%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU761^1 : 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 : n010.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:36:59 AM UTC 2026
% Result : Theorem 2.95s 1.87s
% Output : Refutation 2.95s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 6
% Syntax : Number of formulae : 32 ( 9 unt; 0 typ; 0 def)
% Number of atoms : 4633 (1230 equ; 0 cnn)
% Maximal formula atoms : 283 ( 144 avg)
% Number of connectives : 2445 ( 50 ~; 28 |; 883 &; 343 @)
% ( 3 <=>;1138 =>; 0 <=; 0 <~>)
% Maximal formula depth : 291 ( 66 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 475 ( 472 usr; 286 con; 0-4 aty)
% Number of variables : 70 ( 0 ^; 43 !; 27 ?; 70 :)
% 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_329,type,
sP0: $i > $i > $i > $o > $o ).
thf(func_def_332,type,
sP3: $i > $i > $i > $i > $o ).
thf(func_def_335,type,
sP6: $i > $i > $o ).
thf(func_def_336,type,
sP7: $i > $o ).
thf(func_def_337,type,
sP8: $i > $i > $o ).
thf(func_def_338,type,
sP9: $i > $i > $o ).
thf(func_def_339,type,
sP10: $i > $i > $i > $o ).
thf(func_def_345,type,
sK16: $i > $o ).
thf(func_def_347,type,
sK18: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_348,type,
sK19: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_355,type,
sK26: $i > $o ).
thf(func_def_359,type,
sK30: ( $i > $i ) > $i > $i > $i ).
thf(func_def_362,type,
sK33: $i > $i ).
thf(func_def_365,type,
sK36: $i > $i > $o ).
thf(func_def_386,type,
sK57: $i > $i > $o ).
thf(func_def_427,type,
sK98: $i > $i ).
thf(func_def_432,type,
sK103: $i > $o ).
thf(func_def_434,type,
sK105: $i > $o ).
thf(func_def_439,type,
sK110: $i > $i > $o ).
thf(func_def_472,type,
sK143: $i > $o ).
thf(func_def_474,type,
sK145: $i > ( $i > $o ) > $i > $i ).
thf(func_def_483,type,
sK154: $i > $i > $i ).
thf(func_def_498,type,
sK169: $i > ( $i > $o ) > $i ).
thf(func_def_499,type,
sK170: $i > $o ).
thf(func_def_502,type,
sK173: $i > ( $i > $o ) > $i ).
thf(func_def_503,type,
sK174: $i > $o ).
thf(func_def_508,type,
sK179: $i > $i > $i > $i ).
thf(func_def_509,type,
sK180: $i > $i > $i > $i ).
thf(func_def_516,type,
sK187: $i > $o ).
thf(func_def_522,type,
sK193: $i > $i > $o ).
thf(func_def_524,type,
sK195: $i > $i ).
thf(func_def_525,type,
sK196: $i > $i ).
thf(func_def_526,type,
sK197: $i > ( $i > $i > $o ) > $i ).
thf(func_def_527,type,
sK198: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_528,type,
sK199: $i > ( $i > $i > $o ) > $i ).
thf(func_def_548,type,
sK219: ( $i > $i ) > $i > $i > $i ).
thf(func_def_551,type,
sK222: $i > $i ).
thf(func_def_561,type,
sK232: $i > $o ).
thf(func_def_564,type,
sK235: $i > ( $i > $o ) > $i ).
thf(func_def_565,type,
sK236: $i > ( $i > $o ) > $i ).
thf(func_def_576,type,
sK247: $i > $i > $i > $i > $i ).
thf(func_def_577,type,
sK248: $i > $i > $i > $i > $i ).
thf(func_def_578,type,
sK249: $i > $i > $i > $i > $i ).
thf(func_def_579,type,
sK250: $i > $i > $i > $i > $i ).
thf(func_def_588,type,
sK259: $i > $o ).
thf(func_def_590,type,
sK261: ( $i > $o ) > $i ).
thf(func_def_591,type,
sK262: ( $i > $o ) > $i ).
thf(func_def_615,type,
sK286: $i > $o ).
thf(func_def_616,type,
sK287: $i > ( $i > $o ) > $i ).
thf(func_def_617,type,
sK288: $i > $o ).
thf(func_def_622,type,
sK293: $i > $i > $i > $i ).
thf(func_def_630,type,
sK301: $i > ( $i > $o ) > $i ).
thf(func_def_631,type,
sK302: $i > $o ).
thf(func_def_703,type,
sK374: $i > $o ).
thf(func_def_704,type,
sK375: $i > $i ).
thf(func_def_705,type,
sK376: ( $i > $o ) > $i ).
thf(func_def_709,type,
sK380: $i > $i ).
thf(func_def_727,type,
sK398: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_728,type,
sK399: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_731,type,
sK402: $i > $o ).
thf(func_def_732,type,
sK403: $i > $o ).
thf(func_def_735,type,
sK406: $i > $i ).
thf(func_def_739,type,
sK410: $i > $i > $i ).
thf(func_def_743,type,
sK414: $i > $i ).
thf(func_def_746,type,
sK417: $i > $o ).
thf(func_def_755,type,
sK426: $i > $i > $i ).
thf(func_def_758,type,
sK429: $i > $i > $o ).
thf(func_def_766,type,
sK437: $i > $o ).
thf(func_def_768,type,
sK439: ( $i > $o ) > $i > $i ).
thf(func_def_769,type,
sK440: ( $i > $o ) > $i > $i ).
thf(func_def_770,type,
sK441: $i > $o ).
thf(func_def_774,type,
sK445: $i > $o ).
thf(func_def_779,type,
sK450: $i > $i > $i ).
thf(func_def_780,type,
sK451: $i > $i > $i ).
thf(func_def_786,type,
sK457: $i > $i > $o ).
thf(func_def_788,type,
sK459: $i > $o ).
thf(func_def_789,type,
sK460: $i > $o ).
thf(func_def_790,type,
sK461: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_791,type,
sK462: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_803,type,
sK474: $i > $o ).
thf(func_def_813,type,
sK484: $i > $o ).
thf(func_def_815,type,
sK486: $i > $o ).
thf(func_def_816,type,
sK487: $i > $i ).
thf(func_def_817,type,
sK488: ( $i > $o ) > $i ).
thf(func_def_821,type,
sK492: $i > $o ).
thf(func_def_823,type,
sK494: $i > ( $i > $o ) > $i ).
thf(func_def_827,type,
sK498: $i > $i > $o ).
thf(func_def_844,type,
sK515: $i > $o ).
thf(func_def_850,type,
sK521: $i > $i > $i ).
thf(func_def_875,type,
sK546: $i > $o ).
thf(func_def_886,type,
sK557: $i > $o ).
thf(func_def_887,type,
sK558: $i > $i > $i ).
thf(func_def_888,type,
sK559: $i > $i > $i ).
thf(func_def_889,type,
sK560: $i > $i > $i ).
thf(func_def_890,type,
sK561: $i > $i > $i ).
thf(func_def_891,type,
sK562: $i > $i > $i > $i ).
thf(func_def_892,type,
sK563: $i > $i ).
thf(func_def_893,type,
sK564: $i > $i ).
thf(func_def_894,type,
sK565: $i > $i ).
thf(func_def_895,type,
sK566: $i > $i ).
thf(func_def_896,type,
sK567: $i > $i > $i ).
thf(func_def_897,type,
sK568: $i > $i > $i > $i ).
thf(func_def_898,type,
sK569: $i > $i > $i > $i ).
thf(func_def_899,type,
sK570: $i > $i > $i ).
thf(func_def_900,type,
sK571: $i > $i > $i ).
thf(func_def_902,type,
sK573: $i > $i ).
thf(func_def_903,type,
sK574: $i > $i ).
thf(func_def_904,type,
sK575: $i > $i ).
thf(func_def_905,type,
sK576: $i > $i > $i ).
thf(func_def_906,type,
sK577: $i > $i > $i ).
thf(func_def_916,type,
sK587: $o > $i > $i > $i ).
thf(func_def_932,type,
sK603: $i > ( $i > $o ) > $i ).
thf(func_def_933,type,
sK604: $i > $o ).
thf(func_def_938,type,
sK609: $i > $o ).
thf(func_def_941,type,
sK612: $i > ( $i > $o ) > $i ).
thf(func_def_967,type,
sK638: $i > $i > $i > $i ).
thf(func_def_968,type,
sK639: $i > $i > $i > $i ).
thf(func_def_978,type,
sK649: $i > $i > $i > $i > $i ).
thf(func_def_982,type,
sK653: $i > $i > $i > $i ).
thf(func_def_988,type,
sK659: $i > $o ).
thf(func_def_989,type,
sK660: $i > $o ).
thf(func_def_990,type,
sK661: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_991,type,
sK662: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1011,type,
sK682: $i > $o ).
thf(func_def_1013,type,
sK684: ( $i > $o ) > $i > $i ).
thf(func_def_1028,type,
sK699: $i > $i > $i > $i > $i ).
thf(func_def_1029,type,
sK700: $i > $i > $i > $i > $i ).
thf(func_def_1066,type,
sK737: $i > $i ).
thf(func_def_1068,type,
sK739: $i > ( $i > $i ) > $i > $i ).
thf(func_def_1084,type,
sK755: $i > $i ).
thf(func_def_1085,type,
sK756: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1086,type,
sK757: $i > $o ).
thf(func_def_1089,type,
sK760: $i > ( $i > $o ) > $i ).
thf(func_def_1090,type,
sK761: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1091,type,
sK762: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1095,type,
sK766: $i > $o ).
thf(func_def_1106,type,
sK777: $i > $i > $i ).
thf(func_def_1114,type,
sK785: $i > $i ).
thf(func_def_1115,type,
sK786: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1139,type,
sK810: $i > $i > $i ).
thf(func_def_1144,type,
sK815: $i > $i > $i > $i > $i ).
thf(func_def_1146,type,
sK817: $i > $i ).
thf(func_def_1154,type,
sK825: $i > $i > $i ).
thf(func_def_1161,type,
sK832: $i > $i > $i > $i > $i ).
thf(func_def_1162,type,
sK833: $i > $o ).
thf(func_def_1163,type,
sK834: ( $i > $o ) > $i ).
thf(func_def_1195,type,
sK866: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1198,type,
sK869: $i > $i ).
thf(func_def_1210,type,
sK881: $i > $i > $o ).
thf(func_def_1216,type,
sK887: $i > $i ).
thf(func_def_1219,type,
sK890: $i > $i ).
thf(func_def_1230,type,
ph901:
!>[X0: $tType] : X0 ).
thf(f4548,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f4539]) ).
thf(f4539,plain,
$true != $true,
inference(superposition,[status(thm)],[f4433,f3743]) ).
thf(f3743,plain,
( ( subset @ sK687 @ sK690 )
= $true ),
inference(cnf_transformation,[status(thm)],[f2296]) ).
thf(f2296,plain,
( ( binintersectSubset2 = $true )
& ( notdexE = $true )
& ( setukpairinjL = $true )
& ( dpsetconstrEL2 = $true )
& ( doubleComplementE1 = $true )
& ( ap2apEq2 = $true )
& ( binintersectSubset3 = $true )
& ( setunionsingleton = $true )
& ( binunionEcases = $true )
& ( doubleComplementSub1 = $true )
& ( powersetsubset = $true )
& ( binunionE = $true )
& ( binintersectI = $true )
& ( notsubsetI = $true )
& ( cartprodmempair = $true )
& ( binunionTILcontra = $true )
& ( dsetconstrEL = $true )
& ( complementTnotintersectT = $true )
& ( lam2lamEq = $true )
& ( woz13rule1 = $true )
& ( setminusIRneg = $true )
& ( eqimpsubset2 = $true )
& ( contrasubsetT1 = $true )
& ( setminusSubset2 = $true )
& ( demorgan2 = $true )
& ( uniqinunit = $true )
& ( kpairiskpair = $true )
& ( notdallE = $true )
& ( complementTE1 = $true )
& ( omegaIndAx = $true )
& ( dsetconstrI = $true )
& ( powersetE = $true )
& ( emptysetsubset = $true )
& ( beta2 = $true )
& ( ubforcartprodlem2 = $true )
& ( descr__Cong = $true )
& ( contraSubsetComplement = $true )
& ( noeltsimpempty = $true )
& ( ex1I = $true )
& ( exuE3u = $true )
& ( vacuousDall = $true )
& ( setukpairinjR11 = $true )
& ( emptysetE = $true )
& ( replAx = $true )
& ( exuE2 = $true )
& ( ap2p = $true )
& ( setadjoinIR = $true )
& ( dpsetconstrI = $true )
& ( setunionE2 = $true )
& ( binunionT_lem = $true )
& ( setunionAx = $true )
& ( setukpairinjR = $true )
& ( cartprodpairin = $true )
& ( binintersectLsub = $true )
& ( setoftrueEq = $true )
& ( iftrueProp1 = $true )
& ( ubforcartprodlem3 = $true )
& ( dpsetconstrEL1 = $true )
& ( setbeta = $true )
& ( exuI2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( nonemptyE1 = $true )
& ( upairset2IR = $true )
& ( in__Cong = $true )
& ( cartprodsndpairEq = $true )
& ( disjointsetsI1 = $true )
& ( upairequniteq = $true )
& ( complementT_lem = $true )
& ( exuI1 = $true )
& ( symdiffI2 = $true )
& ( complementTcontraSubset = $true )
& ( inIntersectImpInUnion = $true )
& ( binunionLsub = $true )
& ( upairset2E = $true )
& ( cartprodfstin = $true )
& ( nonemptyImpWitness = $true )
& ( binunionIL = $true )
& ( funcext = $true )
& ( infuncsetfunc = $true )
& ( binintersectTERcontra = $true )
& ( doubleComplementSub2 = $true )
& ( inPowerset = $true )
& ( exuEu = $true )
& ( $true
= ( in @ sK689 @ ( powerset @ sK688 ) ) )
& ( $true
= ( in @ sK690 @ ( powerset @ sK688 ) ) )
& ( $true
!= ( subset @ sK687 @ ( binintersect @ sK689 @ sK690 ) ) )
& ( $true
= ( subset @ sK687 @ sK689 ) )
& ( ( subset @ sK687 @ sK690 )
= $true )
& ( ( in @ sK687 @ ( powerset @ sK688 ) )
= $true )
& ( funcinfuncset = $true )
& ( setunionI = $true )
& ( eta2 = $true )
& ( ubforcartprodlem1 = $true )
& ( setunionE = $true )
& ( upairinpowunion = $true )
& ( dsetconstrER = $true )
& ( complementImpComplementIntersect = $true )
& ( setextT = $true )
& ( setminusILneg = $true )
& ( powersetT_lem = $true )
& ( kpairp = $true )
& ( singletonsswitch = $true )
& ( binintersectSubset5 = $true )
& ( dsetconstr__Cong = $true )
& ( woz13rule2 = $true )
& ( brelnall2 = $true )
& ( binunionIR = $true )
& ( demorgan2b = $true )
& ( setminusELneg = $true )
& ( emptyinunitempty = $true )
& ( doubleComplementI1 = $true )
& ( demorgan2a1 = $true )
& ( lamProp = $true )
& ( secondinupair = $true )
& ( funcGraphProp4 = $true )
& ( dpsetconstrSub = $true )
& ( subsetemptysetimpeq = $true )
& ( funcext2 = $true )
& ( setminusT_lem = $true )
& ( demorgan1a = $true )
& ( funcImageSingleton = $true )
& ( cartprodpairmemEL = $true )
& ( setminusLsub = $true )
& ( omegaSAx = $true )
& ( setukpairinjR12 = $true )
& ( doubleComplementEq = $true )
& ( contrasubsetT = $true )
& ( setadjoin__Cong = $true )
& ( exuI3 = $true )
& ( setminusSubset1 = $true )
& ( binunionRsub = $true )
& ( powersetTI1 = $true )
& ( omega0Ax = $true )
& ( notequalI1 = $true )
& ( emptyInPowerset = $true )
& ( notequalI2 = $true )
& ( symdiffE = $true )
& ( subsetI1 = $true )
& ( lam2p = $true )
& ( sepInPowerset = $true )
& ( binintersectTELcontra = $true )
& ( emptyI = $true )
& ( symdiffIneg1 = $true )
& ( demorgan2a2 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( setunionsingleton2 = $true )
& ( powersetAx = $true )
& ( symdiffI1 = $true )
& ( kfstsingleton = $true )
& ( binintersectER = $true )
& ( eqinunit = $true )
& ( nonemptyI1 = $true )
& ( upairsubunion = $true )
& ( contrasubsetT2 = $true )
& ( exu__Cong = $true )
& ( iffalseProp2 = $true )
& ( contrasubsetT3 = $true )
& ( ifp = $true )
& ( singletoninpowunion = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( setminusERneg = $true )
& ( complementTI1 = $true )
& ( ifSingleton = $true )
& ( cartprodpairmemER = $true )
& ( subsetTI = $true )
& ( subPowSU = $true )
& ( ex1I2 = $true )
& ( singletoninpowerset = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( notinsingleton = $true )
& ( demorgan2a = $true )
& ( demorgan2b2 = $true )
& ( subsetTrans = $true )
& ( powersetI1 = $true )
& ( prop2set2propI = $true )
& ( inIntersectImpInUnion2 = $true )
& ( cartprodmempair1 = $true )
& ( setadjoinOr = $true )
& ( ex1E2 = $true )
& ( setOfPairsIsBReln = $true )
& ( setukpairinjL2 = $true )
& ( upairsetIL = $true )
& ( subsetE2 = $true )
& ( subsetI2 = $true )
& ( apProp = $true )
& ( theprop = $true )
& ( binunionTEcontra = $true )
& ( demorgan1 = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodmempaircEq = $true )
& ( kfstpairEq = $true )
& ( cartprodsndin = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( setminusER = $true )
& ( binintersectSubset4 = $true )
& ( powersetTE1 = $true )
& ( subset2powerset = $true )
& ( dpsetconstrERa = $true )
& ( brelnall1 = $true )
& ( eta1 = $true )
& ( funcGraphProp1 = $true )
& ( setukpairIR = $true )
& ( setadjoinIL = $true )
& ( app = $true )
& ( dpsetconstrER = $true )
& ( beta1 = $true )
& ( emptysetAx = $true )
& ( upairsetE = $true )
& ( funcGraphProp3 = $true )
& ( notinemptyset = $true )
& ( setadjoinSub2 = $true )
& ( binunionTE = $true )
& ( binintersectRsub = $true )
& ( setextsub = $true )
& ( binintersectT_lem = $true )
& ( setadjoinE = $true )
& ( binintersectSubset1 = $true )
& ( emptyE1 = $true )
& ( nonemptyI = $true )
& ( demorgan1b = $true )
& ( emptyinPowerset = $true )
& ( ap2apEq1 = $true )
& ( singletonsuniq = $true )
& ( exuE3e = $true )
& ( prop2setI = $true )
& ( setextAx = $true )
& ( upairsetIR = $true )
& ( prop2setE = $true )
& ( setunionsingleton1 = $true )
& ( setunion__Cong = $true )
& ( complementSubsetComplementIntersect = $true )
& ( powerset__Cong = $true )
& ( eqimpsubset1 = $true )
& ( sepSubset = $true )
& ( binunionTIRcontra = $true )
& ( quantDeMorgan1 = $true )
& ( emptysetimpfalse = $true )
& ( ex1E1 = $true )
& ( setadjoinSub = $true )
& ( quantDeMorgan2 = $true )
& ( setadjoinAx = $true )
& ( foundationAx = $true )
& ( iftrueorfalse = $true )
& ( powersetE1 = $true )
& ( setukpairinjR2 = $true )
& ( woz13rule0 = $true )
& ( singletonprop = $true )
& ( funcGraphProp2 = $true )
& ( setext = $true )
& ( setukpairIL = $true )
& ( quantDeMorgan3 = $true )
& ( emptyset__Cong = $true )
& ( setminusEL = $true )
& ( setukpairinjR1 = $true )
& ( setminusI = $true )
& ( theeq = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( subbreln = $true )
& ( wellorderingAx = $true )
& ( funcextLem = $true )
& ( subsetRefl = $true )
& ( iftrueProp2 = $true )
& ( descrp = $true )
& ( kpairsurjEq = $true )
& ( bs114d = $true )
& ( symdiffIneg2 = $true )
& ( quantDeMorgan4 = $true )
& ( setukpairinjL1 = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( lamp = $true )
& ( eqbreln = $true )
& ( singletonsubset = $true )
& ( cartprodfstpairEq = $true )
& ( omega__Cong = $true )
& ( binintersectEL = $true )
& ( iffalseProp1 = $true )
& ( subsetE = $true )
& ( inCongP = $true )
& ( exuE1 = $true )
& ( powersetI = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK687,sK688,sK689,sK690])],[f1109,f2295,f2294,f2293]) ).
thf(f2293,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( ( in @ X2 @ ( powerset @ X1 ) )
= $true )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ ( powerset @ X1 ) ) )
& ( ( subset @ X0 @ ( binintersect @ X2 @ X3 ) )
!= $true )
& ( $true
= ( subset @ X0 @ X2 ) )
& ( ( subset @ X0 @ X3 )
= $true ) ) )
& ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) )
=> ( ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ sK688 ) ) )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ ( powerset @ sK688 ) ) )
& ( $true
!= ( subset @ sK687 @ ( binintersect @ X2 @ X3 ) ) )
& ( $true
= ( subset @ sK687 @ X2 ) )
& ( $true
= ( subset @ sK687 @ X3 ) ) ) )
& ( ( in @ sK687 @ ( powerset @ sK688 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2294,plain,
( ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ sK688 ) ) )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ ( powerset @ sK688 ) ) )
& ( $true
!= ( subset @ sK687 @ ( binintersect @ X2 @ X3 ) ) )
& ( $true
= ( subset @ sK687 @ X2 ) )
& ( $true
= ( subset @ sK687 @ X3 ) ) ) )
=> ( ( $true
= ( in @ sK689 @ ( powerset @ sK688 ) ) )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ ( powerset @ sK688 ) ) )
& ( ( subset @ sK687 @ ( binintersect @ sK689 @ X3 ) )
!= $true )
& ( $true
= ( subset @ sK687 @ sK689 ) )
& ( $true
= ( subset @ sK687 @ X3 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2295,plain,
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ ( powerset @ sK688 ) ) )
& ( ( subset @ sK687 @ ( binintersect @ sK689 @ X3 ) )
!= $true )
& ( $true
= ( subset @ sK687 @ sK689 ) )
& ( $true
= ( subset @ sK687 @ X3 ) ) )
=> ( ( $true
= ( in @ sK690 @ ( powerset @ sK688 ) ) )
& ( $true
!= ( subset @ sK687 @ ( binintersect @ sK689 @ sK690 ) ) )
& ( $true
= ( subset @ sK687 @ sK689 ) )
& ( ( subset @ sK687 @ sK690 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1109,plain,
( ( binintersectSubset2 = $true )
& ( notdexE = $true )
& ( setukpairinjL = $true )
& ( dpsetconstrEL2 = $true )
& ( doubleComplementE1 = $true )
& ( ap2apEq2 = $true )
& ( binintersectSubset3 = $true )
& ( setunionsingleton = $true )
& ( binunionEcases = $true )
& ( doubleComplementSub1 = $true )
& ( powersetsubset = $true )
& ( binunionE = $true )
& ( binintersectI = $true )
& ( notsubsetI = $true )
& ( cartprodmempair = $true )
& ( binunionTILcontra = $true )
& ( dsetconstrEL = $true )
& ( complementTnotintersectT = $true )
& ( lam2lamEq = $true )
& ( woz13rule1 = $true )
& ( setminusIRneg = $true )
& ( eqimpsubset2 = $true )
& ( contrasubsetT1 = $true )
& ( setminusSubset2 = $true )
& ( demorgan2 = $true )
& ( uniqinunit = $true )
& ( kpairiskpair = $true )
& ( notdallE = $true )
& ( complementTE1 = $true )
& ( omegaIndAx = $true )
& ( dsetconstrI = $true )
& ( powersetE = $true )
& ( emptysetsubset = $true )
& ( beta2 = $true )
& ( ubforcartprodlem2 = $true )
& ( descr__Cong = $true )
& ( contraSubsetComplement = $true )
& ( noeltsimpempty = $true )
& ( ex1I = $true )
& ( exuE3u = $true )
& ( vacuousDall = $true )
& ( setukpairinjR11 = $true )
& ( emptysetE = $true )
& ( replAx = $true )
& ( exuE2 = $true )
& ( ap2p = $true )
& ( setadjoinIR = $true )
& ( dpsetconstrI = $true )
& ( setunionE2 = $true )
& ( binunionT_lem = $true )
& ( setunionAx = $true )
& ( setukpairinjR = $true )
& ( cartprodpairin = $true )
& ( binintersectLsub = $true )
& ( setoftrueEq = $true )
& ( iftrueProp1 = $true )
& ( ubforcartprodlem3 = $true )
& ( dpsetconstrEL1 = $true )
& ( setbeta = $true )
& ( exuI2 = $true )
& ( complementUnionInPowersetComplement = $true )
& ( nonemptyE1 = $true )
& ( upairset2IR = $true )
& ( in__Cong = $true )
& ( cartprodsndpairEq = $true )
& ( disjointsetsI1 = $true )
& ( upairequniteq = $true )
& ( complementT_lem = $true )
& ( exuI1 = $true )
& ( symdiffI2 = $true )
& ( complementTcontraSubset = $true )
& ( inIntersectImpInUnion = $true )
& ( binunionLsub = $true )
& ( upairset2E = $true )
& ( cartprodfstin = $true )
& ( nonemptyImpWitness = $true )
& ( binunionIL = $true )
& ( funcext = $true )
& ( infuncsetfunc = $true )
& ( binintersectTERcontra = $true )
& ( doubleComplementSub2 = $true )
& ( inPowerset = $true )
& ( exuEu = $true )
& ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( ( in @ X2 @ ( powerset @ X1 ) )
= $true )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ ( powerset @ X1 ) ) )
& ( ( subset @ X0 @ ( binintersect @ X2 @ X3 ) )
!= $true )
& ( $true
= ( subset @ X0 @ X2 ) )
& ( ( subset @ X0 @ X3 )
= $true ) ) )
& ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) )
& ( funcinfuncset = $true )
& ( setunionI = $true )
& ( eta2 = $true )
& ( ubforcartprodlem1 = $true )
& ( setunionE = $true )
& ( upairinpowunion = $true )
& ( dsetconstrER = $true )
& ( complementImpComplementIntersect = $true )
& ( setextT = $true )
& ( setminusILneg = $true )
& ( powersetT_lem = $true )
& ( kpairp = $true )
& ( singletonsswitch = $true )
& ( binintersectSubset5 = $true )
& ( dsetconstr__Cong = $true )
& ( woz13rule2 = $true )
& ( brelnall2 = $true )
& ( binunionIR = $true )
& ( demorgan2b = $true )
& ( setminusELneg = $true )
& ( emptyinunitempty = $true )
& ( doubleComplementI1 = $true )
& ( demorgan2a1 = $true )
& ( lamProp = $true )
& ( secondinupair = $true )
& ( funcGraphProp4 = $true )
& ( dpsetconstrSub = $true )
& ( subsetemptysetimpeq = $true )
& ( funcext2 = $true )
& ( setminusT_lem = $true )
& ( demorgan1a = $true )
& ( funcImageSingleton = $true )
& ( cartprodpairmemEL = $true )
& ( setminusLsub = $true )
& ( omegaSAx = $true )
& ( setukpairinjR12 = $true )
& ( doubleComplementEq = $true )
& ( contrasubsetT = $true )
& ( setadjoin__Cong = $true )
& ( exuI3 = $true )
& ( setminusSubset1 = $true )
& ( binunionRsub = $true )
& ( powersetTI1 = $true )
& ( omega0Ax = $true )
& ( notequalI1 = $true )
& ( emptyInPowerset = $true )
& ( notequalI2 = $true )
& ( symdiffE = $true )
& ( subsetI1 = $true )
& ( lam2p = $true )
& ( sepInPowerset = $true )
& ( binintersectTELcontra = $true )
& ( emptyI = $true )
& ( symdiffIneg1 = $true )
& ( demorgan2a2 = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( setunionsingleton2 = $true )
& ( powersetAx = $true )
& ( symdiffI1 = $true )
& ( kfstsingleton = $true )
& ( binintersectER = $true )
& ( eqinunit = $true )
& ( nonemptyI1 = $true )
& ( upairsubunion = $true )
& ( contrasubsetT2 = $true )
& ( exu__Cong = $true )
& ( iffalseProp2 = $true )
& ( contrasubsetT3 = $true )
& ( ifp = $true )
& ( singletoninpowunion = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( setminusERneg = $true )
& ( complementTI1 = $true )
& ( ifSingleton = $true )
& ( cartprodpairmemER = $true )
& ( subsetTI = $true )
& ( subPowSU = $true )
& ( ex1I2 = $true )
& ( singletoninpowerset = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( notinsingleton = $true )
& ( demorgan2a = $true )
& ( demorgan2b2 = $true )
& ( subsetTrans = $true )
& ( powersetI1 = $true )
& ( prop2set2propI = $true )
& ( inIntersectImpInUnion2 = $true )
& ( cartprodmempair1 = $true )
& ( setadjoinOr = $true )
& ( ex1E2 = $true )
& ( setOfPairsIsBReln = $true )
& ( setukpairinjL2 = $true )
& ( upairsetIL = $true )
& ( subsetE2 = $true )
& ( subsetI2 = $true )
& ( apProp = $true )
& ( theprop = $true )
& ( binunionTEcontra = $true )
& ( demorgan1 = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodmempaircEq = $true )
& ( kfstpairEq = $true )
& ( cartprodsndin = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( setminusER = $true )
& ( binintersectSubset4 = $true )
& ( powersetTE1 = $true )
& ( subset2powerset = $true )
& ( dpsetconstrERa = $true )
& ( brelnall1 = $true )
& ( eta1 = $true )
& ( funcGraphProp1 = $true )
& ( setukpairIR = $true )
& ( setadjoinIL = $true )
& ( app = $true )
& ( dpsetconstrER = $true )
& ( beta1 = $true )
& ( emptysetAx = $true )
& ( upairsetE = $true )
& ( funcGraphProp3 = $true )
& ( notinemptyset = $true )
& ( setadjoinSub2 = $true )
& ( binunionTE = $true )
& ( binintersectRsub = $true )
& ( setextsub = $true )
& ( binintersectT_lem = $true )
& ( setadjoinE = $true )
& ( binintersectSubset1 = $true )
& ( emptyE1 = $true )
& ( nonemptyI = $true )
& ( demorgan1b = $true )
& ( emptyinPowerset = $true )
& ( ap2apEq1 = $true )
& ( singletonsuniq = $true )
& ( exuE3e = $true )
& ( prop2setI = $true )
& ( setextAx = $true )
& ( upairsetIR = $true )
& ( prop2setE = $true )
& ( setunionsingleton1 = $true )
& ( setunion__Cong = $true )
& ( complementSubsetComplementIntersect = $true )
& ( powerset__Cong = $true )
& ( eqimpsubset1 = $true )
& ( sepSubset = $true )
& ( binunionTIRcontra = $true )
& ( quantDeMorgan1 = $true )
& ( emptysetimpfalse = $true )
& ( ex1E1 = $true )
& ( setadjoinSub = $true )
& ( quantDeMorgan2 = $true )
& ( setadjoinAx = $true )
& ( foundationAx = $true )
& ( iftrueorfalse = $true )
& ( powersetE1 = $true )
& ( setukpairinjR2 = $true )
& ( woz13rule0 = $true )
& ( singletonprop = $true )
& ( funcGraphProp2 = $true )
& ( setext = $true )
& ( setukpairIL = $true )
& ( quantDeMorgan3 = $true )
& ( emptyset__Cong = $true )
& ( setminusEL = $true )
& ( setukpairinjR1 = $true )
& ( setminusI = $true )
& ( theeq = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( subbreln = $true )
& ( wellorderingAx = $true )
& ( funcextLem = $true )
& ( subsetRefl = $true )
& ( iftrueProp2 = $true )
& ( descrp = $true )
& ( kpairsurjEq = $true )
& ( bs114d = $true )
& ( symdiffIneg2 = $true )
& ( quantDeMorgan4 = $true )
& ( setukpairinjL1 = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( lamp = $true )
& ( eqbreln = $true )
& ( singletonsubset = $true )
& ( cartprodfstpairEq = $true )
& ( omega__Cong = $true )
& ( binintersectEL = $true )
& ( iffalseProp1 = $true )
& ( subsetE = $true )
& ( inCongP = $true )
& ( exuE1 = $true )
& ( powersetI = $true ) ),
inference(flattening,[status(thm)],[f1108]) ).
thf(f1108,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( subset @ X0 @ ( binintersect @ X2 @ X3 ) )
!= $true )
& ( ( subset @ X0 @ X3 )
= $true )
& ( $true
= ( subset @ X0 @ X2 ) )
& ( $true
= ( in @ X3 @ ( powerset @ X1 ) ) ) )
& ( ( in @ X2 @ ( powerset @ X1 ) )
= $true ) )
& ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) )
& ( 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)],[f654]) ).
thf(f654,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 )
=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
=> ! [X2: $i] :
( ( ( in @ X2 @ ( powerset @ X1 ) )
= $true )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ ( powerset @ X1 ) ) )
=> ( ( $true
= ( subset @ X0 @ X2 ) )
=> ( ( ( subset @ X0 @ X3 )
= $true )
=> ( ( subset @ X0 @ ( binintersect @ X2 @ X3 ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f653]) ).
thf(f653,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
=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( powerset @ X1 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X1 ) )
=> ( ( subset @ X0 @ X2 )
=> ( ( subset @ X0 @ X3 )
=> ( subset @ X0 @ ( binintersect @ X2 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f283]) ).
thf(f283,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
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X17: $i] :
( ( in @ X17 @ ( powerset @ X3 ) )
=> ( ( subset @ X11 @ X16 )
=> ( ( subset @ X11 @ X17 )
=> ( subset @ X11 @ ( binintersect @ X16 @ X17 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f282]) ).
thf(f282,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
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X17: $i] :
( ( in @ X17 @ ( powerset @ X3 ) )
=> ( ( subset @ X11 @ X16 )
=> ( ( subset @ X11 @ X17 )
=> ( subset @ X11 @ ( binintersect @ X16 @ X17 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.gJXxL4HX93/DTF2THF_18132.p',woz13rule3) ).
thf(f4433,plain,
( ( subset @ sK687 @ sK690 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f4432]) ).
thf(f4432,plain,
( ( ( subset @ sK687 @ sK690 )
!= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f4431,f3744]) ).
thf(f3744,plain,
( $true
= ( subset @ sK687 @ sK689 ) ),
inference(cnf_transformation,[status(thm)],[f2296]) ).
thf(f4431,plain,
( ( $true
!= ( subset @ sK687 @ sK689 ) )
| ( ( subset @ sK687 @ sK690 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4430]) ).
thf(f4430,plain,
( ( $true != $true )
| ( $true
!= ( subset @ sK687 @ sK689 ) )
| ( ( subset @ sK687 @ sK690 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f4412,f3728]) ).
thf(f3728,plain,
binintersectSubset5 = $true,
inference(cnf_transformation,[status(thm)],[f2296]) ).
thf(f4412,plain,
( ( ( subset @ sK687 @ sK690 )
!= $true )
| ( binintersectSubset5 != $true )
| ( $true
!= ( subset @ sK687 @ sK689 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f4406]) ).
thf(f4406,plain,
( ( $true != $true )
| ( binintersectSubset5 != $true )
| ( ( subset @ sK687 @ sK690 )
!= $true )
| ( $true
!= ( subset @ sK687 @ sK689 ) ) ),
inference(superposition,[status(thm)],[f3745,f3331]) ).
thf(f3331,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( subset @ X4 @ ( binintersect @ X3 @ X5 ) ) )
| ( ( subset @ X4 @ X5 )
!= $true )
| ( ( subset @ X4 @ X3 )
!= $true )
| ( binintersectSubset5 != $true ) ),
inference(cnf_transformation,[status(thm)],[f2068]) ).
thf(f2068,plain,
( ( ( binintersectSubset5 = $true )
| ( ( $true
= ( subset @ sK544 @ sK543 ) )
& ( $true
= ( subset @ sK544 @ sK545 ) )
& ( ( subset @ sK544 @ ( binintersect @ sK543 @ sK545 ) )
!= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X4 @ X3 )
!= $true )
| ( ( subset @ X4 @ X5 )
!= $true )
| ( $true
= ( subset @ X4 @ ( binintersect @ X3 @ X5 ) ) ) )
| ( binintersectSubset5 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK543,sK544,sK545])],[f2066,f2067]) ).
thf(f2067,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( subset @ X1 @ X0 ) )
& ( $true
= ( subset @ X1 @ X2 ) )
& ( $true
!= ( subset @ X1 @ ( binintersect @ X0 @ X2 ) ) ) )
=> ( ( $true
= ( subset @ sK544 @ sK543 ) )
& ( $true
= ( subset @ sK544 @ sK545 ) )
& ( ( subset @ sK544 @ ( binintersect @ sK543 @ sK545 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2066,plain,
( ( ( binintersectSubset5 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( subset @ X1 @ X0 ) )
& ( $true
= ( subset @ X1 @ X2 ) )
& ( $true
!= ( subset @ X1 @ ( binintersect @ X0 @ X2 ) ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X4 @ X3 )
!= $true )
| ( ( subset @ X4 @ X5 )
!= $true )
| ( $true
= ( subset @ X4 @ ( binintersect @ X3 @ X5 ) ) ) )
| ( binintersectSubset5 != $true ) ) ),
inference(rectify,[status(thm)],[f2065]) ).
thf(f2065,plain,
( ( ( binintersectSubset5 = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( subset @ X1 @ X0 ) )
& ( $true
= ( subset @ X1 @ X2 ) )
& ( $true
!= ( subset @ X1 @ ( binintersect @ X0 @ X2 ) ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( subset @ X1 @ X0 ) )
| ( $true
!= ( subset @ X1 @ X2 ) )
| ( $true
= ( subset @ X1 @ ( binintersect @ X0 @ X2 ) ) ) )
| ( binintersectSubset5 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1021]) ).
thf(f1021,plain,
( ( binintersectSubset5 = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( subset @ X1 @ X0 ) )
| ( $true
!= ( subset @ X1 @ X2 ) )
| ( $true
= ( subset @ X1 @ ( binintersect @ X0 @ X2 ) ) ) ) ),
inference(flattening,[status(thm)],[f1020]) ).
thf(f1020,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( subset @ X1 @ ( binintersect @ X0 @ X2 ) ) )
| ( $true
!= ( subset @ X1 @ X2 ) )
| ( $true
!= ( subset @ X1 @ X0 ) ) )
<=> ( binintersectSubset5 = $true ) ),
inference(ennf_transformation,[status(thm)],[f624]) ).
thf(f624,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( subset @ X1 @ X0 ) )
=> ( ( $true
= ( subset @ X1 @ X2 ) )
=> ( $true
= ( subset @ X1 @ ( binintersect @ X0 @ X2 ) ) ) ) )
<=> ( binintersectSubset5 = $true ) ),
inference(fool_elimination,[status(thm)],[f623]) ).
thf(f623,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X1 @ X0 )
=> ( ( subset @ X1 @ X2 )
=> ( subset @ X1 @ ( binintersect @ X0 @ X2 ) ) ) )
= binintersectSubset5 ),
inference(rectify,[status(thm)],[f109]) ).
thf(f109,axiom,
! [X3: $i,X5: $i,X4: $i] :
( ( ( subset @ X5 @ X3 )
=> ( ( subset @ X5 @ X4 )
=> ( subset @ X5 @ ( binintersect @ X3 @ X4 ) ) ) )
= binintersectSubset5 ),
file('/export/starexec/sandbox/tmp/tmp.gJXxL4HX93/DTF2THF_18132.p',binintersectSubset5) ).
thf(f3745,plain,
( $true
!= ( subset @ sK687 @ ( binintersect @ sK689 @ sK690 ) ) ),
inference(cnf_transformation,[status(thm)],[f2296]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU761^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n010.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:32:48 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.46 thf(spec,logic,$$dhol).
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % File : SEU761^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.46 % Domain : Set Theory
% 0.19/0.46 % Problem : Typed Set Theory - First Wizard of Oz Examples - WoZ1 Lemmas
% 0.19/0.46 % Version : Especial.
% 0.19/0.46 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.19/0.46 % (! Z:i.in Z (powerset A) -> subset X Y -> subset X Z ->
% 0.19/0.46 % subset X (binintersect Y Z))))
% 0.19/0.46
% 0.19/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.46 % Source : [Bro08]
% 0.19/0.46 % Names : ZFC263g [Bro08]
% 0.19/0.46
% 0.19/0.46 % Status : Theorem
% 0.19/0.46 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 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
% 0.19/0.46 % Syntax : Number of formulae : 597 ( 281 unt; 315 typ; 281 def)
% 0.19/0.46 % Number of atoms : 1674 ( 409 equ; 0 cnn)
% 0.19/0.46 % Maximal formula atoms : 283 ( 5 avg)
% 0.19/0.46 % Number of connectives : 3529 ( 73 ~; 12 |; 44 &;2543 @)
% 0.19/0.46 % ( 14 <=>; 843 =>; 0 <=; 0 <~>)
% 0.19/0.46 % Maximal formula depth : 290 ( 2 avg)
% 0.19/0.46 % Number of types : 2 ( 0 usr)
% 0.19/0.46 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.19/0.46 % Number of symbols : 318 ( 315 usr; 281 con; 0-4 aty)
% 0.19/0.46 % Number of variables : 960 ( 74 ^; 851 !; 35 ?; 960 :)
% 0.19/0.46 % SPC : TH0_THM_EQU_NAR
% 0.19/0.46
% 0.19/0.46 % Comments : http://mathgate.info/detsetitem.php?id=322
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 thf(in_type,type,
% 0.19/0.46 in: $i > $i > $o ).
% 0.19/0.46
% 0.19/0.46 thf(exu_type,type,
% 0.19/0.46 exu: ( $i > $o ) > $o ).
% 0.19/0.46
% 0.19/0.46 thf(exu,definition,
% 0.19/0.46 ( exu
% 0.19/0.46 = ( ^ [Xphi: $i > $o] :
% 0.19/0.46 ? [Xx: $i] :
% 0.19/0.46 ( ( Xphi @ Xx )
% 0.19/0.46 & ! [Xy: $i] :
% 0.19/0.46 ( ( Xphi @ Xy )
% 0.19/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(setextAx_type,type,
% 0.19/0.46 setextAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setextAx,definition,
% 0.19/0.47 ( setextAx
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 <=> ( in @ Xx @ B ) )
% 0.19/0.47 => ( A = B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyset_type,type,
% 0.19/0.47 emptyset: $i ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetAx_type,type,
% 0.19/0.47 emptysetAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetAx,definition,
% 0.19/0.47 ( emptysetAx
% 0.19/0.47 = ( ! [Xx: $i] :
% 0.19/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoin_type,type,
% 0.19/0.47 setadjoin: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinAx_type,type,
% 0.19/0.47 setadjoinAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinAx,definition,
% 0.19/0.47 ( setadjoinAx
% 0.19/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.47 <=> ( ( Xy = Xx )
% 0.19/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(powerset_type,type,
% 0.19/0.47 powerset: $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(powersetAx_type,type,
% 0.19/0.47 powersetAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(powersetAx,definition,
% 0.19/0.47 ( powersetAx
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.47 <=> ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setunion_type,type,
% 0.19/0.47 setunion: $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(setunionAx_type,type,
% 0.19/0.47 setunionAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setunionAx,definition,
% 0.19/0.47 ( setunionAx
% 0.19/0.47 = ( ! [A: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.47 <=> ? [B: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 & ( in @ B @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(omega_type,type,
% 0.19/0.47 omega: $i ).
% 0.19/0.47
% 0.19/0.47 thf(omega0Ax_type,type,
% 0.19/0.47 omega0Ax: $o ).
% 0.19/0.47
% 0.19/0.47 thf(omega0Ax,definition,
% 0.19/0.47 ( omega0Ax
% 0.19/0.47 = ( in @ emptyset @ omega ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(omegaSAx_type,type,
% 0.19/0.47 omegaSAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(omegaSAx,definition,
% 0.19/0.47 ( omegaSAx
% 0.19/0.47 = ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ omega )
% 0.19/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(omegaIndAx_type,type,
% 0.19/0.47 omegaIndAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(omegaIndAx,definition,
% 0.19/0.47 ( omegaIndAx
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ( ( in @ emptyset @ A )
% 0.19/0.47 & ! [Xx: $i] :
% 0.19/0.47 ( ( ( in @ Xx @ omega )
% 0.19/0.47 & ( in @ Xx @ A ) )
% 0.19/0.47 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ omega )
% 0.19/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(replAx_type,type,
% 0.19/0.47 replAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(replAx,definition,
% 0.19/0.47 ( replAx
% 0.19/0.47 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( exu
% 0.19/0.47 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.19/0.47 => ? [B: $i] :
% 0.19/0.47 ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 <=> ? [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ A )
% 0.19/0.47 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(foundationAx_type,type,
% 0.19/0.47 foundationAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(foundationAx,definition,
% 0.19/0.47 ( foundationAx
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.47 => ? [B: $i] :
% 0.19/0.47 ( ( in @ B @ A )
% 0.19/0.47 & ~ ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(wellorderingAx_type,type,
% 0.19/0.47 wellorderingAx: $o ).
% 0.19/0.47
% 0.19/0.47 thf(wellorderingAx,definition,
% 0.19/0.47 ( wellorderingAx
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ? [B: $i] :
% 0.19/0.47 ( ! [C: $i] :
% 0.19/0.47 ( ( in @ C @ B )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ C )
% 0.19/0.47 => ( in @ Xx @ A ) ) )
% 0.19/0.47 & ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( ( in @ Xx @ A )
% 0.19/0.47 & ( in @ Xy @ A ) )
% 0.19/0.47 => ( ! [C: $i] :
% 0.19/0.47 ( ( in @ C @ B )
% 0.19/0.47 => ( ( in @ Xx @ C )
% 0.19/0.47 <=> ( in @ Xy @ C ) ) )
% 0.19/0.47 => ( Xx = Xy ) ) )
% 0.19/0.47 & ! [C: $i,D: $i] :
% 0.19/0.47 ( ( ( in @ C @ B )
% 0.19/0.47 & ( in @ D @ B ) )
% 0.19/0.47 => ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ C )
% 0.19/0.47 => ( in @ Xx @ D ) )
% 0.19/0.47 | ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ D )
% 0.19/0.47 => ( in @ Xx @ C ) ) ) )
% 0.19/0.47 & ! [C: $i] :
% 0.19/0.47 ( ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ C )
% 0.19/0.47 => ( in @ Xx @ A ) )
% 0.19/0.47 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.19/0.47 => ? [D: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ D @ B )
% 0.19/0.47 & ( in @ Xx @ C )
% 0.19/0.47 & ~ ? [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ D )
% 0.19/0.47 & ( in @ Xy @ C ) )
% 0.19/0.47 & ! [E: $i] :
% 0.19/0.47 ( ( in @ E @ B )
% 0.19/0.47 => ( ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ E )
% 0.19/0.47 => ( in @ Xy @ D ) )
% 0.19/0.47 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(descr_type,type,
% 0.19/0.47 descr: ( $i > $o ) > $i ).
% 0.19/0.47
% 0.19/0.47 thf(descrp_type,type,
% 0.19/0.47 descrp: $o ).
% 0.19/0.47
% 0.19/0.47 thf(descrp,definition,
% 0.19/0.47 ( descrp
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ( Xphi
% 0.19/0.47 @ ( descr
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstr_type,type,
% 0.19/0.47 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstrI_type,type,
% 0.19/0.47 dsetconstrI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstrI,definition,
% 0.19/0.47 ( dsetconstrI
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( Xphi @ Xx )
% 0.19/0.47 => ( in @ Xx
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstrEL_type,type,
% 0.19/0.47 dsetconstrEL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstrEL,definition,
% 0.19/0.47 ( dsetconstrEL
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstrER_type,type,
% 0.19/0.47 dsetconstrER: $o ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstrER,definition,
% 0.19/0.47 ( dsetconstrER
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuE1_type,type,
% 0.19/0.47 exuE1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuE1,definition,
% 0.19/0.47 ( exuE1
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ? [Xx: $i] :
% 0.19/0.47 ( ( Xphi @ Xx )
% 0.19/0.47 & ! [Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(prop2set_type,type,
% 0.19/0.47 prop2set: $o > $i ).
% 0.19/0.47
% 0.19/0.47 thf(prop2setE_type,type,
% 0.19/0.47 prop2setE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(prop2setE,definition,
% 0.19/0.47 ( prop2setE
% 0.19/0.47 = ( ! [Xphi: $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.19/0.47 => Xphi ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetE_type,type,
% 0.19/0.47 emptysetE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetE,definition,
% 0.19/0.47 ( emptysetE
% 0.19/0.47 = ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ emptyset )
% 0.19/0.47 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetimpfalse_type,type,
% 0.19/0.47 emptysetimpfalse: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetimpfalse,definition,
% 0.19/0.47 ( emptysetimpfalse
% 0.19/0.47 = ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ emptyset )
% 0.19/0.47 => $false ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notinemptyset_type,type,
% 0.19/0.47 notinemptyset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notinemptyset,definition,
% 0.19/0.47 ( notinemptyset
% 0.19/0.47 = ( ! [Xx: $i] :
% 0.19/0.47 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuE3e_type,type,
% 0.19/0.47 exuE3e: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuE3e,definition,
% 0.19/0.47 ( exuE3e
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setext_type,type,
% 0.19/0.47 setext: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setext,definition,
% 0.19/0.47 ( setext
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xx @ B ) )
% 0.19/0.47 => ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ A ) )
% 0.19/0.47 => ( A = B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyI_type,type,
% 0.19/0.47 emptyI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyI,definition,
% 0.19/0.47 ( emptyI
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ~ ( in @ Xx @ A )
% 0.19/0.47 => ( A = emptyset ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(noeltsimpempty_type,type,
% 0.19/0.47 noeltsimpempty: $o ).
% 0.19/0.47
% 0.19/0.47 thf(noeltsimpempty,definition,
% 0.19/0.47 ( noeltsimpempty
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ~ ( in @ Xx @ A )
% 0.19/0.47 => ( A = emptyset ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setbeta_type,type,
% 0.19/0.47 setbeta: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setbeta,definition,
% 0.19/0.47 ( setbeta
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( in @ Xx
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.19/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(nonempty_type,type,
% 0.19/0.47 nonempty: $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(nonempty,definition,
% 0.19/0.47 ( nonempty
% 0.19/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyE1_type,type,
% 0.19/0.47 nonemptyE1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyE1,definition,
% 0.19/0.47 ( nonemptyE1
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ( nonempty @ A )
% 0.19/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyI_type,type,
% 0.19/0.47 nonemptyI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyI,definition,
% 0.19/0.47 ( nonemptyI
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( Xphi @ Xx )
% 0.19/0.47 => ( nonempty
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyI1_type,type,
% 0.19/0.47 nonemptyI1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyI1,definition,
% 0.19/0.47 ( nonemptyI1
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.19/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinIL_type,type,
% 0.19/0.47 setadjoinIL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinIL,definition,
% 0.19/0.47 ( setadjoinIL
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyinunitempty_type,type,
% 0.19/0.47 emptyinunitempty: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyinunitempty,definition,
% 0.19/0.47 ( emptyinunitempty
% 0.19/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinIR_type,type,
% 0.19/0.47 setadjoinIR: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinIR,definition,
% 0.19/0.47 ( setadjoinIR
% 0.19/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ A )
% 0.19/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinE_type,type,
% 0.19/0.47 setadjoinE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinE,definition,
% 0.19/0.47 ( setadjoinE
% 0.19/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.47 => ! [Xphi: $o] :
% 0.19/0.47 ( ( ( Xy = Xx )
% 0.19/0.47 => Xphi )
% 0.19/0.47 => ( ( ( in @ Xy @ A )
% 0.19/0.47 => Xphi )
% 0.19/0.47 => Xphi ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinOr_type,type,
% 0.19/0.47 setadjoinOr: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinOr,definition,
% 0.19/0.47 ( setadjoinOr
% 0.19/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.19/0.47 => ( ( Xy = Xx )
% 0.19/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setoftrueEq_type,type,
% 0.19/0.47 setoftrueEq: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setoftrueEq,definition,
% 0.19/0.47 ( setoftrueEq
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xx: $i] : $true )
% 0.19/0.47 = A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(powersetI_type,type,
% 0.19/0.47 powersetI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(powersetI,definition,
% 0.19/0.47 ( powersetI
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ A ) )
% 0.19/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyinPowerset_type,type,
% 0.19/0.47 emptyinPowerset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyinPowerset,definition,
% 0.19/0.47 ( emptyinPowerset
% 0.19/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyInPowerset_type,type,
% 0.19/0.47 emptyInPowerset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyInPowerset,definition,
% 0.19/0.47 ( emptyInPowerset
% 0.19/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(powersetE_type,type,
% 0.19/0.47 powersetE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(powersetE,definition,
% 0.19/0.47 ( powersetE
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.47 => ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setunionI_type,type,
% 0.19/0.47 setunionI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setunionI,definition,
% 0.19/0.47 ( setunionI
% 0.19/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ( ( in @ B @ A )
% 0.19/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setunionE_type,type,
% 0.19/0.47 setunionE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setunionE,definition,
% 0.19/0.47 ( setunionE
% 0.19/0.47 = ( ! [A: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.47 => ! [Xphi: $o] :
% 0.19/0.47 ( ! [B: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ( ( in @ B @ A )
% 0.19/0.47 => Xphi ) )
% 0.19/0.47 => Xphi ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subPowSU_type,type,
% 0.19/0.47 subPowSU: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subPowSU,definition,
% 0.19/0.47 ( subPowSU
% 0.19/0.47 = ( ! [A: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuE2_type,type,
% 0.19/0.47 exuE2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuE2,definition,
% 0.19/0.47 ( exuE2
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ? [Xx: $i] :
% 0.19/0.47 ! [Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xy )
% 0.19/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyImpWitness_type,type,
% 0.19/0.47 nonemptyImpWitness: $o ).
% 0.19/0.47
% 0.19/0.47 thf(nonemptyImpWitness,definition,
% 0.19/0.47 ( nonemptyImpWitness
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ( nonempty @ A )
% 0.19/0.47 => ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & $true ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(uniqinunit_type,type,
% 0.19/0.47 uniqinunit: $o ).
% 0.19/0.47
% 0.19/0.47 thf(uniqinunit,definition,
% 0.19/0.47 ( uniqinunit
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notinsingleton_type,type,
% 0.19/0.47 notinsingleton: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notinsingleton,definition,
% 0.19/0.47 ( notinsingleton
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx != Xy )
% 0.19/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(eqinunit_type,type,
% 0.19/0.47 eqinunit: $o ).
% 0.19/0.47
% 0.19/0.47 thf(eqinunit,definition,
% 0.19/0.47 ( eqinunit
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(singletonsswitch_type,type,
% 0.19/0.47 singletonsswitch: $o ).
% 0.19/0.47
% 0.19/0.47 thf(singletonsswitch,definition,
% 0.19/0.47 ( singletonsswitch
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetE_type,type,
% 0.19/0.47 upairsetE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetE,definition,
% 0.19/0.47 ( upairsetE
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.19/0.47 => ( ( Xz = Xx )
% 0.19/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetIL_type,type,
% 0.19/0.47 upairsetIL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetIL,definition,
% 0.19/0.47 ( upairsetIL
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetIR_type,type,
% 0.19/0.47 upairsetIR: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairsetIR,definition,
% 0.19/0.47 ( upairsetIR
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyE1_type,type,
% 0.19/0.47 emptyE1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyE1,definition,
% 0.19/0.47 ( emptyE1
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ( Xphi @ Xx ) )
% 0.19/0.47 => ( ( ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 = emptyset )
% 0.19/0.47 => $false ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(vacuousDall_type,type,
% 0.19/0.47 vacuousDall: $o ).
% 0.19/0.47
% 0.19/0.47 thf(vacuousDall,definition,
% 0.19/0.47 ( vacuousDall
% 0.19/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ emptyset )
% 0.19/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan1_type,type,
% 0.19/0.47 quantDeMorgan1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan1,definition,
% 0.19/0.47 ( quantDeMorgan1
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ~ ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( Xphi @ Xx ) )
% 0.19/0.47 => ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan2_type,type,
% 0.19/0.47 quantDeMorgan2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan2,definition,
% 0.19/0.47 ( quantDeMorgan2
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ~ ( Xphi @ Xx ) )
% 0.19/0.47 => ~ ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan3_type,type,
% 0.19/0.47 quantDeMorgan3: $o ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan3,definition,
% 0.19/0.47 ( quantDeMorgan3
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ~ ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ( Xphi @ Xx ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan4_type,type,
% 0.19/0.47 quantDeMorgan4: $o ).
% 0.19/0.47
% 0.19/0.47 thf(quantDeMorgan4,definition,
% 0.19/0.47 ( quantDeMorgan4
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ~ ( Xphi @ Xx ) )
% 0.19/0.47 => ~ ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(prop2setI_type,type,
% 0.19/0.47 prop2setI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(prop2setI,definition,
% 0.19/0.47 ( prop2setI
% 0.19/0.47 = ( ! [Xphi: $o] :
% 0.19/0.47 ( Xphi
% 0.19/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(set2prop_type,type,
% 0.19/0.47 set2prop: $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(prop2set2propI_type,type,
% 0.19/0.47 prop2set2propI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(prop2set2propI,definition,
% 0.19/0.47 ( prop2set2propI
% 0.19/0.47 = ( ! [Xphi: $o] :
% 0.19/0.47 ( Xphi
% 0.19/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notdexE_type,type,
% 0.19/0.47 notdexE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notdexE,definition,
% 0.19/0.47 ( notdexE
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ~ ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ( Xphi @ Xx ) )
% 0.19/0.47 => ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notdallE_type,type,
% 0.19/0.47 notdallE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notdallE,definition,
% 0.19/0.47 ( notdallE
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( ~ ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( Xphi @ Xx ) )
% 0.19/0.47 => ? [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuI1_type,type,
% 0.19/0.47 exuI1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuI1,definition,
% 0.19/0.47 ( exuI1
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ? [Xx: $i] :
% 0.19/0.47 ( ( Xphi @ Xx )
% 0.19/0.47 & ! [Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xx = Xy ) ) )
% 0.19/0.47 => ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuI3_type,type,
% 0.19/0.47 exuI3: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuI3,definition,
% 0.19/0.47 ( exuI3
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.19/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xx )
% 0.19/0.47 => ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xx = Xy ) ) )
% 0.19/0.47 => ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuI2_type,type,
% 0.19/0.47 exuI2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuI2,definition,
% 0.19/0.47 ( exuI2
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ? [Xx: $i] :
% 0.19/0.47 ! [Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xy )
% 0.19/0.47 <=> ( Xy = Xx ) )
% 0.19/0.47 => ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(inCongP_type,type,
% 0.19/0.47 inCongP: $o ).
% 0.19/0.47
% 0.19/0.47 thf(inCongP,definition,
% 0.19/0.47 ( inCongP
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(in__Cong_type,type,
% 0.19/0.47 in__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(in__Cong,definition,
% 0.19/0.47 ( in__Cong
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ( ( in @ Xx @ A )
% 0.19/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuE3u_type,type,
% 0.19/0.47 exuE3u: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuE3u,definition,
% 0.19/0.47 ( exuE3u
% 0.19/0.47 = ( ! [Xphi: $i > $o] :
% 0.19/0.47 ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 => ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xphi @ Xx )
% 0.19/0.47 => ( ( Xphi @ Xy )
% 0.19/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exu__Cong_type,type,
% 0.19/0.47 exu__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exu__Cong,definition,
% 0.19/0.47 ( exu__Cong
% 0.19/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ( ( Xphi @ Xx )
% 0.19/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.19/0.47 => ( ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 <=> ( exu
% 0.19/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptyset__Cong_type,type,
% 0.19/0.47 emptyset__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptyset__Cong,definition,
% 0.19/0.47 ( emptyset__Cong
% 0.19/0.47 = ( emptyset = emptyset ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoin__Cong_type,type,
% 0.19/0.47 setadjoin__Cong: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoin__Cong,definition,
% 0.31/0.47 ( setadjoin__Cong
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx = Xy )
% 0.31/0.47 => ! [Xz: $i,Xu: $i] :
% 0.31/0.47 ( ( Xz = Xu )
% 0.31/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.31/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powerset__Cong_type,type,
% 0.31/0.47 powerset__Cong: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powerset__Cong,definition,
% 0.31/0.47 ( powerset__Cong
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( A = B )
% 0.31/0.47 => ( ( powerset @ A )
% 0.31/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunion__Cong_type,type,
% 0.31/0.47 setunion__Cong: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunion__Cong,definition,
% 0.31/0.47 ( setunion__Cong
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( A = B )
% 0.31/0.47 => ( ( setunion @ A )
% 0.31/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(omega__Cong_type,type,
% 0.31/0.47 omega__Cong: $o ).
% 0.31/0.47
% 0.31/0.47 thf(omega__Cong,definition,
% 0.31/0.47 ( omega__Cong
% 0.31/0.47 = ( omega = omega ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(exuEu_type,type,
% 0.31/0.47 exuEu: $o ).
% 0.31/0.47
% 0.31/0.47 thf(exuEu,definition,
% 0.31/0.47 ( exuEu
% 0.31/0.47 = ( ! [Xphi: $i > $o] :
% 0.31/0.47 ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xphi @ Xx )
% 0.31/0.47 => ( ( Xphi @ Xy )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(descr__Cong_type,type,
% 0.31/0.47 descr__Cong: $o ).
% 0.31/0.47
% 0.31/0.47 thf(descr__Cong,definition,
% 0.31/0.47 ( descr__Cong
% 0.31/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx = Xy )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.31/0.47 => ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ( ( exu
% 0.31/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.31/0.47 => ( ( descr
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 = ( descr
% 0.31/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstr__Cong_type,type,
% 0.31/0.47 dsetconstr__Cong: $o ).
% 0.31/0.47
% 0.31/0.47 thf(dsetconstr__Cong,definition,
% 0.31/0.47 ( dsetconstr__Cong
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( A = B )
% 0.31/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( ( Xx = Xy )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.31/0.47 => ( ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 = ( dsetconstr @ B
% 0.31/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subset_type,type,
% 0.31/0.47 subset: $i > $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(disjoint_type,type,
% 0.31/0.47 disjoint: $i > $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(setsmeet_type,type,
% 0.31/0.47 setsmeet: $i > $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetI1_type,type,
% 0.31/0.47 subsetI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetI1,definition,
% 0.31/0.47 ( subsetI1
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ B ) )
% 0.31/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(eqimpsubset2_type,type,
% 0.31/0.47 eqimpsubset2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(eqimpsubset2,definition,
% 0.31/0.47 ( eqimpsubset2
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( A = B )
% 0.31/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(eqimpsubset1_type,type,
% 0.31/0.47 eqimpsubset1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(eqimpsubset1,definition,
% 0.31/0.47 ( eqimpsubset1
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( A = B )
% 0.31/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subsetI2_type,type,
% 0.31/0.47 subsetI2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetI2,definition,
% 0.31/0.47 ( subsetI2
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ B ) )
% 0.31/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetsubset_type,type,
% 0.31/0.47 emptysetsubset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(emptysetsubset,definition,
% 0.31/0.47 ( emptysetsubset
% 0.31/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subsetE_type,type,
% 0.31/0.47 subsetE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetE,definition,
% 0.31/0.47 ( subsetE
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subsetE2_type,type,
% 0.31/0.47 subsetE2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetE2,definition,
% 0.31/0.47 ( subsetE2
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( ~ ( in @ Xx @ B )
% 0.31/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(notsubsetI_type,type,
% 0.31/0.47 notsubsetI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(notsubsetI,definition,
% 0.31/0.47 ( notsubsetI
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ B )
% 0.31/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(notequalI1_type,type,
% 0.31/0.47 notequalI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(notequalI1,definition,
% 0.31/0.47 ( notequalI1
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ~ ( subset @ A @ B )
% 0.31/0.47 => ( A != B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(notequalI2_type,type,
% 0.31/0.47 notequalI2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(notequalI2,definition,
% 0.31/0.47 ( notequalI2
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ B )
% 0.31/0.47 => ( A != B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subsetRefl_type,type,
% 0.31/0.47 subsetRefl: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetRefl,definition,
% 0.31/0.47 ( subsetRefl
% 0.31/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subsetTrans_type,type,
% 0.31/0.47 subsetTrans: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetTrans,definition,
% 0.31/0.47 ( subsetTrans
% 0.31/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( ( subset @ B @ C )
% 0.31/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinSub_type,type,
% 0.31/0.47 setadjoinSub: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinSub,definition,
% 0.31/0.47 ( setadjoinSub
% 0.31/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinSub2_type,type,
% 0.31/0.47 setadjoinSub2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setadjoinSub2,definition,
% 0.31/0.47 ( setadjoinSub2
% 0.31/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subset2powerset_type,type,
% 0.31/0.47 subset2powerset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subset2powerset,definition,
% 0.31/0.47 ( subset2powerset
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setextsub_type,type,
% 0.31/0.47 setextsub: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setextsub,definition,
% 0.31/0.47 ( setextsub
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( ( subset @ B @ A )
% 0.31/0.47 => ( A = B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subsetemptysetimpeq_type,type,
% 0.31/0.47 subsetemptysetimpeq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetemptysetimpeq,definition,
% 0.31/0.47 ( subsetemptysetimpeq
% 0.31/0.47 = ( ! [A: $i] :
% 0.31/0.47 ( ( subset @ A @ emptyset )
% 0.31/0.47 => ( A = emptyset ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powersetI1_type,type,
% 0.31/0.47 powersetI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetI1,definition,
% 0.31/0.47 ( powersetI1
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( subset @ B @ A )
% 0.31/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powersetE1_type,type,
% 0.31/0.47 powersetE1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetE1,definition,
% 0.31/0.47 ( powersetE1
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.31/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(inPowerset_type,type,
% 0.31/0.47 inPowerset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(inPowerset,definition,
% 0.31/0.47 ( inPowerset
% 0.31/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powersetsubset_type,type,
% 0.31/0.47 powersetsubset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetsubset,definition,
% 0.31/0.47 ( powersetsubset
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sepInPowerset_type,type,
% 0.31/0.47 sepInPowerset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(sepInPowerset,definition,
% 0.31/0.47 ( sepInPowerset
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( in
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sepSubset_type,type,
% 0.31/0.47 sepSubset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(sepSubset,definition,
% 0.31/0.47 ( sepSubset
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( subset
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunion_type,type,
% 0.31/0.47 binunion: $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(binunionIL_type,type,
% 0.31/0.47 binunionIL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionIL,definition,
% 0.31/0.47 ( binunionIL
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairset2IR_type,type,
% 0.31/0.47 upairset2IR: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairset2IR,definition,
% 0.31/0.47 ( upairset2IR
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunionIR_type,type,
% 0.31/0.47 binunionIR: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionIR,definition,
% 0.31/0.47 ( binunionIR
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunionEcases_type,type,
% 0.31/0.47 binunionEcases: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionEcases,definition,
% 0.31/0.47 ( binunionEcases
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.31/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/0.47 => ( ( ( in @ Xx @ A )
% 0.31/0.47 => Xphi )
% 0.31/0.47 => ( ( ( in @ Xx @ B )
% 0.31/0.47 => Xphi )
% 0.31/0.47 => Xphi ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunionE_type,type,
% 0.31/0.47 binunionE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionE,definition,
% 0.31/0.47 ( binunionE
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/0.47 => ( ( in @ Xx @ A )
% 0.31/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunionLsub_type,type,
% 0.31/0.47 binunionLsub: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionLsub,definition,
% 0.31/0.47 ( binunionLsub
% 0.31/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunionRsub_type,type,
% 0.31/0.47 binunionRsub: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionRsub,definition,
% 0.31/0.47 ( binunionRsub
% 0.31/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersect_type,type,
% 0.31/0.47 binintersect: $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectI_type,type,
% 0.31/0.47 binintersectI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectI,definition,
% 0.31/0.47 ( binintersectI
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset5_type,type,
% 0.31/0.47 binintersectSubset5: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset5,definition,
% 0.31/0.47 ( binintersectSubset5
% 0.31/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.47 ( ( subset @ C @ A )
% 0.31/0.47 => ( ( subset @ C @ B )
% 0.31/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectEL_type,type,
% 0.31/0.47 binintersectEL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectEL,definition,
% 0.31/0.47 ( binintersectEL
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectLsub_type,type,
% 0.31/0.47 binintersectLsub: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectLsub,definition,
% 0.31/0.47 ( binintersectLsub
% 0.31/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset2_type,type,
% 0.31/0.47 binintersectSubset2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset2,definition,
% 0.31/0.47 ( binintersectSubset2
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( ( binintersect @ A @ B )
% 0.31/0.47 = A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset3_type,type,
% 0.31/0.47 binintersectSubset3: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset3,definition,
% 0.31/0.47 ( binintersectSubset3
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( ( binintersect @ A @ B )
% 0.31/0.47 = B )
% 0.31/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectER_type,type,
% 0.31/0.47 binintersectER: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectER,definition,
% 0.31/0.47 ( binintersectER
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(disjointsetsI1_type,type,
% 0.31/0.47 disjointsetsI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(disjointsetsI1,definition,
% 0.31/0.47 ( disjointsetsI1
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ~ ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ( in @ Xx @ B ) )
% 0.31/0.47 => ( ( binintersect @ A @ B )
% 0.31/0.47 = emptyset ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectRsub_type,type,
% 0.31/0.47 binintersectRsub: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectRsub,definition,
% 0.31/0.47 ( binintersectRsub
% 0.31/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset4_type,type,
% 0.31/0.47 binintersectSubset4: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset4,definition,
% 0.31/0.47 ( binintersectSubset4
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( subset @ B @ A )
% 0.31/0.47 => ( ( binintersect @ A @ B )
% 0.31/0.47 = B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset1_type,type,
% 0.31/0.47 binintersectSubset1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectSubset1,definition,
% 0.31/0.47 ( binintersectSubset1
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( ( binintersect @ A @ B )
% 0.31/0.47 = A )
% 0.31/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(bs114d_type,type,
% 0.31/0.47 bs114d: $o ).
% 0.31/0.47
% 0.31/0.47 thf(bs114d,definition,
% 0.31/0.47 ( bs114d
% 0.31/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.31/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(regular_type,type,
% 0.31/0.47 regular: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminus_type,type,
% 0.31/0.47 setminus: $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(setminusI_type,type,
% 0.31/0.47 setminusI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusI,definition,
% 0.31/0.47 ( setminusI
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusEL_type,type,
% 0.31/0.47 setminusEL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusEL,definition,
% 0.31/0.47 ( setminusEL
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusER_type,type,
% 0.31/0.47 setminusER: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusER,definition,
% 0.31/0.47 ( setminusER
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusSubset2_type,type,
% 0.31/0.47 setminusSubset2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusSubset2,definition,
% 0.31/0.47 ( setminusSubset2
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( subset @ A @ B )
% 0.31/0.47 => ( ( setminus @ A @ B )
% 0.31/0.47 = emptyset ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusERneg_type,type,
% 0.31/0.47 setminusERneg: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusERneg,definition,
% 0.31/0.47 ( setminusERneg
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.47 => ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusELneg_type,type,
% 0.31/0.47 setminusELneg: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusELneg,definition,
% 0.31/0.47 ( setminusELneg
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/0.47 => ( ~ ( in @ Xx @ B )
% 0.31/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusILneg_type,type,
% 0.31/0.47 setminusILneg: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusILneg,definition,
% 0.31/0.47 ( setminusILneg
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ~ ( in @ Xx @ A )
% 0.31/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusIRneg_type,type,
% 0.31/0.47 setminusIRneg: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusIRneg,definition,
% 0.31/0.47 ( setminusIRneg
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ B )
% 0.31/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusLsub_type,type,
% 0.31/0.47 setminusLsub: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusLsub,definition,
% 0.31/0.47 ( setminusLsub
% 0.31/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusSubset1_type,type,
% 0.31/0.47 setminusSubset1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusSubset1,definition,
% 0.31/0.47 ( setminusSubset1
% 0.31/0.47 = ( ! [A: $i,B: $i] :
% 0.31/0.47 ( ( ( setminus @ A @ B )
% 0.31/0.47 = emptyset )
% 0.31/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(symdiff_type,type,
% 0.31/0.47 symdiff: $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffE_type,type,
% 0.31/0.47 symdiffE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffE,definition,
% 0.31/0.47 ( symdiffE
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.31/0.47 => ! [Xphi: $o] :
% 0.31/0.47 ( ( ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ B )
% 0.31/0.47 => Xphi ) )
% 0.31/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ B )
% 0.31/0.47 => Xphi ) )
% 0.31/0.47 => Xphi ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffI1_type,type,
% 0.31/0.47 symdiffI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffI1,definition,
% 0.31/0.47 ( symdiffI1
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffI2_type,type,
% 0.31/0.47 symdiffI2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffI2,definition,
% 0.31/0.47 ( symdiffI2
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ~ ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ B )
% 0.31/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffIneg1_type,type,
% 0.31/0.47 symdiffIneg1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffIneg1,definition,
% 0.31/0.47 ( symdiffIneg1
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ B )
% 0.31/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffIneg2_type,type,
% 0.31/0.47 symdiffIneg2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(symdiffIneg2,definition,
% 0.31/0.47 ( symdiffIneg2
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ~ ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ B )
% 0.31/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(iskpair_type,type,
% 0.31/0.47 iskpair: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(secondinupair_type,type,
% 0.31/0.47 secondinupair: $o ).
% 0.31/0.47
% 0.31/0.47 thf(secondinupair,definition,
% 0.31/0.47 ( secondinupair
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairIL_type,type,
% 0.31/0.47 setukpairIL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairIL,definition,
% 0.31/0.47 ( setukpairIL
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairIR_type,type,
% 0.31/0.47 setukpairIR: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairIR,definition,
% 0.31/0.47 ( setukpairIR
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(kpairiskpair_type,type,
% 0.31/0.47 kpairiskpair: $o ).
% 0.31/0.47
% 0.31/0.47 thf(kpairiskpair,definition,
% 0.31/0.47 ( kpairiskpair
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(kpair_type,type,
% 0.31/0.47 kpair: $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(kpairp_type,type,
% 0.31/0.47 kpairp: $o ).
% 0.31/0.47
% 0.31/0.47 thf(kpairp,definition,
% 0.31/0.47 ( kpairp
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprod_type,type,
% 0.31/0.47 cartprod: $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(singletonsubset_type,type,
% 0.31/0.47 singletonsubset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(singletonsubset,definition,
% 0.31/0.47 ( singletonsubset
% 0.31/0.47 = ( ! [A: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(singletoninpowerset_type,type,
% 0.31/0.47 singletoninpowerset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(singletoninpowerset,definition,
% 0.31/0.47 ( singletoninpowerset
% 0.31/0.47 = ( ! [A: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(singletoninpowunion_type,type,
% 0.31/0.47 singletoninpowunion: $o ).
% 0.31/0.47
% 0.31/0.47 thf(singletoninpowunion,definition,
% 0.31/0.47 ( singletoninpowunion
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairset2E_type,type,
% 0.31/0.47 upairset2E: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairset2E,definition,
% 0.31/0.47 ( upairset2E
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/0.47 => ( ( Xz = Xx )
% 0.31/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairsubunion_type,type,
% 0.31/0.47 upairsubunion: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairsubunion,definition,
% 0.31/0.47 ( upairsubunion
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairinpowunion_type,type,
% 0.31/0.47 upairinpowunion: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairinpowunion,definition,
% 0.31/0.47 ( upairinpowunion
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ubforcartprodlem1_type,type,
% 0.31/0.47 ubforcartprodlem1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ubforcartprodlem1,definition,
% 0.31/0.47 ( ubforcartprodlem1
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ubforcartprodlem2_type,type,
% 0.31/0.47 ubforcartprodlem2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ubforcartprodlem2,definition,
% 0.31/0.47 ( ubforcartprodlem2
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ubforcartprodlem3_type,type,
% 0.31/0.47 ubforcartprodlem3: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ubforcartprodlem3,definition,
% 0.31/0.47 ( ubforcartprodlem3
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodpairin_type,type,
% 0.31/0.47 cartprodpairin: $o ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodpairin,definition,
% 0.31/0.47 ( cartprodpairin
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodmempair1_type,type,
% 0.31/0.47 cartprodmempair1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodmempair1,definition,
% 0.31/0.47 ( cartprodmempair1
% 0.31/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ? [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 & ( Xu
% 0.31/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodmempair_type,type,
% 0.31/0.47 cartprodmempair: $o ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodmempair,definition,
% 0.31/0.47 ( cartprodmempair
% 0.31/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.47 => ( iskpair @ Xu ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunionE2_type,type,
% 0.31/0.47 setunionE2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunionE2,definition,
% 0.31/0.47 ( setunionE2
% 0.31/0.47 = ( ! [A: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/0.47 => ? [X: $i] :
% 0.31/0.47 ( ( in @ X @ A )
% 0.31/0.47 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunionsingleton1_type,type,
% 0.31/0.47 setunionsingleton1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunionsingleton1,definition,
% 0.31/0.47 ( setunionsingleton1
% 0.31/0.47 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunionsingleton2_type,type,
% 0.31/0.47 setunionsingleton2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunionsingleton2,definition,
% 0.31/0.47 ( setunionsingleton2
% 0.31/0.47 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setunionsingleton_type,type,
% 0.31/0.47 setunionsingleton: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setunionsingleton,definition,
% 0.31/0.47 ( setunionsingleton
% 0.31/0.47 = ( ! [Xx: $i] :
% 0.31/0.47 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.31/0.47 = Xx ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(singleton_type,type,
% 0.31/0.47 singleton: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(singletonprop_type,type,
% 0.31/0.47 singletonprop: $o ).
% 0.31/0.47
% 0.31/0.47 thf(singletonprop,definition,
% 0.31/0.47 ( singletonprop
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 => ( ( Xphi @ Xy )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) )
% 0.31/0.47 => ( ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ( Xphi @ Xx ) )
% 0.31/0.47 => ( singleton
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ex1_type,type,
% 0.31/0.47 ex1: $i > ( $i > $o ) > $o ).
% 0.31/0.47
% 0.31/0.47 thf(ex1,definition,
% 0.31/0.47 ( ex1
% 0.31/0.47 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( singleton
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ex1E1_type,type,
% 0.31/0.47 ex1E1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ex1E1,definition,
% 0.31/0.47 ( ex1E1
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ( ex1 @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ex1I_type,type,
% 0.31/0.47 ex1I: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ex1I,definition,
% 0.31/0.47 ( ex1I
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 => ( ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( ( Xphi @ Xy )
% 0.31/0.47 => ( Xy = Xx ) ) )
% 0.31/0.47 => ( ex1 @ A
% 0.31/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ex1I2_type,type,
% 0.31/0.47 ex1I2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ex1I2,definition,
% 0.31/0.47 ( ex1I2
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 => ( ( Xphi @ Xy )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) )
% 0.31/0.47 => ( ? [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 & ( Xphi @ Xx ) )
% 0.31/0.47 => ( ex1 @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(singletonsuniq_type,type,
% 0.31/0.47 singletonsuniq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(singletonsuniq,definition,
% 0.31/0.47 ( singletonsuniq
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( ( setadjoin @ Xx @ emptyset )
% 0.31/0.47 = ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(atmost1p_type,type,
% 0.31/0.47 atmost1p: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(atleast2p_type,type,
% 0.31/0.47 atleast2p: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(atmost2p_type,type,
% 0.31/0.47 atmost2p: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairsetp_type,type,
% 0.31/0.47 upairsetp: $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjL1_type,type,
% 0.31/0.47 setukpairinjL1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjL1,definition,
% 0.31/0.47 ( setukpairinjL1
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.47 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(kfstsingleton_type,type,
% 0.31/0.47 kfstsingleton: $o ).
% 0.31/0.47
% 0.31/0.47 thf(kfstsingleton,definition,
% 0.31/0.47 ( kfstsingleton
% 0.31/0.47 = ( ! [Xu: $i] :
% 0.31/0.47 ( ( iskpair @ Xu )
% 0.31/0.47 => ( singleton
% 0.31/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.31/0.47 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(theprop_type,type,
% 0.31/0.47 theprop: $o ).
% 0.31/0.47
% 0.31/0.47 thf(theprop,definition,
% 0.31/0.47 ( theprop
% 0.31/0.47 = ( ! [X: $i] :
% 0.31/0.47 ( ( singleton @ X )
% 0.31/0.47 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(kfst_type,type,
% 0.31/0.47 kfst: $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(kfstpairEq_type,type,
% 0.31/0.47 kfstpairEq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(kfstpairEq,definition,
% 0.31/0.47 ( kfstpairEq
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.31/0.47 = Xx ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodfstin_type,type,
% 0.31/0.47 cartprodfstin: $o ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodfstin,definition,
% 0.31/0.47 ( cartprodfstin
% 0.31/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.47 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjL2_type,type,
% 0.31/0.47 setukpairinjL2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjL2,definition,
% 0.31/0.47 ( setukpairinjL2
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjL_type,type,
% 0.31/0.47 setukpairinjL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjL,definition,
% 0.31/0.47 ( setukpairinjL
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.31/0.47 = ( kpair @ Xz @ Xu ) )
% 0.31/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjR11_type,type,
% 0.31/0.47 setukpairinjR11: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjR11,definition,
% 0.31/0.47 ( setukpairinjR11
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx = Xy )
% 0.31/0.47 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjR12_type,type,
% 0.31/0.47 setukpairinjR12: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjR12,definition,
% 0.31/0.47 ( setukpairinjR12
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xx = Xy )
% 0.31/0.47 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjR1_type,type,
% 0.31/0.47 setukpairinjR1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjR1,definition,
% 0.31/0.47 ( setukpairinjR1
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.47 => ( ( Xz = Xu )
% 0.31/0.47 => ( Xy = Xu ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(upairequniteq_type,type,
% 0.31/0.47 upairequniteq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(upairequniteq,definition,
% 0.31/0.47 ( upairequniteq
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/0.47 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/0.47 = ( setadjoin @ Xz @ emptyset ) )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjR2_type,type,
% 0.31/0.47 setukpairinjR2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjR2,definition,
% 0.31/0.47 ( setukpairinjR2
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjR_type,type,
% 0.31/0.47 setukpairinjR: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setukpairinjR,definition,
% 0.31/0.47 ( setukpairinjR
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.31/0.47 = ( kpair @ Xz @ Xu ) )
% 0.31/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ksndsingleton_type,type,
% 0.31/0.47 ksndsingleton: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ksndsingleton,definition,
% 0.31/0.47 ( ksndsingleton
% 0.31/0.47 = ( ! [Xu: $i] :
% 0.31/0.47 ( ( iskpair @ Xu )
% 0.31/0.47 => ( singleton
% 0.31/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.31/0.47 @ ^ [Xx: $i] :
% 0.31/0.47 ( Xu
% 0.31/0.47 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ksnd_type,type,
% 0.31/0.47 ksnd: $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(ksndpairEq_type,type,
% 0.31/0.47 ksndpairEq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ksndpairEq,definition,
% 0.31/0.47 ( ksndpairEq
% 0.31/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.31/0.47 = Xy ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(kpairsurjEq_type,type,
% 0.31/0.47 kpairsurjEq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(kpairsurjEq,definition,
% 0.31/0.47 ( kpairsurjEq
% 0.31/0.47 = ( ! [Xu: $i] :
% 0.31/0.47 ( ( iskpair @ Xu )
% 0.31/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.31/0.47 = Xu ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodsndin_type,type,
% 0.31/0.47 cartprodsndin: $o ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodsndin,definition,
% 0.31/0.47 ( cartprodsndin
% 0.31/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.47 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodpairmemEL_type,type,
% 0.31/0.47 cartprodpairmemEL: $o ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodpairmemEL,definition,
% 0.31/0.47 ( cartprodpairmemEL
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.31/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodpairmemER_type,type,
% 0.31/0.47 cartprodpairmemER: $o ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodpairmemER,definition,
% 0.31/0.47 ( cartprodpairmemER
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.31/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodmempaircEq_type,type,
% 0.31/0.47 cartprodmempaircEq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodmempaircEq,definition,
% 0.31/0.47 ( cartprodmempaircEq
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( ( kpair @ Xx @ Xy )
% 0.31/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodfstpairEq_type,type,
% 0.31/0.47 cartprodfstpairEq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodfstpairEq,definition,
% 0.31/0.47 ( cartprodfstpairEq
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.31/0.47 = Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodsndpairEq_type,type,
% 0.31/0.47 cartprodsndpairEq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodsndpairEq,definition,
% 0.31/0.47 ( cartprodsndpairEq
% 0.31/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.31/0.47 = Xy ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodpairsurjEq_type,type,
% 0.31/0.47 cartprodpairsurjEq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(cartprodpairsurjEq,definition,
% 0.31/0.47 ( cartprodpairsurjEq
% 0.31/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.31/0.47 = Xu ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(breln_type,type,
% 0.31/0.47 breln: $i > $i > $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(breln,definition,
% 0.31/0.47 ( breln
% 0.31/0.47 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(dpsetconstr_type,type,
% 0.31/0.47 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.31/0.47
% 0.31/0.47 thf(dpsetconstrI_type,type,
% 0.31/0.47 dpsetconstrI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(dpsetconstrI,definition,
% 0.31/0.47 ( dpsetconstrI
% 0.31/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( ( Xphi @ Xx @ Xy )
% 0.31/0.47 => ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.47 @ ( dpsetconstr @ A @ B
% 0.31/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(dpsetconstrSub_type,type,
% 0.31/0.47 dpsetconstrSub: $o ).
% 0.31/0.47
% 0.31/0.47 thf(dpsetconstrSub,definition,
% 0.31/0.47 ( dpsetconstrSub
% 0.31/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/0.47 ( subset
% 0.31/0.47 @ ( dpsetconstr @ A @ B
% 0.31/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.31/0.47 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setOfPairsIsBReln_type,type,
% 0.31/0.47 setOfPairsIsBReln: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setOfPairsIsBReln,definition,
% 0.31/0.47 ( setOfPairsIsBReln
% 0.31/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/0.47 ( breln @ A @ B
% 0.31/0.47 @ ( dpsetconstr @ A @ B
% 0.31/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(dpsetconstrERa_type,type,
% 0.31/0.47 dpsetconstrERa: $o ).
% 0.31/0.47
% 0.31/0.47 thf(dpsetconstrERa,definition,
% 0.31/0.47 ( dpsetconstrERa
% 0.31/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.47 @ ( dpsetconstr @ A @ B
% 0.31/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(dpsetconstrEL1_type,type,
% 0.31/0.47 dpsetconstrEL1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(dpsetconstrEL1,definition,
% 0.31/0.47 ( dpsetconstrEL1
% 0.31/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.47 @ ( dpsetconstr @ A @ B
% 0.31/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(dpsetconstrEL2_type,type,
% 0.31/0.47 dpsetconstrEL2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(dpsetconstrEL2,definition,
% 0.31/0.47 ( dpsetconstrEL2
% 0.31/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.47 @ ( dpsetconstr @ A @ B
% 0.31/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(dpsetconstrER_type,type,
% 0.31/0.47 dpsetconstrER: $o ).
% 0.31/0.47
% 0.31/0.47 thf(dpsetconstrER,definition,
% 0.31/0.47 ( dpsetconstrER
% 0.31/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/0.47 @ ( dpsetconstr @ A @ B
% 0.31/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(func_type,type,
% 0.31/0.47 func: $i > $i > $i > $o ).
% 0.31/0.47
% 0.31/0.47 thf(funcSet_type,type,
% 0.31/0.47 funcSet: $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(funcImageSingleton_type,type,
% 0.31/0.47 funcImageSingleton: $o ).
% 0.31/0.47
% 0.31/0.47 thf(funcImageSingleton,definition,
% 0.31/0.47 ( funcImageSingleton
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( func @ A @ B @ Xf )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( singleton
% 0.31/0.47 @ ( dsetconstr @ B
% 0.31/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(apProp_type,type,
% 0.31/0.47 apProp: $o ).
% 0.31/0.47
% 0.31/0.47 thf(apProp,definition,
% 0.31/0.47 ( apProp
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( func @ A @ B @ Xf )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in
% 0.31/0.47 @ ( setunion
% 0.31/0.47 @ ( dsetconstr @ B
% 0.31/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.31/0.47 @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ap_type,type,
% 0.31/0.47 ap: $i > $i > $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(app_type,type,
% 0.31/0.47 app: $o ).
% 0.31/0.47
% 0.31/0.47 thf(app,definition,
% 0.31/0.47 ( app
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( func @ A @ B @ Xf )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(infuncsetfunc_type,type,
% 0.31/0.47 infuncsetfunc: $o ).
% 0.31/0.47
% 0.31/0.47 thf(infuncsetfunc,definition,
% 0.31/0.47 ( infuncsetfunc
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.47 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ap2p_type,type,
% 0.31/0.47 ap2p: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ap2p,definition,
% 0.31/0.47 ( ap2p
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(funcinfuncset_type,type,
% 0.31/0.47 funcinfuncset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(funcinfuncset,definition,
% 0.31/0.47 ( funcinfuncset
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( func @ A @ B @ Xf )
% 0.31/0.47 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(lamProp_type,type,
% 0.31/0.47 lamProp: $o ).
% 0.31/0.47
% 0.31/0.47 thf(lamProp,definition,
% 0.31/0.47 ( lamProp
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.47 => ( func @ A @ B
% 0.31/0.47 @ ( dpsetconstr @ A @ B
% 0.31/0.47 @ ^ [Xx: $i,Xy: $i] :
% 0.31/0.47 ( ( Xf @ Xx )
% 0.31/0.47 = Xy ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(lam_type,type,
% 0.31/0.47 lam: $i > $i > ( $i > $i ) > $i ).
% 0.31/0.47
% 0.31/0.47 thf(lamp_type,type,
% 0.31/0.47 lamp: $o ).
% 0.31/0.47
% 0.31/0.47 thf(lamp,definition,
% 0.31/0.47 ( lamp
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.47 => ( func @ A @ B
% 0.31/0.47 @ ( lam @ A @ B
% 0.31/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(lam2p_type,type,
% 0.31/0.47 lam2p: $o ).
% 0.31/0.47
% 0.31/0.47 thf(lam2p,definition,
% 0.31/0.47 ( lam2p
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.47 => ( in
% 0.31/0.47 @ ( lam @ A @ B
% 0.31/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.31/0.47 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(brelnall1_type,type,
% 0.31/0.47 brelnall1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(brelnall1,definition,
% 0.31/0.47 ( brelnall1
% 0.31/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/0.47 ( ( breln @ A @ B @ R )
% 0.31/0.47 => ! [Xphi: $i > $o] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ R )
% 0.31/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(brelnall2_type,type,
% 0.31/0.47 brelnall2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(brelnall2,definition,
% 0.31/0.47 ( brelnall2
% 0.31/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/0.47 ( ( breln @ A @ B @ R )
% 0.31/0.47 => ! [Xphi: $i > $o] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ R )
% 0.31/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ex1E2_type,type,
% 0.31/0.47 ex1E2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ex1E2,definition,
% 0.31/0.47 ( ex1E2
% 0.31/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/0.47 ( ( ex1 @ A
% 0.31/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( ( Xphi @ Xx )
% 0.31/0.47 => ( ( Xphi @ Xy )
% 0.31/0.47 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(funcGraphProp1_type,type,
% 0.31/0.47 funcGraphProp1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(funcGraphProp1,definition,
% 0.31/0.47 ( funcGraphProp1
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( func @ A @ B @ Xf )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(funcGraphProp3_type,type,
% 0.31/0.47 funcGraphProp3: $o ).
% 0.31/0.47
% 0.31/0.47 thf(funcGraphProp3,definition,
% 0.31/0.47 ( funcGraphProp3
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(funcGraphProp2_type,type,
% 0.31/0.47 funcGraphProp2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(funcGraphProp2,definition,
% 0.31/0.47 ( funcGraphProp2
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( func @ A @ B @ Xf )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.31/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(funcextLem_type,type,
% 0.31/0.47 funcextLem: $o ).
% 0.31/0.47
% 0.31/0.47 thf(funcextLem,definition,
% 0.31/0.47 ( funcextLem
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( func @ A @ B @ Xf )
% 0.31/0.47 => ! [Xg: $i] :
% 0.31/0.47 ( ( func @ A @ B @ Xg )
% 0.31/0.47 => ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.31/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(funcGraphProp4_type,type,
% 0.31/0.47 funcGraphProp4: $o ).
% 0.31/0.47
% 0.31/0.47 thf(funcGraphProp4,definition,
% 0.31/0.47 ( funcGraphProp4
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.31/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subbreln_type,type,
% 0.31/0.47 subbreln: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subbreln,definition,
% 0.31/0.47 ( subbreln
% 0.31/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/0.47 ( ( breln @ A @ B @ R )
% 0.31/0.47 => ! [S: $i] :
% 0.31/0.47 ( ( breln @ A @ B @ S )
% 0.31/0.47 => ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.31/0.47 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(eqbreln_type,type,
% 0.31/0.47 eqbreln: $o ).
% 0.31/0.47
% 0.31/0.47 thf(eqbreln,definition,
% 0.31/0.47 ( eqbreln
% 0.31/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/0.47 ( ( breln @ A @ B @ R )
% 0.31/0.47 => ! [S: $i] :
% 0.31/0.47 ( ( breln @ A @ B @ S )
% 0.31/0.47 => ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.31/0.47 => ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ B )
% 0.31/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.31/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.31/0.47 => ( R = S ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(funcext_type,type,
% 0.31/0.47 funcext: $o ).
% 0.31/0.47
% 0.31/0.47 thf(funcext,definition,
% 0.31/0.47 ( funcext
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( func @ A @ B @ Xf )
% 0.31/0.47 => ! [Xg: $i] :
% 0.31/0.47 ( ( func @ A @ B @ Xg )
% 0.31/0.47 => ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(funcext2_type,type,
% 0.31/0.47 funcext2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(funcext2,definition,
% 0.31/0.47 ( funcext2
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.47 => ! [Xg: $i] :
% 0.31/0.47 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.31/0.47 => ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ap2apEq1_type,type,
% 0.31/0.47 ap2apEq1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ap2apEq1,definition,
% 0.31/0.47 ( ap2apEq1
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ap2apEq2_type,type,
% 0.31/0.47 ap2apEq2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ap2apEq2,definition,
% 0.31/0.47 ( ap2apEq2
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( func @ A @ B @ Xf )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(beta1_type,type,
% 0.31/0.47 beta1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(beta1,definition,
% 0.31/0.47 ( beta1
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( ap @ A @ B
% 0.31/0.47 @ ( lam @ A @ B
% 0.31/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.31/0.47 @ Xx )
% 0.31/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(eta1_type,type,
% 0.31/0.47 eta1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(eta1,definition,
% 0.31/0.47 ( eta1
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( func @ A @ B @ Xf )
% 0.31/0.47 => ( ( lam @ A @ B
% 0.31/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.31/0.47 = Xf ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(lam2lamEq_type,type,
% 0.31/0.47 lam2lamEq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(lam2lamEq,definition,
% 0.31/0.47 ( lam2lamEq
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.47 => ( ( lam @ A @ B
% 0.31/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.31/0.47 = ( lam @ A @ B
% 0.31/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(beta2_type,type,
% 0.31/0.47 beta2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(beta2,definition,
% 0.31/0.47 ( beta2
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/0.47 ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( ap @ A @ B
% 0.31/0.47 @ ( lam @ A @ B
% 0.31/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.31/0.47 @ Xx )
% 0.31/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(eta2_type,type,
% 0.31/0.47 eta2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(eta2,definition,
% 0.31/0.47 ( eta2
% 0.31/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/0.47 => ( ( lam @ A @ B
% 0.31/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.31/0.47 = Xf ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(iffalseProp1_type,type,
% 0.31/0.47 iffalseProp1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(iffalseProp1,definition,
% 0.31/0.47 ( iffalseProp1
% 0.31/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( ~ Xphi
% 0.31/0.47 => ( in @ Xy
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xz: $i] :
% 0.31/0.47 ( ( Xphi
% 0.31/0.47 & ( Xz = Xx ) )
% 0.31/0.47 | ( ~ Xphi
% 0.31/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(iffalseProp2_type,type,
% 0.31/0.47 iffalseProp2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(iffalseProp2,definition,
% 0.31/0.47 ( iffalseProp2
% 0.31/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( ~ Xphi
% 0.31/0.47 => ( ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xz: $i] :
% 0.31/0.47 ( ( Xphi
% 0.31/0.47 & ( Xz = Xx ) )
% 0.31/0.47 | ( ~ Xphi
% 0.31/0.47 & ( Xz = Xy ) ) ) )
% 0.31/0.47 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(iftrueProp1_type,type,
% 0.31/0.47 iftrueProp1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(iftrueProp1,definition,
% 0.31/0.47 ( iftrueProp1
% 0.31/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( Xphi
% 0.31/0.47 => ( in @ Xx
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xz: $i] :
% 0.31/0.47 ( ( Xphi
% 0.31/0.47 & ( Xz = Xx ) )
% 0.31/0.47 | ( ~ Xphi
% 0.31/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(iftrueProp2_type,type,
% 0.31/0.47 iftrueProp2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(iftrueProp2,definition,
% 0.31/0.47 ( iftrueProp2
% 0.31/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( Xphi
% 0.31/0.47 => ( ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xz: $i] :
% 0.31/0.47 ( ( Xphi
% 0.31/0.47 & ( Xz = Xx ) )
% 0.31/0.47 | ( ~ Xphi
% 0.31/0.47 & ( Xz = Xy ) ) ) )
% 0.31/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(ifSingleton_type,type,
% 0.31/0.47 ifSingleton: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ifSingleton,definition,
% 0.31/0.47 ( ifSingleton
% 0.31/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( singleton
% 0.31/0.47 @ ( dsetconstr @ A
% 0.31/0.47 @ ^ [Xz: $i] :
% 0.31/0.47 ( ( Xphi
% 0.31/0.47 & ( Xz = Xx ) )
% 0.31/0.47 | ( ~ Xphi
% 0.31/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(if_type,type,
% 0.31/0.47 if: $i > $o > $i > $i > $i ).
% 0.31/0.47
% 0.31/0.47 thf(ifp_type,type,
% 0.31/0.47 ifp: $o ).
% 0.31/0.47
% 0.31/0.47 thf(ifp,definition,
% 0.31/0.47 ( ifp
% 0.31/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(theeq_type,type,
% 0.31/0.47 theeq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(theeq,definition,
% 0.31/0.47 ( theeq
% 0.31/0.47 = ( ! [X: $i] :
% 0.31/0.47 ( ( singleton @ X )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ X )
% 0.31/0.47 => ( ( setunion @ X )
% 0.31/0.47 = Xx ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(iftrue_type,type,
% 0.31/0.47 iftrue: $o ).
% 0.31/0.47
% 0.31/0.47 thf(iftrue,definition,
% 0.31/0.47 ( iftrue
% 0.31/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( Xphi
% 0.31/0.47 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.31/0.47 = Xx ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(iffalse_type,type,
% 0.31/0.47 iffalse: $o ).
% 0.31/0.47
% 0.31/0.47 thf(iffalse,definition,
% 0.31/0.47 ( iffalse
% 0.31/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( ~ Xphi
% 0.31/0.47 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.31/0.47 = Xy ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(iftrueorfalse_type,type,
% 0.31/0.47 iftrueorfalse: $o ).
% 0.31/0.47
% 0.31/0.47 thf(iftrueorfalse,definition,
% 0.31/0.47 ( iftrueorfalse
% 0.31/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ! [Xy: $i] :
% 0.31/0.47 ( ( in @ Xy @ A )
% 0.31/0.47 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectT_lem_type,type,
% 0.31/0.47 binintersectT_lem: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectT_lem,definition,
% 0.31/0.47 ( binintersectT_lem
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunionT_lem_type,type,
% 0.31/0.47 binunionT_lem: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionT_lem,definition,
% 0.31/0.47 ( binunionT_lem
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powersetT_lem_type,type,
% 0.31/0.47 powersetT_lem: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetT_lem,definition,
% 0.31/0.47 ( powersetT_lem
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setminusT_lem_type,type,
% 0.31/0.47 setminusT_lem: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setminusT_lem,definition,
% 0.31/0.47 ( setminusT_lem
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(complementT_lem_type,type,
% 0.31/0.47 complementT_lem: $o ).
% 0.31/0.47
% 0.31/0.47 thf(complementT_lem,definition,
% 0.31/0.47 ( complementT_lem
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(setextT_type,type,
% 0.31/0.47 setextT: $o ).
% 0.31/0.47
% 0.31/0.47 thf(setextT,definition,
% 0.31/0.47 ( setextT
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ X )
% 0.31/0.47 => ( in @ Xx @ Y ) ) )
% 0.31/0.47 => ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ Y )
% 0.31/0.47 => ( in @ Xx @ X ) ) )
% 0.31/0.47 => ( X = Y ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subsetTI_type,type,
% 0.31/0.47 subsetTI: $o ).
% 0.31/0.47
% 0.31/0.47 thf(subsetTI,definition,
% 0.31/0.47 ( subsetTI
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ X )
% 0.31/0.47 => ( in @ Xx @ Y ) ) )
% 0.31/0.47 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powersetTI1_type,type,
% 0.31/0.47 powersetTI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetTI1,definition,
% 0.31/0.47 ( powersetTI1
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ( ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ X )
% 0.31/0.47 => ( in @ Xx @ Y ) ) )
% 0.31/0.47 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(powersetTE1_type,type,
% 0.31/0.47 powersetTE1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(powersetTE1,definition,
% 0.31/0.47 ( powersetTE1
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ X @ ( powerset @ Y ) )
% 0.31/0.47 => ( ( in @ Xx @ X )
% 0.31/0.47 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(complementTI1_type,type,
% 0.31/0.47 complementTI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(complementTI1,definition,
% 0.31/0.47 ( complementTI1
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ X )
% 0.31/0.47 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(complementTE1_type,type,
% 0.31/0.47 complementTE1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(complementTE1,definition,
% 0.31/0.47 ( complementTE1
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.31/0.47 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectTELcontra_type,type,
% 0.31/0.47 binintersectTELcontra: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectTELcontra,definition,
% 0.31/0.47 ( binintersectTELcontra
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ X )
% 0.31/0.47 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectTERcontra_type,type,
% 0.31/0.47 binintersectTERcontra: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binintersectTERcontra,definition,
% 0.31/0.47 ( binintersectTERcontra
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ Y )
% 0.31/0.47 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(contrasubsetT_type,type,
% 0.31/0.47 contrasubsetT: $o ).
% 0.31/0.47
% 0.31/0.47 thf(contrasubsetT,definition,
% 0.31/0.47 ( contrasubsetT
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.31/0.47 => ( ( in @ Xx @ Y )
% 0.31/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(contrasubsetT1_type,type,
% 0.31/0.47 contrasubsetT1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(contrasubsetT1,definition,
% 0.31/0.47 ( contrasubsetT1
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( subset @ X @ Y )
% 0.31/0.47 => ( ~ ( in @ Xx @ Y )
% 0.31/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(contrasubsetT2_type,type,
% 0.31/0.47 contrasubsetT2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(contrasubsetT2,definition,
% 0.31/0.47 ( contrasubsetT2
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ( ( subset @ X @ Y )
% 0.31/0.47 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(contrasubsetT3_type,type,
% 0.31/0.47 contrasubsetT3: $o ).
% 0.31/0.47
% 0.31/0.47 thf(contrasubsetT3,definition,
% 0.31/0.47 ( contrasubsetT3
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.31/0.47 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(doubleComplementI1_type,type,
% 0.31/0.47 doubleComplementI1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(doubleComplementI1,definition,
% 0.31/0.47 ( doubleComplementI1
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ X )
% 0.31/0.47 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(doubleComplementE1_type,type,
% 0.31/0.47 doubleComplementE1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(doubleComplementE1,definition,
% 0.31/0.47 ( doubleComplementE1
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.31/0.47 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(doubleComplementSub1_type,type,
% 0.31/0.47 doubleComplementSub1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(doubleComplementSub1,definition,
% 0.31/0.47 ( doubleComplementSub1
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(doubleComplementSub2_type,type,
% 0.31/0.47 doubleComplementSub2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(doubleComplementSub2,definition,
% 0.31/0.47 ( doubleComplementSub2
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(doubleComplementEq_type,type,
% 0.31/0.47 doubleComplementEq: $o ).
% 0.31/0.47
% 0.31/0.47 thf(doubleComplementEq,definition,
% 0.31/0.47 ( doubleComplementEq
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ( X
% 0.31/0.47 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(complementTnotintersectT_type,type,
% 0.31/0.47 complementTnotintersectT: $o ).
% 0.31/0.47
% 0.31/0.47 thf(complementTnotintersectT,definition,
% 0.31/0.47 ( complementTnotintersectT
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.31/0.47 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(complementImpComplementIntersect_type,type,
% 0.31/0.47 complementImpComplementIntersect: $o ).
% 0.31/0.47
% 0.31/0.47 thf(complementImpComplementIntersect,definition,
% 0.31/0.47 ( complementImpComplementIntersect
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.31/0.47 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(complementSubsetComplementIntersect_type,type,
% 0.31/0.47 complementSubsetComplementIntersect: $o ).
% 0.31/0.47
% 0.31/0.47 thf(complementSubsetComplementIntersect,definition,
% 0.31/0.47 ( complementSubsetComplementIntersect
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(complementInPowersetComplementIntersect_type,type,
% 0.31/0.47 complementInPowersetComplementIntersect: $o ).
% 0.31/0.47
% 0.31/0.47 thf(complementInPowersetComplementIntersect,definition,
% 0.31/0.47 ( complementInPowersetComplementIntersect
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(contraSubsetComplement_type,type,
% 0.31/0.47 contraSubsetComplement: $o ).
% 0.31/0.47
% 0.31/0.47 thf(contraSubsetComplement,definition,
% 0.31/0.47 ( contraSubsetComplement
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ Y )
% 0.31/0.47 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(complementTcontraSubset_type,type,
% 0.31/0.47 complementTcontraSubset: $o ).
% 0.31/0.47
% 0.31/0.47 thf(complementTcontraSubset,definition,
% 0.31/0.47 ( complementTcontraSubset
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.31/0.47 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunionTILcontra_type,type,
% 0.31/0.47 binunionTILcontra: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionTILcontra,definition,
% 0.31/0.47 ( binunionTILcontra
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.31/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunionTIRcontra_type,type,
% 0.31/0.47 binunionTIRcontra: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionTIRcontra,definition,
% 0.31/0.47 ( binunionTIRcontra
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.31/0.47 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(inIntersectImpInUnion_type,type,
% 0.31/0.47 inIntersectImpInUnion: $o ).
% 0.31/0.47
% 0.31/0.47 thf(inIntersectImpInUnion,definition,
% 0.31/0.47 ( inIntersectImpInUnion
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Z: $i] :
% 0.31/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.31/0.47 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(inIntersectImpInUnion2_type,type,
% 0.31/0.47 inIntersectImpInUnion2: $o ).
% 0.31/0.47
% 0.31/0.47 thf(inIntersectImpInUnion2,definition,
% 0.31/0.47 ( inIntersectImpInUnion2
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Z: $i] :
% 0.31/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.31/0.47 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(inIntersectImpInIntersectUnions_type,type,
% 0.31/0.47 inIntersectImpInIntersectUnions: $o ).
% 0.31/0.47
% 0.31/0.47 thf(inIntersectImpInIntersectUnions,definition,
% 0.31/0.47 ( inIntersectImpInIntersectUnions
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Z: $i] :
% 0.31/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.31/0.47 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(intersectInPowersetIntersectUnions_type,type,
% 0.31/0.47 intersectInPowersetIntersectUnions: $o ).
% 0.31/0.47
% 0.31/0.47 thf(intersectInPowersetIntersectUnions,definition,
% 0.31/0.47 ( intersectInPowersetIntersectUnions
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Z: $i] :
% 0.31/0.47 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.47 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(inComplementUnionImpNotIn1_type,type,
% 0.31/0.47 inComplementUnionImpNotIn1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(inComplementUnionImpNotIn1,definition,
% 0.31/0.47 ( inComplementUnionImpNotIn1
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.31/0.47 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(inComplementUnionImpInComplement1_type,type,
% 0.31/0.47 inComplementUnionImpInComplement1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(inComplementUnionImpInComplement1,definition,
% 0.31/0.47 ( inComplementUnionImpInComplement1
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.31/0.47 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunionTE_type,type,
% 0.31/0.47 binunionTE: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionTE,definition,
% 0.31/0.47 ( binunionTE
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xphi: $o,Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.31/0.47 => ( ( ( in @ Xx @ X )
% 0.31/0.47 => Xphi )
% 0.31/0.47 => ( ( ( in @ Xx @ Y )
% 0.31/0.47 => Xphi )
% 0.31/0.47 => Xphi ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(binunionTEcontra_type,type,
% 0.31/0.47 binunionTEcontra: $o ).
% 0.31/0.47
% 0.31/0.47 thf(binunionTEcontra,definition,
% 0.31/0.47 ( binunionTEcontra
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.47 => ! [Y: $i] :
% 0.31/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.47 => ! [Xx: $i] :
% 0.31/0.47 ( ( in @ Xx @ A )
% 0.31/0.47 => ( ~ ( in @ Xx @ X )
% 0.31/0.47 => ( ~ ( in @ Xx @ Y )
% 0.31/0.47 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(demorgan2a1_type,type,
% 0.31/0.47 demorgan2a1: $o ).
% 0.31/0.47
% 0.31/0.47 thf(demorgan2a1,definition,
% 0.31/0.47 ( demorgan2a1
% 0.31/0.47 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.31/0.48 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(complementUnionInPowersetComplement_type,type,
% 0.31/0.48 complementUnionInPowersetComplement: $o ).
% 0.31/0.48
% 0.31/0.48 thf(complementUnionInPowersetComplement,definition,
% 0.31/0.48 ( complementUnionInPowersetComplement
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan2a2_type,type,
% 0.31/0.48 demorgan2a2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan2a2,definition,
% 0.31/0.48 ( demorgan2a2
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.31/0.48 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan1a_type,type,
% 0.31/0.48 demorgan1a: $o ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan1a,definition,
% 0.31/0.48 ( demorgan1a
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.31/0.48 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan1b_type,type,
% 0.31/0.48 demorgan1b: $o ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan1b,definition,
% 0.31/0.48 ( demorgan1b
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.31/0.48 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan1_type,type,
% 0.31/0.48 demorgan1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan1,definition,
% 0.31/0.48 ( demorgan1
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.31/0.48 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan2a_type,type,
% 0.31/0.48 demorgan2a: $o ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan2a,definition,
% 0.31/0.48 ( demorgan2a
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.31/0.48 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan2b2_type,type,
% 0.31/0.48 demorgan2b2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan2b2,definition,
% 0.31/0.48 ( demorgan2b2
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.31/0.48 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.31/0.48 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan2b_type,type,
% 0.31/0.48 demorgan2b: $o ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan2b,definition,
% 0.31/0.48 ( demorgan2b
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.31/0.48 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan2_type,type,
% 0.31/0.48 demorgan2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan2,definition,
% 0.31/0.48 ( demorgan2
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 0.31/0.48 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(woz13rule0_type,type,
% 0.31/0.48 woz13rule0: $o ).
% 0.31/0.48
% 0.31/0.48 thf(woz13rule0,definition,
% 0.31/0.48 ( woz13rule0
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.31/0.48 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(woz13rule1_type,type,
% 0.31/0.48 woz13rule1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(woz13rule1,definition,
% 0.31/0.48 ( woz13rule1
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ! [Z: $i] :
% 0.31/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.48 => ( ( subset @ X @ Z )
% 0.31/0.48 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(woz13rule2_type,type,
% 0.31/0.48 woz13rule2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(woz13rule2,definition,
% 0.31/0.48 ( woz13rule2
% 0.31/0.48 = ( ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ! [Z: $i] :
% 0.31/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.48 => ( ( subset @ Y @ Z )
% 0.31/0.48 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(woz13rule3,conjecture,
% 0.31/0.48 ( setextAx
% 0.31/0.48 => ( emptysetAx
% 0.31/0.48 => ( setadjoinAx
% 0.31/0.48 => ( powersetAx
% 0.31/0.48 => ( setunionAx
% 0.31/0.48 => ( omega0Ax
% 0.31/0.48 => ( omegaSAx
% 0.31/0.48 => ( omegaIndAx
% 0.31/0.48 => ( replAx
% 0.31/0.48 => ( foundationAx
% 0.31/0.48 => ( wellorderingAx
% 0.31/0.48 => ( descrp
% 0.31/0.48 => ( dsetconstrI
% 0.31/0.48 => ( dsetconstrEL
% 0.31/0.48 => ( dsetconstrER
% 0.31/0.48 => ( exuE1
% 0.31/0.48 => ( prop2setE
% 0.31/0.48 => ( emptysetE
% 0.31/0.48 => ( emptysetimpfalse
% 0.31/0.48 => ( notinemptyset
% 0.31/0.48 => ( exuE3e
% 0.31/0.48 => ( setext
% 0.31/0.48 => ( emptyI
% 0.31/0.48 => ( noeltsimpempty
% 0.31/0.48 => ( setbeta
% 0.31/0.48 => ( nonemptyE1
% 0.31/0.48 => ( nonemptyI
% 0.31/0.48 => ( nonemptyI1
% 0.31/0.48 => ( setadjoinIL
% 0.31/0.48 => ( emptyinunitempty
% 0.31/0.48 => ( setadjoinIR
% 0.31/0.48 => ( setadjoinE
% 0.31/0.48 => ( setadjoinOr
% 0.31/0.48 => ( setoftrueEq
% 0.31/0.48 => ( powersetI
% 0.31/0.48 => ( emptyinPowerset
% 0.31/0.48 => ( emptyInPowerset
% 0.31/0.48 => ( powersetE
% 0.31/0.48 => ( setunionI
% 0.31/0.48 => ( setunionE
% 0.31/0.48 => ( subPowSU
% 0.31/0.48 => ( exuE2
% 0.31/0.48 => ( nonemptyImpWitness
% 0.31/0.48 => ( uniqinunit
% 0.31/0.48 => ( notinsingleton
% 0.31/0.48 => ( eqinunit
% 0.31/0.48 => ( singletonsswitch
% 0.31/0.48 => ( upairsetE
% 0.31/0.48 => ( upairsetIL
% 0.31/0.48 => ( upairsetIR
% 0.31/0.48 => ( emptyE1
% 0.31/0.48 => ( vacuousDall
% 0.31/0.48 => ( quantDeMorgan1
% 0.31/0.48 => ( quantDeMorgan2
% 0.31/0.48 => ( quantDeMorgan3
% 0.31/0.48 => ( quantDeMorgan4
% 0.31/0.48 => ( prop2setI
% 0.31/0.48 => ( prop2set2propI
% 0.31/0.48 => ( notdexE
% 0.31/0.48 => ( notdallE
% 0.31/0.48 => ( exuI1
% 0.31/0.48 => ( exuI3
% 0.31/0.48 => ( exuI2
% 0.31/0.48 => ( inCongP
% 0.31/0.48 => ( in__Cong
% 0.31/0.48 => ( exuE3u
% 0.31/0.48 => ( exu__Cong
% 0.31/0.48 => ( emptyset__Cong
% 0.31/0.48 => ( setadjoin__Cong
% 0.31/0.48 => ( powerset__Cong
% 0.31/0.48 => ( setunion__Cong
% 0.31/0.48 => ( omega__Cong
% 0.31/0.48 => ( exuEu
% 0.31/0.48 => ( descr__Cong
% 0.31/0.48 => ( dsetconstr__Cong
% 0.31/0.48 => ( subsetI1
% 0.31/0.48 => ( eqimpsubset2
% 0.31/0.48 => ( eqimpsubset1
% 0.31/0.48 => ( subsetI2
% 0.31/0.48 => ( emptysetsubset
% 0.31/0.48 => ( subsetE
% 0.31/0.48 => ( subsetE2
% 0.31/0.48 => ( notsubsetI
% 0.31/0.48 => ( notequalI1
% 0.31/0.48 => ( notequalI2
% 0.31/0.48 => ( subsetRefl
% 0.31/0.48 => ( subsetTrans
% 0.31/0.48 => ( setadjoinSub
% 0.31/0.48 => ( setadjoinSub2
% 0.31/0.48 => ( subset2powerset
% 0.31/0.48 => ( setextsub
% 0.31/0.48 => ( subsetemptysetimpeq
% 0.31/0.48 => ( powersetI1
% 0.31/0.48 => ( powersetE1
% 0.31/0.48 => ( inPowerset
% 0.31/0.48 => ( powersetsubset
% 0.31/0.48 => ( sepInPowerset
% 0.31/0.48 => ( sepSubset
% 0.31/0.48 => ( binunionIL
% 0.31/0.48 => ( upairset2IR
% 0.31/0.48 => ( binunionIR
% 0.31/0.48 => ( binunionEcases
% 0.31/0.48 => ( binunionE
% 0.31/0.48 => ( binunionLsub
% 0.31/0.48 => ( binunionRsub
% 0.31/0.48 => ( binintersectI
% 0.31/0.48 => ( binintersectSubset5
% 0.31/0.48 => ( binintersectEL
% 0.31/0.48 => ( binintersectLsub
% 0.31/0.48 => ( binintersectSubset2
% 0.31/0.48 => ( binintersectSubset3
% 0.31/0.48 => ( binintersectER
% 0.31/0.48 => ( disjointsetsI1
% 0.31/0.48 => ( binintersectRsub
% 0.31/0.48 => ( binintersectSubset4
% 0.31/0.48 => ( binintersectSubset1
% 0.31/0.48 => ( bs114d
% 0.31/0.48 => ( setminusI
% 0.31/0.48 => ( setminusEL
% 0.31/0.48 => ( setminusER
% 0.31/0.48 => ( setminusSubset2
% 0.31/0.48 => ( setminusERneg
% 0.31/0.48 => ( setminusELneg
% 0.31/0.48 => ( setminusILneg
% 0.31/0.48 => ( setminusIRneg
% 0.31/0.48 => ( setminusLsub
% 0.31/0.48 => ( setminusSubset1
% 0.31/0.48 => ( symdiffE
% 0.31/0.48 => ( symdiffI1
% 0.31/0.48 => ( symdiffI2
% 0.31/0.48 => ( symdiffIneg1
% 0.31/0.48 => ( symdiffIneg2
% 0.31/0.48 => ( secondinupair
% 0.31/0.48 => ( setukpairIL
% 0.31/0.48 => ( setukpairIR
% 0.31/0.48 => ( kpairiskpair
% 0.31/0.48 => ( kpairp
% 0.31/0.48 => ( singletonsubset
% 0.31/0.48 => ( singletoninpowerset
% 0.31/0.48 => ( singletoninpowunion
% 0.31/0.48 => ( upairset2E
% 0.31/0.48 => ( upairsubunion
% 0.31/0.48 => ( upairinpowunion
% 0.31/0.48 => ( ubforcartprodlem1
% 0.31/0.48 => ( ubforcartprodlem2
% 0.31/0.48 => ( ubforcartprodlem3
% 0.31/0.48 => ( cartprodpairin
% 0.31/0.48 => ( cartprodmempair1
% 0.31/0.48 => ( cartprodmempair
% 0.31/0.48 => ( setunionE2
% 0.31/0.48 => ( setunionsingleton1
% 0.31/0.48 => ( setunionsingleton2
% 0.31/0.48 => ( setunionsingleton
% 0.31/0.48 => ( singletonprop
% 0.31/0.48 => ( ex1E1
% 0.31/0.48 => ( ex1I
% 0.31/0.48 => ( ex1I2
% 0.31/0.48 => ( singletonsuniq
% 0.31/0.48 => ( setukpairinjL1
% 0.31/0.48 => ( kfstsingleton
% 0.31/0.48 => ( theprop
% 0.31/0.48 => ( kfstpairEq
% 0.31/0.48 => ( cartprodfstin
% 0.31/0.48 => ( setukpairinjL2
% 0.31/0.48 => ( setukpairinjL
% 0.31/0.48 => ( setukpairinjR11
% 0.31/0.48 => ( setukpairinjR12
% 0.31/0.48 => ( setukpairinjR1
% 0.31/0.48 => ( upairequniteq
% 0.31/0.48 => ( setukpairinjR2
% 0.31/0.48 => ( setukpairinjR
% 0.31/0.48 => ( ksndsingleton
% 0.31/0.48 => ( ksndpairEq
% 0.31/0.48 => ( kpairsurjEq
% 0.31/0.48 => ( cartprodsndin
% 0.31/0.48 => ( cartprodpairmemEL
% 0.31/0.48 => ( cartprodpairmemER
% 0.31/0.48 => ( cartprodmempaircEq
% 0.31/0.48 => ( cartprodfstpairEq
% 0.31/0.48 => ( cartprodsndpairEq
% 0.31/0.48 => ( cartprodpairsurjEq
% 0.31/0.48 => ( dpsetconstrI
% 0.31/0.48 => ( dpsetconstrSub
% 0.31/0.48 => ( setOfPairsIsBReln
% 0.31/0.48 => ( dpsetconstrERa
% 0.31/0.48 => ( dpsetconstrEL1
% 0.31/0.48 => ( dpsetconstrEL2
% 0.31/0.48 => ( dpsetconstrER
% 0.31/0.48 => ( funcImageSingleton
% 0.31/0.48 => ( apProp
% 0.31/0.48 => ( app
% 0.31/0.48 => ( infuncsetfunc
% 0.31/0.48 => ( ap2p
% 0.31/0.48 => ( funcinfuncset
% 0.31/0.48 => ( lamProp
% 0.31/0.48 => ( lamp
% 0.31/0.48 => ( lam2p
% 0.31/0.48 => ( brelnall1
% 0.31/0.48 => ( brelnall2
% 0.31/0.48 => ( ex1E2
% 0.31/0.48 => ( funcGraphProp1
% 0.31/0.48 => ( funcGraphProp3
% 0.31/0.48 => ( funcGraphProp2
% 0.31/0.48 => ( funcextLem
% 0.31/0.48 => ( funcGraphProp4
% 0.31/0.48 => ( subbreln
% 0.31/0.48 => ( eqbreln
% 0.31/0.48 => ( funcext
% 0.31/0.48 => ( funcext2
% 0.31/0.48 => ( ap2apEq1
% 0.31/0.48 => ( ap2apEq2
% 0.31/0.48 => ( beta1
% 0.31/0.48 => ( eta1
% 0.31/0.48 => ( lam2lamEq
% 0.31/0.48 => ( beta2
% 0.31/0.48 => ( eta2
% 0.31/0.48 => ( iffalseProp1
% 0.31/0.48 => ( iffalseProp2
% 0.31/0.48 => ( iftrueProp1
% 0.31/0.48 => ( iftrueProp2
% 0.31/0.48 => ( ifSingleton
% 0.31/0.48 => ( ifp
% 0.31/0.48 => ( theeq
% 0.31/0.48 => ( iftrue
% 0.31/0.48 => ( iffalse
% 0.31/0.48 => ( iftrueorfalse
% 0.31/0.48 => ( binintersectT_lem
% 0.31/0.48 => ( binunionT_lem
% 0.31/0.48 => ( powersetT_lem
% 0.31/0.48 => ( setminusT_lem
% 0.31/0.48 => ( complementT_lem
% 0.31/0.48 => ( setextT
% 0.31/0.48 => ( subsetTI
% 0.31/0.48 => ( powersetTI1
% 0.31/0.48 => ( powersetTE1
% 0.31/0.48 => ( complementTI1
% 0.31/0.48 => ( complementTE1
% 0.31/0.48 => ( binintersectTELcontra
% 0.31/0.48 => ( binintersectTERcontra
% 0.31/0.48 => ( contrasubsetT
% 0.31/0.48 => ( contrasubsetT1
% 0.31/0.48 => ( contrasubsetT2
% 0.31/0.48 => ( contrasubsetT3
% 0.31/0.48 => ( doubleComplementI1
% 0.31/0.48 => ( doubleComplementE1
% 0.31/0.48 => ( doubleComplementSub1
% 0.31/0.48 => ( doubleComplementSub2
% 0.31/0.48 => ( doubleComplementEq
% 0.31/0.48 => ( complementTnotintersectT
% 0.31/0.48 => ( complementImpComplementIntersect
% 0.31/0.48 => ( complementSubsetComplementIntersect
% 0.31/0.48 => ( complementInPowersetComplementIntersect
% 0.31/0.48 => ( contraSubsetComplement
% 0.31/0.48 => ( complementTcontraSubset
% 0.31/0.48 => ( binunionTILcontra
% 0.31/0.48 => ( binunionTIRcontra
% 0.31/0.48 => ( inIntersectImpInUnion
% 0.31/0.48 => ( inIntersectImpInUnion2
% 0.31/0.48 => ( inIntersectImpInIntersectUnions
% 0.31/0.48 => ( intersectInPowersetIntersectUnions
% 0.31/0.48 => ( inComplementUnionImpNotIn1
% 0.31/0.48 => ( inComplementUnionImpInComplement1
% 0.31/0.48 => ( binunionTE
% 0.31/0.48 => ( binunionTEcontra
% 0.31/0.48 => ( demorgan2a1
% 0.31/0.48 => ( complementUnionInPowersetComplement
% 0.31/0.48 => ( demorgan2a2
% 0.31/0.48 => ( demorgan1a
% 0.31/0.48 => ( demorgan1b
% 0.31/0.48 => ( demorgan1
% 0.31/0.48 => ( demorgan2a
% 0.31/0.48 => ( demorgan2b2
% 0.31/0.48 => ( demorgan2b
% 0.31/0.48 => ( demorgan2
% 0.31/0.48 => ( woz13rule0
% 0.31/0.48 => ( woz13rule1
% 0.31/0.48 => ( woz13rule2
% 0.31/0.48 => ! [A: $i,X: $i] :
% 0.31/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.31/0.48 => ! [Y: $i] :
% 0.31/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/0.48 => ! [Z: $i] :
% 0.31/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/0.48 => ( ( subset @ X @ Y )
% 0.31/0.48 => ( ( subset @ X @ Z )
% 0.31/0.48 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 %------------------------------------------------------------------------------
% 0.31/0.48 ------------------------
% 2.73/1.58 ---- Embedded in THF ---
% 2.73/1.58 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.73/1.58 %%% Generated on Tue Feb 24 12:32:49 EST 2026
% 2.73/1.58 %%% using '$$dhol' embedding, version 1.3.0.
% 2.73/1.58 %%% Logic specification used:
% 2.73/1.58 %%% thf(spec, logic, $$dhol).
% 2.73/1.58
% 2.73/1.58 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.gJXxL4HX93/DTF2DTF18132.p
% See solution above
% 2.73/1.58 ------------------------
% 2.92/1.62 ---- Cleaned THF ---
% 2.92/1.62 thf(in_type,type,
% 2.92/1.62 in: $i > $i > $o ).
% 2.92/1.62
% 2.92/1.62 thf(exu_type,type,
% 2.92/1.62 exu: ( $i > $o ) > $o ).
% 2.92/1.62
% 2.92/1.62 thf(exu,definition,
% 2.92/1.62 ( exu
% 2.92/1.62 = ( ^ [Xphi: $i > $o] :
% 2.92/1.62 ? [Xx: $i] :
% 2.92/1.62 ( ( Xphi @ Xx )
% 2.92/1.62 & ! [Xy: $i] :
% 2.92/1.62 ( ( Xphi @ Xy )
% 2.92/1.62 => ( Xx = Xy ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(setextAx_type,type,
% 2.92/1.62 setextAx: $o ).
% 2.92/1.62
% 2.92/1.62 thf(setextAx,definition,
% 2.92/1.62 ( setextAx
% 2.92/1.62 = ( ! [A: $i,B: $i] :
% 2.92/1.62 ( ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 <=> ( in @ Xx @ B ) )
% 2.92/1.62 => ( A = B ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(emptyset_type,type,
% 2.92/1.62 emptyset: $i ).
% 2.92/1.62
% 2.92/1.62 thf(emptysetAx_type,type,
% 2.92/1.62 emptysetAx: $o ).
% 2.92/1.62
% 2.92/1.62 thf(emptysetAx,definition,
% 2.92/1.62 ( emptysetAx
% 2.92/1.62 = ( ! [Xx: $i] :
% 2.92/1.62 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(setadjoin_type,type,
% 2.92/1.62 setadjoin: $i > $i > $i ).
% 2.92/1.62
% 2.92/1.62 thf(setadjoinAx_type,type,
% 2.92/1.62 setadjoinAx: $o ).
% 2.92/1.62
% 2.92/1.62 thf(setadjoinAx,definition,
% 2.92/1.62 ( setadjoinAx
% 2.92/1.62 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.92/1.62 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.92/1.62 <=> ( ( Xy = Xx )
% 2.92/1.62 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(powerset_type,type,
% 2.92/1.62 powerset: $i > $i ).
% 2.92/1.62
% 2.92/1.62 thf(powersetAx_type,type,
% 2.92/1.62 powersetAx: $o ).
% 2.92/1.62
% 2.92/1.62 thf(powersetAx,definition,
% 2.92/1.62 ( powersetAx
% 2.92/1.62 = ( ! [A: $i,B: $i] :
% 2.92/1.62 ( ( in @ B @ ( powerset @ A ) )
% 2.92/1.62 <=> ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ B )
% 2.92/1.62 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(setunion_type,type,
% 2.92/1.62 setunion: $i > $i ).
% 2.92/1.62
% 2.92/1.62 thf(setunionAx_type,type,
% 2.92/1.62 setunionAx: $o ).
% 2.92/1.62
% 2.92/1.62 thf(setunionAx,definition,
% 2.92/1.62 ( setunionAx
% 2.92/1.62 = ( ! [A: $i,Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ ( setunion @ A ) )
% 2.92/1.62 <=> ? [B: $i] :
% 2.92/1.62 ( ( in @ Xx @ B )
% 2.92/1.62 & ( in @ B @ A ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(omega_type,type,
% 2.92/1.62 omega: $i ).
% 2.92/1.62
% 2.92/1.62 thf(omega0Ax_type,type,
% 2.92/1.62 omega0Ax: $o ).
% 2.92/1.62
% 2.92/1.62 thf(omega0Ax,definition,
% 2.92/1.62 ( omega0Ax
% 2.92/1.62 = ( in @ emptyset @ omega ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(omegaSAx_type,type,
% 2.92/1.62 omegaSAx: $o ).
% 2.92/1.62
% 2.92/1.62 thf(omegaSAx,definition,
% 2.92/1.62 ( omegaSAx
% 2.92/1.62 = ( ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ omega )
% 2.92/1.62 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(omegaIndAx_type,type,
% 2.92/1.62 omegaIndAx: $o ).
% 2.92/1.62
% 2.92/1.62 thf(omegaIndAx,definition,
% 2.92/1.62 ( omegaIndAx
% 2.92/1.62 = ( ! [A: $i] :
% 2.92/1.62 ( ( ( in @ emptyset @ A )
% 2.92/1.62 & ! [Xx: $i] :
% 2.92/1.62 ( ( ( in @ Xx @ omega )
% 2.92/1.62 & ( in @ Xx @ A ) )
% 2.92/1.62 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.92/1.62 => ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ omega )
% 2.92/1.62 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(replAx_type,type,
% 2.92/1.62 replAx: $o ).
% 2.92/1.62
% 2.92/1.62 thf(replAx,definition,
% 2.92/1.62 ( replAx
% 2.92/1.62 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.92/1.62 ( ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 => ( exu
% 2.92/1.62 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.92/1.62 => ? [B: $i] :
% 2.92/1.62 ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ B )
% 2.92/1.62 <=> ? [Xy: $i] :
% 2.92/1.62 ( ( in @ Xy @ A )
% 2.92/1.62 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(foundationAx_type,type,
% 2.92/1.62 foundationAx: $o ).
% 2.92/1.62
% 2.92/1.62 thf(foundationAx,definition,
% 2.92/1.62 ( foundationAx
% 2.92/1.62 = ( ! [A: $i] :
% 2.92/1.62 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.92/1.62 => ? [B: $i] :
% 2.92/1.62 ( ( in @ B @ A )
% 2.92/1.62 & ~ ? [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ B )
% 2.92/1.62 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(wellorderingAx_type,type,
% 2.92/1.62 wellorderingAx: $o ).
% 2.92/1.62
% 2.92/1.62 thf(wellorderingAx,definition,
% 2.92/1.62 ( wellorderingAx
% 2.92/1.62 = ( ! [A: $i] :
% 2.92/1.62 ? [B: $i] :
% 2.92/1.62 ( ! [C: $i] :
% 2.92/1.62 ( ( in @ C @ B )
% 2.92/1.62 => ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ C )
% 2.92/1.62 => ( in @ Xx @ A ) ) )
% 2.92/1.62 & ! [Xx: $i,Xy: $i] :
% 2.92/1.62 ( ( ( in @ Xx @ A )
% 2.92/1.62 & ( in @ Xy @ A ) )
% 2.92/1.62 => ( ! [C: $i] :
% 2.92/1.62 ( ( in @ C @ B )
% 2.92/1.62 => ( ( in @ Xx @ C )
% 2.92/1.62 <=> ( in @ Xy @ C ) ) )
% 2.92/1.62 => ( Xx = Xy ) ) )
% 2.92/1.62 & ! [C: $i,D: $i] :
% 2.92/1.62 ( ( ( in @ C @ B )
% 2.92/1.62 & ( in @ D @ B ) )
% 2.92/1.62 => ( ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ C )
% 2.92/1.62 => ( in @ Xx @ D ) )
% 2.92/1.62 | ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ D )
% 2.92/1.62 => ( in @ Xx @ C ) ) ) )
% 2.92/1.62 & ! [C: $i] :
% 2.92/1.62 ( ( ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ C )
% 2.92/1.62 => ( in @ Xx @ A ) )
% 2.92/1.62 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.92/1.62 => ? [D: $i,Xx: $i] :
% 2.92/1.62 ( ( in @ D @ B )
% 2.92/1.62 & ( in @ Xx @ C )
% 2.92/1.62 & ~ ? [Xy: $i] :
% 2.92/1.62 ( ( in @ Xy @ D )
% 2.92/1.62 & ( in @ Xy @ C ) )
% 2.92/1.62 & ! [E: $i] :
% 2.92/1.62 ( ( in @ E @ B )
% 2.92/1.62 => ( ! [Xy: $i] :
% 2.92/1.62 ( ( in @ Xy @ E )
% 2.92/1.62 => ( in @ Xy @ D ) )
% 2.92/1.62 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(descr_type,type,
% 2.92/1.62 descr: ( $i > $o ) > $i ).
% 2.92/1.62
% 2.92/1.62 thf(descrp_type,type,
% 2.92/1.62 descrp: $o ).
% 2.92/1.62
% 2.92/1.62 thf(descrp,definition,
% 2.92/1.62 ( descrp
% 2.92/1.62 = ( ! [Xphi: $i > $o] :
% 2.92/1.62 ( ( exu
% 2.92/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.92/1.62 => ( Xphi
% 2.92/1.62 @ ( descr
% 2.92/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(dsetconstr_type,type,
% 2.92/1.62 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.92/1.62
% 2.92/1.62 thf(dsetconstrI_type,type,
% 2.92/1.62 dsetconstrI: $o ).
% 2.92/1.62
% 2.92/1.62 thf(dsetconstrI,definition,
% 2.92/1.62 ( dsetconstrI
% 2.92/1.62 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 => ( ( Xphi @ Xx )
% 2.92/1.62 => ( in @ Xx
% 2.92/1.62 @ ( dsetconstr @ A
% 2.92/1.62 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(dsetconstrEL_type,type,
% 2.92/1.62 dsetconstrEL: $o ).
% 2.92/1.62
% 2.92/1.62 thf(dsetconstrEL,definition,
% 2.92/1.62 ( dsetconstrEL
% 2.92/1.62 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.92/1.62 ( ( in @ Xx
% 2.92/1.62 @ ( dsetconstr @ A
% 2.92/1.62 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.92/1.62 => ( in @ Xx @ A ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(dsetconstrER_type,type,
% 2.92/1.62 dsetconstrER: $o ).
% 2.92/1.62
% 2.92/1.62 thf(dsetconstrER,definition,
% 2.92/1.62 ( dsetconstrER
% 2.92/1.62 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.92/1.62 ( ( in @ Xx
% 2.92/1.62 @ ( dsetconstr @ A
% 2.92/1.62 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.92/1.62 => ( Xphi @ Xx ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(exuE1_type,type,
% 2.92/1.62 exuE1: $o ).
% 2.92/1.62
% 2.92/1.62 thf(exuE1,definition,
% 2.92/1.62 ( exuE1
% 2.92/1.62 = ( ! [Xphi: $i > $o] :
% 2.92/1.62 ( ( exu
% 2.92/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.92/1.62 => ? [Xx: $i] :
% 2.92/1.62 ( ( Xphi @ Xx )
% 2.92/1.62 & ! [Xy: $i] :
% 2.92/1.62 ( ( Xphi @ Xy )
% 2.92/1.62 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(prop2set_type,type,
% 2.92/1.62 prop2set: $o > $i ).
% 2.92/1.62
% 2.92/1.62 thf(prop2setE_type,type,
% 2.92/1.62 prop2setE: $o ).
% 2.92/1.62
% 2.92/1.62 thf(prop2setE,definition,
% 2.92/1.62 ( prop2setE
% 2.92/1.62 = ( ! [Xphi: $o,Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.92/1.62 => Xphi ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(emptysetE_type,type,
% 2.92/1.62 emptysetE: $o ).
% 2.92/1.62
% 2.92/1.62 thf(emptysetE,definition,
% 2.92/1.62 ( emptysetE
% 2.92/1.62 = ( ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ emptyset )
% 2.92/1.62 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(emptysetimpfalse_type,type,
% 2.92/1.62 emptysetimpfalse: $o ).
% 2.92/1.62
% 2.92/1.62 thf(emptysetimpfalse,definition,
% 2.92/1.62 ( emptysetimpfalse
% 2.92/1.62 = ( ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ emptyset )
% 2.92/1.62 => $false ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(notinemptyset_type,type,
% 2.92/1.62 notinemptyset: $o ).
% 2.92/1.62
% 2.92/1.62 thf(notinemptyset,definition,
% 2.92/1.62 ( notinemptyset
% 2.92/1.62 = ( ! [Xx: $i] :
% 2.92/1.62 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(exuE3e_type,type,
% 2.92/1.62 exuE3e: $o ).
% 2.92/1.62
% 2.92/1.62 thf(exuE3e,definition,
% 2.92/1.62 ( exuE3e
% 2.92/1.62 = ( ! [Xphi: $i > $o] :
% 2.92/1.62 ( ( exu
% 2.92/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.92/1.62 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(setext_type,type,
% 2.92/1.62 setext: $o ).
% 2.92/1.62
% 2.92/1.62 thf(setext,definition,
% 2.92/1.62 ( setext
% 2.92/1.62 = ( ! [A: $i,B: $i] :
% 2.92/1.62 ( ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 => ( in @ Xx @ B ) )
% 2.92/1.62 => ( ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ B )
% 2.92/1.62 => ( in @ Xx @ A ) )
% 2.92/1.62 => ( A = B ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(emptyI_type,type,
% 2.92/1.62 emptyI: $o ).
% 2.92/1.62
% 2.92/1.62 thf(emptyI,definition,
% 2.92/1.62 ( emptyI
% 2.92/1.62 = ( ! [A: $i] :
% 2.92/1.62 ( ! [Xx: $i] :
% 2.92/1.62 ~ ( in @ Xx @ A )
% 2.92/1.62 => ( A = emptyset ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(noeltsimpempty_type,type,
% 2.92/1.62 noeltsimpempty: $o ).
% 2.92/1.62
% 2.92/1.62 thf(noeltsimpempty,definition,
% 2.92/1.62 ( noeltsimpempty
% 2.92/1.62 = ( ! [A: $i] :
% 2.92/1.62 ( ! [Xx: $i] :
% 2.92/1.62 ~ ( in @ Xx @ A )
% 2.92/1.62 => ( A = emptyset ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(setbeta_type,type,
% 2.92/1.62 setbeta: $o ).
% 2.92/1.62
% 2.92/1.62 thf(setbeta,definition,
% 2.92/1.62 ( setbeta
% 2.92/1.62 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 => ( ( in @ Xx
% 2.92/1.62 @ ( dsetconstr @ A
% 2.92/1.62 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.92/1.62 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(nonempty_type,type,
% 2.92/1.62 nonempty: $i > $o ).
% 2.92/1.62
% 2.92/1.62 thf(nonempty,definition,
% 2.92/1.62 ( nonempty
% 2.92/1.62 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(nonemptyE1_type,type,
% 2.92/1.62 nonemptyE1: $o ).
% 2.92/1.62
% 2.92/1.62 thf(nonemptyE1,definition,
% 2.92/1.62 ( nonemptyE1
% 2.92/1.62 = ( ! [A: $i] :
% 2.92/1.62 ( ( nonempty @ A )
% 2.92/1.62 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(nonemptyI_type,type,
% 2.92/1.62 nonemptyI: $o ).
% 2.92/1.62
% 2.92/1.62 thf(nonemptyI,definition,
% 2.92/1.62 ( nonemptyI
% 2.92/1.62 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 => ( ( Xphi @ Xx )
% 2.92/1.62 => ( nonempty
% 2.92/1.62 @ ( dsetconstr @ A
% 2.92/1.62 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(nonemptyI1_type,type,
% 2.92/1.62 nonemptyI1: $o ).
% 2.92/1.62
% 2.92/1.62 thf(nonemptyI1,definition,
% 2.92/1.62 ( nonemptyI1
% 2.92/1.62 = ( ! [A: $i] :
% 2.92/1.62 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.92/1.62 => ( nonempty @ A ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(setadjoinIL_type,type,
% 2.92/1.62 setadjoinIL: $o ).
% 2.92/1.62
% 2.92/1.62 thf(setadjoinIL,definition,
% 2.92/1.62 ( setadjoinIL
% 2.92/1.62 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(emptyinunitempty_type,type,
% 2.92/1.62 emptyinunitempty: $o ).
% 2.92/1.62
% 2.92/1.62 thf(emptyinunitempty,definition,
% 2.92/1.62 ( emptyinunitempty
% 2.92/1.62 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(setadjoinIR_type,type,
% 2.92/1.62 setadjoinIR: $o ).
% 2.92/1.62
% 2.92/1.62 thf(setadjoinIR,definition,
% 2.92/1.62 ( setadjoinIR
% 2.92/1.62 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.92/1.62 ( ( in @ Xy @ A )
% 2.92/1.62 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(setadjoinE_type,type,
% 2.92/1.62 setadjoinE: $o ).
% 2.92/1.62
% 2.92/1.62 thf(setadjoinE,definition,
% 2.92/1.62 ( setadjoinE
% 2.92/1.62 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.92/1.62 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.92/1.62 => ! [Xphi: $o] :
% 2.92/1.62 ( ( ( Xy = Xx )
% 2.92/1.62 => Xphi )
% 2.92/1.62 => ( ( ( in @ Xy @ A )
% 2.92/1.62 => Xphi )
% 2.92/1.62 => Xphi ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(setadjoinOr_type,type,
% 2.92/1.62 setadjoinOr: $o ).
% 2.92/1.62
% 2.92/1.62 thf(setadjoinOr,definition,
% 2.92/1.62 ( setadjoinOr
% 2.92/1.62 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.92/1.62 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.92/1.62 => ( ( Xy = Xx )
% 2.92/1.62 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(setoftrueEq_type,type,
% 2.92/1.62 setoftrueEq: $o ).
% 2.92/1.62
% 2.92/1.62 thf(setoftrueEq,definition,
% 2.92/1.62 ( setoftrueEq
% 2.92/1.62 = ( ! [A: $i] :
% 2.92/1.62 ( ( dsetconstr @ A
% 2.92/1.62 @ ^ [Xx: $i] : $true )
% 2.92/1.62 = A ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(powersetI_type,type,
% 2.92/1.62 powersetI: $o ).
% 2.92/1.62
% 2.92/1.62 thf(powersetI,definition,
% 2.92/1.62 ( powersetI
% 2.92/1.62 = ( ! [A: $i,B: $i] :
% 2.92/1.62 ( ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ B )
% 2.92/1.62 => ( in @ Xx @ A ) )
% 2.92/1.62 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(emptyinPowerset_type,type,
% 2.92/1.62 emptyinPowerset: $o ).
% 2.92/1.62
% 2.92/1.62 thf(emptyinPowerset,definition,
% 2.92/1.62 ( emptyinPowerset
% 2.92/1.62 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(emptyInPowerset_type,type,
% 2.92/1.62 emptyInPowerset: $o ).
% 2.92/1.62
% 2.92/1.62 thf(emptyInPowerset,definition,
% 2.92/1.62 ( emptyInPowerset
% 2.92/1.62 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(powersetE_type,type,
% 2.92/1.62 powersetE: $o ).
% 2.92/1.62
% 2.92/1.62 thf(powersetE,definition,
% 2.92/1.62 ( powersetE
% 2.92/1.62 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.62 ( ( in @ B @ ( powerset @ A ) )
% 2.92/1.62 => ( ( in @ Xx @ B )
% 2.92/1.62 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(setunionI_type,type,
% 2.92/1.62 setunionI: $o ).
% 2.92/1.62
% 2.92/1.62 thf(setunionI,definition,
% 2.92/1.62 ( setunionI
% 2.92/1.62 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.92/1.62 ( ( in @ Xx @ B )
% 2.92/1.62 => ( ( in @ B @ A )
% 2.92/1.62 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(setunionE_type,type,
% 2.92/1.62 setunionE: $o ).
% 2.92/1.62
% 2.92/1.62 thf(setunionE,definition,
% 2.92/1.62 ( setunionE
% 2.92/1.62 = ( ! [A: $i,Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ ( setunion @ A ) )
% 2.92/1.62 => ! [Xphi: $o] :
% 2.92/1.62 ( ! [B: $i] :
% 2.92/1.62 ( ( in @ Xx @ B )
% 2.92/1.62 => ( ( in @ B @ A )
% 2.92/1.62 => Xphi ) )
% 2.92/1.62 => Xphi ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(subPowSU_type,type,
% 2.92/1.62 subPowSU: $o ).
% 2.92/1.62
% 2.92/1.62 thf(subPowSU,definition,
% 2.92/1.62 ( subPowSU
% 2.92/1.62 = ( ! [A: $i,Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(exuE2_type,type,
% 2.92/1.62 exuE2: $o ).
% 2.92/1.62
% 2.92/1.62 thf(exuE2,definition,
% 2.92/1.62 ( exuE2
% 2.92/1.62 = ( ! [Xphi: $i > $o] :
% 2.92/1.62 ( ( exu
% 2.92/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.92/1.62 => ? [Xx: $i] :
% 2.92/1.62 ! [Xy: $i] :
% 2.92/1.62 ( ( Xphi @ Xy )
% 2.92/1.62 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(nonemptyImpWitness_type,type,
% 2.92/1.62 nonemptyImpWitness: $o ).
% 2.92/1.62
% 2.92/1.62 thf(nonemptyImpWitness,definition,
% 2.92/1.62 ( nonemptyImpWitness
% 2.92/1.62 = ( ! [A: $i] :
% 2.92/1.62 ( ( nonempty @ A )
% 2.92/1.62 => ? [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 & $true ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(uniqinunit_type,type,
% 2.92/1.62 uniqinunit: $o ).
% 2.92/1.62
% 2.92/1.62 thf(uniqinunit,definition,
% 2.92/1.62 ( uniqinunit
% 2.92/1.62 = ( ! [Xx: $i,Xy: $i] :
% 2.92/1.62 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.92/1.62 => ( Xx = Xy ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(notinsingleton_type,type,
% 2.92/1.62 notinsingleton: $o ).
% 2.92/1.62
% 2.92/1.62 thf(notinsingleton,definition,
% 2.92/1.62 ( notinsingleton
% 2.92/1.62 = ( ! [Xx: $i,Xy: $i] :
% 2.92/1.62 ( ( Xx != Xy )
% 2.92/1.62 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(eqinunit_type,type,
% 2.92/1.62 eqinunit: $o ).
% 2.92/1.62
% 2.92/1.62 thf(eqinunit,definition,
% 2.92/1.62 ( eqinunit
% 2.92/1.62 = ( ! [Xx: $i,Xy: $i] :
% 2.92/1.62 ( ( Xx = Xy )
% 2.92/1.62 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(singletonsswitch_type,type,
% 2.92/1.62 singletonsswitch: $o ).
% 2.92/1.62
% 2.92/1.62 thf(singletonsswitch,definition,
% 2.92/1.62 ( singletonsswitch
% 2.92/1.62 = ( ! [Xx: $i,Xy: $i] :
% 2.92/1.62 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.92/1.62 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(upairsetE_type,type,
% 2.92/1.62 upairsetE: $o ).
% 2.92/1.62
% 2.92/1.62 thf(upairsetE,definition,
% 2.92/1.62 ( upairsetE
% 2.92/1.62 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.92/1.62 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.92/1.62 => ( ( Xz = Xx )
% 2.92/1.62 | ( Xz = Xy ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(upairsetIL_type,type,
% 2.92/1.62 upairsetIL: $o ).
% 2.92/1.62
% 2.92/1.62 thf(upairsetIL,definition,
% 2.92/1.62 ( upairsetIL
% 2.92/1.62 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(upairsetIR_type,type,
% 2.92/1.62 upairsetIR: $o ).
% 2.92/1.62
% 2.92/1.62 thf(upairsetIR,definition,
% 2.92/1.62 ( upairsetIR
% 2.92/1.62 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(emptyE1_type,type,
% 2.92/1.62 emptyE1: $o ).
% 2.92/1.62
% 2.92/1.62 thf(emptyE1,definition,
% 2.92/1.62 ( emptyE1
% 2.92/1.62 = ( ! [A: $i,Xphi: $i > $o] :
% 2.92/1.62 ( ? [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 & ( Xphi @ Xx ) )
% 2.92/1.62 => ( ( ( dsetconstr @ A
% 2.92/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.92/1.62 = emptyset )
% 2.92/1.62 => $false ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(vacuousDall_type,type,
% 2.92/1.62 vacuousDall: $o ).
% 2.92/1.62
% 2.92/1.62 thf(vacuousDall,definition,
% 2.92/1.62 ( vacuousDall
% 2.92/1.62 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ emptyset )
% 2.92/1.62 => ( Xphi @ Xx ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(quantDeMorgan1_type,type,
% 2.92/1.62 quantDeMorgan1: $o ).
% 2.92/1.62
% 2.92/1.62 thf(quantDeMorgan1,definition,
% 2.92/1.62 ( quantDeMorgan1
% 2.92/1.62 = ( ! [A: $i,Xphi: $i > $o] :
% 2.92/1.62 ( ~ ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 => ( Xphi @ Xx ) )
% 2.92/1.62 => ? [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(quantDeMorgan2_type,type,
% 2.92/1.62 quantDeMorgan2: $o ).
% 2.92/1.62
% 2.92/1.62 thf(quantDeMorgan2,definition,
% 2.92/1.62 ( quantDeMorgan2
% 2.92/1.62 = ( ! [A: $i,Xphi: $i > $o] :
% 2.92/1.62 ( ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 => ~ ( Xphi @ Xx ) )
% 2.92/1.62 => ~ ? [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(quantDeMorgan3_type,type,
% 2.92/1.62 quantDeMorgan3: $o ).
% 2.92/1.62
% 2.92/1.62 thf(quantDeMorgan3,definition,
% 2.92/1.62 ( quantDeMorgan3
% 2.92/1.62 = ( ! [A: $i,Xphi: $i > $o] :
% 2.92/1.62 ( ~ ? [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 & ( Xphi @ Xx ) )
% 2.92/1.62 => ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(quantDeMorgan4_type,type,
% 2.92/1.62 quantDeMorgan4: $o ).
% 2.92/1.62
% 2.92/1.62 thf(quantDeMorgan4,definition,
% 2.92/1.62 ( quantDeMorgan4
% 2.92/1.62 = ( ! [A: $i,Xphi: $i > $o] :
% 2.92/1.62 ( ? [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 & ~ ( Xphi @ Xx ) )
% 2.92/1.62 => ~ ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(prop2setI_type,type,
% 2.92/1.62 prop2setI: $o ).
% 2.92/1.62
% 2.92/1.62 thf(prop2setI,definition,
% 2.92/1.62 ( prop2setI
% 2.92/1.62 = ( ! [Xphi: $o] :
% 2.92/1.62 ( Xphi
% 2.92/1.62 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(set2prop_type,type,
% 2.92/1.62 set2prop: $i > $o ).
% 2.92/1.62
% 2.92/1.62 thf(prop2set2propI_type,type,
% 2.92/1.62 prop2set2propI: $o ).
% 2.92/1.62
% 2.92/1.62 thf(prop2set2propI,definition,
% 2.92/1.62 ( prop2set2propI
% 2.92/1.62 = ( ! [Xphi: $o] :
% 2.92/1.62 ( Xphi
% 2.92/1.62 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(notdexE_type,type,
% 2.92/1.62 notdexE: $o ).
% 2.92/1.62
% 2.92/1.62 thf(notdexE,definition,
% 2.92/1.62 ( notdexE
% 2.92/1.62 = ( ! [A: $i,Xphi: $i > $o] :
% 2.92/1.62 ( ~ ? [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 & ( Xphi @ Xx ) )
% 2.92/1.62 => ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(notdallE_type,type,
% 2.92/1.62 notdallE: $o ).
% 2.92/1.62
% 2.92/1.62 thf(notdallE,definition,
% 2.92/1.62 ( notdallE
% 2.92/1.62 = ( ! [A: $i,Xphi: $i > $o] :
% 2.92/1.62 ( ~ ! [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 => ( Xphi @ Xx ) )
% 2.92/1.62 => ? [Xx: $i] :
% 2.92/1.62 ( ( in @ Xx @ A )
% 2.92/1.62 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(exuI1_type,type,
% 2.92/1.62 exuI1: $o ).
% 2.92/1.62
% 2.92/1.62 thf(exuI1,definition,
% 2.92/1.62 ( exuI1
% 2.92/1.62 = ( ! [Xphi: $i > $o] :
% 2.92/1.62 ( ? [Xx: $i] :
% 2.92/1.62 ( ( Xphi @ Xx )
% 2.92/1.62 & ! [Xy: $i] :
% 2.92/1.62 ( ( Xphi @ Xy )
% 2.92/1.62 => ( Xx = Xy ) ) )
% 2.92/1.62 => ( exu
% 2.92/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(exuI3_type,type,
% 2.92/1.62 exuI3: $o ).
% 2.92/1.62
% 2.92/1.62 thf(exuI3,definition,
% 2.92/1.62 ( exuI3
% 2.92/1.62 = ( ! [Xphi: $i > $o] :
% 2.92/1.62 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.92/1.62 => ( ! [Xx: $i,Xy: $i] :
% 2.92/1.62 ( ( Xphi @ Xx )
% 2.92/1.62 => ( ( Xphi @ Xy )
% 2.92/1.62 => ( Xx = Xy ) ) )
% 2.92/1.62 => ( exu
% 2.92/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(exuI2_type,type,
% 2.92/1.62 exuI2: $o ).
% 2.92/1.62
% 2.92/1.62 thf(exuI2,definition,
% 2.92/1.62 ( exuI2
% 2.92/1.62 = ( ! [Xphi: $i > $o] :
% 2.92/1.62 ( ? [Xx: $i] :
% 2.92/1.62 ! [Xy: $i] :
% 2.92/1.62 ( ( Xphi @ Xy )
% 2.92/1.62 <=> ( Xy = Xx ) )
% 2.92/1.62 => ( exu
% 2.92/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(inCongP_type,type,
% 2.92/1.62 inCongP: $o ).
% 2.92/1.62
% 2.92/1.62 thf(inCongP,definition,
% 2.92/1.62 ( inCongP
% 2.92/1.62 = ( ! [A: $i,B: $i] :
% 2.92/1.62 ( ( A = B )
% 2.92/1.62 => ! [Xx: $i,Xy: $i] :
% 2.92/1.62 ( ( Xx = Xy )
% 2.92/1.62 => ( ( in @ Xx @ A )
% 2.92/1.62 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(in__Cong_type,type,
% 2.92/1.62 in__Cong: $o ).
% 2.92/1.62
% 2.92/1.62 thf(in__Cong,definition,
% 2.92/1.62 ( in__Cong
% 2.92/1.62 = ( ! [A: $i,B: $i] :
% 2.92/1.62 ( ( A = B )
% 2.92/1.62 => ! [Xx: $i,Xy: $i] :
% 2.92/1.62 ( ( Xx = Xy )
% 2.92/1.62 => ( ( in @ Xx @ A )
% 2.92/1.62 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(exuE3u_type,type,
% 2.92/1.62 exuE3u: $o ).
% 2.92/1.62
% 2.92/1.62 thf(exuE3u,definition,
% 2.92/1.62 ( exuE3u
% 2.92/1.62 = ( ! [Xphi: $i > $o] :
% 2.92/1.62 ( ( exu
% 2.92/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.92/1.62 => ! [Xx: $i,Xy: $i] :
% 2.92/1.62 ( ( Xphi @ Xx )
% 2.92/1.62 => ( ( Xphi @ Xy )
% 2.92/1.62 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.92/1.62
% 2.92/1.62 thf(exu__Cong_type,type,
% 2.92/1.62 exu__Cong: $o ).
% 2.92/1.62
% 2.92/1.62 thf(exu__Cong,definition,
% 2.92/1.62 ( exu__Cong
% 2.92/1.62 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.92/1.62 ( ! [Xx: $i,Xy: $i] :
% 2.92/1.62 ( ( Xx = Xy )
% 2.92/1.62 => ( ( Xphi @ Xx )
% 2.92/1.62 <=> ( Xpsi @ Xy ) ) )
% 2.92/1.63 => ( ( exu
% 2.92/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.92/1.63 <=> ( exu
% 2.92/1.63 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(emptyset__Cong_type,type,
% 2.92/1.63 emptyset__Cong: $o ).
% 2.92/1.63
% 2.92/1.63 thf(emptyset__Cong,definition,
% 2.92/1.63 ( emptyset__Cong
% 2.92/1.63 = ( emptyset = emptyset ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setadjoin__Cong_type,type,
% 2.92/1.63 setadjoin__Cong: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setadjoin__Cong,definition,
% 2.92/1.63 ( setadjoin__Cong
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.92/1.63 ( ( Xx = Xy )
% 2.92/1.63 => ! [Xz: $i,Xu: $i] :
% 2.92/1.63 ( ( Xz = Xu )
% 2.92/1.63 => ( ( setadjoin @ Xx @ Xz )
% 2.92/1.63 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(powerset__Cong_type,type,
% 2.92/1.63 powerset__Cong: $o ).
% 2.92/1.63
% 2.92/1.63 thf(powerset__Cong,definition,
% 2.92/1.63 ( powerset__Cong
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ( A = B )
% 2.92/1.63 => ( ( powerset @ A )
% 2.92/1.63 = ( powerset @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setunion__Cong_type,type,
% 2.92/1.63 setunion__Cong: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setunion__Cong,definition,
% 2.92/1.63 ( setunion__Cong
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ( A = B )
% 2.92/1.63 => ( ( setunion @ A )
% 2.92/1.63 = ( setunion @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(omega__Cong_type,type,
% 2.92/1.63 omega__Cong: $o ).
% 2.92/1.63
% 2.92/1.63 thf(omega__Cong,definition,
% 2.92/1.63 ( omega__Cong
% 2.92/1.63 = ( omega = omega ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(exuEu_type,type,
% 2.92/1.63 exuEu: $o ).
% 2.92/1.63
% 2.92/1.63 thf(exuEu,definition,
% 2.92/1.63 ( exuEu
% 2.92/1.63 = ( ! [Xphi: $i > $o] :
% 2.92/1.63 ( ( exu
% 2.92/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.92/1.63 => ! [Xx: $i,Xy: $i] :
% 2.92/1.63 ( ( Xphi @ Xx )
% 2.92/1.63 => ( ( Xphi @ Xy )
% 2.92/1.63 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(descr__Cong_type,type,
% 2.92/1.63 descr__Cong: $o ).
% 2.92/1.63
% 2.92/1.63 thf(descr__Cong,definition,
% 2.92/1.63 ( descr__Cong
% 2.92/1.63 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.92/1.63 ( ! [Xx: $i,Xy: $i] :
% 2.92/1.63 ( ( Xx = Xy )
% 2.92/1.63 => ( ( Xphi @ Xx )
% 2.92/1.63 <=> ( Xpsi @ Xy ) ) )
% 2.92/1.63 => ( ( exu
% 2.92/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.92/1.63 => ( ( exu
% 2.92/1.63 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.92/1.63 => ( ( descr
% 2.92/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.92/1.63 = ( descr
% 2.92/1.63 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(dsetconstr__Cong_type,type,
% 2.92/1.63 dsetconstr__Cong: $o ).
% 2.92/1.63
% 2.92/1.63 thf(dsetconstr__Cong,definition,
% 2.92/1.63 ( dsetconstr__Cong
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ( A = B )
% 2.92/1.63 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.92/1.63 ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( ( Xx = Xy )
% 2.92/1.63 => ( ( Xphi @ Xx )
% 2.92/1.63 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.92/1.63 => ( ( dsetconstr @ A
% 2.92/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.92/1.63 = ( dsetconstr @ B
% 2.92/1.63 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(subset_type,type,
% 2.92/1.63 subset: $i > $i > $o ).
% 2.92/1.63
% 2.92/1.63 thf(disjoint_type,type,
% 2.92/1.63 disjoint: $i > $i > $o ).
% 2.92/1.63
% 2.92/1.63 thf(setsmeet_type,type,
% 2.92/1.63 setsmeet: $i > $i > $o ).
% 2.92/1.63
% 2.92/1.63 thf(subsetI1_type,type,
% 2.92/1.63 subsetI1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(subsetI1,definition,
% 2.92/1.63 ( subsetI1
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ Xx @ B ) )
% 2.92/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(eqimpsubset2_type,type,
% 2.92/1.63 eqimpsubset2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(eqimpsubset2,definition,
% 2.92/1.63 ( eqimpsubset2
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ( A = B )
% 2.92/1.63 => ( subset @ B @ A ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(eqimpsubset1_type,type,
% 2.92/1.63 eqimpsubset1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(eqimpsubset1,definition,
% 2.92/1.63 ( eqimpsubset1
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ( A = B )
% 2.92/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(subsetI2_type,type,
% 2.92/1.63 subsetI2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(subsetI2,definition,
% 2.92/1.63 ( subsetI2
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ Xx @ B ) )
% 2.92/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(emptysetsubset_type,type,
% 2.92/1.63 emptysetsubset: $o ).
% 2.92/1.63
% 2.92/1.63 thf(emptysetsubset,definition,
% 2.92/1.63 ( emptysetsubset
% 2.92/1.63 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(subsetE_type,type,
% 2.92/1.63 subsetE: $o ).
% 2.92/1.63
% 2.92/1.63 thf(subsetE,definition,
% 2.92/1.63 ( subsetE
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( subset @ A @ B )
% 2.92/1.63 => ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(subsetE2_type,type,
% 2.92/1.63 subsetE2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(subsetE2,definition,
% 2.92/1.63 ( subsetE2
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( subset @ A @ B )
% 2.92/1.63 => ( ~ ( in @ Xx @ B )
% 2.92/1.63 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(notsubsetI_type,type,
% 2.92/1.63 notsubsetI: $o ).
% 2.92/1.63
% 2.92/1.63 thf(notsubsetI,definition,
% 2.92/1.63 ( notsubsetI
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ~ ( in @ Xx @ B )
% 2.92/1.63 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(notequalI1_type,type,
% 2.92/1.63 notequalI1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(notequalI1,definition,
% 2.92/1.63 ( notequalI1
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ~ ( subset @ A @ B )
% 2.92/1.63 => ( A != B ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(notequalI2_type,type,
% 2.92/1.63 notequalI2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(notequalI2,definition,
% 2.92/1.63 ( notequalI2
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ~ ( in @ Xx @ B )
% 2.92/1.63 => ( A != B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(subsetRefl_type,type,
% 2.92/1.63 subsetRefl: $o ).
% 2.92/1.63
% 2.92/1.63 thf(subsetRefl,definition,
% 2.92/1.63 ( subsetRefl
% 2.92/1.63 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(subsetTrans_type,type,
% 2.92/1.63 subsetTrans: $o ).
% 2.92/1.63
% 2.92/1.63 thf(subsetTrans,definition,
% 2.92/1.63 ( subsetTrans
% 2.92/1.63 = ( ! [A: $i,B: $i,C: $i] :
% 2.92/1.63 ( ( subset @ A @ B )
% 2.92/1.63 => ( ( subset @ B @ C )
% 2.92/1.63 => ( subset @ A @ C ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setadjoinSub_type,type,
% 2.92/1.63 setadjoinSub: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setadjoinSub,definition,
% 2.92/1.63 ( setadjoinSub
% 2.92/1.63 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setadjoinSub2_type,type,
% 2.92/1.63 setadjoinSub2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setadjoinSub2,definition,
% 2.92/1.63 ( setadjoinSub2
% 2.92/1.63 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.92/1.63 ( ( subset @ A @ B )
% 2.92/1.63 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(subset2powerset_type,type,
% 2.92/1.63 subset2powerset: $o ).
% 2.92/1.63
% 2.92/1.63 thf(subset2powerset,definition,
% 2.92/1.63 ( subset2powerset
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ( subset @ A @ B )
% 2.92/1.63 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setextsub_type,type,
% 2.92/1.63 setextsub: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setextsub,definition,
% 2.92/1.63 ( setextsub
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ( subset @ A @ B )
% 2.92/1.63 => ( ( subset @ B @ A )
% 2.92/1.63 => ( A = B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(subsetemptysetimpeq_type,type,
% 2.92/1.63 subsetemptysetimpeq: $o ).
% 2.92/1.63
% 2.92/1.63 thf(subsetemptysetimpeq,definition,
% 2.92/1.63 ( subsetemptysetimpeq
% 2.92/1.63 = ( ! [A: $i] :
% 2.92/1.63 ( ( subset @ A @ emptyset )
% 2.92/1.63 => ( A = emptyset ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(powersetI1_type,type,
% 2.92/1.63 powersetI1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(powersetI1,definition,
% 2.92/1.63 ( powersetI1
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ( subset @ B @ A )
% 2.92/1.63 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(powersetE1_type,type,
% 2.92/1.63 powersetE1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(powersetE1,definition,
% 2.92/1.63 ( powersetE1
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ( in @ B @ ( powerset @ A ) )
% 2.92/1.63 => ( subset @ B @ A ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(inPowerset_type,type,
% 2.92/1.63 inPowerset: $o ).
% 2.92/1.63
% 2.92/1.63 thf(inPowerset,definition,
% 2.92/1.63 ( inPowerset
% 2.92/1.63 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(powersetsubset_type,type,
% 2.92/1.63 powersetsubset: $o ).
% 2.92/1.63
% 2.92/1.63 thf(powersetsubset,definition,
% 2.92/1.63 ( powersetsubset
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ( subset @ A @ B )
% 2.92/1.63 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(sepInPowerset_type,type,
% 2.92/1.63 sepInPowerset: $o ).
% 2.92/1.63
% 2.92/1.63 thf(sepInPowerset,definition,
% 2.92/1.63 ( sepInPowerset
% 2.92/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.92/1.63 ( in
% 2.92/1.63 @ ( dsetconstr @ A
% 2.92/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.92/1.63 @ ( powerset @ A ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(sepSubset_type,type,
% 2.92/1.63 sepSubset: $o ).
% 2.92/1.63
% 2.92/1.63 thf(sepSubset,definition,
% 2.92/1.63 ( sepSubset
% 2.92/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.92/1.63 ( subset
% 2.92/1.63 @ ( dsetconstr @ A
% 2.92/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.92/1.63 @ A ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binunion_type,type,
% 2.92/1.63 binunion: $i > $i > $i ).
% 2.92/1.63
% 2.92/1.63 thf(binunionIL_type,type,
% 2.92/1.63 binunionIL: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binunionIL,definition,
% 2.92/1.63 ( binunionIL
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(upairset2IR_type,type,
% 2.92/1.63 upairset2IR: $o ).
% 2.92/1.63
% 2.92/1.63 thf(upairset2IR,definition,
% 2.92/1.63 ( upairset2IR
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binunionIR_type,type,
% 2.92/1.63 binunionIR: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binunionIR,definition,
% 2.92/1.63 ( binunionIR
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ B )
% 2.92/1.63 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binunionEcases_type,type,
% 2.92/1.63 binunionEcases: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binunionEcases,definition,
% 2.92/1.63 ( binunionEcases
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.92/1.63 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.92/1.63 => ( ( ( in @ Xx @ A )
% 2.92/1.63 => Xphi )
% 2.92/1.63 => ( ( ( in @ Xx @ B )
% 2.92/1.63 => Xphi )
% 2.92/1.63 => Xphi ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binunionE_type,type,
% 2.92/1.63 binunionE: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binunionE,definition,
% 2.92/1.63 ( binunionE
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.92/1.63 => ( ( in @ Xx @ A )
% 2.92/1.63 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binunionLsub_type,type,
% 2.92/1.63 binunionLsub: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binunionLsub,definition,
% 2.92/1.63 ( binunionLsub
% 2.92/1.63 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binunionRsub_type,type,
% 2.92/1.63 binunionRsub: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binunionRsub,definition,
% 2.92/1.63 ( binunionRsub
% 2.92/1.63 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binintersect_type,type,
% 2.92/1.63 binintersect: $i > $i > $i ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectI_type,type,
% 2.92/1.63 binintersectI: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectI,definition,
% 2.92/1.63 ( binintersectI
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ B )
% 2.92/1.63 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectSubset5_type,type,
% 2.92/1.63 binintersectSubset5: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectSubset5,definition,
% 2.92/1.63 ( binintersectSubset5
% 2.92/1.63 = ( ! [A: $i,B: $i,C: $i] :
% 2.92/1.63 ( ( subset @ C @ A )
% 2.92/1.63 => ( ( subset @ C @ B )
% 2.92/1.63 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectEL_type,type,
% 2.92/1.63 binintersectEL: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectEL,definition,
% 2.92/1.63 ( binintersectEL
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.92/1.63 => ( in @ Xx @ A ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectLsub_type,type,
% 2.92/1.63 binintersectLsub: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectLsub,definition,
% 2.92/1.63 ( binintersectLsub
% 2.92/1.63 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectSubset2_type,type,
% 2.92/1.63 binintersectSubset2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectSubset2,definition,
% 2.92/1.63 ( binintersectSubset2
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ( subset @ A @ B )
% 2.92/1.63 => ( ( binintersect @ A @ B )
% 2.92/1.63 = A ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectSubset3_type,type,
% 2.92/1.63 binintersectSubset3: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectSubset3,definition,
% 2.92/1.63 ( binintersectSubset3
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ( ( binintersect @ A @ B )
% 2.92/1.63 = B )
% 2.92/1.63 => ( subset @ B @ A ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectER_type,type,
% 2.92/1.63 binintersectER: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectER,definition,
% 2.92/1.63 ( binintersectER
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.92/1.63 => ( in @ Xx @ B ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(disjointsetsI1_type,type,
% 2.92/1.63 disjointsetsI1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(disjointsetsI1,definition,
% 2.92/1.63 ( disjointsetsI1
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ~ ? [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 & ( in @ Xx @ B ) )
% 2.92/1.63 => ( ( binintersect @ A @ B )
% 2.92/1.63 = emptyset ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectRsub_type,type,
% 2.92/1.63 binintersectRsub: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectRsub,definition,
% 2.92/1.63 ( binintersectRsub
% 2.92/1.63 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectSubset4_type,type,
% 2.92/1.63 binintersectSubset4: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectSubset4,definition,
% 2.92/1.63 ( binintersectSubset4
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ( subset @ B @ A )
% 2.92/1.63 => ( ( binintersect @ A @ B )
% 2.92/1.63 = B ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectSubset1_type,type,
% 2.92/1.63 binintersectSubset1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectSubset1,definition,
% 2.92/1.63 ( binintersectSubset1
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ( ( binintersect @ A @ B )
% 2.92/1.63 = A )
% 2.92/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(bs114d_type,type,
% 2.92/1.63 bs114d: $o ).
% 2.92/1.63
% 2.92/1.63 thf(bs114d,definition,
% 2.92/1.63 ( bs114d
% 2.92/1.63 = ( ! [A: $i,B: $i,C: $i] :
% 2.92/1.63 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.92/1.63 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(regular_type,type,
% 2.92/1.63 regular: $i > $o ).
% 2.92/1.63
% 2.92/1.63 thf(setminus_type,type,
% 2.92/1.63 setminus: $i > $i > $i ).
% 2.92/1.63
% 2.92/1.63 thf(setminusI_type,type,
% 2.92/1.63 setminusI: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setminusI,definition,
% 2.92/1.63 ( setminusI
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ~ ( in @ Xx @ B )
% 2.92/1.63 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setminusEL_type,type,
% 2.92/1.63 setminusEL: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setminusEL,definition,
% 2.92/1.63 ( setminusEL
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.92/1.63 => ( in @ Xx @ A ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setminusER_type,type,
% 2.92/1.63 setminusER: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setminusER,definition,
% 2.92/1.63 ( setminusER
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.92/1.63 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setminusSubset2_type,type,
% 2.92/1.63 setminusSubset2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setminusSubset2,definition,
% 2.92/1.63 ( setminusSubset2
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ( subset @ A @ B )
% 2.92/1.63 => ( ( setminus @ A @ B )
% 2.92/1.63 = emptyset ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setminusERneg_type,type,
% 2.92/1.63 setminusERneg: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setminusERneg,definition,
% 2.92/1.63 ( setminusERneg
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.92/1.63 => ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setminusELneg_type,type,
% 2.92/1.63 setminusELneg: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setminusELneg,definition,
% 2.92/1.63 ( setminusELneg
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.92/1.63 => ( ~ ( in @ Xx @ B )
% 2.92/1.63 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setminusILneg_type,type,
% 2.92/1.63 setminusILneg: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setminusILneg,definition,
% 2.92/1.63 ( setminusILneg
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ~ ( in @ Xx @ A )
% 2.92/1.63 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setminusIRneg_type,type,
% 2.92/1.63 setminusIRneg: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setminusIRneg,definition,
% 2.92/1.63 ( setminusIRneg
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ B )
% 2.92/1.63 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setminusLsub_type,type,
% 2.92/1.63 setminusLsub: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setminusLsub,definition,
% 2.92/1.63 ( setminusLsub
% 2.92/1.63 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setminusSubset1_type,type,
% 2.92/1.63 setminusSubset1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setminusSubset1,definition,
% 2.92/1.63 ( setminusSubset1
% 2.92/1.63 = ( ! [A: $i,B: $i] :
% 2.92/1.63 ( ( ( setminus @ A @ B )
% 2.92/1.63 = emptyset )
% 2.92/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(symdiff_type,type,
% 2.92/1.63 symdiff: $i > $i > $i ).
% 2.92/1.63
% 2.92/1.63 thf(symdiffE_type,type,
% 2.92/1.63 symdiffE: $o ).
% 2.92/1.63
% 2.92/1.63 thf(symdiffE,definition,
% 2.92/1.63 ( symdiffE
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.92/1.63 => ! [Xphi: $o] :
% 2.92/1.63 ( ( ( in @ Xx @ A )
% 2.92/1.63 => ( ~ ( in @ Xx @ B )
% 2.92/1.63 => Xphi ) )
% 2.92/1.63 => ( ( ~ ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ B )
% 2.92/1.63 => Xphi ) )
% 2.92/1.63 => Xphi ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(symdiffI1_type,type,
% 2.92/1.63 symdiffI1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(symdiffI1,definition,
% 2.92/1.63 ( symdiffI1
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ~ ( in @ Xx @ B )
% 2.92/1.63 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(symdiffI2_type,type,
% 2.92/1.63 symdiffI2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(symdiffI2,definition,
% 2.92/1.63 ( symdiffI2
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ~ ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ B )
% 2.92/1.63 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(symdiffIneg1_type,type,
% 2.92/1.63 symdiffIneg1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(symdiffIneg1,definition,
% 2.92/1.63 ( symdiffIneg1
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ B )
% 2.92/1.63 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(symdiffIneg2_type,type,
% 2.92/1.63 symdiffIneg2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(symdiffIneg2,definition,
% 2.92/1.63 ( symdiffIneg2
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ~ ( in @ Xx @ A )
% 2.92/1.63 => ( ~ ( in @ Xx @ B )
% 2.92/1.63 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(iskpair_type,type,
% 2.92/1.63 iskpair: $i > $o ).
% 2.92/1.63
% 2.92/1.63 thf(secondinupair_type,type,
% 2.92/1.63 secondinupair: $o ).
% 2.92/1.63
% 2.92/1.63 thf(secondinupair,definition,
% 2.92/1.63 ( secondinupair
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairIL_type,type,
% 2.92/1.63 setukpairIL: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairIL,definition,
% 2.92/1.63 ( setukpairIL
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairIR_type,type,
% 2.92/1.63 setukpairIR: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairIR,definition,
% 2.92/1.63 ( setukpairIR
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(kpairiskpair_type,type,
% 2.92/1.63 kpairiskpair: $o ).
% 2.92/1.63
% 2.92/1.63 thf(kpairiskpair,definition,
% 2.92/1.63 ( kpairiskpair
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(kpair_type,type,
% 2.92/1.63 kpair: $i > $i > $i ).
% 2.92/1.63
% 2.92/1.63 thf(kpairp_type,type,
% 2.92/1.63 kpairp: $o ).
% 2.92/1.63
% 2.92/1.63 thf(kpairp,definition,
% 2.92/1.63 ( kpairp
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(cartprod_type,type,
% 2.92/1.63 cartprod: $i > $i > $i ).
% 2.92/1.63
% 2.92/1.63 thf(singletonsubset_type,type,
% 2.92/1.63 singletonsubset: $o ).
% 2.92/1.63
% 2.92/1.63 thf(singletonsubset,definition,
% 2.92/1.63 ( singletonsubset
% 2.92/1.63 = ( ! [A: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(singletoninpowerset_type,type,
% 2.92/1.63 singletoninpowerset: $o ).
% 2.92/1.63
% 2.92/1.63 thf(singletoninpowerset,definition,
% 2.92/1.63 ( singletoninpowerset
% 2.92/1.63 = ( ! [A: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(singletoninpowunion_type,type,
% 2.92/1.63 singletoninpowunion: $o ).
% 2.92/1.63
% 2.92/1.63 thf(singletoninpowunion,definition,
% 2.92/1.63 ( singletoninpowunion
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(upairset2E_type,type,
% 2.92/1.63 upairset2E: $o ).
% 2.92/1.63
% 2.92/1.63 thf(upairset2E,definition,
% 2.92/1.63 ( upairset2E
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.92/1.63 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.92/1.63 => ( ( Xz = Xx )
% 2.92/1.63 | ( Xz = Xy ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(upairsubunion_type,type,
% 2.92/1.63 upairsubunion: $o ).
% 2.92/1.63
% 2.92/1.63 thf(upairsubunion,definition,
% 2.92/1.63 ( upairsubunion
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(upairinpowunion_type,type,
% 2.92/1.63 upairinpowunion: $o ).
% 2.92/1.63
% 2.92/1.63 thf(upairinpowunion,definition,
% 2.92/1.63 ( upairinpowunion
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(ubforcartprodlem1_type,type,
% 2.92/1.63 ubforcartprodlem1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(ubforcartprodlem1,definition,
% 2.92/1.63 ( ubforcartprodlem1
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(ubforcartprodlem2_type,type,
% 2.92/1.63 ubforcartprodlem2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(ubforcartprodlem2,definition,
% 2.92/1.63 ( ubforcartprodlem2
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(ubforcartprodlem3_type,type,
% 2.92/1.63 ubforcartprodlem3: $o ).
% 2.92/1.63
% 2.92/1.63 thf(ubforcartprodlem3,definition,
% 2.92/1.63 ( ubforcartprodlem3
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodpairin_type,type,
% 2.92/1.63 cartprodpairin: $o ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodpairin,definition,
% 2.92/1.63 ( cartprodpairin
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodmempair1_type,type,
% 2.92/1.63 cartprodmempair1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodmempair1,definition,
% 2.92/1.63 ( cartprodmempair1
% 2.92/1.63 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.92/1.63 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.92/1.63 => ? [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 & ? [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 & ( Xu
% 2.92/1.63 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodmempair_type,type,
% 2.92/1.63 cartprodmempair: $o ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodmempair,definition,
% 2.92/1.63 ( cartprodmempair
% 2.92/1.63 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.92/1.63 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.92/1.63 => ( iskpair @ Xu ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setunionE2_type,type,
% 2.92/1.63 setunionE2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setunionE2,definition,
% 2.92/1.63 ( setunionE2
% 2.92/1.63 = ( ! [A: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ ( setunion @ A ) )
% 2.92/1.63 => ? [X: $i] :
% 2.92/1.63 ( ( in @ X @ A )
% 2.92/1.63 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setunionsingleton1_type,type,
% 2.92/1.63 setunionsingleton1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setunionsingleton1,definition,
% 2.92/1.63 ( setunionsingleton1
% 2.92/1.63 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setunionsingleton2_type,type,
% 2.92/1.63 setunionsingleton2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setunionsingleton2,definition,
% 2.92/1.63 ( setunionsingleton2
% 2.92/1.63 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setunionsingleton_type,type,
% 2.92/1.63 setunionsingleton: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setunionsingleton,definition,
% 2.92/1.63 ( setunionsingleton
% 2.92/1.63 = ( ! [Xx: $i] :
% 2.92/1.63 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.92/1.63 = Xx ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(singleton_type,type,
% 2.92/1.63 singleton: $i > $o ).
% 2.92/1.63
% 2.92/1.63 thf(singletonprop_type,type,
% 2.92/1.63 singletonprop: $o ).
% 2.92/1.63
% 2.92/1.63 thf(singletonprop,definition,
% 2.92/1.63 ( singletonprop
% 2.92/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.92/1.63 ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ A )
% 2.92/1.63 => ( ( Xphi @ Xx )
% 2.92/1.63 => ( ( Xphi @ Xy )
% 2.92/1.63 => ( Xx = Xy ) ) ) ) )
% 2.92/1.63 => ( ? [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 & ( Xphi @ Xx ) )
% 2.92/1.63 => ( singleton
% 2.92/1.63 @ ( dsetconstr @ A
% 2.92/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(ex1_type,type,
% 2.92/1.63 ex1: $i > ( $i > $o ) > $o ).
% 2.92/1.63
% 2.92/1.63 thf(ex1,definition,
% 2.92/1.63 ( ex1
% 2.92/1.63 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.92/1.63 ( singleton
% 2.92/1.63 @ ( dsetconstr @ A
% 2.92/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(ex1E1_type,type,
% 2.92/1.63 ex1E1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(ex1E1,definition,
% 2.92/1.63 ( ex1E1
% 2.92/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.92/1.63 ( ( ex1 @ A
% 2.92/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.92/1.63 => ? [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(ex1I_type,type,
% 2.92/1.63 ex1I: $o ).
% 2.92/1.63
% 2.92/1.63 thf(ex1I,definition,
% 2.92/1.63 ( ex1I
% 2.92/1.63 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( Xphi @ Xx )
% 2.92/1.63 => ( ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ A )
% 2.92/1.63 => ( ( Xphi @ Xy )
% 2.92/1.63 => ( Xy = Xx ) ) )
% 2.92/1.63 => ( ex1 @ A
% 2.92/1.63 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(ex1I2_type,type,
% 2.92/1.63 ex1I2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(ex1I2,definition,
% 2.92/1.63 ( ex1I2
% 2.92/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.92/1.63 ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ A )
% 2.92/1.63 => ( ( Xphi @ Xx )
% 2.92/1.63 => ( ( Xphi @ Xy )
% 2.92/1.63 => ( Xx = Xy ) ) ) ) )
% 2.92/1.63 => ( ? [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 & ( Xphi @ Xx ) )
% 2.92/1.63 => ( ex1 @ A
% 2.92/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(singletonsuniq_type,type,
% 2.92/1.63 singletonsuniq: $o ).
% 2.92/1.63
% 2.92/1.63 thf(singletonsuniq,definition,
% 2.92/1.63 ( singletonsuniq
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.92/1.63 ( ( ( setadjoin @ Xx @ emptyset )
% 2.92/1.63 = ( setadjoin @ Xy @ emptyset ) )
% 2.92/1.63 => ( Xx = Xy ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(atmost1p_type,type,
% 2.92/1.63 atmost1p: $i > $o ).
% 2.92/1.63
% 2.92/1.63 thf(atleast2p_type,type,
% 2.92/1.63 atleast2p: $i > $o ).
% 2.92/1.63
% 2.92/1.63 thf(atmost2p_type,type,
% 2.92/1.63 atmost2p: $i > $o ).
% 2.92/1.63
% 2.92/1.63 thf(upairsetp_type,type,
% 2.92/1.63 upairsetp: $i > $o ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairinjL1_type,type,
% 2.92/1.63 setukpairinjL1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairinjL1,definition,
% 2.92/1.63 ( setukpairinjL1
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.92/1.63 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.92/1.63 => ( Xx = Xz ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(kfstsingleton_type,type,
% 2.92/1.63 kfstsingleton: $o ).
% 2.92/1.63
% 2.92/1.63 thf(kfstsingleton,definition,
% 2.92/1.63 ( kfstsingleton
% 2.92/1.63 = ( ! [Xu: $i] :
% 2.92/1.63 ( ( iskpair @ Xu )
% 2.92/1.63 => ( singleton
% 2.92/1.63 @ ( dsetconstr @ ( setunion @ Xu )
% 2.92/1.63 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(theprop_type,type,
% 2.92/1.63 theprop: $o ).
% 2.92/1.63
% 2.92/1.63 thf(theprop,definition,
% 2.92/1.63 ( theprop
% 2.92/1.63 = ( ! [X: $i] :
% 2.92/1.63 ( ( singleton @ X )
% 2.92/1.63 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(kfst_type,type,
% 2.92/1.63 kfst: $i > $i ).
% 2.92/1.63
% 2.92/1.63 thf(kfstpairEq_type,type,
% 2.92/1.63 kfstpairEq: $o ).
% 2.92/1.63
% 2.92/1.63 thf(kfstpairEq,definition,
% 2.92/1.63 ( kfstpairEq
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.92/1.63 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.92/1.63 = Xx ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodfstin_type,type,
% 2.92/1.63 cartprodfstin: $o ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodfstin,definition,
% 2.92/1.63 ( cartprodfstin
% 2.92/1.63 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.92/1.63 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.92/1.63 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairinjL2_type,type,
% 2.92/1.63 setukpairinjL2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairinjL2,definition,
% 2.92/1.63 ( setukpairinjL2
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.92/1.63 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.92/1.63 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.92/1.63 => ( Xx = Xz ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairinjL_type,type,
% 2.92/1.63 setukpairinjL: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairinjL,definition,
% 2.92/1.63 ( setukpairinjL
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.92/1.63 ( ( ( kpair @ Xx @ Xy )
% 2.92/1.63 = ( kpair @ Xz @ Xu ) )
% 2.92/1.63 => ( Xx = Xz ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairinjR11_type,type,
% 2.92/1.63 setukpairinjR11: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairinjR11,definition,
% 2.92/1.63 ( setukpairinjR11
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.92/1.63 ( ( Xx = Xy )
% 2.92/1.63 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.92/1.63 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairinjR12_type,type,
% 2.92/1.63 setukpairinjR12: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairinjR12,definition,
% 2.92/1.63 ( setukpairinjR12
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.92/1.63 ( ( Xx = Xy )
% 2.92/1.63 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.92/1.63 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairinjR1_type,type,
% 2.92/1.63 setukpairinjR1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairinjR1,definition,
% 2.92/1.63 ( setukpairinjR1
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.92/1.63 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.92/1.63 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.92/1.63 => ( ( Xz = Xu )
% 2.92/1.63 => ( Xy = Xu ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(upairequniteq_type,type,
% 2.92/1.63 upairequniteq: $o ).
% 2.92/1.63
% 2.92/1.63 thf(upairequniteq,definition,
% 2.92/1.63 ( upairequniteq
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.92/1.63 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.92/1.63 = ( setadjoin @ Xz @ emptyset ) )
% 2.92/1.63 => ( Xx = Xy ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairinjR2_type,type,
% 2.92/1.63 setukpairinjR2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairinjR2,definition,
% 2.92/1.63 ( setukpairinjR2
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.92/1.63 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.92/1.63 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.92/1.63 => ( Xy = Xu ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairinjR_type,type,
% 2.92/1.63 setukpairinjR: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setukpairinjR,definition,
% 2.92/1.63 ( setukpairinjR
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.92/1.63 ( ( ( kpair @ Xx @ Xy )
% 2.92/1.63 = ( kpair @ Xz @ Xu ) )
% 2.92/1.63 => ( Xy = Xu ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(ksndsingleton_type,type,
% 2.92/1.63 ksndsingleton: $o ).
% 2.92/1.63
% 2.92/1.63 thf(ksndsingleton,definition,
% 2.92/1.63 ( ksndsingleton
% 2.92/1.63 = ( ! [Xu: $i] :
% 2.92/1.63 ( ( iskpair @ Xu )
% 2.92/1.63 => ( singleton
% 2.92/1.63 @ ( dsetconstr @ ( setunion @ Xu )
% 2.92/1.63 @ ^ [Xx: $i] :
% 2.92/1.63 ( Xu
% 2.92/1.63 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(ksnd_type,type,
% 2.92/1.63 ksnd: $i > $i ).
% 2.92/1.63
% 2.92/1.63 thf(ksndpairEq_type,type,
% 2.92/1.63 ksndpairEq: $o ).
% 2.92/1.63
% 2.92/1.63 thf(ksndpairEq,definition,
% 2.92/1.63 ( ksndpairEq
% 2.92/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.92/1.63 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.92/1.63 = Xy ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(kpairsurjEq_type,type,
% 2.92/1.63 kpairsurjEq: $o ).
% 2.92/1.63
% 2.92/1.63 thf(kpairsurjEq,definition,
% 2.92/1.63 ( kpairsurjEq
% 2.92/1.63 = ( ! [Xu: $i] :
% 2.92/1.63 ( ( iskpair @ Xu )
% 2.92/1.63 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.92/1.63 = Xu ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodsndin_type,type,
% 2.92/1.63 cartprodsndin: $o ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodsndin,definition,
% 2.92/1.63 ( cartprodsndin
% 2.92/1.63 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.92/1.63 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.92/1.63 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodpairmemEL_type,type,
% 2.92/1.63 cartprodpairmemEL: $o ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodpairmemEL,definition,
% 2.92/1.63 ( cartprodpairmemEL
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.92/1.63 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.92/1.63 => ( in @ Xx @ A ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodpairmemER_type,type,
% 2.92/1.63 cartprodpairmemER: $o ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodpairmemER,definition,
% 2.92/1.63 ( cartprodpairmemER
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.92/1.63 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.92/1.63 => ( in @ Xy @ B ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodmempaircEq_type,type,
% 2.92/1.63 cartprodmempaircEq: $o ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodmempaircEq,definition,
% 2.92/1.63 ( cartprodmempaircEq
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( ( kpair @ Xx @ Xy )
% 2.92/1.63 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodfstpairEq_type,type,
% 2.92/1.63 cartprodfstpairEq: $o ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodfstpairEq,definition,
% 2.92/1.63 ( cartprodfstpairEq
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.92/1.63 = Xx ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodsndpairEq_type,type,
% 2.92/1.63 cartprodsndpairEq: $o ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodsndpairEq,definition,
% 2.92/1.63 ( cartprodsndpairEq
% 2.92/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.92/1.63 = Xy ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodpairsurjEq_type,type,
% 2.92/1.63 cartprodpairsurjEq: $o ).
% 2.92/1.63
% 2.92/1.63 thf(cartprodpairsurjEq,definition,
% 2.92/1.63 ( cartprodpairsurjEq
% 2.92/1.63 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.92/1.63 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.92/1.63 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.92/1.63 = Xu ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(breln_type,type,
% 2.92/1.63 breln: $i > $i > $i > $o ).
% 2.92/1.63
% 2.92/1.63 thf(breln,definition,
% 2.92/1.63 ( breln
% 2.92/1.63 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(dpsetconstr_type,type,
% 2.92/1.63 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.92/1.63
% 2.92/1.63 thf(dpsetconstrI_type,type,
% 2.92/1.63 dpsetconstrI: $o ).
% 2.92/1.63
% 2.92/1.63 thf(dpsetconstrI,definition,
% 2.92/1.63 ( dpsetconstrI
% 2.92/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( ( Xphi @ Xx @ Xy )
% 2.92/1.63 => ( in @ ( kpair @ Xx @ Xy )
% 2.92/1.63 @ ( dpsetconstr @ A @ B
% 2.92/1.63 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(dpsetconstrSub_type,type,
% 2.92/1.63 dpsetconstrSub: $o ).
% 2.92/1.63
% 2.92/1.63 thf(dpsetconstrSub,definition,
% 2.92/1.63 ( dpsetconstrSub
% 2.92/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.92/1.63 ( subset
% 2.92/1.63 @ ( dpsetconstr @ A @ B
% 2.92/1.63 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.92/1.63 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setOfPairsIsBReln_type,type,
% 2.92/1.63 setOfPairsIsBReln: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setOfPairsIsBReln,definition,
% 2.92/1.63 ( setOfPairsIsBReln
% 2.92/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.92/1.63 ( breln @ A @ B
% 2.92/1.63 @ ( dpsetconstr @ A @ B
% 2.92/1.63 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(dpsetconstrERa_type,type,
% 2.92/1.63 dpsetconstrERa: $o ).
% 2.92/1.63
% 2.92/1.63 thf(dpsetconstrERa,definition,
% 2.92/1.63 ( dpsetconstrERa
% 2.92/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.92/1.63 @ ( dpsetconstr @ A @ B
% 2.92/1.63 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.92/1.63 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(dpsetconstrEL1_type,type,
% 2.92/1.63 dpsetconstrEL1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(dpsetconstrEL1,definition,
% 2.92/1.63 ( dpsetconstrEL1
% 2.92/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.92/1.63 ( ( in @ ( kpair @ Xx @ Xy )
% 2.92/1.63 @ ( dpsetconstr @ A @ B
% 2.92/1.63 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.92/1.63 => ( in @ Xx @ A ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(dpsetconstrEL2_type,type,
% 2.92/1.63 dpsetconstrEL2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(dpsetconstrEL2,definition,
% 2.92/1.63 ( dpsetconstrEL2
% 2.92/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.92/1.63 ( ( in @ ( kpair @ Xx @ Xy )
% 2.92/1.63 @ ( dpsetconstr @ A @ B
% 2.92/1.63 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.92/1.63 => ( in @ Xy @ B ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(dpsetconstrER_type,type,
% 2.92/1.63 dpsetconstrER: $o ).
% 2.92/1.63
% 2.92/1.63 thf(dpsetconstrER,definition,
% 2.92/1.63 ( dpsetconstrER
% 2.92/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.92/1.63 ( ( in @ ( kpair @ Xx @ Xy )
% 2.92/1.63 @ ( dpsetconstr @ A @ B
% 2.92/1.63 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.92/1.63 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(func_type,type,
% 2.92/1.63 func: $i > $i > $i > $o ).
% 2.92/1.63
% 2.92/1.63 thf(funcSet_type,type,
% 2.92/1.63 funcSet: $i > $i > $i ).
% 2.92/1.63
% 2.92/1.63 thf(funcImageSingleton_type,type,
% 2.92/1.63 funcImageSingleton: $o ).
% 2.92/1.63
% 2.92/1.63 thf(funcImageSingleton,definition,
% 2.92/1.63 ( funcImageSingleton
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( func @ A @ B @ Xf )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( singleton
% 2.92/1.63 @ ( dsetconstr @ B
% 2.92/1.63 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(apProp_type,type,
% 2.92/1.63 apProp: $o ).
% 2.92/1.63
% 2.92/1.63 thf(apProp,definition,
% 2.92/1.63 ( apProp
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( func @ A @ B @ Xf )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( in
% 2.92/1.63 @ ( setunion
% 2.92/1.63 @ ( dsetconstr @ B
% 2.92/1.63 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.92/1.63 @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(ap_type,type,
% 2.92/1.63 ap: $i > $i > $i > $i > $i ).
% 2.92/1.63
% 2.92/1.63 thf(app_type,type,
% 2.92/1.63 app: $o ).
% 2.92/1.63
% 2.92/1.63 thf(app,definition,
% 2.92/1.63 ( app
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( func @ A @ B @ Xf )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(infuncsetfunc_type,type,
% 2.92/1.63 infuncsetfunc: $o ).
% 2.92/1.63
% 2.92/1.63 thf(infuncsetfunc,definition,
% 2.92/1.63 ( infuncsetfunc
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.92/1.63 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(ap2p_type,type,
% 2.92/1.63 ap2p: $o ).
% 2.92/1.63
% 2.92/1.63 thf(ap2p,definition,
% 2.92/1.63 ( ap2p
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(funcinfuncset_type,type,
% 2.92/1.63 funcinfuncset: $o ).
% 2.92/1.63
% 2.92/1.63 thf(funcinfuncset,definition,
% 2.92/1.63 ( funcinfuncset
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( func @ A @ B @ Xf )
% 2.92/1.63 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(lamProp_type,type,
% 2.92/1.63 lamProp: $o ).
% 2.92/1.63
% 2.92/1.63 thf(lamProp,definition,
% 2.92/1.63 ( lamProp
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.92/1.63 ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.92/1.63 => ( func @ A @ B
% 2.92/1.63 @ ( dpsetconstr @ A @ B
% 2.92/1.63 @ ^ [Xx: $i,Xy: $i] :
% 2.92/1.63 ( ( Xf @ Xx )
% 2.92/1.63 = Xy ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(lam_type,type,
% 2.92/1.63 lam: $i > $i > ( $i > $i ) > $i ).
% 2.92/1.63
% 2.92/1.63 thf(lamp_type,type,
% 2.92/1.63 lamp: $o ).
% 2.92/1.63
% 2.92/1.63 thf(lamp,definition,
% 2.92/1.63 ( lamp
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.92/1.63 ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.92/1.63 => ( func @ A @ B
% 2.92/1.63 @ ( lam @ A @ B
% 2.92/1.63 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(lam2p_type,type,
% 2.92/1.63 lam2p: $o ).
% 2.92/1.63
% 2.92/1.63 thf(lam2p,definition,
% 2.92/1.63 ( lam2p
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.92/1.63 ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.92/1.63 => ( in
% 2.92/1.63 @ ( lam @ A @ B
% 2.92/1.63 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.92/1.63 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(brelnall1_type,type,
% 2.92/1.63 brelnall1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(brelnall1,definition,
% 2.92/1.63 ( brelnall1
% 2.92/1.63 = ( ! [A: $i,B: $i,R: $i] :
% 2.92/1.63 ( ( breln @ A @ B @ R )
% 2.92/1.63 => ! [Xphi: $i > $o] :
% 2.92/1.63 ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.92/1.63 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ R )
% 2.92/1.63 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(brelnall2_type,type,
% 2.92/1.63 brelnall2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(brelnall2,definition,
% 2.92/1.63 ( brelnall2
% 2.92/1.63 = ( ! [A: $i,B: $i,R: $i] :
% 2.92/1.63 ( ( breln @ A @ B @ R )
% 2.92/1.63 => ! [Xphi: $i > $o] :
% 2.92/1.63 ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.92/1.63 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ R )
% 2.92/1.63 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(ex1E2_type,type,
% 2.92/1.63 ex1E2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(ex1E2,definition,
% 2.92/1.63 ( ex1E2
% 2.92/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.92/1.63 ( ( ex1 @ A
% 2.92/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ A )
% 2.92/1.63 => ( ( Xphi @ Xx )
% 2.92/1.63 => ( ( Xphi @ Xy )
% 2.92/1.63 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(funcGraphProp1_type,type,
% 2.92/1.63 funcGraphProp1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(funcGraphProp1,definition,
% 2.92/1.63 ( funcGraphProp1
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( func @ A @ B @ Xf )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(funcGraphProp3_type,type,
% 2.92/1.63 funcGraphProp3: $o ).
% 2.92/1.63
% 2.92/1.63 thf(funcGraphProp3,definition,
% 2.92/1.63 ( funcGraphProp3
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(funcGraphProp2_type,type,
% 2.92/1.63 funcGraphProp2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(funcGraphProp2,definition,
% 2.92/1.63 ( funcGraphProp2
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( func @ A @ B @ Xf )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.92/1.63 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.92/1.63 = Xy ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(funcextLem_type,type,
% 2.92/1.63 funcextLem: $o ).
% 2.92/1.63
% 2.92/1.63 thf(funcextLem,definition,
% 2.92/1.63 ( funcextLem
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( func @ A @ B @ Xf )
% 2.92/1.63 => ! [Xg: $i] :
% 2.92/1.63 ( ( func @ A @ B @ Xg )
% 2.92/1.63 => ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.92/1.63 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.92/1.63 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(funcGraphProp4_type,type,
% 2.92/1.63 funcGraphProp4: $o ).
% 2.92/1.63
% 2.92/1.63 thf(funcGraphProp4,definition,
% 2.92/1.63 ( funcGraphProp4
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.92/1.63 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.92/1.63 = Xy ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(subbreln_type,type,
% 2.92/1.63 subbreln: $o ).
% 2.92/1.63
% 2.92/1.63 thf(subbreln,definition,
% 2.92/1.63 ( subbreln
% 2.92/1.63 = ( ! [A: $i,B: $i,R: $i] :
% 2.92/1.63 ( ( breln @ A @ B @ R )
% 2.92/1.63 => ! [S: $i] :
% 2.92/1.63 ( ( breln @ A @ B @ S )
% 2.92/1.63 => ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.92/1.63 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.92/1.63 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(eqbreln_type,type,
% 2.92/1.63 eqbreln: $o ).
% 2.92/1.63
% 2.92/1.63 thf(eqbreln,definition,
% 2.92/1.63 ( eqbreln
% 2.92/1.63 = ( ! [A: $i,B: $i,R: $i] :
% 2.92/1.63 ( ( breln @ A @ B @ R )
% 2.92/1.63 => ! [S: $i] :
% 2.92/1.63 ( ( breln @ A @ B @ S )
% 2.92/1.63 => ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.92/1.63 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.92/1.63 => ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ B )
% 2.92/1.63 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.92/1.63 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.92/1.63 => ( R = S ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(funcext_type,type,
% 2.92/1.63 funcext: $o ).
% 2.92/1.63
% 2.92/1.63 thf(funcext,definition,
% 2.92/1.63 ( funcext
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( func @ A @ B @ Xf )
% 2.92/1.63 => ! [Xg: $i] :
% 2.92/1.63 ( ( func @ A @ B @ Xg )
% 2.92/1.63 => ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.92/1.63 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.92/1.63 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(funcext2_type,type,
% 2.92/1.63 funcext2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(funcext2,definition,
% 2.92/1.63 ( funcext2
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.92/1.63 => ! [Xg: $i] :
% 2.92/1.63 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.92/1.63 => ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.92/1.63 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.92/1.63 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(ap2apEq1_type,type,
% 2.92/1.63 ap2apEq1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(ap2apEq1,definition,
% 2.92/1.63 ( ap2apEq1
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.92/1.63 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(ap2apEq2_type,type,
% 2.92/1.63 ap2apEq2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(ap2apEq2,definition,
% 2.92/1.63 ( ap2apEq2
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( func @ A @ B @ Xf )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.92/1.63 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(beta1_type,type,
% 2.92/1.63 beta1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(beta1,definition,
% 2.92/1.63 ( beta1
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.92/1.63 ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( ap @ A @ B
% 2.92/1.63 @ ( lam @ A @ B
% 2.92/1.63 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.92/1.63 @ Xx )
% 2.92/1.63 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(eta1_type,type,
% 2.92/1.63 eta1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(eta1,definition,
% 2.92/1.63 ( eta1
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( func @ A @ B @ Xf )
% 2.92/1.63 => ( ( lam @ A @ B
% 2.92/1.63 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.92/1.63 = Xf ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(lam2lamEq_type,type,
% 2.92/1.63 lam2lamEq: $o ).
% 2.92/1.63
% 2.92/1.63 thf(lam2lamEq,definition,
% 2.92/1.63 ( lam2lamEq
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.92/1.63 ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.92/1.63 => ( ( lam @ A @ B
% 2.92/1.63 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.92/1.63 = ( lam @ A @ B
% 2.92/1.63 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(beta2_type,type,
% 2.92/1.63 beta2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(beta2,definition,
% 2.92/1.63 ( beta2
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.92/1.63 ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( ap @ A @ B
% 2.92/1.63 @ ( lam @ A @ B
% 2.92/1.63 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.92/1.63 @ Xx )
% 2.92/1.63 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(eta2_type,type,
% 2.92/1.63 eta2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(eta2,definition,
% 2.92/1.63 ( eta2
% 2.92/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.92/1.63 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.92/1.63 => ( ( lam @ A @ B
% 2.92/1.63 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.92/1.63 = Xf ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(iffalseProp1_type,type,
% 2.92/1.63 iffalseProp1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(iffalseProp1,definition,
% 2.92/1.63 ( iffalseProp1
% 2.92/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ A )
% 2.92/1.63 => ( ~ Xphi
% 2.92/1.63 => ( in @ Xy
% 2.92/1.63 @ ( dsetconstr @ A
% 2.92/1.63 @ ^ [Xz: $i] :
% 2.92/1.63 ( ( Xphi
% 2.92/1.63 & ( Xz = Xx ) )
% 2.92/1.63 | ( ~ Xphi
% 2.92/1.63 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(iffalseProp2_type,type,
% 2.92/1.63 iffalseProp2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(iffalseProp2,definition,
% 2.92/1.63 ( iffalseProp2
% 2.92/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ A )
% 2.92/1.63 => ( ~ Xphi
% 2.92/1.63 => ( ( dsetconstr @ A
% 2.92/1.63 @ ^ [Xz: $i] :
% 2.92/1.63 ( ( Xphi
% 2.92/1.63 & ( Xz = Xx ) )
% 2.92/1.63 | ( ~ Xphi
% 2.92/1.63 & ( Xz = Xy ) ) ) )
% 2.92/1.63 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(iftrueProp1_type,type,
% 2.92/1.63 iftrueProp1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(iftrueProp1,definition,
% 2.92/1.63 ( iftrueProp1
% 2.92/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ A )
% 2.92/1.63 => ( Xphi
% 2.92/1.63 => ( in @ Xx
% 2.92/1.63 @ ( dsetconstr @ A
% 2.92/1.63 @ ^ [Xz: $i] :
% 2.92/1.63 ( ( Xphi
% 2.92/1.63 & ( Xz = Xx ) )
% 2.92/1.63 | ( ~ Xphi
% 2.92/1.63 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(iftrueProp2_type,type,
% 2.92/1.63 iftrueProp2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(iftrueProp2,definition,
% 2.92/1.63 ( iftrueProp2
% 2.92/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ A )
% 2.92/1.63 => ( Xphi
% 2.92/1.63 => ( ( dsetconstr @ A
% 2.92/1.63 @ ^ [Xz: $i] :
% 2.92/1.63 ( ( Xphi
% 2.92/1.63 & ( Xz = Xx ) )
% 2.92/1.63 | ( ~ Xphi
% 2.92/1.63 & ( Xz = Xy ) ) ) )
% 2.92/1.63 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(ifSingleton_type,type,
% 2.92/1.63 ifSingleton: $o ).
% 2.92/1.63
% 2.92/1.63 thf(ifSingleton,definition,
% 2.92/1.63 ( ifSingleton
% 2.92/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ A )
% 2.92/1.63 => ( singleton
% 2.92/1.63 @ ( dsetconstr @ A
% 2.92/1.63 @ ^ [Xz: $i] :
% 2.92/1.63 ( ( Xphi
% 2.92/1.63 & ( Xz = Xx ) )
% 2.92/1.63 | ( ~ Xphi
% 2.92/1.63 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(if_type,type,
% 2.92/1.63 if: $i > $o > $i > $i > $i ).
% 2.92/1.63
% 2.92/1.63 thf(ifp_type,type,
% 2.92/1.63 ifp: $o ).
% 2.92/1.63
% 2.92/1.63 thf(ifp,definition,
% 2.92/1.63 ( ifp
% 2.92/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ A )
% 2.92/1.63 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(theeq_type,type,
% 2.92/1.63 theeq: $o ).
% 2.92/1.63
% 2.92/1.63 thf(theeq,definition,
% 2.92/1.63 ( theeq
% 2.92/1.63 = ( ! [X: $i] :
% 2.92/1.63 ( ( singleton @ X )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ X )
% 2.92/1.63 => ( ( setunion @ X )
% 2.92/1.63 = Xx ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(iftrue_type,type,
% 2.92/1.63 iftrue: $o ).
% 2.92/1.63
% 2.92/1.63 thf(iftrue,definition,
% 2.92/1.63 ( iftrue
% 2.92/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ A )
% 2.92/1.63 => ( Xphi
% 2.92/1.63 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.92/1.63 = Xx ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(iffalse_type,type,
% 2.92/1.63 iffalse: $o ).
% 2.92/1.63
% 2.92/1.63 thf(iffalse,definition,
% 2.92/1.63 ( iffalse
% 2.92/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ A )
% 2.92/1.63 => ( ~ Xphi
% 2.92/1.63 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.92/1.63 = Xy ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(iftrueorfalse_type,type,
% 2.92/1.63 iftrueorfalse: $o ).
% 2.92/1.63
% 2.92/1.63 thf(iftrueorfalse,definition,
% 2.92/1.63 ( iftrueorfalse
% 2.92/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ! [Xy: $i] :
% 2.92/1.63 ( ( in @ Xy @ A )
% 2.92/1.63 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectT_lem_type,type,
% 2.92/1.63 binintersectT_lem: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectT_lem,definition,
% 2.92/1.63 ( binintersectT_lem
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binunionT_lem_type,type,
% 2.92/1.63 binunionT_lem: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binunionT_lem,definition,
% 2.92/1.63 ( binunionT_lem
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(powersetT_lem_type,type,
% 2.92/1.63 powersetT_lem: $o ).
% 2.92/1.63
% 2.92/1.63 thf(powersetT_lem,definition,
% 2.92/1.63 ( powersetT_lem
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setminusT_lem_type,type,
% 2.92/1.63 setminusT_lem: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setminusT_lem,definition,
% 2.92/1.63 ( setminusT_lem
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(complementT_lem_type,type,
% 2.92/1.63 complementT_lem: $o ).
% 2.92/1.63
% 2.92/1.63 thf(complementT_lem,definition,
% 2.92/1.63 ( complementT_lem
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(setextT_type,type,
% 2.92/1.63 setextT: $o ).
% 2.92/1.63
% 2.92/1.63 thf(setextT,definition,
% 2.92/1.63 ( setextT
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ X )
% 2.92/1.63 => ( in @ Xx @ Y ) ) )
% 2.92/1.63 => ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ Y )
% 2.92/1.63 => ( in @ Xx @ X ) ) )
% 2.92/1.63 => ( X = Y ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(subsetTI_type,type,
% 2.92/1.63 subsetTI: $o ).
% 2.92/1.63
% 2.92/1.63 thf(subsetTI,definition,
% 2.92/1.63 ( subsetTI
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ X )
% 2.92/1.63 => ( in @ Xx @ Y ) ) )
% 2.92/1.63 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(powersetTI1_type,type,
% 2.92/1.63 powersetTI1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(powersetTI1,definition,
% 2.92/1.63 ( powersetTI1
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ( ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ X )
% 2.92/1.63 => ( in @ Xx @ Y ) ) )
% 2.92/1.63 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(powersetTE1_type,type,
% 2.92/1.63 powersetTE1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(powersetTE1,definition,
% 2.92/1.63 ( powersetTE1
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ X @ ( powerset @ Y ) )
% 2.92/1.63 => ( ( in @ Xx @ X )
% 2.92/1.63 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(complementTI1_type,type,
% 2.92/1.63 complementTI1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(complementTI1,definition,
% 2.92/1.63 ( complementTI1
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ X )
% 2.92/1.63 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(complementTE1_type,type,
% 2.92/1.63 complementTE1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(complementTE1,definition,
% 2.92/1.63 ( complementTE1
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.92/1.63 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectTELcontra_type,type,
% 2.92/1.63 binintersectTELcontra: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectTELcontra,definition,
% 2.92/1.63 ( binintersectTELcontra
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ~ ( in @ Xx @ X )
% 2.92/1.63 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectTERcontra_type,type,
% 2.92/1.63 binintersectTERcontra: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binintersectTERcontra,definition,
% 2.92/1.63 ( binintersectTERcontra
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ~ ( in @ Xx @ Y )
% 2.92/1.63 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(contrasubsetT_type,type,
% 2.92/1.63 contrasubsetT: $o ).
% 2.92/1.63
% 2.92/1.63 thf(contrasubsetT,definition,
% 2.92/1.63 ( contrasubsetT
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.92/1.63 => ( ( in @ Xx @ Y )
% 2.92/1.63 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(contrasubsetT1_type,type,
% 2.92/1.63 contrasubsetT1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(contrasubsetT1,definition,
% 2.92/1.63 ( contrasubsetT1
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( subset @ X @ Y )
% 2.92/1.63 => ( ~ ( in @ Xx @ Y )
% 2.92/1.63 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(contrasubsetT2_type,type,
% 2.92/1.63 contrasubsetT2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(contrasubsetT2,definition,
% 2.92/1.63 ( contrasubsetT2
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ( ( subset @ X @ Y )
% 2.92/1.63 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(contrasubsetT3_type,type,
% 2.92/1.63 contrasubsetT3: $o ).
% 2.92/1.63
% 2.92/1.63 thf(contrasubsetT3,definition,
% 2.92/1.63 ( contrasubsetT3
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.92/1.63 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(doubleComplementI1_type,type,
% 2.92/1.63 doubleComplementI1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(doubleComplementI1,definition,
% 2.92/1.63 ( doubleComplementI1
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ X )
% 2.92/1.63 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(doubleComplementE1_type,type,
% 2.92/1.63 doubleComplementE1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(doubleComplementE1,definition,
% 2.92/1.63 ( doubleComplementE1
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 2.92/1.63 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(doubleComplementSub1_type,type,
% 2.92/1.63 doubleComplementSub1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(doubleComplementSub1,definition,
% 2.92/1.63 ( doubleComplementSub1
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(doubleComplementSub2_type,type,
% 2.92/1.63 doubleComplementSub2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(doubleComplementSub2,definition,
% 2.92/1.63 ( doubleComplementSub2
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(doubleComplementEq_type,type,
% 2.92/1.63 doubleComplementEq: $o ).
% 2.92/1.63
% 2.92/1.63 thf(doubleComplementEq,definition,
% 2.92/1.63 ( doubleComplementEq
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ( X
% 2.92/1.63 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(complementTnotintersectT_type,type,
% 2.92/1.63 complementTnotintersectT: $o ).
% 2.92/1.63
% 2.92/1.63 thf(complementTnotintersectT,definition,
% 2.92/1.63 ( complementTnotintersectT
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.92/1.63 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(complementImpComplementIntersect_type,type,
% 2.92/1.63 complementImpComplementIntersect: $o ).
% 2.92/1.63
% 2.92/1.63 thf(complementImpComplementIntersect,definition,
% 2.92/1.63 ( complementImpComplementIntersect
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.92/1.63 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(complementSubsetComplementIntersect_type,type,
% 2.92/1.63 complementSubsetComplementIntersect: $o ).
% 2.92/1.63
% 2.92/1.63 thf(complementSubsetComplementIntersect,definition,
% 2.92/1.63 ( complementSubsetComplementIntersect
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(complementInPowersetComplementIntersect_type,type,
% 2.92/1.63 complementInPowersetComplementIntersect: $o ).
% 2.92/1.63
% 2.92/1.63 thf(complementInPowersetComplementIntersect,definition,
% 2.92/1.63 ( complementInPowersetComplementIntersect
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(contraSubsetComplement_type,type,
% 2.92/1.63 contraSubsetComplement: $o ).
% 2.92/1.63
% 2.92/1.63 thf(contraSubsetComplement,definition,
% 2.92/1.63 ( contraSubsetComplement
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ Y )
% 2.92/1.63 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(complementTcontraSubset_type,type,
% 2.92/1.63 complementTcontraSubset: $o ).
% 2.92/1.63
% 2.92/1.63 thf(complementTcontraSubset,definition,
% 2.92/1.63 ( complementTcontraSubset
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.92/1.63 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binunionTILcontra_type,type,
% 2.92/1.63 binunionTILcontra: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binunionTILcontra,definition,
% 2.92/1.63 ( binunionTILcontra
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.92/1.63 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binunionTIRcontra_type,type,
% 2.92/1.63 binunionTIRcontra: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binunionTIRcontra,definition,
% 2.92/1.63 ( binunionTIRcontra
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.92/1.63 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(inIntersectImpInUnion_type,type,
% 2.92/1.63 inIntersectImpInUnion: $o ).
% 2.92/1.63
% 2.92/1.63 thf(inIntersectImpInUnion,definition,
% 2.92/1.63 ( inIntersectImpInUnion
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Z: $i] :
% 2.92/1.63 ( ( in @ Z @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.92/1.63 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(inIntersectImpInUnion2_type,type,
% 2.92/1.63 inIntersectImpInUnion2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(inIntersectImpInUnion2,definition,
% 2.92/1.63 ( inIntersectImpInUnion2
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Z: $i] :
% 2.92/1.63 ( ( in @ Z @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.92/1.63 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(inIntersectImpInIntersectUnions_type,type,
% 2.92/1.63 inIntersectImpInIntersectUnions: $o ).
% 2.92/1.63
% 2.92/1.63 thf(inIntersectImpInIntersectUnions,definition,
% 2.92/1.63 ( inIntersectImpInIntersectUnions
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Z: $i] :
% 2.92/1.63 ( ( in @ Z @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.92/1.63 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(intersectInPowersetIntersectUnions_type,type,
% 2.92/1.63 intersectInPowersetIntersectUnions: $o ).
% 2.92/1.63
% 2.92/1.63 thf(intersectInPowersetIntersectUnions,definition,
% 2.92/1.63 ( intersectInPowersetIntersectUnions
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Z: $i] :
% 2.92/1.63 ( ( in @ Z @ ( powerset @ A ) )
% 2.92/1.63 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(inComplementUnionImpNotIn1_type,type,
% 2.92/1.63 inComplementUnionImpNotIn1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(inComplementUnionImpNotIn1,definition,
% 2.92/1.63 ( inComplementUnionImpNotIn1
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.92/1.63 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(inComplementUnionImpInComplement1_type,type,
% 2.92/1.63 inComplementUnionImpInComplement1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(inComplementUnionImpInComplement1,definition,
% 2.92/1.63 ( inComplementUnionImpInComplement1
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.92/1.63 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binunionTE_type,type,
% 2.92/1.63 binunionTE: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binunionTE,definition,
% 2.92/1.63 ( binunionTE
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xphi: $o,Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 2.92/1.63 => ( ( ( in @ Xx @ X )
% 2.92/1.63 => Xphi )
% 2.92/1.63 => ( ( ( in @ Xx @ Y )
% 2.92/1.63 => Xphi )
% 2.92/1.63 => Xphi ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(binunionTEcontra_type,type,
% 2.92/1.63 binunionTEcontra: $o ).
% 2.92/1.63
% 2.92/1.63 thf(binunionTEcontra,definition,
% 2.92/1.63 ( binunionTEcontra
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ~ ( in @ Xx @ X )
% 2.92/1.63 => ( ~ ( in @ Xx @ Y )
% 2.92/1.63 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan2a1_type,type,
% 2.92/1.63 demorgan2a1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan2a1,definition,
% 2.92/1.63 ( demorgan2a1
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.92/1.63 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(complementUnionInPowersetComplement_type,type,
% 2.92/1.63 complementUnionInPowersetComplement: $o ).
% 2.92/1.63
% 2.92/1.63 thf(complementUnionInPowersetComplement,definition,
% 2.92/1.63 ( complementUnionInPowersetComplement
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan2a2_type,type,
% 2.92/1.63 demorgan2a2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan2a2,definition,
% 2.92/1.63 ( demorgan2a2
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.92/1.63 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan1a_type,type,
% 2.92/1.63 demorgan1a: $o ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan1a,definition,
% 2.92/1.63 ( demorgan1a
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 2.92/1.63 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan1b_type,type,
% 2.92/1.63 demorgan1b: $o ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan1b,definition,
% 2.92/1.63 ( demorgan1b
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.92/1.63 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan1_type,type,
% 2.92/1.63 demorgan1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan1,definition,
% 2.92/1.63 ( demorgan1
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 2.92/1.63 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan2a_type,type,
% 2.92/1.63 demorgan2a: $o ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan2a,definition,
% 2.92/1.63 ( demorgan2a
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 2.92/1.63 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan2b2_type,type,
% 2.92/1.63 demorgan2b2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan2b2,definition,
% 2.92/1.63 ( demorgan2b2
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.92/1.63 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 2.92/1.63 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan2b_type,type,
% 2.92/1.63 demorgan2b: $o ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan2b,definition,
% 2.92/1.63 ( demorgan2b
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ A )
% 2.92/1.63 => ( ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 2.92/1.63 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan2_type,type,
% 2.92/1.63 demorgan2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(demorgan2,definition,
% 2.92/1.63 ( demorgan2
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ( ( setminus @ A @ ( binunion @ X @ Y ) )
% 2.92/1.63 = ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(woz13rule0_type,type,
% 2.92/1.63 woz13rule0: $o ).
% 2.92/1.63
% 2.92/1.63 thf(woz13rule0,definition,
% 2.92/1.63 ( woz13rule0
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Xx: $i] :
% 2.92/1.63 ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 2.92/1.63 => ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(woz13rule1_type,type,
% 2.92/1.63 woz13rule1: $o ).
% 2.92/1.63
% 2.92/1.63 thf(woz13rule1,definition,
% 2.92/1.63 ( woz13rule1
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Z: $i] :
% 2.92/1.63 ( ( in @ Z @ ( powerset @ A ) )
% 2.92/1.63 => ( ( subset @ X @ Z )
% 2.92/1.63 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(woz13rule2_type,type,
% 2.92/1.63 woz13rule2: $o ).
% 2.92/1.63
% 2.92/1.63 thf(woz13rule2,definition,
% 2.92/1.63 ( woz13rule2
% 2.92/1.63 = ( ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Z: $i] :
% 2.92/1.63 ( ( in @ Z @ ( powerset @ A ) )
% 2.92/1.63 => ( ( subset @ Y @ Z )
% 2.92/1.63 => ( subset @ ( binintersect @ X @ Y ) @ Z ) ) ) ) ) ) ) ).
% 2.92/1.63
% 2.92/1.63 thf(woz13rule3,conjecture,
% 2.92/1.63 ( setextAx
% 2.92/1.63 => ( emptysetAx
% 2.92/1.63 => ( setadjoinAx
% 2.92/1.63 => ( powersetAx
% 2.92/1.63 => ( setunionAx
% 2.92/1.63 => ( omega0Ax
% 2.92/1.63 => ( omegaSAx
% 2.92/1.63 => ( omegaIndAx
% 2.92/1.63 => ( replAx
% 2.92/1.63 => ( foundationAx
% 2.92/1.63 => ( wellorderingAx
% 2.92/1.63 => ( descrp
% 2.92/1.63 => ( dsetconstrI
% 2.92/1.63 => ( dsetconstrEL
% 2.92/1.63 => ( dsetconstrER
% 2.92/1.63 => ( exuE1
% 2.92/1.63 => ( prop2setE
% 2.92/1.63 => ( emptysetE
% 2.92/1.63 => ( emptysetimpfalse
% 2.92/1.63 => ( notinemptyset
% 2.92/1.63 => ( exuE3e
% 2.92/1.63 => ( setext
% 2.92/1.63 => ( emptyI
% 2.92/1.63 => ( noeltsimpempty
% 2.92/1.63 => ( setbeta
% 2.92/1.63 => ( nonemptyE1
% 2.92/1.63 => ( nonemptyI
% 2.92/1.63 => ( nonemptyI1
% 2.92/1.63 => ( setadjoinIL
% 2.92/1.63 => ( emptyinunitempty
% 2.92/1.63 => ( setadjoinIR
% 2.92/1.63 => ( setadjoinE
% 2.92/1.63 => ( setadjoinOr
% 2.92/1.63 => ( setoftrueEq
% 2.92/1.63 => ( powersetI
% 2.92/1.63 => ( emptyinPowerset
% 2.92/1.63 => ( emptyInPowerset
% 2.92/1.63 => ( powersetE
% 2.92/1.63 => ( setunionI
% 2.92/1.63 => ( setunionE
% 2.92/1.63 => ( subPowSU
% 2.92/1.63 => ( exuE2
% 2.92/1.63 => ( nonemptyImpWitness
% 2.92/1.63 => ( uniqinunit
% 2.92/1.63 => ( notinsingleton
% 2.92/1.63 => ( eqinunit
% 2.92/1.63 => ( singletonsswitch
% 2.92/1.63 => ( upairsetE
% 2.92/1.63 => ( upairsetIL
% 2.92/1.63 => ( upairsetIR
% 2.92/1.63 => ( emptyE1
% 2.92/1.63 => ( vacuousDall
% 2.92/1.63 => ( quantDeMorgan1
% 2.92/1.63 => ( quantDeMorgan2
% 2.92/1.63 => ( quantDeMorgan3
% 2.92/1.63 => ( quantDeMorgan4
% 2.92/1.63 => ( prop2setI
% 2.92/1.63 => ( prop2set2propI
% 2.92/1.63 => ( notdexE
% 2.92/1.63 => ( notdallE
% 2.92/1.63 => ( exuI1
% 2.92/1.63 => ( exuI3
% 2.92/1.63 => ( exuI2
% 2.92/1.63 => ( inCongP
% 2.92/1.63 => ( in__Cong
% 2.92/1.63 => ( exuE3u
% 2.92/1.63 => ( exu__Cong
% 2.92/1.63 => ( emptyset__Cong
% 2.92/1.63 => ( setadjoin__Cong
% 2.92/1.63 => ( powerset__Cong
% 2.92/1.63 => ( setunion__Cong
% 2.92/1.63 => ( omega__Cong
% 2.92/1.63 => ( exuEu
% 2.92/1.63 => ( descr__Cong
% 2.92/1.63 => ( dsetconstr__Cong
% 2.92/1.63 => ( subsetI1
% 2.92/1.63 => ( eqimpsubset2
% 2.92/1.63 => ( eqimpsubset1
% 2.92/1.63 => ( subsetI2
% 2.92/1.63 => ( emptysetsubset
% 2.92/1.63 => ( subsetE
% 2.92/1.63 => ( subsetE2
% 2.92/1.63 => ( notsubsetI
% 2.92/1.63 => ( notequalI1
% 2.92/1.63 => ( notequalI2
% 2.92/1.63 => ( subsetRefl
% 2.92/1.63 => ( subsetTrans
% 2.92/1.63 => ( setadjoinSub
% 2.92/1.63 => ( setadjoinSub2
% 2.92/1.63 => ( subset2powerset
% 2.92/1.63 => ( setextsub
% 2.92/1.63 => ( subsetemptysetimpeq
% 2.92/1.63 => ( powersetI1
% 2.92/1.63 => ( powersetE1
% 2.92/1.63 => ( inPowerset
% 2.92/1.63 => ( powersetsubset
% 2.92/1.63 => ( sepInPowerset
% 2.92/1.63 => ( sepSubset
% 2.92/1.63 => ( binunionIL
% 2.92/1.63 => ( upairset2IR
% 2.92/1.63 => ( binunionIR
% 2.92/1.63 => ( binunionEcases
% 2.92/1.63 => ( binunionE
% 2.92/1.63 => ( binunionLsub
% 2.92/1.63 => ( binunionRsub
% 2.92/1.63 => ( binintersectI
% 2.92/1.63 => ( binintersectSubset5
% 2.92/1.63 => ( binintersectEL
% 2.92/1.63 => ( binintersectLsub
% 2.92/1.63 => ( binintersectSubset2
% 2.92/1.63 => ( binintersectSubset3
% 2.92/1.63 => ( binintersectER
% 2.92/1.63 => ( disjointsetsI1
% 2.92/1.63 => ( binintersectRsub
% 2.92/1.63 => ( binintersectSubset4
% 2.92/1.63 => ( binintersectSubset1
% 2.92/1.63 => ( bs114d
% 2.92/1.63 => ( setminusI
% 2.92/1.63 => ( setminusEL
% 2.92/1.63 => ( setminusER
% 2.92/1.63 => ( setminusSubset2
% 2.92/1.63 => ( setminusERneg
% 2.92/1.63 => ( setminusELneg
% 2.92/1.63 => ( setminusILneg
% 2.92/1.63 => ( setminusIRneg
% 2.92/1.63 => ( setminusLsub
% 2.92/1.63 => ( setminusSubset1
% 2.92/1.63 => ( symdiffE
% 2.92/1.63 => ( symdiffI1
% 2.92/1.63 => ( symdiffI2
% 2.92/1.63 => ( symdiffIneg1
% 2.92/1.63 => ( symdiffIneg2
% 2.92/1.63 => ( secondinupair
% 2.92/1.63 => ( setukpairIL
% 2.92/1.63 => ( setukpairIR
% 2.92/1.63 => ( kpairiskpair
% 2.92/1.63 => ( kpairp
% 2.92/1.63 => ( singletonsubset
% 2.92/1.63 => ( singletoninpowerset
% 2.92/1.63 => ( singletoninpowunion
% 2.92/1.63 => ( upairset2E
% 2.92/1.63 => ( upairsubunion
% 2.92/1.63 => ( upairinpowunion
% 2.92/1.63 => ( ubforcartprodlem1
% 2.92/1.63 => ( ubforcartprodlem2
% 2.92/1.63 => ( ubforcartprodlem3
% 2.92/1.63 => ( cartprodpairin
% 2.92/1.63 => ( cartprodmempair1
% 2.92/1.63 => ( cartprodmempair
% 2.92/1.63 => ( setunionE2
% 2.92/1.63 => ( setunionsingleton1
% 2.92/1.63 => ( setunionsingleton2
% 2.92/1.63 => ( setunionsingleton
% 2.92/1.63 => ( singletonprop
% 2.92/1.63 => ( ex1E1
% 2.92/1.63 => ( ex1I
% 2.92/1.63 => ( ex1I2
% 2.92/1.63 => ( singletonsuniq
% 2.92/1.63 => ( setukpairinjL1
% 2.92/1.63 => ( kfstsingleton
% 2.92/1.63 => ( theprop
% 2.92/1.63 => ( kfstpairEq
% 2.92/1.63 => ( cartprodfstin
% 2.92/1.63 => ( setukpairinjL2
% 2.92/1.63 => ( setukpairinjL
% 2.92/1.63 => ( setukpairinjR11
% 2.92/1.63 => ( setukpairinjR12
% 2.92/1.63 => ( setukpairinjR1
% 2.92/1.63 => ( upairequniteq
% 2.92/1.63 => ( setukpairinjR2
% 2.92/1.63 => ( setukpairinjR
% 2.92/1.63 => ( ksndsingleton
% 2.92/1.63 => ( ksndpairEq
% 2.92/1.63 => ( kpairsurjEq
% 2.92/1.63 => ( cartprodsndin
% 2.92/1.63 => ( cartprodpairmemEL
% 2.92/1.63 => ( cartprodpairmemER
% 2.92/1.63 => ( cartprodmempaircEq
% 2.92/1.63 => ( cartprodfstpairEq
% 2.92/1.63 => ( cartprodsndpairEq
% 2.92/1.63 => ( cartprodpairsurjEq
% 2.92/1.63 => ( dpsetconstrI
% 2.92/1.63 => ( dpsetconstrSub
% 2.92/1.63 => ( setOfPairsIsBReln
% 2.92/1.63 => ( dpsetconstrERa
% 2.92/1.63 => ( dpsetconstrEL1
% 2.92/1.63 => ( dpsetconstrEL2
% 2.92/1.63 => ( dpsetconstrER
% 2.92/1.63 => ( funcImageSingleton
% 2.92/1.63 => ( apProp
% 2.92/1.63 => ( app
% 2.92/1.63 => ( infuncsetfunc
% 2.92/1.63 => ( ap2p
% 2.92/1.63 => ( funcinfuncset
% 2.92/1.63 => ( lamProp
% 2.92/1.63 => ( lamp
% 2.92/1.63 => ( lam2p
% 2.92/1.63 => ( brelnall1
% 2.92/1.63 => ( brelnall2
% 2.92/1.63 => ( ex1E2
% 2.92/1.63 => ( funcGraphProp1
% 2.92/1.63 => ( funcGraphProp3
% 2.92/1.63 => ( funcGraphProp2
% 2.92/1.63 => ( funcextLem
% 2.92/1.63 => ( funcGraphProp4
% 2.92/1.63 => ( subbreln
% 2.92/1.63 => ( eqbreln
% 2.92/1.63 => ( funcext
% 2.92/1.63 => ( funcext2
% 2.92/1.63 => ( ap2apEq1
% 2.92/1.63 => ( ap2apEq2
% 2.92/1.63 => ( beta1
% 2.92/1.63 => ( eta1
% 2.92/1.63 => ( lam2lamEq
% 2.92/1.63 => ( beta2
% 2.92/1.63 => ( eta2
% 2.92/1.63 => ( iffalseProp1
% 2.92/1.63 => ( iffalseProp2
% 2.92/1.63 => ( iftrueProp1
% 2.92/1.63 => ( iftrueProp2
% 2.92/1.63 => ( ifSingleton
% 2.92/1.63 => ( ifp
% 2.92/1.63 => ( theeq
% 2.92/1.63 => ( iftrue
% 2.92/1.63 => ( iffalse
% 2.92/1.63 => ( iftrueorfalse
% 2.92/1.63 => ( binintersectT_lem
% 2.92/1.63 => ( binunionT_lem
% 2.92/1.63 => ( powersetT_lem
% 2.92/1.63 => ( setminusT_lem
% 2.92/1.63 => ( complementT_lem
% 2.92/1.63 => ( setextT
% 2.92/1.63 => ( subsetTI
% 2.92/1.63 => ( powersetTI1
% 2.92/1.63 => ( powersetTE1
% 2.92/1.63 => ( complementTI1
% 2.92/1.63 => ( complementTE1
% 2.92/1.63 => ( binintersectTELcontra
% 2.92/1.63 => ( binintersectTERcontra
% 2.92/1.63 => ( contrasubsetT
% 2.92/1.63 => ( contrasubsetT1
% 2.92/1.63 => ( contrasubsetT2
% 2.92/1.63 => ( contrasubsetT3
% 2.92/1.63 => ( doubleComplementI1
% 2.92/1.63 => ( doubleComplementE1
% 2.92/1.63 => ( doubleComplementSub1
% 2.92/1.63 => ( doubleComplementSub2
% 2.92/1.63 => ( doubleComplementEq
% 2.92/1.63 => ( complementTnotintersectT
% 2.92/1.63 => ( complementImpComplementIntersect
% 2.92/1.63 => ( complementSubsetComplementIntersect
% 2.92/1.63 => ( complementInPowersetComplementIntersect
% 2.92/1.63 => ( contraSubsetComplement
% 2.92/1.63 => ( complementTcontraSubset
% 2.92/1.63 => ( binunionTILcontra
% 2.92/1.63 => ( binunionTIRcontra
% 2.92/1.63 => ( inIntersectImpInUnion
% 2.92/1.63 => ( inIntersectImpInUnion2
% 2.92/1.63 => ( inIntersectImpInIntersectUnions
% 2.92/1.63 => ( intersectInPowersetIntersectUnions
% 2.92/1.63 => ( inComplementUnionImpNotIn1
% 2.92/1.63 => ( inComplementUnionImpInComplement1
% 2.92/1.63 => ( binunionTE
% 2.92/1.63 => ( binunionTEcontra
% 2.92/1.63 => ( demorgan2a1
% 2.92/1.63 => ( complementUnionInPowersetComplement
% 2.92/1.63 => ( demorgan2a2
% 2.92/1.63 => ( demorgan1a
% 2.92/1.63 => ( demorgan1b
% 2.92/1.63 => ( demorgan1
% 2.92/1.63 => ( demorgan2a
% 2.92/1.63 => ( demorgan2b2
% 2.92/1.63 => ( demorgan2b
% 2.92/1.63 => ( demorgan2
% 2.92/1.63 => ( woz13rule0
% 2.92/1.63 => ( woz13rule1
% 2.92/1.63 => ( woz13rule2
% 2.92/1.63 => ! [A: $i,X: $i] :
% 2.92/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.92/1.63 => ! [Y: $i] :
% 2.92/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.92/1.63 => ! [Z: $i] :
% 2.92/1.63 ( ( in @ Z @ ( powerset @ A ) )
% 2.92/1.63 => ( ( subset @ X @ Y )
% 2.92/1.63 => ( ( subset @ X @ Z )
% 2.92/1.63 => ( subset @ X @ ( binintersect @ Y @ Z ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.92/1.63 ------------------------
% 2.92/1.67 % (18462)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_18132 for (2999ds/183Mi)
% 2.92/1.67 % (18463)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_18132 for (2999ds/4Mi)
% 2.92/1.67 % (18464)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_18132 for (2999ds/27Mi)
% 2.92/1.67 % (18463)Instruction limit reached!
% 2.92/1.67 % (18463)------------------------------
% 2.92/1.67 % (18463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.67 % (18463)Termination reason: Unknown
% 2.92/1.67 % (18463)Termination phase: shuffling
% 2.92/1.67
% 2.92/1.67 % (18463)Memory used [KB]: 1535
% 2.92/1.67 % (18463)Time elapsed: 0.003 s
% 2.92/1.67 % (18463)Instructions burned: 6 (million)
% 2.92/1.67 % (18463)------------------------------
% 2.92/1.67 % (18463)------------------------------
% 2.92/1.68 % (18465)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_18132 for (2999ds/2Mi)
% 2.92/1.68 % (18466)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_18132 for (2999ds/2Mi)
% 2.92/1.68 % (18467)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_18132 for (2999ds/275Mi)
% 2.92/1.68 % (18468)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_18132 for (2999ds/18Mi)
% 2.92/1.68 % (18464)Instruction limit reached!
% 2.92/1.68 % (18464)------------------------------
% 2.92/1.68 % (18464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.68 % (18464)Termination reason: Unknown
% 2.92/1.68 % (18464)Termination phase: shuffling
% 2.92/1.68
% 2.92/1.68 % (18464)Memory used [KB]: 1918
% 2.92/1.68 % (18464)Time elapsed: 0.009 s
% 2.92/1.68 % (18464)Instructions burned: 27 (million)
% 2.92/1.68 % (18464)------------------------------
% 2.92/1.68 % (18464)------------------------------
% 2.92/1.68 % (18465)Instruction limit reached!
% 2.92/1.68 % (18465)------------------------------
% 2.92/1.68 % (18465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.68 % (18465)Termination reason: Unknown
% 2.92/1.68 % (18465)Termination phase: shuffling
% 2.92/1.68
% 2.92/1.68 % (18465)Memory used [KB]: 1535
% 2.92/1.68 % (18465)Time elapsed: 0.004 s
% 2.92/1.68 % (18465)Instructions burned: 3 (million)
% 2.92/1.68 % (18465)------------------------------
% 2.92/1.68 % (18465)------------------------------
% 2.92/1.68 % (18466)Instruction limit reached!
% 2.92/1.68 % (18466)------------------------------
% 2.92/1.68 % (18466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.68 % (18466)Termination reason: Unknown
% 2.92/1.68 % (18466)Termination phase: shuffling
% 2.92/1.68
% 2.92/1.68 % (18466)Memory used [KB]: 1535
% 2.92/1.68 % (18466)Time elapsed: 0.003 s
% 2.92/1.68 % (18466)Instructions burned: 3 (million)
% 2.92/1.68 % (18466)------------------------------
% 2.92/1.68 % (18466)------------------------------
% 2.92/1.68 % (18469)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_18132 for (2999ds/3Mi)
% 2.92/1.68 % (18469)Instruction limit reached!
% 2.92/1.68 % (18469)------------------------------
% 2.92/1.68 % (18469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.68 % (18469)Termination reason: Unknown
% 2.92/1.68 % (18469)Termination phase: shuffling
% 2.92/1.68
% 2.92/1.68 % (18469)Memory used [KB]: 1535
% 2.92/1.68 % (18469)Time elapsed: 0.002 s
% 2.92/1.68 % (18469)Instructions burned: 4 (million)
% 2.92/1.68 % (18469)------------------------------
% 2.92/1.68 % (18469)------------------------------
% 2.92/1.69 % (18470)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_18132 for (2999ds/37Mi)
% 2.92/1.69 % (18468)Instruction limit reached!
% 2.92/1.69 % (18468)------------------------------
% 2.92/1.69 % (18468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.92/1.69 % (18468)Termination reason: Unknown
% 2.92/1.69 % (18468)Termination phase: shuffling
% 2.92/1.69
% 2.92/1.69 % (18468)Memory used [KB]: 1918
% 2.92/1.69 % (18468)Time elapsed: 0.013 s
% 2.92/1.69 % (18468)Instructions burned: 19 (million)
% 2.92/1.69 % (18468)------------------------------
% 2.92/1.69 % (18468)------------------------------
% 2.92/1.69 % (18473)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_18132 for (2999ds/1041Mi)
% 2.95/1.70 % (18471)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_18132 for (2999ds/15Mi)
% 2.95/1.70 % (18472)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_18132 for (2999ds/3Mi)
% 2.95/1.70 % (18472)Instruction limit reached!
% 2.95/1.70 % (18472)------------------------------
% 2.95/1.70 % (18472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.70 % (18472)Termination reason: Unknown
% 2.95/1.70 % (18472)Termination phase: shuffling
% 2.95/1.70
% 2.95/1.70 % (18472)Memory used [KB]: 1535
% 2.95/1.70 % (18472)Time elapsed: 0.004 s
% 2.95/1.70 % (18472)Instructions burned: 3 (million)
% 2.95/1.70 % (18472)------------------------------
% 2.95/1.70 % (18472)------------------------------
% 2.95/1.70 % (18470)Instruction limit reached!
% 2.95/1.70 % (18470)------------------------------
% 2.95/1.70 % (18470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.70 % (18470)Termination reason: Unknown
% 2.95/1.70 % (18470)Termination phase: shuffling
% 2.95/1.70
% 2.95/1.70 % (18470)Memory used [KB]: 2302
% 2.95/1.70 % (18470)Time elapsed: 0.012 s
% 2.95/1.70 % (18470)Instructions burned: 39 (million)
% 2.95/1.70 % (18470)------------------------------
% 2.95/1.70 % (18470)------------------------------
% 2.95/1.70 % (18471)Instruction limit reached!
% 2.95/1.70 % (18471)------------------------------
% 2.95/1.70 % (18471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.70 % (18471)Termination reason: Unknown
% 2.95/1.70 % (18471)Termination phase: shuffling
% 2.95/1.70
% 2.95/1.70 % (18471)Memory used [KB]: 1791
% 2.95/1.70 % (18471)Time elapsed: 0.011 s
% 2.95/1.70 % (18471)Instructions burned: 15 (million)
% 2.95/1.70 % (18471)------------------------------
% 2.95/1.70 % (18471)------------------------------
% 2.95/1.71 % (18474)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_18132 for (2999ds/7Mi)
% 2.95/1.71 % (18476)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_18132 for (2999ds/3Mi)
% 2.95/1.71 % (18474)Instruction limit reached!
% 2.95/1.71 % (18474)------------------------------
% 2.95/1.71 % (18474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.71 % (18474)Termination reason: Unknown
% 2.95/1.71 % (18474)Termination phase: shuffling
% 2.95/1.71
% 2.95/1.71 % (18474)Memory used [KB]: 1535
% 2.95/1.71 % (18474)Time elapsed: 0.005 s
% 2.95/1.71 % (18474)Instructions burned: 7 (million)
% 2.95/1.71 % (18474)------------------------------
% 2.95/1.71 % (18474)------------------------------
% 2.95/1.71 % (18476)Instruction limit reached!
% 2.95/1.71 % (18476)------------------------------
% 2.95/1.71 % (18476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.71 % (18476)Termination reason: Unknown
% 2.95/1.71 % (18476)Termination phase: shuffling
% 2.95/1.71
% 2.95/1.71 % (18476)Memory used [KB]: 1535
% 2.95/1.71 % (18476)Time elapsed: 0.002 s
% 2.95/1.71 % (18476)Instructions burned: 4 (million)
% 2.95/1.71 % (18476)------------------------------
% 2.95/1.71 % (18476)------------------------------
% 2.95/1.71 % (18475)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_18132 for (2999ds/16Mi)
% 2.95/1.72 % (18462)Instruction limit reached!
% 2.95/1.72 % (18462)------------------------------
% 2.95/1.72 % (18462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.72 % (18462)Termination reason: Unknown
% 2.95/1.72 % (18462)Termination phase: Saturation
% 2.95/1.72
% 2.95/1.72 % (18462)Memory used [KB]: 7419
% 2.95/1.72 % (18462)Time elapsed: 0.049 s
% 2.95/1.72 % (18462)Instructions burned: 184 (million)
% 2.95/1.72 % (18462)------------------------------
% 2.95/1.72 % (18462)------------------------------
% 2.95/1.72 % (18479)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_18132 for (2999ds/3Mi)
% 2.95/1.72 % (18478)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_18132 for (2999ds/7Mi)
% 2.95/1.72 % (18479)Instruction limit reached!
% 2.95/1.72 % (18479)------------------------------
% 2.95/1.72 % (18479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.72 % (18479)Termination reason: Unknown
% 2.95/1.72 % (18479)Termination phase: shuffling
% 2.95/1.72
% 2.95/1.72 % (18479)Memory used [KB]: 1535
% 2.95/1.72 % (18479)Time elapsed: 0.003 s
% 2.95/1.72 % (18479)Instructions burned: 4 (million)
% 2.95/1.72 % (18479)------------------------------
% 2.95/1.72 % (18479)------------------------------
% 2.95/1.72 % (18475)Instruction limit reached!
% 2.95/1.72 % (18475)------------------------------
% 2.95/1.72 % (18475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.72 % (18475)Termination reason: Unknown
% 2.95/1.72 % (18475)Termination phase: shuffling
% 2.95/1.72
% 2.95/1.72 % (18475)Memory used [KB]: 1918
% 2.95/1.72 % (18475)Time elapsed: 0.009 s
% 2.95/1.72 % (18475)Instructions burned: 17 (million)
% 2.95/1.72 % (18475)------------------------------
% 2.95/1.72 % (18475)------------------------------
% 2.95/1.72 % (18477)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_18132 for (2999ds/3Mi)
% 2.95/1.72 % (18478)Instruction limit reached!
% 2.95/1.72 % (18478)------------------------------
% 2.95/1.72 % (18478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.72 % (18478)Termination reason: Unknown
% 2.95/1.72 % (18478)Termination phase: shuffling
% 2.95/1.72
% 2.95/1.72 % (18478)Memory used [KB]: 1663
% 2.95/1.72 % (18478)Time elapsed: 0.004 s
% 2.95/1.72 % (18478)Instructions burned: 8 (million)
% 2.95/1.72 % (18478)------------------------------
% 2.95/1.72 % (18478)------------------------------
% 2.95/1.72 % (18477)Instruction limit reached!
% 2.95/1.72 % (18477)------------------------------
% 2.95/1.72 % (18477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.72 % (18477)Termination reason: Unknown
% 2.95/1.72 % (18477)Termination phase: shuffling
% 2.95/1.72
% 2.95/1.72 % (18477)Memory used [KB]: 1535
% 2.95/1.72 % (18477)Time elapsed: 0.004 s
% 2.95/1.72 % (18477)Instructions burned: 4 (million)
% 2.95/1.72 % (18477)------------------------------
% 2.95/1.72 % (18477)------------------------------
% 2.95/1.73 % (18480)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_18132 for (2999ds/4Mi)
% 2.95/1.73 % (18480)Instruction limit reached!
% 2.95/1.73 % (18480)------------------------------
% 2.95/1.73 % (18480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.73 % (18480)Termination reason: Unknown
% 2.95/1.73 % (18480)Termination phase: shuffling
% 2.95/1.73
% 2.95/1.73 % (18480)Memory used [KB]: 1535
% 2.95/1.73 % (18480)Time elapsed: 0.003 s
% 2.95/1.73 % (18480)Instructions burned: 7 (million)
% 2.95/1.73 % (18480)------------------------------
% 2.95/1.73 % (18480)------------------------------
% 2.95/1.73 % (18481)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_18132 for (2999ds/18Mi)
% 2.95/1.73 % (18482)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_18132 for (2999ds/710Mi)
% 2.95/1.73 % (18483)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_18132 for (2999ds/6Mi)
% 2.95/1.74 % (18483)Instruction limit reached!
% 2.95/1.74 % (18483)------------------------------
% 2.95/1.74 % (18483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.74 % (18483)Termination reason: Unknown
% 2.95/1.74 % (18483)Termination phase: shuffling
% 2.95/1.74
% 2.95/1.74 % (18483)Memory used [KB]: 1535
% 2.95/1.74 % (18483)Time elapsed: 0.003 s
% 2.95/1.74 % (18483)Instructions burned: 7 (million)
% 2.95/1.74 % (18483)------------------------------
% 2.95/1.74 % (18483)------------------------------
% 2.95/1.74 % (18481)Instruction limit reached!
% 2.95/1.74 % (18481)------------------------------
% 2.95/1.74 % (18481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.74 % (18481)Termination reason: Unknown
% 2.95/1.74 % (18481)Termination phase: shuffling
% 2.95/1.74
% 2.95/1.74 % (18481)Memory used [KB]: 1918
% 2.95/1.74 % (18481)Time elapsed: 0.006 s
% 2.95/1.74 % (18481)Instructions burned: 18 (million)
% 2.95/1.74 % (18481)------------------------------
% 2.95/1.74 % (18481)------------------------------
% 2.95/1.74 % (18485)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_18132 for (2999ds/21Mi)
% 2.95/1.74 % (18484)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_18132 for (2999ds/902Mi)
% 2.95/1.74 % (18485)Instruction limit reached!
% 2.95/1.74 % (18485)------------------------------
% 2.95/1.74 % (18485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.74 % (18485)Termination reason: Unknown
% 2.95/1.74 % (18485)Termination phase: shuffling
% 2.95/1.74
% 2.95/1.74 % (18485)Memory used [KB]: 1918
% 2.95/1.74 % (18485)Time elapsed: 0.007 s
% 2.95/1.74 % (18485)Instructions burned: 22 (million)
% 2.95/1.74 % (18485)------------------------------
% 2.95/1.74 % (18485)------------------------------
% 2.95/1.75 % (18486)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_18132 for (2999ds/5Mi)
% 2.95/1.75 % (18486)Instruction limit reached!
% 2.95/1.75 % (18486)------------------------------
% 2.95/1.75 % (18486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.75 % (18486)Termination reason: Unknown
% 2.95/1.75 % (18486)Termination phase: shuffling
% 2.95/1.75
% 2.95/1.75 % (18486)Memory used [KB]: 1663
% 2.95/1.75 % (18486)Time elapsed: 0.003 s
% 2.95/1.75 % (18486)Instructions burned: 8 (million)
% 2.95/1.75 % (18486)------------------------------
% 2.95/1.75 % (18486)------------------------------
% 2.95/1.75 % (18488)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_18132 for (2999ds/377Mi)
% 2.95/1.75 % (18487)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_18132 for (2999ds/6Mi)
% 2.95/1.76 % (18487)Instruction limit reached!
% 2.95/1.76 % (18487)------------------------------
% 2.95/1.76 % (18487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.76 % (18487)Termination reason: Unknown
% 2.95/1.76 % (18487)Termination phase: shuffling
% 2.95/1.76
% 2.95/1.76 % (18487)Memory used [KB]: 1663
% 2.95/1.76 % (18487)Time elapsed: 0.006 s
% 2.95/1.76 % (18487)Instructions burned: 7 (million)
% 2.95/1.76 % (18487)------------------------------
% 2.95/1.76 % (18487)------------------------------
% 2.95/1.76 % (18489)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_18132 for (2998ds/779Mi)
% 2.95/1.78 % (18490)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_18132 for (2998ds/19Mi)
% 2.95/1.79 % (18490)Instruction limit reached!
% 2.95/1.79 % (18490)------------------------------
% 2.95/1.79 % (18490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.79 % (18490)Termination reason: Unknown
% 2.95/1.79 % (18490)Termination phase: shuffling
% 2.95/1.79
% 2.95/1.79 % (18490)Memory used [KB]: 1918
% 2.95/1.79 % (18490)Time elapsed: 0.012 s
% 2.95/1.79 % (18490)Instructions burned: 19 (million)
% 2.95/1.79 % (18490)------------------------------
% 2.95/1.79 % (18490)------------------------------
% 2.95/1.80 % (18467)Instruction limit reached!
% 2.95/1.80 % (18467)------------------------------
% 2.95/1.80 % (18467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.80 % (18467)Termination reason: Unknown
% 2.95/1.80 % (18467)Termination phase: Saturation
% 2.95/1.80
% 2.95/1.80 % (18467)Memory used [KB]: 9978
% 2.95/1.80 % (18467)Time elapsed: 0.127 s
% 2.95/1.80 % (18467)Instructions burned: 275 (million)
% 2.95/1.80 % (18467)------------------------------
% 2.95/1.80 % (18467)------------------------------
% 2.95/1.80 % (18491)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_18132 for (2998ds/879Mi)
% 2.95/1.81 % (18492)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_18132 for (2998ds/17Mi)
% 2.95/1.82 % (18492)Instruction limit reached!
% 2.95/1.82 % (18492)------------------------------
% 2.95/1.82 % (18492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.82 % (18492)Termination reason: Unknown
% 2.95/1.82 % (18492)Termination phase: shuffling
% 2.95/1.82
% 2.95/1.82 % (18492)Memory used [KB]: 1791
% 2.95/1.82 % (18492)Time elapsed: 0.006 s
% 2.95/1.82 % (18492)Instructions burned: 17 (million)
% 2.95/1.82 % (18492)------------------------------
% 2.95/1.82 % (18492)------------------------------
% 2.95/1.83 % (18493)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_18132 for (2998ds/3Mi)
% 2.95/1.83 % (18493)Instruction limit reached!
% 2.95/1.83 % (18493)------------------------------
% 2.95/1.83 % (18493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.83 % (18493)Termination reason: Unknown
% 2.95/1.83 % (18493)Termination phase: shuffling
% 2.95/1.83
% 2.95/1.83 % (18493)Memory used [KB]: 1535
% 2.95/1.83 % (18493)Time elapsed: 0.002 s
% 2.95/1.83 % (18493)Instructions burned: 3 (million)
% 2.95/1.83 % (18493)------------------------------
% 2.95/1.83 % (18493)------------------------------
% 2.95/1.84 % (18494)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_18132 for (2998ds/30Mi)
% 2.95/1.85 % (18494)Instruction limit reached!
% 2.95/1.85 % (18494)------------------------------
% 2.95/1.85 % (18494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.85 % (18494)Termination reason: Unknown
% 2.95/1.85 % (18494)Termination phase: shuffling
% 2.95/1.85
% 2.95/1.85 % (18494)Memory used [KB]: 2046
% 2.95/1.85 % (18494)Time elapsed: 0.011 s
% 2.95/1.85 % (18494)Instructions burned: 31 (million)
% 2.95/1.85 % (18494)------------------------------
% 2.95/1.85 % (18494)------------------------------
% 2.95/1.86 % (18488)Instruction limit reached!
% 2.95/1.86 % (18488)------------------------------
% 2.95/1.86 % (18488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.86 % (18488)Termination reason: Unknown
% 2.95/1.86 % (18488)Termination phase: Saturation
% 2.95/1.86
% 2.95/1.86 % (18488)Memory used [KB]: 9466
% 2.95/1.86 % (18488)Time elapsed: 0.104 s
% 2.95/1.86 % (18488)Instructions burned: 379 (million)
% 2.95/1.86 % (18488)------------------------------
% 2.95/1.86 % (18488)------------------------------
% 2.95/1.86 % (18495)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_18132 for (2997ds/127Mi)
% 2.95/1.87 % (18496)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_18132 for (2997ds/100Mi)
% 2.95/1.87 % (18484)First to succeed.
% 2.95/1.87 % (18484)Refutation found. Thanks to Tanya!
% 2.95/1.87 % SZS status Theorem for DTF2THF_18132
% 2.95/1.87 % SZS output start Proof for DTF2THF_18132
% 2.95/1.87 thf(func_def_0, type, in: $i > $i > $o).
% 2.95/1.87 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.95/1.87 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.95/1.87 thf(func_def_8, type, powerset: $i > $i).
% 2.95/1.87 thf(func_def_10, type, setunion: $i > $i).
% 2.95/1.87 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.95/1.87 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.95/1.87 thf(func_def_26, type, prop2set: $o > $i).
% 2.95/1.87 thf(func_def_36, type, nonempty: $i > $o).
% 2.95/1.87 thf(func_def_69, type, set2prop: $i > $o).
% 2.95/1.87 thf(func_def_88, type, subset: $i > $i > $o).
% 2.95/1.87 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.95/1.87 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.95/1.87 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.95/1.87 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.95/1.87 thf(func_def_135, type, regular: $i > $o).
% 2.95/1.87 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.95/1.87 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.95/1.87 thf(func_def_153, type, iskpair: $i > $o).
% 2.95/1.87 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.95/1.87 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.95/1.87 thf(func_def_177, type, singleton: $i > $o).
% 2.95/1.87 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.95/1.87 thf(func_def_184, type, atmost1p: $i > $o).
% 2.95/1.87 thf(func_def_185, type, atleast2p: $i > $o).
% 2.95/1.87 thf(func_def_186, type, atmost2p: $i > $o).
% 2.95/1.87 thf(func_def_187, type, upairsetp: $i > $o).
% 2.95/1.87 thf(func_def_191, type, kfst: $i > $i).
% 2.95/1.87 thf(func_def_203, type, ksnd: $i > $i).
% 2.95/1.87 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.95/1.87 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.95/1.87 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.95/1.87 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.95/1.87 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.95/1.87 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.95/1.87 thf(func_def_329, type, sP0: $i > $i > $i > $o > $o).
% 2.95/1.87 thf(func_def_332, type, sP3: $i > $i > $i > $i > $o).
% 2.95/1.87 thf(func_def_335, type, sP6: $i > $i > $o).
% 2.95/1.87 thf(func_def_336, type, sP7: $i > $o).
% 2.95/1.87 thf(func_def_337, type, sP8: $i > $i > $o).
% 2.95/1.87 thf(func_def_338, type, sP9: $i > $i > $o).
% 2.95/1.87 thf(func_def_339, type, sP10: $i > $i > $i > $o).
% 2.95/1.87 thf(func_def_345, type, sK16: $i > $o).
% 2.95/1.87 thf(func_def_347, type, sK18: ($i > $o) > $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_348, type, sK19: ($i > $o) > $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_355, type, sK26: $i > $o).
% 2.95/1.87 thf(func_def_359, type, sK30: ($i > $i) > $i > $i > $i).
% 2.95/1.87 thf(func_def_362, type, sK33: $i > $i).
% 2.95/1.87 thf(func_def_365, type, sK36: $i > $i > $o).
% 2.95/1.87 thf(func_def_386, type, sK57: $i > $i > $o).
% 2.95/1.87 thf(func_def_427, type, sK98: $i > $i).
% 2.95/1.87 thf(func_def_432, type, sK103: $i > $o).
% 2.95/1.87 thf(func_def_434, type, sK105: $i > $o).
% 2.95/1.87 thf(func_def_439, type, sK110: $i > $i > $o).
% 2.95/1.87 thf(func_def_472, type, sK143: $i > $o).
% 2.95/1.87 thf(func_def_474, type, sK145: $i > ($i > $o) > $i > $i).
% 2.95/1.87 thf(func_def_483, type, sK154: $i > $i > $i).
% 2.95/1.87 thf(func_def_498, type, sK169: $i > ($i > $o) > $i).
% 2.95/1.87 thf(func_def_499, type, sK170: $i > $o).
% 2.95/1.87 thf(func_def_502, type, sK173: $i > ($i > $o) > $i).
% 2.95/1.87 thf(func_def_503, type, sK174: $i > $o).
% 2.95/1.87 thf(func_def_508, type, sK179: $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_509, type, sK180: $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_516, type, sK187: $i > $o).
% 2.95/1.87 thf(func_def_522, type, sK193: $i > $i > $o).
% 2.95/1.87 thf(func_def_524, type, sK195: $i > $i).
% 2.95/1.87 thf(func_def_525, type, sK196: $i > $i).
% 2.95/1.87 thf(func_def_526, type, sK197: $i > ($i > $i > $o) > $i).
% 2.95/1.87 thf(func_def_527, type, sK198: $i > $i > ($i > $i > $o) > $i).
% 2.95/1.87 thf(func_def_528, type, sK199: $i > ($i > $i > $o) > $i).
% 2.95/1.87 thf(func_def_548, type, sK219: ($i > $i) > $i > $i > $i).
% 2.95/1.87 thf(func_def_551, type, sK222: $i > $i).
% 2.95/1.87 thf(func_def_561, type, sK232: $i > $o).
% 2.95/1.87 thf(func_def_564, type, sK235: $i > ($i > $o) > $i).
% 2.95/1.87 thf(func_def_565, type, sK236: $i > ($i > $o) > $i).
% 2.95/1.87 thf(func_def_576, type, sK247: $i > $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_577, type, sK248: $i > $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_578, type, sK249: $i > $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_579, type, sK250: $i > $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_588, type, sK259: $i > $o).
% 2.95/1.87 thf(func_def_590, type, sK261: ($i > $o) > $i).
% 2.95/1.87 thf(func_def_591, type, sK262: ($i > $o) > $i).
% 2.95/1.87 thf(func_def_615, type, sK286: $i > $o).
% 2.95/1.87 thf(func_def_616, type, sK287: $i > ($i > $o) > $i).
% 2.95/1.87 thf(func_def_617, type, sK288: $i > $o).
% 2.95/1.87 thf(func_def_622, type, sK293: $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_630, type, sK301: $i > ($i > $o) > $i).
% 2.95/1.87 thf(func_def_631, type, sK302: $i > $o).
% 2.95/1.87 thf(func_def_703, type, sK374: $i > $o).
% 2.95/1.87 thf(func_def_704, type, sK375: $i > $i).
% 2.95/1.87 thf(func_def_705, type, sK376: ($i > $o) > $i).
% 2.95/1.87 thf(func_def_709, type, sK380: $i > $i).
% 2.95/1.87 thf(func_def_727, type, sK398: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.95/1.87 thf(func_def_728, type, sK399: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.95/1.87 thf(func_def_731, type, sK402: $i > $o).
% 2.95/1.87 thf(func_def_732, type, sK403: $i > $o).
% 2.95/1.87 thf(func_def_735, type, sK406: $i > $i).
% 2.95/1.87 thf(func_def_739, type, sK410: $i > $i > $i).
% 2.95/1.87 thf(func_def_743, type, sK414: $i > $i).
% 2.95/1.87 thf(func_def_746, type, sK417: $i > $o).
% 2.95/1.87 thf(func_def_755, type, sK426: $i > $i > $i).
% 2.95/1.87 thf(func_def_758, type, sK429: $i > $i > $o).
% 2.95/1.87 thf(func_def_766, type, sK437: $i > $o).
% 2.95/1.87 thf(func_def_768, type, sK439: ($i > $o) > $i > $i).
% 2.95/1.87 thf(func_def_769, type, sK440: ($i > $o) > $i > $i).
% 2.95/1.87 thf(func_def_770, type, sK441: $i > $o).
% 2.95/1.87 thf(func_def_774, type, sK445: $i > $o).
% 2.95/1.87 thf(func_def_779, type, sK450: $i > $i > $i).
% 2.95/1.87 thf(func_def_780, type, sK451: $i > $i > $i).
% 2.95/1.87 thf(func_def_786, type, sK457: $i > $i > $o).
% 2.95/1.87 thf(func_def_788, type, sK459: $i > $o).
% 2.95/1.87 thf(func_def_789, type, sK460: $i > $o).
% 2.95/1.87 thf(func_def_790, type, sK461: ($i > $o) > ($i > $o) > $i).
% 2.95/1.87 thf(func_def_791, type, sK462: ($i > $o) > ($i > $o) > $i).
% 2.95/1.87 thf(func_def_803, type, sK474: $i > $o).
% 2.95/1.87 thf(func_def_813, type, sK484: $i > $o).
% 2.95/1.87 thf(func_def_815, type, sK486: $i > $o).
% 2.95/1.87 thf(func_def_816, type, sK487: $i > $i).
% 2.95/1.87 thf(func_def_817, type, sK488: ($i > $o) > $i).
% 2.95/1.87 thf(func_def_821, type, sK492: $i > $o).
% 2.95/1.87 thf(func_def_823, type, sK494: $i > ($i > $o) > $i).
% 2.95/1.87 thf(func_def_827, type, sK498: $i > $i > $o).
% 2.95/1.87 thf(func_def_844, type, sK515: $i > $o).
% 2.95/1.87 thf(func_def_850, type, sK521: $i > $i > $i).
% 2.95/1.87 thf(func_def_875, type, sK546: $i > $o).
% 2.95/1.87 thf(func_def_886, type, sK557: $i > $o).
% 2.95/1.87 thf(func_def_887, type, sK558: $i > $i > $i).
% 2.95/1.87 thf(func_def_888, type, sK559: $i > $i > $i).
% 2.95/1.87 thf(func_def_889, type, sK560: $i > $i > $i).
% 2.95/1.87 thf(func_def_890, type, sK561: $i > $i > $i).
% 2.95/1.87 thf(func_def_891, type, sK562: $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_892, type, sK563: $i > $i).
% 2.95/1.87 thf(func_def_893, type, sK564: $i > $i).
% 2.95/1.87 thf(func_def_894, type, sK565: $i > $i).
% 2.95/1.87 thf(func_def_895, type, sK566: $i > $i).
% 2.95/1.87 thf(func_def_896, type, sK567: $i > $i > $i).
% 2.95/1.87 thf(func_def_897, type, sK568: $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_898, type, sK569: $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_899, type, sK570: $i > $i > $i).
% 2.95/1.87 thf(func_def_900, type, sK571: $i > $i > $i).
% 2.95/1.87 thf(func_def_902, type, sK573: $i > $i).
% 2.95/1.87 thf(func_def_903, type, sK574: $i > $i).
% 2.95/1.87 thf(func_def_904, type, sK575: $i > $i).
% 2.95/1.87 thf(func_def_905, type, sK576: $i > $i > $i).
% 2.95/1.87 thf(func_def_906, type, sK577: $i > $i > $i).
% 2.95/1.87 thf(func_def_916, type, sK587: $o > $i > $i > $i).
% 2.95/1.87 thf(func_def_932, type, sK603: $i > ($i > $o) > $i).
% 2.95/1.87 thf(func_def_933, type, sK604: $i > $o).
% 2.95/1.87 thf(func_def_938, type, sK609: $i > $o).
% 2.95/1.87 thf(func_def_941, type, sK612: $i > ($i > $o) > $i).
% 2.95/1.87 thf(func_def_967, type, sK638: $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_968, type, sK639: $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_978, type, sK649: $i > $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_982, type, sK653: $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_988, type, sK659: $i > $o).
% 2.95/1.87 thf(func_def_989, type, sK660: $i > $o).
% 2.95/1.87 thf(func_def_990, type, sK661: ($i > $o) > ($i > $o) > $i).
% 2.95/1.87 thf(func_def_991, type, sK662: ($i > $o) > ($i > $o) > $i).
% 2.95/1.87 thf(func_def_1011, type, sK682: $i > $o).
% 2.95/1.87 thf(func_def_1013, type, sK684: ($i > $o) > $i > $i).
% 2.95/1.87 thf(func_def_1028, type, sK699: $i > $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_1029, type, sK700: $i > $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_1066, type, sK737: $i > $i).
% 2.95/1.87 thf(func_def_1068, type, sK739: $i > ($i > $i) > $i > $i).
% 2.95/1.87 thf(func_def_1084, type, sK755: $i > $i).
% 2.95/1.87 thf(func_def_1085, type, sK756: ($i > $i) > $i > $i > $i).
% 2.95/1.87 thf(func_def_1086, type, sK757: $i > $o).
% 2.95/1.87 thf(func_def_1089, type, sK760: $i > ($i > $o) > $i).
% 2.95/1.87 thf(func_def_1090, type, sK761: ($i > $o) > $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_1091, type, sK762: ($i > $o) > $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_1095, type, sK766: $i > $o).
% 2.95/1.87 thf(func_def_1106, type, sK777: $i > $i > $i).
% 2.95/1.87 thf(func_def_1114, type, sK785: $i > $i).
% 2.95/1.87 thf(func_def_1115, type, sK786: ($i > $i) > $i > $i > $i).
% 2.95/1.87 thf(func_def_1139, type, sK810: $i > $i > $i).
% 2.95/1.87 thf(func_def_1144, type, sK815: $i > $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_1146, type, sK817: $i > $i).
% 2.95/1.87 thf(func_def_1154, type, sK825: $i > $i > $i).
% 2.95/1.87 thf(func_def_1161, type, sK832: $i > $i > $i > $i > $i).
% 2.95/1.87 thf(func_def_1162, type, sK833: $i > $o).
% 2.95/1.87 thf(func_def_1163, type, sK834: ($i > $o) > $i).
% 2.95/1.87 thf(func_def_1195, type, sK866: ($i > $i) > $i > $i > $i).
% 2.95/1.87 thf(func_def_1198, type, sK869: $i > $i).
% 2.95/1.87 thf(func_def_1210, type, sK881: $i > $i > $o).
% 2.95/1.87 thf(func_def_1216, type, sK887: $i > $i).
% 2.95/1.87 thf(func_def_1219, type, sK890: $i > $i).
% 2.95/1.87 thf(func_def_1230, type, ph901: !>[X0: $tType]:(X0)).
% 2.95/1.87 thf(f4548,plain,(
% 2.95/1.87 $false),
% 2.95/1.87 inference(trivial_inequality_removal,[status(thm)],[f4539])).
% 2.95/1.87 thf(f4539,plain,(
% 2.95/1.87 ($true != $true)),
% 2.95/1.87 inference(superposition,[status(thm)],[f4433,f3743])).
% 2.95/1.87 thf(f3743,plain,(
% 2.95/1.87 ((subset @ sK687 @ sK690) = $true)),
% 2.95/1.87 inference(cnf_transformation,[status(thm)],[f2296])).
% 2.95/1.87 thf(f2296,plain,(
% 2.95/1.87 (binintersectSubset2 = $true) & (notdexE = $true) & (setukpairinjL = $true) & (dpsetconstrEL2 = $true) & (doubleComplementE1 = $true) & (ap2apEq2 = $true) & (binintersectSubset3 = $true) & (setunionsingleton = $true) & (binunionEcases = $true) & (doubleComplementSub1 = $true) & (powersetsubset = $true) & (binunionE = $true) & (binintersectI = $true) & (notsubsetI = $true) & (cartprodmempair = $true) & (binunionTILcontra = $true) & (dsetconstrEL = $true) & (complementTnotintersectT = $true) & (lam2lamEq = $true) & (woz13rule1 = $true) & (setminusIRneg = $true) & (eqimpsubset2 = $true) & (contrasubsetT1 = $true) & (setminusSubset2 = $true) & (demorgan2 = $true) & (uniqinunit = $true) & (kpairiskpair = $true) & (notdallE = $true) & (complementTE1 = $true) & (omegaIndAx = $true) & (dsetconstrI = $true) & (powersetE = $true) & (emptysetsubset = $true) & (beta2 = $true) & (ubforcartprodlem2 = $true) & (descr__Cong = $true) & (contraSubsetComplement = $true) & (noeltsimpempty = $true) & (ex1I = $true) & (exuE3u = $true) & (vacuousDall = $true) & (setukpairinjR11 = $true) & (emptysetE = $true) & (replAx = $true) & (exuE2 = $true) & (ap2p = $true) & (setadjoinIR = $true) & (dpsetconstrI = $true) & (setunionE2 = $true) & (binunionT_lem = $true) & (setunionAx = $true) & (setukpairinjR = $true) & (cartprodpairin = $true) & (binintersectLsub = $true) & (setoftrueEq = $true) & (iftrueProp1 = $true) & (ubforcartprodlem3 = $true) & (dpsetconstrEL1 = $true) & (setbeta = $true) & (exuI2 = $true) & (complementUnionInPowersetComplement = $true) & (nonemptyE1 = $true) & (upairset2IR = $true) & (in__Cong = $true) & (cartprodsndpairEq = $true) & (disjointsetsI1 = $true) & (upairequniteq = $true) & (complementT_lem = $true) & (exuI1 = $true) & (symdiffI2 = $true) & (complementTcontraSubset = $true) & (inIntersectImpInUnion = $true) & (binunionLsub = $true) & (upairset2E = $true) & (cartprodfstin = $true) & (nonemptyImpWitness = $true) & (binunionIL = $true) & (funcext = $true) & (infuncsetfunc = $true) & (binintersectTERcontra = $true) & (doubleComplementSub2 = $true) & (inPowerset = $true) & (exuEu = $true) & ((($true = (in @ sK689 @ (powerset @ sK688))) & (($true = (in @ sK690 @ (powerset @ sK688))) & ($true != (subset @ sK687 @ (binintersect @ sK689 @ sK690))) & ($true = (subset @ sK687 @ sK689)) & ((subset @ sK687 @ sK690) = $true))) & ((in @ sK687 @ (powerset @ sK688)) = $true)) & (funcinfuncset = $true) & (setunionI = $true) & (eta2 = $true) & (ubforcartprodlem1 = $true) & (setunionE = $true) & (upairinpowunion = $true) & (dsetconstrER = $true) & (complementImpComplementIntersect = $true) & (setextT = $true) & (setminusILneg = $true) & (powersetT_lem = $true) & (kpairp = $true) & (singletonsswitch = $true) & (binintersectSubset5 = $true) & (dsetconstr__Cong = $true) & (woz13rule2 = $true) & (brelnall2 = $true) & (binunionIR = $true) & (demorgan2b = $true) & (setminusELneg = $true) & (emptyinunitempty = $true) & (doubleComplementI1 = $true) & (demorgan2a1 = $true) & (lamProp = $true) & (secondinupair = $true) & (funcGraphProp4 = $true) & (dpsetconstrSub = $true) & (subsetemptysetimpeq = $true) & (funcext2 = $true) & (setminusT_lem = $true) & (demorgan1a = $true) & (funcImageSingleton = $true) & (cartprodpairmemEL = $true) & (setminusLsub = $true) & (omegaSAx = $true) & (setukpairinjR12 = $true) & (doubleComplementEq = $true) & (contrasubsetT = $true) & (setadjoin__Cong = $true) & (exuI3 = $true) & (setminusSubset1 = $true) & (binunionRsub = $true) & (powersetTI1 = $true) & (omega0Ax = $true) & (notequalI1 = $true) & (emptyInPowerset = $true) & (notequalI2 = $true) & (symdiffE = $true) & (subsetI1 = $true) & (lam2p = $true) & (sepInPowerset = $true) & (binintersectTELcontra = $true) & (emptyI = $true) & (symdiffIneg1 = $true) & (demorgan2a2 = $true) & (inComplementUnionImpInComplement1 = $true) & (setunionsingleton2 = $true) & (powersetAx = $true) & (symdiffI1 = $true) & (kfstsingleton = $true) & (binintersectER = $true) & (eqinunit = $true) & (nonemptyI1 = $true) & (upairsubunion = $true) & (contrasubsetT2 = $true) & (exu__Cong = $true) & (iffalseProp2 = $true) & (contrasubsetT3 = $true) & (ifp = $true) & (singletoninpowunion = $true) & (inComplementUnionImpNotIn1 = $true) & (complementInPowersetComplementIntersect = $true) & (setminusERneg = $true) & (complementTI1 = $true) & (ifSingleton = $true) & (cartprodpairmemER = $true) & (subsetTI = $true) & (subPowSU = $true) & (ex1I2 = $true) & (singletoninpowerset = $true) & (iffalse = $true) & (iftrue = $true) & (notinsingleton = $true) & (demorgan2a = $true) & (demorgan2b2 = $true) & (subsetTrans = $true) & (powersetI1 = $true) & (prop2set2propI = $true) & (inIntersectImpInUnion2 = $true) & (cartprodmempair1 = $true) & (setadjoinOr = $true) & (ex1E2 = $true) & (setOfPairsIsBReln = $true) & (setukpairinjL2 = $true) & (upairsetIL = $true) & (subsetE2 = $true) & (subsetI2 = $true) & (apProp = $true) & (theprop = $true) & (binunionTEcontra = $true) & (demorgan1 = $true) & (cartprodpairsurjEq = $true) & (cartprodmempaircEq = $true) & (kfstpairEq = $true) & (cartprodsndin = $true) & (intersectInPowersetIntersectUnions = $true) & (setminusER = $true) & (binintersectSubset4 = $true) & (powersetTE1 = $true) & (subset2powerset = $true) & (dpsetconstrERa = $true) & (brelnall1 = $true) & (eta1 = $true) & (funcGraphProp1 = $true) & (setukpairIR = $true) & (setadjoinIL = $true) & (app = $true) & (dpsetconstrER = $true) & (beta1 = $true) & (emptysetAx = $true) & (upairsetE = $true) & (funcGraphProp3 = $true) & (notinemptyset = $true) & (setadjoinSub2 = $true) & (binunionTE = $true) & (binintersectRsub = $true) & (setextsub = $true) & (binintersectT_lem = $true) & (setadjoinE = $true) & (binintersectSubset1 = $true) & (emptyE1 = $true) & (nonemptyI = $true) & (demorgan1b = $true) & (emptyinPowerset = $true) & (ap2apEq1 = $true) & (singletonsuniq = $true) & (exuE3e = $true) & (prop2setI = $true) & (setextAx = $true) & (upairsetIR = $true) & (prop2setE = $true) & (setunionsingleton1 = $true) & (setunion__Cong = $true) & (complementSubsetComplementIntersect = $true) & (powerset__Cong = $true) & (eqimpsubset1 = $true) & (sepSubset = $true) & (binunionTIRcontra = $true) & (quantDeMorgan1 = $true) & (emptysetimpfalse = $true) & (ex1E1 = $true) & (setadjoinSub = $true) & (quantDeMorgan2 = $true) & (setadjoinAx = $true) & (foundationAx = $true) & (iftrueorfalse = $true) & (powersetE1 = $true) & (setukpairinjR2 = $true) & (woz13rule0 = $true) & (singletonprop = $true) & (funcGraphProp2 = $true) & (setext = $true) & (setukpairIL = $true) & (quantDeMorgan3 = $true) & (emptyset__Cong = $true) & (setminusEL = $true) & (setukpairinjR1 = $true) & (setminusI = $true) & (theeq = $true) & (inIntersectImpInIntersectUnions = $true) & (subbreln = $true) & (wellorderingAx = $true) & (funcextLem = $true) & (subsetRefl = $true) & (iftrueProp2 = $true) & (descrp = $true) & (kpairsurjEq = $true) & (bs114d = $true) & (symdiffIneg2 = $true) & (quantDeMorgan4 = $true) & (setukpairinjL1 = $true) & (ksndpairEq = $true) & (ksndsingleton = $true) & (lamp = $true) & (eqbreln = $true) & (singletonsubset = $true) & (cartprodfstpairEq = $true) & (omega__Cong = $true) & (binintersectEL = $true) & (iffalseProp1 = $true) & (subsetE = $true) & (inCongP = $true) & (exuE1 = $true) & (powersetI = $true)),
% 2.95/1.87 inference(skolemisation,[status(esa),new_symbols(skolem,[sK687,sK688,sK689,sK690])],[f1109,f2295,f2294,f2293])).
% 2.95/1.87 thf(f2293,plain,(
% 2.95/1.87 ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (((in @ X2 @ (powerset @ X1)) = $true) & ? [X3 : $i] : (($true = (in @ X3 @ (powerset @ X1))) & ((subset @ X0 @ (binintersect @ X2 @ X3)) != $true) & ($true = (subset @ X0 @ X2)) & ((subset @ X0 @ X3) = $true))) & ($true = (in @ X0 @ (powerset @ X1)))) => (? [X2 : $i] : (($true = (in @ X2 @ (powerset @ sK688))) & ? [X3 : $i] : (($true = (in @ X3 @ (powerset @ sK688))) & ($true != (subset @ sK687 @ (binintersect @ X2 @ X3))) & ($true = (subset @ sK687 @ X2)) & ($true = (subset @ sK687 @ X3)))) & ((in @ sK687 @ (powerset @ sK688)) = $true))),
% 2.95/1.87 introduced(definition,[],[choice_axiom])).
% 2.95/1.87 thf(f2294,plain,(
% 2.95/1.87 ? [X2 : $i] : (($true = (in @ X2 @ (powerset @ sK688))) & ? [X3 : $i] : (($true = (in @ X3 @ (powerset @ sK688))) & ($true != (subset @ sK687 @ (binintersect @ X2 @ X3))) & ($true = (subset @ sK687 @ X2)) & ($true = (subset @ sK687 @ X3)))) => (($true = (in @ sK689 @ (powerset @ sK688))) & ? [X3 : $i] : (($true = (in @ X3 @ (powerset @ sK688))) & ((subset @ sK687 @ (binintersect @ sK689 @ X3)) != $true) & ($true = (subset @ sK687 @ sK689)) & ($true = (subset @ sK687 @ X3))))),
% 2.95/1.87 introduced(definition,[],[choice_axiom])).
% 2.95/1.87 thf(f2295,plain,(
% 2.95/1.87 ? [X3 : $i] : (($true = (in @ X3 @ (powerset @ sK688))) & ((subset @ sK687 @ (binintersect @ sK689 @ X3)) != $true) & ($true = (subset @ sK687 @ sK689)) & ($true = (subset @ sK687 @ X3))) => (($true = (in @ sK690 @ (powerset @ sK688))) & ($true != (subset @ sK687 @ (binintersect @ sK689 @ sK690))) & ($true = (subset @ sK687 @ sK689)) & ((subset @ sK687 @ sK690) = $true))),
% 2.95/1.87 introduced(definition,[],[choice_axiom])).
% 2.95/1.87 thf(f1109,plain,(
% 2.95/1.87 (binintersectSubset2 = $true) & (notdexE = $true) & (setukpairinjL = $true) & (dpsetconstrEL2 = $true) & (doubleComplementE1 = $true) & (ap2apEq2 = $true) & (binintersectSubset3 = $true) & (setunionsingleton = $true) & (binunionEcases = $true) & (doubleComplementSub1 = $true) & (powersetsubset = $true) & (binunionE = $true) & (binintersectI = $true) & (notsubsetI = $true) & (cartprodmempair = $true) & (binunionTILcontra = $true) & (dsetconstrEL = $true) & (complementTnotintersectT = $true) & (lam2lamEq = $true) & (woz13rule1 = $true) & (setminusIRneg = $true) & (eqimpsubset2 = $true) & (contrasubsetT1 = $true) & (setminusSubset2 = $true) & (demorgan2 = $true) & (uniqinunit = $true) & (kpairiskpair = $true) & (notdallE = $true) & (complementTE1 = $true) & (omegaIndAx = $true) & (dsetconstrI = $true) & (powersetE = $true) & (emptysetsubset = $true) & (beta2 = $true) & (ubforcartprodlem2 = $true) & (descr__Cong = $true) & (contraSubsetComplement = $true) & (noeltsimpempty = $true) & (ex1I = $true) & (exuE3u = $true) & (vacuousDall = $true) & (setukpairinjR11 = $true) & (emptysetE = $true) & (replAx = $true) & (exuE2 = $true) & (ap2p = $true) & (setadjoinIR = $true) & (dpsetconstrI = $true) & (setunionE2 = $true) & (binunionT_lem = $true) & (setunionAx = $true) & (setukpairinjR = $true) & (cartprodpairin = $true) & (binintersectLsub = $true) & (setoftrueEq = $true) & (iftrueProp1 = $true) & (ubforcartprodlem3 = $true) & (dpsetconstrEL1 = $true) & (setbeta = $true) & (exuI2 = $true) & (complementUnionInPowersetComplement = $true) & (nonemptyE1 = $true) & (upairset2IR = $true) & (in__Cong = $true) & (cartprodsndpairEq = $true) & (disjointsetsI1 = $true) & (upairequniteq = $true) & (complementT_lem = $true) & (exuI1 = $true) & (symdiffI2 = $true) & (complementTcontraSubset = $true) & (inIntersectImpInUnion = $true) & (binunionLsub = $true) & (upairset2E = $true) & (cartprodfstin = $true) & (nonemptyImpWitness = $true) & (binunionIL = $true) & (funcext = $true) & (infuncsetfunc = $true) & (binintersectTERcontra = $true) & (doubleComplementSub2 = $true) & (inPowerset = $true) & (exuEu = $true) & ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (((in @ X2 @ (powerset @ X1)) = $true) & ? [X3 : $i] : (($true = (in @ X3 @ (powerset @ X1))) & ((subset @ X0 @ (binintersect @ X2 @ X3)) != $true) & ($true = (subset @ X0 @ X2)) & ((subset @ X0 @ X3) = $true))) & ($true = (in @ X0 @ (powerset @ X1)))) & (funcinfuncset = $true) & (setunionI = $true) & (eta2 = $true) & (ubforcartprodlem1 = $true) & (setunionE = $true) & (upairinpowunion = $true) & (dsetconstrER = $true) & (complementImpComplementIntersect = $true) & (setextT = $true) & (setminusILneg = $true) & (powersetT_lem = $true) & (kpairp = $true) & (singletonsswitch = $true) & (binintersectSubset5 = $true) & (dsetconstr__Cong = $true) & (woz13rule2 = $true) & (brelnall2 = $true) & (binunionIR = $true) & (demorgan2b = $true) & (setminusELneg = $true) & (emptyinunitempty = $true) & (doubleComplementI1 = $true) & (demorgan2a1 = $true) & (lamProp = $true) & (secondinupair = $true) & (funcGraphProp4 = $true) & (dpsetconstrSub = $true) & (subsetemptysetimpeq = $true) & (funcext2 = $true) & (setminusT_lem = $true) & (demorgan1a = $true) & (funcImageSingleton = $true) & (cartprodpairmemEL = $true) & (setminusLsub = $true) & (omegaSAx = $true) & (setukpairinjR12 = $true) & (doubleComplementEq = $true) & (contrasubsetT = $true) & (setadjoin__Cong = $true) & (exuI3 = $true) & (setminusSubset1 = $true) & (binunionRsub = $true) & (powersetTI1 = $true) & (omega0Ax = $true) & (notequalI1 = $true) & (emptyInPowerset = $true) & (notequalI2 = $true) & (symdiffE = $true) & (subsetI1 = $true) & (lam2p = $true) & (sepInPowerset = $true) & (binintersectTELcontra = $true) & (emptyI = $true) & (symdiffIneg1 = $true) & (demorgan2a2 = $true) & (inComplementUnionImpInComplement1 = $true) & (setunionsingleton2 = $true) & (powersetAx = $true) & (symdiffI1 = $true) & (kfstsingleton = $true) & (binintersectER = $true) & (eqinunit = $true) & (nonemptyI1 = $true) & (upairsubunion = $true) & (contrasubsetT2 = $true) & (exu__Cong = $true) & (iffalseProp2 = $true) & (contrasubsetT3 = $true) & (ifp = $true) & (singletoninpowunion = $true) & (inComplementUnionImpNotIn1 = $true) & (complementInPowersetComplementIntersect = $true) & (setminusERneg = $true) & (complementTI1 = $true) & (ifSingleton = $true) & (cartprodpairmemER = $true) & (subsetTI = $true) & (subPowSU = $true) & (ex1I2 = $true) & (singletoninpowerset = $true) & (iffalse = $true) & (iftrue = $true) & (notinsingleton = $true) & (demorgan2a = $true) & (demorgan2b2 = $true) & (subsetTrans = $true) & (powersetI1 = $true) & (prop2set2propI = $true) & (inIntersectImpInUnion2 = $true) & (cartprodmempair1 = $true) & (setadjoinOr = $true) & (ex1E2 = $true) & (setOfPairsIsBReln = $true) & (setukpairinjL2 = $true) & (upairsetIL = $true) & (subsetE2 = $true) & (subsetI2 = $true) & (apProp = $true) & (theprop = $true) & (binunionTEcontra = $true) & (demorgan1 = $true) & (cartprodpairsurjEq = $true) & (cartprodmempaircEq = $true) & (kfstpairEq = $true) & (cartprodsndin = $true) & (intersectInPowersetIntersectUnions = $true) & (setminusER = $true) & (binintersectSubset4 = $true) & (powersetTE1 = $true) & (subset2powerset = $true) & (dpsetconstrERa = $true) & (brelnall1 = $true) & (eta1 = $true) & (funcGraphProp1 = $true) & (setukpairIR = $true) & (setadjoinIL = $true) & (app = $true) & (dpsetconstrER = $true) & (beta1 = $true) & (emptysetAx = $true) & (upairsetE = $true) & (funcGraphProp3 = $true) & (notinemptyset = $true) & (setadjoinSub2 = $true) & (binunionTE = $true) & (binintersectRsub = $true) & (setextsub = $true) & (binintersectT_lem = $true) & (setadjoinE = $true) & (binintersectSubset1 = $true) & (emptyE1 = $true) & (nonemptyI = $true) & (demorgan1b = $true) & (emptyinPowerset = $true) & (ap2apEq1 = $true) & (singletonsuniq = $true) & (exuE3e = $true) & (prop2setI = $true) & (setextAx = $true) & (upairsetIR = $true) & (prop2setE = $true) & (setunionsingleton1 = $true) & (setunion__Cong = $true) & (complementSubsetComplementIntersect = $true) & (powerset__Cong = $true) & (eqimpsubset1 = $true) & (sepSubset = $true) & (binunionTIRcontra = $true) & (quantDeMorgan1 = $true) & (emptysetimpfalse = $true) & (ex1E1 = $true) & (setadjoinSub = $true) & (quantDeMorgan2 = $true) & (setadjoinAx = $true) & (foundationAx = $true) & (iftrueorfalse = $true) & (powersetE1 = $true) & (setukpairinjR2 = $true) & (woz13rule0 = $true) & (singletonprop = $true) & (funcGraphProp2 = $true) & (setext = $true) & (setukpairIL = $true) & (quantDeMorgan3 = $true) & (emptyset__Cong = $true) & (setminusEL = $true) & (setukpairinjR1 = $true) & (setminusI = $true) & (theeq = $true) & (inIntersectImpInIntersectUnions = $true) & (subbreln = $true) & (wellorderingAx = $true) & (funcextLem = $true) & (subsetRefl = $true) & (iftrueProp2 = $true) & (descrp = $true) & (kpairsurjEq = $true) & (bs114d = $true) & (symdiffIneg2 = $true) & (quantDeMorgan4 = $true) & (setukpairinjL1 = $true) & (ksndpairEq = $true) & (ksndsingleton = $true) & (lamp = $true) & (eqbreln = $true) & (singletonsubset = $true) & (cartprodfstpairEq = $true) & (omega__Cong = $true) & (binintersectEL = $true) & (iffalseProp1 = $true) & (subsetE = $true) & (inCongP = $true) & (exuE1 = $true) & (powersetI = $true)),
% 2.95/1.87 inference(flattening,[status(thm)],[f1108])).
% 2.95/1.87 thf(f1108,plain,(
% 2.95/1.87 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (((((subset @ X0 @ (binintersect @ X2 @ X3)) != $true) & ((subset @ X0 @ X3) = $true)) & ($true = (subset @ X0 @ X2))) & ($true = (in @ X3 @ (powerset @ X1)))) & ((in @ X2 @ (powerset @ X1)) = $true)) & ($true = (in @ X0 @ (powerset @ X1)))) & (woz13rule2 = $true)) & (woz13rule1 = $true)) & (woz13rule0 = $true)) & (demorgan2 = $true)) & (demorgan2b = $true)) & (demorgan2b2 = $true)) & (demorgan2a = $true)) & (demorgan1 = $true)) & (demorgan1b = $true)) & (demorgan1a = $true)) & (demorgan2a2 = $true)) & (complementUnionInPowersetComplement = $true)) & (demorgan2a1 = $true)) & (binunionTEcontra = $true)) & (binunionTE = $true)) & (inComplementUnionImpInComplement1 = $true)) & (inComplementUnionImpNotIn1 = $true)) & (intersectInPowersetIntersectUnions = $true)) & (inIntersectImpInIntersectUnions = $true)) & (inIntersectImpInUnion2 = $true)) & (inIntersectImpInUnion = $true)) & (binunionTIRcontra = $true)) & (binunionTILcontra = $true)) & (complementTcontraSubset = $true)) & (contraSubsetComplement = $true)) & (complementInPowersetComplementIntersect = $true)) & (complementSubsetComplementIntersect = $true)) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.95/1.87 inference(ennf_transformation,[status(thm)],[f654])).
% 2.95/1.87 thf(f654,plain,(
% 2.95/1.87 ~((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) => ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) => ! [X2 : $i] : (((in @ X2 @ (powerset @ X1)) = $true) => ! [X3 : $i] : (($true = (in @ X3 @ (powerset @ X1))) => (($true = (subset @ X0 @ X2)) => (((subset @ X0 @ X3) = $true) => ((subset @ X0 @ (binintersect @ X2 @ X3)) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.95/1.87 inference(fool_elimination,[status(thm)],[f653])).
% 2.95/1.87 thf(f653,plain,(
% 2.95/1.87 ~(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 => ! [X0 : $i,X1 : $i] : ((in @ X0 @ (powerset @ X1)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X1)) => ! [X3 : $i] : ((in @ X3 @ (powerset @ X1)) => ((subset @ X0 @ X2) => ((subset @ X0 @ X3) => (subset @ X0 @ (binintersect @ X2 @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.95/1.87 inference(rectify,[status(thm)],[f283])).
% 2.95/1.87 thf(f283,negated_conjecture,(
% 2.95/1.87 ~(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 => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X17 : $i] : ((in @ X17 @ (powerset @ X3)) => ((subset @ X11 @ X16) => ((subset @ X11 @ X17) => (subset @ X11 @ (binintersect @ X16 @ X17))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.95/1.87 inference(negated_conjecture,[status(cth)],[f282])).
% 2.95/1.87 thf(f282,conjecture,(
% 2.95/1.87 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 => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X17 : $i] : ((in @ X17 @ (powerset @ X3)) => ((subset @ X11 @ X16) => ((subset @ X11 @ X17) => (subset @ X11 @ (binintersect @ X16 @ X17)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.95/1.87 file('/export/starexec/sandbox/tmp/tmp.gJXxL4HX93/DTF2THF_18132.p',woz13rule3)).
% 2.95/1.87 thf(f4433,plain,(
% 2.95/1.87 ((subset @ sK687 @ sK690) != $true)),
% 2.95/1.87 inference(trivial_inequality_removal,[status(thm)],[f4432])).
% 2.95/1.87 thf(f4432,plain,(
% 2.95/1.87 ((subset @ sK687 @ sK690) != $true) | ($true != $true)),
% 2.95/1.87 inference(forward_demodulation,[status(thm)],[f4431,f3744])).
% 2.95/1.87 thf(f3744,plain,(
% 2.95/1.87 ($true = (subset @ sK687 @ sK689))),
% 2.95/1.87 inference(cnf_transformation,[status(thm)],[f2296])).
% 2.95/1.87 thf(f4431,plain,(
% 2.95/1.87 ($true != (subset @ sK687 @ sK689)) | ((subset @ sK687 @ sK690) != $true)),
% 2.95/1.87 inference(trivial_inequality_removal,[status(thm)],[f4430])).
% 2.95/1.87 thf(f4430,plain,(
% 2.95/1.87 ($true != $true) | ($true != (subset @ sK687 @ sK689)) | ((subset @ sK687 @ sK690) != $true)),
% 2.95/1.87 inference(forward_demodulation,[status(thm)],[f4412,f3728])).
% 2.95/1.87 thf(f3728,plain,(
% 2.95/1.87 (binintersectSubset5 = $true)),
% 2.95/1.87 inference(cnf_transformation,[status(thm)],[f2296])).
% 2.95/1.87 thf(f4412,plain,(
% 2.95/1.87 ((subset @ sK687 @ sK690) != $true) | (binintersectSubset5 != $true) | ($true != (subset @ sK687 @ sK689))),
% 2.95/1.87 inference(trivial_inequality_removal,[status(thm)],[f4406])).
% 2.95/1.87 thf(f4406,plain,(
% 2.95/1.87 ($true != $true) | (binintersectSubset5 != $true) | ((subset @ sK687 @ sK690) != $true) | ($true != (subset @ sK687 @ sK689))),
% 2.95/1.87 inference(superposition,[status(thm)],[f3745,f3331])).
% 2.95/1.87 thf(f3331,plain,(
% 2.95/1.87 ( ! [X3 : $i,X4 : $i,X5 : $i] : (($true = (subset @ X4 @ (binintersect @ X3 @ X5))) | ((subset @ X4 @ X5) != $true) | ((subset @ X4 @ X3) != $true) | (binintersectSubset5 != $true)) )),
% 2.95/1.87 inference(cnf_transformation,[status(thm)],[f2068])).
% 2.95/1.87 thf(f2068,plain,(
% 2.95/1.87 ((binintersectSubset5 = $true) | (($true = (subset @ sK544 @ sK543)) & ($true = (subset @ sK544 @ sK545)) & ((subset @ sK544 @ (binintersect @ sK543 @ sK545)) != $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X4 @ X3) != $true) | ((subset @ X4 @ X5) != $true) | ($true = (subset @ X4 @ (binintersect @ X3 @ X5)))) | (binintersectSubset5 != $true))),
% 2.95/1.87 inference(skolemisation,[status(esa),new_symbols(skolem,[sK543,sK544,sK545])],[f2066,f2067])).
% 2.95/1.87 thf(f2067,plain,(
% 2.95/1.87 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (subset @ X1 @ X0)) & ($true = (subset @ X1 @ X2)) & ($true != (subset @ X1 @ (binintersect @ X0 @ X2)))) => (($true = (subset @ sK544 @ sK543)) & ($true = (subset @ sK544 @ sK545)) & ((subset @ sK544 @ (binintersect @ sK543 @ sK545)) != $true))),
% 2.95/1.87 introduced(definition,[],[choice_axiom])).
% 2.95/1.87 thf(f2066,plain,(
% 2.95/1.87 ((binintersectSubset5 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (subset @ X1 @ X0)) & ($true = (subset @ X1 @ X2)) & ($true != (subset @ X1 @ (binintersect @ X0 @ X2))))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X4 @ X3) != $true) | ((subset @ X4 @ X5) != $true) | ($true = (subset @ X4 @ (binintersect @ X3 @ X5)))) | (binintersectSubset5 != $true))),
% 2.95/1.87 inference(rectify,[status(thm)],[f2065])).
% 2.95/1.87 thf(f2065,plain,(
% 2.95/1.87 ((binintersectSubset5 = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (subset @ X1 @ X0)) & ($true = (subset @ X1 @ X2)) & ($true != (subset @ X1 @ (binintersect @ X0 @ X2))))) & (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (subset @ X1 @ X0)) | ($true != (subset @ X1 @ X2)) | ($true = (subset @ X1 @ (binintersect @ X0 @ X2)))) | (binintersectSubset5 != $true))),
% 2.95/1.87 inference(nnf_transformation,[status(thm)],[f1021])).
% 2.95/1.87 thf(f1021,plain,(
% 2.95/1.87 (binintersectSubset5 = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true != (subset @ X1 @ X0)) | ($true != (subset @ X1 @ X2)) | ($true = (subset @ X1 @ (binintersect @ X0 @ X2))))),
% 2.95/1.87 inference(flattening,[status(thm)],[f1020])).
% 2.95/1.87 thf(f1020,plain,(
% 2.95/1.87 ! [X0 : $i,X1 : $i,X2 : $i] : ((($true = (subset @ X1 @ (binintersect @ X0 @ X2))) | ($true != (subset @ X1 @ X2))) | ($true != (subset @ X1 @ X0))) <=> (binintersectSubset5 = $true)),
% 2.95/1.87 inference(ennf_transformation,[status(thm)],[f624])).
% 2.95/1.87 thf(f624,plain,(
% 2.95/1.87 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (subset @ X1 @ X0)) => (($true = (subset @ X1 @ X2)) => ($true = (subset @ X1 @ (binintersect @ X0 @ X2))))) <=> (binintersectSubset5 = $true)),
% 2.95/1.87 inference(fool_elimination,[status(thm)],[f623])).
% 2.95/1.87 thf(f623,plain,(
% 2.95/1.87 (! [X0 : $i,X1 : $i,X2 : $i] : ((subset @ X1 @ X0) => ((subset @ X1 @ X2) => (subset @ X1 @ (binintersect @ X0 @ X2)))) = binintersectSubset5)),
% 2.95/1.87 inference(rectify,[status(thm)],[f109])).
% 2.95/1.87 thf(f109,axiom,(
% 2.95/1.87 (! [X3 : $i,X5 : $i,X4 : $i] : ((subset @ X5 @ X3) => ((subset @ X5 @ X4) => (subset @ X5 @ (binintersect @ X3 @ X4)))) = binintersectSubset5)),
% 2.95/1.87 file('/export/starexec/sandbox/tmp/tmp.gJXxL4HX93/DTF2THF_18132.p',binintersectSubset5)).
% 2.95/1.87 thf(f3745,plain,(
% 2.95/1.87 ($true != (subset @ sK687 @ (binintersect @ sK689 @ sK690)))),
% 2.95/1.87 inference(cnf_transformation,[status(thm)],[f2296])).
% 2.95/1.87 % SZS output end Proof for DTF2THF_18132
% 2.95/1.87 % (18484)------------------------------
% 2.95/1.87 % (18484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.95/1.87 % (18484)Termination reason: Refutation
% 2.95/1.87
% 2.95/1.87 % (18484)Memory used [KB]: 10874
% 2.95/1.87 % (18484)Time elapsed: 0.133 s
% 2.95/1.87 % (18484)Instructions burned: 396 (million)
% 2.95/1.87 % (18484)------------------------------
% 2.95/1.87 % (18484)------------------------------
% 2.95/1.87 % (18460)Success in time 0.223 s
%------------------------------------------------------------------------------