%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU767^1 : 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 : n024.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 2.26s 1.36s
% Output : Refutation 2.26s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 6
% Syntax : Number of formulae : 31 ( 9 unt; 0 typ; 0 def)
% Number of atoms : 4610 (1222 equ; 0 cnn)
% Maximal formula atoms : 286 ( 148 avg)
% Number of connectives : 2519 ( 48 ~; 26 |; 876 &; 417 @)
% ( 2 <=>;1150 =>; 0 <=; 0 <~>)
% Maximal formula depth : 295 ( 69 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 480 ( 477 usr; 291 con; 0-4 aty)
% Number of variables : 58 ( 0 ^; 36 !; 22 ?; 58 :)
% 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_337,type,
sP2: $i > $i > $i > $o ).
thf(func_def_341,type,
sP6: $i > $i > $i > $i > $o ).
thf(func_def_343,type,
sP8: $i > $o > $i > $i > $o ).
thf(func_def_344,type,
sP9: $i > $i > $o ).
thf(func_def_345,type,
sP10: $i > $o ).
thf(func_def_346,type,
sP11: $i > $i > $o ).
thf(func_def_347,type,
sP12: $i > $i > $o ).
thf(func_def_354,type,
sK19: $i > $i > $i > $i > $i ).
thf(func_def_358,type,
sK23: $o > $i > $i > $i ).
thf(func_def_363,type,
sK28: $i > $i > $i ).
thf(func_def_383,type,
sK48: $i > $i > $i > $i ).
thf(func_def_384,type,
sK49: $i > $i > $i > $i ).
thf(func_def_412,type,
sK77: $i > $i > $o ).
thf(func_def_414,type,
sK79: $i > $i ).
thf(func_def_415,type,
sK80: $i > $i ).
thf(func_def_416,type,
sK81: $i > ( $i > $i > $o ) > $i ).
thf(func_def_417,type,
sK82: $i > ( $i > $i > $o ) > $i ).
thf(func_def_418,type,
sK83: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_443,type,
sK108: $i > $i ).
thf(func_def_445,type,
sK110: $i > ( $i > $i ) > $i > $i ).
thf(func_def_457,type,
sK122: $i > $i > $i ).
thf(func_def_462,type,
sK127: $i > $i > $o ).
thf(func_def_488,type,
sK153: $i > $o ).
thf(func_def_495,type,
sK160: $i > $i > $o ).
thf(func_def_501,type,
sK166: $i > $i > $i > $i ).
thf(func_def_506,type,
sK171: $i > $i ).
thf(func_def_517,type,
sK182: $i > $i ).
thf(func_def_542,type,
sK207: $i > $i ).
thf(func_def_543,type,
sK208: $i > $i ).
thf(func_def_551,type,
sK216: ( $i > $o ) > $i > $i ).
thf(func_def_553,type,
sK218: $i > $o ).
thf(func_def_563,type,
sK228: $i > $i > $i ).
thf(func_def_566,type,
sK231: $i > $o ).
thf(func_def_567,type,
sK232: $i > $o ).
thf(func_def_568,type,
sK233: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_569,type,
sK234: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_589,type,
sK254: $i > $i > $i > $i ).
thf(func_def_590,type,
sK255: $i > $i > $i > $i ).
thf(func_def_599,type,
sK264: $i > $i > $i > $i ).
thf(func_def_605,type,
sK270: $i > $o ).
thf(func_def_607,type,
sK272: $i > $o ).
thf(func_def_609,type,
sK274: ( $i > $o ) > $i ).
thf(func_def_610,type,
sK275: ( $i > $o ) > $i ).
thf(func_def_621,type,
sK286: $i > $i > $o ).
thf(func_def_638,type,
sK303: $i > $i ).
thf(func_def_644,type,
sK309: ( $i > $o ) > $i > $i ).
thf(func_def_646,type,
sK311: $i > $o ).
thf(func_def_660,type,
sK325: ( $i > $o ) > $i > $i ).
thf(func_def_662,type,
sK327: $i > $o ).
thf(func_def_686,type,
sK351: $i > $i > $o ).
thf(func_def_708,type,
sK373: $i > $o ).
thf(func_def_713,type,
sK378: $i > $i > $i > $i > $i ).
thf(func_def_714,type,
sK379: $i > $i > $i > $i > $i ).
thf(func_def_727,type,
sK392: $i > $o ).
thf(func_def_730,type,
sK395: $i > $i > $o ).
thf(func_def_732,type,
sK397: $i > $i ).
thf(func_def_739,type,
sK404: $i > $o ).
thf(func_def_743,type,
sK408: $i > $i > $i ).
thf(func_def_749,type,
sK414: $i > ( $i > $o ) > $i > $i ).
thf(func_def_751,type,
sK416: $i > $o ).
thf(func_def_757,type,
sK422: $i > $o ).
thf(func_def_762,type,
sK427: $i > ( $i > $i ) > $i > $i ).
thf(func_def_764,type,
sK429: $i > $i ).
thf(func_def_768,type,
sK433: $i > $o ).
thf(func_def_777,type,
sK442: $i > $o ).
thf(func_def_795,type,
sK460: $i > $i > $o ).
thf(func_def_808,type,
sK473: $i > $i > $i ).
thf(func_def_809,type,
sK474: $i > $o ).
thf(func_def_824,type,
sK489: $i > $i > $i ).
thf(func_def_825,type,
sK490: $i > $i > $i ).
thf(func_def_874,type,
sK539: $i > $i > $i > $i > $i ).
thf(func_def_895,type,
sK560: $i > ( $i > $o ) > $i ).
thf(func_def_896,type,
sK561: $i > $o ).
thf(func_def_900,type,
sK565: $i > $o ).
thf(func_def_901,type,
sK566: $i > $o ).
thf(func_def_902,type,
sK567: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_903,type,
sK568: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_905,type,
sK570: $i > $o ).
thf(func_def_907,type,
sK572: ( $i > $o ) > $i > $i ).
thf(func_def_908,type,
sK573: ( $i > $o ) > $i > $i ).
thf(func_def_909,type,
sK574: $i > ( $i > $i ) > $i > $i ).
thf(func_def_911,type,
sK576: $i > $i ).
thf(func_def_915,type,
sK580: $i > $o ).
thf(func_def_923,type,
sK588: $i > $o ).
thf(func_def_924,type,
sK589: $i > $o ).
thf(func_def_925,type,
sK590: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_926,type,
sK591: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_929,type,
sK594: $i > $o ).
thf(func_def_932,type,
sK597: $i > $o ).
thf(func_def_933,type,
sK598: $i > $i ).
thf(func_def_934,type,
sK599: ( $i > $o ) > $i ).
thf(func_def_943,type,
sK608: $i > $o ).
thf(func_def_946,type,
sK611: $i > ( $i > $o ) > $i ).
thf(func_def_947,type,
sK612: $i > ( $i > $o ) > $i ).
thf(func_def_948,type,
sK613: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_949,type,
sK614: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_953,type,
sK618: $i > $o ).
thf(func_def_958,type,
sK623: $i > $o ).
thf(func_def_989,type,
sK654: $i > ( $i > $o ) > $i ).
thf(func_def_990,type,
sK655: $i > $o ).
thf(func_def_1005,type,
sK670: $i > $i > $i ).
thf(func_def_1036,type,
sK701: $i > $o ).
thf(func_def_1038,type,
sK703: $i > ( $i > $o ) > $i ).
thf(func_def_1039,type,
sK704: $i > $i > $i ).
thf(func_def_1043,type,
sK708: $i > $i ).
thf(func_def_1068,type,
sK733: ( $i > $o ) > $i ).
thf(func_def_1069,type,
sK734: $i > $o ).
thf(func_def_1077,type,
sK742: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1079,type,
sK744: $i > $i ).
thf(func_def_1084,type,
sK749: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1085,type,
sK750: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1089,type,
sK754: $i > $o ).
thf(func_def_1105,type,
sK770: $i > $i > $i > $i > $i ).
thf(func_def_1106,type,
sK771: $i > $i > $i > $i > $i ).
thf(func_def_1107,type,
sK772: $i > $i > $i > $i > $i ).
thf(func_def_1108,type,
sK773: $i > $i > $i > $i > $i ).
thf(func_def_1115,type,
sK780: $i > $o ).
thf(func_def_1117,type,
sK782: ( $i > $o ) > $i > $i ).
thf(func_def_1121,type,
sK786: $i > $i > $i ).
thf(func_def_1123,type,
sK788: $i > $o ).
thf(func_def_1124,type,
sK789: $i > $i ).
thf(func_def_1125,type,
sK790: ( $i > $o ) > $i ).
thf(func_def_1130,type,
sK795: $i > $i > $i > $i > $i ).
thf(func_def_1139,type,
sK804: $i > $i ).
thf(func_def_1140,type,
sK805: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1156,type,
sK821: $i > $o ).
thf(func_def_1201,type,
sK866: ( $i > $o ) > $i > $i ).
thf(func_def_1203,type,
sK868: $i > $o ).
thf(func_def_1209,type,
sK874: $i > $i ).
thf(func_def_1211,type,
sK876: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1214,type,
sK879: $i > $i > $o ).
thf(func_def_1219,type,
sK884: $i > $i > $i ).
thf(func_def_1220,type,
sK885: $i > $i > $i ).
thf(func_def_1221,type,
sK886: $i > $i > $i ).
thf(func_def_1222,type,
sK887: $i > $i > $i ).
thf(func_def_1223,type,
sK888: $i > $i > $i > $i ).
thf(func_def_1224,type,
sK889: $i > $i ).
thf(func_def_1225,type,
sK890: $i > $i ).
thf(func_def_1226,type,
sK891: $i > $i ).
thf(func_def_1227,type,
sK892: $i > $i ).
thf(func_def_1228,type,
sK893: $i > $i > $i > $i ).
thf(func_def_1229,type,
sK894: $i > $i > $i > $i ).
thf(func_def_1230,type,
sK895: $i > $i > $i ).
thf(func_def_1231,type,
sK896: $i > $i > $i ).
thf(func_def_1232,type,
sK897: $i > $i > $i ).
thf(func_def_1233,type,
sK898: $i > $i ).
thf(func_def_1234,type,
sK899: $i > $i > $i ).
thf(func_def_1236,type,
sK901: $i > $i ).
thf(func_def_1237,type,
sK902: $i > $i ).
thf(func_def_1246,type,
sK911: $i > ( $i > $o ) > $i ).
thf(func_def_1247,type,
sK912: $i > $o ).
thf(func_def_1259,type,
ph924:
!>[X0: $tType] : X0 ).
thf(f4561,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f4550]) ).
thf(f4550,plain,
$true != $true,
inference(superposition,[status(thm)],[f4479,f3991]) ).
thf(f3991,plain,
( $true
= ( in @ sK916 @ ( powerset @ sK915 ) ) ),
inference(cnf_transformation,[status(thm)],[f2668]) ).
thf(f2668,plain,
( ( nonemptyImpWitness = $true )
& ( iffalse = $true )
& ( demorgan1a = $true )
& ( cartprodpairin = $true )
& ( binunionRsub = $true )
& ( binintersectSubset2 = $true )
& ( binintersectSubset1 = $true )
& ( emptysetimpfalse = $true )
& ( infuncsetfunc = $true )
& ( subsetE2 = $true )
& ( prop2setI = $true )
& ( setminusILneg = $true )
& ( emptyset__Cong = $true )
& ( eqimpsubset1 = $true )
& ( powersetsubset = $true )
& ( cartprodpairmemEL = $true )
& ( complementImpComplementIntersect = $true )
& ( bs114d = $true )
& ( binunionTEcontra = $true )
& ( powersetE1 = $true )
& ( setext = $true )
& ( setukpairinjR1 = $true )
& ( setukpairIR = $true )
& ( iftrueProp2 = $true )
& ( powersetI = $true )
& ( notinsingleton = $true )
& ( setminusER = $true )
& ( setminusSubset2 = $true )
& ( exuE2 = $true )
& ( setukpairinjL2 = $true )
& ( ubforcartprodlem2 = $true )
& ( eqbreln = $true )
& ( cartprodmempair1 = $true )
& ( setadjoinE = $true )
& ( setukpairinjL1 = $true )
& ( notequalI2 = $true )
& ( inPowerset = $true )
& ( setadjoinOr = $true )
& ( setunionsingleton1 = $true )
& ( eqimpsubset2 = $true )
& ( powersetTI1 = $true )
& ( setminusERneg = $true )
& ( woz1_3 = $true )
& ( binintersectI = $true )
& ( upairsetIR = $true )
& ( woz1_4 = $true )
& ( subsetI2 = $true )
& ( singletonsuniq = $true )
& ( sepInPowerset = $true )
& ( setextT = $true )
& ( descrp = $true )
& ( inIntersectImpInUnion = $true )
& ( cartprodmempaircEq = $true )
& ( emptyE1 = $true )
& ( subbreln = $true )
& ( setextsub = $true )
& ( binunionIL = $true )
& ( setminusLsub = $true )
& ( contraSubsetComplement = $true )
& ( funcGraphProp2 = $true )
& ( binintersectT_lem = $true )
& ( setukpairinjL = $true )
& ( subsetRefl = $true )
& ( ap2apEq2 = $true )
& ( setminusELneg = $true )
& ( setextAx = $true )
& ( cartprodmempair = $true )
& ( exuI2 = $true )
& ( ex1E1 = $true )
& ( contrasubsetT1 = $true )
& ( secondinupair = $true )
& ( doubleComplementI1 = $true )
& ( subset2powerset = $true )
& ( setukpairinjR = $true )
& ( doubleComplementSub2 = $true )
& ( funcImageSingleton = $true )
& ( kpairp = $true )
& ( subsetI1 = $true )
& ( woz13rule3 = $true )
& ( iftrueorfalse = $true )
& ( lamp = $true )
& ( contrasubsetT = $true )
& ( omegaSAx = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( symdiffE = $true )
& ( kfstpairEq = $true )
& ( kpairsurjEq = $true )
& ( cartprodfstin = $true )
& ( notdexE = $true )
& ( funcinfuncset = $true )
& ( setadjoin__Cong = $true )
& ( complementSubsetComplementIntersect = $true )
& ( ifp = $true )
& ( singletonprop = $true )
& ( nonemptyE1 = $true )
& ( funcext2 = $true )
& ( cartprodsndpairEq = $true )
& ( complementTI1 = $true )
& ( binunionIR = $true )
& ( woz13rule0 = $true )
& ( iftrueProp1 = $true )
& ( dsetconstrER = $true )
& ( setunionE2 = $true )
& ( iftrue = $true )
& ( singletoninpowunion = $true )
& ( lam2lamEq = $true )
& ( exuE3e = $true )
& ( iffalseProp2 = $true )
& ( woz13rule2 = $true )
& ( setminusEL = $true )
& ( singletonsubset = $true )
& ( binintersectLsub = $true )
& ( setminusIRneg = $true )
& ( nonemptyI = $true )
& ( demorgan2a = $true )
& ( apProp = $true )
& ( funcGraphProp3 = $true )
& ( setminusT_lem = $true )
& ( uniqinunit = $true )
& ( inCongP = $true )
& ( quantDeMorgan2 = $true )
& ( woz1_2 = $true )
& ( demorgan2 = $true )
& ( emptyinunitempty = $true )
& ( emptysetE = $true )
& ( upairsetE = $true )
& ( binunionTIRcontra = $true )
& ( setadjoinIL = $true )
& ( funcGraphProp4 = $true )
& ( upairsetIL = $true )
& ( contrasubsetT2 = $true )
& ( noeltsimpempty = $true )
& ( powersetT_lem = $true )
& ( kfstsingleton = $true )
& ( exuI1 = $true )
& ( binunionLsub = $true )
& ( setunionI = $true )
& ( subsetTI = $true )
& ( setukpairIL = $true )
& ( upairset2IR = $true )
& ( funcext = $true )
& ( upairequniteq = $true )
& ( upairinpowunion = $true )
& ( eta2 = $true )
& ( binintersectSubset3 = $true )
& ( kpairiskpair = $true )
& ( inIntersectImpInUnion2 = $true )
& ( setunionAx = $true )
& ( dpsetconstrSub = $true )
& ( binintersectEL = $true )
& ( funcextLem = $true )
& ( notdallE = $true )
& ( ex1I = $true )
& ( omega__Cong = $true )
& ( exuEu = $true )
& ( complementTcontraSubset = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( setadjoinAx = $true )
& ( woz13rule1 = $true )
& ( subPowSU = $true )
& ( foundationAx = $true )
& ( doubleComplementSub1 = $true )
& ( theprop = $true )
& ( binintersectER = $true )
& ( ap2p = $true )
& ( complementTE1 = $true )
& ( app = $true )
& ( contrasubsetT3 = $true )
& ( setunionsingleton = $true )
& ( quantDeMorgan1 = $true )
& ( binunionTILcontra = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( prop2set2propI = $true )
& ( setadjoinSub2 = $true )
& ( demorgan2b2 = $true )
& ( setadjoinSub = $true )
& ( nonemptyI1 = $true )
& ( upairsubunion = $true )
& ( demorgan1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setminusI = $true )
& ( setunionE = $true )
& ( theeq = $true )
& ( notsubsetI = $true )
& ( singletonsswitch = $true )
& ( powersetE = $true )
& ( beta2 = $true )
& ( binunionEcases = $true )
& ( sepSubset = $true )
& ( lamProp = $true )
& ( binintersectTERcontra = $true )
& ( ubforcartprodlem1 = $true )
& ( ksndsingleton = $true )
& ( brelnall2 = $true )
& ( symdiffIneg1 = $true )
& ( demorgan1b = $true )
& ( dpsetconstrEL2 = $true )
& ( emptyI = $true )
& ( quantDeMorgan3 = $true )
& ( powersetAx = $true )
& ( ex1I2 = $true )
& ( $true
= ( in @ sK914 @ ( powerset @ sK915 ) ) )
& ( $true
!= ( in @ ( setminus @ sK915 @ ( binunion @ sK914 @ sK916 ) ) @ ( powerset @ ( setminus @ sK915 @ sK914 ) ) ) )
& ( $true
= ( in @ sK916 @ ( powerset @ sK915 ) ) )
& ( ap2apEq1 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( woz1_1 = $true )
& ( emptyInPowerset = $true )
& ( powersetI1 = $true )
& ( binunionTE = $true )
& ( dpsetconstrEL1 = $true )
& ( setunionsingleton2 = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( prop2setE = $true )
& ( vacuousDall = $true )
& ( setunion__Cong = $true )
& ( wellorderingAx = $true )
& ( cartprodfstpairEq = $true )
& ( funcGraphProp1 = $true )
& ( replAx = $true )
& ( subsetE = $true )
& ( omega0Ax = $true )
& ( exuE3u = $true )
& ( setadjoinIR = $true )
& ( complementTnotintersectT = $true )
& ( eta1 = $true )
& ( ubforcartprodlem3 = $true )
& ( dsetconstrI = $true )
& ( exuE1 = $true )
& ( setukpairinjR2 = $true )
& ( omegaIndAx = $true )
& ( woz13rule4 = $true )
& ( singletoninpowerset = $true )
& ( descr__Cong = $true )
& ( dpsetconstrER = $true )
& ( binintersectSubset5 = $true )
& ( upairset2E = $true )
& ( setminusSubset1 = $true )
& ( iffalseProp1 = $true )
& ( symdiffI1 = $true )
& ( brelnall1 = $true )
& ( symdiffI2 = $true )
& ( ex1E2 = $true )
& ( dsetconstr__Cong = $true )
& ( binintersectSubset4 = $true )
& ( demorgan2a2 = $true )
& ( exu__Cong = $true )
& ( setukpairinjR11 = $true )
& ( emptysetsubset = $true )
& ( ksndpairEq = $true )
& ( setoftrueEq = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( powersetTE1 = $true )
& ( binunionT_lem = $true )
& ( symdiffIneg2 = $true )
& ( dsetconstrEL = $true )
& ( complementT_lem = $true )
& ( cartprodpairsurjEq = $true )
& ( emptysetAx = $true )
& ( doubleComplementE1 = $true )
& ( demorgan2b = $true )
& ( subsetTrans = $true )
& ( lam2p = $true )
& ( eqinunit = $true )
& ( in__Cong = $true )
& ( setOfPairsIsBReln = $true )
& ( cartprodpairmemER = $true )
& ( ifSingleton = $true )
& ( disjointsetsI1 = $true )
& ( exuI3 = $true )
& ( binunionE = $true )
& ( binintersectRsub = $true )
& ( dpsetconstrERa = $true )
& ( dpsetconstrI = $true )
& ( setukpairinjR12 = $true )
& ( setbeta = $true )
& ( powerset__Cong = $true )
& ( binintersectTELcontra = $true )
& ( doubleComplementEq = $true )
& ( beta1 = $true )
& ( notinemptyset = $true )
& ( quantDeMorgan4 = $true )
& ( notequalI1 = $true )
& ( emptyinPowerset = $true )
& ( cartprodsndin = $true )
& ( demorgan2a1 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK914,sK915,sK916])],[f1167,f2667,f2666]) ).
thf(f2666,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
& ? [X2: $i] :
( ( $true
!= ( in @ ( setminus @ X1 @ ( binunion @ X0 @ X2 ) ) @ ( powerset @ ( setminus @ X1 @ X0 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) ) ) )
=> ( ( $true
= ( in @ sK914 @ ( powerset @ sK915 ) ) )
& ? [X2: $i] :
( ( $true
!= ( in @ ( setminus @ sK915 @ ( binunion @ sK914 @ X2 ) ) @ ( powerset @ ( setminus @ sK915 @ sK914 ) ) ) )
& ( ( in @ X2 @ ( powerset @ sK915 ) )
= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2667,plain,
( ? [X2: $i] :
( ( $true
!= ( in @ ( setminus @ sK915 @ ( binunion @ sK914 @ X2 ) ) @ ( powerset @ ( setminus @ sK915 @ sK914 ) ) ) )
& ( ( in @ X2 @ ( powerset @ sK915 ) )
= $true ) )
=> ( ( $true
!= ( in @ ( setminus @ sK915 @ ( binunion @ sK914 @ sK916 ) ) @ ( powerset @ ( setminus @ sK915 @ sK914 ) ) ) )
& ( $true
= ( in @ sK916 @ ( powerset @ sK915 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1167,plain,
( ( nonemptyImpWitness = $true )
& ( iffalse = $true )
& ( demorgan1a = $true )
& ( cartprodpairin = $true )
& ( binunionRsub = $true )
& ( binintersectSubset2 = $true )
& ( binintersectSubset1 = $true )
& ( emptysetimpfalse = $true )
& ( infuncsetfunc = $true )
& ( subsetE2 = $true )
& ( prop2setI = $true )
& ( setminusILneg = $true )
& ( emptyset__Cong = $true )
& ( eqimpsubset1 = $true )
& ( powersetsubset = $true )
& ( cartprodpairmemEL = $true )
& ( complementImpComplementIntersect = $true )
& ( bs114d = $true )
& ( binunionTEcontra = $true )
& ( powersetE1 = $true )
& ( setext = $true )
& ( setukpairinjR1 = $true )
& ( setukpairIR = $true )
& ( iftrueProp2 = $true )
& ( powersetI = $true )
& ( notinsingleton = $true )
& ( setminusER = $true )
& ( setminusSubset2 = $true )
& ( exuE2 = $true )
& ( setukpairinjL2 = $true )
& ( ubforcartprodlem2 = $true )
& ( eqbreln = $true )
& ( cartprodmempair1 = $true )
& ( setadjoinE = $true )
& ( setukpairinjL1 = $true )
& ( notequalI2 = $true )
& ( inPowerset = $true )
& ( setadjoinOr = $true )
& ( setunionsingleton1 = $true )
& ( eqimpsubset2 = $true )
& ( powersetTI1 = $true )
& ( setminusERneg = $true )
& ( woz1_3 = $true )
& ( binintersectI = $true )
& ( upairsetIR = $true )
& ( woz1_4 = $true )
& ( subsetI2 = $true )
& ( singletonsuniq = $true )
& ( sepInPowerset = $true )
& ( setextT = $true )
& ( descrp = $true )
& ( inIntersectImpInUnion = $true )
& ( cartprodmempaircEq = $true )
& ( emptyE1 = $true )
& ( subbreln = $true )
& ( setextsub = $true )
& ( binunionIL = $true )
& ( setminusLsub = $true )
& ( contraSubsetComplement = $true )
& ( funcGraphProp2 = $true )
& ( binintersectT_lem = $true )
& ( setukpairinjL = $true )
& ( subsetRefl = $true )
& ( ap2apEq2 = $true )
& ( setminusELneg = $true )
& ( setextAx = $true )
& ( cartprodmempair = $true )
& ( exuI2 = $true )
& ( ex1E1 = $true )
& ( contrasubsetT1 = $true )
& ( secondinupair = $true )
& ( doubleComplementI1 = $true )
& ( subset2powerset = $true )
& ( setukpairinjR = $true )
& ( doubleComplementSub2 = $true )
& ( funcImageSingleton = $true )
& ( kpairp = $true )
& ( subsetI1 = $true )
& ( woz13rule3 = $true )
& ( iftrueorfalse = $true )
& ( lamp = $true )
& ( contrasubsetT = $true )
& ( omegaSAx = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( symdiffE = $true )
& ( kfstpairEq = $true )
& ( kpairsurjEq = $true )
& ( cartprodfstin = $true )
& ( notdexE = $true )
& ( funcinfuncset = $true )
& ( setadjoin__Cong = $true )
& ( complementSubsetComplementIntersect = $true )
& ( ifp = $true )
& ( singletonprop = $true )
& ( nonemptyE1 = $true )
& ( funcext2 = $true )
& ( cartprodsndpairEq = $true )
& ( complementTI1 = $true )
& ( binunionIR = $true )
& ( woz13rule0 = $true )
& ( iftrueProp1 = $true )
& ( dsetconstrER = $true )
& ( setunionE2 = $true )
& ( iftrue = $true )
& ( singletoninpowunion = $true )
& ( lam2lamEq = $true )
& ( exuE3e = $true )
& ( iffalseProp2 = $true )
& ( woz13rule2 = $true )
& ( setminusEL = $true )
& ( singletonsubset = $true )
& ( binintersectLsub = $true )
& ( setminusIRneg = $true )
& ( nonemptyI = $true )
& ( demorgan2a = $true )
& ( apProp = $true )
& ( funcGraphProp3 = $true )
& ( setminusT_lem = $true )
& ( uniqinunit = $true )
& ( inCongP = $true )
& ( quantDeMorgan2 = $true )
& ( woz1_2 = $true )
& ( demorgan2 = $true )
& ( emptyinunitempty = $true )
& ( emptysetE = $true )
& ( upairsetE = $true )
& ( binunionTIRcontra = $true )
& ( setadjoinIL = $true )
& ( funcGraphProp4 = $true )
& ( upairsetIL = $true )
& ( contrasubsetT2 = $true )
& ( noeltsimpempty = $true )
& ( powersetT_lem = $true )
& ( kfstsingleton = $true )
& ( exuI1 = $true )
& ( binunionLsub = $true )
& ( setunionI = $true )
& ( subsetTI = $true )
& ( setukpairIL = $true )
& ( upairset2IR = $true )
& ( funcext = $true )
& ( upairequniteq = $true )
& ( upairinpowunion = $true )
& ( eta2 = $true )
& ( binintersectSubset3 = $true )
& ( kpairiskpair = $true )
& ( inIntersectImpInUnion2 = $true )
& ( setunionAx = $true )
& ( dpsetconstrSub = $true )
& ( binintersectEL = $true )
& ( funcextLem = $true )
& ( notdallE = $true )
& ( ex1I = $true )
& ( omega__Cong = $true )
& ( exuEu = $true )
& ( complementTcontraSubset = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( setadjoinAx = $true )
& ( woz13rule1 = $true )
& ( subPowSU = $true )
& ( foundationAx = $true )
& ( doubleComplementSub1 = $true )
& ( theprop = $true )
& ( binintersectER = $true )
& ( ap2p = $true )
& ( complementTE1 = $true )
& ( app = $true )
& ( contrasubsetT3 = $true )
& ( setunionsingleton = $true )
& ( quantDeMorgan1 = $true )
& ( binunionTILcontra = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( prop2set2propI = $true )
& ( setadjoinSub2 = $true )
& ( demorgan2b2 = $true )
& ( setadjoinSub = $true )
& ( nonemptyI1 = $true )
& ( upairsubunion = $true )
& ( demorgan1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setminusI = $true )
& ( setunionE = $true )
& ( theeq = $true )
& ( notsubsetI = $true )
& ( singletonsswitch = $true )
& ( powersetE = $true )
& ( beta2 = $true )
& ( binunionEcases = $true )
& ( sepSubset = $true )
& ( lamProp = $true )
& ( binintersectTERcontra = $true )
& ( ubforcartprodlem1 = $true )
& ( ksndsingleton = $true )
& ( brelnall2 = $true )
& ( symdiffIneg1 = $true )
& ( demorgan1b = $true )
& ( dpsetconstrEL2 = $true )
& ( emptyI = $true )
& ( quantDeMorgan3 = $true )
& ( powersetAx = $true )
& ( ex1I2 = $true )
& ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
& ? [X2: $i] :
( ( $true
!= ( in @ ( setminus @ X1 @ ( binunion @ X0 @ X2 ) ) @ ( powerset @ ( setminus @ X1 @ X0 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) ) ) )
& ( ap2apEq1 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( woz1_1 = $true )
& ( emptyInPowerset = $true )
& ( powersetI1 = $true )
& ( binunionTE = $true )
& ( dpsetconstrEL1 = $true )
& ( setunionsingleton2 = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( prop2setE = $true )
& ( vacuousDall = $true )
& ( setunion__Cong = $true )
& ( wellorderingAx = $true )
& ( cartprodfstpairEq = $true )
& ( funcGraphProp1 = $true )
& ( replAx = $true )
& ( subsetE = $true )
& ( omega0Ax = $true )
& ( exuE3u = $true )
& ( setadjoinIR = $true )
& ( complementTnotintersectT = $true )
& ( eta1 = $true )
& ( ubforcartprodlem3 = $true )
& ( dsetconstrI = $true )
& ( exuE1 = $true )
& ( setukpairinjR2 = $true )
& ( omegaIndAx = $true )
& ( woz13rule4 = $true )
& ( singletoninpowerset = $true )
& ( descr__Cong = $true )
& ( dpsetconstrER = $true )
& ( binintersectSubset5 = $true )
& ( upairset2E = $true )
& ( setminusSubset1 = $true )
& ( iffalseProp1 = $true )
& ( symdiffI1 = $true )
& ( brelnall1 = $true )
& ( symdiffI2 = $true )
& ( ex1E2 = $true )
& ( dsetconstr__Cong = $true )
& ( binintersectSubset4 = $true )
& ( demorgan2a2 = $true )
& ( exu__Cong = $true )
& ( setukpairinjR11 = $true )
& ( emptysetsubset = $true )
& ( ksndpairEq = $true )
& ( setoftrueEq = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( powersetTE1 = $true )
& ( binunionT_lem = $true )
& ( symdiffIneg2 = $true )
& ( dsetconstrEL = $true )
& ( complementT_lem = $true )
& ( cartprodpairsurjEq = $true )
& ( emptysetAx = $true )
& ( doubleComplementE1 = $true )
& ( demorgan2b = $true )
& ( subsetTrans = $true )
& ( lam2p = $true )
& ( eqinunit = $true )
& ( in__Cong = $true )
& ( setOfPairsIsBReln = $true )
& ( cartprodpairmemER = $true )
& ( ifSingleton = $true )
& ( disjointsetsI1 = $true )
& ( exuI3 = $true )
& ( binunionE = $true )
& ( binintersectRsub = $true )
& ( dpsetconstrERa = $true )
& ( dpsetconstrI = $true )
& ( setukpairinjR12 = $true )
& ( setbeta = $true )
& ( powerset__Cong = $true )
& ( binintersectTELcontra = $true )
& ( doubleComplementEq = $true )
& ( beta1 = $true )
& ( notinemptyset = $true )
& ( quantDeMorgan4 = $true )
& ( notequalI1 = $true )
& ( emptyinPowerset = $true )
& ( cartprodsndin = $true )
& ( demorgan2a1 = $true ) ),
inference(flattening,[status(thm)],[f1166]) ).
thf(f1166,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
& ? [X2: $i] :
( ( $true
!= ( in @ ( setminus @ X1 @ ( binunion @ X0 @ X2 ) ) @ ( powerset @ ( setminus @ X1 @ X0 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) ) ) )
& ( woz1_4 = $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)],[f516]) ).
thf(f516,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 )
=> ( ( woz1_4 = $true )
=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) )
=> ( $true
= ( in @ ( setminus @ X1 @ ( binunion @ X0 @ X2 ) ) @ ( powerset @ ( setminus @ X1 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f515]) ).
thf(f515,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
=> ( woz1_4
=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( powerset @ X1 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ( in @ ( setminus @ X1 @ ( binunion @ X0 @ X2 ) ) @ ( powerset @ ( setminus @ X1 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f289]) ).
thf(f289,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
=> ( woz1_4
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( in @ ( setminus @ X3 @ ( binunion @ X11 @ X16 ) ) @ ( powerset @ ( setminus @ X3 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f288]) ).
thf(f288,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
=> ( woz1_4
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( in @ ( setminus @ X3 @ ( binunion @ X11 @ X16 ) ) @ ( powerset @ ( setminus @ X3 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.7mug10HeGF/DTF2THF_7573.p',woz1_5) ).
thf(f4479,plain,
( $true
!= ( in @ sK916 @ ( powerset @ sK915 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f4478]) ).
thf(f4478,plain,
( ( $true
!= ( in @ sK916 @ ( powerset @ sK915 ) ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f4477,f3989]) ).
thf(f3989,plain,
complementUnionInPowersetComplement = $true,
inference(cnf_transformation,[status(thm)],[f2668]) ).
thf(f4477,plain,
( ( $true
!= ( in @ sK916 @ ( powerset @ sK915 ) ) )
| ( complementUnionInPowersetComplement != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4476]) ).
thf(f4476,plain,
( ( $true != $true )
| ( complementUnionInPowersetComplement != $true )
| ( $true
!= ( in @ sK916 @ ( powerset @ sK915 ) ) ) ),
inference(forward_demodulation,[status(thm)],[f4464,f3993]) ).
thf(f3993,plain,
( $true
= ( in @ sK914 @ ( powerset @ sK915 ) ) ),
inference(cnf_transformation,[status(thm)],[f2668]) ).
thf(f4464,plain,
( ( $true
!= ( in @ sK916 @ ( powerset @ sK915 ) ) )
| ( $true
!= ( in @ sK914 @ ( powerset @ sK915 ) ) )
| ( complementUnionInPowersetComplement != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4447]) ).
thf(f4447,plain,
( ( $true
!= ( in @ sK916 @ ( powerset @ sK915 ) ) )
| ( $true
!= ( in @ sK914 @ ( powerset @ sK915 ) ) )
| ( $true != $true )
| ( complementUnionInPowersetComplement != $true ) ),
inference(superposition,[status(thm)],[f3992,f3805]) ).
thf(f3805,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
= ( in @ ( setminus @ X0 @ ( binunion @ X1 @ X2 ) ) @ ( powerset @ ( setminus @ X0 @ X1 ) ) ) )
| ( complementUnionInPowersetComplement != $true )
| ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) )
| ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f2550]) ).
thf(f2550,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) )
| ! [X2: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) )
| ( $true
= ( in @ ( setminus @ X0 @ ( binunion @ X1 @ X2 ) ) @ ( powerset @ ( setminus @ X0 @ X1 ) ) ) ) ) )
| ( complementUnionInPowersetComplement != $true ) )
& ( ( complementUnionInPowersetComplement = $true )
| ( ( $true
= ( in @ sK840 @ ( powerset @ sK839 ) ) )
& ( $true
= ( in @ sK841 @ ( powerset @ sK839 ) ) )
& ( ( in @ ( setminus @ sK839 @ ( binunion @ sK840 @ sK841 ) ) @ ( powerset @ ( setminus @ sK839 @ sK840 ) ) )
!= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK839,sK840,sK841])],[f2547,f2549,f2548]) ).
thf(f2548,plain,
( ? [X3: $i,X4: $i] :
( ( ( in @ X4 @ ( powerset @ X3 ) )
= $true )
& ? [X5: $i] :
( ( $true
= ( in @ X5 @ ( powerset @ X3 ) ) )
& ( $true
!= ( in @ ( setminus @ X3 @ ( binunion @ X4 @ X5 ) ) @ ( powerset @ ( setminus @ X3 @ X4 ) ) ) ) ) )
=> ( ( $true
= ( in @ sK840 @ ( powerset @ sK839 ) ) )
& ? [X5: $i] :
( ( $true
= ( in @ X5 @ ( powerset @ sK839 ) ) )
& ( ( in @ ( setminus @ sK839 @ ( binunion @ sK840 @ X5 ) ) @ ( powerset @ ( setminus @ sK839 @ sK840 ) ) )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2549,plain,
( ? [X5: $i] :
( ( $true
= ( in @ X5 @ ( powerset @ sK839 ) ) )
& ( ( in @ ( setminus @ sK839 @ ( binunion @ sK840 @ X5 ) ) @ ( powerset @ ( setminus @ sK839 @ sK840 ) ) )
!= $true ) )
=> ( ( $true
= ( in @ sK841 @ ( powerset @ sK839 ) ) )
& ( ( in @ ( setminus @ sK839 @ ( binunion @ sK840 @ sK841 ) ) @ ( powerset @ ( setminus @ sK839 @ sK840 ) ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2547,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) )
| ! [X2: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) )
| ( $true
= ( in @ ( setminus @ X0 @ ( binunion @ X1 @ X2 ) ) @ ( powerset @ ( setminus @ X0 @ X1 ) ) ) ) ) )
| ( complementUnionInPowersetComplement != $true ) )
& ( ( complementUnionInPowersetComplement = $true )
| ? [X3: $i,X4: $i] :
( ( ( in @ X4 @ ( powerset @ X3 ) )
= $true )
& ? [X5: $i] :
( ( $true
= ( in @ X5 @ ( powerset @ X3 ) ) )
& ( $true
!= ( in @ ( setminus @ X3 @ ( binunion @ X4 @ X5 ) ) @ ( powerset @ ( setminus @ X3 @ X4 ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f2546]) ).
thf(f2546,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) )
| ! [X2: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) )
| ( $true
= ( in @ ( setminus @ X0 @ ( binunion @ X1 @ X2 ) ) @ ( powerset @ ( setminus @ X0 @ X1 ) ) ) ) ) )
| ( complementUnionInPowersetComplement != $true ) )
& ( ( complementUnionInPowersetComplement = $true )
| ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
& ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
& ( $true
!= ( in @ ( setminus @ X0 @ ( binunion @ X1 @ X2 ) ) @ ( powerset @ ( setminus @ X0 @ X1 ) ) ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1213]) ).
thf(f1213,plain,
( ! [X0: $i,X1: $i] :
( ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) )
| ! [X2: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) )
| ( $true
= ( in @ ( setminus @ X0 @ ( binunion @ X1 @ X2 ) ) @ ( powerset @ ( setminus @ X0 @ X1 ) ) ) ) ) )
<=> ( complementUnionInPowersetComplement = $true ) ),
inference(ennf_transformation,[status(thm)],[f584]) ).
thf(f584,plain,
( ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
=> ( $true
= ( in @ ( setminus @ X0 @ ( binunion @ X1 @ X2 ) ) @ ( powerset @ ( setminus @ X0 @ X1 ) ) ) ) ) )
<=> ( complementUnionInPowersetComplement = $true ) ),
inference(fool_elimination,[status(thm)],[f583]) ).
thf(f583,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X0 ) )
=> ( in @ ( setminus @ X0 @ ( binunion @ X1 @ X2 ) ) @ ( powerset @ ( setminus @ X0 @ X1 ) ) ) ) )
= complementUnionInPowersetComplement ),
inference(rectify,[status(thm)],[f270]) ).
thf(f270,axiom,
! [X3: $i,X11: $i] :
( ( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( in @ ( setminus @ X3 @ ( binunion @ X11 @ X16 ) ) @ ( powerset @ ( setminus @ X3 @ X11 ) ) ) ) )
= complementUnionInPowersetComplement ),
file('/export/starexec/sandbox/tmp/tmp.7mug10HeGF/DTF2THF_7573.p',complementUnionInPowersetComplement) ).
thf(f3992,plain,
( $true
!= ( in @ ( setminus @ sK915 @ ( binunion @ sK914 @ sK916 ) ) @ ( powerset @ ( setminus @ sK915 @ sK914 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f2668]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : SEU767^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.07 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.07/0.25 % Computer : n024.cluster.edu
% 0.07/0.25 % Model : x86_64 x86_64
% 0.07/0.25 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.25 % Memory : 8042.1875MB
% 0.07/0.25 % OS : Linux 3.10.0-693.el7.x86_64
% 0.07/0.25 % CPULimit : 300
% 0.07/0.25 % WCLimit : 300
% 0.07/0.25 % DateTime : Tue Feb 24 07:33:10 EST 2026
% 0.07/0.25 % CPUTime :
% 0.07/0.25 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.07/0.31 ---- Original DTF file ---
% 0.07/0.31 thf(spec,logic,$$dhol).
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 % File : SEU767^1 : TPTP v9.2.1. Released v3.7.0.
% 0.07/0.31 % Domain : Set Theory
% 0.07/0.31 % Problem : Typed Set Theory - First Wizard of Oz Examples - WoZ1 Problems
% 0.07/0.31 % Version : Especial.
% 0.07/0.31 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.07/0.31 % in (setminus A (binunion X Y)) (powerset (setminus A X))))
% 0.07/0.31
% 0.07/0.31 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.07/0.31 % Source : [Bro08]
% 0.07/0.31 % Names : ZFC269g [Bro08]
% 0.07/0.31
% 0.07/0.31 % Status : Theorem
% 0.07/0.31 % Rating : 0.11 v9.1.0, 0.12 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
% 0.07/0.31 % Syntax : Number of formulae : 609 ( 287 unt; 321 typ; 287 def)
% 0.07/0.31 % Number of atoms : 1718 ( 416 equ; 0 cnn)
% 0.07/0.31 % Maximal formula atoms : 286 ( 5 avg)
% 0.07/0.31 % Number of connectives : 3675 ( 73 ~; 12 |; 44 &;2663 @)
% 0.07/0.31 % ( 14 <=>; 869 =>; 0 <=; 0 <~>)
% 0.07/0.31 % Maximal formula depth : 294 ( 2 avg)
% 0.07/0.31 % Number of types : 2 ( 0 usr)
% 0.07/0.31 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.07/0.31 % Number of symbols : 324 ( 321 usr; 287 con; 0-4 aty)
% 0.07/0.31 % Number of variables : 983 ( 74 ^; 874 !; 35 ?; 983 :)
% 0.07/0.31 % SPC : TH0_THM_EQU_NAR
% 0.07/0.31
% 0.07/0.31 % Comments : http://mathgate.info/detsetitem.php?id=328
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 thf(in_type,type,
% 0.07/0.31 in: $i > $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(exu_type,type,
% 0.07/0.31 exu: ( $i > $o ) > $o ).
% 0.07/0.31
% 0.07/0.31 thf(exu,definition,
% 0.07/0.31 ( exu
% 0.07/0.31 = ( ^ [Xphi: $i > $o] :
% 0.07/0.31 ? [Xx: $i] :
% 0.07/0.31 ( ( Xphi @ Xx )
% 0.07/0.31 & ! [Xy: $i] :
% 0.07/0.31 ( ( Xphi @ Xy )
% 0.07/0.31 => ( Xx = Xy ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setextAx_type,type,
% 0.07/0.31 setextAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setextAx,definition,
% 0.07/0.31 ( setextAx
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 <=> ( in @ Xx @ B ) )
% 0.07/0.31 => ( A = B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(emptyset_type,type,
% 0.07/0.31 emptyset: $i ).
% 0.07/0.31
% 0.07/0.31 thf(emptysetAx_type,type,
% 0.07/0.31 emptysetAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(emptysetAx,definition,
% 0.07/0.31 ( emptysetAx
% 0.07/0.31 = ( ! [Xx: $i] :
% 0.07/0.31 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoin_type,type,
% 0.07/0.31 setadjoin: $i > $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinAx_type,type,
% 0.07/0.31 setadjoinAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinAx,definition,
% 0.07/0.31 ( setadjoinAx
% 0.07/0.31 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.07/0.31 <=> ( ( Xy = Xx )
% 0.07/0.31 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(powerset_type,type,
% 0.07/0.31 powerset: $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(powersetAx_type,type,
% 0.07/0.31 powersetAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(powersetAx,definition,
% 0.07/0.31 ( powersetAx
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( in @ B @ ( powerset @ A ) )
% 0.07/0.31 <=> ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setunion_type,type,
% 0.07/0.31 setunion: $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(setunionAx_type,type,
% 0.07/0.31 setunionAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setunionAx,definition,
% 0.07/0.31 ( setunionAx
% 0.07/0.31 = ( ! [A: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ ( setunion @ A ) )
% 0.07/0.31 <=> ? [B: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 & ( in @ B @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(omega_type,type,
% 0.07/0.31 omega: $i ).
% 0.07/0.31
% 0.07/0.31 thf(omega0Ax_type,type,
% 0.07/0.31 omega0Ax: $o ).
% 0.07/0.31
% 0.07/0.31 thf(omega0Ax,definition,
% 0.07/0.31 ( omega0Ax
% 0.07/0.31 = ( in @ emptyset @ omega ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(omegaSAx_type,type,
% 0.07/0.31 omegaSAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(omegaSAx,definition,
% 0.07/0.31 ( omegaSAx
% 0.07/0.31 = ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ omega )
% 0.07/0.31 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(omegaIndAx_type,type,
% 0.07/0.31 omegaIndAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(omegaIndAx,definition,
% 0.07/0.31 ( omegaIndAx
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ( ( ( in @ emptyset @ A )
% 0.07/0.31 & ! [Xx: $i] :
% 0.07/0.31 ( ( ( in @ Xx @ omega )
% 0.07/0.31 & ( in @ Xx @ A ) )
% 0.07/0.31 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.07/0.31 => ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ omega )
% 0.07/0.31 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(replAx_type,type,
% 0.07/0.31 replAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(replAx,definition,
% 0.07/0.31 ( replAx
% 0.07/0.31 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( exu
% 0.07/0.31 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.07/0.31 => ? [B: $i] :
% 0.07/0.31 ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 <=> ? [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ A )
% 0.07/0.31 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(foundationAx_type,type,
% 0.07/0.31 foundationAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(foundationAx,definition,
% 0.07/0.31 ( foundationAx
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.07/0.31 => ? [B: $i] :
% 0.07/0.31 ( ( in @ B @ A )
% 0.07/0.31 & ~ ? [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(wellorderingAx_type,type,
% 0.07/0.31 wellorderingAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(wellorderingAx,definition,
% 0.07/0.31 ( wellorderingAx
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ? [B: $i] :
% 0.07/0.31 ( ! [C: $i] :
% 0.07/0.31 ( ( in @ C @ B )
% 0.07/0.31 => ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ C )
% 0.07/0.31 => ( in @ Xx @ A ) ) )
% 0.07/0.31 & ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( ( in @ Xx @ A )
% 0.07/0.31 & ( in @ Xy @ A ) )
% 0.07/0.31 => ( ! [C: $i] :
% 0.07/0.31 ( ( in @ C @ B )
% 0.07/0.31 => ( ( in @ Xx @ C )
% 0.07/0.31 <=> ( in @ Xy @ C ) ) )
% 0.07/0.31 => ( Xx = Xy ) ) )
% 0.07/0.31 & ! [C: $i,D: $i] :
% 0.07/0.31 ( ( ( in @ C @ B )
% 0.07/0.31 & ( in @ D @ B ) )
% 0.07/0.31 => ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ C )
% 0.07/0.31 => ( in @ Xx @ D ) )
% 0.07/0.31 | ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ D )
% 0.07/0.31 => ( in @ Xx @ C ) ) ) )
% 0.07/0.31 & ! [C: $i] :
% 0.07/0.31 ( ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ C )
% 0.07/0.31 => ( in @ Xx @ A ) )
% 0.07/0.31 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.07/0.31 => ? [D: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ D @ B )
% 0.07/0.31 & ( in @ Xx @ C )
% 0.07/0.31 & ~ ? [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ D )
% 0.07/0.31 & ( in @ Xy @ C ) )
% 0.07/0.31 & ! [E: $i] :
% 0.07/0.31 ( ( in @ E @ B )
% 0.07/0.31 => ( ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ E )
% 0.07/0.31 => ( in @ Xy @ D ) )
% 0.07/0.31 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(descr_type,type,
% 0.07/0.31 descr: ( $i > $o ) > $i ).
% 0.07/0.31
% 0.07/0.31 thf(descrp_type,type,
% 0.07/0.31 descrp: $o ).
% 0.07/0.31
% 0.07/0.31 thf(descrp,definition,
% 0.07/0.31 ( descrp
% 0.07/0.31 = ( ! [Xphi: $i > $o] :
% 0.07/0.31 ( ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 => ( Xphi
% 0.07/0.31 @ ( descr
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstr_type,type,
% 0.07/0.31 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrI_type,type,
% 0.07/0.31 dsetconstrI: $o ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrI,definition,
% 0.07/0.31 ( dsetconstrI
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( ( Xphi @ Xx )
% 0.07/0.31 => ( in @ Xx
% 0.07/0.31 @ ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrEL_type,type,
% 0.07/0.31 dsetconstrEL: $o ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrEL,definition,
% 0.07/0.31 ( dsetconstrEL
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx
% 0.07/0.31 @ ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.07/0.31 => ( in @ Xx @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrER_type,type,
% 0.07/0.31 dsetconstrER: $o ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrER,definition,
% 0.07/0.31 ( dsetconstrER
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx
% 0.07/0.31 @ ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.07/0.31 => ( Xphi @ Xx ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(exuE1_type,type,
% 0.07/0.31 exuE1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(exuE1,definition,
% 0.07/0.31 ( exuE1
% 0.07/0.31 = ( ! [Xphi: $i > $o] :
% 0.07/0.31 ( ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 => ? [Xx: $i] :
% 0.07/0.31 ( ( Xphi @ Xx )
% 0.07/0.31 & ! [Xy: $i] :
% 0.07/0.31 ( ( Xphi @ Xy )
% 0.07/0.31 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(prop2set_type,type,
% 0.07/0.31 prop2set: $o > $i ).
% 0.07/0.31
% 0.07/0.31 thf(prop2setE_type,type,
% 0.07/0.31 prop2setE: $o ).
% 0.07/0.31
% 0.07/0.31 thf(prop2setE,definition,
% 0.07/0.31 ( prop2setE
% 0.07/0.31 = ( ! [Xphi: $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.07/0.31 => Xphi ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(emptysetE_type,type,
% 0.07/0.31 emptysetE: $o ).
% 0.07/0.31
% 0.07/0.31 thf(emptysetE,definition,
% 0.07/0.31 ( emptysetE
% 0.07/0.31 = ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ emptyset )
% 0.07/0.31 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(emptysetimpfalse_type,type,
% 0.07/0.31 emptysetimpfalse: $o ).
% 0.07/0.31
% 0.07/0.31 thf(emptysetimpfalse,definition,
% 0.07/0.31 ( emptysetimpfalse
% 0.07/0.31 = ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ emptyset )
% 0.07/0.31 => $false ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(notinemptyset_type,type,
% 0.07/0.31 notinemptyset: $o ).
% 0.07/0.31
% 0.07/0.31 thf(notinemptyset,definition,
% 0.07/0.31 ( notinemptyset
% 0.07/0.31 = ( ! [Xx: $i] :
% 0.07/0.31 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(exuE3e_type,type,
% 0.07/0.31 exuE3e: $o ).
% 0.07/0.31
% 0.07/0.31 thf(exuE3e,definition,
% 0.07/0.31 ( exuE3e
% 0.07/0.31 = ( ! [Xphi: $i > $o] :
% 0.07/0.31 ( ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setext_type,type,
% 0.07/0.31 setext: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setext,definition,
% 0.07/0.31 ( setext
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( in @ Xx @ B ) )
% 0.07/0.31 => ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 => ( in @ Xx @ A ) )
% 0.07/0.31 => ( A = B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(emptyI_type,type,
% 0.07/0.31 emptyI: $o ).
% 0.07/0.31
% 0.07/0.31 thf(emptyI,definition,
% 0.07/0.31 ( emptyI
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ~ ( in @ Xx @ A )
% 0.07/0.31 => ( A = emptyset ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(noeltsimpempty_type,type,
% 0.07/0.31 noeltsimpempty: $o ).
% 0.07/0.31
% 0.07/0.31 thf(noeltsimpempty,definition,
% 0.07/0.31 ( noeltsimpempty
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ~ ( in @ Xx @ A )
% 0.07/0.31 => ( A = emptyset ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setbeta_type,type,
% 0.07/0.31 setbeta: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setbeta,definition,
% 0.07/0.31 ( setbeta
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( ( in @ Xx
% 0.07/0.31 @ ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.07/0.31 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(nonempty_type,type,
% 0.07/0.31 nonempty: $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(nonempty,definition,
% 0.07/0.31 ( nonempty
% 0.07/0.31 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(nonemptyE1_type,type,
% 0.07/0.31 nonemptyE1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(nonemptyE1,definition,
% 0.07/0.31 ( nonemptyE1
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ( ( nonempty @ A )
% 0.07/0.31 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(nonemptyI_type,type,
% 0.07/0.31 nonemptyI: $o ).
% 0.07/0.31
% 0.07/0.31 thf(nonemptyI,definition,
% 0.07/0.31 ( nonemptyI
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( ( Xphi @ Xx )
% 0.07/0.31 => ( nonempty
% 0.07/0.31 @ ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(nonemptyI1_type,type,
% 0.07/0.31 nonemptyI1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(nonemptyI1,definition,
% 0.07/0.31 ( nonemptyI1
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.07/0.31 => ( nonempty @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinIL_type,type,
% 0.07/0.31 setadjoinIL: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinIL,definition,
% 0.07/0.31 ( setadjoinIL
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(emptyinunitempty_type,type,
% 0.07/0.31 emptyinunitempty: $o ).
% 0.07/0.31
% 0.07/0.31 thf(emptyinunitempty,definition,
% 0.07/0.31 ( emptyinunitempty
% 0.07/0.31 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinIR_type,type,
% 0.07/0.31 setadjoinIR: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinIR,definition,
% 0.07/0.31 ( setadjoinIR
% 0.07/0.31 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ A )
% 0.07/0.31 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinE_type,type,
% 0.07/0.31 setadjoinE: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinE,definition,
% 0.07/0.31 ( setadjoinE
% 0.07/0.31 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.07/0.31 => ! [Xphi: $o] :
% 0.07/0.31 ( ( ( Xy = Xx )
% 0.07/0.31 => Xphi )
% 0.07/0.31 => ( ( ( in @ Xy @ A )
% 0.07/0.31 => Xphi )
% 0.07/0.31 => Xphi ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinOr_type,type,
% 0.07/0.31 setadjoinOr: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinOr,definition,
% 0.07/0.31 ( setadjoinOr
% 0.07/0.31 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.07/0.31 => ( ( Xy = Xx )
% 0.07/0.31 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setoftrueEq_type,type,
% 0.07/0.31 setoftrueEq: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setoftrueEq,definition,
% 0.07/0.31 ( setoftrueEq
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ( ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xx: $i] : $true )
% 0.07/0.31 = A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(powersetI_type,type,
% 0.07/0.31 powersetI: $o ).
% 0.07/0.31
% 0.07/0.31 thf(powersetI,definition,
% 0.07/0.31 ( powersetI
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 => ( in @ Xx @ A ) )
% 0.07/0.31 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(emptyinPowerset_type,type,
% 0.07/0.31 emptyinPowerset: $o ).
% 0.07/0.31
% 0.07/0.31 thf(emptyinPowerset,definition,
% 0.07/0.31 ( emptyinPowerset
% 0.07/0.31 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(emptyInPowerset_type,type,
% 0.07/0.31 emptyInPowerset: $o ).
% 0.07/0.31
% 0.07/0.31 thf(emptyInPowerset,definition,
% 0.07/0.31 ( emptyInPowerset
% 0.07/0.31 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(powersetE_type,type,
% 0.07/0.31 powersetE: $o ).
% 0.07/0.31
% 0.07/0.31 thf(powersetE,definition,
% 0.07/0.31 ( powersetE
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ B @ ( powerset @ A ) )
% 0.07/0.31 => ( ( in @ Xx @ B )
% 0.07/0.31 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setunionI_type,type,
% 0.07/0.31 setunionI: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setunionI,definition,
% 0.07/0.31 ( setunionI
% 0.07/0.31 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 => ( ( in @ B @ A )
% 0.07/0.31 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setunionE_type,type,
% 0.07/0.31 setunionE: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setunionE,definition,
% 0.07/0.31 ( setunionE
% 0.07/0.31 = ( ! [A: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ ( setunion @ A ) )
% 0.07/0.31 => ! [Xphi: $o] :
% 0.07/0.31 ( ! [B: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 => ( ( in @ B @ A )
% 0.07/0.31 => Xphi ) )
% 0.07/0.31 => Xphi ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(subPowSU_type,type,
% 0.07/0.31 subPowSU: $o ).
% 0.07/0.31
% 0.07/0.31 thf(subPowSU,definition,
% 0.07/0.31 ( subPowSU
% 0.07/0.31 = ( ! [A: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(exuE2_type,type,
% 0.07/0.31 exuE2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(exuE2,definition,
% 0.07/0.31 ( exuE2
% 0.07/0.31 = ( ! [Xphi: $i > $o] :
% 0.07/0.31 ( ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 => ? [Xx: $i] :
% 0.07/0.31 ! [Xy: $i] :
% 0.07/0.31 ( ( Xphi @ Xy )
% 0.07/0.31 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(nonemptyImpWitness_type,type,
% 0.07/0.31 nonemptyImpWitness: $o ).
% 0.07/0.31
% 0.07/0.31 thf(nonemptyImpWitness,definition,
% 0.07/0.31 ( nonemptyImpWitness
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ( ( nonempty @ A )
% 0.07/0.31 => ? [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 & $true ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(uniqinunit_type,type,
% 0.07/0.31 uniqinunit: $o ).
% 0.07/0.31
% 0.07/0.31 thf(uniqinunit,definition,
% 0.07/0.31 ( uniqinunit
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.07/0.31 => ( Xx = Xy ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(notinsingleton_type,type,
% 0.07/0.31 notinsingleton: $o ).
% 0.07/0.31
% 0.07/0.31 thf(notinsingleton,definition,
% 0.07/0.31 ( notinsingleton
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( Xx != Xy )
% 0.07/0.31 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(eqinunit_type,type,
% 0.07/0.31 eqinunit: $o ).
% 0.07/0.31
% 0.07/0.31 thf(eqinunit,definition,
% 0.07/0.31 ( eqinunit
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( Xx = Xy )
% 0.07/0.31 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(singletonsswitch_type,type,
% 0.07/0.31 singletonsswitch: $o ).
% 0.07/0.31
% 0.07/0.31 thf(singletonsswitch,definition,
% 0.07/0.31 ( singletonsswitch
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.07/0.31 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(upairsetE_type,type,
% 0.07/0.31 upairsetE: $o ).
% 0.07/0.31
% 0.07/0.31 thf(upairsetE,definition,
% 0.07/0.31 ( upairsetE
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.07/0.31 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.07/0.31 => ( ( Xz = Xx )
% 0.07/0.31 | ( Xz = Xy ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(upairsetIL_type,type,
% 0.07/0.31 upairsetIL: $o ).
% 0.07/0.31
% 0.07/0.31 thf(upairsetIL,definition,
% 0.07/0.31 ( upairsetIL
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(upairsetIR_type,type,
% 0.07/0.31 upairsetIR: $o ).
% 0.07/0.31
% 0.07/0.31 thf(upairsetIR,definition,
% 0.07/0.31 ( upairsetIR
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(emptyE1_type,type,
% 0.07/0.31 emptyE1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(emptyE1,definition,
% 0.07/0.31 ( emptyE1
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.31 ( ? [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 & ( Xphi @ Xx ) )
% 0.07/0.31 => ( ( ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 = emptyset )
% 0.07/0.31 => $false ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(vacuousDall_type,type,
% 0.07/0.31 vacuousDall: $o ).
% 0.07/0.31
% 0.07/0.31 thf(vacuousDall,definition,
% 0.07/0.31 ( vacuousDall
% 0.07/0.31 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ emptyset )
% 0.07/0.31 => ( Xphi @ Xx ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(quantDeMorgan1_type,type,
% 0.07/0.31 quantDeMorgan1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(quantDeMorgan1,definition,
% 0.07/0.31 ( quantDeMorgan1
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.31 ( ~ ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( Xphi @ Xx ) )
% 0.07/0.31 => ? [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(quantDeMorgan2_type,type,
% 0.07/0.31 quantDeMorgan2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(quantDeMorgan2,definition,
% 0.07/0.31 ( quantDeMorgan2
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ~ ( Xphi @ Xx ) )
% 0.07/0.31 => ~ ? [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(quantDeMorgan3_type,type,
% 0.07/0.31 quantDeMorgan3: $o ).
% 0.07/0.31
% 0.07/0.31 thf(quantDeMorgan3,definition,
% 0.07/0.31 ( quantDeMorgan3
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.31 ( ~ ? [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 & ( Xphi @ Xx ) )
% 0.07/0.31 => ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(quantDeMorgan4_type,type,
% 0.07/0.31 quantDeMorgan4: $o ).
% 0.07/0.31
% 0.07/0.31 thf(quantDeMorgan4,definition,
% 0.07/0.31 ( quantDeMorgan4
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.31 ( ? [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 & ~ ( Xphi @ Xx ) )
% 0.07/0.31 => ~ ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(prop2setI_type,type,
% 0.07/0.31 prop2setI: $o ).
% 0.07/0.31
% 0.07/0.31 thf(prop2setI,definition,
% 0.07/0.31 ( prop2setI
% 0.07/0.31 = ( ! [Xphi: $o] :
% 0.07/0.31 ( Xphi
% 0.07/0.31 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(set2prop_type,type,
% 0.07/0.31 set2prop: $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(prop2set2propI_type,type,
% 0.07/0.31 prop2set2propI: $o ).
% 0.07/0.31
% 0.07/0.31 thf(prop2set2propI,definition,
% 0.07/0.31 ( prop2set2propI
% 0.07/0.31 = ( ! [Xphi: $o] :
% 0.07/0.31 ( Xphi
% 0.07/0.31 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(notdexE_type,type,
% 0.07/0.31 notdexE: $o ).
% 0.07/0.31
% 0.07/0.31 thf(notdexE,definition,
% 0.07/0.31 ( notdexE
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.31 ( ~ ? [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 & ( Xphi @ Xx ) )
% 0.07/0.31 => ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(notdallE_type,type,
% 0.07/0.31 notdallE: $o ).
% 0.07/0.31
% 0.07/0.31 thf(notdallE,definition,
% 0.07/0.31 ( notdallE
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.31 ( ~ ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( Xphi @ Xx ) )
% 0.07/0.31 => ? [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(exuI1_type,type,
% 0.07/0.31 exuI1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(exuI1,definition,
% 0.07/0.31 ( exuI1
% 0.07/0.31 = ( ! [Xphi: $i > $o] :
% 0.07/0.31 ( ? [Xx: $i] :
% 0.07/0.31 ( ( Xphi @ Xx )
% 0.07/0.31 & ! [Xy: $i] :
% 0.07/0.31 ( ( Xphi @ Xy )
% 0.07/0.31 => ( Xx = Xy ) ) )
% 0.07/0.31 => ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(exuI3_type,type,
% 0.07/0.31 exuI3: $o ).
% 0.07/0.31
% 0.07/0.31 thf(exuI3,definition,
% 0.07/0.31 ( exuI3
% 0.07/0.31 = ( ! [Xphi: $i > $o] :
% 0.07/0.31 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.07/0.31 => ( ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( Xphi @ Xx )
% 0.07/0.31 => ( ( Xphi @ Xy )
% 0.07/0.31 => ( Xx = Xy ) ) )
% 0.07/0.31 => ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(exuI2_type,type,
% 0.07/0.31 exuI2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(exuI2,definition,
% 0.07/0.31 ( exuI2
% 0.07/0.31 = ( ! [Xphi: $i > $o] :
% 0.07/0.31 ( ? [Xx: $i] :
% 0.07/0.31 ! [Xy: $i] :
% 0.07/0.31 ( ( Xphi @ Xy )
% 0.07/0.31 <=> ( Xy = Xx ) )
% 0.07/0.31 => ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(inCongP_type,type,
% 0.07/0.31 inCongP: $o ).
% 0.07/0.31
% 0.07/0.31 thf(inCongP,definition,
% 0.07/0.31 ( inCongP
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( A = B )
% 0.07/0.31 => ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( Xx = Xy )
% 0.07/0.31 => ( ( in @ Xx @ A )
% 0.07/0.31 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(in__Cong_type,type,
% 0.07/0.31 in__Cong: $o ).
% 0.07/0.31
% 0.07/0.31 thf(in__Cong,definition,
% 0.07/0.31 ( in__Cong
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( A = B )
% 0.07/0.31 => ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( Xx = Xy )
% 0.07/0.31 => ( ( in @ Xx @ A )
% 0.07/0.31 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(exuE3u_type,type,
% 0.07/0.31 exuE3u: $o ).
% 0.07/0.31
% 0.07/0.31 thf(exuE3u,definition,
% 0.07/0.31 ( exuE3u
% 0.07/0.31 = ( ! [Xphi: $i > $o] :
% 0.07/0.31 ( ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 => ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( Xphi @ Xx )
% 0.07/0.31 => ( ( Xphi @ Xy )
% 0.07/0.31 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(exu__Cong_type,type,
% 0.07/0.31 exu__Cong: $o ).
% 0.07/0.31
% 0.07/0.31 thf(exu__Cong,definition,
% 0.07/0.31 ( exu__Cong
% 0.07/0.31 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.07/0.31 ( ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( Xx = Xy )
% 0.07/0.31 => ( ( Xphi @ Xx )
% 0.07/0.31 <=> ( Xpsi @ Xy ) ) )
% 0.07/0.31 => ( ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 <=> ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(emptyset__Cong_type,type,
% 0.07/0.31 emptyset__Cong: $o ).
% 0.07/0.31
% 0.07/0.31 thf(emptyset__Cong,definition,
% 0.07/0.31 ( emptyset__Cong
% 0.07/0.31 = ( emptyset = emptyset ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoin__Cong_type,type,
% 0.07/0.31 setadjoin__Cong: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoin__Cong,definition,
% 0.07/0.31 ( setadjoin__Cong
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( Xx = Xy )
% 0.07/0.31 => ! [Xz: $i,Xu: $i] :
% 0.07/0.31 ( ( Xz = Xu )
% 0.07/0.31 => ( ( setadjoin @ Xx @ Xz )
% 0.07/0.31 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(powerset__Cong_type,type,
% 0.07/0.31 powerset__Cong: $o ).
% 0.07/0.31
% 0.07/0.31 thf(powerset__Cong,definition,
% 0.07/0.31 ( powerset__Cong
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( A = B )
% 0.07/0.31 => ( ( powerset @ A )
% 0.07/0.31 = ( powerset @ B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setunion__Cong_type,type,
% 0.07/0.31 setunion__Cong: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setunion__Cong,definition,
% 0.07/0.31 ( setunion__Cong
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( A = B )
% 0.07/0.31 => ( ( setunion @ A )
% 0.07/0.31 = ( setunion @ B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(omega__Cong_type,type,
% 0.07/0.31 omega__Cong: $o ).
% 0.07/0.31
% 0.07/0.31 thf(omega__Cong,definition,
% 0.07/0.31 ( omega__Cong
% 0.07/0.31 = ( omega = omega ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(exuEu_type,type,
% 0.07/0.31 exuEu: $o ).
% 0.07/0.31
% 0.07/0.31 thf(exuEu,definition,
% 0.07/0.31 ( exuEu
% 0.07/0.31 = ( ! [Xphi: $i > $o] :
% 0.07/0.31 ( ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 => ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( Xphi @ Xx )
% 0.07/0.31 => ( ( Xphi @ Xy )
% 0.07/0.31 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(descr__Cong_type,type,
% 0.07/0.31 descr__Cong: $o ).
% 0.07/0.31
% 0.07/0.31 thf(descr__Cong,definition,
% 0.07/0.31 ( descr__Cong
% 0.07/0.31 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.07/0.31 ( ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( Xx = Xy )
% 0.07/0.31 => ( ( Xphi @ Xx )
% 0.07/0.31 <=> ( Xpsi @ Xy ) ) )
% 0.07/0.31 => ( ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 => ( ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.07/0.31 => ( ( descr
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 = ( descr
% 0.07/0.31 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstr__Cong_type,type,
% 0.07/0.31 dsetconstr__Cong: $o ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstr__Cong,definition,
% 0.07/0.31 ( dsetconstr__Cong
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( A = B )
% 0.07/0.31 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ B )
% 0.07/0.31 => ( ( Xx = Xy )
% 0.07/0.31 => ( ( Xphi @ Xx )
% 0.07/0.31 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.07/0.31 => ( ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 = ( dsetconstr @ B
% 0.07/0.31 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(subset_type,type,
% 0.07/0.31 subset: $i > $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(disjoint_type,type,
% 0.07/0.31 disjoint: $i > $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(setsmeet_type,type,
% 0.07/0.31 setsmeet: $i > $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(subsetI1_type,type,
% 0.07/0.31 subsetI1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(subsetI1,definition,
% 0.07/0.31 ( subsetI1
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( in @ Xx @ B ) )
% 0.07/0.31 => ( subset @ A @ B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(eqimpsubset2_type,type,
% 0.07/0.31 eqimpsubset2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(eqimpsubset2,definition,
% 0.07/0.31 ( eqimpsubset2
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( A = B )
% 0.07/0.31 => ( subset @ B @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(eqimpsubset1_type,type,
% 0.07/0.31 eqimpsubset1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(eqimpsubset1,definition,
% 0.07/0.31 ( eqimpsubset1
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( A = B )
% 0.07/0.31 => ( subset @ A @ B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(subsetI2_type,type,
% 0.07/0.31 subsetI2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(subsetI2,definition,
% 0.07/0.31 ( subsetI2
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( in @ Xx @ B ) )
% 0.07/0.31 => ( subset @ A @ B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(emptysetsubset_type,type,
% 0.07/0.31 emptysetsubset: $o ).
% 0.07/0.31
% 0.07/0.31 thf(emptysetsubset,definition,
% 0.07/0.31 ( emptysetsubset
% 0.07/0.31 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(subsetE_type,type,
% 0.07/0.31 subsetE: $o ).
% 0.07/0.31
% 0.07/0.31 thf(subsetE,definition,
% 0.07/0.31 ( subsetE
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( subset @ A @ B )
% 0.07/0.31 => ( ( in @ Xx @ A )
% 0.07/0.31 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(subsetE2_type,type,
% 0.07/0.31 subsetE2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(subsetE2,definition,
% 0.07/0.31 ( subsetE2
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( subset @ A @ B )
% 0.07/0.31 => ( ~ ( in @ Xx @ B )
% 0.07/0.31 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(notsubsetI_type,type,
% 0.07/0.31 notsubsetI: $o ).
% 0.07/0.31
% 0.07/0.31 thf(notsubsetI,definition,
% 0.07/0.31 ( notsubsetI
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( ~ ( in @ Xx @ B )
% 0.07/0.31 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(notequalI1_type,type,
% 0.07/0.31 notequalI1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(notequalI1,definition,
% 0.07/0.31 ( notequalI1
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ~ ( subset @ A @ B )
% 0.07/0.31 => ( A != B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(notequalI2_type,type,
% 0.07/0.31 notequalI2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(notequalI2,definition,
% 0.07/0.31 ( notequalI2
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( ~ ( in @ Xx @ B )
% 0.07/0.31 => ( A != B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(subsetRefl_type,type,
% 0.07/0.31 subsetRefl: $o ).
% 0.07/0.31
% 0.07/0.31 thf(subsetRefl,definition,
% 0.07/0.31 ( subsetRefl
% 0.07/0.31 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(subsetTrans_type,type,
% 0.07/0.31 subsetTrans: $o ).
% 0.07/0.31
% 0.07/0.31 thf(subsetTrans,definition,
% 0.07/0.31 ( subsetTrans
% 0.07/0.31 = ( ! [A: $i,B: $i,C: $i] :
% 0.07/0.31 ( ( subset @ A @ B )
% 0.07/0.31 => ( ( subset @ B @ C )
% 0.07/0.31 => ( subset @ A @ C ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinSub_type,type,
% 0.07/0.31 setadjoinSub: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinSub,definition,
% 0.07/0.31 ( setadjoinSub
% 0.07/0.31 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinSub2_type,type,
% 0.07/0.31 setadjoinSub2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinSub2,definition,
% 0.07/0.31 ( setadjoinSub2
% 0.07/0.31 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.07/0.31 ( ( subset @ A @ B )
% 0.07/0.31 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(subset2powerset_type,type,
% 0.07/0.31 subset2powerset: $o ).
% 0.07/0.31
% 0.07/0.31 thf(subset2powerset,definition,
% 0.07/0.31 ( subset2powerset
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( subset @ A @ B )
% 0.07/0.31 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setextsub_type,type,
% 0.07/0.31 setextsub: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setextsub,definition,
% 0.07/0.31 ( setextsub
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( subset @ A @ B )
% 0.07/0.31 => ( ( subset @ B @ A )
% 0.07/0.31 => ( A = B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(subsetemptysetimpeq_type,type,
% 0.07/0.31 subsetemptysetimpeq: $o ).
% 0.07/0.31
% 0.07/0.31 thf(subsetemptysetimpeq,definition,
% 0.07/0.31 ( subsetemptysetimpeq
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ( ( subset @ A @ emptyset )
% 0.07/0.31 => ( A = emptyset ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(powersetI1_type,type,
% 0.07/0.31 powersetI1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(powersetI1,definition,
% 0.07/0.31 ( powersetI1
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( subset @ B @ A )
% 0.07/0.31 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(powersetE1_type,type,
% 0.07/0.31 powersetE1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(powersetE1,definition,
% 0.07/0.31 ( powersetE1
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( in @ B @ ( powerset @ A ) )
% 0.07/0.31 => ( subset @ B @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(inPowerset_type,type,
% 0.07/0.31 inPowerset: $o ).
% 0.07/0.31
% 0.07/0.31 thf(inPowerset,definition,
% 0.07/0.31 ( inPowerset
% 0.07/0.31 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(powersetsubset_type,type,
% 0.07/0.31 powersetsubset: $o ).
% 0.07/0.31
% 0.07/0.31 thf(powersetsubset,definition,
% 0.07/0.31 ( powersetsubset
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( subset @ A @ B )
% 0.07/0.31 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(sepInPowerset_type,type,
% 0.07/0.31 sepInPowerset: $o ).
% 0.07/0.31
% 0.07/0.31 thf(sepInPowerset,definition,
% 0.07/0.31 ( sepInPowerset
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.31 ( in
% 0.07/0.31 @ ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 @ ( powerset @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(sepSubset_type,type,
% 0.07/0.31 sepSubset: $o ).
% 0.07/0.31
% 0.07/0.31 thf(sepSubset,definition,
% 0.07/0.31 ( sepSubset
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.31 ( subset
% 0.07/0.31 @ ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(binunion_type,type,
% 0.07/0.31 binunion: $i > $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(binunionIL_type,type,
% 0.07/0.31 binunionIL: $o ).
% 0.07/0.31
% 0.07/0.31 thf(binunionIL,definition,
% 0.07/0.31 ( binunionIL
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(upairset2IR_type,type,
% 0.07/0.31 upairset2IR: $o ).
% 0.07/0.31
% 0.07/0.31 thf(upairset2IR,definition,
% 0.07/0.31 ( upairset2IR
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(binunionIR_type,type,
% 0.07/0.31 binunionIR: $o ).
% 0.07/0.31
% 0.07/0.31 thf(binunionIR,definition,
% 0.07/0.31 ( binunionIR
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(binunionEcases_type,type,
% 0.07/0.31 binunionEcases: $o ).
% 0.07/0.31
% 0.07/0.31 thf(binunionEcases,definition,
% 0.07/0.31 ( binunionEcases
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.07/0.31 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.07/0.31 => ( ( ( in @ Xx @ A )
% 0.07/0.31 => Xphi )
% 0.07/0.31 => ( ( ( in @ Xx @ B )
% 0.07/0.31 => Xphi )
% 0.07/0.31 => Xphi ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(binunionE_type,type,
% 0.07/0.31 binunionE: $o ).
% 0.07/0.31
% 0.07/0.31 thf(binunionE,definition,
% 0.07/0.31 ( binunionE
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.07/0.31 => ( ( in @ Xx @ A )
% 0.07/0.31 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(binunionLsub_type,type,
% 0.07/0.31 binunionLsub: $o ).
% 0.07/0.31
% 0.07/0.31 thf(binunionLsub,definition,
% 0.07/0.31 ( binunionLsub
% 0.07/0.31 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(binunionRsub_type,type,
% 0.07/0.31 binunionRsub: $o ).
% 0.07/0.31
% 0.07/0.31 thf(binunionRsub,definition,
% 0.07/0.31 ( binunionRsub
% 0.07/0.31 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(binintersect_type,type,
% 0.07/0.31 binintersect: $i > $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectI_type,type,
% 0.07/0.31 binintersectI: $o ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectI,definition,
% 0.07/0.31 ( binintersectI
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( ( in @ Xx @ B )
% 0.07/0.31 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectSubset5_type,type,
% 0.07/0.31 binintersectSubset5: $o ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectSubset5,definition,
% 0.07/0.31 ( binintersectSubset5
% 0.07/0.31 = ( ! [A: $i,B: $i,C: $i] :
% 0.07/0.31 ( ( subset @ C @ A )
% 0.07/0.31 => ( ( subset @ C @ B )
% 0.07/0.31 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectEL_type,type,
% 0.07/0.31 binintersectEL: $o ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectEL,definition,
% 0.07/0.31 ( binintersectEL
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.07/0.31 => ( in @ Xx @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectLsub_type,type,
% 0.07/0.31 binintersectLsub: $o ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectLsub,definition,
% 0.07/0.31 ( binintersectLsub
% 0.07/0.31 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectSubset2_type,type,
% 0.07/0.31 binintersectSubset2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectSubset2,definition,
% 0.07/0.31 ( binintersectSubset2
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( subset @ A @ B )
% 0.07/0.31 => ( ( binintersect @ A @ B )
% 0.07/0.31 = A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectSubset3_type,type,
% 0.07/0.31 binintersectSubset3: $o ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectSubset3,definition,
% 0.07/0.31 ( binintersectSubset3
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( ( binintersect @ A @ B )
% 0.07/0.31 = B )
% 0.07/0.31 => ( subset @ B @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectER_type,type,
% 0.07/0.31 binintersectER: $o ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectER,definition,
% 0.07/0.31 ( binintersectER
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.07/0.31 => ( in @ Xx @ B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(disjointsetsI1_type,type,
% 0.07/0.31 disjointsetsI1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(disjointsetsI1,definition,
% 0.07/0.31 ( disjointsetsI1
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ~ ? [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 & ( in @ Xx @ B ) )
% 0.07/0.31 => ( ( binintersect @ A @ B )
% 0.07/0.31 = emptyset ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectRsub_type,type,
% 0.07/0.31 binintersectRsub: $o ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectRsub,definition,
% 0.07/0.31 ( binintersectRsub
% 0.07/0.31 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectSubset4_type,type,
% 0.07/0.31 binintersectSubset4: $o ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectSubset4,definition,
% 0.07/0.31 ( binintersectSubset4
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( subset @ B @ A )
% 0.07/0.31 => ( ( binintersect @ A @ B )
% 0.07/0.31 = B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectSubset1_type,type,
% 0.07/0.31 binintersectSubset1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(binintersectSubset1,definition,
% 0.07/0.31 ( binintersectSubset1
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( ( binintersect @ A @ B )
% 0.07/0.31 = A )
% 0.07/0.31 => ( subset @ A @ B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(bs114d_type,type,
% 0.07/0.31 bs114d: $o ).
% 0.07/0.31
% 0.07/0.31 thf(bs114d,definition,
% 0.07/0.31 ( bs114d
% 0.07/0.31 = ( ! [A: $i,B: $i,C: $i] :
% 0.07/0.31 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.07/0.31 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(regular_type,type,
% 0.07/0.31 regular: $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(setminus_type,type,
% 0.07/0.31 setminus: $i > $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(setminusI_type,type,
% 0.07/0.31 setminusI: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setminusI,definition,
% 0.07/0.31 ( setminusI
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( ~ ( in @ Xx @ B )
% 0.07/0.31 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setminusEL_type,type,
% 0.07/0.31 setminusEL: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setminusEL,definition,
% 0.07/0.31 ( setminusEL
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.07/0.31 => ( in @ Xx @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setminusER_type,type,
% 0.07/0.31 setminusER: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setminusER,definition,
% 0.07/0.31 ( setminusER
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.07/0.31 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setminusSubset2_type,type,
% 0.07/0.31 setminusSubset2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setminusSubset2,definition,
% 0.07/0.31 ( setminusSubset2
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( subset @ A @ B )
% 0.07/0.31 => ( ( setminus @ A @ B )
% 0.07/0.31 = emptyset ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setminusERneg_type,type,
% 0.07/0.31 setminusERneg: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setminusERneg,definition,
% 0.07/0.31 ( setminusERneg
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.07/0.31 => ( ( in @ Xx @ A )
% 0.07/0.31 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setminusELneg_type,type,
% 0.07/0.31 setminusELneg: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setminusELneg,definition,
% 0.07/0.31 ( setminusELneg
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.07/0.31 => ( ~ ( in @ Xx @ B )
% 0.07/0.31 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setminusILneg_type,type,
% 0.07/0.31 setminusILneg: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setminusILneg,definition,
% 0.07/0.31 ( setminusILneg
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ~ ( in @ Xx @ A )
% 0.07/0.31 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setminusIRneg_type,type,
% 0.07/0.31 setminusIRneg: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setminusIRneg,definition,
% 0.07/0.31 ( setminusIRneg
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setminusLsub_type,type,
% 0.07/0.31 setminusLsub: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setminusLsub,definition,
% 0.07/0.31 ( setminusLsub
% 0.07/0.31 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setminusSubset1_type,type,
% 0.07/0.31 setminusSubset1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setminusSubset1,definition,
% 0.07/0.31 ( setminusSubset1
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( ( setminus @ A @ B )
% 0.07/0.31 = emptyset )
% 0.07/0.31 => ( subset @ A @ B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(symdiff_type,type,
% 0.07/0.31 symdiff: $i > $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(symdiffE_type,type,
% 0.07/0.31 symdiffE: $o ).
% 0.07/0.31
% 0.07/0.31 thf(symdiffE,definition,
% 0.07/0.31 ( symdiffE
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.07/0.31 => ! [Xphi: $o] :
% 0.07/0.31 ( ( ( in @ Xx @ A )
% 0.07/0.31 => ( ~ ( in @ Xx @ B )
% 0.07/0.31 => Xphi ) )
% 0.07/0.31 => ( ( ~ ( in @ Xx @ A )
% 0.07/0.31 => ( ( in @ Xx @ B )
% 0.07/0.31 => Xphi ) )
% 0.07/0.31 => Xphi ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(symdiffI1_type,type,
% 0.07/0.31 symdiffI1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(symdiffI1,definition,
% 0.07/0.31 ( symdiffI1
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( ~ ( in @ Xx @ B )
% 0.07/0.31 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(symdiffI2_type,type,
% 0.07/0.31 symdiffI2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(symdiffI2,definition,
% 0.07/0.31 ( symdiffI2
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ~ ( in @ Xx @ A )
% 0.07/0.31 => ( ( in @ Xx @ B )
% 0.07/0.31 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(symdiffIneg1_type,type,
% 0.07/0.31 symdiffIneg1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(symdiffIneg1,definition,
% 0.07/0.31 ( symdiffIneg1
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( ( in @ Xx @ B )
% 0.07/0.31 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(symdiffIneg2_type,type,
% 0.07/0.31 symdiffIneg2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(symdiffIneg2,definition,
% 0.07/0.31 ( symdiffIneg2
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ~ ( in @ Xx @ A )
% 0.07/0.31 => ( ~ ( in @ Xx @ B )
% 0.07/0.31 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(iskpair_type,type,
% 0.07/0.31 iskpair: $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(secondinupair_type,type,
% 0.07/0.31 secondinupair: $o ).
% 0.07/0.31
% 0.07/0.31 thf(secondinupair,definition,
% 0.07/0.31 ( secondinupair
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairIL_type,type,
% 0.07/0.31 setukpairIL: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairIL,definition,
% 0.07/0.31 ( setukpairIL
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairIR_type,type,
% 0.07/0.31 setukpairIR: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairIR,definition,
% 0.07/0.31 ( setukpairIR
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(kpairiskpair_type,type,
% 0.07/0.31 kpairiskpair: $o ).
% 0.07/0.31
% 0.07/0.31 thf(kpairiskpair,definition,
% 0.07/0.31 ( kpairiskpair
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(kpair_type,type,
% 0.07/0.31 kpair: $i > $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(kpairp_type,type,
% 0.07/0.31 kpairp: $o ).
% 0.07/0.31
% 0.07/0.31 thf(kpairp,definition,
% 0.07/0.31 ( kpairp
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(cartprod_type,type,
% 0.07/0.31 cartprod: $i > $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(singletonsubset_type,type,
% 0.07/0.31 singletonsubset: $o ).
% 0.07/0.31
% 0.07/0.31 thf(singletonsubset,definition,
% 0.07/0.31 ( singletonsubset
% 0.07/0.31 = ( ! [A: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(singletoninpowerset_type,type,
% 0.07/0.31 singletoninpowerset: $o ).
% 0.07/0.31
% 0.07/0.31 thf(singletoninpowerset,definition,
% 0.07/0.31 ( singletoninpowerset
% 0.07/0.31 = ( ! [A: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(singletoninpowunion_type,type,
% 0.07/0.31 singletoninpowunion: $o ).
% 0.07/0.31
% 0.07/0.31 thf(singletoninpowunion,definition,
% 0.07/0.31 ( singletoninpowunion
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(upairset2E_type,type,
% 0.07/0.31 upairset2E: $o ).
% 0.07/0.31
% 0.07/0.31 thf(upairset2E,definition,
% 0.07/0.31 ( upairset2E
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.07/0.31 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.07/0.31 => ( ( Xz = Xx )
% 0.07/0.31 | ( Xz = Xy ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(upairsubunion_type,type,
% 0.07/0.31 upairsubunion: $o ).
% 0.07/0.31
% 0.07/0.31 thf(upairsubunion,definition,
% 0.07/0.31 ( upairsubunion
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ B )
% 0.07/0.31 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(upairinpowunion_type,type,
% 0.07/0.31 upairinpowunion: $o ).
% 0.07/0.31
% 0.07/0.31 thf(upairinpowunion,definition,
% 0.07/0.31 ( upairinpowunion
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ B )
% 0.07/0.31 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(ubforcartprodlem1_type,type,
% 0.07/0.31 ubforcartprodlem1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(ubforcartprodlem1,definition,
% 0.07/0.31 ( ubforcartprodlem1
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ B )
% 0.07/0.31 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(ubforcartprodlem2_type,type,
% 0.07/0.31 ubforcartprodlem2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(ubforcartprodlem2,definition,
% 0.07/0.31 ( ubforcartprodlem2
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ B )
% 0.07/0.31 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(ubforcartprodlem3_type,type,
% 0.07/0.31 ubforcartprodlem3: $o ).
% 0.07/0.31
% 0.07/0.31 thf(ubforcartprodlem3,definition,
% 0.07/0.31 ( ubforcartprodlem3
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ B )
% 0.07/0.31 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodpairin_type,type,
% 0.07/0.31 cartprodpairin: $o ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodpairin,definition,
% 0.07/0.31 ( cartprodpairin
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ B )
% 0.07/0.31 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodmempair1_type,type,
% 0.07/0.31 cartprodmempair1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodmempair1,definition,
% 0.07/0.31 ( cartprodmempair1
% 0.07/0.31 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.07/0.31 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.07/0.31 => ? [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 & ? [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ B )
% 0.07/0.31 & ( Xu
% 0.07/0.31 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodmempair_type,type,
% 0.07/0.31 cartprodmempair: $o ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodmempair,definition,
% 0.07/0.31 ( cartprodmempair
% 0.07/0.31 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.07/0.31 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.07/0.31 => ( iskpair @ Xu ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setunionE2_type,type,
% 0.07/0.31 setunionE2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setunionE2,definition,
% 0.07/0.31 ( setunionE2
% 0.07/0.31 = ( ! [A: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ ( setunion @ A ) )
% 0.07/0.31 => ? [X: $i] :
% 0.07/0.31 ( ( in @ X @ A )
% 0.07/0.31 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setunionsingleton1_type,type,
% 0.07/0.31 setunionsingleton1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setunionsingleton1,definition,
% 0.07/0.31 ( setunionsingleton1
% 0.07/0.31 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setunionsingleton2_type,type,
% 0.07/0.31 setunionsingleton2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setunionsingleton2,definition,
% 0.07/0.31 ( setunionsingleton2
% 0.07/0.31 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setunionsingleton_type,type,
% 0.07/0.31 setunionsingleton: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setunionsingleton,definition,
% 0.07/0.31 ( setunionsingleton
% 0.07/0.31 = ( ! [Xx: $i] :
% 0.07/0.31 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.07/0.31 = Xx ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(singleton_type,type,
% 0.07/0.31 singleton: $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(singletonprop_type,type,
% 0.07/0.31 singletonprop: $o ).
% 0.07/0.31
% 0.07/0.31 thf(singletonprop,definition,
% 0.07/0.31 ( singletonprop
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ A )
% 0.07/0.31 => ( ( Xphi @ Xx )
% 0.07/0.31 => ( ( Xphi @ Xy )
% 0.07/0.31 => ( Xx = Xy ) ) ) ) )
% 0.07/0.31 => ( ? [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 & ( Xphi @ Xx ) )
% 0.07/0.31 => ( singleton
% 0.07/0.31 @ ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(ex1_type,type,
% 0.07/0.31 ex1: $i > ( $i > $o ) > $o ).
% 0.07/0.31
% 0.07/0.31 thf(ex1,definition,
% 0.07/0.31 ( ex1
% 0.07/0.31 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.07/0.31 ( singleton
% 0.07/0.31 @ ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(ex1E1_type,type,
% 0.07/0.31 ex1E1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(ex1E1,definition,
% 0.07/0.31 ( ex1E1
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.31 ( ( ex1 @ A
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 => ? [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(ex1I_type,type,
% 0.07/0.31 ex1I: $o ).
% 0.07/0.31
% 0.07/0.31 thf(ex1I,definition,
% 0.07/0.31 ( ex1I
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( ( Xphi @ Xx )
% 0.07/0.31 => ( ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ A )
% 0.07/0.31 => ( ( Xphi @ Xy )
% 0.07/0.31 => ( Xy = Xx ) ) )
% 0.07/0.31 => ( ex1 @ A
% 0.07/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(ex1I2_type,type,
% 0.07/0.31 ex1I2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(ex1I2,definition,
% 0.07/0.31 ( ex1I2
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ A )
% 0.07/0.31 => ( ( Xphi @ Xx )
% 0.07/0.31 => ( ( Xphi @ Xy )
% 0.07/0.31 => ( Xx = Xy ) ) ) ) )
% 0.07/0.31 => ( ? [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 & ( Xphi @ Xx ) )
% 0.07/0.31 => ( ex1 @ A
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(singletonsuniq_type,type,
% 0.07/0.31 singletonsuniq: $o ).
% 0.07/0.31
% 0.07/0.31 thf(singletonsuniq,definition,
% 0.07/0.31 ( singletonsuniq
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( ( setadjoin @ Xx @ emptyset )
% 0.07/0.31 = ( setadjoin @ Xy @ emptyset ) )
% 0.07/0.31 => ( Xx = Xy ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(atmost1p_type,type,
% 0.07/0.31 atmost1p: $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(atleast2p_type,type,
% 0.07/0.31 atleast2p: $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(atmost2p_type,type,
% 0.07/0.31 atmost2p: $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(upairsetp_type,type,
% 0.07/0.31 upairsetp: $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairinjL1_type,type,
% 0.07/0.31 setukpairinjL1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairinjL1,definition,
% 0.07/0.31 ( setukpairinjL1
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.07/0.31 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.07/0.31 => ( Xx = Xz ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(kfstsingleton_type,type,
% 0.07/0.31 kfstsingleton: $o ).
% 0.07/0.31
% 0.07/0.31 thf(kfstsingleton,definition,
% 0.07/0.31 ( kfstsingleton
% 0.07/0.31 = ( ! [Xu: $i] :
% 0.07/0.31 ( ( iskpair @ Xu )
% 0.07/0.31 => ( singleton
% 0.07/0.31 @ ( dsetconstr @ ( setunion @ Xu )
% 0.07/0.31 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(theprop_type,type,
% 0.07/0.31 theprop: $o ).
% 0.07/0.31
% 0.07/0.31 thf(theprop,definition,
% 0.07/0.31 ( theprop
% 0.07/0.31 = ( ! [X: $i] :
% 0.07/0.31 ( ( singleton @ X )
% 0.07/0.31 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(kfst_type,type,
% 0.07/0.31 kfst: $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(kfstpairEq_type,type,
% 0.07/0.31 kfstpairEq: $o ).
% 0.07/0.31
% 0.07/0.31 thf(kfstpairEq,definition,
% 0.07/0.31 ( kfstpairEq
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.07/0.31 = Xx ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodfstin_type,type,
% 0.07/0.31 cartprodfstin: $o ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodfstin,definition,
% 0.07/0.31 ( cartprodfstin
% 0.07/0.31 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.07/0.31 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.07/0.31 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairinjL2_type,type,
% 0.07/0.31 setukpairinjL2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairinjL2,definition,
% 0.07/0.31 ( setukpairinjL2
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.07/0.31 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.07/0.31 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.07/0.31 => ( Xx = Xz ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairinjL_type,type,
% 0.07/0.31 setukpairinjL: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairinjL,definition,
% 0.07/0.31 ( setukpairinjL
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.07/0.31 ( ( ( kpair @ Xx @ Xy )
% 0.07/0.31 = ( kpair @ Xz @ Xu ) )
% 0.07/0.31 => ( Xx = Xz ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairinjR11_type,type,
% 0.07/0.31 setukpairinjR11: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairinjR11,definition,
% 0.07/0.31 ( setukpairinjR11
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( Xx = Xy )
% 0.07/0.31 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.07/0.31 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairinjR12_type,type,
% 0.07/0.31 setukpairinjR12: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairinjR12,definition,
% 0.07/0.31 ( setukpairinjR12
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( Xx = Xy )
% 0.07/0.31 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.07/0.31 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairinjR1_type,type,
% 0.07/0.31 setukpairinjR1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairinjR1,definition,
% 0.07/0.31 ( setukpairinjR1
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.07/0.31 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.07/0.31 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.07/0.31 => ( ( Xz = Xu )
% 0.07/0.31 => ( Xy = Xu ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(upairequniteq_type,type,
% 0.07/0.31 upairequniteq: $o ).
% 0.07/0.31
% 0.07/0.31 thf(upairequniteq,definition,
% 0.07/0.31 ( upairequniteq
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.07/0.31 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.07/0.31 = ( setadjoin @ Xz @ emptyset ) )
% 0.07/0.31 => ( Xx = Xy ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairinjR2_type,type,
% 0.07/0.31 setukpairinjR2: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairinjR2,definition,
% 0.07/0.31 ( setukpairinjR2
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.07/0.31 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.07/0.31 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.07/0.31 => ( Xy = Xu ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairinjR_type,type,
% 0.07/0.31 setukpairinjR: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setukpairinjR,definition,
% 0.07/0.31 ( setukpairinjR
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.07/0.31 ( ( ( kpair @ Xx @ Xy )
% 0.07/0.31 = ( kpair @ Xz @ Xu ) )
% 0.07/0.31 => ( Xy = Xu ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(ksndsingleton_type,type,
% 0.07/0.31 ksndsingleton: $o ).
% 0.07/0.31
% 0.07/0.31 thf(ksndsingleton,definition,
% 0.07/0.31 ( ksndsingleton
% 0.07/0.31 = ( ! [Xu: $i] :
% 0.07/0.31 ( ( iskpair @ Xu )
% 0.07/0.31 => ( singleton
% 0.07/0.31 @ ( dsetconstr @ ( setunion @ Xu )
% 0.07/0.31 @ ^ [Xx: $i] :
% 0.07/0.31 ( Xu
% 0.07/0.31 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(ksnd_type,type,
% 0.07/0.31 ksnd: $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(ksndpairEq_type,type,
% 0.07/0.31 ksndpairEq: $o ).
% 0.07/0.31
% 0.07/0.31 thf(ksndpairEq,definition,
% 0.07/0.31 ( ksndpairEq
% 0.07/0.31 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.07/0.31 = Xy ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(kpairsurjEq_type,type,
% 0.07/0.31 kpairsurjEq: $o ).
% 0.07/0.31
% 0.07/0.31 thf(kpairsurjEq,definition,
% 0.07/0.31 ( kpairsurjEq
% 0.07/0.31 = ( ! [Xu: $i] :
% 0.07/0.31 ( ( iskpair @ Xu )
% 0.07/0.31 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.07/0.31 = Xu ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodsndin_type,type,
% 0.07/0.31 cartprodsndin: $o ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodsndin,definition,
% 0.07/0.31 ( cartprodsndin
% 0.07/0.31 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.07/0.31 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.07/0.31 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodpairmemEL_type,type,
% 0.07/0.31 cartprodpairmemEL: $o ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodpairmemEL,definition,
% 0.07/0.31 ( cartprodpairmemEL
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.07/0.31 => ( in @ Xx @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodpairmemER_type,type,
% 0.07/0.31 cartprodpairmemER: $o ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodpairmemER,definition,
% 0.07/0.31 ( cartprodpairmemER
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.07/0.31 => ( in @ Xy @ B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodmempaircEq_type,type,
% 0.07/0.31 cartprodmempaircEq: $o ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodmempaircEq,definition,
% 0.07/0.31 ( cartprodmempaircEq
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ B )
% 0.07/0.31 => ( ( kpair @ Xx @ Xy )
% 0.07/0.31 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodfstpairEq_type,type,
% 0.07/0.31 cartprodfstpairEq: $o ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodfstpairEq,definition,
% 0.07/0.31 ( cartprodfstpairEq
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ B )
% 0.07/0.31 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.07/0.31 = Xx ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodsndpairEq_type,type,
% 0.07/0.31 cartprodsndpairEq: $o ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodsndpairEq,definition,
% 0.07/0.31 ( cartprodsndpairEq
% 0.07/0.31 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ B )
% 0.07/0.31 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.07/0.31 = Xy ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodpairsurjEq_type,type,
% 0.07/0.31 cartprodpairsurjEq: $o ).
% 0.07/0.31
% 0.07/0.31 thf(cartprodpairsurjEq,definition,
% 0.07/0.31 ( cartprodpairsurjEq
% 0.07/0.31 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.07/0.31 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.07/0.31 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.07/0.31 = Xu ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(breln_type,type,
% 0.07/0.31 breln: $i > $i > $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(breln,definition,
% 0.07/0.31 ( breln
% 0.07/0.31 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(dpsetconstr_type,type,
% 0.07/0.31 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.07/0.31
% 0.07/0.31 thf(dpsetconstrI_type,type,
% 0.07/0.31 dpsetconstrI: $o ).
% 0.07/0.31
% 0.07/0.31 thf(dpsetconstrI,definition,
% 0.07/0.31 ( dpsetconstrI
% 0.07/0.31 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ B )
% 0.07/0.31 => ( ( Xphi @ Xx @ Xy )
% 0.07/0.31 => ( in @ ( kpair @ Xx @ Xy )
% 0.07/0.31 @ ( dpsetconstr @ A @ B
% 0.07/0.31 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(dpsetconstrSub_type,type,
% 0.07/0.31 dpsetconstrSub: $o ).
% 0.07/0.31
% 0.07/0.31 thf(dpsetconstrSub,definition,
% 0.07/0.31 ( dpsetconstrSub
% 0.07/0.31 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.07/0.31 ( subset
% 0.07/0.31 @ ( dpsetconstr @ A @ B
% 0.07/0.31 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.07/0.31 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setOfPairsIsBReln_type,type,
% 0.07/0.31 setOfPairsIsBReln: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setOfPairsIsBReln,definition,
% 0.07/0.31 ( setOfPairsIsBReln
% 0.07/0.31 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.07/0.31 ( breln @ A @ B
% 0.07/0.31 @ ( dpsetconstr @ A @ B
% 0.07/0.31 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(dpsetconstrERa_type,type,
% 0.07/0.31 dpsetconstrERa: $o ).
% 0.07/0.31
% 0.07/0.31 thf(dpsetconstrERa,definition,
% 0.07/0.31 ( dpsetconstrERa
% 0.07/0.31 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ B )
% 0.07/0.31 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.07/0.31 @ ( dpsetconstr @ A @ B
% 0.07/0.31 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.07/0.31 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(dpsetconstrEL1_type,type,
% 0.07/0.31 dpsetconstrEL1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(dpsetconstrEL1,definition,
% 0.07/0.31 ( dpsetconstrEL1
% 0.07/0.31 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( in @ ( kpair @ Xx @ Xy )
% 0.07/0.31 @ ( dpsetconstr @ A @ B
% 0.07/0.31 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.07/0.31 => ( in @ Xx @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(dpsetconstrEL2_type,type,
% 0.07/0.31 dpsetconstrEL2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(dpsetconstrEL2,definition,
% 0.07/0.32 ( dpsetconstrEL2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ ( kpair @ Xx @ Xy )
% 0.07/0.32 @ ( dpsetconstr @ A @ B
% 0.07/0.32 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.07/0.32 => ( in @ Xy @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(dpsetconstrER_type,type,
% 0.07/0.32 dpsetconstrER: $o ).
% 0.07/0.32
% 0.07/0.32 thf(dpsetconstrER,definition,
% 0.07/0.32 ( dpsetconstrER
% 0.07/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ ( kpair @ Xx @ Xy )
% 0.07/0.32 @ ( dpsetconstr @ A @ B
% 0.07/0.32 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.07/0.32 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(func_type,type,
% 0.07/0.32 func: $i > $i > $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcSet_type,type,
% 0.07/0.32 funcSet: $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(funcImageSingleton_type,type,
% 0.07/0.32 funcImageSingleton: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcImageSingleton,definition,
% 0.07/0.32 ( funcImageSingleton
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( singleton
% 0.07/0.32 @ ( dsetconstr @ B
% 0.07/0.32 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(apProp_type,type,
% 0.07/0.32 apProp: $o ).
% 0.07/0.32
% 0.07/0.32 thf(apProp,definition,
% 0.07/0.32 ( apProp
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in
% 0.07/0.32 @ ( setunion
% 0.07/0.32 @ ( dsetconstr @ B
% 0.07/0.32 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.07/0.32 @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ap_type,type,
% 0.07/0.32 ap: $i > $i > $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(app_type,type,
% 0.07/0.32 app: $o ).
% 0.07/0.32
% 0.07/0.32 thf(app,definition,
% 0.07/0.32 ( app
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(infuncsetfunc_type,type,
% 0.07/0.32 infuncsetfunc: $o ).
% 0.07/0.32
% 0.07/0.32 thf(infuncsetfunc,definition,
% 0.07/0.32 ( infuncsetfunc
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.07/0.32 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ap2p_type,type,
% 0.07/0.32 ap2p: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ap2p,definition,
% 0.07/0.32 ( ap2p
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(funcinfuncset_type,type,
% 0.07/0.32 funcinfuncset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcinfuncset,definition,
% 0.07/0.32 ( funcinfuncset
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(lamProp_type,type,
% 0.07/0.32 lamProp: $o ).
% 0.07/0.32
% 0.07/0.32 thf(lamProp,definition,
% 0.07/0.32 ( lamProp
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.07/0.32 => ( func @ A @ B
% 0.07/0.32 @ ( dpsetconstr @ A @ B
% 0.07/0.32 @ ^ [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xf @ Xx )
% 0.07/0.32 = Xy ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(lam_type,type,
% 0.07/0.32 lam: $i > $i > ( $i > $i ) > $i ).
% 0.07/0.32
% 0.07/0.32 thf(lamp_type,type,
% 0.07/0.32 lamp: $o ).
% 0.07/0.32
% 0.07/0.32 thf(lamp,definition,
% 0.07/0.32 ( lamp
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.07/0.32 => ( func @ A @ B
% 0.07/0.32 @ ( lam @ A @ B
% 0.07/0.32 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(lam2p_type,type,
% 0.07/0.32 lam2p: $o ).
% 0.07/0.32
% 0.07/0.32 thf(lam2p,definition,
% 0.07/0.32 ( lam2p
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.07/0.32 => ( in
% 0.07/0.32 @ ( lam @ A @ B
% 0.07/0.32 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.07/0.32 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(brelnall1_type,type,
% 0.07/0.32 brelnall1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(brelnall1,definition,
% 0.07/0.32 ( brelnall1
% 0.07/0.32 = ( ! [A: $i,B: $i,R: $i] :
% 0.07/0.32 ( ( breln @ A @ B @ R )
% 0.07/0.32 => ! [Xphi: $i > $o] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.07/0.32 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ R )
% 0.07/0.32 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(brelnall2_type,type,
% 0.07/0.32 brelnall2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(brelnall2,definition,
% 0.07/0.32 ( brelnall2
% 0.07/0.32 = ( ! [A: $i,B: $i,R: $i] :
% 0.07/0.32 ( ( breln @ A @ B @ R )
% 0.07/0.32 => ! [Xphi: $i > $o] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.07/0.32 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ R )
% 0.07/0.32 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ex1E2_type,type,
% 0.07/0.32 ex1E2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ex1E2,definition,
% 0.07/0.32 ( ex1E2
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ( ex1 @ A
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( ( Xphi @ Xx )
% 0.07/0.32 => ( ( Xphi @ Xy )
% 0.07/0.32 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(funcGraphProp1_type,type,
% 0.07/0.32 funcGraphProp1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcGraphProp1,definition,
% 0.07/0.32 ( funcGraphProp1
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(funcGraphProp3_type,type,
% 0.07/0.32 funcGraphProp3: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcGraphProp3,definition,
% 0.07/0.32 ( funcGraphProp3
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(funcGraphProp2_type,type,
% 0.07/0.32 funcGraphProp2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcGraphProp2,definition,
% 0.07/0.32 ( funcGraphProp2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.07/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.07/0.32 = Xy ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(funcextLem_type,type,
% 0.07/0.32 funcextLem: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcextLem,definition,
% 0.07/0.32 ( funcextLem
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ! [Xg: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xg )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.07/0.32 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.07/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(funcGraphProp4_type,type,
% 0.07/0.32 funcGraphProp4: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcGraphProp4,definition,
% 0.07/0.32 ( funcGraphProp4
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.07/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.07/0.32 = Xy ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subbreln_type,type,
% 0.07/0.32 subbreln: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subbreln,definition,
% 0.07/0.32 ( subbreln
% 0.07/0.32 = ( ! [A: $i,B: $i,R: $i] :
% 0.07/0.32 ( ( breln @ A @ B @ R )
% 0.07/0.32 => ! [S: $i] :
% 0.07/0.32 ( ( breln @ A @ B @ S )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.07/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.07/0.32 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(eqbreln_type,type,
% 0.07/0.32 eqbreln: $o ).
% 0.07/0.32
% 0.07/0.32 thf(eqbreln,definition,
% 0.07/0.32 ( eqbreln
% 0.07/0.32 = ( ! [A: $i,B: $i,R: $i] :
% 0.07/0.32 ( ( breln @ A @ B @ R )
% 0.07/0.32 => ! [S: $i] :
% 0.07/0.32 ( ( breln @ A @ B @ S )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.07/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.07/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.07/0.32 => ( R = S ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(funcext_type,type,
% 0.07/0.32 funcext: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcext,definition,
% 0.07/0.32 ( funcext
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ! [Xg: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xg )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.07/0.32 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.07/0.32 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(funcext2_type,type,
% 0.07/0.32 funcext2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcext2,definition,
% 0.07/0.32 ( funcext2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.07/0.32 => ! [Xg: $i] :
% 0.07/0.32 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.07/0.32 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.07/0.32 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ap2apEq1_type,type,
% 0.07/0.32 ap2apEq1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ap2apEq1,definition,
% 0.07/0.32 ( ap2apEq1
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.07/0.32 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ap2apEq2_type,type,
% 0.07/0.32 ap2apEq2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ap2apEq2,definition,
% 0.07/0.32 ( ap2apEq2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.07/0.32 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(beta1_type,type,
% 0.07/0.32 beta1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(beta1,definition,
% 0.07/0.32 ( beta1
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( ap @ A @ B
% 0.07/0.32 @ ( lam @ A @ B
% 0.07/0.32 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.07/0.32 @ Xx )
% 0.07/0.32 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(eta1_type,type,
% 0.07/0.32 eta1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(eta1,definition,
% 0.07/0.32 ( eta1
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ( ( lam @ A @ B
% 0.07/0.32 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.07/0.32 = Xf ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(lam2lamEq_type,type,
% 0.07/0.32 lam2lamEq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(lam2lamEq,definition,
% 0.07/0.32 ( lam2lamEq
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.07/0.32 => ( ( lam @ A @ B
% 0.07/0.32 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.07/0.32 = ( lam @ A @ B
% 0.07/0.32 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(beta2_type,type,
% 0.07/0.32 beta2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(beta2,definition,
% 0.07/0.32 ( beta2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( ap @ A @ B
% 0.07/0.32 @ ( lam @ A @ B
% 0.07/0.32 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.07/0.32 @ Xx )
% 0.07/0.32 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(eta2_type,type,
% 0.07/0.32 eta2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(eta2,definition,
% 0.07/0.32 ( eta2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.07/0.32 => ( ( lam @ A @ B
% 0.07/0.32 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.07/0.32 = Xf ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(iffalseProp1_type,type,
% 0.07/0.32 iffalseProp1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(iffalseProp1,definition,
% 0.07/0.32 ( iffalseProp1
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( ~ Xphi
% 0.07/0.32 => ( in @ Xy
% 0.07/0.32 @ ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xz: $i] :
% 0.07/0.32 ( ( Xphi
% 0.07/0.32 & ( Xz = Xx ) )
% 0.07/0.32 | ( ~ Xphi
% 0.07/0.32 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(iffalseProp2_type,type,
% 0.07/0.32 iffalseProp2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(iffalseProp2,definition,
% 0.07/0.32 ( iffalseProp2
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( ~ Xphi
% 0.07/0.32 => ( ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xz: $i] :
% 0.07/0.32 ( ( Xphi
% 0.07/0.32 & ( Xz = Xx ) )
% 0.07/0.32 | ( ~ Xphi
% 0.07/0.32 & ( Xz = Xy ) ) ) )
% 0.07/0.32 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(iftrueProp1_type,type,
% 0.07/0.32 iftrueProp1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(iftrueProp1,definition,
% 0.07/0.32 ( iftrueProp1
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( Xphi
% 0.07/0.32 => ( in @ Xx
% 0.07/0.32 @ ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xz: $i] :
% 0.07/0.32 ( ( Xphi
% 0.07/0.32 & ( Xz = Xx ) )
% 0.07/0.32 | ( ~ Xphi
% 0.07/0.32 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(iftrueProp2_type,type,
% 0.07/0.32 iftrueProp2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(iftrueProp2,definition,
% 0.07/0.32 ( iftrueProp2
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( Xphi
% 0.07/0.32 => ( ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xz: $i] :
% 0.07/0.32 ( ( Xphi
% 0.07/0.32 & ( Xz = Xx ) )
% 0.07/0.32 | ( ~ Xphi
% 0.07/0.32 & ( Xz = Xy ) ) ) )
% 0.07/0.32 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ifSingleton_type,type,
% 0.07/0.32 ifSingleton: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ifSingleton,definition,
% 0.07/0.32 ( ifSingleton
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( singleton
% 0.07/0.32 @ ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xz: $i] :
% 0.07/0.32 ( ( Xphi
% 0.07/0.32 & ( Xz = Xx ) )
% 0.07/0.32 | ( ~ Xphi
% 0.07/0.32 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(if_type,type,
% 0.07/0.32 if: $i > $o > $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(ifp_type,type,
% 0.07/0.32 ifp: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ifp,definition,
% 0.07/0.32 ( ifp
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(theeq_type,type,
% 0.07/0.32 theeq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(theeq,definition,
% 0.07/0.32 ( theeq
% 0.07/0.32 = ( ! [X: $i] :
% 0.07/0.32 ( ( singleton @ X )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ X )
% 0.07/0.32 => ( ( setunion @ X )
% 0.07/0.32 = Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(iftrue_type,type,
% 0.07/0.32 iftrue: $o ).
% 0.07/0.32
% 0.07/0.32 thf(iftrue,definition,
% 0.07/0.32 ( iftrue
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( Xphi
% 0.07/0.32 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.07/0.32 = Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(iffalse_type,type,
% 0.07/0.32 iffalse: $o ).
% 0.07/0.32
% 0.07/0.32 thf(iffalse,definition,
% 0.07/0.32 ( iffalse
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( ~ Xphi
% 0.07/0.32 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.07/0.32 = Xy ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(iftrueorfalse_type,type,
% 0.07/0.32 iftrueorfalse: $o ).
% 0.07/0.32
% 0.07/0.32 thf(iftrueorfalse,definition,
% 0.07/0.32 ( iftrueorfalse
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectT_lem_type,type,
% 0.07/0.32 binintersectT_lem: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectT_lem,definition,
% 0.07/0.32 ( binintersectT_lem
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunionT_lem_type,type,
% 0.07/0.32 binunionT_lem: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionT_lem,definition,
% 0.07/0.32 ( binunionT_lem
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powersetT_lem_type,type,
% 0.07/0.32 powersetT_lem: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powersetT_lem,definition,
% 0.07/0.32 ( powersetT_lem
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusT_lem_type,type,
% 0.07/0.32 setminusT_lem: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusT_lem,definition,
% 0.07/0.32 ( setminusT_lem
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(complementT_lem_type,type,
% 0.07/0.32 complementT_lem: $o ).
% 0.07/0.32
% 0.07/0.32 thf(complementT_lem,definition,
% 0.07/0.32 ( complementT_lem
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setextT_type,type,
% 0.07/0.32 setextT: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setextT,definition,
% 0.07/0.32 ( setextT
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ X )
% 0.07/0.32 => ( in @ Xx @ Y ) ) )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ Y )
% 0.07/0.32 => ( in @ Xx @ X ) ) )
% 0.07/0.32 => ( X = Y ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subsetTI_type,type,
% 0.07/0.32 subsetTI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetTI,definition,
% 0.07/0.32 ( subsetTI
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ X )
% 0.07/0.32 => ( in @ Xx @ Y ) ) )
% 0.07/0.32 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powersetTI1_type,type,
% 0.07/0.32 powersetTI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powersetTI1,definition,
% 0.07/0.32 ( powersetTI1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ X )
% 0.07/0.32 => ( in @ Xx @ Y ) ) )
% 0.07/0.32 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powersetTE1_type,type,
% 0.07/0.32 powersetTE1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powersetTE1,definition,
% 0.07/0.32 ( powersetTE1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ X @ ( powerset @ Y ) )
% 0.07/0.32 => ( ( in @ Xx @ X )
% 0.07/0.32 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(complementTI1_type,type,
% 0.07/0.32 complementTI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(complementTI1,definition,
% 0.07/0.32 ( complementTI1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ X )
% 0.07/0.32 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(complementTE1_type,type,
% 0.07/0.32 complementTE1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(complementTE1,definition,
% 0.07/0.32 ( complementTE1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.07/0.32 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectTELcontra_type,type,
% 0.07/0.32 binintersectTELcontra: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectTELcontra,definition,
% 0.07/0.32 ( binintersectTELcontra
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ X )
% 0.07/0.32 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectTERcontra_type,type,
% 0.07/0.32 binintersectTERcontra: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectTERcontra,definition,
% 0.07/0.32 ( binintersectTERcontra
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ Y )
% 0.07/0.32 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(contrasubsetT_type,type,
% 0.07/0.32 contrasubsetT: $o ).
% 0.07/0.32
% 0.07/0.32 thf(contrasubsetT,definition,
% 0.07/0.32 ( contrasubsetT
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.07/0.32 => ( ( in @ Xx @ Y )
% 0.07/0.32 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(contrasubsetT1_type,type,
% 0.07/0.32 contrasubsetT1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(contrasubsetT1,definition,
% 0.07/0.32 ( contrasubsetT1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( subset @ X @ Y )
% 0.07/0.32 => ( ~ ( in @ Xx @ Y )
% 0.07/0.32 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(contrasubsetT2_type,type,
% 0.07/0.32 contrasubsetT2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(contrasubsetT2,definition,
% 0.07/0.32 ( contrasubsetT2
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( ( subset @ X @ Y )
% 0.07/0.32 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(contrasubsetT3_type,type,
% 0.07/0.32 contrasubsetT3: $o ).
% 0.07/0.32
% 0.07/0.32 thf(contrasubsetT3,definition,
% 0.07/0.32 ( contrasubsetT3
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.07/0.32 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementI1_type,type,
% 0.07/0.32 doubleComplementI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementI1,definition,
% 0.07/0.32 ( doubleComplementI1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ X )
% 0.07/0.32 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementE1_type,type,
% 0.07/0.32 doubleComplementE1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementE1,definition,
% 0.07/0.32 ( doubleComplementE1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.07/0.32 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementSub1_type,type,
% 0.07/0.32 doubleComplementSub1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementSub1,definition,
% 0.07/0.32 ( doubleComplementSub1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementSub2_type,type,
% 0.07/0.32 doubleComplementSub2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementSub2,definition,
% 0.07/0.32 ( doubleComplementSub2
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementEq_type,type,
% 0.07/0.32 doubleComplementEq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementEq,definition,
% 0.07/0.32 ( doubleComplementEq
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ( X
% 0.07/0.32 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(complementTnotintersectT_type,type,
% 0.07/0.32 complementTnotintersectT: $o ).
% 0.07/0.32
% 0.07/0.32 thf(complementTnotintersectT,definition,
% 0.07/0.32 ( complementTnotintersectT
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.07/0.32 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(complementImpComplementIntersect_type,type,
% 0.07/0.32 complementImpComplementIntersect: $o ).
% 0.07/0.32
% 0.07/0.32 thf(complementImpComplementIntersect,definition,
% 0.07/0.32 ( complementImpComplementIntersect
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.07/0.32 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(complementSubsetComplementIntersect_type,type,
% 0.07/0.32 complementSubsetComplementIntersect: $o ).
% 0.07/0.32
% 0.07/0.32 thf(complementSubsetComplementIntersect,definition,
% 0.07/0.32 ( complementSubsetComplementIntersect
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(complementInPowersetComplementIntersect_type,type,
% 0.07/0.32 complementInPowersetComplementIntersect: $o ).
% 0.07/0.32
% 0.07/0.32 thf(complementInPowersetComplementIntersect,definition,
% 0.07/0.32 ( complementInPowersetComplementIntersect
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(contraSubsetComplement_type,type,
% 0.07/0.32 contraSubsetComplement: $o ).
% 0.07/0.32
% 0.07/0.32 thf(contraSubsetComplement,definition,
% 0.07/0.32 ( contraSubsetComplement
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ Y )
% 0.07/0.32 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(complementTcontraSubset_type,type,
% 0.07/0.32 complementTcontraSubset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(complementTcontraSubset,definition,
% 0.07/0.32 ( complementTcontraSubset
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.07/0.32 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunionTILcontra_type,type,
% 0.07/0.32 binunionTILcontra: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionTILcontra,definition,
% 0.07/0.32 ( binunionTILcontra
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.07/0.32 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunionTIRcontra_type,type,
% 0.07/0.32 binunionTIRcontra: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionTIRcontra,definition,
% 0.07/0.32 ( binunionTIRcontra
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.07/0.32 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(inIntersectImpInUnion_type,type,
% 0.07/0.32 inIntersectImpInUnion: $o ).
% 0.07/0.32
% 0.07/0.32 thf(inIntersectImpInUnion,definition,
% 0.07/0.32 ( inIntersectImpInUnion
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Z: $i] :
% 0.07/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.07/0.32 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(inIntersectImpInUnion2_type,type,
% 0.07/0.32 inIntersectImpInUnion2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(inIntersectImpInUnion2,definition,
% 0.07/0.32 ( inIntersectImpInUnion2
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Z: $i] :
% 0.07/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.07/0.32 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(inIntersectImpInIntersectUnions_type,type,
% 0.07/0.32 inIntersectImpInIntersectUnions: $o ).
% 0.07/0.32
% 0.07/0.32 thf(inIntersectImpInIntersectUnions,definition,
% 0.07/0.32 ( inIntersectImpInIntersectUnions
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Z: $i] :
% 0.07/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.07/0.32 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(intersectInPowersetIntersectUnions_type,type,
% 0.07/0.32 intersectInPowersetIntersectUnions: $o ).
% 0.07/0.32
% 0.07/0.32 thf(intersectInPowersetIntersectUnions,definition,
% 0.07/0.32 ( intersectInPowersetIntersectUnions
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Z: $i] :
% 0.07/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.07/0.32 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(inComplementUnionImpNotIn1_type,type,
% 0.07/0.32 inComplementUnionImpNotIn1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(inComplementUnionImpNotIn1,definition,
% 0.07/0.32 ( inComplementUnionImpNotIn1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.07/0.32 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(inComplementUnionImpInComplement1_type,type,
% 0.07/0.32 inComplementUnionImpInComplement1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(inComplementUnionImpInComplement1,definition,
% 0.07/0.32 ( inComplementUnionImpInComplement1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.07/0.32 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunionTE_type,type,
% 0.07/0.32 binunionTE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionTE,definition,
% 0.07/0.32 ( binunionTE
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.07/0.32 => ( ( ( in @ Xx @ X )
% 0.07/0.32 => Xphi )
% 0.07/0.32 => ( ( ( in @ Xx @ Y )
% 0.07/0.32 => Xphi )
% 0.07/0.32 => Xphi ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunionTEcontra_type,type,
% 0.07/0.32 binunionTEcontra: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionTEcontra,definition,
% 0.07/0.32 ( binunionTEcontra
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ X )
% 0.07/0.32 => ( ~ ( in @ Xx @ Y )
% 0.07/0.32 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan2a1_type,type,
% 0.07/0.32 demorgan2a1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan2a1,definition,
% 0.07/0.32 ( demorgan2a1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.07/0.32 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(complementUnionInPowersetComplement_type,type,
% 0.07/0.32 complementUnionInPowersetComplement: $o ).
% 0.07/0.32
% 0.07/0.32 thf(complementUnionInPowersetComplement,definition,
% 0.07/0.32 ( complementUnionInPowersetComplement
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan2a2_type,type,
% 0.07/0.32 demorgan2a2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan2a2,definition,
% 0.07/0.32 ( demorgan2a2
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.07/0.32 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan1a_type,type,
% 0.07/0.32 demorgan1a: $o ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan1a,definition,
% 0.07/0.32 ( demorgan1a
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.07/0.32 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan1b_type,type,
% 0.07/0.32 demorgan1b: $o ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan1b,definition,
% 0.07/0.32 ( demorgan1b
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.07/0.32 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan1_type,type,
% 0.07/0.32 demorgan1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan1,definition,
% 0.07/0.32 ( demorgan1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.07/0.32 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan2a_type,type,
% 0.07/0.32 demorgan2a: $o ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan2a,definition,
% 0.07/0.32 ( demorgan2a
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.07/0.32 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan2b2_type,type,
% 0.07/0.32 demorgan2b2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan2b2,definition,
% 0.07/0.32 ( demorgan2b2
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.07/0.32 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.07/0.32 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan2b_type,type,
% 0.07/0.32 demorgan2b: $o ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan2b,definition,
% 0.07/0.32 ( demorgan2b
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.07/0.32 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan2_type,type,
% 0.07/0.32 demorgan2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(demorgan2,definition,
% 0.07/0.32 ( demorgan2
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.07/0.32 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(woz13rule0_type,type,
% 0.07/0.32 woz13rule0: $o ).
% 0.07/0.32
% 0.07/0.32 thf(woz13rule0,definition,
% 0.07/0.32 ( woz13rule0
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.07/0.32 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(woz13rule1_type,type,
% 0.07/0.32 woz13rule1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(woz13rule1,definition,
% 0.07/0.32 ( woz13rule1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Z: $i] :
% 0.07/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.07/0.32 => ( ( subset @ X @ Z )
% 0.07/0.32 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(woz13rule2_type,type,
% 0.07/0.32 woz13rule2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(woz13rule2,definition,
% 0.07/0.32 ( woz13rule2
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Z: $i] :
% 0.07/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.07/0.32 => ( ( subset @ Y @ Z )
% 0.07/0.32 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(woz13rule3_type,type,
% 0.07/0.32 woz13rule3: $o ).
% 0.07/0.32
% 0.07/0.32 thf(woz13rule3,definition,
% 0.07/0.32 ( woz13rule3
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Z: $i] :
% 0.07/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.07/0.32 => ( ( subset @ X @ Y )
% 0.07/0.32 => ( ( subset @ X @ Z )
% 0.07/0.32 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(woz13rule4_type,type,
% 0.07/0.32 woz13rule4: $o ).
% 0.07/0.32
% 0.07/0.32 thf(woz13rule4,definition,
% 0.07/0.32 ( woz13rule4
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Z: $i] :
% 0.07/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.07/0.32 => ! [W: $i] :
% 0.07/0.32 ( ( in @ W @ ( powerset @ A ) )
% 0.07/0.32 => ( ( subset @ X @ Z )
% 0.07/0.32 => ( ( subset @ Y @ W )
% 0.07/0.32 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(woz1_1_type,type,
% 0.07/0.32 woz1_1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(woz1_1,definition,
% 0.07/0.32 ( woz1_1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(woz1_2_type,type,
% 0.07/0.32 woz1_2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(woz1_2,definition,
% 0.07/0.32 ( woz1_2
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Z: $i] :
% 0.07/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.07/0.32 => ! [W: $i] :
% 0.07/0.32 ( ( in @ W @ ( powerset @ A ) )
% 0.07/0.32 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 0.07/0.32 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(woz1_3_type,type,
% 0.07/0.32 woz1_3: $o ).
% 0.07/0.32
% 0.07/0.32 thf(woz1_3,definition,
% 0.07/0.32 ( woz1_3
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Z: $i] :
% 0.07/0.32 ( ( in @ Z @ ( powerset @ A ) )
% 0.07/0.32 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(woz1_4_type,type,
% 0.07/0.32 woz1_4: $o ).
% 0.07/0.32
% 0.07/0.32 thf(woz1_4,definition,
% 0.07/0.32 ( woz1_4
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.07/0.32 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(woz1_5,conjecture,
% 0.07/0.32 ( setextAx
% 0.07/0.32 => ( emptysetAx
% 0.07/0.32 => ( setadjoinAx
% 0.07/0.32 => ( powersetAx
% 0.07/0.32 => ( setunionAx
% 0.07/0.32 => ( omega0Ax
% 0.07/0.32 => ( omegaSAx
% 0.07/0.32 => ( omegaIndAx
% 0.07/0.32 => ( replAx
% 0.07/0.32 => ( foundationAx
% 0.07/0.32 => ( wellorderingAx
% 0.07/0.32 => ( descrp
% 0.07/0.32 => ( dsetconstrI
% 0.07/0.32 => ( dsetconstrEL
% 0.07/0.32 => ( dsetconstrER
% 0.07/0.32 => ( exuE1
% 0.07/0.32 => ( prop2setE
% 0.07/0.32 => ( emptysetE
% 0.07/0.32 => ( emptysetimpfalse
% 0.07/0.32 => ( notinemptyset
% 0.07/0.32 => ( exuE3e
% 0.07/0.32 => ( setext
% 0.07/0.32 => ( emptyI
% 0.07/0.32 => ( noeltsimpempty
% 0.07/0.32 => ( setbeta
% 0.07/0.32 => ( nonemptyE1
% 0.07/0.32 => ( nonemptyI
% 0.07/0.32 => ( nonemptyI1
% 0.07/0.32 => ( setadjoinIL
% 0.07/0.32 => ( emptyinunitempty
% 0.07/0.32 => ( setadjoinIR
% 0.07/0.32 => ( setadjoinE
% 0.07/0.32 => ( setadjoinOr
% 0.07/0.32 => ( setoftrueEq
% 0.07/0.32 => ( powersetI
% 0.07/0.32 => ( emptyinPowerset
% 0.07/0.32 => ( emptyInPowerset
% 0.07/0.32 => ( powersetE
% 0.07/0.32 => ( setunionI
% 0.07/0.32 => ( setunionE
% 0.07/0.32 => ( subPowSU
% 0.07/0.32 => ( exuE2
% 0.07/0.32 => ( nonemptyImpWitness
% 0.07/0.32 => ( uniqinunit
% 0.07/0.32 => ( notinsingleton
% 0.07/0.32 => ( eqinunit
% 0.07/0.32 => ( singletonsswitch
% 0.07/0.32 => ( upairsetE
% 0.07/0.32 => ( upairsetIL
% 0.07/0.32 => ( upairsetIR
% 0.07/0.32 => ( emptyE1
% 0.07/0.32 => ( vacuousDall
% 0.07/0.32 => ( quantDeMorgan1
% 0.07/0.32 => ( quantDeMorgan2
% 0.07/0.32 => ( quantDeMorgan3
% 0.07/0.32 => ( quantDeMorgan4
% 0.07/0.32 => ( prop2setI
% 0.07/0.32 => ( prop2set2propI
% 0.07/0.32 => ( notdexE
% 0.07/0.32 => ( notdallE
% 0.07/0.32 => ( exuI1
% 0.07/0.32 => ( exuI3
% 0.07/0.32 => ( exuI2
% 0.07/0.32 => ( inCongP
% 0.07/0.32 => ( in__Cong
% 0.07/0.32 => ( exuE3u
% 0.07/0.32 => ( exu__Cong
% 0.07/0.32 => ( emptyset__Cong
% 0.07/0.32 => ( setadjoin__Cong
% 0.07/0.32 => ( powerset__Cong
% 0.07/0.32 => ( setunion__Cong
% 0.07/0.32 => ( omega__Cong
% 0.07/0.32 => ( exuEu
% 0.07/0.32 => ( descr__Cong
% 0.07/0.32 => ( dsetconstr__Cong
% 0.07/0.32 => ( subsetI1
% 0.07/0.32 => ( eqimpsubset2
% 0.07/0.32 => ( eqimpsubset1
% 0.07/0.32 => ( subsetI2
% 0.07/0.32 => ( emptysetsubset
% 0.07/0.32 => ( subsetE
% 0.07/0.32 => ( subsetE2
% 0.07/0.32 => ( notsubsetI
% 0.07/0.32 => ( notequalI1
% 0.07/0.32 => ( notequalI2
% 0.07/0.32 => ( subsetRefl
% 0.07/0.32 => ( subsetTrans
% 0.07/0.32 => ( setadjoinSub
% 0.07/0.32 => ( setadjoinSub2
% 0.07/0.32 => ( subset2powerset
% 0.07/0.32 => ( setextsub
% 0.07/0.32 => ( subsetemptysetimpeq
% 0.07/0.32 => ( powersetI1
% 0.07/0.32 => ( powersetE1
% 0.07/0.32 => ( inPowerset
% 0.07/0.32 => ( powersetsubset
% 0.07/0.32 => ( sepInPowerset
% 0.07/0.32 => ( sepSubset
% 0.07/0.32 => ( binunionIL
% 0.07/0.32 => ( upairset2IR
% 0.07/0.32 => ( binunionIR
% 0.07/0.32 => ( binunionEcases
% 0.07/0.32 => ( binunionE
% 0.07/0.32 => ( binunionLsub
% 0.07/0.32 => ( binunionRsub
% 0.07/0.32 => ( binintersectI
% 0.07/0.32 => ( binintersectSubset5
% 0.07/0.32 => ( binintersectEL
% 0.07/0.32 => ( binintersectLsub
% 0.07/0.32 => ( binintersectSubset2
% 0.07/0.32 => ( binintersectSubset3
% 0.07/0.32 => ( binintersectER
% 0.07/0.32 => ( disjointsetsI1
% 0.07/0.32 => ( binintersectRsub
% 0.07/0.32 => ( binintersectSubset4
% 0.07/0.32 => ( binintersectSubset1
% 0.07/0.32 => ( bs114d
% 0.07/0.32 => ( setminusI
% 0.07/0.32 => ( setminusEL
% 0.07/0.32 => ( setminusER
% 0.07/0.32 => ( setminusSubset2
% 0.07/0.32 => ( setminusERneg
% 0.07/0.32 => ( setminusELneg
% 0.07/0.32 => ( setminusILneg
% 0.07/0.32 => ( setminusIRneg
% 0.07/0.32 => ( setminusLsub
% 0.07/0.32 => ( setminusSubset1
% 0.07/0.32 => ( symdiffE
% 0.07/0.32 => ( symdiffI1
% 0.07/0.32 => ( symdiffI2
% 0.07/0.32 => ( symdiffIneg1
% 0.07/0.32 => ( symdiffIneg2
% 0.07/0.32 => ( secondinupair
% 0.07/0.32 => ( setukpairIL
% 0.07/0.32 => ( setukpairIR
% 0.07/0.32 => ( kpairiskpair
% 0.07/0.32 => ( kpairp
% 0.07/0.32 => ( singletonsubset
% 0.07/0.32 => ( singletoninpowerset
% 0.07/0.32 => ( singletoninpowunion
% 0.07/0.32 => ( upairset2E
% 0.07/0.32 => ( upairsubunion
% 0.07/0.32 => ( upairinpowunion
% 0.07/0.32 => ( ubforcartprodlem1
% 0.07/0.32 => ( ubforcartprodlem2
% 0.07/0.32 => ( ubforcartprodlem3
% 0.07/0.32 => ( cartprodpairin
% 0.07/0.32 => ( cartprodmempair1
% 0.07/0.32 => ( cartprodmempair
% 0.07/0.32 => ( setunionE2
% 0.07/0.32 => ( setunionsingleton1
% 0.07/0.32 => ( setunionsingleton2
% 0.07/0.32 => ( setunionsingleton
% 0.07/0.32 => ( singletonprop
% 0.07/0.32 => ( ex1E1
% 0.07/0.32 => ( ex1I
% 0.07/0.32 => ( ex1I2
% 0.07/0.32 => ( singletonsuniq
% 0.07/0.32 => ( setukpairinjL1
% 0.07/0.32 => ( kfstsingleton
% 0.07/0.32 => ( theprop
% 0.07/0.32 => ( kfstpairEq
% 0.07/0.32 => ( cartprodfstin
% 0.07/0.32 => ( setukpairinjL2
% 0.07/0.32 => ( setukpairinjL
% 0.07/0.32 => ( setukpairinjR11
% 0.07/0.32 => ( setukpairinjR12
% 0.07/0.32 => ( setukpairinjR1
% 0.07/0.32 => ( upairequniteq
% 0.07/0.32 => ( setukpairinjR2
% 0.07/0.32 => ( setukpairinjR
% 0.07/0.32 => ( ksndsingleton
% 0.07/0.32 => ( ksndpairEq
% 0.07/0.32 => ( kpairsurjEq
% 0.07/0.32 => ( cartprodsndin
% 0.07/0.32 => ( cartprodpairmemEL
% 0.07/0.32 => ( cartprodpairmemER
% 0.07/0.32 => ( cartprodmempaircEq
% 0.07/0.32 => ( cartprodfstpairEq
% 0.07/0.32 => ( cartprodsndpairEq
% 0.07/0.32 => ( cartprodpairsurjEq
% 0.07/0.32 => ( dpsetconstrI
% 0.07/0.32 => ( dpsetconstrSub
% 0.07/0.32 => ( setOfPairsIsBReln
% 0.07/0.32 => ( dpsetconstrERa
% 0.07/0.32 => ( dpsetconstrEL1
% 0.07/0.32 => ( dpsetconstrEL2
% 0.07/0.32 => ( dpsetconstrER
% 0.07/0.32 => ( funcImageSingleton
% 0.07/0.32 => ( apProp
% 0.07/0.32 => ( app
% 0.07/0.32 => ( infuncsetfunc
% 0.07/0.32 => ( ap2p
% 0.07/0.32 => ( funcinfuncset
% 0.07/0.32 => ( lamProp
% 0.07/0.32 => ( lamp
% 0.07/0.32 => ( lam2p
% 0.07/0.32 => ( brelnall1
% 0.07/0.32 => ( brelnall2
% 0.07/0.32 => ( ex1E2
% 0.07/0.32 => ( funcGraphProp1
% 0.07/0.32 => ( funcGraphProp3
% 0.07/0.32 => ( funcGraphProp2
% 0.07/0.32 => ( funcextLem
% 0.07/0.32 => ( funcGraphProp4
% 0.07/0.32 => ( subbreln
% 0.07/0.32 => ( eqbreln
% 0.07/0.32 => ( funcext
% 0.07/0.32 => ( funcext2
% 0.07/0.32 => ( ap2apEq1
% 0.07/0.32 => ( ap2apEq2
% 0.07/0.32 => ( beta1
% 0.07/0.32 => ( eta1
% 0.07/0.32 => ( lam2lamEq
% 0.07/0.32 => ( beta2
% 0.07/0.32 => ( eta2
% 0.07/0.32 => ( iffalseProp1
% 0.07/0.32 => ( iffalseProp2
% 0.07/0.32 => ( iftrueProp1
% 0.07/0.32 => ( iftrueProp2
% 0.07/0.32 => ( ifSingleton
% 0.07/0.32 => ( ifp
% 0.07/0.32 => ( theeq
% 0.07/0.32 => ( iftrue
% 0.07/0.32 => ( iffalse
% 0.07/0.32 => ( iftrueorfalse
% 0.07/0.32 => ( binintersectT_lem
% 0.07/0.32 => ( binunionT_lem
% 0.07/0.32 => ( powersetT_lem
% 0.07/0.32 => ( setminusT_lem
% 0.07/0.32 => ( complementT_lem
% 0.07/0.32 => ( setextT
% 0.07/0.32 => ( subsetTI
% 0.07/0.32 => ( powersetTI1
% 0.07/0.32 => ( powersetTE1
% 0.07/0.32 => ( complementTI1
% 0.07/0.32 => ( complementTE1
% 0.07/0.32 => ( binintersectTELcontra
% 0.07/0.32 => ( binintersectTERcontra
% 0.07/0.32 => ( contrasubsetT
% 0.07/0.32 => ( contrasubsetT1
% 0.07/0.32 => ( contrasubsetT2
% 0.07/0.32 => ( contrasubsetT3
% 0.07/0.32 => ( doubleComplementI1
% 0.07/0.32 => ( doubleComplementE1
% 0.07/0.32 => ( doubleComplementSub1
% 0.07/0.32 => ( doubleComplementSub2
% 0.07/0.32 => ( doubleComplementEq
% 0.07/0.32 => ( complementTnotintersectT
% 0.07/0.32 => ( complementImpComplementIntersect
% 0.07/0.32 => ( complementSubsetComplementIntersect
% 0.07/0.32 => ( complementInPowersetComplementIntersect
% 0.07/0.32 => ( contraSubsetComplement
% 0.07/0.32 => ( complementTcontraSubset
% 0.07/0.32 => ( binunionTILcontra
% 0.07/0.32 => ( binunionTIRcontra
% 0.07/0.32 => ( inIntersectImpInUnion
% 0.07/0.32 => ( inIntersectImpInUnion2
% 0.07/0.32 => ( inIntersectImpInIntersectUnions
% 0.07/0.32 => ( intersectInPowersetIntersectUnions
% 0.07/0.32 => ( inComplementUnionImpNotIn1
% 0.07/0.32 => ( inComplementUnionImpInComplement1
% 0.07/0.32 => ( binunionTE
% 0.07/0.32 => ( binunionTEcontra
% 0.07/0.32 => ( demorgan2a1
% 0.07/0.32 => ( complementUnionInPowersetComplement
% 0.07/0.32 => ( demorgan2a2
% 0.07/0.32 => ( demorgan1a
% 0.07/0.32 => ( demorgan1b
% 0.07/0.32 => ( demorgan1
% 0.07/0.32 => ( demorgan2a
% 0.07/0.32 => ( demorgan2b2
% 0.07/0.32 => ( demorgan2b
% 0.07/0.32 => ( demorgan2
% 0.07/0.32 => ( woz13rule0
% 0.07/0.32 => ( woz13rule1
% 0.07/0.32 => ( woz13rule2
% 0.07/0.32 => ( woz13rule3
% 0.07/0.32 => ( woz13rule4
% 0.07/0.32 => ( woz1_1
% 0.07/0.32 => ( woz1_2
% 0.07/0.32 => ( woz1_3
% 0.07/0.32 => ( woz1_4
% 0.07/0.32 => ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 %------------------------------------------------------------------------------
% 0.07/0.32 ------------------------
% 1.84/1.12 ---- Embedded in THF ---
% 2.23/1.12 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.23/1.12 %%% Generated on Tue Feb 24 07:33:11 EST 2026
% 2.23/1.12 %%% using '$$dhol' embedding, version 1.3.0.
% 2.23/1.12 %%% Logic specification used:
% 2.23/1.12 %%% thf(spec, logic, $$dhol).
% 2.23/1.12
% 2.23/1.12 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.7mug10HeGF/DTF2DTF7573.p
% See solution above
% 2.23/1.12 ------------------------
% 2.23/1.15 ---- Cleaned THF ---
% 2.23/1.15 thf(in_type,type,
% 2.23/1.15 in: $i > $i > $o ).
% 2.23/1.15
% 2.23/1.15 thf(exu_type,type,
% 2.23/1.15 exu: ( $i > $o ) > $o ).
% 2.23/1.15
% 2.23/1.15 thf(exu,definition,
% 2.23/1.15 ( exu
% 2.23/1.15 = ( ^ [Xphi: $i > $o] :
% 2.23/1.15 ? [Xx: $i] :
% 2.23/1.15 ( ( Xphi @ Xx )
% 2.23/1.15 & ! [Xy: $i] :
% 2.23/1.15 ( ( Xphi @ Xy )
% 2.23/1.15 => ( Xx = Xy ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setextAx_type,type,
% 2.23/1.15 setextAx: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setextAx,definition,
% 2.23/1.15 ( setextAx
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 <=> ( in @ Xx @ B ) )
% 2.23/1.15 => ( A = B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(emptyset_type,type,
% 2.23/1.15 emptyset: $i ).
% 2.23/1.15
% 2.23/1.15 thf(emptysetAx_type,type,
% 2.23/1.15 emptysetAx: $o ).
% 2.23/1.15
% 2.23/1.15 thf(emptysetAx,definition,
% 2.23/1.15 ( emptysetAx
% 2.23/1.15 = ( ! [Xx: $i] :
% 2.23/1.15 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoin_type,type,
% 2.23/1.15 setadjoin: $i > $i > $i ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoinAx_type,type,
% 2.23/1.15 setadjoinAx: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoinAx,definition,
% 2.23/1.15 ( setadjoinAx
% 2.23/1.15 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.23/1.15 <=> ( ( Xy = Xx )
% 2.23/1.15 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(powerset_type,type,
% 2.23/1.15 powerset: $i > $i ).
% 2.23/1.15
% 2.23/1.15 thf(powersetAx_type,type,
% 2.23/1.15 powersetAx: $o ).
% 2.23/1.15
% 2.23/1.15 thf(powersetAx,definition,
% 2.23/1.15 ( powersetAx
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( in @ B @ ( powerset @ A ) )
% 2.23/1.15 <=> ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ B )
% 2.23/1.15 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setunion_type,type,
% 2.23/1.15 setunion: $i > $i ).
% 2.23/1.15
% 2.23/1.15 thf(setunionAx_type,type,
% 2.23/1.15 setunionAx: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setunionAx,definition,
% 2.23/1.15 ( setunionAx
% 2.23/1.15 = ( ! [A: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ ( setunion @ A ) )
% 2.23/1.15 <=> ? [B: $i] :
% 2.23/1.15 ( ( in @ Xx @ B )
% 2.23/1.15 & ( in @ B @ A ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(omega_type,type,
% 2.23/1.15 omega: $i ).
% 2.23/1.15
% 2.23/1.15 thf(omega0Ax_type,type,
% 2.23/1.15 omega0Ax: $o ).
% 2.23/1.15
% 2.23/1.15 thf(omega0Ax,definition,
% 2.23/1.15 ( omega0Ax
% 2.23/1.15 = ( in @ emptyset @ omega ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(omegaSAx_type,type,
% 2.23/1.15 omegaSAx: $o ).
% 2.23/1.15
% 2.23/1.15 thf(omegaSAx,definition,
% 2.23/1.15 ( omegaSAx
% 2.23/1.15 = ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ omega )
% 2.23/1.15 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(omegaIndAx_type,type,
% 2.23/1.15 omegaIndAx: $o ).
% 2.23/1.15
% 2.23/1.15 thf(omegaIndAx,definition,
% 2.23/1.15 ( omegaIndAx
% 2.23/1.15 = ( ! [A: $i] :
% 2.23/1.15 ( ( ( in @ emptyset @ A )
% 2.23/1.15 & ! [Xx: $i] :
% 2.23/1.15 ( ( ( in @ Xx @ omega )
% 2.23/1.15 & ( in @ Xx @ A ) )
% 2.23/1.15 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ omega )
% 2.23/1.15 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(replAx_type,type,
% 2.23/1.15 replAx: $o ).
% 2.23/1.15
% 2.23/1.15 thf(replAx,definition,
% 2.23/1.15 ( replAx
% 2.23/1.15 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( exu
% 2.23/1.15 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.23/1.15 => ? [B: $i] :
% 2.23/1.15 ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ B )
% 2.23/1.15 <=> ? [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ A )
% 2.23/1.15 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(foundationAx_type,type,
% 2.23/1.15 foundationAx: $o ).
% 2.23/1.15
% 2.23/1.15 thf(foundationAx,definition,
% 2.23/1.15 ( foundationAx
% 2.23/1.15 = ( ! [A: $i] :
% 2.23/1.15 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.23/1.15 => ? [B: $i] :
% 2.23/1.15 ( ( in @ B @ A )
% 2.23/1.15 & ~ ? [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ B )
% 2.23/1.15 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(wellorderingAx_type,type,
% 2.23/1.15 wellorderingAx: $o ).
% 2.23/1.15
% 2.23/1.15 thf(wellorderingAx,definition,
% 2.23/1.15 ( wellorderingAx
% 2.23/1.15 = ( ! [A: $i] :
% 2.23/1.15 ? [B: $i] :
% 2.23/1.15 ( ! [C: $i] :
% 2.23/1.15 ( ( in @ C @ B )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ C )
% 2.23/1.15 => ( in @ Xx @ A ) ) )
% 2.23/1.15 & ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( ( in @ Xx @ A )
% 2.23/1.15 & ( in @ Xy @ A ) )
% 2.23/1.15 => ( ! [C: $i] :
% 2.23/1.15 ( ( in @ C @ B )
% 2.23/1.15 => ( ( in @ Xx @ C )
% 2.23/1.15 <=> ( in @ Xy @ C ) ) )
% 2.23/1.15 => ( Xx = Xy ) ) )
% 2.23/1.15 & ! [C: $i,D: $i] :
% 2.23/1.15 ( ( ( in @ C @ B )
% 2.23/1.15 & ( in @ D @ B ) )
% 2.23/1.15 => ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ C )
% 2.23/1.15 => ( in @ Xx @ D ) )
% 2.23/1.15 | ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ D )
% 2.23/1.15 => ( in @ Xx @ C ) ) ) )
% 2.23/1.15 & ! [C: $i] :
% 2.23/1.15 ( ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ C )
% 2.23/1.15 => ( in @ Xx @ A ) )
% 2.23/1.15 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.23/1.15 => ? [D: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ D @ B )
% 2.23/1.15 & ( in @ Xx @ C )
% 2.23/1.15 & ~ ? [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ D )
% 2.23/1.15 & ( in @ Xy @ C ) )
% 2.23/1.15 & ! [E: $i] :
% 2.23/1.15 ( ( in @ E @ B )
% 2.23/1.15 => ( ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ E )
% 2.23/1.15 => ( in @ Xy @ D ) )
% 2.23/1.15 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(descr_type,type,
% 2.23/1.15 descr: ( $i > $o ) > $i ).
% 2.23/1.15
% 2.23/1.15 thf(descrp_type,type,
% 2.23/1.15 descrp: $o ).
% 2.23/1.15
% 2.23/1.15 thf(descrp,definition,
% 2.23/1.15 ( descrp
% 2.23/1.15 = ( ! [Xphi: $i > $o] :
% 2.23/1.15 ( ( exu
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.23/1.15 => ( Xphi
% 2.23/1.15 @ ( descr
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(dsetconstr_type,type,
% 2.23/1.15 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.23/1.15
% 2.23/1.15 thf(dsetconstrI_type,type,
% 2.23/1.15 dsetconstrI: $o ).
% 2.23/1.15
% 2.23/1.15 thf(dsetconstrI,definition,
% 2.23/1.15 ( dsetconstrI
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( ( Xphi @ Xx )
% 2.23/1.15 => ( in @ Xx
% 2.23/1.15 @ ( dsetconstr @ A
% 2.23/1.15 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(dsetconstrEL_type,type,
% 2.23/1.15 dsetconstrEL: $o ).
% 2.23/1.15
% 2.23/1.15 thf(dsetconstrEL,definition,
% 2.23/1.15 ( dsetconstrEL
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx
% 2.23/1.15 @ ( dsetconstr @ A
% 2.23/1.15 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.23/1.15 => ( in @ Xx @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(dsetconstrER_type,type,
% 2.23/1.15 dsetconstrER: $o ).
% 2.23/1.15
% 2.23/1.15 thf(dsetconstrER,definition,
% 2.23/1.15 ( dsetconstrER
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx
% 2.23/1.15 @ ( dsetconstr @ A
% 2.23/1.15 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.23/1.15 => ( Xphi @ Xx ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(exuE1_type,type,
% 2.23/1.15 exuE1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(exuE1,definition,
% 2.23/1.15 ( exuE1
% 2.23/1.15 = ( ! [Xphi: $i > $o] :
% 2.23/1.15 ( ( exu
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.23/1.15 => ? [Xx: $i] :
% 2.23/1.15 ( ( Xphi @ Xx )
% 2.23/1.15 & ! [Xy: $i] :
% 2.23/1.15 ( ( Xphi @ Xy )
% 2.23/1.15 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(prop2set_type,type,
% 2.23/1.15 prop2set: $o > $i ).
% 2.23/1.15
% 2.23/1.15 thf(prop2setE_type,type,
% 2.23/1.15 prop2setE: $o ).
% 2.23/1.15
% 2.23/1.15 thf(prop2setE,definition,
% 2.23/1.15 ( prop2setE
% 2.23/1.15 = ( ! [Xphi: $o,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.23/1.15 => Xphi ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(emptysetE_type,type,
% 2.23/1.15 emptysetE: $o ).
% 2.23/1.15
% 2.23/1.15 thf(emptysetE,definition,
% 2.23/1.15 ( emptysetE
% 2.23/1.15 = ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ emptyset )
% 2.23/1.15 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(emptysetimpfalse_type,type,
% 2.23/1.15 emptysetimpfalse: $o ).
% 2.23/1.15
% 2.23/1.15 thf(emptysetimpfalse,definition,
% 2.23/1.15 ( emptysetimpfalse
% 2.23/1.15 = ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ emptyset )
% 2.23/1.15 => $false ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(notinemptyset_type,type,
% 2.23/1.15 notinemptyset: $o ).
% 2.23/1.15
% 2.23/1.15 thf(notinemptyset,definition,
% 2.23/1.15 ( notinemptyset
% 2.23/1.15 = ( ! [Xx: $i] :
% 2.23/1.15 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(exuE3e_type,type,
% 2.23/1.15 exuE3e: $o ).
% 2.23/1.15
% 2.23/1.15 thf(exuE3e,definition,
% 2.23/1.15 ( exuE3e
% 2.23/1.15 = ( ! [Xphi: $i > $o] :
% 2.23/1.15 ( ( exu
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.23/1.15 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setext_type,type,
% 2.23/1.15 setext: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setext,definition,
% 2.23/1.15 ( setext
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ Xx @ B ) )
% 2.23/1.15 => ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ B )
% 2.23/1.15 => ( in @ Xx @ A ) )
% 2.23/1.15 => ( A = B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(emptyI_type,type,
% 2.23/1.15 emptyI: $o ).
% 2.23/1.15
% 2.23/1.15 thf(emptyI,definition,
% 2.23/1.15 ( emptyI
% 2.23/1.15 = ( ! [A: $i] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ~ ( in @ Xx @ A )
% 2.23/1.15 => ( A = emptyset ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(noeltsimpempty_type,type,
% 2.23/1.15 noeltsimpempty: $o ).
% 2.23/1.15
% 2.23/1.15 thf(noeltsimpempty,definition,
% 2.23/1.15 ( noeltsimpempty
% 2.23/1.15 = ( ! [A: $i] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ~ ( in @ Xx @ A )
% 2.23/1.15 => ( A = emptyset ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setbeta_type,type,
% 2.23/1.15 setbeta: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setbeta,definition,
% 2.23/1.15 ( setbeta
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( ( in @ Xx
% 2.23/1.15 @ ( dsetconstr @ A
% 2.23/1.15 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.23/1.15 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(nonempty_type,type,
% 2.23/1.15 nonempty: $i > $o ).
% 2.23/1.15
% 2.23/1.15 thf(nonempty,definition,
% 2.23/1.15 ( nonempty
% 2.23/1.15 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(nonemptyE1_type,type,
% 2.23/1.15 nonemptyE1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(nonemptyE1,definition,
% 2.23/1.15 ( nonemptyE1
% 2.23/1.15 = ( ! [A: $i] :
% 2.23/1.15 ( ( nonempty @ A )
% 2.23/1.15 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(nonemptyI_type,type,
% 2.23/1.15 nonemptyI: $o ).
% 2.23/1.15
% 2.23/1.15 thf(nonemptyI,definition,
% 2.23/1.15 ( nonemptyI
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( ( Xphi @ Xx )
% 2.23/1.15 => ( nonempty
% 2.23/1.15 @ ( dsetconstr @ A
% 2.23/1.15 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(nonemptyI1_type,type,
% 2.23/1.15 nonemptyI1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(nonemptyI1,definition,
% 2.23/1.15 ( nonemptyI1
% 2.23/1.15 = ( ! [A: $i] :
% 2.23/1.15 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.23/1.15 => ( nonempty @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoinIL_type,type,
% 2.23/1.15 setadjoinIL: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoinIL,definition,
% 2.23/1.15 ( setadjoinIL
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(emptyinunitempty_type,type,
% 2.23/1.15 emptyinunitempty: $o ).
% 2.23/1.15
% 2.23/1.15 thf(emptyinunitempty,definition,
% 2.23/1.15 ( emptyinunitempty
% 2.23/1.15 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoinIR_type,type,
% 2.23/1.15 setadjoinIR: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoinIR,definition,
% 2.23/1.15 ( setadjoinIR
% 2.23/1.15 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ A )
% 2.23/1.15 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoinE_type,type,
% 2.23/1.15 setadjoinE: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoinE,definition,
% 2.23/1.15 ( setadjoinE
% 2.23/1.15 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.23/1.15 => ! [Xphi: $o] :
% 2.23/1.15 ( ( ( Xy = Xx )
% 2.23/1.15 => Xphi )
% 2.23/1.15 => ( ( ( in @ Xy @ A )
% 2.23/1.15 => Xphi )
% 2.23/1.15 => Xphi ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoinOr_type,type,
% 2.23/1.15 setadjoinOr: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoinOr,definition,
% 2.23/1.15 ( setadjoinOr
% 2.23/1.15 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.23/1.15 => ( ( Xy = Xx )
% 2.23/1.15 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setoftrueEq_type,type,
% 2.23/1.15 setoftrueEq: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setoftrueEq,definition,
% 2.23/1.15 ( setoftrueEq
% 2.23/1.15 = ( ! [A: $i] :
% 2.23/1.15 ( ( dsetconstr @ A
% 2.23/1.15 @ ^ [Xx: $i] : $true )
% 2.23/1.15 = A ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(powersetI_type,type,
% 2.23/1.15 powersetI: $o ).
% 2.23/1.15
% 2.23/1.15 thf(powersetI,definition,
% 2.23/1.15 ( powersetI
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ B )
% 2.23/1.15 => ( in @ Xx @ A ) )
% 2.23/1.15 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(emptyinPowerset_type,type,
% 2.23/1.15 emptyinPowerset: $o ).
% 2.23/1.15
% 2.23/1.15 thf(emptyinPowerset,definition,
% 2.23/1.15 ( emptyinPowerset
% 2.23/1.15 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(emptyInPowerset_type,type,
% 2.23/1.15 emptyInPowerset: $o ).
% 2.23/1.15
% 2.23/1.15 thf(emptyInPowerset,definition,
% 2.23/1.15 ( emptyInPowerset
% 2.23/1.15 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(powersetE_type,type,
% 2.23/1.15 powersetE: $o ).
% 2.23/1.15
% 2.23/1.15 thf(powersetE,definition,
% 2.23/1.15 ( powersetE
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ B @ ( powerset @ A ) )
% 2.23/1.15 => ( ( in @ Xx @ B )
% 2.23/1.15 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setunionI_type,type,
% 2.23/1.15 setunionI: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setunionI,definition,
% 2.23/1.15 ( setunionI
% 2.23/1.15 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.23/1.15 ( ( in @ Xx @ B )
% 2.23/1.15 => ( ( in @ B @ A )
% 2.23/1.15 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setunionE_type,type,
% 2.23/1.15 setunionE: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setunionE,definition,
% 2.23/1.15 ( setunionE
% 2.23/1.15 = ( ! [A: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ ( setunion @ A ) )
% 2.23/1.15 => ! [Xphi: $o] :
% 2.23/1.15 ( ! [B: $i] :
% 2.23/1.15 ( ( in @ Xx @ B )
% 2.23/1.15 => ( ( in @ B @ A )
% 2.23/1.15 => Xphi ) )
% 2.23/1.15 => Xphi ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(subPowSU_type,type,
% 2.23/1.15 subPowSU: $o ).
% 2.23/1.15
% 2.23/1.15 thf(subPowSU,definition,
% 2.23/1.15 ( subPowSU
% 2.23/1.15 = ( ! [A: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(exuE2_type,type,
% 2.23/1.15 exuE2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(exuE2,definition,
% 2.23/1.15 ( exuE2
% 2.23/1.15 = ( ! [Xphi: $i > $o] :
% 2.23/1.15 ( ( exu
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.23/1.15 => ? [Xx: $i] :
% 2.23/1.15 ! [Xy: $i] :
% 2.23/1.15 ( ( Xphi @ Xy )
% 2.23/1.15 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(nonemptyImpWitness_type,type,
% 2.23/1.15 nonemptyImpWitness: $o ).
% 2.23/1.15
% 2.23/1.15 thf(nonemptyImpWitness,definition,
% 2.23/1.15 ( nonemptyImpWitness
% 2.23/1.15 = ( ! [A: $i] :
% 2.23/1.15 ( ( nonempty @ A )
% 2.23/1.15 => ? [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 & $true ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(uniqinunit_type,type,
% 2.23/1.15 uniqinunit: $o ).
% 2.23/1.15
% 2.23/1.15 thf(uniqinunit,definition,
% 2.23/1.15 ( uniqinunit
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.23/1.15 => ( Xx = Xy ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(notinsingleton_type,type,
% 2.23/1.15 notinsingleton: $o ).
% 2.23/1.15
% 2.23/1.15 thf(notinsingleton,definition,
% 2.23/1.15 ( notinsingleton
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( Xx != Xy )
% 2.23/1.15 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(eqinunit_type,type,
% 2.23/1.15 eqinunit: $o ).
% 2.23/1.15
% 2.23/1.15 thf(eqinunit,definition,
% 2.23/1.15 ( eqinunit
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( Xx = Xy )
% 2.23/1.15 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(singletonsswitch_type,type,
% 2.23/1.15 singletonsswitch: $o ).
% 2.23/1.15
% 2.23/1.15 thf(singletonsswitch,definition,
% 2.23/1.15 ( singletonsswitch
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.23/1.15 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(upairsetE_type,type,
% 2.23/1.15 upairsetE: $o ).
% 2.23/1.15
% 2.23/1.15 thf(upairsetE,definition,
% 2.23/1.15 ( upairsetE
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.23/1.15 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.23/1.15 => ( ( Xz = Xx )
% 2.23/1.15 | ( Xz = Xy ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(upairsetIL_type,type,
% 2.23/1.15 upairsetIL: $o ).
% 2.23/1.15
% 2.23/1.15 thf(upairsetIL,definition,
% 2.23/1.15 ( upairsetIL
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(upairsetIR_type,type,
% 2.23/1.15 upairsetIR: $o ).
% 2.23/1.15
% 2.23/1.15 thf(upairsetIR,definition,
% 2.23/1.15 ( upairsetIR
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(emptyE1_type,type,
% 2.23/1.15 emptyE1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(emptyE1,definition,
% 2.23/1.15 ( emptyE1
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o] :
% 2.23/1.15 ( ? [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 & ( Xphi @ Xx ) )
% 2.23/1.15 => ( ( ( dsetconstr @ A
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.23/1.15 = emptyset )
% 2.23/1.15 => $false ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(vacuousDall_type,type,
% 2.23/1.15 vacuousDall: $o ).
% 2.23/1.15
% 2.23/1.15 thf(vacuousDall,definition,
% 2.23/1.15 ( vacuousDall
% 2.23/1.15 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ emptyset )
% 2.23/1.15 => ( Xphi @ Xx ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(quantDeMorgan1_type,type,
% 2.23/1.15 quantDeMorgan1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(quantDeMorgan1,definition,
% 2.23/1.15 ( quantDeMorgan1
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o] :
% 2.23/1.15 ( ~ ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( Xphi @ Xx ) )
% 2.23/1.15 => ? [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(quantDeMorgan2_type,type,
% 2.23/1.15 quantDeMorgan2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(quantDeMorgan2,definition,
% 2.23/1.15 ( quantDeMorgan2
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ~ ( Xphi @ Xx ) )
% 2.23/1.15 => ~ ? [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(quantDeMorgan3_type,type,
% 2.23/1.15 quantDeMorgan3: $o ).
% 2.23/1.15
% 2.23/1.15 thf(quantDeMorgan3,definition,
% 2.23/1.15 ( quantDeMorgan3
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o] :
% 2.23/1.15 ( ~ ? [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 & ( Xphi @ Xx ) )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(quantDeMorgan4_type,type,
% 2.23/1.15 quantDeMorgan4: $o ).
% 2.23/1.15
% 2.23/1.15 thf(quantDeMorgan4,definition,
% 2.23/1.15 ( quantDeMorgan4
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o] :
% 2.23/1.15 ( ? [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 & ~ ( Xphi @ Xx ) )
% 2.23/1.15 => ~ ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(prop2setI_type,type,
% 2.23/1.15 prop2setI: $o ).
% 2.23/1.15
% 2.23/1.15 thf(prop2setI,definition,
% 2.23/1.15 ( prop2setI
% 2.23/1.15 = ( ! [Xphi: $o] :
% 2.23/1.15 ( Xphi
% 2.23/1.15 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(set2prop_type,type,
% 2.23/1.15 set2prop: $i > $o ).
% 2.23/1.15
% 2.23/1.15 thf(prop2set2propI_type,type,
% 2.23/1.15 prop2set2propI: $o ).
% 2.23/1.15
% 2.23/1.15 thf(prop2set2propI,definition,
% 2.23/1.15 ( prop2set2propI
% 2.23/1.15 = ( ! [Xphi: $o] :
% 2.23/1.15 ( Xphi
% 2.23/1.15 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(notdexE_type,type,
% 2.23/1.15 notdexE: $o ).
% 2.23/1.15
% 2.23/1.15 thf(notdexE,definition,
% 2.23/1.15 ( notdexE
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o] :
% 2.23/1.15 ( ~ ? [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 & ( Xphi @ Xx ) )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(notdallE_type,type,
% 2.23/1.15 notdallE: $o ).
% 2.23/1.15
% 2.23/1.15 thf(notdallE,definition,
% 2.23/1.15 ( notdallE
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o] :
% 2.23/1.15 ( ~ ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( Xphi @ Xx ) )
% 2.23/1.15 => ? [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(exuI1_type,type,
% 2.23/1.15 exuI1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(exuI1,definition,
% 2.23/1.15 ( exuI1
% 2.23/1.15 = ( ! [Xphi: $i > $o] :
% 2.23/1.15 ( ? [Xx: $i] :
% 2.23/1.15 ( ( Xphi @ Xx )
% 2.23/1.15 & ! [Xy: $i] :
% 2.23/1.15 ( ( Xphi @ Xy )
% 2.23/1.15 => ( Xx = Xy ) ) )
% 2.23/1.15 => ( exu
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(exuI3_type,type,
% 2.23/1.15 exuI3: $o ).
% 2.23/1.15
% 2.23/1.15 thf(exuI3,definition,
% 2.23/1.15 ( exuI3
% 2.23/1.15 = ( ! [Xphi: $i > $o] :
% 2.23/1.15 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.23/1.15 => ( ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( Xphi @ Xx )
% 2.23/1.15 => ( ( Xphi @ Xy )
% 2.23/1.15 => ( Xx = Xy ) ) )
% 2.23/1.15 => ( exu
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(exuI2_type,type,
% 2.23/1.15 exuI2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(exuI2,definition,
% 2.23/1.15 ( exuI2
% 2.23/1.15 = ( ! [Xphi: $i > $o] :
% 2.23/1.15 ( ? [Xx: $i] :
% 2.23/1.15 ! [Xy: $i] :
% 2.23/1.15 ( ( Xphi @ Xy )
% 2.23/1.15 <=> ( Xy = Xx ) )
% 2.23/1.15 => ( exu
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(inCongP_type,type,
% 2.23/1.15 inCongP: $o ).
% 2.23/1.15
% 2.23/1.15 thf(inCongP,definition,
% 2.23/1.15 ( inCongP
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( A = B )
% 2.23/1.15 => ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( Xx = Xy )
% 2.23/1.15 => ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(in__Cong_type,type,
% 2.23/1.15 in__Cong: $o ).
% 2.23/1.15
% 2.23/1.15 thf(in__Cong,definition,
% 2.23/1.15 ( in__Cong
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( A = B )
% 2.23/1.15 => ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( Xx = Xy )
% 2.23/1.15 => ( ( in @ Xx @ A )
% 2.23/1.15 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(exuE3u_type,type,
% 2.23/1.15 exuE3u: $o ).
% 2.23/1.15
% 2.23/1.15 thf(exuE3u,definition,
% 2.23/1.15 ( exuE3u
% 2.23/1.15 = ( ! [Xphi: $i > $o] :
% 2.23/1.15 ( ( exu
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.23/1.15 => ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( Xphi @ Xx )
% 2.23/1.15 => ( ( Xphi @ Xy )
% 2.23/1.15 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(exu__Cong_type,type,
% 2.23/1.15 exu__Cong: $o ).
% 2.23/1.15
% 2.23/1.15 thf(exu__Cong,definition,
% 2.23/1.15 ( exu__Cong
% 2.23/1.15 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.23/1.15 ( ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( Xx = Xy )
% 2.23/1.15 => ( ( Xphi @ Xx )
% 2.23/1.15 <=> ( Xpsi @ Xy ) ) )
% 2.23/1.15 => ( ( exu
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.23/1.15 <=> ( exu
% 2.23/1.15 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(emptyset__Cong_type,type,
% 2.23/1.15 emptyset__Cong: $o ).
% 2.23/1.15
% 2.23/1.15 thf(emptyset__Cong,definition,
% 2.23/1.15 ( emptyset__Cong
% 2.23/1.15 = ( emptyset = emptyset ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoin__Cong_type,type,
% 2.23/1.15 setadjoin__Cong: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoin__Cong,definition,
% 2.23/1.15 ( setadjoin__Cong
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( Xx = Xy )
% 2.23/1.15 => ! [Xz: $i,Xu: $i] :
% 2.23/1.15 ( ( Xz = Xu )
% 2.23/1.15 => ( ( setadjoin @ Xx @ Xz )
% 2.23/1.15 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(powerset__Cong_type,type,
% 2.23/1.15 powerset__Cong: $o ).
% 2.23/1.15
% 2.23/1.15 thf(powerset__Cong,definition,
% 2.23/1.15 ( powerset__Cong
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( A = B )
% 2.23/1.15 => ( ( powerset @ A )
% 2.23/1.15 = ( powerset @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setunion__Cong_type,type,
% 2.23/1.15 setunion__Cong: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setunion__Cong,definition,
% 2.23/1.15 ( setunion__Cong
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( A = B )
% 2.23/1.15 => ( ( setunion @ A )
% 2.23/1.15 = ( setunion @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(omega__Cong_type,type,
% 2.23/1.15 omega__Cong: $o ).
% 2.23/1.15
% 2.23/1.15 thf(omega__Cong,definition,
% 2.23/1.15 ( omega__Cong
% 2.23/1.15 = ( omega = omega ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(exuEu_type,type,
% 2.23/1.15 exuEu: $o ).
% 2.23/1.15
% 2.23/1.15 thf(exuEu,definition,
% 2.23/1.15 ( exuEu
% 2.23/1.15 = ( ! [Xphi: $i > $o] :
% 2.23/1.15 ( ( exu
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.23/1.15 => ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( Xphi @ Xx )
% 2.23/1.15 => ( ( Xphi @ Xy )
% 2.23/1.15 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(descr__Cong_type,type,
% 2.23/1.15 descr__Cong: $o ).
% 2.23/1.15
% 2.23/1.15 thf(descr__Cong,definition,
% 2.23/1.15 ( descr__Cong
% 2.23/1.15 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.23/1.15 ( ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( Xx = Xy )
% 2.23/1.15 => ( ( Xphi @ Xx )
% 2.23/1.15 <=> ( Xpsi @ Xy ) ) )
% 2.23/1.15 => ( ( exu
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.23/1.15 => ( ( exu
% 2.23/1.15 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.23/1.15 => ( ( descr
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.23/1.15 = ( descr
% 2.23/1.15 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(dsetconstr__Cong_type,type,
% 2.23/1.15 dsetconstr__Cong: $o ).
% 2.23/1.15
% 2.23/1.15 thf(dsetconstr__Cong,definition,
% 2.23/1.15 ( dsetconstr__Cong
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( A = B )
% 2.23/1.15 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( ( Xx = Xy )
% 2.23/1.15 => ( ( Xphi @ Xx )
% 2.23/1.15 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.23/1.15 => ( ( dsetconstr @ A
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.23/1.15 = ( dsetconstr @ B
% 2.23/1.15 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(subset_type,type,
% 2.23/1.15 subset: $i > $i > $o ).
% 2.23/1.15
% 2.23/1.15 thf(disjoint_type,type,
% 2.23/1.15 disjoint: $i > $i > $o ).
% 2.23/1.15
% 2.23/1.15 thf(setsmeet_type,type,
% 2.23/1.15 setsmeet: $i > $i > $o ).
% 2.23/1.15
% 2.23/1.15 thf(subsetI1_type,type,
% 2.23/1.15 subsetI1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(subsetI1,definition,
% 2.23/1.15 ( subsetI1
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ Xx @ B ) )
% 2.23/1.15 => ( subset @ A @ B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(eqimpsubset2_type,type,
% 2.23/1.15 eqimpsubset2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(eqimpsubset2,definition,
% 2.23/1.15 ( eqimpsubset2
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( A = B )
% 2.23/1.15 => ( subset @ B @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(eqimpsubset1_type,type,
% 2.23/1.15 eqimpsubset1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(eqimpsubset1,definition,
% 2.23/1.15 ( eqimpsubset1
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( A = B )
% 2.23/1.15 => ( subset @ A @ B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(subsetI2_type,type,
% 2.23/1.15 subsetI2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(subsetI2,definition,
% 2.23/1.15 ( subsetI2
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ Xx @ B ) )
% 2.23/1.15 => ( subset @ A @ B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(emptysetsubset_type,type,
% 2.23/1.15 emptysetsubset: $o ).
% 2.23/1.15
% 2.23/1.15 thf(emptysetsubset,definition,
% 2.23/1.15 ( emptysetsubset
% 2.23/1.15 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(subsetE_type,type,
% 2.23/1.15 subsetE: $o ).
% 2.23/1.15
% 2.23/1.15 thf(subsetE,definition,
% 2.23/1.15 ( subsetE
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( subset @ A @ B )
% 2.23/1.15 => ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(subsetE2_type,type,
% 2.23/1.15 subsetE2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(subsetE2,definition,
% 2.23/1.15 ( subsetE2
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( subset @ A @ B )
% 2.23/1.15 => ( ~ ( in @ Xx @ B )
% 2.23/1.15 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(notsubsetI_type,type,
% 2.23/1.15 notsubsetI: $o ).
% 2.23/1.15
% 2.23/1.15 thf(notsubsetI,definition,
% 2.23/1.15 ( notsubsetI
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( ~ ( in @ Xx @ B )
% 2.23/1.15 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(notequalI1_type,type,
% 2.23/1.15 notequalI1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(notequalI1,definition,
% 2.23/1.15 ( notequalI1
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ~ ( subset @ A @ B )
% 2.23/1.15 => ( A != B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(notequalI2_type,type,
% 2.23/1.15 notequalI2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(notequalI2,definition,
% 2.23/1.15 ( notequalI2
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( ~ ( in @ Xx @ B )
% 2.23/1.15 => ( A != B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(subsetRefl_type,type,
% 2.23/1.15 subsetRefl: $o ).
% 2.23/1.15
% 2.23/1.15 thf(subsetRefl,definition,
% 2.23/1.15 ( subsetRefl
% 2.23/1.15 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(subsetTrans_type,type,
% 2.23/1.15 subsetTrans: $o ).
% 2.23/1.15
% 2.23/1.15 thf(subsetTrans,definition,
% 2.23/1.15 ( subsetTrans
% 2.23/1.15 = ( ! [A: $i,B: $i,C: $i] :
% 2.23/1.15 ( ( subset @ A @ B )
% 2.23/1.15 => ( ( subset @ B @ C )
% 2.23/1.15 => ( subset @ A @ C ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoinSub_type,type,
% 2.23/1.15 setadjoinSub: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoinSub,definition,
% 2.23/1.15 ( setadjoinSub
% 2.23/1.15 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoinSub2_type,type,
% 2.23/1.15 setadjoinSub2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setadjoinSub2,definition,
% 2.23/1.15 ( setadjoinSub2
% 2.23/1.15 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.23/1.15 ( ( subset @ A @ B )
% 2.23/1.15 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(subset2powerset_type,type,
% 2.23/1.15 subset2powerset: $o ).
% 2.23/1.15
% 2.23/1.15 thf(subset2powerset,definition,
% 2.23/1.15 ( subset2powerset
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( subset @ A @ B )
% 2.23/1.15 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setextsub_type,type,
% 2.23/1.15 setextsub: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setextsub,definition,
% 2.23/1.15 ( setextsub
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( subset @ A @ B )
% 2.23/1.15 => ( ( subset @ B @ A )
% 2.23/1.15 => ( A = B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(subsetemptysetimpeq_type,type,
% 2.23/1.15 subsetemptysetimpeq: $o ).
% 2.23/1.15
% 2.23/1.15 thf(subsetemptysetimpeq,definition,
% 2.23/1.15 ( subsetemptysetimpeq
% 2.23/1.15 = ( ! [A: $i] :
% 2.23/1.15 ( ( subset @ A @ emptyset )
% 2.23/1.15 => ( A = emptyset ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(powersetI1_type,type,
% 2.23/1.15 powersetI1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(powersetI1,definition,
% 2.23/1.15 ( powersetI1
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( subset @ B @ A )
% 2.23/1.15 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(powersetE1_type,type,
% 2.23/1.15 powersetE1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(powersetE1,definition,
% 2.23/1.15 ( powersetE1
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( in @ B @ ( powerset @ A ) )
% 2.23/1.15 => ( subset @ B @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(inPowerset_type,type,
% 2.23/1.15 inPowerset: $o ).
% 2.23/1.15
% 2.23/1.15 thf(inPowerset,definition,
% 2.23/1.15 ( inPowerset
% 2.23/1.15 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(powersetsubset_type,type,
% 2.23/1.15 powersetsubset: $o ).
% 2.23/1.15
% 2.23/1.15 thf(powersetsubset,definition,
% 2.23/1.15 ( powersetsubset
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( subset @ A @ B )
% 2.23/1.15 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(sepInPowerset_type,type,
% 2.23/1.15 sepInPowerset: $o ).
% 2.23/1.15
% 2.23/1.15 thf(sepInPowerset,definition,
% 2.23/1.15 ( sepInPowerset
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o] :
% 2.23/1.15 ( in
% 2.23/1.15 @ ( dsetconstr @ A
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.23/1.15 @ ( powerset @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(sepSubset_type,type,
% 2.23/1.15 sepSubset: $o ).
% 2.23/1.15
% 2.23/1.15 thf(sepSubset,definition,
% 2.23/1.15 ( sepSubset
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o] :
% 2.23/1.15 ( subset
% 2.23/1.15 @ ( dsetconstr @ A
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.23/1.15 @ A ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(binunion_type,type,
% 2.23/1.15 binunion: $i > $i > $i ).
% 2.23/1.15
% 2.23/1.15 thf(binunionIL_type,type,
% 2.23/1.15 binunionIL: $o ).
% 2.23/1.15
% 2.23/1.15 thf(binunionIL,definition,
% 2.23/1.15 ( binunionIL
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(upairset2IR_type,type,
% 2.23/1.15 upairset2IR: $o ).
% 2.23/1.15
% 2.23/1.15 thf(upairset2IR,definition,
% 2.23/1.15 ( upairset2IR
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(binunionIR_type,type,
% 2.23/1.15 binunionIR: $o ).
% 2.23/1.15
% 2.23/1.15 thf(binunionIR,definition,
% 2.23/1.15 ( binunionIR
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ B )
% 2.23/1.15 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(binunionEcases_type,type,
% 2.23/1.15 binunionEcases: $o ).
% 2.23/1.15
% 2.23/1.15 thf(binunionEcases,definition,
% 2.23/1.15 ( binunionEcases
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.23/1.15 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.23/1.15 => ( ( ( in @ Xx @ A )
% 2.23/1.15 => Xphi )
% 2.23/1.15 => ( ( ( in @ Xx @ B )
% 2.23/1.15 => Xphi )
% 2.23/1.15 => Xphi ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(binunionE_type,type,
% 2.23/1.15 binunionE: $o ).
% 2.23/1.15
% 2.23/1.15 thf(binunionE,definition,
% 2.23/1.15 ( binunionE
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.23/1.15 => ( ( in @ Xx @ A )
% 2.23/1.15 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(binunionLsub_type,type,
% 2.23/1.15 binunionLsub: $o ).
% 2.23/1.15
% 2.23/1.15 thf(binunionLsub,definition,
% 2.23/1.15 ( binunionLsub
% 2.23/1.15 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(binunionRsub_type,type,
% 2.23/1.15 binunionRsub: $o ).
% 2.23/1.15
% 2.23/1.15 thf(binunionRsub,definition,
% 2.23/1.15 ( binunionRsub
% 2.23/1.15 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(binintersect_type,type,
% 2.23/1.15 binintersect: $i > $i > $i ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectI_type,type,
% 2.23/1.15 binintersectI: $o ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectI,definition,
% 2.23/1.15 ( binintersectI
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( ( in @ Xx @ B )
% 2.23/1.15 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectSubset5_type,type,
% 2.23/1.15 binintersectSubset5: $o ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectSubset5,definition,
% 2.23/1.15 ( binintersectSubset5
% 2.23/1.15 = ( ! [A: $i,B: $i,C: $i] :
% 2.23/1.15 ( ( subset @ C @ A )
% 2.23/1.15 => ( ( subset @ C @ B )
% 2.23/1.15 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectEL_type,type,
% 2.23/1.15 binintersectEL: $o ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectEL,definition,
% 2.23/1.15 ( binintersectEL
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.23/1.15 => ( in @ Xx @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectLsub_type,type,
% 2.23/1.15 binintersectLsub: $o ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectLsub,definition,
% 2.23/1.15 ( binintersectLsub
% 2.23/1.15 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectSubset2_type,type,
% 2.23/1.15 binintersectSubset2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectSubset2,definition,
% 2.23/1.15 ( binintersectSubset2
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( subset @ A @ B )
% 2.23/1.15 => ( ( binintersect @ A @ B )
% 2.23/1.15 = A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectSubset3_type,type,
% 2.23/1.15 binintersectSubset3: $o ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectSubset3,definition,
% 2.23/1.15 ( binintersectSubset3
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( ( binintersect @ A @ B )
% 2.23/1.15 = B )
% 2.23/1.15 => ( subset @ B @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectER_type,type,
% 2.23/1.15 binintersectER: $o ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectER,definition,
% 2.23/1.15 ( binintersectER
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.23/1.15 => ( in @ Xx @ B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(disjointsetsI1_type,type,
% 2.23/1.15 disjointsetsI1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(disjointsetsI1,definition,
% 2.23/1.15 ( disjointsetsI1
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ~ ? [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 & ( in @ Xx @ B ) )
% 2.23/1.15 => ( ( binintersect @ A @ B )
% 2.23/1.15 = emptyset ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectRsub_type,type,
% 2.23/1.15 binintersectRsub: $o ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectRsub,definition,
% 2.23/1.15 ( binintersectRsub
% 2.23/1.15 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectSubset4_type,type,
% 2.23/1.15 binintersectSubset4: $o ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectSubset4,definition,
% 2.23/1.15 ( binintersectSubset4
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( subset @ B @ A )
% 2.23/1.15 => ( ( binintersect @ A @ B )
% 2.23/1.15 = B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectSubset1_type,type,
% 2.23/1.15 binintersectSubset1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(binintersectSubset1,definition,
% 2.23/1.15 ( binintersectSubset1
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( ( binintersect @ A @ B )
% 2.23/1.15 = A )
% 2.23/1.15 => ( subset @ A @ B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(bs114d_type,type,
% 2.23/1.15 bs114d: $o ).
% 2.23/1.15
% 2.23/1.15 thf(bs114d,definition,
% 2.23/1.15 ( bs114d
% 2.23/1.15 = ( ! [A: $i,B: $i,C: $i] :
% 2.23/1.15 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.23/1.15 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(regular_type,type,
% 2.23/1.15 regular: $i > $o ).
% 2.23/1.15
% 2.23/1.15 thf(setminus_type,type,
% 2.23/1.15 setminus: $i > $i > $i ).
% 2.23/1.15
% 2.23/1.15 thf(setminusI_type,type,
% 2.23/1.15 setminusI: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setminusI,definition,
% 2.23/1.15 ( setminusI
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( ~ ( in @ Xx @ B )
% 2.23/1.15 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setminusEL_type,type,
% 2.23/1.15 setminusEL: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setminusEL,definition,
% 2.23/1.15 ( setminusEL
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.23/1.15 => ( in @ Xx @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setminusER_type,type,
% 2.23/1.15 setminusER: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setminusER,definition,
% 2.23/1.15 ( setminusER
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.23/1.15 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setminusSubset2_type,type,
% 2.23/1.15 setminusSubset2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setminusSubset2,definition,
% 2.23/1.15 ( setminusSubset2
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( subset @ A @ B )
% 2.23/1.15 => ( ( setminus @ A @ B )
% 2.23/1.15 = emptyset ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setminusERneg_type,type,
% 2.23/1.15 setminusERneg: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setminusERneg,definition,
% 2.23/1.15 ( setminusERneg
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.23/1.15 => ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setminusELneg_type,type,
% 2.23/1.15 setminusELneg: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setminusELneg,definition,
% 2.23/1.15 ( setminusELneg
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.23/1.15 => ( ~ ( in @ Xx @ B )
% 2.23/1.15 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setminusILneg_type,type,
% 2.23/1.15 setminusILneg: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setminusILneg,definition,
% 2.23/1.15 ( setminusILneg
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ~ ( in @ Xx @ A )
% 2.23/1.15 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setminusIRneg_type,type,
% 2.23/1.15 setminusIRneg: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setminusIRneg,definition,
% 2.23/1.15 ( setminusIRneg
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ B )
% 2.23/1.15 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setminusLsub_type,type,
% 2.23/1.15 setminusLsub: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setminusLsub,definition,
% 2.23/1.15 ( setminusLsub
% 2.23/1.15 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setminusSubset1_type,type,
% 2.23/1.15 setminusSubset1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setminusSubset1,definition,
% 2.23/1.15 ( setminusSubset1
% 2.23/1.15 = ( ! [A: $i,B: $i] :
% 2.23/1.15 ( ( ( setminus @ A @ B )
% 2.23/1.15 = emptyset )
% 2.23/1.15 => ( subset @ A @ B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(symdiff_type,type,
% 2.23/1.15 symdiff: $i > $i > $i ).
% 2.23/1.15
% 2.23/1.15 thf(symdiffE_type,type,
% 2.23/1.15 symdiffE: $o ).
% 2.23/1.15
% 2.23/1.15 thf(symdiffE,definition,
% 2.23/1.15 ( symdiffE
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.23/1.15 => ! [Xphi: $o] :
% 2.23/1.15 ( ( ( in @ Xx @ A )
% 2.23/1.15 => ( ~ ( in @ Xx @ B )
% 2.23/1.15 => Xphi ) )
% 2.23/1.15 => ( ( ~ ( in @ Xx @ A )
% 2.23/1.15 => ( ( in @ Xx @ B )
% 2.23/1.15 => Xphi ) )
% 2.23/1.15 => Xphi ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(symdiffI1_type,type,
% 2.23/1.15 symdiffI1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(symdiffI1,definition,
% 2.23/1.15 ( symdiffI1
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( ~ ( in @ Xx @ B )
% 2.23/1.15 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(symdiffI2_type,type,
% 2.23/1.15 symdiffI2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(symdiffI2,definition,
% 2.23/1.15 ( symdiffI2
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ~ ( in @ Xx @ A )
% 2.23/1.15 => ( ( in @ Xx @ B )
% 2.23/1.15 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(symdiffIneg1_type,type,
% 2.23/1.15 symdiffIneg1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(symdiffIneg1,definition,
% 2.23/1.15 ( symdiffIneg1
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( ( in @ Xx @ B )
% 2.23/1.15 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(symdiffIneg2_type,type,
% 2.23/1.15 symdiffIneg2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(symdiffIneg2,definition,
% 2.23/1.15 ( symdiffIneg2
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ~ ( in @ Xx @ A )
% 2.23/1.15 => ( ~ ( in @ Xx @ B )
% 2.23/1.15 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(iskpair_type,type,
% 2.23/1.15 iskpair: $i > $o ).
% 2.23/1.15
% 2.23/1.15 thf(secondinupair_type,type,
% 2.23/1.15 secondinupair: $o ).
% 2.23/1.15
% 2.23/1.15 thf(secondinupair,definition,
% 2.23/1.15 ( secondinupair
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairIL_type,type,
% 2.23/1.15 setukpairIL: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairIL,definition,
% 2.23/1.15 ( setukpairIL
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairIR_type,type,
% 2.23/1.15 setukpairIR: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairIR,definition,
% 2.23/1.15 ( setukpairIR
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(kpairiskpair_type,type,
% 2.23/1.15 kpairiskpair: $o ).
% 2.23/1.15
% 2.23/1.15 thf(kpairiskpair,definition,
% 2.23/1.15 ( kpairiskpair
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(kpair_type,type,
% 2.23/1.15 kpair: $i > $i > $i ).
% 2.23/1.15
% 2.23/1.15 thf(kpairp_type,type,
% 2.23/1.15 kpairp: $o ).
% 2.23/1.15
% 2.23/1.15 thf(kpairp,definition,
% 2.23/1.15 ( kpairp
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(cartprod_type,type,
% 2.23/1.15 cartprod: $i > $i > $i ).
% 2.23/1.15
% 2.23/1.15 thf(singletonsubset_type,type,
% 2.23/1.15 singletonsubset: $o ).
% 2.23/1.15
% 2.23/1.15 thf(singletonsubset,definition,
% 2.23/1.15 ( singletonsubset
% 2.23/1.15 = ( ! [A: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(singletoninpowerset_type,type,
% 2.23/1.15 singletoninpowerset: $o ).
% 2.23/1.15
% 2.23/1.15 thf(singletoninpowerset,definition,
% 2.23/1.15 ( singletoninpowerset
% 2.23/1.15 = ( ! [A: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(singletoninpowunion_type,type,
% 2.23/1.15 singletoninpowunion: $o ).
% 2.23/1.15
% 2.23/1.15 thf(singletoninpowunion,definition,
% 2.23/1.15 ( singletoninpowunion
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(upairset2E_type,type,
% 2.23/1.15 upairset2E: $o ).
% 2.23/1.15
% 2.23/1.15 thf(upairset2E,definition,
% 2.23/1.15 ( upairset2E
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.23/1.15 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.23/1.15 => ( ( Xz = Xx )
% 2.23/1.15 | ( Xz = Xy ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(upairsubunion_type,type,
% 2.23/1.15 upairsubunion: $o ).
% 2.23/1.15
% 2.23/1.15 thf(upairsubunion,definition,
% 2.23/1.15 ( upairsubunion
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(upairinpowunion_type,type,
% 2.23/1.15 upairinpowunion: $o ).
% 2.23/1.15
% 2.23/1.15 thf(upairinpowunion,definition,
% 2.23/1.15 ( upairinpowunion
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(ubforcartprodlem1_type,type,
% 2.23/1.15 ubforcartprodlem1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(ubforcartprodlem1,definition,
% 2.23/1.15 ( ubforcartprodlem1
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(ubforcartprodlem2_type,type,
% 2.23/1.15 ubforcartprodlem2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(ubforcartprodlem2,definition,
% 2.23/1.15 ( ubforcartprodlem2
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(ubforcartprodlem3_type,type,
% 2.23/1.15 ubforcartprodlem3: $o ).
% 2.23/1.15
% 2.23/1.15 thf(ubforcartprodlem3,definition,
% 2.23/1.15 ( ubforcartprodlem3
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodpairin_type,type,
% 2.23/1.15 cartprodpairin: $o ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodpairin,definition,
% 2.23/1.15 ( cartprodpairin
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodmempair1_type,type,
% 2.23/1.15 cartprodmempair1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodmempair1,definition,
% 2.23/1.15 ( cartprodmempair1
% 2.23/1.15 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.23/1.15 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.23/1.15 => ? [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 & ? [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 & ( Xu
% 2.23/1.15 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodmempair_type,type,
% 2.23/1.15 cartprodmempair: $o ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodmempair,definition,
% 2.23/1.15 ( cartprodmempair
% 2.23/1.15 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.23/1.15 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.23/1.15 => ( iskpair @ Xu ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setunionE2_type,type,
% 2.23/1.15 setunionE2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setunionE2,definition,
% 2.23/1.15 ( setunionE2
% 2.23/1.15 = ( ! [A: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ ( setunion @ A ) )
% 2.23/1.15 => ? [X: $i] :
% 2.23/1.15 ( ( in @ X @ A )
% 2.23/1.15 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setunionsingleton1_type,type,
% 2.23/1.15 setunionsingleton1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setunionsingleton1,definition,
% 2.23/1.15 ( setunionsingleton1
% 2.23/1.15 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setunionsingleton2_type,type,
% 2.23/1.15 setunionsingleton2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setunionsingleton2,definition,
% 2.23/1.15 ( setunionsingleton2
% 2.23/1.15 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setunionsingleton_type,type,
% 2.23/1.15 setunionsingleton: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setunionsingleton,definition,
% 2.23/1.15 ( setunionsingleton
% 2.23/1.15 = ( ! [Xx: $i] :
% 2.23/1.15 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.23/1.15 = Xx ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(singleton_type,type,
% 2.23/1.15 singleton: $i > $o ).
% 2.23/1.15
% 2.23/1.15 thf(singletonprop_type,type,
% 2.23/1.15 singletonprop: $o ).
% 2.23/1.15
% 2.23/1.15 thf(singletonprop,definition,
% 2.23/1.15 ( singletonprop
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ A )
% 2.23/1.15 => ( ( Xphi @ Xx )
% 2.23/1.15 => ( ( Xphi @ Xy )
% 2.23/1.15 => ( Xx = Xy ) ) ) ) )
% 2.23/1.15 => ( ? [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 & ( Xphi @ Xx ) )
% 2.23/1.15 => ( singleton
% 2.23/1.15 @ ( dsetconstr @ A
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(ex1_type,type,
% 2.23/1.15 ex1: $i > ( $i > $o ) > $o ).
% 2.23/1.15
% 2.23/1.15 thf(ex1,definition,
% 2.23/1.15 ( ex1
% 2.23/1.15 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.23/1.15 ( singleton
% 2.23/1.15 @ ( dsetconstr @ A
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(ex1E1_type,type,
% 2.23/1.15 ex1E1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(ex1E1,definition,
% 2.23/1.15 ( ex1E1
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o] :
% 2.23/1.15 ( ( ex1 @ A
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.23/1.15 => ? [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(ex1I_type,type,
% 2.23/1.15 ex1I: $o ).
% 2.23/1.15
% 2.23/1.15 thf(ex1I,definition,
% 2.23/1.15 ( ex1I
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( ( Xphi @ Xx )
% 2.23/1.15 => ( ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ A )
% 2.23/1.15 => ( ( Xphi @ Xy )
% 2.23/1.15 => ( Xy = Xx ) ) )
% 2.23/1.15 => ( ex1 @ A
% 2.23/1.15 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(ex1I2_type,type,
% 2.23/1.15 ex1I2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(ex1I2,definition,
% 2.23/1.15 ( ex1I2
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ A )
% 2.23/1.15 => ( ( Xphi @ Xx )
% 2.23/1.15 => ( ( Xphi @ Xy )
% 2.23/1.15 => ( Xx = Xy ) ) ) ) )
% 2.23/1.15 => ( ? [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 & ( Xphi @ Xx ) )
% 2.23/1.15 => ( ex1 @ A
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(singletonsuniq_type,type,
% 2.23/1.15 singletonsuniq: $o ).
% 2.23/1.15
% 2.23/1.15 thf(singletonsuniq,definition,
% 2.23/1.15 ( singletonsuniq
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( ( setadjoin @ Xx @ emptyset )
% 2.23/1.15 = ( setadjoin @ Xy @ emptyset ) )
% 2.23/1.15 => ( Xx = Xy ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(atmost1p_type,type,
% 2.23/1.15 atmost1p: $i > $o ).
% 2.23/1.15
% 2.23/1.15 thf(atleast2p_type,type,
% 2.23/1.15 atleast2p: $i > $o ).
% 2.23/1.15
% 2.23/1.15 thf(atmost2p_type,type,
% 2.23/1.15 atmost2p: $i > $o ).
% 2.23/1.15
% 2.23/1.15 thf(upairsetp_type,type,
% 2.23/1.15 upairsetp: $i > $o ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairinjL1_type,type,
% 2.23/1.15 setukpairinjL1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairinjL1,definition,
% 2.23/1.15 ( setukpairinjL1
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.23/1.15 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.23/1.15 => ( Xx = Xz ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(kfstsingleton_type,type,
% 2.23/1.15 kfstsingleton: $o ).
% 2.23/1.15
% 2.23/1.15 thf(kfstsingleton,definition,
% 2.23/1.15 ( kfstsingleton
% 2.23/1.15 = ( ! [Xu: $i] :
% 2.23/1.15 ( ( iskpair @ Xu )
% 2.23/1.15 => ( singleton
% 2.23/1.15 @ ( dsetconstr @ ( setunion @ Xu )
% 2.23/1.15 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(theprop_type,type,
% 2.23/1.15 theprop: $o ).
% 2.23/1.15
% 2.23/1.15 thf(theprop,definition,
% 2.23/1.15 ( theprop
% 2.23/1.15 = ( ! [X: $i] :
% 2.23/1.15 ( ( singleton @ X )
% 2.23/1.15 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(kfst_type,type,
% 2.23/1.15 kfst: $i > $i ).
% 2.23/1.15
% 2.23/1.15 thf(kfstpairEq_type,type,
% 2.23/1.15 kfstpairEq: $o ).
% 2.23/1.15
% 2.23/1.15 thf(kfstpairEq,definition,
% 2.23/1.15 ( kfstpairEq
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.23/1.15 = Xx ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodfstin_type,type,
% 2.23/1.15 cartprodfstin: $o ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodfstin,definition,
% 2.23/1.15 ( cartprodfstin
% 2.23/1.15 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.23/1.15 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.23/1.15 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairinjL2_type,type,
% 2.23/1.15 setukpairinjL2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairinjL2,definition,
% 2.23/1.15 ( setukpairinjL2
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.23/1.15 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.23/1.15 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.23/1.15 => ( Xx = Xz ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairinjL_type,type,
% 2.23/1.15 setukpairinjL: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairinjL,definition,
% 2.23/1.15 ( setukpairinjL
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.23/1.15 ( ( ( kpair @ Xx @ Xy )
% 2.23/1.15 = ( kpair @ Xz @ Xu ) )
% 2.23/1.15 => ( Xx = Xz ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairinjR11_type,type,
% 2.23/1.15 setukpairinjR11: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairinjR11,definition,
% 2.23/1.15 ( setukpairinjR11
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( Xx = Xy )
% 2.23/1.15 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.23/1.15 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairinjR12_type,type,
% 2.23/1.15 setukpairinjR12: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairinjR12,definition,
% 2.23/1.15 ( setukpairinjR12
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( Xx = Xy )
% 2.23/1.15 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.23/1.15 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairinjR1_type,type,
% 2.23/1.15 setukpairinjR1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairinjR1,definition,
% 2.23/1.15 ( setukpairinjR1
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.23/1.15 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.23/1.15 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.23/1.15 => ( ( Xz = Xu )
% 2.23/1.15 => ( Xy = Xu ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(upairequniteq_type,type,
% 2.23/1.15 upairequniteq: $o ).
% 2.23/1.15
% 2.23/1.15 thf(upairequniteq,definition,
% 2.23/1.15 ( upairequniteq
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.23/1.15 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.23/1.15 = ( setadjoin @ Xz @ emptyset ) )
% 2.23/1.15 => ( Xx = Xy ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairinjR2_type,type,
% 2.23/1.15 setukpairinjR2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairinjR2,definition,
% 2.23/1.15 ( setukpairinjR2
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.23/1.15 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.23/1.15 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.23/1.15 => ( Xy = Xu ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairinjR_type,type,
% 2.23/1.15 setukpairinjR: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setukpairinjR,definition,
% 2.23/1.15 ( setukpairinjR
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.23/1.15 ( ( ( kpair @ Xx @ Xy )
% 2.23/1.15 = ( kpair @ Xz @ Xu ) )
% 2.23/1.15 => ( Xy = Xu ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(ksndsingleton_type,type,
% 2.23/1.15 ksndsingleton: $o ).
% 2.23/1.15
% 2.23/1.15 thf(ksndsingleton,definition,
% 2.23/1.15 ( ksndsingleton
% 2.23/1.15 = ( ! [Xu: $i] :
% 2.23/1.15 ( ( iskpair @ Xu )
% 2.23/1.15 => ( singleton
% 2.23/1.15 @ ( dsetconstr @ ( setunion @ Xu )
% 2.23/1.15 @ ^ [Xx: $i] :
% 2.23/1.15 ( Xu
% 2.23/1.15 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(ksnd_type,type,
% 2.23/1.15 ksnd: $i > $i ).
% 2.23/1.15
% 2.23/1.15 thf(ksndpairEq_type,type,
% 2.23/1.15 ksndpairEq: $o ).
% 2.23/1.15
% 2.23/1.15 thf(ksndpairEq,definition,
% 2.23/1.15 ( ksndpairEq
% 2.23/1.15 = ( ! [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.23/1.15 = Xy ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(kpairsurjEq_type,type,
% 2.23/1.15 kpairsurjEq: $o ).
% 2.23/1.15
% 2.23/1.15 thf(kpairsurjEq,definition,
% 2.23/1.15 ( kpairsurjEq
% 2.23/1.15 = ( ! [Xu: $i] :
% 2.23/1.15 ( ( iskpair @ Xu )
% 2.23/1.15 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.23/1.15 = Xu ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodsndin_type,type,
% 2.23/1.15 cartprodsndin: $o ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodsndin,definition,
% 2.23/1.15 ( cartprodsndin
% 2.23/1.15 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.23/1.15 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.23/1.15 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodpairmemEL_type,type,
% 2.23/1.15 cartprodpairmemEL: $o ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodpairmemEL,definition,
% 2.23/1.15 ( cartprodpairmemEL
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.23/1.15 => ( in @ Xx @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodpairmemER_type,type,
% 2.23/1.15 cartprodpairmemER: $o ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodpairmemER,definition,
% 2.23/1.15 ( cartprodpairmemER
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.23/1.15 => ( in @ Xy @ B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodmempaircEq_type,type,
% 2.23/1.15 cartprodmempaircEq: $o ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodmempaircEq,definition,
% 2.23/1.15 ( cartprodmempaircEq
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( ( kpair @ Xx @ Xy )
% 2.23/1.15 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodfstpairEq_type,type,
% 2.23/1.15 cartprodfstpairEq: $o ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodfstpairEq,definition,
% 2.23/1.15 ( cartprodfstpairEq
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.23/1.15 = Xx ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodsndpairEq_type,type,
% 2.23/1.15 cartprodsndpairEq: $o ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodsndpairEq,definition,
% 2.23/1.15 ( cartprodsndpairEq
% 2.23/1.15 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.23/1.15 = Xy ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodpairsurjEq_type,type,
% 2.23/1.15 cartprodpairsurjEq: $o ).
% 2.23/1.15
% 2.23/1.15 thf(cartprodpairsurjEq,definition,
% 2.23/1.15 ( cartprodpairsurjEq
% 2.23/1.15 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.23/1.15 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.23/1.15 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.23/1.15 = Xu ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(breln_type,type,
% 2.23/1.15 breln: $i > $i > $i > $o ).
% 2.23/1.15
% 2.23/1.15 thf(breln,definition,
% 2.23/1.15 ( breln
% 2.23/1.15 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(dpsetconstr_type,type,
% 2.23/1.15 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.23/1.15
% 2.23/1.15 thf(dpsetconstrI_type,type,
% 2.23/1.15 dpsetconstrI: $o ).
% 2.23/1.15
% 2.23/1.15 thf(dpsetconstrI,definition,
% 2.23/1.15 ( dpsetconstrI
% 2.23/1.15 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( ( Xphi @ Xx @ Xy )
% 2.23/1.15 => ( in @ ( kpair @ Xx @ Xy )
% 2.23/1.15 @ ( dpsetconstr @ A @ B
% 2.23/1.15 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(dpsetconstrSub_type,type,
% 2.23/1.15 dpsetconstrSub: $o ).
% 2.23/1.15
% 2.23/1.15 thf(dpsetconstrSub,definition,
% 2.23/1.15 ( dpsetconstrSub
% 2.23/1.15 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.23/1.15 ( subset
% 2.23/1.15 @ ( dpsetconstr @ A @ B
% 2.23/1.15 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.23/1.15 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(setOfPairsIsBReln_type,type,
% 2.23/1.15 setOfPairsIsBReln: $o ).
% 2.23/1.15
% 2.23/1.15 thf(setOfPairsIsBReln,definition,
% 2.23/1.15 ( setOfPairsIsBReln
% 2.23/1.15 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.23/1.15 ( breln @ A @ B
% 2.23/1.15 @ ( dpsetconstr @ A @ B
% 2.23/1.15 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(dpsetconstrERa_type,type,
% 2.23/1.15 dpsetconstrERa: $o ).
% 2.23/1.15
% 2.23/1.15 thf(dpsetconstrERa,definition,
% 2.23/1.15 ( dpsetconstrERa
% 2.23/1.15 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.23/1.15 @ ( dpsetconstr @ A @ B
% 2.23/1.15 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.23/1.15 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(dpsetconstrEL1_type,type,
% 2.23/1.15 dpsetconstrEL1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(dpsetconstrEL1,definition,
% 2.23/1.15 ( dpsetconstrEL1
% 2.23/1.15 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( in @ ( kpair @ Xx @ Xy )
% 2.23/1.15 @ ( dpsetconstr @ A @ B
% 2.23/1.15 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.23/1.15 => ( in @ Xx @ A ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(dpsetconstrEL2_type,type,
% 2.23/1.15 dpsetconstrEL2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(dpsetconstrEL2,definition,
% 2.23/1.15 ( dpsetconstrEL2
% 2.23/1.15 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( in @ ( kpair @ Xx @ Xy )
% 2.23/1.15 @ ( dpsetconstr @ A @ B
% 2.23/1.15 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.23/1.15 => ( in @ Xy @ B ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(dpsetconstrER_type,type,
% 2.23/1.15 dpsetconstrER: $o ).
% 2.23/1.15
% 2.23/1.15 thf(dpsetconstrER,definition,
% 2.23/1.15 ( dpsetconstrER
% 2.23/1.15 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( in @ ( kpair @ Xx @ Xy )
% 2.23/1.15 @ ( dpsetconstr @ A @ B
% 2.23/1.15 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.23/1.15 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(func_type,type,
% 2.23/1.15 func: $i > $i > $i > $o ).
% 2.23/1.15
% 2.23/1.15 thf(funcSet_type,type,
% 2.23/1.15 funcSet: $i > $i > $i ).
% 2.23/1.15
% 2.23/1.15 thf(funcImageSingleton_type,type,
% 2.23/1.15 funcImageSingleton: $o ).
% 2.23/1.15
% 2.23/1.15 thf(funcImageSingleton,definition,
% 2.23/1.15 ( funcImageSingleton
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.15 ( ( func @ A @ B @ Xf )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( singleton
% 2.23/1.15 @ ( dsetconstr @ B
% 2.23/1.15 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(apProp_type,type,
% 2.23/1.15 apProp: $o ).
% 2.23/1.15
% 2.23/1.15 thf(apProp,definition,
% 2.23/1.15 ( apProp
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.15 ( ( func @ A @ B @ Xf )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( in
% 2.23/1.15 @ ( setunion
% 2.23/1.15 @ ( dsetconstr @ B
% 2.23/1.15 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.23/1.15 @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(ap_type,type,
% 2.23/1.15 ap: $i > $i > $i > $i > $i ).
% 2.23/1.15
% 2.23/1.15 thf(app_type,type,
% 2.23/1.15 app: $o ).
% 2.23/1.15
% 2.23/1.15 thf(app,definition,
% 2.23/1.15 ( app
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.15 ( ( func @ A @ B @ Xf )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(infuncsetfunc_type,type,
% 2.23/1.15 infuncsetfunc: $o ).
% 2.23/1.15
% 2.23/1.15 thf(infuncsetfunc,definition,
% 2.23/1.15 ( infuncsetfunc
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.15 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.23/1.15 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(ap2p_type,type,
% 2.23/1.15 ap2p: $o ).
% 2.23/1.15
% 2.23/1.15 thf(ap2p,definition,
% 2.23/1.15 ( ap2p
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.15 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(funcinfuncset_type,type,
% 2.23/1.15 funcinfuncset: $o ).
% 2.23/1.15
% 2.23/1.15 thf(funcinfuncset,definition,
% 2.23/1.15 ( funcinfuncset
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.15 ( ( func @ A @ B @ Xf )
% 2.23/1.15 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(lamProp_type,type,
% 2.23/1.15 lamProp: $o ).
% 2.23/1.15
% 2.23/1.15 thf(lamProp,definition,
% 2.23/1.15 ( lamProp
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.23/1.15 => ( func @ A @ B
% 2.23/1.15 @ ( dpsetconstr @ A @ B
% 2.23/1.15 @ ^ [Xx: $i,Xy: $i] :
% 2.23/1.15 ( ( Xf @ Xx )
% 2.23/1.15 = Xy ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(lam_type,type,
% 2.23/1.15 lam: $i > $i > ( $i > $i ) > $i ).
% 2.23/1.15
% 2.23/1.15 thf(lamp_type,type,
% 2.23/1.15 lamp: $o ).
% 2.23/1.15
% 2.23/1.15 thf(lamp,definition,
% 2.23/1.15 ( lamp
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.23/1.15 => ( func @ A @ B
% 2.23/1.15 @ ( lam @ A @ B
% 2.23/1.15 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(lam2p_type,type,
% 2.23/1.15 lam2p: $o ).
% 2.23/1.15
% 2.23/1.15 thf(lam2p,definition,
% 2.23/1.15 ( lam2p
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.23/1.15 => ( in
% 2.23/1.15 @ ( lam @ A @ B
% 2.23/1.15 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.23/1.15 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(brelnall1_type,type,
% 2.23/1.15 brelnall1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(brelnall1,definition,
% 2.23/1.15 ( brelnall1
% 2.23/1.15 = ( ! [A: $i,B: $i,R: $i] :
% 2.23/1.15 ( ( breln @ A @ B @ R )
% 2.23/1.15 => ! [Xphi: $i > $o] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.23/1.15 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ R )
% 2.23/1.15 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(brelnall2_type,type,
% 2.23/1.15 brelnall2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(brelnall2,definition,
% 2.23/1.15 ( brelnall2
% 2.23/1.15 = ( ! [A: $i,B: $i,R: $i] :
% 2.23/1.15 ( ( breln @ A @ B @ R )
% 2.23/1.15 => ! [Xphi: $i > $o] :
% 2.23/1.15 ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.23/1.15 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ R )
% 2.23/1.15 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(ex1E2_type,type,
% 2.23/1.15 ex1E2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(ex1E2,definition,
% 2.23/1.15 ( ex1E2
% 2.23/1.15 = ( ! [A: $i,Xphi: $i > $o] :
% 2.23/1.15 ( ( ex1 @ A
% 2.23/1.15 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ A )
% 2.23/1.15 => ( ( Xphi @ Xx )
% 2.23/1.15 => ( ( Xphi @ Xy )
% 2.23/1.15 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(funcGraphProp1_type,type,
% 2.23/1.15 funcGraphProp1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(funcGraphProp1,definition,
% 2.23/1.15 ( funcGraphProp1
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.15 ( ( func @ A @ B @ Xf )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(funcGraphProp3_type,type,
% 2.23/1.15 funcGraphProp3: $o ).
% 2.23/1.15
% 2.23/1.15 thf(funcGraphProp3,definition,
% 2.23/1.15 ( funcGraphProp3
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.15 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(funcGraphProp2_type,type,
% 2.23/1.15 funcGraphProp2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(funcGraphProp2,definition,
% 2.23/1.15 ( funcGraphProp2
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.15 ( ( func @ A @ B @ Xf )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.23/1.15 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.23/1.15 = Xy ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(funcextLem_type,type,
% 2.23/1.15 funcextLem: $o ).
% 2.23/1.15
% 2.23/1.15 thf(funcextLem,definition,
% 2.23/1.15 ( funcextLem
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.15 ( ( func @ A @ B @ Xf )
% 2.23/1.15 => ! [Xg: $i] :
% 2.23/1.15 ( ( func @ A @ B @ Xg )
% 2.23/1.15 => ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.23/1.15 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.23/1.15 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(funcGraphProp4_type,type,
% 2.23/1.15 funcGraphProp4: $o ).
% 2.23/1.15
% 2.23/1.15 thf(funcGraphProp4,definition,
% 2.23/1.15 ( funcGraphProp4
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.15 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.23/1.15 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.23/1.15 = Xy ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(subbreln_type,type,
% 2.23/1.15 subbreln: $o ).
% 2.23/1.15
% 2.23/1.15 thf(subbreln,definition,
% 2.23/1.15 ( subbreln
% 2.23/1.15 = ( ! [A: $i,B: $i,R: $i] :
% 2.23/1.15 ( ( breln @ A @ B @ R )
% 2.23/1.15 => ! [S: $i] :
% 2.23/1.15 ( ( breln @ A @ B @ S )
% 2.23/1.15 => ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.23/1.15 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.23/1.15 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(eqbreln_type,type,
% 2.23/1.15 eqbreln: $o ).
% 2.23/1.15
% 2.23/1.15 thf(eqbreln,definition,
% 2.23/1.15 ( eqbreln
% 2.23/1.15 = ( ! [A: $i,B: $i,R: $i] :
% 2.23/1.15 ( ( breln @ A @ B @ R )
% 2.23/1.15 => ! [S: $i] :
% 2.23/1.15 ( ( breln @ A @ B @ S )
% 2.23/1.15 => ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.23/1.15 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.23/1.15 => ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ! [Xy: $i] :
% 2.23/1.15 ( ( in @ Xy @ B )
% 2.23/1.15 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.23/1.15 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.23/1.15 => ( R = S ) ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(funcext_type,type,
% 2.23/1.15 funcext: $o ).
% 2.23/1.15
% 2.23/1.15 thf(funcext,definition,
% 2.23/1.15 ( funcext
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.15 ( ( func @ A @ B @ Xf )
% 2.23/1.15 => ! [Xg: $i] :
% 2.23/1.15 ( ( func @ A @ B @ Xg )
% 2.23/1.15 => ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.23/1.15 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.23/1.15 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(funcext2_type,type,
% 2.23/1.15 funcext2: $o ).
% 2.23/1.15
% 2.23/1.15 thf(funcext2,definition,
% 2.23/1.15 ( funcext2
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.15 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.23/1.15 => ! [Xg: $i] :
% 2.23/1.15 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.23/1.15 => ( ! [Xx: $i] :
% 2.23/1.15 ( ( in @ Xx @ A )
% 2.23/1.15 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.23/1.15 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.23/1.15 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.23/1.15
% 2.23/1.15 thf(ap2apEq1_type,type,
% 2.23/1.15 ap2apEq1: $o ).
% 2.23/1.15
% 2.23/1.15 thf(ap2apEq1,definition,
% 2.23/1.15 ( ap2apEq1
% 2.23/1.15 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.15 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.23/1.15 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.23/1.16 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(ap2apEq2_type,type,
% 2.23/1.16 ap2apEq2: $o ).
% 2.23/1.16
% 2.23/1.16 thf(ap2apEq2,definition,
% 2.23/1.16 ( ap2apEq2
% 2.23/1.16 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.16 ( ( func @ A @ B @ Xf )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.23/1.16 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(beta1_type,type,
% 2.23/1.16 beta1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(beta1,definition,
% 2.23/1.16 ( beta1
% 2.23/1.16 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.23/1.16 ( ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( ap @ A @ B
% 2.23/1.16 @ ( lam @ A @ B
% 2.23/1.16 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.23/1.16 @ Xx )
% 2.23/1.16 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(eta1_type,type,
% 2.23/1.16 eta1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(eta1,definition,
% 2.23/1.16 ( eta1
% 2.23/1.16 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.16 ( ( func @ A @ B @ Xf )
% 2.23/1.16 => ( ( lam @ A @ B
% 2.23/1.16 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.23/1.16 = Xf ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(lam2lamEq_type,type,
% 2.23/1.16 lam2lamEq: $o ).
% 2.23/1.16
% 2.23/1.16 thf(lam2lamEq,definition,
% 2.23/1.16 ( lam2lamEq
% 2.23/1.16 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.23/1.16 ( ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.23/1.16 => ( ( lam @ A @ B
% 2.23/1.16 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.23/1.16 = ( lam @ A @ B
% 2.23/1.16 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(beta2_type,type,
% 2.23/1.16 beta2: $o ).
% 2.23/1.16
% 2.23/1.16 thf(beta2,definition,
% 2.23/1.16 ( beta2
% 2.23/1.16 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.23/1.16 ( ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( ap @ A @ B
% 2.23/1.16 @ ( lam @ A @ B
% 2.23/1.16 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.23/1.16 @ Xx )
% 2.23/1.16 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(eta2_type,type,
% 2.23/1.16 eta2: $o ).
% 2.23/1.16
% 2.23/1.16 thf(eta2,definition,
% 2.23/1.16 ( eta2
% 2.23/1.16 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.23/1.16 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.23/1.16 => ( ( lam @ A @ B
% 2.23/1.16 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.23/1.16 = Xf ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(iffalseProp1_type,type,
% 2.23/1.16 iffalseProp1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(iffalseProp1,definition,
% 2.23/1.16 ( iffalseProp1
% 2.23/1.16 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ! [Xy: $i] :
% 2.23/1.16 ( ( in @ Xy @ A )
% 2.23/1.16 => ( ~ Xphi
% 2.23/1.16 => ( in @ Xy
% 2.23/1.16 @ ( dsetconstr @ A
% 2.23/1.16 @ ^ [Xz: $i] :
% 2.23/1.16 ( ( Xphi
% 2.23/1.16 & ( Xz = Xx ) )
% 2.23/1.16 | ( ~ Xphi
% 2.23/1.16 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(iffalseProp2_type,type,
% 2.23/1.16 iffalseProp2: $o ).
% 2.23/1.16
% 2.23/1.16 thf(iffalseProp2,definition,
% 2.23/1.16 ( iffalseProp2
% 2.23/1.16 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ! [Xy: $i] :
% 2.23/1.16 ( ( in @ Xy @ A )
% 2.23/1.16 => ( ~ Xphi
% 2.23/1.16 => ( ( dsetconstr @ A
% 2.23/1.16 @ ^ [Xz: $i] :
% 2.23/1.16 ( ( Xphi
% 2.23/1.16 & ( Xz = Xx ) )
% 2.23/1.16 | ( ~ Xphi
% 2.23/1.16 & ( Xz = Xy ) ) ) )
% 2.23/1.16 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(iftrueProp1_type,type,
% 2.23/1.16 iftrueProp1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(iftrueProp1,definition,
% 2.23/1.16 ( iftrueProp1
% 2.23/1.16 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ! [Xy: $i] :
% 2.23/1.16 ( ( in @ Xy @ A )
% 2.23/1.16 => ( Xphi
% 2.23/1.16 => ( in @ Xx
% 2.23/1.16 @ ( dsetconstr @ A
% 2.23/1.16 @ ^ [Xz: $i] :
% 2.23/1.16 ( ( Xphi
% 2.23/1.16 & ( Xz = Xx ) )
% 2.23/1.16 | ( ~ Xphi
% 2.23/1.16 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(iftrueProp2_type,type,
% 2.23/1.16 iftrueProp2: $o ).
% 2.23/1.16
% 2.23/1.16 thf(iftrueProp2,definition,
% 2.23/1.16 ( iftrueProp2
% 2.23/1.16 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ! [Xy: $i] :
% 2.23/1.16 ( ( in @ Xy @ A )
% 2.23/1.16 => ( Xphi
% 2.23/1.16 => ( ( dsetconstr @ A
% 2.23/1.16 @ ^ [Xz: $i] :
% 2.23/1.16 ( ( Xphi
% 2.23/1.16 & ( Xz = Xx ) )
% 2.23/1.16 | ( ~ Xphi
% 2.23/1.16 & ( Xz = Xy ) ) ) )
% 2.23/1.16 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(ifSingleton_type,type,
% 2.23/1.16 ifSingleton: $o ).
% 2.23/1.16
% 2.23/1.16 thf(ifSingleton,definition,
% 2.23/1.16 ( ifSingleton
% 2.23/1.16 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ! [Xy: $i] :
% 2.23/1.16 ( ( in @ Xy @ A )
% 2.23/1.16 => ( singleton
% 2.23/1.16 @ ( dsetconstr @ A
% 2.23/1.16 @ ^ [Xz: $i] :
% 2.23/1.16 ( ( Xphi
% 2.23/1.16 & ( Xz = Xx ) )
% 2.23/1.16 | ( ~ Xphi
% 2.23/1.16 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(if_type,type,
% 2.23/1.16 if: $i > $o > $i > $i > $i ).
% 2.23/1.16
% 2.23/1.16 thf(ifp_type,type,
% 2.23/1.16 ifp: $o ).
% 2.23/1.16
% 2.23/1.16 thf(ifp,definition,
% 2.23/1.16 ( ifp
% 2.23/1.16 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ! [Xy: $i] :
% 2.23/1.16 ( ( in @ Xy @ A )
% 2.23/1.16 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(theeq_type,type,
% 2.23/1.16 theeq: $o ).
% 2.23/1.16
% 2.23/1.16 thf(theeq,definition,
% 2.23/1.16 ( theeq
% 2.23/1.16 = ( ! [X: $i] :
% 2.23/1.16 ( ( singleton @ X )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ X )
% 2.23/1.16 => ( ( setunion @ X )
% 2.23/1.16 = Xx ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(iftrue_type,type,
% 2.23/1.16 iftrue: $o ).
% 2.23/1.16
% 2.23/1.16 thf(iftrue,definition,
% 2.23/1.16 ( iftrue
% 2.23/1.16 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ! [Xy: $i] :
% 2.23/1.16 ( ( in @ Xy @ A )
% 2.23/1.16 => ( Xphi
% 2.23/1.16 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.23/1.16 = Xx ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(iffalse_type,type,
% 2.23/1.16 iffalse: $o ).
% 2.23/1.16
% 2.23/1.16 thf(iffalse,definition,
% 2.23/1.16 ( iffalse
% 2.23/1.16 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ! [Xy: $i] :
% 2.23/1.16 ( ( in @ Xy @ A )
% 2.23/1.16 => ( ~ Xphi
% 2.23/1.16 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.23/1.16 = Xy ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(iftrueorfalse_type,type,
% 2.23/1.16 iftrueorfalse: $o ).
% 2.23/1.16
% 2.23/1.16 thf(iftrueorfalse,definition,
% 2.23/1.16 ( iftrueorfalse
% 2.23/1.16 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ! [Xy: $i] :
% 2.23/1.16 ( ( in @ Xy @ A )
% 2.23/1.16 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(binintersectT_lem_type,type,
% 2.23/1.16 binintersectT_lem: $o ).
% 2.23/1.16
% 2.23/1.16 thf(binintersectT_lem,definition,
% 2.23/1.16 ( binintersectT_lem
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(binunionT_lem_type,type,
% 2.23/1.16 binunionT_lem: $o ).
% 2.23/1.16
% 2.23/1.16 thf(binunionT_lem,definition,
% 2.23/1.16 ( binunionT_lem
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(powersetT_lem_type,type,
% 2.23/1.16 powersetT_lem: $o ).
% 2.23/1.16
% 2.23/1.16 thf(powersetT_lem,definition,
% 2.23/1.16 ( powersetT_lem
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(setminusT_lem_type,type,
% 2.23/1.16 setminusT_lem: $o ).
% 2.23/1.16
% 2.23/1.16 thf(setminusT_lem,definition,
% 2.23/1.16 ( setminusT_lem
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(complementT_lem_type,type,
% 2.23/1.16 complementT_lem: $o ).
% 2.23/1.16
% 2.23/1.16 thf(complementT_lem,definition,
% 2.23/1.16 ( complementT_lem
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(setextT_type,type,
% 2.23/1.16 setextT: $o ).
% 2.23/1.16
% 2.23/1.16 thf(setextT,definition,
% 2.23/1.16 ( setextT
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ X )
% 2.23/1.16 => ( in @ Xx @ Y ) ) )
% 2.23/1.16 => ( ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ Y )
% 2.23/1.16 => ( in @ Xx @ X ) ) )
% 2.23/1.16 => ( X = Y ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(subsetTI_type,type,
% 2.23/1.16 subsetTI: $o ).
% 2.23/1.16
% 2.23/1.16 thf(subsetTI,definition,
% 2.23/1.16 ( subsetTI
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ X )
% 2.23/1.16 => ( in @ Xx @ Y ) ) )
% 2.23/1.16 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(powersetTI1_type,type,
% 2.23/1.16 powersetTI1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(powersetTI1,definition,
% 2.23/1.16 ( powersetTI1
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ X )
% 2.23/1.16 => ( in @ Xx @ Y ) ) )
% 2.23/1.16 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(powersetTE1_type,type,
% 2.23/1.16 powersetTE1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(powersetTE1,definition,
% 2.23/1.16 ( powersetTE1
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ X @ ( powerset @ Y ) )
% 2.23/1.16 => ( ( in @ Xx @ X )
% 2.23/1.16 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(complementTI1_type,type,
% 2.23/1.16 complementTI1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(complementTI1,definition,
% 2.23/1.16 ( complementTI1
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ X )
% 2.23/1.16 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(complementTE1_type,type,
% 2.23/1.16 complementTE1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(complementTE1,definition,
% 2.23/1.16 ( complementTE1
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.23/1.16 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(binintersectTELcontra_type,type,
% 2.23/1.16 binintersectTELcontra: $o ).
% 2.23/1.16
% 2.23/1.16 thf(binintersectTELcontra,definition,
% 2.23/1.16 ( binintersectTELcontra
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ~ ( in @ Xx @ X )
% 2.23/1.16 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(binintersectTERcontra_type,type,
% 2.23/1.16 binintersectTERcontra: $o ).
% 2.23/1.16
% 2.23/1.16 thf(binintersectTERcontra,definition,
% 2.23/1.16 ( binintersectTERcontra
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ~ ( in @ Xx @ Y )
% 2.23/1.16 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(contrasubsetT_type,type,
% 2.23/1.16 contrasubsetT: $o ).
% 2.23/1.16
% 2.23/1.16 thf(contrasubsetT,definition,
% 2.23/1.16 ( contrasubsetT
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.23/1.16 => ( ( in @ Xx @ Y )
% 2.23/1.16 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(contrasubsetT1_type,type,
% 2.23/1.16 contrasubsetT1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(contrasubsetT1,definition,
% 2.23/1.16 ( contrasubsetT1
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( subset @ X @ Y )
% 2.23/1.16 => ( ~ ( in @ Xx @ Y )
% 2.23/1.16 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(contrasubsetT2_type,type,
% 2.23/1.16 contrasubsetT2: $o ).
% 2.23/1.16
% 2.23/1.16 thf(contrasubsetT2,definition,
% 2.23/1.16 ( contrasubsetT2
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( ( subset @ X @ Y )
% 2.23/1.16 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(contrasubsetT3_type,type,
% 2.23/1.16 contrasubsetT3: $o ).
% 2.23/1.16
% 2.23/1.16 thf(contrasubsetT3,definition,
% 2.23/1.16 ( contrasubsetT3
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.23/1.16 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(doubleComplementI1_type,type,
% 2.23/1.16 doubleComplementI1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(doubleComplementI1,definition,
% 2.23/1.16 ( doubleComplementI1
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ X )
% 2.23/1.16 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(doubleComplementE1_type,type,
% 2.23/1.16 doubleComplementE1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(doubleComplementE1,definition,
% 2.23/1.16 ( doubleComplementE1
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 2.23/1.16 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(doubleComplementSub1_type,type,
% 2.23/1.16 doubleComplementSub1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(doubleComplementSub1,definition,
% 2.23/1.16 ( doubleComplementSub1
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(doubleComplementSub2_type,type,
% 2.23/1.16 doubleComplementSub2: $o ).
% 2.23/1.16
% 2.23/1.16 thf(doubleComplementSub2,definition,
% 2.23/1.16 ( doubleComplementSub2
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(doubleComplementEq_type,type,
% 2.23/1.16 doubleComplementEq: $o ).
% 2.23/1.16
% 2.23/1.16 thf(doubleComplementEq,definition,
% 2.23/1.16 ( doubleComplementEq
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ( X
% 2.23/1.16 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(complementTnotintersectT_type,type,
% 2.23/1.16 complementTnotintersectT: $o ).
% 2.23/1.16
% 2.23/1.16 thf(complementTnotintersectT,definition,
% 2.23/1.16 ( complementTnotintersectT
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.23/1.16 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(complementImpComplementIntersect_type,type,
% 2.23/1.16 complementImpComplementIntersect: $o ).
% 2.23/1.16
% 2.23/1.16 thf(complementImpComplementIntersect,definition,
% 2.23/1.16 ( complementImpComplementIntersect
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.23/1.16 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(complementSubsetComplementIntersect_type,type,
% 2.23/1.16 complementSubsetComplementIntersect: $o ).
% 2.23/1.16
% 2.23/1.16 thf(complementSubsetComplementIntersect,definition,
% 2.23/1.16 ( complementSubsetComplementIntersect
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(complementInPowersetComplementIntersect_type,type,
% 2.23/1.16 complementInPowersetComplementIntersect: $o ).
% 2.23/1.16
% 2.23/1.16 thf(complementInPowersetComplementIntersect,definition,
% 2.23/1.16 ( complementInPowersetComplementIntersect
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(contraSubsetComplement_type,type,
% 2.23/1.16 contraSubsetComplement: $o ).
% 2.23/1.16
% 2.23/1.16 thf(contraSubsetComplement,definition,
% 2.23/1.16 ( contraSubsetComplement
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ Y )
% 2.23/1.16 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(complementTcontraSubset_type,type,
% 2.23/1.16 complementTcontraSubset: $o ).
% 2.23/1.16
% 2.23/1.16 thf(complementTcontraSubset,definition,
% 2.23/1.16 ( complementTcontraSubset
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.23/1.16 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(binunionTILcontra_type,type,
% 2.23/1.16 binunionTILcontra: $o ).
% 2.23/1.16
% 2.23/1.16 thf(binunionTILcontra,definition,
% 2.23/1.16 ( binunionTILcontra
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.23/1.16 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(binunionTIRcontra_type,type,
% 2.23/1.16 binunionTIRcontra: $o ).
% 2.23/1.16
% 2.23/1.16 thf(binunionTIRcontra,definition,
% 2.23/1.16 ( binunionTIRcontra
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.23/1.16 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(inIntersectImpInUnion_type,type,
% 2.23/1.16 inIntersectImpInUnion: $o ).
% 2.23/1.16
% 2.23/1.16 thf(inIntersectImpInUnion,definition,
% 2.23/1.16 ( inIntersectImpInUnion
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Z: $i] :
% 2.23/1.16 ( ( in @ Z @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.23/1.16 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(inIntersectImpInUnion2_type,type,
% 2.23/1.16 inIntersectImpInUnion2: $o ).
% 2.23/1.16
% 2.23/1.16 thf(inIntersectImpInUnion2,definition,
% 2.23/1.16 ( inIntersectImpInUnion2
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Z: $i] :
% 2.23/1.16 ( ( in @ Z @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.23/1.16 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(inIntersectImpInIntersectUnions_type,type,
% 2.23/1.16 inIntersectImpInIntersectUnions: $o ).
% 2.23/1.16
% 2.23/1.16 thf(inIntersectImpInIntersectUnions,definition,
% 2.23/1.16 ( inIntersectImpInIntersectUnions
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Z: $i] :
% 2.23/1.16 ( ( in @ Z @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.23/1.16 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(intersectInPowersetIntersectUnions_type,type,
% 2.23/1.16 intersectInPowersetIntersectUnions: $o ).
% 2.23/1.16
% 2.23/1.16 thf(intersectInPowersetIntersectUnions,definition,
% 2.23/1.16 ( intersectInPowersetIntersectUnions
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Z: $i] :
% 2.23/1.16 ( ( in @ Z @ ( powerset @ A ) )
% 2.23/1.16 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(inComplementUnionImpNotIn1_type,type,
% 2.23/1.16 inComplementUnionImpNotIn1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(inComplementUnionImpNotIn1,definition,
% 2.23/1.16 ( inComplementUnionImpNotIn1
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.23/1.16 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(inComplementUnionImpInComplement1_type,type,
% 2.23/1.16 inComplementUnionImpInComplement1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(inComplementUnionImpInComplement1,definition,
% 2.23/1.16 ( inComplementUnionImpInComplement1
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.23/1.16 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(binunionTE_type,type,
% 2.23/1.16 binunionTE: $o ).
% 2.23/1.16
% 2.23/1.16 thf(binunionTE,definition,
% 2.23/1.16 ( binunionTE
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xphi: $o,Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.23/1.16 => ( ( ( in @ Xx @ X )
% 2.23/1.16 => Xphi )
% 2.23/1.16 => ( ( ( in @ Xx @ Y )
% 2.23/1.16 => Xphi )
% 2.23/1.16 => Xphi ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(binunionTEcontra_type,type,
% 2.23/1.16 binunionTEcontra: $o ).
% 2.23/1.16
% 2.23/1.16 thf(binunionTEcontra,definition,
% 2.23/1.16 ( binunionTEcontra
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ~ ( in @ Xx @ X )
% 2.23/1.16 => ( ~ ( in @ Xx @ Y )
% 2.23/1.16 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan2a1_type,type,
% 2.23/1.16 demorgan2a1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan2a1,definition,
% 2.23/1.16 ( demorgan2a1
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.23/1.16 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(complementUnionInPowersetComplement_type,type,
% 2.23/1.16 complementUnionInPowersetComplement: $o ).
% 2.23/1.16
% 2.23/1.16 thf(complementUnionInPowersetComplement,definition,
% 2.23/1.16 ( complementUnionInPowersetComplement
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan2a2_type,type,
% 2.23/1.16 demorgan2a2: $o ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan2a2,definition,
% 2.23/1.16 ( demorgan2a2
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.23/1.16 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan1a_type,type,
% 2.23/1.16 demorgan1a: $o ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan1a,definition,
% 2.23/1.16 ( demorgan1a
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 2.23/1.16 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan1b_type,type,
% 2.23/1.16 demorgan1b: $o ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan1b,definition,
% 2.23/1.16 ( demorgan1b
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.23/1.16 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan1_type,type,
% 2.23/1.16 demorgan1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan1,definition,
% 2.23/1.16 ( demorgan1
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 2.23/1.16 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan2a_type,type,
% 2.23/1.16 demorgan2a: $o ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan2a,definition,
% 2.23/1.16 ( demorgan2a
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.23/1.16 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan2b2_type,type,
% 2.23/1.16 demorgan2b2: $o ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan2b2,definition,
% 2.23/1.16 ( demorgan2b2
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.23/1.16 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 2.23/1.16 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan2b_type,type,
% 2.23/1.16 demorgan2b: $o ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan2b,definition,
% 2.23/1.16 ( demorgan2b
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ A )
% 2.23/1.16 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.23/1.16 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan2_type,type,
% 2.23/1.16 demorgan2: $o ).
% 2.23/1.16
% 2.23/1.16 thf(demorgan2,definition,
% 2.23/1.16 ( demorgan2
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 2.23/1.16 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(woz13rule0_type,type,
% 2.23/1.16 woz13rule0: $o ).
% 2.23/1.16
% 2.23/1.16 thf(woz13rule0,definition,
% 2.23/1.16 ( woz13rule0
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Xx: $i] :
% 2.23/1.16 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.23/1.16 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(woz13rule1_type,type,
% 2.23/1.16 woz13rule1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(woz13rule1,definition,
% 2.23/1.16 ( woz13rule1
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Z: $i] :
% 2.23/1.16 ( ( in @ Z @ ( powerset @ A ) )
% 2.23/1.16 => ( ( subset @ X @ Z )
% 2.23/1.16 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(woz13rule2_type,type,
% 2.23/1.16 woz13rule2: $o ).
% 2.23/1.16
% 2.23/1.16 thf(woz13rule2,definition,
% 2.23/1.16 ( woz13rule2
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Z: $i] :
% 2.23/1.16 ( ( in @ Z @ ( powerset @ A ) )
% 2.23/1.16 => ( ( subset @ Y @ Z )
% 2.23/1.16 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(woz13rule3_type,type,
% 2.23/1.16 woz13rule3: $o ).
% 2.23/1.16
% 2.23/1.16 thf(woz13rule3,definition,
% 2.23/1.16 ( woz13rule3
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Z: $i] :
% 2.23/1.16 ( ( in @ Z @ ( powerset @ A ) )
% 2.23/1.16 => ( ( subset @ X @ Y )
% 2.23/1.16 => ( ( subset @ X @ Z )
% 2.23/1.16 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(woz13rule4_type,type,
% 2.23/1.16 woz13rule4: $o ).
% 2.23/1.16
% 2.23/1.16 thf(woz13rule4,definition,
% 2.23/1.16 ( woz13rule4
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Z: $i] :
% 2.23/1.16 ( ( in @ Z @ ( powerset @ A ) )
% 2.23/1.16 => ! [W: $i] :
% 2.23/1.16 ( ( in @ W @ ( powerset @ A ) )
% 2.23/1.16 => ( ( subset @ X @ Z )
% 2.23/1.16 => ( ( subset @ Y @ W )
% 2.23/1.16 => ( subset @ ( binintersect @ X @ Y ) @ ( binintersect @ Z @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(woz1_1_type,type,
% 2.23/1.16 woz1_1: $o ).
% 2.23/1.16
% 2.23/1.16 thf(woz1_1,definition,
% 2.23/1.16 ( woz1_1
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(woz1_2_type,type,
% 2.23/1.16 woz1_2: $o ).
% 2.23/1.16
% 2.23/1.16 thf(woz1_2,definition,
% 2.23/1.16 ( woz1_2
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Z: $i] :
% 2.23/1.16 ( ( in @ Z @ ( powerset @ A ) )
% 2.23/1.16 => ! [W: $i] :
% 2.23/1.16 ( ( in @ W @ ( powerset @ A ) )
% 2.23/1.16 => ( ( setminus @ A @ ( binintersect @ ( binunion @ X @ Y ) @ ( binunion @ Z @ W ) ) )
% 2.23/1.16 = ( binunion @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) @ ( binintersect @ ( setminus @ A @ Z ) @ ( setminus @ A @ W ) ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(woz1_3_type,type,
% 2.23/1.16 woz1_3: $o ).
% 2.23/1.16
% 2.23/1.16 thf(woz1_3,definition,
% 2.23/1.16 ( woz1_3
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ! [Z: $i] :
% 2.23/1.16 ( ( in @ Z @ ( powerset @ A ) )
% 2.23/1.16 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(woz1_4_type,type,
% 2.23/1.16 woz1_4: $o ).
% 2.23/1.16
% 2.23/1.16 thf(woz1_4,definition,
% 2.23/1.16 ( woz1_4
% 2.23/1.16 = ( ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.23/1.16 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.23/1.16
% 2.23/1.16 thf(woz1_5,conjecture,
% 2.23/1.16 ( setextAx
% 2.23/1.16 => ( emptysetAx
% 2.23/1.16 => ( setadjoinAx
% 2.23/1.16 => ( powersetAx
% 2.23/1.16 => ( setunionAx
% 2.23/1.16 => ( omega0Ax
% 2.23/1.16 => ( omegaSAx
% 2.23/1.16 => ( omegaIndAx
% 2.23/1.16 => ( replAx
% 2.23/1.16 => ( foundationAx
% 2.23/1.16 => ( wellorderingAx
% 2.23/1.16 => ( descrp
% 2.23/1.16 => ( dsetconstrI
% 2.23/1.16 => ( dsetconstrEL
% 2.23/1.16 => ( dsetconstrER
% 2.23/1.16 => ( exuE1
% 2.23/1.16 => ( prop2setE
% 2.23/1.16 => ( emptysetE
% 2.23/1.16 => ( emptysetimpfalse
% 2.23/1.16 => ( notinemptyset
% 2.23/1.16 => ( exuE3e
% 2.23/1.16 => ( setext
% 2.23/1.16 => ( emptyI
% 2.23/1.16 => ( noeltsimpempty
% 2.23/1.16 => ( setbeta
% 2.23/1.16 => ( nonemptyE1
% 2.23/1.16 => ( nonemptyI
% 2.23/1.16 => ( nonemptyI1
% 2.23/1.16 => ( setadjoinIL
% 2.23/1.16 => ( emptyinunitempty
% 2.23/1.16 => ( setadjoinIR
% 2.23/1.16 => ( setadjoinE
% 2.23/1.16 => ( setadjoinOr
% 2.23/1.16 => ( setoftrueEq
% 2.23/1.16 => ( powersetI
% 2.23/1.16 => ( emptyinPowerset
% 2.23/1.16 => ( emptyInPowerset
% 2.23/1.16 => ( powersetE
% 2.23/1.16 => ( setunionI
% 2.23/1.16 => ( setunionE
% 2.23/1.16 => ( subPowSU
% 2.23/1.16 => ( exuE2
% 2.23/1.16 => ( nonemptyImpWitness
% 2.23/1.16 => ( uniqinunit
% 2.23/1.16 => ( notinsingleton
% 2.23/1.16 => ( eqinunit
% 2.23/1.16 => ( singletonsswitch
% 2.23/1.16 => ( upairsetE
% 2.23/1.16 => ( upairsetIL
% 2.23/1.16 => ( upairsetIR
% 2.23/1.16 => ( emptyE1
% 2.23/1.16 => ( vacuousDall
% 2.23/1.16 => ( quantDeMorgan1
% 2.23/1.16 => ( quantDeMorgan2
% 2.23/1.16 => ( quantDeMorgan3
% 2.23/1.16 => ( quantDeMorgan4
% 2.23/1.16 => ( prop2setI
% 2.23/1.16 => ( prop2set2propI
% 2.23/1.16 => ( notdexE
% 2.23/1.16 => ( notdallE
% 2.23/1.16 => ( exuI1
% 2.23/1.16 => ( exuI3
% 2.23/1.16 => ( exuI2
% 2.23/1.16 => ( inCongP
% 2.23/1.16 => ( in__Cong
% 2.23/1.16 => ( exuE3u
% 2.23/1.16 => ( exu__Cong
% 2.23/1.16 => ( emptyset__Cong
% 2.23/1.16 => ( setadjoin__Cong
% 2.23/1.16 => ( powerset__Cong
% 2.23/1.16 => ( setunion__Cong
% 2.23/1.16 => ( omega__Cong
% 2.23/1.16 => ( exuEu
% 2.23/1.16 => ( descr__Cong
% 2.23/1.16 => ( dsetconstr__Cong
% 2.23/1.16 => ( subsetI1
% 2.23/1.16 => ( eqimpsubset2
% 2.23/1.16 => ( eqimpsubset1
% 2.23/1.16 => ( subsetI2
% 2.23/1.16 => ( emptysetsubset
% 2.23/1.16 => ( subsetE
% 2.23/1.16 => ( subsetE2
% 2.23/1.16 => ( notsubsetI
% 2.23/1.16 => ( notequalI1
% 2.23/1.16 => ( notequalI2
% 2.23/1.16 => ( subsetRefl
% 2.23/1.16 => ( subsetTrans
% 2.23/1.16 => ( setadjoinSub
% 2.23/1.16 => ( setadjoinSub2
% 2.23/1.16 => ( subset2powerset
% 2.23/1.16 => ( setextsub
% 2.23/1.16 => ( subsetemptysetimpeq
% 2.23/1.16 => ( powersetI1
% 2.23/1.16 => ( powersetE1
% 2.23/1.16 => ( inPowerset
% 2.23/1.16 => ( powersetsubset
% 2.23/1.16 => ( sepInPowerset
% 2.23/1.16 => ( sepSubset
% 2.23/1.16 => ( binunionIL
% 2.23/1.16 => ( upairset2IR
% 2.23/1.16 => ( binunionIR
% 2.23/1.16 => ( binunionEcases
% 2.23/1.16 => ( binunionE
% 2.23/1.16 => ( binunionLsub
% 2.23/1.16 => ( binunionRsub
% 2.23/1.16 => ( binintersectI
% 2.23/1.16 => ( binintersectSubset5
% 2.23/1.16 => ( binintersectEL
% 2.23/1.16 => ( binintersectLsub
% 2.23/1.16 => ( binintersectSubset2
% 2.23/1.16 => ( binintersectSubset3
% 2.23/1.16 => ( binintersectER
% 2.23/1.16 => ( disjointsetsI1
% 2.23/1.16 => ( binintersectRsub
% 2.23/1.16 => ( binintersectSubset4
% 2.23/1.16 => ( binintersectSubset1
% 2.23/1.16 => ( bs114d
% 2.23/1.16 => ( setminusI
% 2.23/1.16 => ( setminusEL
% 2.23/1.16 => ( setminusER
% 2.23/1.16 => ( setminusSubset2
% 2.23/1.16 => ( setminusERneg
% 2.23/1.16 => ( setminusELneg
% 2.23/1.16 => ( setminusILneg
% 2.23/1.16 => ( setminusIRneg
% 2.23/1.16 => ( setminusLsub
% 2.23/1.16 => ( setminusSubset1
% 2.23/1.16 => ( symdiffE
% 2.23/1.16 => ( symdiffI1
% 2.23/1.16 => ( symdiffI2
% 2.23/1.16 => ( symdiffIneg1
% 2.23/1.16 => ( symdiffIneg2
% 2.23/1.16 => ( secondinupair
% 2.23/1.16 => ( setukpairIL
% 2.23/1.16 => ( setukpairIR
% 2.23/1.16 => ( kpairiskpair
% 2.23/1.16 => ( kpairp
% 2.23/1.16 => ( singletonsubset
% 2.23/1.16 => ( singletoninpowerset
% 2.23/1.16 => ( singletoninpowunion
% 2.23/1.16 => ( upairset2E
% 2.23/1.16 => ( upairsubunion
% 2.23/1.16 => ( upairinpowunion
% 2.23/1.16 => ( ubforcartprodlem1
% 2.23/1.16 => ( ubforcartprodlem2
% 2.23/1.16 => ( ubforcartprodlem3
% 2.23/1.16 => ( cartprodpairin
% 2.23/1.16 => ( cartprodmempair1
% 2.23/1.16 => ( cartprodmempair
% 2.23/1.16 => ( setunionE2
% 2.23/1.16 => ( setunionsingleton1
% 2.23/1.16 => ( setunionsingleton2
% 2.23/1.16 => ( setunionsingleton
% 2.23/1.16 => ( singletonprop
% 2.23/1.16 => ( ex1E1
% 2.23/1.16 => ( ex1I
% 2.23/1.16 => ( ex1I2
% 2.23/1.16 => ( singletonsuniq
% 2.23/1.16 => ( setukpairinjL1
% 2.23/1.16 => ( kfstsingleton
% 2.23/1.16 => ( theprop
% 2.23/1.16 => ( kfstpairEq
% 2.23/1.16 => ( cartprodfstin
% 2.23/1.16 => ( setukpairinjL2
% 2.23/1.16 => ( setukpairinjL
% 2.23/1.16 => ( setukpairinjR11
% 2.23/1.16 => ( setukpairinjR12
% 2.23/1.16 => ( setukpairinjR1
% 2.23/1.16 => ( upairequniteq
% 2.23/1.16 => ( setukpairinjR2
% 2.23/1.16 => ( setukpairinjR
% 2.23/1.16 => ( ksndsingleton
% 2.23/1.16 => ( ksndpairEq
% 2.23/1.16 => ( kpairsurjEq
% 2.23/1.16 => ( cartprodsndin
% 2.23/1.16 => ( cartprodpairmemEL
% 2.23/1.16 => ( cartprodpairmemER
% 2.23/1.16 => ( cartprodmempaircEq
% 2.23/1.16 => ( cartprodfstpairEq
% 2.23/1.16 => ( cartprodsndpairEq
% 2.23/1.16 => ( cartprodpairsurjEq
% 2.23/1.16 => ( dpsetconstrI
% 2.23/1.16 => ( dpsetconstrSub
% 2.23/1.16 => ( setOfPairsIsBReln
% 2.23/1.16 => ( dpsetconstrERa
% 2.23/1.16 => ( dpsetconstrEL1
% 2.23/1.16 => ( dpsetconstrEL2
% 2.23/1.16 => ( dpsetconstrER
% 2.23/1.16 => ( funcImageSingleton
% 2.23/1.16 => ( apProp
% 2.23/1.16 => ( app
% 2.23/1.16 => ( infuncsetfunc
% 2.23/1.16 => ( ap2p
% 2.23/1.16 => ( funcinfuncset
% 2.23/1.16 => ( lamProp
% 2.23/1.16 => ( lamp
% 2.23/1.16 => ( lam2p
% 2.23/1.16 => ( brelnall1
% 2.23/1.16 => ( brelnall2
% 2.23/1.16 => ( ex1E2
% 2.23/1.16 => ( funcGraphProp1
% 2.23/1.16 => ( funcGraphProp3
% 2.23/1.16 => ( funcGraphProp2
% 2.23/1.16 => ( funcextLem
% 2.23/1.16 => ( funcGraphProp4
% 2.23/1.16 => ( subbreln
% 2.23/1.16 => ( eqbreln
% 2.23/1.16 => ( funcext
% 2.23/1.16 => ( funcext2
% 2.23/1.16 => ( ap2apEq1
% 2.23/1.16 => ( ap2apEq2
% 2.23/1.16 => ( beta1
% 2.23/1.16 => ( eta1
% 2.23/1.16 => ( lam2lamEq
% 2.23/1.16 => ( beta2
% 2.23/1.16 => ( eta2
% 2.23/1.16 => ( iffalseProp1
% 2.23/1.16 => ( iffalseProp2
% 2.23/1.16 => ( iftrueProp1
% 2.23/1.16 => ( iftrueProp2
% 2.23/1.16 => ( ifSingleton
% 2.23/1.16 => ( ifp
% 2.23/1.16 => ( theeq
% 2.23/1.16 => ( iftrue
% 2.23/1.16 => ( iffalse
% 2.23/1.16 => ( iftrueorfalse
% 2.23/1.16 => ( binintersectT_lem
% 2.23/1.16 => ( binunionT_lem
% 2.23/1.16 => ( powersetT_lem
% 2.23/1.16 => ( setminusT_lem
% 2.23/1.16 => ( complementT_lem
% 2.23/1.16 => ( setextT
% 2.23/1.16 => ( subsetTI
% 2.23/1.16 => ( powersetTI1
% 2.23/1.16 => ( powersetTE1
% 2.23/1.16 => ( complementTI1
% 2.23/1.16 => ( complementTE1
% 2.23/1.16 => ( binintersectTELcontra
% 2.23/1.16 => ( binintersectTERcontra
% 2.23/1.16 => ( contrasubsetT
% 2.23/1.16 => ( contrasubsetT1
% 2.23/1.16 => ( contrasubsetT2
% 2.23/1.16 => ( contrasubsetT3
% 2.23/1.16 => ( doubleComplementI1
% 2.23/1.16 => ( doubleComplementE1
% 2.23/1.16 => ( doubleComplementSub1
% 2.23/1.16 => ( doubleComplementSub2
% 2.23/1.16 => ( doubleComplementEq
% 2.23/1.16 => ( complementTnotintersectT
% 2.23/1.16 => ( complementImpComplementIntersect
% 2.23/1.16 => ( complementSubsetComplementIntersect
% 2.23/1.16 => ( complementInPowersetComplementIntersect
% 2.23/1.16 => ( contraSubsetComplement
% 2.23/1.16 => ( complementTcontraSubset
% 2.23/1.16 => ( binunionTILcontra
% 2.23/1.16 => ( binunionTIRcontra
% 2.23/1.16 => ( inIntersectImpInUnion
% 2.23/1.16 => ( inIntersectImpInUnion2
% 2.23/1.16 => ( inIntersectImpInIntersectUnions
% 2.23/1.16 => ( intersectInPowersetIntersectUnions
% 2.23/1.16 => ( inComplementUnionImpNotIn1
% 2.23/1.16 => ( inComplementUnionImpInComplement1
% 2.23/1.16 => ( binunionTE
% 2.23/1.16 => ( binunionTEcontra
% 2.23/1.16 => ( demorgan2a1
% 2.23/1.16 => ( complementUnionInPowersetComplement
% 2.23/1.16 => ( demorgan2a2
% 2.23/1.16 => ( demorgan1a
% 2.23/1.16 => ( demorgan1b
% 2.23/1.16 => ( demorgan1
% 2.23/1.16 => ( demorgan2a
% 2.23/1.16 => ( demorgan2b2
% 2.23/1.16 => ( demorgan2b
% 2.23/1.16 => ( demorgan2
% 2.23/1.16 => ( woz13rule0
% 2.23/1.16 => ( woz13rule1
% 2.23/1.16 => ( woz13rule2
% 2.23/1.16 => ( woz13rule3
% 2.23/1.16 => ( woz13rule4
% 2.23/1.16 => ( woz1_1
% 2.23/1.16 => ( woz1_2
% 2.23/1.16 => ( woz1_3
% 2.23/1.16 => ( woz1_4
% 2.23/1.16 => ! [A: $i,X: $i] :
% 2.23/1.16 ( ( in @ X @ ( powerset @ A ) )
% 2.23/1.16 => ! [Y: $i] :
% 2.23/1.16 ( ( in @ Y @ ( powerset @ A ) )
% 2.23/1.16 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.23/1.16 ------------------------
% 2.23/1.18 % (7830)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_7573 for (2999ds/2Mi)
% 2.23/1.18 % (7831)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_7573 for (2999ds/275Mi)
% 2.23/1.18 % (7829)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_7573 for (2999ds/2Mi)
% 2.23/1.18 % (7828)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_7573 for (2999ds/27Mi)
% 2.23/1.18 % (7827)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_7573 for (2999ds/4Mi)
% 2.23/1.18 % (7826)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_7573 for (2999ds/183Mi)
% 2.23/1.18 % (7832)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_7573 for (2999ds/18Mi)
% 2.23/1.18 % (7827)Instruction limit reached!
% 2.23/1.18 % (7827)------------------------------
% 2.23/1.18 % (7827)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.18 % (7827)Termination reason: Unknown
% 2.23/1.18 % (7827)Termination phase: shuffling
% 2.23/1.18
% 2.23/1.18 % (7827)Memory used [KB]: 1535
% 2.23/1.18 % (7827)Time elapsed: 0.003 s
% 2.23/1.18 % (7827)Instructions burned: 4 (million)
% 2.23/1.18 % (7827)------------------------------
% 2.23/1.18 % (7827)------------------------------
% 2.23/1.18 % (7829)Instruction limit reached!
% 2.23/1.18 % (7829)------------------------------
% 2.23/1.18 % (7829)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.18 % (7829)Termination reason: Unknown
% 2.23/1.18 % (7829)Termination phase: shuffling
% 2.23/1.18
% 2.23/1.18 % (7829)Memory used [KB]: 1535
% 2.23/1.18 % (7829)Time elapsed: 0.003 s
% 2.23/1.18 % (7829)Instructions burned: 5 (million)
% 2.23/1.18 % (7829)------------------------------
% 2.23/1.18 % (7829)------------------------------
% 2.23/1.18 % (7830)Instruction limit reached!
% 2.23/1.18 % (7830)------------------------------
% 2.23/1.18 % (7830)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.18 % (7830)Termination reason: Unknown
% 2.23/1.18 % (7830)Termination phase: shuffling
% 2.23/1.18
% 2.23/1.18 % (7830)Memory used [KB]: 1535
% 2.23/1.18 % (7830)Time elapsed: 0.003 s
% 2.23/1.18 % (7830)Instructions burned: 5 (million)
% 2.23/1.18 % (7830)------------------------------
% 2.23/1.18 % (7830)------------------------------
% 2.23/1.19 % (7832)Instruction limit reached!
% 2.23/1.19 % (7832)------------------------------
% 2.23/1.19 % (7832)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.19 % (7832)Termination reason: Unknown
% 2.23/1.19 % (7832)Termination phase: shuffling
% 2.23/1.19
% 2.23/1.19 % (7832)Memory used [KB]: 1918
% 2.23/1.19 % (7832)Time elapsed: 0.008 s
% 2.23/1.19 % (7832)Instructions burned: 20 (million)
% 2.23/1.19 % (7832)------------------------------
% 2.23/1.19 % (7832)------------------------------
% 2.23/1.19 % (7828)Instruction limit reached!
% 2.23/1.19 % (7828)------------------------------
% 2.23/1.19 % (7828)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.19 % (7828)Termination reason: Unknown
% 2.23/1.19 % (7828)Termination phase: shuffling
% 2.23/1.19
% 2.23/1.19 % (7828)Memory used [KB]: 2046
% 2.23/1.19 % (7828)Time elapsed: 0.010 s
% 2.23/1.19 % (7828)Instructions burned: 30 (million)
% 2.23/1.19 % (7828)------------------------------
% 2.23/1.19 % (7828)------------------------------
% 2.23/1.19 % (7842)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_7573 for (2999ds/3Mi)
% 2.23/1.19 % (7843)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_7573 for (2999ds/37Mi)
% 2.23/1.19 % (7844)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_7573 for (2999ds/15Mi)
% 2.23/1.19 % (7842)Instruction limit reached!
% 2.23/1.19 % (7842)------------------------------
% 2.23/1.19 % (7842)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.23/1.19 % (7842)Termination reason: Unknown
% 2.23/1.19 % (7842)Termination phase: shuffling
% 2.23/1.19
% 2.23/1.19 % (7842)Memory used [KB]: 1535
% 2.23/1.19 % (7842)Time elapsed: 0.003 s
% 2.23/1.19 % (7842)Instructions burned: 6 (million)
% 2.23/1.19 % (7842)------------------------------
% 2.23/1.19 % (7842)------------------------------
% 2.26/1.20 % (7847)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_7573 for (2999ds/3Mi)
% 2.26/1.20 % (7844)Instruction limit reached!
% 2.26/1.20 % (7844)------------------------------
% 2.26/1.20 % (7844)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.20 % (7844)Termination reason: Unknown
% 2.26/1.20 % (7844)Termination phase: shuffling
% 2.26/1.20
% 2.26/1.20 % (7844)Memory used [KB]: 1791
% 2.26/1.20 % (7844)Time elapsed: 0.007 s
% 2.26/1.20 % (7844)Instructions burned: 17 (million)
% 2.26/1.20 % (7844)------------------------------
% 2.26/1.20 % (7844)------------------------------
% 2.26/1.20 % (7847)Instruction limit reached!
% 2.26/1.20 % (7847)------------------------------
% 2.26/1.20 % (7847)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.20 % (7847)Termination reason: Unknown
% 2.26/1.20 % (7847)Termination phase: shuffling
% 2.26/1.20
% 2.26/1.20 % (7847)Memory used [KB]: 1535
% 2.26/1.20 % (7847)Time elapsed: 0.002 s
% 2.26/1.20 % (7847)Instructions burned: 3 (million)
% 2.26/1.20 % (7847)------------------------------
% 2.26/1.20 % (7847)------------------------------
% 2.26/1.20 % (7848)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_7573 for (2999ds/1041Mi)
% 2.26/1.20 % (7843)Instruction limit reached!
% 2.26/1.20 % (7843)------------------------------
% 2.26/1.20 % (7843)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.20 % (7843)Termination reason: Unknown
% 2.26/1.20 % (7843)Termination phase: shuffling
% 2.26/1.20
% 2.26/1.20 % (7843)Memory used [KB]: 2302
% 2.26/1.20 % (7843)Time elapsed: 0.012 s
% 2.26/1.20 % (7843)Instructions burned: 40 (million)
% 2.26/1.20 % (7843)------------------------------
% 2.26/1.20 % (7843)------------------------------
% 2.26/1.20 % (7853)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_7573 for (2999ds/7Mi)
% 2.26/1.21 % (7853)Instruction limit reached!
% 2.26/1.21 % (7853)------------------------------
% 2.26/1.21 % (7853)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.21 % (7853)Termination reason: Unknown
% 2.26/1.21 % (7853)Termination phase: shuffling
% 2.26/1.21
% 2.26/1.21 % (7853)Memory used [KB]: 1535
% 2.26/1.21 % (7853)Time elapsed: 0.003 s
% 2.26/1.21 % (7853)Instructions burned: 7 (million)
% 2.26/1.21 % (7853)------------------------------
% 2.26/1.21 % (7853)------------------------------
% 2.26/1.21 % (7859)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_7573 for (2999ds/16Mi)
% 2.26/1.21 % (7860)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_7573 for (2999ds/3Mi)
% 2.26/1.21 % (7860)Instruction limit reached!
% 2.26/1.21 % (7860)------------------------------
% 2.26/1.21 % (7860)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.21 % (7860)Termination reason: Unknown
% 2.26/1.21 % (7860)Termination phase: shuffling
% 2.26/1.21
% 2.26/1.21 % (7860)Memory used [KB]: 1535
% 2.26/1.21 % (7860)Time elapsed: 0.002 s
% 2.26/1.21 % (7860)Instructions burned: 3 (million)
% 2.26/1.21 % (7860)------------------------------
% 2.26/1.21 % (7860)------------------------------
% 2.26/1.21 % (7859)Instruction limit reached!
% 2.26/1.21 % (7859)------------------------------
% 2.26/1.21 % (7859)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.21 % (7859)Termination reason: Unknown
% 2.26/1.21 % (7859)Termination phase: shuffling
% 2.26/1.21
% 2.26/1.21 % (7859)Memory used [KB]: 1791
% 2.26/1.21 % (7859)Time elapsed: 0.006 s
% 2.26/1.21 % (7859)Instructions burned: 18 (million)
% 2.26/1.21 % (7859)------------------------------
% 2.26/1.21 % (7859)------------------------------
% 2.26/1.21 % (7861)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_7573 for (2999ds/3Mi)
% 2.26/1.21 % (7861)Instruction limit reached!
% 2.26/1.21 % (7861)------------------------------
% 2.26/1.21 % (7861)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.21 % (7861)Termination reason: Unknown
% 2.26/1.21 % (7861)Termination phase: shuffling
% 2.26/1.21
% 2.26/1.21 % (7861)Memory used [KB]: 1535
% 2.26/1.21 % (7861)Time elapsed: 0.002 s
% 2.26/1.21 % (7861)Instructions burned: 3 (million)
% 2.26/1.21 % (7861)------------------------------
% 2.26/1.21 % (7861)------------------------------
% 2.26/1.22 % (7862)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_7573 for (2999ds/7Mi)
% 2.26/1.22 % (7862)Instruction limit reached!
% 2.26/1.22 % (7862)------------------------------
% 2.26/1.22 % (7862)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.22 % (7862)Termination reason: Unknown
% 2.26/1.22 % (7862)Termination phase: shuffling
% 2.26/1.22
% 2.26/1.22 % (7862)Memory used [KB]: 1535
% 2.26/1.22 % (7862)Time elapsed: 0.003 s
% 2.26/1.22 % (7862)Instructions burned: 7 (million)
% 2.26/1.22 % (7862)------------------------------
% 2.26/1.22 % (7862)------------------------------
% 2.26/1.22 % (7868)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_7573 for (2999ds/3Mi)
% 2.26/1.22 % (7868)Instruction limit reached!
% 2.26/1.22 % (7868)------------------------------
% 2.26/1.22 % (7868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.22 % (7868)Termination reason: Unknown
% 2.26/1.22 % (7868)Termination phase: shuffling
% 2.26/1.22
% 2.26/1.22 % (7868)Memory used [KB]: 1535
% 2.26/1.22 % (7868)Time elapsed: 0.002 s
% 2.26/1.22 % (7868)Instructions burned: 3 (million)
% 2.26/1.22 % (7868)------------------------------
% 2.26/1.22 % (7868)------------------------------
% 2.26/1.22 % (7875)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_7573 for (2999ds/4Mi)
% 2.26/1.22 % (7877)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_7573 for (2999ds/18Mi)
% 2.26/1.22 % (7875)Instruction limit reached!
% 2.26/1.22 % (7875)------------------------------
% 2.26/1.22 % (7875)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.22 % (7875)Termination reason: Unknown
% 2.26/1.22 % (7875)Termination phase: shuffling
% 2.26/1.22
% 2.26/1.22 % (7875)Memory used [KB]: 1535
% 2.26/1.22 % (7875)Time elapsed: 0.003 s
% 2.26/1.22 % (7875)Instructions burned: 6 (million)
% 2.26/1.22 % (7875)------------------------------
% 2.26/1.22 % (7875)------------------------------
% 2.26/1.23 % (7881)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_7573 for (2999ds/710Mi)
% 2.26/1.23 % (7877)Instruction limit reached!
% 2.26/1.23 % (7877)------------------------------
% 2.26/1.23 % (7877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.23 % (7877)Termination reason: Unknown
% 2.26/1.23 % (7877)Termination phase: shuffling
% 2.26/1.23
% 2.26/1.23 % (7877)Memory used [KB]: 1791
% 2.26/1.23 % (7877)Time elapsed: 0.006 s
% 2.26/1.23 % (7877)Instructions burned: 18 (million)
% 2.26/1.23 % (7877)------------------------------
% 2.26/1.23 % (7877)------------------------------
% 2.26/1.23 % (7885)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_7573 for (2999ds/6Mi)
% 2.26/1.23 % (7826)Instruction limit reached!
% 2.26/1.23 % (7826)------------------------------
% 2.26/1.23 % (7826)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.23 % (7826)Termination reason: Unknown
% 2.26/1.23 % (7826)Termination phase: Saturation
% 2.26/1.23
% 2.26/1.23 % (7826)Memory used [KB]: 7419
% 2.26/1.23 % (7826)Time elapsed: 0.049 s
% 2.26/1.23 % (7826)Instructions burned: 184 (million)
% 2.26/1.23 % (7826)------------------------------
% 2.26/1.23 % (7826)------------------------------
% 2.26/1.23 % (7885)Instruction limit reached!
% 2.26/1.23 % (7885)------------------------------
% 2.26/1.23 % (7885)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.23 % (7885)Termination reason: Unknown
% 2.26/1.23 % (7885)Termination phase: shuffling
% 2.26/1.23
% 2.26/1.23 % (7885)Memory used [KB]: 1663
% 2.26/1.23 % (7885)Time elapsed: 0.003 s
% 2.26/1.23 % (7885)Instructions burned: 6 (million)
% 2.26/1.23 % (7885)------------------------------
% 2.26/1.23 % (7885)------------------------------
% 2.26/1.23 % (7888)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_7573 for (2999ds/902Mi)
% 2.26/1.24 % (7889)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_7573 for (2999ds/21Mi)
% 2.26/1.24 % (7890)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_7573 for (2999ds/5Mi)
% 2.26/1.24 % (7893)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_7573 for (2999ds/6Mi)
% 2.26/1.24 % (7890)Instruction limit reached!
% 2.26/1.24 % (7890)------------------------------
% 2.26/1.24 % (7890)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.24 % (7890)Termination reason: Unknown
% 2.26/1.24 % (7890)Termination phase: shuffling
% 2.26/1.24
% 2.26/1.24 % (7890)Memory used [KB]: 1535
% 2.26/1.24 % (7890)Time elapsed: 0.003 s
% 2.26/1.24 % (7890)Instructions burned: 7 (million)
% 2.26/1.24 % (7890)------------------------------
% 2.26/1.24 % (7890)------------------------------
% 2.26/1.24 % (7893)Instruction limit reached!
% 2.26/1.24 % (7893)------------------------------
% 2.26/1.24 % (7893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.24 % (7893)Termination reason: Unknown
% 2.26/1.24 % (7893)Termination phase: shuffling
% 2.26/1.24
% 2.26/1.24 % (7893)Memory used [KB]: 1535
% 2.26/1.24 % (7893)Time elapsed: 0.003 s
% 2.26/1.24 % (7893)Instructions burned: 7 (million)
% 2.26/1.24 % (7893)------------------------------
% 2.26/1.24 % (7893)------------------------------
% 2.26/1.24 % (7889)Instruction limit reached!
% 2.26/1.24 % (7889)------------------------------
% 2.26/1.24 % (7889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.24 % (7889)Termination reason: Unknown
% 2.26/1.24 % (7889)Termination phase: shuffling
% 2.26/1.24
% 2.26/1.24 % (7889)Memory used [KB]: 1918
% 2.26/1.24 % (7889)Time elapsed: 0.007 s
% 2.26/1.24 % (7889)Instructions burned: 21 (million)
% 2.26/1.24 % (7889)------------------------------
% 2.26/1.24 % (7889)------------------------------
% 2.26/1.25 % (7909)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_7573 for (2999ds/377Mi)
% 2.26/1.25 % (7912)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_7573 for (2999ds/779Mi)
% 2.26/1.25 % (7916)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_7573 for (2999ds/19Mi)
% 2.26/1.26 % (7916)Instruction limit reached!
% 2.26/1.26 % (7916)------------------------------
% 2.26/1.26 % (7916)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.26 % (7916)Termination reason: Unknown
% 2.26/1.26 % (7916)Termination phase: shuffling
% 2.26/1.26
% 2.26/1.26 % (7916)Memory used [KB]: 1918
% 2.26/1.26 % (7916)Time elapsed: 0.007 s
% 2.26/1.26 % (7916)Instructions burned: 22 (million)
% 2.26/1.26 % (7916)------------------------------
% 2.26/1.26 % (7916)------------------------------
% 2.26/1.26 % (7831)Instruction limit reached!
% 2.26/1.26 % (7831)------------------------------
% 2.26/1.26 % (7831)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.26 % (7831)Termination reason: Unknown
% 2.26/1.26 % (7831)Termination phase: Saturation
% 2.26/1.26
% 2.26/1.26 % (7831)Memory used [KB]: 10106
% 2.26/1.26 % (7831)Time elapsed: 0.080 s
% 2.26/1.26 % (7831)Instructions burned: 277 (million)
% 2.26/1.26 % (7831)------------------------------
% 2.26/1.26 % (7831)------------------------------
% 2.26/1.27 % (7929)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_7573 for (2998ds/879Mi)
% 2.26/1.27 % (7931)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_7573 for (2998ds/17Mi)
% 2.26/1.28 % (7931)Instruction limit reached!
% 2.26/1.28 % (7931)------------------------------
% 2.26/1.28 % (7931)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.28 % (7931)Termination reason: Unknown
% 2.26/1.28 % (7931)Termination phase: shuffling
% 2.26/1.28
% 2.26/1.28 % (7931)Memory used [KB]: 1791
% 2.26/1.28 % (7931)Time elapsed: 0.006 s
% 2.26/1.28 % (7931)Instructions burned: 18 (million)
% 2.26/1.28 % (7931)------------------------------
% 2.26/1.28 % (7931)------------------------------
% 2.26/1.28 % (7943)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_7573 for (2998ds/3Mi)
% 2.26/1.29 % (7943)Instruction limit reached!
% 2.26/1.29 % (7943)------------------------------
% 2.26/1.29 % (7943)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.29 % (7943)Termination reason: Unknown
% 2.26/1.29 % (7943)Termination phase: shuffling
% 2.26/1.29
% 2.26/1.29 % (7943)Memory used [KB]: 1535
% 2.26/1.29 % (7943)Time elapsed: 0.003 s
% 2.26/1.29 % (7943)Instructions burned: 5 (million)
% 2.26/1.29 % (7943)------------------------------
% 2.26/1.29 % (7943)------------------------------
% 2.26/1.30 % (7953)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_7573 for (2998ds/30Mi)
% 2.26/1.31 % (7953)Instruction limit reached!
% 2.26/1.31 % (7953)------------------------------
% 2.26/1.31 % (7953)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.31 % (7953)Termination reason: Unknown
% 2.26/1.31 % (7953)Termination phase: shuffling
% 2.26/1.31
% 2.26/1.31 % (7953)Memory used [KB]: 2174
% 2.26/1.31 % (7953)Time elapsed: 0.010 s
% 2.26/1.31 % (7953)Instructions burned: 32 (million)
% 2.26/1.31 % (7953)------------------------------
% 2.26/1.31 % (7953)------------------------------
% 2.26/1.32 % (7961)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_7573 for (2998ds/127Mi)
% 2.26/1.35 % (7961)Instruction limit reached!
% 2.26/1.35 % (7961)------------------------------
% 2.26/1.35 % (7961)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.35 % (7961)Termination reason: Unknown
% 2.26/1.35 % (7961)Termination phase: Property scanning
% 2.26/1.35
% 2.26/1.35 % (7961)Memory used [KB]: 2686
% 2.26/1.35 % (7961)Time elapsed: 0.034 s
% 2.26/1.35 % (7961)Instructions burned: 130 (million)
% 2.26/1.35 % (7961)------------------------------
% 2.26/1.35 % (7961)------------------------------
% 2.26/1.35 % (7909)Instruction limit reached!
% 2.26/1.35 % (7909)------------------------------
% 2.26/1.35 % (7909)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.35 % (7909)Termination reason: Unknown
% 2.26/1.35 % (7909)Termination phase: Saturation
% 2.26/1.35
% 2.26/1.35 % (7909)Memory used [KB]: 9466
% 2.26/1.35 % (7909)Time elapsed: 0.102 s
% 2.26/1.35 % (7909)Instructions burned: 378 (million)
% 2.26/1.35 % (7909)------------------------------
% 2.26/1.35 % (7909)------------------------------
% 2.26/1.35 % (7888)First to succeed.
% 2.26/1.36 % (7985)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_7573 for (2998ds/100Mi)
% 2.26/1.36 % (7888)Refutation found. Thanks to Tanya!
% 2.26/1.36 % SZS status Theorem for DTF2THF_7573
% 2.26/1.36 % SZS output start Proof for DTF2THF_7573
% 2.26/1.36 thf(func_def_0, type, in: $i > $i > $o).
% 2.26/1.36 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.26/1.36 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.26/1.36 thf(func_def_8, type, powerset: $i > $i).
% 2.26/1.36 thf(func_def_10, type, setunion: $i > $i).
% 2.26/1.36 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.26/1.36 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.26/1.36 thf(func_def_26, type, prop2set: $o > $i).
% 2.26/1.36 thf(func_def_36, type, nonempty: $i > $o).
% 2.26/1.36 thf(func_def_69, type, set2prop: $i > $o).
% 2.26/1.36 thf(func_def_88, type, subset: $i > $i > $o).
% 2.26/1.36 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.26/1.36 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.26/1.36 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.26/1.36 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.26/1.36 thf(func_def_135, type, regular: $i > $o).
% 2.26/1.36 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.26/1.36 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.26/1.36 thf(func_def_153, type, iskpair: $i > $o).
% 2.26/1.36 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.26/1.36 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.26/1.36 thf(func_def_177, type, singleton: $i > $o).
% 2.26/1.36 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.26/1.36 thf(func_def_184, type, atmost1p: $i > $o).
% 2.26/1.36 thf(func_def_185, type, atleast2p: $i > $o).
% 2.26/1.36 thf(func_def_186, type, atmost2p: $i > $o).
% 2.26/1.36 thf(func_def_187, type, upairsetp: $i > $o).
% 2.26/1.36 thf(func_def_191, type, kfst: $i > $i).
% 2.26/1.36 thf(func_def_203, type, ksnd: $i > $i).
% 2.26/1.36 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.26/1.36 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.26/1.36 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.26/1.36 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.26/1.36 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.26/1.36 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.26/1.36 thf(func_def_337, type, sP2: $i > $i > $i > $o).
% 2.26/1.36 thf(func_def_341, type, sP6: $i > $i > $i > $i > $o).
% 2.26/1.36 thf(func_def_343, type, sP8: $i > $o > $i > $i > $o).
% 2.26/1.36 thf(func_def_344, type, sP9: $i > $i > $o).
% 2.26/1.36 thf(func_def_345, type, sP10: $i > $o).
% 2.26/1.36 thf(func_def_346, type, sP11: $i > $i > $o).
% 2.26/1.36 thf(func_def_347, type, sP12: $i > $i > $o).
% 2.26/1.36 thf(func_def_354, type, sK19: $i > $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_358, type, sK23: $o > $i > $i > $i).
% 2.26/1.36 thf(func_def_363, type, sK28: $i > $i > $i).
% 2.26/1.36 thf(func_def_383, type, sK48: $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_384, type, sK49: $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_412, type, sK77: $i > $i > $o).
% 2.26/1.36 thf(func_def_414, type, sK79: $i > $i).
% 2.26/1.36 thf(func_def_415, type, sK80: $i > $i).
% 2.26/1.36 thf(func_def_416, type, sK81: $i > ($i > $i > $o) > $i).
% 2.26/1.36 thf(func_def_417, type, sK82: $i > ($i > $i > $o) > $i).
% 2.26/1.36 thf(func_def_418, type, sK83: $i > $i > ($i > $i > $o) > $i).
% 2.26/1.36 thf(func_def_443, type, sK108: $i > $i).
% 2.26/1.36 thf(func_def_445, type, sK110: $i > ($i > $i) > $i > $i).
% 2.26/1.36 thf(func_def_457, type, sK122: $i > $i > $i).
% 2.26/1.36 thf(func_def_462, type, sK127: $i > $i > $o).
% 2.26/1.36 thf(func_def_488, type, sK153: $i > $o).
% 2.26/1.36 thf(func_def_495, type, sK160: $i > $i > $o).
% 2.26/1.36 thf(func_def_501, type, sK166: $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_506, type, sK171: $i > $i).
% 2.26/1.36 thf(func_def_517, type, sK182: $i > $i).
% 2.26/1.36 thf(func_def_542, type, sK207: $i > $i).
% 2.26/1.36 thf(func_def_543, type, sK208: $i > $i).
% 2.26/1.36 thf(func_def_551, type, sK216: ($i > $o) > $i > $i).
% 2.26/1.36 thf(func_def_553, type, sK218: $i > $o).
% 2.26/1.36 thf(func_def_563, type, sK228: $i > $i > $i).
% 2.26/1.36 thf(func_def_566, type, sK231: $i > $o).
% 2.26/1.36 thf(func_def_567, type, sK232: $i > $o).
% 2.26/1.36 thf(func_def_568, type, sK233: ($i > $o) > ($i > $o) > $i).
% 2.26/1.36 thf(func_def_569, type, sK234: ($i > $o) > ($i > $o) > $i).
% 2.26/1.36 thf(func_def_589, type, sK254: $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_590, type, sK255: $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_599, type, sK264: $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_605, type, sK270: $i > $o).
% 2.26/1.36 thf(func_def_607, type, sK272: $i > $o).
% 2.26/1.36 thf(func_def_609, type, sK274: ($i > $o) > $i).
% 2.26/1.36 thf(func_def_610, type, sK275: ($i > $o) > $i).
% 2.26/1.36 thf(func_def_621, type, sK286: $i > $i > $o).
% 2.26/1.36 thf(func_def_638, type, sK303: $i > $i).
% 2.26/1.36 thf(func_def_644, type, sK309: ($i > $o) > $i > $i).
% 2.26/1.36 thf(func_def_646, type, sK311: $i > $o).
% 2.26/1.36 thf(func_def_660, type, sK325: ($i > $o) > $i > $i).
% 2.26/1.36 thf(func_def_662, type, sK327: $i > $o).
% 2.26/1.36 thf(func_def_686, type, sK351: $i > $i > $o).
% 2.26/1.36 thf(func_def_708, type, sK373: $i > $o).
% 2.26/1.36 thf(func_def_713, type, sK378: $i > $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_714, type, sK379: $i > $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_727, type, sK392: $i > $o).
% 2.26/1.36 thf(func_def_730, type, sK395: $i > $i > $o).
% 2.26/1.36 thf(func_def_732, type, sK397: $i > $i).
% 2.26/1.36 thf(func_def_739, type, sK404: $i > $o).
% 2.26/1.36 thf(func_def_743, type, sK408: $i > $i > $i).
% 2.26/1.36 thf(func_def_749, type, sK414: $i > ($i > $o) > $i > $i).
% 2.26/1.36 thf(func_def_751, type, sK416: $i > $o).
% 2.26/1.36 thf(func_def_757, type, sK422: $i > $o).
% 2.26/1.36 thf(func_def_762, type, sK427: $i > ($i > $i) > $i > $i).
% 2.26/1.36 thf(func_def_764, type, sK429: $i > $i).
% 2.26/1.36 thf(func_def_768, type, sK433: $i > $o).
% 2.26/1.36 thf(func_def_777, type, sK442: $i > $o).
% 2.26/1.36 thf(func_def_795, type, sK460: $i > $i > $o).
% 2.26/1.36 thf(func_def_808, type, sK473: $i > $i > $i).
% 2.26/1.36 thf(func_def_809, type, sK474: $i > $o).
% 2.26/1.36 thf(func_def_824, type, sK489: $i > $i > $i).
% 2.26/1.36 thf(func_def_825, type, sK490: $i > $i > $i).
% 2.26/1.36 thf(func_def_874, type, sK539: $i > $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_895, type, sK560: $i > ($i > $o) > $i).
% 2.26/1.36 thf(func_def_896, type, sK561: $i > $o).
% 2.26/1.36 thf(func_def_900, type, sK565: $i > $o).
% 2.26/1.36 thf(func_def_901, type, sK566: $i > $o).
% 2.26/1.36 thf(func_def_902, type, sK567: ($i > $o) > ($i > $o) > $i).
% 2.26/1.36 thf(func_def_903, type, sK568: ($i > $o) > ($i > $o) > $i).
% 2.26/1.36 thf(func_def_905, type, sK570: $i > $o).
% 2.26/1.36 thf(func_def_907, type, sK572: ($i > $o) > $i > $i).
% 2.26/1.36 thf(func_def_908, type, sK573: ($i > $o) > $i > $i).
% 2.26/1.36 thf(func_def_909, type, sK574: $i > ($i > $i) > $i > $i).
% 2.26/1.36 thf(func_def_911, type, sK576: $i > $i).
% 2.26/1.36 thf(func_def_915, type, sK580: $i > $o).
% 2.26/1.36 thf(func_def_923, type, sK588: $i > $o).
% 2.26/1.36 thf(func_def_924, type, sK589: $i > $o).
% 2.26/1.36 thf(func_def_925, type, sK590: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.26/1.36 thf(func_def_926, type, sK591: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.26/1.36 thf(func_def_929, type, sK594: $i > $o).
% 2.26/1.36 thf(func_def_932, type, sK597: $i > $o).
% 2.26/1.36 thf(func_def_933, type, sK598: $i > $i).
% 2.26/1.36 thf(func_def_934, type, sK599: ($i > $o) > $i).
% 2.26/1.36 thf(func_def_943, type, sK608: $i > $o).
% 2.26/1.36 thf(func_def_946, type, sK611: $i > ($i > $o) > $i).
% 2.26/1.36 thf(func_def_947, type, sK612: $i > ($i > $o) > $i).
% 2.26/1.36 thf(func_def_948, type, sK613: ($i > $o) > $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_949, type, sK614: ($i > $o) > $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_953, type, sK618: $i > $o).
% 2.26/1.36 thf(func_def_958, type, sK623: $i > $o).
% 2.26/1.36 thf(func_def_989, type, sK654: $i > ($i > $o) > $i).
% 2.26/1.36 thf(func_def_990, type, sK655: $i > $o).
% 2.26/1.36 thf(func_def_1005, type, sK670: $i > $i > $i).
% 2.26/1.36 thf(func_def_1036, type, sK701: $i > $o).
% 2.26/1.36 thf(func_def_1038, type, sK703: $i > ($i > $o) > $i).
% 2.26/1.36 thf(func_def_1039, type, sK704: $i > $i > $i).
% 2.26/1.36 thf(func_def_1043, type, sK708: $i > $i).
% 2.26/1.36 thf(func_def_1068, type, sK733: ($i > $o) > $i).
% 2.26/1.36 thf(func_def_1069, type, sK734: $i > $o).
% 2.26/1.36 thf(func_def_1077, type, sK742: $i > ($i > $i) > $i > $i).
% 2.26/1.36 thf(func_def_1079, type, sK744: $i > $i).
% 2.26/1.36 thf(func_def_1084, type, sK749: ($i > $o) > $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_1085, type, sK750: ($i > $o) > $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_1089, type, sK754: $i > $o).
% 2.26/1.36 thf(func_def_1105, type, sK770: $i > $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_1106, type, sK771: $i > $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_1107, type, sK772: $i > $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_1108, type, sK773: $i > $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_1115, type, sK780: $i > $o).
% 2.26/1.36 thf(func_def_1117, type, sK782: ($i > $o) > $i > $i).
% 2.26/1.36 thf(func_def_1121, type, sK786: $i > $i > $i).
% 2.26/1.36 thf(func_def_1123, type, sK788: $i > $o).
% 2.26/1.36 thf(func_def_1124, type, sK789: $i > $i).
% 2.26/1.36 thf(func_def_1125, type, sK790: ($i > $o) > $i).
% 2.26/1.36 thf(func_def_1130, type, sK795: $i > $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_1139, type, sK804: $i > $i).
% 2.26/1.36 thf(func_def_1140, type, sK805: ($i > $i) > $i > $i > $i).
% 2.26/1.36 thf(func_def_1156, type, sK821: $i > $o).
% 2.26/1.36 thf(func_def_1201, type, sK866: ($i > $o) > $i > $i).
% 2.26/1.36 thf(func_def_1203, type, sK868: $i > $o).
% 2.26/1.36 thf(func_def_1209, type, sK874: $i > $i).
% 2.26/1.36 thf(func_def_1211, type, sK876: $i > ($i > $i) > $i > $i).
% 2.26/1.36 thf(func_def_1214, type, sK879: $i > $i > $o).
% 2.26/1.36 thf(func_def_1219, type, sK884: $i > $i > $i).
% 2.26/1.36 thf(func_def_1220, type, sK885: $i > $i > $i).
% 2.26/1.36 thf(func_def_1221, type, sK886: $i > $i > $i).
% 2.26/1.36 thf(func_def_1222, type, sK887: $i > $i > $i).
% 2.26/1.36 thf(func_def_1223, type, sK888: $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_1224, type, sK889: $i > $i).
% 2.26/1.36 thf(func_def_1225, type, sK890: $i > $i).
% 2.26/1.36 thf(func_def_1226, type, sK891: $i > $i).
% 2.26/1.36 thf(func_def_1227, type, sK892: $i > $i).
% 2.26/1.36 thf(func_def_1228, type, sK893: $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_1229, type, sK894: $i > $i > $i > $i).
% 2.26/1.36 thf(func_def_1230, type, sK895: $i > $i > $i).
% 2.26/1.36 thf(func_def_1231, type, sK896: $i > $i > $i).
% 2.26/1.36 thf(func_def_1232, type, sK897: $i > $i > $i).
% 2.26/1.36 thf(func_def_1233, type, sK898: $i > $i).
% 2.26/1.36 thf(func_def_1234, type, sK899: $i > $i > $i).
% 2.26/1.36 thf(func_def_1236, type, sK901: $i > $i).
% 2.26/1.36 thf(func_def_1237, type, sK902: $i > $i).
% 2.26/1.36 thf(func_def_1246, type, sK911: $i > ($i > $o) > $i).
% 2.26/1.36 thf(func_def_1247, type, sK912: $i > $o).
% 2.26/1.36 thf(func_def_1259, type, ph924: !>[X0: $tType]:(X0)).
% 2.26/1.36 thf(f4561,plain,(
% 2.26/1.36 $false),
% 2.26/1.36 inference(trivial_inequality_removal,[status(thm)],[f4550])).
% 2.26/1.36 thf(f4550,plain,(
% 2.26/1.36 ($true != $true)),
% 2.26/1.36 inference(superposition,[status(thm)],[f4479,f3991])).
% 2.26/1.36 thf(f3991,plain,(
% 2.26/1.36 ($true = (in @ sK916 @ (powerset @ sK915)))),
% 2.26/1.36 inference(cnf_transformation,[status(thm)],[f2668])).
% 2.26/1.36 thf(f2668,plain,(
% 2.26/1.36 (nonemptyImpWitness = $true) & (iffalse = $true) & (demorgan1a = $true) & (cartprodpairin = $true) & (binunionRsub = $true) & (binintersectSubset2 = $true) & (binintersectSubset1 = $true) & (emptysetimpfalse = $true) & (infuncsetfunc = $true) & (subsetE2 = $true) & (prop2setI = $true) & (setminusILneg = $true) & (emptyset__Cong = $true) & (eqimpsubset1 = $true) & (powersetsubset = $true) & (cartprodpairmemEL = $true) & (complementImpComplementIntersect = $true) & (bs114d = $true) & (binunionTEcontra = $true) & (powersetE1 = $true) & (setext = $true) & (setukpairinjR1 = $true) & (setukpairIR = $true) & (iftrueProp2 = $true) & (powersetI = $true) & (notinsingleton = $true) & (setminusER = $true) & (setminusSubset2 = $true) & (exuE2 = $true) & (setukpairinjL2 = $true) & (ubforcartprodlem2 = $true) & (eqbreln = $true) & (cartprodmempair1 = $true) & (setadjoinE = $true) & (setukpairinjL1 = $true) & (notequalI2 = $true) & (inPowerset = $true) & (setadjoinOr = $true) & (setunionsingleton1 = $true) & (eqimpsubset2 = $true) & (powersetTI1 = $true) & (setminusERneg = $true) & (woz1_3 = $true) & (binintersectI = $true) & (upairsetIR = $true) & (woz1_4 = $true) & (subsetI2 = $true) & (singletonsuniq = $true) & (sepInPowerset = $true) & (setextT = $true) & (descrp = $true) & (inIntersectImpInUnion = $true) & (cartprodmempaircEq = $true) & (emptyE1 = $true) & (subbreln = $true) & (setextsub = $true) & (binunionIL = $true) & (setminusLsub = $true) & (contraSubsetComplement = $true) & (funcGraphProp2 = $true) & (binintersectT_lem = $true) & (setukpairinjL = $true) & (subsetRefl = $true) & (ap2apEq2 = $true) & (setminusELneg = $true) & (setextAx = $true) & (cartprodmempair = $true) & (exuI2 = $true) & (ex1E1 = $true) & (contrasubsetT1 = $true) & (secondinupair = $true) & (doubleComplementI1 = $true) & (subset2powerset = $true) & (setukpairinjR = $true) & (doubleComplementSub2 = $true) & (funcImageSingleton = $true) & (kpairp = $true) & (subsetI1 = $true) & (woz13rule3 = $true) & (iftrueorfalse = $true) & (lamp = $true) & (contrasubsetT = $true) & (omegaSAx = $true) & (inComplementUnionImpInComplement1 = $true) & (symdiffE = $true) & (kfstpairEq = $true) & (kpairsurjEq = $true) & (cartprodfstin = $true) & (notdexE = $true) & (funcinfuncset = $true) & (setadjoin__Cong = $true) & (complementSubsetComplementIntersect = $true) & (ifp = $true) & (singletonprop = $true) & (nonemptyE1 = $true) & (funcext2 = $true) & (cartprodsndpairEq = $true) & (complementTI1 = $true) & (binunionIR = $true) & (woz13rule0 = $true) & (iftrueProp1 = $true) & (dsetconstrER = $true) & (setunionE2 = $true) & (iftrue = $true) & (singletoninpowunion = $true) & (lam2lamEq = $true) & (exuE3e = $true) & (iffalseProp2 = $true) & (woz13rule2 = $true) & (setminusEL = $true) & (singletonsubset = $true) & (binintersectLsub = $true) & (setminusIRneg = $true) & (nonemptyI = $true) & (demorgan2a = $true) & (apProp = $true) & (funcGraphProp3 = $true) & (setminusT_lem = $true) & (uniqinunit = $true) & (inCongP = $true) & (quantDeMorgan2 = $true) & (woz1_2 = $true) & (demorgan2 = $true) & (emptyinunitempty = $true) & (emptysetE = $true) & (upairsetE = $true) & (binunionTIRcontra = $true) & (setadjoinIL = $true) & (funcGraphProp4 = $true) & (upairsetIL = $true) & (contrasubsetT2 = $true) & (noeltsimpempty = $true) & (powersetT_lem = $true) & (kfstsingleton = $true) & (exuI1 = $true) & (binunionLsub = $true) & (setunionI = $true) & (subsetTI = $true) & (setukpairIL = $true) & (upairset2IR = $true) & (funcext = $true) & (upairequniteq = $true) & (upairinpowunion = $true) & (eta2 = $true) & (binintersectSubset3 = $true) & (kpairiskpair = $true) & (inIntersectImpInUnion2 = $true) & (setunionAx = $true) & (dpsetconstrSub = $true) & (binintersectEL = $true) & (funcextLem = $true) & (notdallE = $true) & (ex1I = $true) & (omega__Cong = $true) & (exuEu = $true) & (complementTcontraSubset = $true) & (complementInPowersetComplementIntersect = $true) & (setadjoinAx = $true) & (woz13rule1 = $true) & (subPowSU = $true) & (foundationAx = $true) & (doubleComplementSub1 = $true) & (theprop = $true) & (binintersectER = $true) & (ap2p = $true) & (complementTE1 = $true) & (app = $true) & (contrasubsetT3 = $true) & (setunionsingleton = $true) & (quantDeMorgan1 = $true) & (binunionTILcontra = $true) & (intersectInPowersetIntersectUnions = $true) & (prop2set2propI = $true) & (setadjoinSub2 = $true) & (demorgan2b2 = $true) & (setadjoinSub = $true) & (nonemptyI1 = $true) & (upairsubunion = $true) & (demorgan1 = $true) & (subsetemptysetimpeq = $true) & (setminusI = $true) & (setunionE = $true) & (theeq = $true) & (notsubsetI = $true) & (singletonsswitch = $true) & (powersetE = $true) & (beta2 = $true) & (binunionEcases = $true) & (sepSubset = $true) & (lamProp = $true) & (binintersectTERcontra = $true) & (ubforcartprodlem1 = $true) & (ksndsingleton = $true) & (brelnall2 = $true) & (symdiffIneg1 = $true) & (demorgan1b = $true) & (dpsetconstrEL2 = $true) & (emptyI = $true) & (quantDeMorgan3 = $true) & (powersetAx = $true) & (ex1I2 = $true) & (($true = (in @ sK914 @ (powerset @ sK915))) & (($true != (in @ (setminus @ sK915 @ (binunion @ sK914 @ sK916)) @ (powerset @ (setminus @ sK915 @ sK914)))) & ($true = (in @ sK916 @ (powerset @ sK915))))) & (ap2apEq1 = $true) & (complementUnionInPowersetComplement = $true) & (woz1_1 = $true) & (emptyInPowerset = $true) & (powersetI1 = $true) & (binunionTE = $true) & (dpsetconstrEL1 = $true) & (setunionsingleton2 = $true) & (inIntersectImpInIntersectUnions = $true) & (prop2setE = $true) & (vacuousDall = $true) & (setunion__Cong = $true) & (wellorderingAx = $true) & (cartprodfstpairEq = $true) & (funcGraphProp1 = $true) & (replAx = $true) & (subsetE = $true) & (omega0Ax = $true) & (exuE3u = $true) & (setadjoinIR = $true) & (complementTnotintersectT = $true) & (eta1 = $true) & (ubforcartprodlem3 = $true) & (dsetconstrI = $true) & (exuE1 = $true) & (setukpairinjR2 = $true) & (omegaIndAx = $true) & (woz13rule4 = $true) & (singletoninpowerset = $true) & (descr__Cong = $true) & (dpsetconstrER = $true) & (binintersectSubset5 = $true) & (upairset2E = $true) & (setminusSubset1 = $true) & (iffalseProp1 = $true) & (symdiffI1 = $true) & (brelnall1 = $true) & (symdiffI2 = $true) & (ex1E2 = $true) & (dsetconstr__Cong = $true) & (binintersectSubset4 = $true) & (demorgan2a2 = $true) & (exu__Cong = $true) & (setukpairinjR11 = $true) & (emptysetsubset = $true) & (ksndpairEq = $true) & (setoftrueEq = $true) & (inComplementUnionImpNotIn1 = $true) & (powersetTE1 = $true) & (binunionT_lem = $true) & (symdiffIneg2 = $true) & (dsetconstrEL = $true) & (complementT_lem = $true) & (cartprodpairsurjEq = $true) & (emptysetAx = $true) & (doubleComplementE1 = $true) & (demorgan2b = $true) & (subsetTrans = $true) & (lam2p = $true) & (eqinunit = $true) & (in__Cong = $true) & (setOfPairsIsBReln = $true) & (cartprodpairmemER = $true) & (ifSingleton = $true) & (disjointsetsI1 = $true) & (exuI3 = $true) & (binunionE = $true) & (binintersectRsub = $true) & (dpsetconstrERa = $true) & (dpsetconstrI = $true) & (setukpairinjR12 = $true) & (setbeta = $true) & (powerset__Cong = $true) & (binintersectTELcontra = $true) & (doubleComplementEq = $true) & (beta1 = $true) & (notinemptyset = $true) & (quantDeMorgan4 = $true) & (notequalI1 = $true) & (emptyinPowerset = $true) & (cartprodsndin = $true) & (demorgan2a1 = $true)),
% 2.26/1.36 inference(skolemisation,[status(esa),new_symbols(skolem,[sK914,sK915,sK916])],[f1167,f2667,f2666])).
% 2.26/1.36 thf(f2666,plain,(
% 2.26/1.36 ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) & ? [X2 : $i] : (($true != (in @ (setminus @ X1 @ (binunion @ X0 @ X2)) @ (powerset @ (setminus @ X1 @ X0)))) & ($true = (in @ X2 @ (powerset @ X1))))) => (($true = (in @ sK914 @ (powerset @ sK915))) & ? [X2 : $i] : (($true != (in @ (setminus @ sK915 @ (binunion @ sK914 @ X2)) @ (powerset @ (setminus @ sK915 @ sK914)))) & ((in @ X2 @ (powerset @ sK915)) = $true)))),
% 2.26/1.36 introduced(definition,[],[choice_axiom])).
% 2.26/1.36 thf(f2667,plain,(
% 2.26/1.36 ? [X2 : $i] : (($true != (in @ (setminus @ sK915 @ (binunion @ sK914 @ X2)) @ (powerset @ (setminus @ sK915 @ sK914)))) & ((in @ X2 @ (powerset @ sK915)) = $true)) => (($true != (in @ (setminus @ sK915 @ (binunion @ sK914 @ sK916)) @ (powerset @ (setminus @ sK915 @ sK914)))) & ($true = (in @ sK916 @ (powerset @ sK915))))),
% 2.26/1.36 introduced(definition,[],[choice_axiom])).
% 2.26/1.36 thf(f1167,plain,(
% 2.26/1.36 (nonemptyImpWitness = $true) & (iffalse = $true) & (demorgan1a = $true) & (cartprodpairin = $true) & (binunionRsub = $true) & (binintersectSubset2 = $true) & (binintersectSubset1 = $true) & (emptysetimpfalse = $true) & (infuncsetfunc = $true) & (subsetE2 = $true) & (prop2setI = $true) & (setminusILneg = $true) & (emptyset__Cong = $true) & (eqimpsubset1 = $true) & (powersetsubset = $true) & (cartprodpairmemEL = $true) & (complementImpComplementIntersect = $true) & (bs114d = $true) & (binunionTEcontra = $true) & (powersetE1 = $true) & (setext = $true) & (setukpairinjR1 = $true) & (setukpairIR = $true) & (iftrueProp2 = $true) & (powersetI = $true) & (notinsingleton = $true) & (setminusER = $true) & (setminusSubset2 = $true) & (exuE2 = $true) & (setukpairinjL2 = $true) & (ubforcartprodlem2 = $true) & (eqbreln = $true) & (cartprodmempair1 = $true) & (setadjoinE = $true) & (setukpairinjL1 = $true) & (notequalI2 = $true) & (inPowerset = $true) & (setadjoinOr = $true) & (setunionsingleton1 = $true) & (eqimpsubset2 = $true) & (powersetTI1 = $true) & (setminusERneg = $true) & (woz1_3 = $true) & (binintersectI = $true) & (upairsetIR = $true) & (woz1_4 = $true) & (subsetI2 = $true) & (singletonsuniq = $true) & (sepInPowerset = $true) & (setextT = $true) & (descrp = $true) & (inIntersectImpInUnion = $true) & (cartprodmempaircEq = $true) & (emptyE1 = $true) & (subbreln = $true) & (setextsub = $true) & (binunionIL = $true) & (setminusLsub = $true) & (contraSubsetComplement = $true) & (funcGraphProp2 = $true) & (binintersectT_lem = $true) & (setukpairinjL = $true) & (subsetRefl = $true) & (ap2apEq2 = $true) & (setminusELneg = $true) & (setextAx = $true) & (cartprodmempair = $true) & (exuI2 = $true) & (ex1E1 = $true) & (contrasubsetT1 = $true) & (secondinupair = $true) & (doubleComplementI1 = $true) & (subset2powerset = $true) & (setukpairinjR = $true) & (doubleComplementSub2 = $true) & (funcImageSingleton = $true) & (kpairp = $true) & (subsetI1 = $true) & (woz13rule3 = $true) & (iftrueorfalse = $true) & (lamp = $true) & (contrasubsetT = $true) & (omegaSAx = $true) & (inComplementUnionImpInComplement1 = $true) & (symdiffE = $true) & (kfstpairEq = $true) & (kpairsurjEq = $true) & (cartprodfstin = $true) & (notdexE = $true) & (funcinfuncset = $true) & (setadjoin__Cong = $true) & (complementSubsetComplementIntersect = $true) & (ifp = $true) & (singletonprop = $true) & (nonemptyE1 = $true) & (funcext2 = $true) & (cartprodsndpairEq = $true) & (complementTI1 = $true) & (binunionIR = $true) & (woz13rule0 = $true) & (iftrueProp1 = $true) & (dsetconstrER = $true) & (setunionE2 = $true) & (iftrue = $true) & (singletoninpowunion = $true) & (lam2lamEq = $true) & (exuE3e = $true) & (iffalseProp2 = $true) & (woz13rule2 = $true) & (setminusEL = $true) & (singletonsubset = $true) & (binintersectLsub = $true) & (setminusIRneg = $true) & (nonemptyI = $true) & (demorgan2a = $true) & (apProp = $true) & (funcGraphProp3 = $true) & (setminusT_lem = $true) & (uniqinunit = $true) & (inCongP = $true) & (quantDeMorgan2 = $true) & (woz1_2 = $true) & (demorgan2 = $true) & (emptyinunitempty = $true) & (emptysetE = $true) & (upairsetE = $true) & (binunionTIRcontra = $true) & (setadjoinIL = $true) & (funcGraphProp4 = $true) & (upairsetIL = $true) & (contrasubsetT2 = $true) & (noeltsimpempty = $true) & (powersetT_lem = $true) & (kfstsingleton = $true) & (exuI1 = $true) & (binunionLsub = $true) & (setunionI = $true) & (subsetTI = $true) & (setukpairIL = $true) & (upairset2IR = $true) & (funcext = $true) & (upairequniteq = $true) & (upairinpowunion = $true) & (eta2 = $true) & (binintersectSubset3 = $true) & (kpairiskpair = $true) & (inIntersectImpInUnion2 = $true) & (setunionAx = $true) & (dpsetconstrSub = $true) & (binintersectEL = $true) & (funcextLem = $true) & (notdallE = $true) & (ex1I = $true) & (omega__Cong = $true) & (exuEu = $true) & (complementTcontraSubset = $true) & (complementInPowersetComplementIntersect = $true) & (setadjoinAx = $true) & (woz13rule1 = $true) & (subPowSU = $true) & (foundationAx = $true) & (doubleComplementSub1 = $true) & (theprop = $true) & (binintersectER = $true) & (ap2p = $true) & (complementTE1 = $true) & (app = $true) & (contrasubsetT3 = $true) & (setunionsingleton = $true) & (quantDeMorgan1 = $true) & (binunionTILcontra = $true) & (intersectInPowersetIntersectUnions = $true) & (prop2set2propI = $true) & (setadjoinSub2 = $true) & (demorgan2b2 = $true) & (setadjoinSub = $true) & (nonemptyI1 = $true) & (upairsubunion = $true) & (demorgan1 = $true) & (subsetemptysetimpeq = $true) & (setminusI = $true) & (setunionE = $true) & (theeq = $true) & (notsubsetI = $true) & (singletonsswitch = $true) & (powersetE = $true) & (beta2 = $true) & (binunionEcases = $true) & (sepSubset = $true) & (lamProp = $true) & (binintersectTERcontra = $true) & (ubforcartprodlem1 = $true) & (ksndsingleton = $true) & (brelnall2 = $true) & (symdiffIneg1 = $true) & (demorgan1b = $true) & (dpsetconstrEL2 = $true) & (emptyI = $true) & (quantDeMorgan3 = $true) & (powersetAx = $true) & (ex1I2 = $true) & ? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) & ? [X2 : $i] : (($true != (in @ (setminus @ X1 @ (binunion @ X0 @ X2)) @ (powerset @ (setminus @ X1 @ X0)))) & ($true = (in @ X2 @ (powerset @ X1))))) & (ap2apEq1 = $true) & (complementUnionInPowersetComplement = $true) & (woz1_1 = $true) & (emptyInPowerset = $true) & (powersetI1 = $true) & (binunionTE = $true) & (dpsetconstrEL1 = $true) & (setunionsingleton2 = $true) & (inIntersectImpInIntersectUnions = $true) & (prop2setE = $true) & (vacuousDall = $true) & (setunion__Cong = $true) & (wellorderingAx = $true) & (cartprodfstpairEq = $true) & (funcGraphProp1 = $true) & (replAx = $true) & (subsetE = $true) & (omega0Ax = $true) & (exuE3u = $true) & (setadjoinIR = $true) & (complementTnotintersectT = $true) & (eta1 = $true) & (ubforcartprodlem3 = $true) & (dsetconstrI = $true) & (exuE1 = $true) & (setukpairinjR2 = $true) & (omegaIndAx = $true) & (woz13rule4 = $true) & (singletoninpowerset = $true) & (descr__Cong = $true) & (dpsetconstrER = $true) & (binintersectSubset5 = $true) & (upairset2E = $true) & (setminusSubset1 = $true) & (iffalseProp1 = $true) & (symdiffI1 = $true) & (brelnall1 = $true) & (symdiffI2 = $true) & (ex1E2 = $true) & (dsetconstr__Cong = $true) & (binintersectSubset4 = $true) & (demorgan2a2 = $true) & (exu__Cong = $true) & (setukpairinjR11 = $true) & (emptysetsubset = $true) & (ksndpairEq = $true) & (setoftrueEq = $true) & (inComplementUnionImpNotIn1 = $true) & (powersetTE1 = $true) & (binunionT_lem = $true) & (symdiffIneg2 = $true) & (dsetconstrEL = $true) & (complementT_lem = $true) & (cartprodpairsurjEq = $true) & (emptysetAx = $true) & (doubleComplementE1 = $true) & (demorgan2b = $true) & (subsetTrans = $true) & (lam2p = $true) & (eqinunit = $true) & (in__Cong = $true) & (setOfPairsIsBReln = $true) & (cartprodpairmemER = $true) & (ifSingleton = $true) & (disjointsetsI1 = $true) & (exuI3 = $true) & (binunionE = $true) & (binintersectRsub = $true) & (dpsetconstrERa = $true) & (dpsetconstrI = $true) & (setukpairinjR12 = $true) & (setbeta = $true) & (powerset__Cong = $true) & (binintersectTELcontra = $true) & (doubleComplementEq = $true) & (beta1 = $true) & (notinemptyset = $true) & (quantDeMorgan4 = $true) & (notequalI1 = $true) & (emptyinPowerset = $true) & (cartprodsndin = $true) & (demorgan2a1 = $true)),
% 2.26/1.36 inference(flattening,[status(thm)],[f1166])).
% 2.26/1.36 thf(f1166,plain,(
% 2.26/1.36 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) & ? [X2 : $i] : (($true != (in @ (setminus @ X1 @ (binunion @ X0 @ X2)) @ (powerset @ (setminus @ X1 @ X0)))) & ($true = (in @ X2 @ (powerset @ X1))))) & (woz1_4 = $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)),
% 2.26/1.36 inference(ennf_transformation,[status(thm)],[f516])).
% 2.26/1.36 thf(f516,plain,(
% 2.26/1.36 ~((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) => ((woz1_4 = $true) => ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X1))) => ($true = (in @ (setminus @ X1 @ (binunion @ X0 @ X2)) @ (powerset @ (setminus @ X1 @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.26/1.36 inference(fool_elimination,[status(thm)],[f515])).
% 2.26/1.36 thf(f515,plain,(
% 2.26/1.36 ~(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 => (woz1_4 => ! [X0 : $i,X1 : $i] : ((in @ X0 @ (powerset @ X1)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X1)) => (in @ (setminus @ X1 @ (binunion @ X0 @ X2)) @ (powerset @ (setminus @ X1 @ X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.26/1.36 inference(rectify,[status(thm)],[f289])).
% 2.26/1.36 thf(f289,negated_conjecture,(
% 2.26/1.36 ~(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 => (woz1_4 => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => (in @ (setminus @ X3 @ (binunion @ X11 @ X16)) @ (powerset @ (setminus @ X3 @ X11))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.26/1.36 inference(negated_conjecture,[status(cth)],[f288])).
% 2.26/1.36 thf(f288,conjecture,(
% 2.26/1.36 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 => (woz1_4 => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => (in @ (setminus @ X3 @ (binunion @ X11 @ X16)) @ (powerset @ (setminus @ X3 @ X11)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.26/1.36 file('/export/starexec/sandbox/tmp/tmp.7mug10HeGF/DTF2THF_7573.p',woz1_5)).
% 2.26/1.36 thf(f4479,plain,(
% 2.26/1.36 ($true != (in @ sK916 @ (powerset @ sK915)))),
% 2.26/1.36 inference(trivial_inequality_removal,[status(thm)],[f4478])).
% 2.26/1.36 thf(f4478,plain,(
% 2.26/1.36 ($true != (in @ sK916 @ (powerset @ sK915))) | ($true != $true)),
% 2.26/1.36 inference(forward_demodulation,[status(thm)],[f4477,f3989])).
% 2.26/1.36 thf(f3989,plain,(
% 2.26/1.36 (complementUnionInPowersetComplement = $true)),
% 2.26/1.36 inference(cnf_transformation,[status(thm)],[f2668])).
% 2.26/1.36 thf(f4477,plain,(
% 2.26/1.36 ($true != (in @ sK916 @ (powerset @ sK915))) | (complementUnionInPowersetComplement != $true)),
% 2.26/1.36 inference(trivial_inequality_removal,[status(thm)],[f4476])).
% 2.26/1.36 thf(f4476,plain,(
% 2.26/1.36 ($true != $true) | (complementUnionInPowersetComplement != $true) | ($true != (in @ sK916 @ (powerset @ sK915)))),
% 2.26/1.36 inference(forward_demodulation,[status(thm)],[f4464,f3993])).
% 2.26/1.36 thf(f3993,plain,(
% 2.26/1.36 ($true = (in @ sK914 @ (powerset @ sK915)))),
% 2.26/1.36 inference(cnf_transformation,[status(thm)],[f2668])).
% 2.26/1.36 thf(f4464,plain,(
% 2.26/1.36 ($true != (in @ sK916 @ (powerset @ sK915))) | ($true != (in @ sK914 @ (powerset @ sK915))) | (complementUnionInPowersetComplement != $true)),
% 2.26/1.36 inference(trivial_inequality_removal,[status(thm)],[f4447])).
% 2.26/1.36 thf(f4447,plain,(
% 2.26/1.36 ($true != (in @ sK916 @ (powerset @ sK915))) | ($true != (in @ sK914 @ (powerset @ sK915))) | ($true != $true) | (complementUnionInPowersetComplement != $true)),
% 2.26/1.36 inference(superposition,[status(thm)],[f3992,f3805])).
% 2.26/1.36 thf(f3805,plain,(
% 2.26/1.36 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true = (in @ (setminus @ X0 @ (binunion @ X1 @ X2)) @ (powerset @ (setminus @ X0 @ X1)))) | (complementUnionInPowersetComplement != $true) | ($true != (in @ X1 @ (powerset @ X0))) | ($true != (in @ X2 @ (powerset @ X0)))) )),
% 2.26/1.36 inference(cnf_transformation,[status(thm)],[f2550])).
% 2.26/1.36 thf(f2550,plain,(
% 2.26/1.36 (! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ (powerset @ X0))) | ! [X2 : $i] : (($true != (in @ X2 @ (powerset @ X0))) | ($true = (in @ (setminus @ X0 @ (binunion @ X1 @ X2)) @ (powerset @ (setminus @ X0 @ X1)))))) | (complementUnionInPowersetComplement != $true)) & ((complementUnionInPowersetComplement = $true) | (($true = (in @ sK840 @ (powerset @ sK839))) & (($true = (in @ sK841 @ (powerset @ sK839))) & ((in @ (setminus @ sK839 @ (binunion @ sK840 @ sK841)) @ (powerset @ (setminus @ sK839 @ sK840))) != $true))))),
% 2.26/1.36 inference(skolemisation,[status(esa),new_symbols(skolem,[sK839,sK840,sK841])],[f2547,f2549,f2548])).
% 2.26/1.36 thf(f2548,plain,(
% 2.26/1.36 ? [X3 : $i,X4 : $i] : (((in @ X4 @ (powerset @ X3)) = $true) & ? [X5 : $i] : (($true = (in @ X5 @ (powerset @ X3))) & ($true != (in @ (setminus @ X3 @ (binunion @ X4 @ X5)) @ (powerset @ (setminus @ X3 @ X4)))))) => (($true = (in @ sK840 @ (powerset @ sK839))) & ? [X5 : $i] : (($true = (in @ X5 @ (powerset @ sK839))) & ((in @ (setminus @ sK839 @ (binunion @ sK840 @ X5)) @ (powerset @ (setminus @ sK839 @ sK840))) != $true)))),
% 2.26/1.36 introduced(definition,[],[choice_axiom])).
% 2.26/1.36 thf(f2549,plain,(
% 2.26/1.36 ? [X5 : $i] : (($true = (in @ X5 @ (powerset @ sK839))) & ((in @ (setminus @ sK839 @ (binunion @ sK840 @ X5)) @ (powerset @ (setminus @ sK839 @ sK840))) != $true)) => (($true = (in @ sK841 @ (powerset @ sK839))) & ((in @ (setminus @ sK839 @ (binunion @ sK840 @ sK841)) @ (powerset @ (setminus @ sK839 @ sK840))) != $true))),
% 2.26/1.36 introduced(definition,[],[choice_axiom])).
% 2.26/1.36 thf(f2547,plain,(
% 2.26/1.36 (! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ (powerset @ X0))) | ! [X2 : $i] : (($true != (in @ X2 @ (powerset @ X0))) | ($true = (in @ (setminus @ X0 @ (binunion @ X1 @ X2)) @ (powerset @ (setminus @ X0 @ X1)))))) | (complementUnionInPowersetComplement != $true)) & ((complementUnionInPowersetComplement = $true) | ? [X3 : $i,X4 : $i] : (((in @ X4 @ (powerset @ X3)) = $true) & ? [X5 : $i] : (($true = (in @ X5 @ (powerset @ X3))) & ($true != (in @ (setminus @ X3 @ (binunion @ X4 @ X5)) @ (powerset @ (setminus @ X3 @ X4)))))))),
% 2.26/1.36 inference(rectify,[status(thm)],[f2546])).
% 2.26/1.36 thf(f2546,plain,(
% 2.26/1.36 (! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ (powerset @ X0))) | ! [X2 : $i] : (($true != (in @ X2 @ (powerset @ X0))) | ($true = (in @ (setminus @ X0 @ (binunion @ X1 @ X2)) @ (powerset @ (setminus @ X0 @ X1)))))) | (complementUnionInPowersetComplement != $true)) & ((complementUnionInPowersetComplement = $true) | ? [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) & ? [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) & ($true != (in @ (setminus @ X0 @ (binunion @ X1 @ X2)) @ (powerset @ (setminus @ X0 @ X1)))))))),
% 2.26/1.36 inference(nnf_transformation,[status(thm)],[f1213])).
% 2.26/1.36 thf(f1213,plain,(
% 2.26/1.36 ! [X0 : $i,X1 : $i] : (($true != (in @ X1 @ (powerset @ X0))) | ! [X2 : $i] : (($true != (in @ X2 @ (powerset @ X0))) | ($true = (in @ (setminus @ X0 @ (binunion @ X1 @ X2)) @ (powerset @ (setminus @ X0 @ X1)))))) <=> (complementUnionInPowersetComplement = $true)),
% 2.26/1.36 inference(ennf_transformation,[status(thm)],[f584])).
% 2.26/1.36 thf(f584,plain,(
% 2.26/1.36 ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) => ($true = (in @ (setminus @ X0 @ (binunion @ X1 @ X2)) @ (powerset @ (setminus @ X0 @ X1)))))) <=> (complementUnionInPowersetComplement = $true)),
% 2.26/1.36 inference(fool_elimination,[status(thm)],[f583])).
% 2.26/1.36 thf(f583,plain,(
% 2.26/1.36 (! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X0)) => (in @ (setminus @ X0 @ (binunion @ X1 @ X2)) @ (powerset @ (setminus @ X0 @ X1))))) = complementUnionInPowersetComplement)),
% 2.26/1.36 inference(rectify,[status(thm)],[f270])).
% 2.26/1.36 thf(f270,axiom,(
% 2.26/1.36 (! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => (in @ (setminus @ X3 @ (binunion @ X11 @ X16)) @ (powerset @ (setminus @ X3 @ X11))))) = complementUnionInPowersetComplement)),
% 2.26/1.36 file('/export/starexec/sandbox/tmp/tmp.7mug10HeGF/DTF2THF_7573.p',complementUnionInPowersetComplement)).
% 2.26/1.36 thf(f3992,plain,(
% 2.26/1.36 ($true != (in @ (setminus @ sK915 @ (binunion @ sK914 @ sK916)) @ (powerset @ (setminus @ sK915 @ sK914))))),
% 2.26/1.36 inference(cnf_transformation,[status(thm)],[f2668])).
% 2.26/1.36 % SZS output end Proof for DTF2THF_7573
% 2.26/1.36 % (7888)------------------------------
% 2.26/1.36 % (7888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.26/1.36 % (7888)Termination reason: Refutation
% 2.26/1.36
% 2.26/1.36 % (7888)Memory used [KB]: 10874
% 2.26/1.36 % (7888)Time elapsed: 0.124 s
% 2.26/1.36 % (7888)Instructions burned: 400 (million)
% 2.26/1.36 % (7888)------------------------------
% 2.26/1.36 % (7888)------------------------------
% 2.26/1.36 % (7819)Success in time 0.206 s
%------------------------------------------------------------------------------