectUnions,definition,
% 2.91/1.61 ( inIntersectImpInIntersectUnions
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Z: $i] :
% 2.91/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.91/1.61 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(intersectInPowersetIntersectUnions_type,type,
% 2.91/1.61 intersectInPowersetIntersectUnions: $o ).
% 2.91/1.61
% 2.91/1.61 thf(intersectInPowersetIntersectUnions,definition,
% 2.91/1.61 ( intersectInPowersetIntersectUnions
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Z: $i] :
% 2.91/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 2.91/1.61 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(inComplementUnionImpNotIn1_type,type,
% 2.91/1.61 inComplementUnionImpNotIn1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(inComplementUnionImpNotIn1,definition,
% 2.91/1.61 ( inComplementUnionImpNotIn1
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.91/1.61 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(inComplementUnionImpInComplement1_type,type,
% 2.91/1.61 inComplementUnionImpInComplement1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(inComplementUnionImpInComplement1,definition,
% 2.91/1.61 ( inComplementUnionImpInComplement1
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.91/1.61 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(binunionTE_type,type,
% 2.91/1.61 binunionTE: $o ).
% 2.91/1.61
% 2.91/1.61 thf(binunionTE,definition,
% 2.91/1.61 ( binunionTE
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xphi: $o,Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.91/1.61 => ( ( ( in @ Xx @ X )
% 2.91/1.61 => Xphi )
% 2.91/1.61 => ( ( ( in @ Xx @ Y )
% 2.91/1.61 => Xphi )
% 2.91/1.61 => Xphi ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(binunionTEcontra_type,type,
% 2.91/1.61 binunionTEcontra: $o ).
% 2.91/1.61
% 2.91/1.61 thf(binunionTEcontra,definition,
% 2.91/1.61 ( binunionTEcontra
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ~ ( in @ Xx @ X )
% 2.91/1.61 => ( ~ ( in @ Xx @ Y )
% 2.91/1.61 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan2a1_type,type,
% 2.91/1.61 demorgan2a1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan2a1,definition,
% 2.91/1.61 ( demorgan2a1
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.91/1.61 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(complementUnionInPowersetComplement_type,type,
% 2.91/1.61 complementUnionInPowersetComplement: $o ).
% 2.91/1.61
% 2.91/1.61 thf(complementUnionInPowersetComplement,definition,
% 2.91/1.61 ( complementUnionInPowersetComplement
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan2a2_type,type,
% 2.91/1.61 demorgan2a2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan2a2,definition,
% 2.91/1.61 ( demorgan2a2
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.91/1.61 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan1a_type,type,
% 2.91/1.61 demorgan1a: $o ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan1a,definition,
% 2.91/1.61 ( demorgan1a
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 2.91/1.61 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan1b_type,type,
% 2.91/1.61 demorgan1b: $o ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan1b,definition,
% 2.91/1.61 ( demorgan1b
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.91/1.61 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan1_type,type,
% 2.91/1.61 demorgan1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan1,definition,
% 2.91/1.61 ( demorgan1
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 2.91/1.61 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan2a_type,type,
% 2.91/1.61 demorgan2a: $o ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan2a,definition,
% 2.91/1.61 ( demorgan2a
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.91/1.61 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan2b2_type,type,
% 2.91/1.61 demorgan2b2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan2b2,definition,
% 2.91/1.61 ( demorgan2b2
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.91/1.61 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 2.91/1.61 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan2b_type,type,
% 2.91/1.61 demorgan2b: $o ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan2b,definition,
% 2.91/1.61 ( demorgan2b
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ A )
% 2.91/1.61 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.91/1.61 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan2_type,type,
% 2.91/1.61 demorgan2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(demorgan2,definition,
% 2.91/1.61 ( demorgan2
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 2.91/1.61 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(woz13rule0_type,type,
% 2.91/1.61 woz13rule0: $o ).
% 2.91/1.61
% 2.91/1.61 thf(woz13rule0,definition,
% 2.91/1.61 ( woz13rule0
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Xx: $i] :
% 2.91/1.61 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.91/1.61 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(woz13rule1_type,type,
% 2.91/1.61 woz13rule1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(woz13rule1,definition,
% 2.91/1.61 ( woz13rule1
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Z: $i] :
% 2.91/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 2.91/1.61 => ( ( subset @ X @ Z )
% 2.91/1.61 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(woz13rule2_type,type,
% 2.91/1.61 woz13rule2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(woz13rule2,definition,
% 2.91/1.61 ( woz13rule2
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Z: $i] :
% 2.91/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 2.91/1.61 => ( ( subset @ Y @ Z )
% 2.91/1.61 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(woz13rule3_type,type,
% 2.91/1.61 woz13rule3: $o ).
% 2.91/1.61
% 2.91/1.61 thf(woz13rule3,definition,
% 2.91/1.61 ( woz13rule3
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Z: $i] :
% 2.91/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 2.91/1.61 => ( ( subset @ X @ Y )
% 2.91/1.61 => ( ( subset @ X @ Z )
% 2.91/1.61 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(woz13rule4_type,type,
% 2.91/1.61 woz13rule4: $o ).
% 2.91/1.61
% 2.91/1.61 thf(woz13rule4,definition,
% 2.91/1.61 ( woz13rule4
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Z: $i] :
% 2.91/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 2.91/1.61 => ! [W: $i] :
% 2.91/1.61 ( ( in @ W @ ( powerset @ A ) )
% 2.91/1.61 => ( ( subset @ X @ Z )
% 2.91/1.61 => ( ( subset @ Y @ W )
% 2.91/1.61 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(woz1_1_type,type,
% 2.91/1.61 woz1_1: $o ).
% 2.91/1.61
% 2.91/1.61 thf(woz1_1,definition,
% 2.91/1.61 ( woz1_1
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(woz1_2_type,type,
% 2.91/1.61 woz1_2: $o ).
% 2.91/1.61
% 2.91/1.61 thf(woz1_2,definition,
% 2.91/1.61 ( woz1_2
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Z: $i] :
% 2.91/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 2.91/1.61 => ! [W: $i] :
% 2.91/1.61 ( ( in @ W @ ( powerset @ A ) )
% 2.91/1.61 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 2.91/1.61 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(woz1_3_type,type,
% 2.91/1.61 woz1_3: $o ).
% 2.91/1.61
% 2.91/1.61 thf(woz1_3,definition,
% 2.91/1.61 ( woz1_3
% 2.91/1.61 = ( ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ! [Z: $i] :
% 2.91/1.61 ( ( in @ Z @ ( powerset @ A ) )
% 2.91/1.61 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61
% 2.91/1.61 thf(woz1_4,conjecture,
% 2.91/1.61 ( setextAx
% 2.91/1.61 => ( emptysetAx
% 2.91/1.61 => ( setadjoinAx
% 2.91/1.61 => ( powersetAx
% 2.91/1.61 => ( setunionAx
% 2.91/1.61 => ( omega0Ax
% 2.91/1.61 => ( omegaSAx
% 2.91/1.61 => ( omegaIndAx
% 2.91/1.61 => ( replAx
% 2.91/1.61 => ( foundationAx
% 2.91/1.61 => ( wellorderingAx
% 2.91/1.61 => ( descrp
% 2.91/1.61 => ( dsetconstrI
% 2.91/1.61 => ( dsetconstrEL
% 2.91/1.61 => ( dsetconstrER
% 2.91/1.61 => ( exuE1
% 2.91/1.61 => ( prop2setE
% 2.91/1.61 => ( emptysetE
% 2.91/1.61 => ( emptysetimpfalse
% 2.91/1.61 => ( notinemptyset
% 2.91/1.61 => ( exuE3e
% 2.91/1.61 => ( setext
% 2.91/1.61 => ( emptyI
% 2.91/1.61 => ( noeltsimpempty
% 2.91/1.61 => ( setbeta
% 2.91/1.61 => ( nonemptyE1
% 2.91/1.61 => ( nonemptyI
% 2.91/1.61 => ( nonemptyI1
% 2.91/1.61 => ( setadjoinIL
% 2.91/1.61 => ( emptyinunitempty
% 2.91/1.61 => ( setadjoinIR
% 2.91/1.61 => ( setadjoinE
% 2.91/1.61 => ( setadjoinOr
% 2.91/1.61 => ( setoftrueEq
% 2.91/1.61 => ( powersetI
% 2.91/1.61 => ( emptyinPowerset
% 2.91/1.61 => ( emptyInPowerset
% 2.91/1.61 => ( powersetE
% 2.91/1.61 => ( setunionI
% 2.91/1.61 => ( setunionE
% 2.91/1.61 => ( subPowSU
% 2.91/1.61 => ( exuE2
% 2.91/1.61 => ( nonemptyImpWitness
% 2.91/1.61 => ( uniqinunit
% 2.91/1.61 => ( notinsingleton
% 2.91/1.61 => ( eqinunit
% 2.91/1.61 => ( singletonsswitch
% 2.91/1.61 => ( upairsetE
% 2.91/1.61 => ( upairsetIL
% 2.91/1.61 => ( upairsetIR
% 2.91/1.61 => ( emptyE1
% 2.91/1.61 => ( vacuousDall
% 2.91/1.61 => ( quantDeMorgan1
% 2.91/1.61 => ( quantDeMorgan2
% 2.91/1.61 => ( quantDeMorgan3
% 2.91/1.61 => ( quantDeMorgan4
% 2.91/1.61 => ( prop2setI
% 2.91/1.61 => ( prop2set2propI
% 2.91/1.61 => ( notdexE
% 2.91/1.61 => ( notdallE
% 2.91/1.61 => ( exuI1
% 2.91/1.61 => ( exuI3
% 2.91/1.61 => ( exuI2
% 2.91/1.61 => ( inCongP
% 2.91/1.61 => ( in__Cong
% 2.91/1.61 => ( exuE3u
% 2.91/1.61 => ( exu__Cong
% 2.91/1.61 => ( emptyset__Cong
% 2.91/1.61 => ( setadjoin__Cong
% 2.91/1.61 => ( powerset__Cong
% 2.91/1.61 => ( setunion__Cong
% 2.91/1.61 => ( omega__Cong
% 2.91/1.61 => ( exuEu
% 2.91/1.61 => ( descr__Cong
% 2.91/1.61 => ( dsetconstr__Cong
% 2.91/1.61 => ( subsetI1
% 2.91/1.61 => ( eqimpsubset2
% 2.91/1.61 => ( eqimpsubset1
% 2.91/1.61 => ( subsetI2
% 2.91/1.61 => ( emptysetsubset
% 2.91/1.61 => ( subsetE
% 2.91/1.61 => ( subsetE2
% 2.91/1.61 => ( notsubsetI
% 2.91/1.61 => ( notequalI1
% 2.91/1.61 => ( notequalI2
% 2.91/1.61 => ( subsetRefl
% 2.91/1.61 => ( subsetTrans
% 2.91/1.61 => ( setadjoinSub
% 2.91/1.61 => ( setadjoinSub2
% 2.91/1.61 => ( subset2powerset
% 2.91/1.61 => ( setextsub
% 2.91/1.61 => ( subsetemptysetimpeq
% 2.91/1.61 => ( powersetI1
% 2.91/1.61 => ( powersetE1
% 2.91/1.61 => ( inPowerset
% 2.91/1.61 => ( powersetsubset
% 2.91/1.61 => ( sepInPowerset
% 2.91/1.61 => ( sepSubset
% 2.91/1.61 => ( binunionIL
% 2.91/1.61 => ( upairset2IR
% 2.91/1.61 => ( binunionIR
% 2.91/1.61 => ( binunionEcases
% 2.91/1.61 => ( binunionE
% 2.91/1.61 => ( binunionLsub
% 2.91/1.61 => ( binunionRsub
% 2.91/1.61 => ( binintersectI
% 2.91/1.61 => ( binintersectSubset5
% 2.91/1.61 => ( binintersectEL
% 2.91/1.61 => ( binintersectLsub
% 2.91/1.61 => ( binintersectSubset2
% 2.91/1.61 => ( binintersectSubset3
% 2.91/1.61 => ( binintersectER
% 2.91/1.61 => ( disjointsetsI1
% 2.91/1.61 => ( binintersectRsub
% 2.91/1.61 => ( binintersectSubset4
% 2.91/1.61 => ( binintersectSubset1
% 2.91/1.61 => ( bs114d
% 2.91/1.61 => ( setminusI
% 2.91/1.61 => ( setminusEL
% 2.91/1.61 => ( setminusER
% 2.91/1.61 => ( setminusSubset2
% 2.91/1.61 => ( setminusERneg
% 2.91/1.61 => ( setminusELneg
% 2.91/1.61 => ( setminusILneg
% 2.91/1.61 => ( setminusIRneg
% 2.91/1.61 => ( setminusLsub
% 2.91/1.61 => ( setminusSubset1
% 2.91/1.61 => ( symdiffE
% 2.91/1.61 => ( symdiffI1
% 2.91/1.61 => ( symdiffI2
% 2.91/1.61 => ( symdiffIneg1
% 2.91/1.61 => ( symdiffIneg2
% 2.91/1.61 => ( secondinupair
% 2.91/1.61 => ( setukpairIL
% 2.91/1.61 => ( setukpairIR
% 2.91/1.61 => ( kpairiskpair
% 2.91/1.61 => ( kpairp
% 2.91/1.61 => ( singletonsubset
% 2.91/1.61 => ( singletoninpowerset
% 2.91/1.61 => ( singletoninpowunion
% 2.91/1.61 => ( upairset2E
% 2.91/1.61 => ( upairsubunion
% 2.91/1.61 => ( upairinpowunion
% 2.91/1.61 => ( ubforcartprodlem1
% 2.91/1.61 => ( ubforcartprodlem2
% 2.91/1.61 => ( ubforcartprodlem3
% 2.91/1.61 => ( cartprodpairin
% 2.91/1.61 => ( cartprodmempair1
% 2.91/1.61 => ( cartprodmempair
% 2.91/1.61 => ( setunionE2
% 2.91/1.61 => ( setunionsingleton1
% 2.91/1.61 => ( setunionsingleton2
% 2.91/1.61 => ( setunionsingleton
% 2.91/1.61 => ( singletonprop
% 2.91/1.61 => ( ex1E1
% 2.91/1.61 => ( ex1I
% 2.91/1.61 => ( ex1I2
% 2.91/1.61 => ( singletonsuniq
% 2.91/1.61 => ( setukpairinjL1
% 2.91/1.61 => ( kfstsingleton
% 2.91/1.61 => ( theprop
% 2.91/1.61 => ( kfstpairEq
% 2.91/1.61 => ( cartprodfstin
% 2.91/1.61 => ( setukpairinjL2
% 2.91/1.61 => ( setukpairinjL
% 2.91/1.61 => ( setukpairinjR11
% 2.91/1.61 => ( setukpairinjR12
% 2.91/1.61 => ( setukpairinjR1
% 2.91/1.61 => ( upairequniteq
% 2.91/1.61 => ( setukpairinjR2
% 2.91/1.61 => ( setukpairinjR
% 2.91/1.61 => ( ksndsingleton
% 2.91/1.61 => ( ksndpairEq
% 2.91/1.61 => ( kpairsurjEq
% 2.91/1.61 => ( cartprodsndin
% 2.91/1.61 => ( cartprodpairmemEL
% 2.91/1.61 => ( cartprodpairmemER
% 2.91/1.61 => ( cartprodmempaircEq
% 2.91/1.61 => ( cartprodfstpairEq
% 2.91/1.61 => ( cartprodsndpairEq
% 2.91/1.61 => ( cartprodpairsurjEq
% 2.91/1.61 => ( dpsetconstrI
% 2.91/1.61 => ( dpsetconstrSub
% 2.91/1.61 => ( setOfPairsIsBReln
% 2.91/1.61 => ( dpsetconstrERa
% 2.91/1.61 => ( dpsetconstrEL1
% 2.91/1.61 => ( dpsetconstrEL2
% 2.91/1.61 => ( dpsetconstrER
% 2.91/1.61 => ( funcImageSingleton
% 2.91/1.61 => ( apProp
% 2.91/1.61 => ( app
% 2.91/1.61 => ( infuncsetfunc
% 2.91/1.61 => ( ap2p
% 2.91/1.61 => ( funcinfuncset
% 2.91/1.61 => ( lamProp
% 2.91/1.61 => ( lamp
% 2.91/1.61 => ( lam2p
% 2.91/1.61 => ( brelnall1
% 2.91/1.61 => ( brelnall2
% 2.91/1.61 => ( ex1E2
% 2.91/1.61 => ( funcGraphProp1
% 2.91/1.61 => ( funcGraphProp3
% 2.91/1.61 => ( funcGraphProp2
% 2.91/1.61 => ( funcextLem
% 2.91/1.61 => ( funcGraphProp4
% 2.91/1.61 => ( subbreln
% 2.91/1.61 => ( eqbreln
% 2.91/1.61 => ( funcext
% 2.91/1.61 => ( funcext2
% 2.91/1.61 => ( ap2apEq1
% 2.91/1.61 => ( ap2apEq2
% 2.91/1.61 => ( beta1
% 2.91/1.61 => ( eta1
% 2.91/1.61 => ( lam2lamEq
% 2.91/1.61 => ( beta2
% 2.91/1.61 => ( eta2
% 2.91/1.61 => ( iffalseProp1
% 2.91/1.61 => ( iffalseProp2
% 2.91/1.61 => ( iftrueProp1
% 2.91/1.61 => ( iftrueProp2
% 2.91/1.61 => ( ifSingleton
% 2.91/1.61 => ( ifp
% 2.91/1.61 => ( theeq
% 2.91/1.61 => ( iftrue
% 2.91/1.61 => ( iffalse
% 2.91/1.61 => ( iftrueorfalse
% 2.91/1.61 => ( binintersectT_lem
% 2.91/1.61 => ( binunionT_lem
% 2.91/1.61 => ( powersetT_lem
% 2.91/1.61 => ( setminusT_lem
% 2.91/1.61 => ( complementT_lem
% 2.91/1.61 => ( setextT
% 2.91/1.61 => ( subsetTI
% 2.91/1.61 => ( powersetTI1
% 2.91/1.61 => ( powersetTE1
% 2.91/1.61 => ( complementTI1
% 2.91/1.61 => ( complementTE1
% 2.91/1.61 => ( binintersectTELcontra
% 2.91/1.61 => ( binintersectTERcontra
% 2.91/1.61 => ( contrasubsetT
% 2.91/1.61 => ( contrasubsetT1
% 2.91/1.61 => ( contrasubsetT2
% 2.91/1.61 => ( contrasubsetT3
% 2.91/1.61 => ( doubleComplementI1
% 2.91/1.61 => ( doubleComplementE1
% 2.91/1.61 => ( doubleComplementSub1
% 2.91/1.61 => ( doubleComplementSub2
% 2.91/1.61 => ( doubleComplementEq
% 2.91/1.61 => ( complementTnotintersectT
% 2.91/1.61 => ( complementImpComplementIntersect
% 2.91/1.61 => ( complementSubsetComplementIntersect
% 2.91/1.61 => ( complementInPowersetComplementIntersect
% 2.91/1.61 => ( contraSubsetComplement
% 2.91/1.61 => ( complementTcontraSubset
% 2.91/1.61 => ( binunionTILcontra
% 2.91/1.61 => ( binunionTIRcontra
% 2.91/1.61 => ( inIntersectImpInUnion
% 2.91/1.61 => ( inIntersectImpInUnion2
% 2.91/1.61 => ( inIntersectImpInIntersectUnions
% 2.91/1.61 => ( intersectInPowersetIntersectUnions
% 2.91/1.61 => ( inComplementUnionImpNotIn1
% 2.91/1.61 => ( inComplementUnionImpInComplement1
% 2.91/1.61 => ( binunionTE
% 2.91/1.61 => ( binunionTEcontra
% 2.91/1.61 => ( demorgan2a1
% 2.91/1.61 => ( complementUnionInPowersetComplement
% 2.91/1.61 => ( demorgan2a2
% 2.91/1.61 => ( demorgan1a
% 2.91/1.61 => ( demorgan1b
% 2.91/1.61 => ( demorgan1
% 2.91/1.61 => ( demorgan2a
% 2.91/1.61 => ( demorgan2b2
% 2.91/1.61 => ( demorgan2b
% 2.91/1.61 => ( demorgan2
% 2.91/1.61 => ( woz13rule0
% 2.91/1.61 => ( woz13rule1
% 2.91/1.61 => ( woz13rule2
% 2.91/1.61 => ( woz13rule3
% 2.91/1.61 => ( woz13rule4
% 2.91/1.61 => ( woz1_1
% 2.91/1.61 => ( woz1_2
% 2.91/1.61 => ( woz1_3
% 2.91/1.61 => ! [A: $i,X: $i] :
% 2.91/1.61 ( ( in @ X @ ( powerset @ A ) )
% 2.91/1.61 => ! [Y: $i] :
% 2.91/1.61 ( ( in @ Y @ ( powerset @ A ) )
% 2.91/1.61 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.91/1.61 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.91/1.61 ------------------------
% 2.91/1.65 % (28818)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_28543 for (2999ds/2Mi)
% 2.91/1.65 % (28815)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_28543 for (2999ds/4Mi)
% 2.91/1.65 % (28818)Instruction limit reached!
% 2.91/1.65 % (28818)------------------------------
% 2.91/1.65 % (28818)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.91/1.65 % (28818)Termination reason: Unknown
% 2.91/1.65 % (28818)Termination phase: shuffling
% 2.91/1.65
% 2.91/1.65 % (28818)Memory used [KB]: 1535
% 2.91/1.65 % (28818)Time elapsed: 0.002 s
% 2.91/1.65 % (28818)Instructions burned: 2 (million)
% 2.91/1.65 % (28818)------------------------------
% 2.91/1.65 % (28818)------------------------------
% 2.91/1.65 % (28819)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_28543 for (2999ds/275Mi)
% 2.91/1.65 % (28815)Instruction limit reached!
% 2.91/1.65 % (28815)------------------------------
% 2.91/1.65 % (28815)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.91/1.65 % (28815)Termination reason: Unknown
% 2.91/1.65 % (28815)Termination phase: shuffling
% 2.91/1.65
% 2.91/1.65 % (28815)Memory used [KB]: 1535
% 2.91/1.65 % (28815)Time elapsed: 0.003 s
% 2.91/1.65 % (28815)Instructions burned: 6 (million)
% 2.91/1.65 % (28815)------------------------------
% 2.91/1.65 % (28815)------------------------------
% 2.91/1.66 % (28814)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_28543 for (2999ds/183Mi)
% 2.91/1.66 % (28816)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_28543 for (2999ds/27Mi)
% 2.91/1.66 % (28817)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_28543 for (2999ds/2Mi)
% 2.91/1.66 % (28820)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_28543 for (2999ds/18Mi)
% 2.91/1.66 % (28817)Instruction limit reached!
% 2.91/1.66 % (28817)------------------------------
% 2.91/1.66 % (28817)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.91/1.66 % (28817)Termination reason: Unknown
% 2.91/1.66 % (28817)Termination phase: shuffling
% 2.91/1.66
% 2.91/1.66 % (28817)Memory used [KB]: 1535
% 2.91/1.66 % (28817)Time elapsed: 0.003 s
% 2.91/1.66 % (28817)Instructions burned: 3 (million)
% 2.91/1.66 % (28817)------------------------------
% 2.91/1.66 % (28817)------------------------------
% 2.91/1.66 % (28821)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_28543 for (2999ds/3Mi)
% 2.91/1.66 % (28821)Instruction limit reached!
% 2.91/1.66 % (28821)------------------------------
% 2.91/1.66 % (28821)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.91/1.66 % (28821)Termination reason: Unknown
% 2.91/1.66 % (28821)Termination phase: shuffling
% 2.91/1.66
% 2.91/1.66 % (28821)Memory used [KB]: 1535
% 2.91/1.66 % (28821)Time elapsed: 0.002 s
% 2.91/1.66 % (28821)Instructions burned: 3 (million)
% 2.91/1.66 % (28821)------------------------------
% 2.91/1.66 % (28821)------------------------------
% 2.91/1.66 % (28822)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_28543 for (2999ds/37Mi)
% 2.91/1.67 % (28820)Instruction limit reached!
% 2.91/1.67 % (28820)------------------------------
% 2.91/1.67 % (28820)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.91/1.67 % (28820)Termination reason: Unknown
% 2.91/1.67 % (28820)Termination phase: shuffling
% 2.91/1.67
% 2.91/1.67 % (28820)Memory used [KB]: 1918
% 2.91/1.67 % (28820)Time elapsed: 0.013 s
% 2.91/1.67 % (28820)Instructions burned: 19 (million)
% 2.91/1.67 % (28820)------------------------------
% 2.91/1.67 % (28820)------------------------------
% 2.91/1.67 % (28824)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_28543 for (2999ds/3Mi)
% 2.91/1.67 % (28824)Instruction limit reached!
% 2.91/1.67 % (28824)------------------------------
% 2.91/1.67 % (28824)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.91/1.67 % (28816)Instruction limit reached!
% 2.91/1.67 % (28816)------------------------------
% 2.91/1.67 % (28816)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.91/1.67 % (28824)Termination reason: Unknown
% 2.91/1.67 % (28816)Termination reason: Unknown
% 2.91/1.67 % (28824)Termination phase: shuffling
% 2.91/1.67
% 2.91/1.67 % (28824)Memory used [KB]: 1535
% 2.91/1.67 % (28824)Time elapsed: 0.003 s
% 2.91/1.67 % (28816)Termination phase: shuffling
% 2.91/1.67 % (28824)Instructions burned: 5 (million)
% 2.91/1.67
% 2.91/1.67 % (28824)------------------------------
% 2.91/1.67 % (28824)------------------------------
% 2.91/1.67 % (28816)Memory used [KB]: 1918
% 2.91/1.67 % (28816)Time elapsed: 0.016 s
% 2.91/1.67 % (28816)Instructions burned: 28 (million)
% 2.91/1.67 % (28816)------------------------------
% 2.91/1.67 % (28816)------------------------------
% 2.91/1.67 % (28822)Instruction limit reached!
% 2.91/1.67 % (28822)------------------------------
% 2.91/1.67 % (28822)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.91/1.67 % (28822)Termination reason: Unknown
% 2.91/1.67 % (28822)Termination phase: shuffling
% 2.91/1.67
% 2.91/1.67 % (28822)Memory used [KB]: 2302
% 2.91/1.67 % (28822)Time elapsed: 0.012 s
% 2.91/1.67 % (28822)Instructions burned: 38 (million)
% 2.91/1.67 % (28822)------------------------------
% 2.91/1.67 % (28822)------------------------------
% 2.91/1.68 % (28823)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_28543 for (2999ds/15Mi)
% 2.91/1.68 % (28826)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_28543 for (2999ds/7Mi)
% 2.91/1.69 % (28828)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_28543 for (2999ds/3Mi)
% 2.91/1.69 % (28826)Instruction limit reached!
% 2.91/1.69 % (28826)------------------------------
% 2.91/1.69 % (28826)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.91/1.69 % (28826)Termination reason: Unknown
% 2.91/1.69 % (28826)Termination phase: shuffling
% 2.91/1.69
% 2.91/1.69 % (28826)Memory used [KB]: 1663
% 2.91/1.69 % (28826)Time elapsed: 0.004 s
% 2.91/1.69 % (28826)Instructions burned: 10 (million)
% 2.91/1.69 % (28826)------------------------------
% 2.91/1.69 % (28826)------------------------------
% 2.91/1.69 % (28828)Instruction limit reached!
% 2.91/1.69 % (28828)------------------------------
% 2.91/1.69 % (28828)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.91/1.69 % (28828)Termination reason: Unknown
% 2.91/1.69 % (28828)Termination phase: shuffling
% 2.91/1.69
% 2.91/1.69 % (28828)Memory used [KB]: 1535
% 2.91/1.69 % (28828)Time elapsed: 0.002 s
% 2.91/1.69 % (28828)Instructions burned: 3 (million)
% 2.91/1.69 % (28828)------------------------------
% 2.91/1.69 % (28828)------------------------------
% 2.91/1.69 % (28823)Instruction limit reached!
% 2.91/1.69 % (28823)------------------------------
% 2.91/1.69 % (28823)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.91/1.69 % (28823)Termination reason: Unknown
% 2.91/1.69 % (28823)Termination phase: shuffling
% 2.91/1.69
% 2.91/1.69 % (28823)Memory used [KB]: 1791
% 2.91/1.69 % (28823)Time elapsed: 0.011 s
% 2.91/1.69 % (28823)Instructions burned: 15 (million)
% 2.91/1.69 % (28823)------------------------------
% 2.91/1.69 % (28823)------------------------------
% 2.91/1.69 % (28825)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_28543 for (2999ds/1041Mi)
% 2.91/1.69 % (28827)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_28543 for (2999ds/16Mi)
% 2.91/1.70 % (28829)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_28543 for (2999ds/3Mi)
% 2.95/1.70 % (28829)Instruction limit reached!
% 2.95/1.70 % (28829)------------------------------
% 2.95/1.70 % (28829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.70 % (28829)Termination reason: Unknown
% 2.95/1.70 % (28829)Termination phase: shuffling
% 2.95/1.70
% 2.95/1.70 % (28829)Memory used [KB]: 1535
% 2.95/1.70 % (28829)Time elapsed: 0.002 s
% 2.95/1.70 % (28829)Instructions burned: 3 (million)
% 2.95/1.70 % (28829)------------------------------
% 2.95/1.70 % (28829)------------------------------
% 2.95/1.70 % (28830)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_28543 for (2999ds/7Mi)
% 2.95/1.70 % (28830)Instruction limit reached!
% 2.95/1.70 % (28830)------------------------------
% 2.95/1.70 % (28830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.70 % (28830)Termination reason: Unknown
% 2.95/1.70 % (28830)Termination phase: shuffling
% 2.95/1.70
% 2.95/1.70 % (28830)Memory used [KB]: 1535
% 2.95/1.70 % (28830)Time elapsed: 0.003 s
% 2.95/1.70 % (28830)Instructions burned: 7 (million)
% 2.95/1.70 % (28830)------------------------------
% 2.95/1.70 % (28830)------------------------------
% 2.95/1.70 % (28827)Instruction limit reached!
% 2.95/1.70 % (28827)------------------------------
% 2.95/1.70 % (28827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.70 % (28827)Termination reason: Unknown
% 2.95/1.70 % (28827)Termination phase: shuffling
% 2.95/1.70
% 2.95/1.70 % (28827)Memory used [KB]: 1918
% 2.95/1.70 % (28827)Time elapsed: 0.011 s
% 2.95/1.70 % (28827)Instructions burned: 17 (million)
% 2.95/1.70 % (28827)------------------------------
% 2.95/1.70 % (28827)------------------------------
% 2.95/1.70 % (28831)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_28543 for (2999ds/3Mi)
% 2.95/1.70 % (28831)Instruction limit reached!
% 2.95/1.70 % (28831)------------------------------
% 2.95/1.70 % (28831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.70 % (28831)Termination reason: Unknown
% 2.95/1.70 % (28831)Termination phase: shuffling
% 2.95/1.70
% 2.95/1.70 % (28831)Memory used [KB]: 1535
% 2.95/1.70 % (28831)Time elapsed: 0.004 s
% 2.95/1.70 % (28831)Instructions burned: 6 (million)
% 2.95/1.70 % (28831)------------------------------
% 2.95/1.70 % (28831)------------------------------
% 2.95/1.71 % (28832)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_28543 for (2999ds/4Mi)
% 2.95/1.71 % (28832)Instruction limit reached!
% 2.95/1.71 % (28832)------------------------------
% 2.95/1.71 % (28832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.71 % (28832)Termination reason: Unknown
% 2.95/1.71 % (28832)Termination phase: shuffling
% 2.95/1.71
% 2.95/1.71 % (28832)Memory used [KB]: 1535
% 2.95/1.71 % (28832)Time elapsed: 0.003 s
% 2.95/1.71 % (28832)Instructions burned: 7 (million)
% 2.95/1.71 % (28832)------------------------------
% 2.95/1.71 % (28832)------------------------------
% 2.95/1.71 % (28833)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_28543 for (2999ds/18Mi)
% 2.95/1.71 % (28833)Instruction limit reached!
% 2.95/1.71 % (28833)------------------------------
% 2.95/1.71 % (28833)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.71 % (28833)Termination reason: Unknown
% 2.95/1.71 % (28833)Termination phase: shuffling
% 2.95/1.71
% 2.95/1.71 % (28833)Memory used [KB]: 1791
% 2.95/1.71 % (28833)Time elapsed: 0.006 s
% 2.95/1.71 % (28833)Instructions burned: 18 (million)
% 2.95/1.71 % (28833)------------------------------
% 2.95/1.71 % (28833)------------------------------
% 2.95/1.71 % (28835)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_28543 for (2999ds/6Mi)
% 2.95/1.72 % (28835)Instruction limit reached!
% 2.95/1.72 % (28835)------------------------------
% 2.95/1.72 % (28835)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.72 % (28835)Termination reason: Unknown
% 2.95/1.72 % (28835)Termination phase: shuffling
% 2.95/1.72
% 2.95/1.72 % (28835)Memory used [KB]: 1663
% 2.95/1.72 % (28835)Time elapsed: 0.004 s
% 2.95/1.72 % (28835)Instructions burned: 7 (million)
% 2.95/1.72 % (28835)------------------------------
% 2.95/1.72 % (28835)------------------------------
% 2.95/1.72 % (28836)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_28543 for (2999ds/902Mi)
% 2.95/1.72 % (28834)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_28543 for (2999ds/710Mi)
% 2.95/1.72 % (28837)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_28543 for (2999ds/21Mi)
% 2.95/1.73 % (28838)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_28543 for (2999ds/5Mi)
% 2.95/1.73 % (28838)Instruction limit reached!
% 2.95/1.73 % (28838)------------------------------
% 2.95/1.73 % (28838)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.73 % (28838)Termination reason: Unknown
% 2.95/1.73 % (28838)Termination phase: shuffling
% 2.95/1.73
% 2.95/1.73 % (28838)Memory used [KB]: 1535
% 2.95/1.73 % (28838)Time elapsed: 0.003 s
% 2.95/1.73 % (28838)Instructions burned: 7 (million)
% 2.95/1.73 % (28838)------------------------------
% 2.95/1.73 % (28838)------------------------------
% 2.95/1.73 % (28819)Instruction limit reached!
% 2.95/1.73 % (28819)------------------------------
% 2.95/1.73 % (28819)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.73 % (28819)Termination reason: Unknown
% 2.95/1.73 % (28819)Termination phase: Saturation
% 2.95/1.73
% 2.95/1.73 % (28819)Memory used [KB]: 10106
% 2.95/1.73 % (28819)Time elapsed: 0.079 s
% 2.95/1.73 % (28819)Instructions burned: 276 (million)
% 2.95/1.73 % (28819)------------------------------
% 2.95/1.73 % (28819)------------------------------
% 2.95/1.73 % (28837)Instruction limit reached!
% 2.95/1.73 % (28837)------------------------------
% 2.95/1.73 % (28837)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.73 % (28837)Termination reason: Unknown
% 2.95/1.73 % (28837)Termination phase: shuffling
% 2.95/1.73
% 2.95/1.73 % (28837)Memory used [KB]: 1918
% 2.95/1.73 % (28837)Time elapsed: 0.007 s
% 2.95/1.73 % (28837)Instructions burned: 21 (million)
% 2.95/1.73 % (28837)------------------------------
% 2.95/1.73 % (28837)------------------------------
% 2.95/1.74 % (28839)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_28543 for (2998ds/6Mi)
% 2.95/1.74 % (28840)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_28543 for (2998ds/377Mi)
% 2.95/1.74 % (28841)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_28543 for (2998ds/779Mi)
% 2.95/1.74 % (28839)Instruction limit reached!
% 2.95/1.74 % (28839)------------------------------
% 2.95/1.74 % (28839)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.74 % (28839)Termination reason: Unknown
% 2.95/1.74 % (28839)Termination phase: shuffling
% 2.95/1.74
% 2.95/1.74 % (28839)Memory used [KB]: 1663
% 2.95/1.74 % (28839)Time elapsed: 0.003 s
% 2.95/1.74 % (28839)Instructions burned: 7 (million)
% 2.95/1.74 % (28839)------------------------------
% 2.95/1.74 % (28839)------------------------------
% 2.95/1.75 % (28814)Instruction limit reached!
% 2.95/1.75 % (28814)------------------------------
% 2.95/1.75 % (28814)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.75 % (28814)Termination reason: Unknown
% 2.95/1.75 % (28814)Termination phase: Saturation
% 2.95/1.75
% 2.95/1.75 % (28814)Memory used [KB]: 7419
% 2.95/1.75 % (28814)Time elapsed: 0.090 s
% 2.95/1.75 % (28814)Instructions burned: 183 (million)
% 2.95/1.75 % (28814)------------------------------
% 2.95/1.75 % (28814)------------------------------
% 2.95/1.75 % (28842)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_28543 for (2998ds/19Mi)
% 2.95/1.76 % (28842)Instruction limit reached!
% 2.95/1.76 % (28842)------------------------------
% 2.95/1.76 % (28842)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.76 % (28842)Termination reason: Unknown
% 2.95/1.76 % (28842)Termination phase: shuffling
% 2.95/1.76
% 2.95/1.76 % (28842)Memory used [KB]: 1918
% 2.95/1.76 % (28842)Time elapsed: 0.007 s
% 2.95/1.76 % (28842)Instructions burned: 21 (million)
% 2.95/1.76 % (28842)------------------------------
% 2.95/1.76 % (28842)------------------------------
% 2.95/1.76 % (28843)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_28543 for (2998ds/879Mi)
% 2.95/1.77 % (28844)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_28543 for (2998ds/17Mi)
% 2.95/1.77 % (28844)Instruction limit reached!
% 2.95/1.77 % (28844)------------------------------
% 2.95/1.77 % (28844)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.77 % (28844)Termination reason: Unknown
% 2.95/1.77 % (28844)Termination phase: shuffling
% 2.95/1.77
% 2.95/1.77 % (28844)Memory used [KB]: 1791
% 2.95/1.77 % (28844)Time elapsed: 0.007 s
% 2.95/1.77 % (28844)Instructions burned: 17 (million)
% 2.95/1.77 % (28844)------------------------------
% 2.95/1.77 % (28844)------------------------------
% 2.95/1.78 % (28845)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_28543 for (2998ds/3Mi)
% 2.95/1.78 % (28845)Instruction limit reached!
% 2.95/1.78 % (28845)------------------------------
% 2.95/1.78 % (28845)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.78 % (28845)Termination reason: Unknown
% 2.95/1.78 % (28845)Termination phase: shuffling
% 2.95/1.78
% 2.95/1.78 % (28845)Memory used [KB]: 1535
% 2.95/1.78 % (28845)Time elapsed: 0.002 s
% 2.95/1.78 % (28845)Instructions burned: 3 (million)
% 2.95/1.78 % (28845)------------------------------
% 2.95/1.78 % (28845)------------------------------
% 2.95/1.80 % (28846)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_28543 for (2998ds/30Mi)
% 2.95/1.80 % (28846)Instruction limit reached!
% 2.95/1.80 % (28846)------------------------------
% 2.95/1.80 % (28846)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.80 % (28846)Termination reason: Unknown
% 2.95/1.80 % (28846)Termination phase: shuffling
% 2.95/1.80
% 2.95/1.80 % (28846)Memory used [KB]: 2046
% 2.95/1.80 % (28846)Time elapsed: 0.010 s
% 2.95/1.80 % (28846)Instructions burned: 31 (million)
% 2.95/1.80 % (28846)------------------------------
% 2.95/1.80 % (28846)------------------------------
% 2.95/1.81 % (28847)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_28543 for (2998ds/127Mi)
% 2.95/1.84 % (28840)Instruction limit reached!
% 2.95/1.84 % (28840)------------------------------
% 2.95/1.84 % (28840)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.84 % (28840)Termination reason: Unknown
% 2.95/1.84 % (28840)Termination phase: Saturation
% 2.95/1.84
% 2.95/1.84 % (28840)Memory used [KB]: 9466
% 2.95/1.84 % (28840)Time elapsed: 0.101 s
% 2.95/1.84 % (28840)Instructions burned: 377 (million)
% 2.95/1.84 % (28840)------------------------------
% 2.95/1.84 % (28840)------------------------------
% 2.95/1.84 % (28836)First to succeed.
% 2.95/1.84 % (28836)Refutation found. Thanks to Tanya!
% 2.95/1.84 % SZS status Theorem for DTF2THF_28543
% 2.95/1.84 % SZS output start Proof for DTF2THF_28543
% 2.95/1.84 thf(func_def_0, type, in: $i > $i > $o).
% 2.95/1.84 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.95/1.84 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.95/1.84 thf(func_def_8, type, powerset: $i > $i).
% 2.95/1.84 thf(func_def_10, type, setunion: $i > $i).
% 2.95/1.84 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.95/1.84 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.95/1.84 thf(func_def_26, type, prop2set: $o > $i).
% 2.95/1.84 thf(func_def_36, type, nonempty: $i > $o).
% 2.95/1.84 thf(func_def_69, type, set2prop: $i > $o).
% 2.95/1.84 thf(func_def_88, type, subset: $i > $i > $o).
% 2.95/1.84 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.95/1.84 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.95/1.84 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.95/1.84 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.95/1.84 thf(func_def_135, type, regular: $i > $o).
% 2.95/1.84 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.95/1.84 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.95/1.84 thf(func_def_153, type, iskpair: $i > $o).
% 2.95/1.84 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.95/1.84 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.95/1.84 thf(func_def_177, type, singleton: $i > $o).
% 2.95/1.84 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.95/1.84 thf(func_def_184, type, atmost1p: $i > $o).
% 2.95/1.84 thf(func_def_185, type, atleast2p: $i > $o).
% 2.95/1.84 thf(func_def_186, type, atmost2p: $i > $o).
% 2.95/1.84 thf(func_def_187, type, upairsetp: $i > $o).
% 2.95/1.84 thf(func_def_191, type, kfst: $i > $i).
% 2.95/1.84 thf(func_def_203, type, ksnd: $i > $i).
% 2.95/1.84 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.95/1.84 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.95/1.84 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.95/1.84 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.95/1.84 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.95/1.84 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.95/1.84 thf(func_def_334, type, sP0: $i > $i > $o).
% 2.95/1.84 thf(func_def_335, type, sP1: $i > $o).
% 2.95/1.84 thf(func_def_336, type, sP2: $i > $i > $o).
% 2.95/1.84 thf(func_def_337, type, sP3: $i > $i > $o).
% 2.95/1.84 thf(func_def_338, type, sP4: $o > $i > $i > $i > $o).
% 2.95/1.84 thf(func_def_340, type, sP6: $i > $i > $i > $o).
% 2.95/1.84 thf(func_def_345, type, sP11: $i > $i > $i > $i > $o).
% 2.95/1.84 thf(func_def_350, type, sK16: ($i > $o) > $i > $i > $i).
% 2.95/1.84 thf(func_def_353, type, sK19: $i > $o).
% 2.95/1.84 thf(func_def_360, type, sK26: $i > ($i > $o) > $i).
% 2.95/1.84 thf(func_def_361, type, sK27: $i > $o).
% 2.95/1.84 thf(func_def_379, type, sK45: $i > $i).
% 2.95/1.84 thf(func_def_391, type, sK57: $i > $i).
% 2.95/1.84 thf(func_def_394, type, sK60: $i > $i > $i).
% 2.95/1.84 thf(func_def_398, type, sK64: $i > $i > $i).
% 2.95/1.84 thf(func_def_399, type, sK65: $i > $i > $i).
% 2.95/1.84 thf(func_def_400, type, sK66: $i > $i > $i).
% 2.95/1.84 thf(func_def_401, type, sK67: $i > $i > $i).
% 2.95/1.84 thf(func_def_402, type, sK68: $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_403, type, sK69: $i > $i).
% 2.95/1.84 thf(func_def_404, type, sK70: $i > $i).
% 2.95/1.84 thf(func_def_405, type, sK71: $i > $i).
% 2.95/1.84 thf(func_def_406, type, sK72: $i > $i).
% 2.95/1.84 thf(func_def_407, type, sK73: $i > $i > $i).
% 2.95/1.84 thf(func_def_408, type, sK74: $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_409, type, sK75: $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_410, type, sK76: $i > $i > $i).
% 2.95/1.84 thf(func_def_411, type, sK77: $i > $i > $i).
% 2.95/1.84 thf(func_def_413, type, sK79: $i > $i).
% 2.95/1.84 thf(func_def_414, type, sK80: $i > $i).
% 2.95/1.84 thf(func_def_415, type, sK81: $i > $i).
% 2.95/1.84 thf(func_def_416, type, sK82: $i > $i > $i).
% 2.95/1.84 thf(func_def_417, type, sK83: $i > ($i > $i) > $i > $i).
% 2.95/1.84 thf(func_def_419, type, sK85: $i > $i).
% 2.95/1.84 thf(func_def_423, type, sK89: $i > $i).
% 2.95/1.84 thf(func_def_424, type, sK90: ($i > $i) > $i > $i > $i).
% 2.95/1.84 thf(func_def_434, type, sK100: $i > $o).
% 2.95/1.84 thf(func_def_435, type, sK101: $i > $i).
% 2.95/1.84 thf(func_def_436, type, sK102: ($i > $o) > $i).
% 2.95/1.84 thf(func_def_437, type, sK103: ($i > $o) > ($i > $o) > $i).
% 2.95/1.84 thf(func_def_438, type, sK104: ($i > $o) > ($i > $o) > $i).
% 2.95/1.84 thf(func_def_439, type, sK105: $i > $o).
% 2.95/1.84 thf(func_def_440, type, sK106: $i > $o).
% 2.95/1.84 thf(func_def_445, type, sK111: $i > $o).
% 2.95/1.84 thf(func_def_448, type, sK114: $i > ($i > $o) > $i).
% 2.95/1.84 thf(func_def_460, type, sK126: $i > $o).
% 2.95/1.84 thf(func_def_475, type, sK141: $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_476, type, sK142: $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_509, type, sK175: $i > $i > $o).
% 2.95/1.84 thf(func_def_517, type, sK183: $i > $i > $i).
% 2.95/1.84 thf(func_def_539, type, sK205: $i > $i > $o).
% 2.95/1.84 thf(func_def_540, type, sK206: $i > $i).
% 2.95/1.84 thf(func_def_541, type, sK207: $i > $i).
% 2.95/1.84 thf(func_def_542, type, sK208: ($i > $i > $o) > $i > $i).
% 2.95/1.84 thf(func_def_543, type, sK209: $i > ($i > $i > $o) > $i > $i).
% 2.95/1.84 thf(func_def_544, type, sK210: ($i > $i > $o) > $i > $i).
% 2.95/1.84 thf(func_def_549, type, sK215: $i > $o).
% 2.95/1.84 thf(func_def_550, type, sK216: $i > $o).
% 2.95/1.84 thf(func_def_551, type, sK217: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.95/1.84 thf(func_def_552, type, sK218: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.95/1.84 thf(func_def_553, type, sK219: $i > $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_563, type, sK229: ($i > $o) > $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_564, type, sK230: ($i > $o) > $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_568, type, sK234: $i > $o).
% 2.95/1.84 thf(func_def_576, type, sK242: $i > $i > $i).
% 2.95/1.84 thf(func_def_584, type, sK250: $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_588, type, sK254: $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_602, type, sK268: $i > $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_613, type, sK279: $i > $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_614, type, sK280: $i > $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_632, type, sK298: $i > $o).
% 2.95/1.84 thf(func_def_652, type, sK318: $i > $o).
% 2.95/1.84 thf(func_def_657, type, sK323: $i > $o).
% 2.95/1.84 thf(func_def_685, type, sK351: $i > $o).
% 2.95/1.84 thf(func_def_686, type, sK352: $i > $i > $o).
% 2.95/1.84 thf(func_def_718, type, sK384: $i > $o).
% 2.95/1.84 thf(func_def_745, type, sK411: $i > $o).
% 2.95/1.84 thf(func_def_746, type, sK412: $i > $o).
% 2.95/1.84 thf(func_def_769, type, sK435: $i > $i).
% 2.95/1.84 thf(func_def_788, type, sK454: $i > $i > $o).
% 2.95/1.84 thf(func_def_805, type, sK471: $i > $i > $i).
% 2.95/1.84 thf(func_def_806, type, sK472: $i > $i > $i).
% 2.95/1.84 thf(func_def_813, type, sK479: $i > $o).
% 2.95/1.84 thf(func_def_815, type, sK481: ($i > $o) > $i > $i).
% 2.95/1.84 thf(func_def_816, type, sK482: $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_820, type, sK486: $i > ($i > $o) > $i).
% 2.95/1.84 thf(func_def_821, type, sK487: $i > $o).
% 2.95/1.84 thf(func_def_832, type, sK498: $i > $i).
% 2.95/1.84 thf(func_def_834, type, sK500: $i > ($i > $i) > $i > $i).
% 2.95/1.84 thf(func_def_836, type, sK502: $i > $i).
% 2.95/1.84 thf(func_def_856, type, sK522: $i > $i > $i).
% 2.95/1.84 thf(func_def_859, type, sK525: $o > $i > $i > $i).
% 2.95/1.84 thf(func_def_864, type, sK530: $i > $o).
% 2.95/1.84 thf(func_def_866, type, sK532: ($i > $o) > $i > $i).
% 2.95/1.84 thf(func_def_890, type, sK556: $i > $i > $o).
% 2.95/1.84 thf(func_def_905, type, sK571: $i > $i > $o).
% 2.95/1.84 thf(func_def_920, type, sK586: ($i > $i) > $i > $i > $i).
% 2.95/1.84 thf(func_def_923, type, sK589: $i > $i).
% 2.95/1.84 thf(func_def_926, type, sK592: $i > $i > $o).
% 2.95/1.84 thf(func_def_928, type, sK594: $i > $o).
% 2.95/1.84 thf(func_def_929, type, sK595: $i > $o).
% 2.95/1.84 thf(func_def_930, type, sK596: ($i > $o) > ($i > $o) > $i).
% 2.95/1.84 thf(func_def_931, type, sK597: ($i > $o) > ($i > $o) > $i).
% 2.95/1.84 thf(func_def_933, type, sK599: $i > $o).
% 2.95/1.84 thf(func_def_935, type, sK601: $i > ($i > $o) > $i).
% 2.95/1.84 thf(func_def_936, type, sK602: $i > $i).
% 2.95/1.84 thf(func_def_947, type, sK613: $i > $o).
% 2.95/1.84 thf(func_def_949, type, sK615: ($i > $o) > $i > $i).
% 2.95/1.84 thf(func_def_950, type, sK616: $i > $o).
% 2.95/1.84 thf(func_def_954, type, sK620: $i > $o).
% 2.95/1.84 thf(func_def_956, type, sK622: ($i > $o) > $i > $i).
% 2.95/1.84 thf(func_def_957, type, sK623: ($i > $o) > $i > $i).
% 2.95/1.84 thf(func_def_980, type, sK646: ($i > $o) > $i).
% 2.95/1.84 thf(func_def_981, type, sK647: ($i > $o) > $i).
% 2.95/1.84 thf(func_def_982, type, sK648: $i > $o).
% 2.95/1.84 thf(func_def_1011, type, sK677: $i > $i > $i).
% 2.95/1.84 thf(func_def_1028, type, sK694: $i > $i).
% 2.95/1.84 thf(func_def_1030, type, sK696: $i > $i).
% 2.95/1.84 thf(func_def_1031, type, sK697: $i > $o).
% 2.95/1.84 thf(func_def_1032, type, sK698: ($i > $o) > $i).
% 2.95/1.84 thf(func_def_1045, type, sK711: $i > $i > $i).
% 2.95/1.84 thf(func_def_1053, type, sK719: $i > $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_1106, type, sK772: $i > $o).
% 2.95/1.84 thf(func_def_1108, type, sK774: ($i > $o) > $i).
% 2.95/1.84 thf(func_def_1109, type, sK775: $i > $o).
% 2.95/1.84 thf(func_def_1110, type, sK776: $i > $i).
% 2.95/1.84 thf(func_def_1111, type, sK777: $i > $o).
% 2.95/1.84 thf(func_def_1134, type, sK800: $i > $o).
% 2.95/1.84 thf(func_def_1136, type, sK802: ($i > $o) > $i > $i).
% 2.95/1.84 thf(func_def_1137, type, sK803: ($i > $o) > $i > $i).
% 2.95/1.84 thf(func_def_1141, type, sK807: ($i > $i) > $i > $i > $i).
% 2.95/1.84 thf(func_def_1144, type, sK810: $i > $i).
% 2.95/1.84 thf(func_def_1162, type, sK828: $i > ($i > $o) > $i).
% 2.95/1.84 thf(func_def_1163, type, sK829: $i > $o).
% 2.95/1.84 thf(func_def_1169, type, sK835: $i > $i > ($i > $i) > $i).
% 2.95/1.84 thf(func_def_1170, type, sK836: $i > $i).
% 2.95/1.84 thf(func_def_1175, type, sK841: $i > $i > $i).
% 2.95/1.84 thf(func_def_1176, type, sK842: $i > $i > $i).
% 2.95/1.84 thf(func_def_1183, type, sK849: $i > $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_1184, type, sK850: $i > $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_1185, type, sK851: $i > $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_1186, type, sK852: $i > $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_1190, type, sK856: $i > $i > $o).
% 2.95/1.84 thf(func_def_1192, type, sK858: $i > ($i > $o) > $i).
% 2.95/1.84 thf(func_def_1193, type, sK859: $i > $o).
% 2.95/1.84 thf(func_def_1196, type, sK862: $i > $o).
% 2.95/1.84 thf(func_def_1205, type, sK871: $i > $o).
% 2.95/1.84 thf(func_def_1207, type, sK873: ($i > $o) > $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_1208, type, sK874: ($i > $o) > $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_1218, type, sK884: $i > $o).
% 2.95/1.84 thf(func_def_1227, type, sK893: $i > $i > $i > $i).
% 2.95/1.84 thf(func_def_1255, type, ph921: !>[X0: $tType]:(X0)).
% 2.95/1.84 thf(f4683,plain,(
% 2.95/1.84 $false),
% 2.95/1.84 inference(trivial_inequality_removal,[status(thm)],[f4672])).
% 2.95/1.84 thf(f4672,plain,(
% 2.95/1.84 ($true != $true)),
% 2.95/1.84 inference(superposition,[status(thm)],[f4491,f3731])).
% 2.95/1.84 thf(f3731,plain,(
% 2.95/1.84 ((subset @ sK740 @ (setminus @ sK739 @ sK741)) = $true)),
% 2.95/1.84 inference(cnf_transformation,[status(thm)],[f2394])).
% 2.95/1.84 thf(f2394,plain,(
% 2.95/1.84 (doubleComplementEq = $true) & (dpsetconstrER = $true) & (funcextLem = $true) & (notdallE = $true) & (emptysetE = $true) & (woz13rule1 = $true) & (complementSubsetComplementIntersect = $true) & (inComplementUnionImpNotIn1 = $true) & (brelnall2 = $true) & (binunionT_lem = $true) & (funcinfuncset = $true) & (subsetTI = $true) & (notdexE = $true) & (prop2setE = $true) & (eqbreln = $true) & (binunionEcases = $true) & (emptyinunitempty = $true) & (quantDeMorgan3 = $true) & (beta2 = $true) & (doubleComplementSub1 = $true) & (prop2set2propI = $true) & (upairsetIL = $true) & (contrasubsetT1 = $true) & (powersetT_lem = $true) & (binintersectI = $true) & (setukpairinjL2 = $true) & (funcGraphProp1 = $true) & (funcGraphProp3 = $true) & (omega0Ax = $true) & (binintersectEL = $true) & (subsetE = $true) & (doubleComplementE1 = $true) & (subPowSU = $true) & (binunionLsub = $true) & (nonemptyE1 = $true) & (setminusIRneg = $true) & (prop2setI = $true) & (woz13rule3 = $true) & (dpsetconstrERa = $true) & (demorgan1a = $true) & (woz1_3 = $true) & (ubforcartprodlem2 = $true) & (emptyset__Cong = $true) & (setminusSubset1 = $true) & (sepInPowerset = $true) & (complementInPowersetComplementIntersect = $true) & (eqimpsubset1 = $true) & (ap2apEq1 = $true) & (vacuousDall = $true) & (iftrue = $true) & (setminusT_lem = $true) & (upairequniteq = $true) & (woz13rule0 = $true) & (funcext = $true) & (emptysetimpfalse = $true) & (setunionsingleton2 = $true) & (exuE3e = $true) & (funcGraphProp4 = $true) & (emptyI = $true) & (ex1E2 = $true) & (setminusLsub = $true) & (notequalI1 = $true) & (setadjoinAx = $true) & (funcImageSingleton = $true) & (notinemptyset = $true) & (eqimpsubset2 = $true) & (notsubsetI = $true) & (binunionTILcontra = $true) & (demorgan2a2 = $true) & (setunionAx = $true) & (setadjoin__Cong = $true) & (setminusERneg = $true) & (setadjoinSub = $true) & (inIntersectImpInIntersectUnions = $true) & (funcext2 = $true) & (setunionE = $true) & (noeltsimpempty = $true) & (cartprodpairsurjEq = $true) & (woz1_2 = $true) & (doubleComplementSub2 = $true) & (setextT = $true) & (subsetRefl = $true) & (setunionE2 = $true) & (cartprodpairin = $true) & (woz13rule2 = $true) & (setadjoinIL = $true) & (setminusSubset2 = $true) & (binunionIL = $true) & (omegaIndAx = $true) & (lam2lamEq = $true) & (theeq = $true) & (iffalseProp2 = $true) & (setOfPairsIsBReln = $true) & (binintersectSubset3 = $true) & (iftrueorfalse = $true) & (inIntersectImpInUnion = $true) & (upairsetIR = $true) & (setukpairinjR2 = $true) & (eta2 = $true) & (brelnall1 = $true) & (binunionRsub = $true) & (funcGraphProp2 = $true) & (subsetemptysetimpeq = $true) & (setadjoinOr = $true) & (lamp = $true) & (cartprodpairmemEL = $true) & (upairsetE = $true) & (upairinpowunion = $true) & (complementT_lem = $true) & (singletonprop = $true) & (kpairsurjEq = $true) & (ap2p = $true) & (setunionI = $true) & (setukpairinjR1 = $true) & (setukpairinjR = $true) & (subbreln = $true) & (notinsingleton = $true) & (upairset2IR = $true) & (powersetTE1 = $true) & (emptyinPowerset = $true) & (in__Cong = $true) & (kfstpairEq = $true) & (ubforcartprodlem1 = $true) & (exuEu = $true) & (setadjoinE = $true) & (setunion__Cong = $true) & (quantDeMorgan4 = $true) & (cartprodmempair1 = $true) & (demorgan2a = $true) & (cartprodmempair = $true) & (singletonsswitch = $true) & (cartprodfstpairEq = $true) & (ubforcartprodlem3 = $true) & (eta1 = $true) & (dpsetconstrSub = $true) & (binintersectSubset4 = $true) & (binintersectSubset1 = $true) & (setunionsingleton1 = $true) & (descr__Cong = $true) & (upairset2E = $true) & (demorgan1 = $true) & (binintersectRsub = $true) & (setminusELneg = $true) & (sepSubset = $true) & (woz1_1 = $true) & (setminusER = $true) & (nonemptyI1 = $true) & (binintersectSubset2 = $true) & (dpsetconstrI = $true) & (binintersectLsub = $true) & (binintersectT_lem = $true) & (doubleComplementI1 = $true) & (powersetTI1 = $true) & (setextAx = $true) & (lamProp = $true) & (exuE2 = $true) & (powersetAx = $true) & (cartprodfstin = $true) & (uniqinunit = $true) & (cartprodsndin = $true) & (contrasubsetT = $true) & (powersetI = $true) & (powerset__Cong = $true) & (symdiffE = $true) & (setminusEL = $true) & (complementImpComplementIntersect = $true) & (setukpairIR = $true) & (iffalseProp1 = $true) & (dpsetconstrEL1 = $true) & (exuI3 = $true) & (setukpairinjL = $true) & (notequalI2 = $true) & (secondinupair = $true) & (complementTE1 = $true) & (setextsub = $true) & (binunionTEcontra = $true) & (contraSubsetComplement = $true) & (dsetconstrEL = $true) & (symdiffIneg1 = $true) & (descrp = $true) & (iftrueProp2 = $true) & (infuncsetfunc = $true) & (cartprodmempaircEq = $true) & (wellorderingAx = $true) & (bs114d = $true) & (symdiffI2 = $true) & (nonemptyI = $true) & (binunionIR = $true) & (demorgan1b = $true) & (emptysetAx = $true) & (singletoninpowunion = $true) & (singletonsuniq = $true) & (iftrueProp1 = $true) & (symdiffIneg2 = $true) & (ifSingleton = $true) & (inCongP = $true) & (emptyE1 = $true) & (subset2powerset = $true) & (setbeta = $true) & (omegaSAx = $true) & (eqinunit = $true) & (setadjoinSub2 = $true) & (ifp = $true) & (contrasubsetT3 = $true) & (inPowerset = $true) & (inComplementUnionImpInComplement1 = $true) & (cartprodpairmemER = $true) & (singletoninpowerset = $true) & (upairsubunion = $true) & (iffalse = $true) & (replAx = $true) & (setoftrueEq = $true) & (ap2apEq2 = $true) & (demorgan2a1 = $true) & (theprop = $true) & ((((subset @ sK741 @ (setminus @ sK739 @ sK740)) != $true) & ((subset @ sK740 @ (setminus @ sK739 @ sK741)) = $true) & ($true = (in @ sK741 @ (powerset @ sK739)))) & ((in @ sK740 @ (powerset @ sK739)) = $true)) & (exu__Cong = $true) & (dsetconstrER = $true) & (ex1I2 = $true) & (powersetI1 = $true) & (dsetconstrI = $true) & (binintersectER = $true) & (emptysetsubset = $true) & (singletonsubset = $true) & (quantDeMorgan1 = $true) & (exuI2 = $true) & (setukpairinjL1 = $true) & (intersectInPowersetIntersectUnions = $true) & (setunionsingleton = $true) & (setukpairIL = $true) & (kpairiskpair = $true) & (setminusILneg = $true) & (powersetE1 = $true) & (setukpairinjR11 = $true) & (demorgan2b = $true) & (demorgan2b2 = $true) & (powersetsubset = $true) & (nonemptyImpWitness = $true) & (setadjoinIR = $true) & (binintersectSubset5 = $true) & (complementUnionInPowersetComplement = $true) & (ex1I = $true) & (quantDeMorgan2 = $true) & (kfstsingleton = $true) & (dsetconstr__Cong = $true) & (contrasubsetT2 = $true) & (dpsetconstrEL2 = $true) & (cartprodsndpairEq = $true) & (binunionTE = $true) & (binintersectTELcontra = $true) & (omega__Cong = $true) & (subsetE2 = $true) & (binintersectTERcontra = $true) & (ex1E1 = $true) & (binunionE = $true) & (complementTnotintersectT = $true) & (exuE1 = $true) & (woz13rule4 = $true) & (disjointsetsI1 = $true) & (complementTcontraSubset = $true) & (exuI1 = $true) & (complementTI1 = $true) & (subsetI1 = $true) & (exuE3u = $true) & (ksndsingleton = $true) & (lam2p = $true) & (app = $true) & (symdiffI1 = $true) & (kpairp = $true) & (subsetTrans = $true) & (setminusI = $true) & (setukpairinjR12 = $true) & (demorgan2 = $true) & (apProp = $true) & (setext = $true) & (inIntersectImpInUnion2 = $true) & (foundationAx = $true) & (emptyInPowerset = $true) & (binunionTIRcontra = $true) & (powersetE = $true) & (beta1 = $true) & (ksndpairEq = $true) & (subsetI2 = $true)),
% 2.95/1.84 inference(skolemisation,[status(esa),new_symbols(skolem,[sK739,sK740,sK741])],[f998,f2393,f2392])).
% 2.95/1.84 thf(f2392,plain,(
% 2.95/1.84 ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (((subset @ X2 @ (setminus @ X0 @ X1)) != $true) & ((subset @ X1 @ (setminus @ X0 @ X2)) = $true) & ($true = (in @ X2 @ (powerset @ X0)))) & ((in @ X1 @ (powerset @ X0)) = $true)) => (? [X2 : $i] : (((subset @ X2 @ (setminus @ sK739 @ sK740)) != $true) & ((subset @ sK740 @ (setminus @ sK739 @ X2)) = $true) & ((in @ X2 @ (powerset @ sK739)) = $true)) & ((in @ sK740 @ (powerset @ sK739)) = $true))),
% 2.95/1.84 introduced(definition,[],[choice_axiom])).
% 2.95/1.84 thf(f2393,plain,(
% 2.95/1.84 ? [X2 : $i] : (((subset @ X2 @ (setminus @ sK739 @ sK740)) != $true) & ((subset @ sK740 @ (setminus @ sK739 @ X2)) = $true) & ((in @ X2 @ (powerset @ sK739)) = $true)) => (((subset @ sK741 @ (setminus @ sK739 @ sK740)) != $true) & ((subset @ sK740 @ (setminus @ sK739 @ sK741)) = $true) & ($true = (in @ sK741 @ (powerset @ sK739))))),
% 2.95/1.84 introduced(definition,[],[choice_axiom])).
% 2.95/1.84 thf(f998,plain,(
% 2.95/1.84 (doubleComplementEq = $true) & (dpsetconstrER = $true) & (funcextLem = $true) & (notdallE = $true) & (emptysetE = $true) & (woz13rule1 = $true) & (complementSubsetComplementIntersect = $true) & (inComplementUnionImpNotIn1 = $true) & (brelnall2 = $true) & (binunionT_lem = $true) & (funcinfuncset = $true) & (subsetTI = $true) & (notdexE = $true) & (prop2setE = $true) & (eqbreln = $true) & (binunionEcases = $true) & (emptyinunitempty = $true) & (quantDeMorgan3 = $true) & (beta2 = $true) & (doubleComplementSub1 = $true) & (prop2set2propI = $true) & (upairsetIL = $true) & (contrasubsetT1 = $true) & (powersetT_lem = $true) & (binintersectI = $true) & (setukpairinjL2 = $true) & (funcGraphProp1 = $true) & (funcGraphProp3 = $true) & (omega0Ax = $true) & (binintersectEL = $true) & (subsetE = $true) & (doubleComplementE1 = $true) & (subPowSU = $true) & (binunionLsub = $true) & (nonemptyE1 = $true) & (setminusIRneg = $true) & (prop2setI = $true) & (woz13rule3 = $true) & (dpsetconstrERa = $true) & (demorgan1a = $true) & (woz1_3 = $true) & (ubforcartprodlem2 = $true) & (emptyset__Cong = $true) & (setminusSubset1 = $true) & (sepInPowerset = $true) & (complementInPowersetComplementIntersect = $true) & (eqimpsubset1 = $true) & (ap2apEq1 = $true) & (vacuousDall = $true) & (iftrue = $true) & (setminusT_lem = $true) & (upairequniteq = $true) & (woz13rule0 = $true) & (funcext = $true) & (emptysetimpfalse = $true) & (setunionsingleton2 = $true) & (exuE3e = $true) & (funcGraphProp4 = $true) & (emptyI = $true) & (ex1E2 = $true) & (setminusLsub = $true) & (notequalI1 = $true) & (setadjoinAx = $true) & (funcImageSingleton = $true) & (notinemptyset = $true) & (eqimpsubset2 = $true) & (notsubsetI = $true) & (binunionTILcontra = $true) & (demorgan2a2 = $true) & (setunionAx = $true) & (setadjoin__Cong = $true) & (setminusERneg = $true) & (setadjoinSub = $true) & (inIntersectImpInIntersectUnions = $true) & (funcext2 = $true) & (setunionE = $true) & (noeltsimpempty = $true) & (cartprodpairsurjEq = $true) & (woz1_2 = $true) & (doubleComplementSub2 = $true) & (setextT = $true) & (subsetRefl = $true) & (setunionE2 = $true) & (cartprodpairin = $true) & (woz13rule2 = $true) & (setadjoinIL = $true) & (setminusSubset2 = $true) & (binunionIL = $true) & (omegaIndAx = $true) & (lam2lamEq = $true) & (theeq = $true) & (iffalseProp2 = $true) & (setOfPairsIsBReln = $true) & (binintersectSubset3 = $true) & (iftrueorfalse = $true) & (inIntersectImpInUnion = $true) & (upairsetIR = $true) & (setukpairinjR2 = $true) & (eta2 = $true) & (brelnall1 = $true) & (binunionRsub = $true) & (funcGraphProp2 = $true) & (subsetemptysetimpeq = $true) & (setadjoinOr = $true) & (lamp = $true) & (cartprodpairmemEL = $true) & (upairsetE = $true) & (upairinpowunion = $true) & (complementT_lem = $true) & (singletonprop = $true) & (kpairsurjEq = $true) & (ap2p = $true) & (setunionI = $true) & (setukpairinjR1 = $true) & (setukpairinjR = $true) & (subbreln = $true) & (notinsingleton = $true) & (upairset2IR = $true) & (powersetTE1 = $true) & (emptyinPowerset = $true) & (in__Cong = $true) & (kfstpairEq = $true) & (ubforcartprodlem1 = $true) & (exuEu = $true) & (setadjoinE = $true) & (setunion__Cong = $true) & (quantDeMorgan4 = $true) & (cartprodmempair1 = $true) & (demorgan2a = $true) & (cartprodmempair = $true) & (singletonsswitch = $true) & (cartprodfstpairEq = $true) & (ubforcartprodlem3 = $true) & (eta1 = $true) & (dpsetconstrSub = $true) & (binintersectSubset4 = $true) & (binintersectSubset1 = $true) & (setunionsingleton1 = $true) & (descr__Cong = $true) & (upairset2E = $true) & (demorgan1 = $true) & (binintersectRsub = $true) & (setminusELneg = $true) & (sepSubset = $true) & (woz1_1 = $true) & (setminusER = $true) & (nonemptyI1 = $true) & (binintersectSubset2 = $true) & (dpsetconstrI = $true) & (binintersectLsub = $true) & (binintersectT_lem = $true) & (doubleComplementI1 = $true) & (powersetTI1 = $true) & (setextAx = $true) & (lamProp = $true) & (exuE2 = $true) & (powersetAx = $true) & (cartprodfstin = $true) & (uniqinunit = $true) & (cartprodsndin = $true) & (contrasubsetT = $true) & (powersetI = $true) & (powerset__Cong = $true) & (symdiffE = $true) & (setminusEL = $true) & (complementImpComplementIntersect = $true) & (setukpairIR = $true) & (iffalseProp1 = $true) & (dpsetconstrEL1 = $true) & (exuI3 = $true) & (setukpairinjL = $true) & (notequalI2 = $true) & (secondinupair = $true) & (complementTE1 = $true) & (setextsub = $true) & (binunionTEcontra = $true) & (contraSubsetComplement = $true) & (dsetconstrEL = $true) & (symdiffIneg1 = $true) & (descrp = $true) & (iftrueProp2 = $true) & (infuncsetfunc = $true) & (cartprodmempaircEq = $true) & (wellorderingAx = $true) & (bs114d = $true) & (symdiffI2 = $true) & (nonemptyI = $true) & (binunionIR = $true) & (demorgan1b = $true) & (emptysetAx = $true) & (singletoninpowunion = $true) & (singletonsuniq = $true) & (iftrueProp1 = $true) & (symdiffIneg2 = $true) & (ifSingleton = $true) & (inCongP = $true) & (emptyE1 = $true) & (subset2powerset = $true) & (setbeta = $true) & (omegaSAx = $true) & (eqinunit = $true) & (setadjoinSub2 = $true) & (ifp = $true) & (contrasubsetT3 = $true) & (inPowerset = $true) & (inComplementUnionImpInComplement1 = $true) & (cartprodpairmemER = $true) & (singletoninpowerset = $true) & (upairsubunion = $true) & (iffalse = $true) & (replAx = $true) & (setoftrueEq = $true) & (ap2apEq2 = $true) & (demorgan2a1 = $true) & (theprop = $true) & ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (((subset @ X2 @ (setminus @ X0 @ X1)) != $true) & ((subset @ X1 @ (setminus @ X0 @ X2)) = $true) & ($true = (in @ X2 @ (powerset @ X0)))) & ((in @ X1 @ (powerset @ X0)) = $true)) & (exu__Cong = $true) & (dsetconstrER = $true) & (ex1I2 = $true) & (powersetI1 = $true) & (dsetconstrI = $true) & (binintersectER = $true) & (emptysetsubset = $true) & (singletonsubset = $true) & (quantDeMorgan1 = $true) & (exuI2 = $true) & (setukpairinjL1 = $true) & (intersectInPowersetIntersectUnions = $true) & (setunionsingleton = $true) & (setukpairIL = $true) & (kpairiskpair = $true) & (setminusILneg = $true) & (powersetE1 = $true) & (setukpairinjR11 = $true) & (demorgan2b = $true) & (demorgan2b2 = $true) & (powersetsubset = $true) & (nonemptyImpWitness = $true) & (setadjoinIR = $true) & (binintersectSubset5 = $true) & (complementUnionInPowersetComplement = $true) & (ex1I = $true) & (quantDeMorgan2 = $true) & (kfstsingleton = $true) & (dsetconstr__Cong = $true) & (contrasubsetT2 = $true) & (dpsetconstrEL2 = $true) & (cartprodsndpairEq = $true) & (binunionTE = $true) & (binintersectTELcontra = $true) & (omega__Cong = $true) & (subsetE2 = $true) & (binintersectTERcontra = $true) & (ex1E1 = $true) & (binunionE = $true) & (complementTnotintersectT = $true) & (exuE1 = $true) & (woz13rule4 = $true) & (disjointsetsI1 = $true) & (complementTcontraSubset = $true) & (exuI1 = $true) & (complementTI1 = $true) & (subsetI1 = $true) & (exuE3u = $true) & (ksndsingleton = $true) & (lam2p = $true) & (app = $true) & (symdiffI1 = $true) & (kpairp = $true) & (subsetTrans = $true) & (setminusI = $true) & (setukpairinjR12 = $true) & (demorgan2 = $true) & (apProp = $true) & (setext = $true) & (inIntersectImpInUnion2 = $true) & (foundationAx = $true) & (emptyInPowerset = $true) & (binunionTIRcontra = $true) & (powersetE = $true) & (beta1 = $true) & (ksndpairEq = $true) & (subsetI2 = $true)),
% 2.95/1.84 inference(flattening,[status(thm)],[f997])).
% 2.95/1.84 thf(f997,plain,(
% 2.95/1.84 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : ((((subset @ X2 @ (setminus @ X0 @ X1)) != $true) & ((subset @ X1 @ (setminus @ X0 @ X2)) = $true)) & ($true = (in @ X2 @ (powerset @ X0)))) & ((in @ X1 @ (powerset @ X0)) = $true)) & (woz1_3 = $true)) & (woz1_2 = $true)) & (woz1_1 = $true)) & (woz13rule4 = $true)) & (woz13rule3 = $true)) & (woz13rule2 = $true)) & (woz13rule1 = $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)),
% 2.95/1.84 inference(ennf_transformation,[status(thm)],[f626])).
% 2.95/1.84 thf(f626,plain,(
% 2.95/1.84 ~((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) => ((woz13rule1 = $true) => ((woz13rule2 = $true) => ((woz13rule3 = $true) => ((woz13rule4 = $true) => ((woz1_1 = $true) => ((woz1_2 = $true) => ((woz1_3 = $true) => ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) => (((subset @ X1 @ (setminus @ X0 @ X2)) = $true) => ((subset @ X2 @ (setminus @ X0 @ X1)) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.95/1.84 inference(fool_elimination,[status(thm)],[f625])).
% 2.95/1.84 thf(f625,plain,(
% 2.95/1.84 ~(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 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => ! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X0)) => ((subset @ X1 @ (setminus @ X0 @ X2)) => (subset @ X2 @ (setminus @ X0 @ X1)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.95/1.84 inference(rectify,[status(thm)],[f288])).
% 2.95/1.84 thf(f288,negated_conjecture,(
% 2.95/1.84 ~(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 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ((subset @ X11 @ (setminus @ X3 @ X16)) => (subset @ X16 @ (setminus @ X3 @ X11)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.95/1.84 inference(negated_conjecture,[status(cth)],[f287])).
% 2.95/1.84 thf(f287,conjecture,(
% 2.95/1.84 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 => (woz13rule1 => (woz13rule2 => (woz13rule3 => (woz13rule4 => (woz1_1 => (woz1_2 => (woz1_3 => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ((subset @ X11 @ (setminus @ X3 @ X16)) => (subset @ X16 @ (setminus @ X3 @ X11))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.95/1.84 file('/export/starexec/sandbox/tmp/tmp.WxwUZkQ1Ii/DTF2THF_28543.p',woz1_4)).
% 2.95/1.84 thf(f4491,plain,(
% 2.95/1.84 ((subset @ sK740 @ (setminus @ sK739 @ sK741)) != $true)),
% 2.95/1.84 inference(trivial_inequality_removal,[status(thm)],[f4490])).
% 2.95/1.84 thf(f4490,plain,(
% 2.95/1.84 ($true != $true) | ((subset @ sK740 @ (setminus @ sK739 @ sK741)) != $true)),
% 2.95/1.84 inference(forward_demodulation,[status(thm)],[f4489,f3729])).
% 2.95/1.84 thf(f3729,plain,(
% 2.95/1.84 ((in @ sK740 @ (powerset @ sK739)) = $true)),
% 2.95/1.84 inference(cnf_transformation,[status(thm)],[f2394])).
% 2.95/1.84 thf(f4489,plain,(
% 2.95/1.84 ((subset @ sK740 @ (setminus @ sK739 @ sK741)) != $true) | ((in @ sK740 @ (powerset @ sK739)) != $true)),
% 2.95/1.84 inference(trivial_inequality_removal,[status(thm)],[f4488])).
% 2.95/1.84 thf(f4488,plain,(
% 2.95/1.84 ((in @ sK740 @ (powerset @ sK739)) != $true) | ((subset @ sK740 @ (setminus @ sK739 @ sK741)) != $true) | ($true != $true)),
% 2.95/1.84 inference(forward_demodulation,[status(thm)],[f4487,f3685])).
% 2.95/1.84 thf(f3685,plain,(
% 2.95/1.84 (complementTcontraSubset = $true)),
% 2.95/1.84 inference(cnf_transformation,[status(thm)],[f2394])).
% 2.95/1.84 thf(f4487,plain,(
% 2.95/1.84 (complementTcontraSubset != $true) | ((in @ sK740 @ (powerset @ sK739)) != $true) | ((subset @ sK740 @ (setminus @ sK739 @ sK741)) != $true)),
% 2.95/1.84 inference(trivial_inequality_removal,[status(thm)],[f4486])).
% 2.95/1.84 thf(f4486,plain,(
% 2.95/1.84 ($true != $true) | (complementTcontraSubset != $true) | ((in @ sK740 @ (powerset @ sK739)) != $true) | ((subset @ sK740 @ (setminus @ sK739 @ sK741)) != $true)),
% 2.95/1.84 inference(forward_demodulation,[status(thm)],[f4476,f3730])).
% 2.95/1.84 thf(f3730,plain,(
% 2.95/1.84 ($true = (in @ sK741 @ (powerset @ sK739)))),
% 2.95/1.84 inference(cnf_transformation,[status(thm)],[f2394])).
% 2.95/1.84 thf(f4476,plain,(
% 2.95/1.84 ($true != (in @ sK741 @ (powerset @ sK739))) | ((in @ sK740 @ (powerset @ sK739)) != $true) | ((subset @ sK740 @ (setminus @ sK739 @ sK741)) != $true) | (complementTcontraSubset != $true)),
% 2.95/1.84 inference(trivial_inequality_removal,[status(thm)],[f4470])).
% 2.95/1.84 thf(f4470,plain,(
% 2.95/1.84 (complementTcontraSubset != $true) | ($true != (in @ sK741 @ (powerset @ sK739))) | ((subset @ sK740 @ (setminus @ sK739 @ sK741)) != $true) | ((in @ sK740 @ (powerset @ sK739)) != $true) | ($true != $true)),
% 2.95/1.84 inference(superposition,[status(thm)],[f3732,f3151])).
% 2.95/1.84 thf(f3151,plain,(
% 2.95/1.84 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((subset @ X2 @ (setminus @ X0 @ X1)) = $true) | ((in @ X1 @ (powerset @ X0)) != $true) | ((subset @ X1 @ (setminus @ X0 @ X2)) != $true) | (complementTcontraSubset != $true) | ($true != (in @ X2 @ (powerset @ X0)))) )),
% 2.95/1.84 inference(cnf_transformation,[status(thm)],[f1810])).
% 2.95/1.84 thf(f1810,plain,(
% 2.95/1.84 (! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true != (in @ X2 @ (powerset @ X0))) | ((subset @ X1 @ (setminus @ X0 @ X2)) != $true) | ((subset @ X2 @ (setminus @ X0 @ X1)) = $true)) | ((in @ X1 @ (powerset @ X0)) != $true)) | (complementTcontraSubset != $true)) & ((complementTcontraSubset = $true) | ((((in @ sK361 @ (powerset @ sK359)) = $true) & ((subset @ sK360 @ (setminus @ sK359 @ sK361)) = $true) & ((subset @ sK361 @ (setminus @ sK359 @ sK360)) != $true)) & ((in @ sK360 @ (powerset @ sK359)) = $true)))),
% 2.95/1.84 inference(skolemisation,[status(esa),new_symbols(skolem,[sK359,sK360,sK361])],[f1807,f1809,f1808])).
% 2.95/1.84 thf(f1808,plain,(
% 2.95/1.84 ? [X3 : $i,X4 : $i] : (? [X5 : $i] : (((in @ X5 @ (powerset @ X3)) = $true) & ((subset @ X4 @ (setminus @ X3 @ X5)) = $true) & ((subset @ X5 @ (setminus @ X3 @ X4)) != $true)) & ((in @ X4 @ (powerset @ X3)) = $true)) => (? [X5 : $i] : (((in @ X5 @ (powerset @ sK359)) = $true) & ((subset @ sK360 @ (setminus @ sK359 @ X5)) = $true) & ((subset @ X5 @ (setminus @ sK359 @ sK360)) != $true)) & ((in @ sK360 @ (powerset @ sK359)) = $true))),
% 2.95/1.84 introduced(definition,[],[choice_axiom])).
% 2.95/1.84 thf(f1809,plain,(
% 2.95/1.84 ? [X5 : $i] : (((in @ X5 @ (powerset @ sK359)) = $true) & ((subset @ sK360 @ (setminus @ sK359 @ X5)) = $true) & ((subset @ X5 @ (setminus @ sK359 @ sK360)) != $true)) => (((in @ sK361 @ (powerset @ sK359)) = $true) & ((subset @ sK360 @ (setminus @ sK359 @ sK361)) = $true) & ((subset @ sK361 @ (setminus @ sK359 @ sK360)) != $true))),
% 2.95/1.84 introduced(definition,[],[choice_axiom])).
% 2.95/1.84 thf(f1807,plain,(
% 2.95/1.84 (! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true != (in @ X2 @ (powerset @ X0))) | ((subset @ X1 @ (setminus @ X0 @ X2)) != $true) | ((subset @ X2 @ (setminus @ X0 @ X1)) = $true)) | ((in @ X1 @ (powerset @ X0)) != $true)) | (complementTcontraSubset != $true)) & ((complementTcontraSubset = $true) | ? [X3 : $i,X4 : $i] : (? [X5 : $i] : (((in @ X5 @ (powerset @ X3)) = $true) & ((subset @ X4 @ (setminus @ X3 @ X5)) = $true) & ((subset @ X5 @ (setminus @ X3 @ X4)) != $true)) & ((in @ X4 @ (powerset @ X3)) = $true)))),
% 2.95/1.84 inference(rectify,[status(thm)],[f1806])).
% 2.95/1.84 thf(f1806,plain,(
% 2.95/1.84 (! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true != (in @ X2 @ (powerset @ X0))) | ((subset @ X1 @ (setminus @ X0 @ X2)) != $true) | ((subset @ X2 @ (setminus @ X0 @ X1)) = $true)) | ((in @ X1 @ (powerset @ X0)) != $true)) | (complementTcontraSubset != $true)) & ((complementTcontraSubset = $true) | ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) & ((subset @ X1 @ (setminus @ X0 @ X2)) = $true) & ((subset @ X2 @ (setminus @ X0 @ X1)) != $true)) & ((in @ X1 @ (powerset @ X0)) = $true)))),
% 2.95/1.84 inference(nnf_transformation,[status(thm)],[f1002])).
% 2.95/1.84 thf(f1002,plain,(
% 2.95/1.84 ! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true != (in @ X2 @ (powerset @ X0))) | ((subset @ X1 @ (setminus @ X0 @ X2)) != $true) | ((subset @ X2 @ (setminus @ X0 @ X1)) = $true)) | ((in @ X1 @ (powerset @ X0)) != $true)) <=> (complementTcontraSubset = $true)),
% 2.95/1.84 inference(flattening,[status(thm)],[f1001])).
% 2.95/1.84 thf(f1001,plain,(
% 2.95/1.84 ! [X0 : $i,X1 : $i] : (! [X2 : $i] : ((((subset @ X2 @ (setminus @ X0 @ X1)) = $true) | ((subset @ X1 @ (setminus @ X0 @ X2)) != $true)) | ($true != (in @ X2 @ (powerset @ X0)))) | ((in @ X1 @ (powerset @ X0)) != $true)) <=> (complementTcontraSubset = $true)),
% 2.95/1.84 inference(ennf_transformation,[status(thm)],[f502])).
% 2.95/1.84 thf(f502,plain,(
% 2.95/1.84 ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) => (((subset @ X1 @ (setminus @ X0 @ X2)) = $true) => ((subset @ X2 @ (setminus @ X0 @ X1)) = $true)))) <=> (complementTcontraSubset = $true)),
% 2.95/1.84 inference(fool_elimination,[status(thm)],[f501])).
% 2.95/1.84 thf(f501,plain,(
% 2.95/1.84 (! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X0)) => ((subset @ X1 @ (setminus @ X0 @ X2)) => (subset @ X2 @ (setminus @ X0 @ X1))))) = complementTcontraSubset)),
% 2.95/1.84 inference(rectify,[status(thm)],[f258])).
% 2.95/1.84 thf(f258,axiom,(
% 2.95/1.84 (! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ((subset @ X11 @ (setminus @ X3 @ X16)) => (subset @ X16 @ (setminus @ X3 @ X11))))) = complementTcontraSubset)),
% 2.95/1.84 file('/export/starexec/sandbox/tmp/tmp.WxwUZkQ1Ii/DTF2THF_28543.p',complementTcontraSubset)).
% 2.95/1.84 thf(f3732,plain,(
% 2.95/1.84 ((subset @ sK741 @ (setminus @ sK739 @ sK740)) != $true)),
% 2.95/1.84 inference(cnf_transformation,[status(thm)],[f2394])).
% 2.95/1.84 % SZS output end Proof for DTF2THF_28543
% 2.95/1.84 % (28836)------------------------------
% 2.95/1.84 % (28836)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.84 % (28836)Termination reason: Refutation
% 2.95/1.84
% 2.95/1.84 % (28836)Memory used [KB]: 11001
% 2.95/1.84 % (28836)Time elapsed: 0.126 s
% 2.95/1.84 % (28836)Instructions burned: 413 (million)
% 2.95/1.84 % (28836)------------------------------
% 2.95/1.84 % (28836)------------------------------
% 2.95/1.84 % (28813)Success in time 0.213 s
%------------------------------------------------------------------------------