%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU755^1 : 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 : n027.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 09:36:56 AM UTC 2026
% Result : Theorem 4.99s 2.21s
% Output : Refutation 4.99s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 15
% Syntax : Number of formulae : 86 ( 23 unt; 0 typ; 3 def)
% Number of atoms : 5143 (1365 equ; 0 cnn)
% Maximal formula atoms : 277 ( 59 avg)
% Number of connectives : 3035 ( 138 ~; 97 |; 894 &; 760 @)
% ( 16 <=>;1130 =>; 0 <=; 0 <~>)
% Maximal formula depth : 286 ( 27 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 481 ( 478 usr; 292 con; 0-4 aty)
% Number of variables : 181 ( 0 ^; 127 !; 54 ?; 181 :)
% 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_323,type,
sP0: $i > $i > $i > $o ).
thf(func_def_324,type,
sP1: $i > $i > $o ).
thf(func_def_325,type,
sP2: $i > $o ).
thf(func_def_326,type,
sP3: $i > $i > $o ).
thf(func_def_327,type,
sP4: $i > $i > $o ).
thf(func_def_329,type,
sP6: $i > $i > $i > $i > $o ).
thf(func_def_335,type,
sP12: $o > $i > $i > $i > $o ).
thf(func_def_339,type,
sK16: $i > $i > $i > $i ).
thf(func_def_357,type,
sK34: $i > $o ).
thf(func_def_362,type,
sK39: $i > $o ).
thf(func_def_363,type,
sK40: $i > $o ).
thf(func_def_364,type,
sK41: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_365,type,
sK42: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_374,type,
sK51: $i > $o ).
thf(func_def_381,type,
sK58: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_382,type,
sK59: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_383,type,
sK60: $i > $o ).
thf(func_def_384,type,
sK61: $i > $o ).
thf(func_def_393,type,
sK70: $i > ( $i > $o ) > $i ).
thf(func_def_394,type,
sK71: $i > $o ).
thf(func_def_405,type,
sK82: ( $i > $o ) > $i ).
thf(func_def_406,type,
sK83: $i > $o ).
thf(func_def_407,type,
sK84: $i > $i ).
thf(func_def_451,type,
sK128: $i > $o ).
thf(func_def_469,type,
sK146: $i > $o ).
thf(func_def_483,type,
sK160: $i > $o ).
thf(func_def_485,type,
sK162: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_486,type,
sK163: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_489,type,
sK166: $i > $i > $o ).
thf(func_def_492,type,
sK169: $i > $i ).
thf(func_def_494,type,
sK171: $i > $i ).
thf(func_def_505,type,
sK182: $i > $i > $i > $i ).
thf(func_def_509,type,
sK186: $i > $i > $i > $i ).
thf(func_def_522,type,
sK199: $i > $o ).
thf(func_def_527,type,
sK204: $i > $i > $o ).
thf(func_def_533,type,
sK210: $i > $o ).
thf(func_def_549,type,
sK226: $i > $o ).
thf(func_def_558,type,
sK235: $i > $o ).
thf(func_def_560,type,
sK237: $i > $i > $i ).
thf(func_def_561,type,
sK238: $i > $i > $i ).
thf(func_def_562,type,
sK239: $i > $i > $i ).
thf(func_def_563,type,
sK240: $i > $i > $i ).
thf(func_def_564,type,
sK241: $i > $i > $i > $i ).
thf(func_def_565,type,
sK242: $i > $i ).
thf(func_def_566,type,
sK243: $i > $i ).
thf(func_def_567,type,
sK244: $i > $i ).
thf(func_def_568,type,
sK245: $i > $i ).
thf(func_def_569,type,
sK246: $i > $i > $i ).
thf(func_def_570,type,
sK247: $i > $i > $i > $i ).
thf(func_def_571,type,
sK248: $i > $i > $i > $i ).
thf(func_def_572,type,
sK249: $i > $i > $i ).
thf(func_def_573,type,
sK250: $i > $i > $i ).
thf(func_def_574,type,
sK251: $i > $i ).
thf(func_def_575,type,
sK252: $i > $i > $i ).
thf(func_def_577,type,
sK254: $i > $i ).
thf(func_def_578,type,
sK255: $i > $i ).
thf(func_def_579,type,
sK256: $i > $i > $i ).
thf(func_def_591,type,
sK268: $i > $i > $i ).
thf(func_def_606,type,
sK283: $i > $i > $o ).
thf(func_def_617,type,
sK294: $i > $i > $i > $i > $i ).
thf(func_def_648,type,
sK325: $i > $o ).
thf(func_def_649,type,
sK326: ( $i > $o ) > $i > $i ).
thf(func_def_666,type,
sK343: $o > $i > $i > $i ).
thf(func_def_686,type,
sK363: $i > $i ).
thf(func_def_694,type,
sK371: $i > $i > $o ).
thf(func_def_695,type,
sK372: ( $i > $o ) > $i ).
thf(func_def_696,type,
sK373: ( $i > $o ) > $i ).
thf(func_def_697,type,
sK374: $i > $o ).
thf(func_def_709,type,
sK386: $i > $i > $i ).
thf(func_def_719,type,
sK396: $i > $i > $i ).
thf(func_def_724,type,
sK401: $i > $i > ( $i > $i ) > $i ).
thf(func_def_725,type,
sK402: $i > $i ).
thf(func_def_747,type,
sK424: ( $i > $o ) > $i ).
thf(func_def_748,type,
sK425: $i > $o ).
thf(func_def_767,type,
sK444: $i > $i > $o ).
thf(func_def_769,type,
sK446: $i > ( $i > $o ) > $i > $i ).
thf(func_def_771,type,
sK448: $i > $o ).
thf(func_def_776,type,
sK453: $i > $o ).
thf(func_def_783,type,
sK460: $i > $i ).
thf(func_def_793,type,
sK470: $i > $i > ( $i > $i ) > $i ).
thf(func_def_794,type,
sK471: $i > $i ).
thf(func_def_801,type,
sK478: $i > $i > $i > $i > $i ).
thf(func_def_802,type,
sK479: $i > $i > $i > $i > $i ).
thf(func_def_803,type,
sK480: $i > $i > $i > $i > $i ).
thf(func_def_804,type,
sK481: $i > $i > $i > $i > $i ).
thf(func_def_811,type,
sK488: $i > $i > $o ).
thf(func_def_812,type,
sK489: $i > $i ).
thf(func_def_813,type,
sK490: $i > $i ).
thf(func_def_814,type,
sK491: ( $i > $i > $o ) > $i > $i ).
thf(func_def_815,type,
sK492: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_816,type,
sK493: ( $i > $i > $o ) > $i > $i ).
thf(func_def_817,type,
sK494: ( $i > $o ) > $i > $i ).
thf(func_def_819,type,
sK496: $i > $o ).
thf(func_def_821,type,
sK498: $i > $o ).
thf(func_def_824,type,
sK501: $i > $i > $i ).
thf(func_def_832,type,
sK509: $i > $o ).
thf(func_def_834,type,
sK511: $i > ( $i > $o ) > $i ).
thf(func_def_835,type,
sK512: ( $i > $i ) > $i > $i > $i ).
thf(func_def_838,type,
sK515: $i > $i ).
thf(func_def_844,type,
sK521: $i > ( $i > $o ) > $i ).
thf(func_def_845,type,
sK522: $i > $o ).
thf(func_def_849,type,
sK526: $i > $i ).
thf(func_def_850,type,
sK527: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_851,type,
sK528: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_855,type,
sK532: $i > $o ).
thf(func_def_868,type,
sK545: $i > $i ).
thf(func_def_885,type,
sK562: $i > $o ).
thf(func_def_888,type,
sK565: $i > ( $i > $o ) > $i ).
thf(func_def_898,type,
sK575: $i > $i > $i > $i > $i ).
thf(func_def_904,type,
sK581: $i > ( $i > $i ) > $i > $i ).
thf(func_def_906,type,
sK583: $i > $i ).
thf(func_def_909,type,
sK586: $i > $i > $i ).
thf(func_def_930,type,
sK607: $i > $i ).
thf(func_def_934,type,
sK611: $i > $o ).
thf(func_def_936,type,
sK613: $i > $i > $i ).
thf(func_def_937,type,
sK614: $i > $i > $i ).
thf(func_def_953,type,
sK630: $i > $i > $i > $i ).
thf(func_def_962,type,
sK639: $i > $i ).
thf(func_def_963,type,
sK640: ( $i > $i ) > $i > $i > $i ).
thf(func_def_981,type,
sK658: $i > $o ).
thf(func_def_984,type,
sK661: $i > ( $i > $o ) > $i ).
thf(func_def_996,type,
sK673: ( $i > $o ) > $i ).
thf(func_def_997,type,
sK674: $i > $o ).
thf(func_def_998,type,
sK675: $i > $i ).
thf(func_def_1021,type,
sK698: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1024,type,
sK701: $i > $i ).
thf(func_def_1029,type,
sK706: $i > $o ).
thf(func_def_1032,type,
sK709: $i > ( $i > $o ) > $i ).
thf(func_def_1033,type,
sK710: $i > $o ).
thf(func_def_1050,type,
sK727: $i > $o ).
thf(func_def_1070,type,
sK747: $i > $i > $o ).
thf(func_def_1077,type,
sK754: $i > $o ).
thf(func_def_1080,type,
sK757: $i > ( $i > $o ) > $i ).
thf(func_def_1083,type,
sK760: $i > $i > $i ).
thf(func_def_1097,type,
sK774: $i > $o ).
thf(func_def_1100,type,
sK777: $i > ( $i > $o ) > $i ).
thf(func_def_1101,type,
sK778: $i > ( $i > $o ) > $i ).
thf(func_def_1106,type,
sK783: $i > $o ).
thf(func_def_1109,type,
sK786: $i > ( $i > $o ) > $i ).
thf(func_def_1110,type,
sK787: $i > ( $i > $o ) > $i ).
thf(func_def_1132,type,
sK809: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_1133,type,
sK810: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_1136,type,
sK813: $i > $o ).
thf(func_def_1137,type,
sK814: $i > $o ).
thf(func_def_1151,type,
sK828: $i > $i > $i > $i ).
thf(func_def_1152,type,
sK829: $i > $i > $i > $i ).
thf(func_def_1163,type,
sK840: $i > $i > $o ).
thf(func_def_1175,type,
sK852: $i > $i > $i ).
thf(func_def_1181,type,
sK858: $i > $i > $i > $i > $i ).
thf(func_def_1182,type,
sK859: $i > $i > $i > $i > $i ).
thf(func_def_1191,type,
sK868: $i > $i > $i > $i > $i ).
thf(func_def_1201,type,
ph878:
!>[X0: $tType] : X0 ).
thf(f4668,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f4328,f4394,f4622,f4643]) ).
thf(f4643,plain,
( spl877_13
| spl877_6
| ~ spl877_18 ),
inference(avatar_split_clause,[status(thm)],[f4642,f4438,f4283,f4365]) ).
thf(f4365,definition,
( spl877_13
<=> ( $true
= ( in @ sK279 @ sK278 ) ) ),
introduced(definition,[new_symbols(naming,[spl877_13])],[avatar_definition]) ).
thf(f4283,definition,
( spl877_6
<=> ( $true
= ( in @ sK279 @ sK276 ) ) ),
introduced(definition,[new_symbols(naming,[spl877_6])],[avatar_definition]) ).
thf(f4438,definition,
( spl877_18
<=> ( ( in @ sK279 @ ( binunion @ sK276 @ sK278 ) )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl877_18])],[avatar_definition]) ).
thf(f4642,plain,
( ( $true
= ( in @ sK279 @ sK278 ) )
| ( $true
= ( in @ sK279 @ sK276 ) )
| ~ spl877_18 ),
inference(trivial_inequality_removal,[status(thm)],[f4641]) ).
thf(f4641,plain,
( ( $true
= ( in @ sK279 @ sK278 ) )
| ( $true != $true )
| ( $true
= ( in @ sK279 @ sK276 ) )
| ~ spl877_18 ),
inference(forward_demodulation,[status(thm)],[f4633,f3182]) ).
thf(f3182,plain,
binunionE = $true,
inference(cnf_transformation,[status(thm)],[f1628]) ).
thf(f1628,plain,
( ( setunionI = $true )
& ( ap2apEq1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( subsetTI = $true )
& ( subsetE2 = $true )
& ( binunionE = $true )
& ( demorgan2a = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( powersetE1 = $true )
& ( binintersectTERcontra = $true )
& ( setOfPairsIsBReln = $true )
& ( iftrue = $true )
& ( ksndpairEq = $true )
& ( powersetTI1 = $true )
& ( app = $true )
& ( notequalI1 = $true )
& ( singletoninpowunion = $true )
& ( binintersectSubset3 = $true )
& ( binunionTILcontra = $true )
& ( funcGraphProp3 = $true )
& ( upairsetIR = $true )
& ( dpsetconstrERa = $true )
& ( binunionEcases = $true )
& ( wellorderingAx = $true )
& ( lamp = $true )
& ( theprop = $true )
& ( binintersectTELcontra = $true )
& ( prop2setI = $true )
& ( setunion__Cong = $true )
& ( iftrueProp2 = $true )
& ( emptysetsubset = $true )
& ( eqinunit = $true )
& ( quantDeMorgan2 = $true )
& ( notdexE = $true )
& ( binunionRsub = $true )
& ( symdiffI2 = $true )
& ( singletonsubset = $true )
& ( exuE2 = $true )
& ( replAx = $true )
& ( kfstpairEq = $true )
& ( contraSubsetComplement = $true )
& ( secondinupair = $true )
& ( beta1 = $true )
& ( setadjoinSub2 = $true )
& ( singletonsswitch = $true )
& ( doubleComplementEq = $true )
& ( complementTI1 = $true )
& ( setminusIRneg = $true )
& ( symdiffIneg1 = $true )
& ( setadjoinIR = $true )
& ( nonemptyI1 = $true )
& ( notinemptyset = $true )
& ( subPowSU = $true )
& ( ex1E1 = $true )
& ( emptyI = $true )
& ( cartprodfstin = $true )
& ( disjointsetsI1 = $true )
& ( emptyinunitempty = $true )
& ( funcext2 = $true )
& ( binunionIL = $true )
& ( powerset__Cong = $true )
& ( contrasubsetT3 = $true )
& ( binunionTE = $true )
& ( binintersectSubset1 = $true )
& ( emptysetE = $true )
& ( setadjoinAx = $true )
& ( binintersectEL = $true )
& ( iftrueProp1 = $true )
& ( ubforcartprodlem2 = $true )
& ( lamProp = $true )
& ( emptysetimpfalse = $true )
& ( setminusERneg = $true )
& ( upairset2IR = $true )
& ( binintersectSubset4 = $true )
& ( kfstsingleton = $true )
& ( complementImpComplementIntersect = $true )
& ( inPowerset = $true )
& ( cartprodsndpairEq = $true )
& ( exuI2 = $true )
& ( symdiffI1 = $true )
& ( powersetI1 = $true )
& ( cartprodpairmemEL = $true )
& ( binunionTIRcontra = $true )
& ( setadjoinE = $true )
& ( powersetAx = $true )
& ( contrasubsetT = $true )
& ( powersetT_lem = $true )
& ( inCongP = $true )
& ( subbreln = $true )
& ( demorgan1b = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( exuEu = $true )
& ( singletoninpowerset = $true )
& ( apProp = $true )
& ( demorgan2a2 = $true )
& ( dsetconstr__Cong = $true )
& ( funcext = $true )
& ( binintersectSubset5 = $true )
& ( upairsubunion = $true )
& ( notinsingleton = $true )
& ( setminusSubset1 = $true )
& ( beta2 = $true )
& ( setadjoinSub = $true )
& ( powersetE = $true )
& ( setunionAx = $true )
& ( demorgan1a = $true )
& ( nonemptyE1 = $true )
& ( funcGraphProp4 = $true )
& ( prop2setE = $true )
& ( emptyInPowerset = $true )
& ( complementUnionInPowersetComplement = $true )
& ( complementTnotintersectT = $true )
& ( setukpairIR = $true )
& ( binintersectER = $true )
& ( exu__Cong = $true )
& ( doubleComplementI1 = $true )
& ( notsubsetI = $true )
& ( setminusI = $true )
& ( ksndsingleton = $true )
& ( powersetTE1 = $true )
& ( doubleComplementSub2 = $true )
& ( exuI3 = $true )
& ( setminusER = $true )
& ( emptyset__Cong = $true )
& ( cartprodpairin = $true )
& ( powersetI = $true )
& ( complementTE1 = $true )
& ( eqimpsubset1 = $true )
& ( symdiffIneg2 = $true )
& ( setminusLsub = $true )
& ( setunionsingleton = $true )
& ( demorgan2a1 = $true )
& ( cartprodmempair1 = $true )
& ( iffalseProp2 = $true )
& ( eqbreln = $true )
& ( emptyE1 = $true )
& ( setextsub = $true )
& ( cartprodsndin = $true )
& ( kpairiskpair = $true )
& ( setbeta = $true )
& ( complementSubsetComplementIntersect = $true )
& ( descrp = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( dpsetconstrI = $true )
& ( iftrueorfalse = $true )
& ( $true
= ( in @ sK276 @ ( powerset @ sK277 ) ) )
& ( $true
!= ( in @ sK279 @ ( setminus @ sK277 @ ( binunion @ sK276 @ sK278 ) ) ) )
& ( $true
= ( in @ sK279 @ sK277 ) )
& ( ( in @ sK279 @ ( setminus @ sK277 @ sK278 ) )
= $true )
& ( ( in @ sK279 @ ( setminus @ sK277 @ sK276 ) )
= $true )
& ( ( in @ sK278 @ ( powerset @ sK277 ) )
= $true )
& ( brelnall1 = $true )
& ( setukpairIL = $true )
& ( binintersectRsub = $true )
& ( setunionE2 = $true )
& ( subsetI1 = $true )
& ( dsetconstrER = $true )
& ( upairequniteq = $true )
& ( binunionTEcontra = $true )
& ( upairsetIL = $true )
& ( eqimpsubset2 = $true )
& ( doubleComplementSub1 = $true )
& ( setext = $true )
& ( brelnall2 = $true )
& ( binunionLsub = $true )
& ( noeltsimpempty = $true )
& ( ap2p = $true )
& ( ubforcartprodlem3 = $true )
& ( kpairsurjEq = $true )
& ( nonemptyI = $true )
& ( cartprodpairmemER = $true )
& ( exuE1 = $true )
& ( nonemptyImpWitness = $true )
& ( exuE3e = $true )
& ( uniqinunit = $true )
& ( quantDeMorgan4 = $true )
& ( ex1I = $true )
& ( upairinpowunion = $true )
& ( binintersectLsub = $true )
& ( bs114d = $true )
& ( dpsetconstrEL1 = $true )
& ( complementT_lem = $true )
& ( inIntersectImpInUnion = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( dpsetconstrER = $true )
& ( setukpairinjL2 = $true )
& ( exuE3u = $true )
& ( sepInPowerset = $true )
& ( dsetconstrI = $true )
& ( complementTcontraSubset = $true )
& ( ifSingleton = $true )
& ( eta2 = $true )
& ( setunionE = $true )
& ( dsetconstrEL = $true )
& ( setminusT_lem = $true )
& ( binunionT_lem = $true )
& ( setextAx = $true )
& ( cartprodpairsurjEq = $true )
& ( subsetRefl = $true )
& ( dpsetconstrSub = $true )
& ( emptysetAx = $true )
& ( ap2apEq2 = $true )
& ( omegaSAx = $true )
& ( descr__Cong = $true )
& ( setukpairinjL1 = $true )
& ( ex1I2 = $true )
& ( iffalse = $true )
& ( contrasubsetT2 = $true )
& ( theeq = $true )
& ( singletonsuniq = $true )
& ( ifp = $true )
& ( kpairp = $true )
& ( powersetsubset = $true )
& ( setunionsingleton2 = $true )
& ( lam2lamEq = $true )
& ( lam2p = $true )
& ( exuI1 = $true )
& ( subset2powerset = $true )
& ( doubleComplementE1 = $true )
& ( binintersectI = $true )
& ( funcinfuncset = $true )
& ( emptyinPowerset = $true )
& ( singletonprop = $true )
& ( cartprodmempair = $true )
& ( sepSubset = $true )
& ( setukpairinjR = $true )
& ( vacuousDall = $true )
& ( binintersectSubset2 = $true )
& ( symdiffE = $true )
& ( setminusILneg = $true )
& ( ex1E2 = $true )
& ( ubforcartprodlem1 = $true )
& ( funcGraphProp1 = $true )
& ( omegaIndAx = $true )
& ( notequalI2 = $true )
& ( omega__Cong = $true )
& ( setukpairinjR2 = $true )
& ( demorgan1 = $true )
& ( quantDeMorgan1 = $true )
& ( setadjoin__Cong = $true )
& ( upairsetE = $true )
& ( setukpairinjL = $true )
& ( setukpairinjR1 = $true )
& ( prop2set2propI = $true )
& ( subsetI2 = $true )
& ( subsetTrans = $true )
& ( in__Cong = $true )
& ( setminusEL = $true )
& ( setadjoinOr = $true )
& ( setextT = $true )
& ( setukpairinjR12 = $true )
& ( inIntersectImpInUnion2 = $true )
& ( setadjoinIL = $true )
& ( setoftrueEq = $true )
& ( quantDeMorgan3 = $true )
& ( setukpairinjR11 = $true )
& ( upairset2E = $true )
& ( dpsetconstrEL2 = $true )
& ( binunionIR = $true )
& ( omega0Ax = $true )
& ( notdallE = $true )
& ( subsetemptysetimpeq = $true )
& ( setunionsingleton1 = $true )
& ( binintersectT_lem = $true )
& ( funcextLem = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodfstpairEq = $true )
& ( setminusSubset2 = $true )
& ( setminusELneg = $true )
& ( eta1 = $true )
& ( subsetE = $true )
& ( contrasubsetT1 = $true )
& ( infuncsetfunc = $true )
& ( funcGraphProp2 = $true )
& ( iffalseProp1 = $true )
& ( foundationAx = $true )
& ( funcImageSingleton = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK276,sK277,sK278,sK279])],[f1107,f1627,f1626,f1625]) ).
thf(f1625,plain,
( ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ ( setminus @ X1 @ ( binunion @ X0 @ X2 ) ) )
!= $true )
& ( ( in @ X3 @ X1 )
= $true )
& ( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
= $true )
& ( $true
= ( in @ X3 @ ( setminus @ X1 @ X0 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) ) ) )
=> ( ( $true
= ( in @ sK276 @ ( powerset @ sK277 ) ) )
& ? [X2: $i] :
( ? [X3: $i] :
( ( $true
!= ( in @ X3 @ ( setminus @ sK277 @ ( binunion @ sK276 @ X2 ) ) ) )
& ( ( in @ X3 @ sK277 )
= $true )
& ( $true
= ( in @ X3 @ ( setminus @ sK277 @ X2 ) ) )
& ( ( in @ X3 @ ( setminus @ sK277 @ sK276 ) )
= $true ) )
& ( $true
= ( in @ X2 @ ( powerset @ sK277 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1626,plain,
( ? [X2: $i] :
( ? [X3: $i] :
( ( $true
!= ( in @ X3 @ ( setminus @ sK277 @ ( binunion @ sK276 @ X2 ) ) ) )
& ( ( in @ X3 @ sK277 )
= $true )
& ( $true
= ( in @ X3 @ ( setminus @ sK277 @ X2 ) ) )
& ( ( in @ X3 @ ( setminus @ sK277 @ sK276 ) )
= $true ) )
& ( $true
= ( in @ X2 @ ( powerset @ sK277 ) ) ) )
=> ( ? [X3: $i] :
( ( $true
!= ( in @ X3 @ ( setminus @ sK277 @ ( binunion @ sK276 @ sK278 ) ) ) )
& ( ( in @ X3 @ sK277 )
= $true )
& ( ( in @ X3 @ ( setminus @ sK277 @ sK278 ) )
= $true )
& ( ( in @ X3 @ ( setminus @ sK277 @ sK276 ) )
= $true ) )
& ( ( in @ sK278 @ ( powerset @ sK277 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1627,plain,
( ? [X3: $i] :
( ( $true
!= ( in @ X3 @ ( setminus @ sK277 @ ( binunion @ sK276 @ sK278 ) ) ) )
& ( ( in @ X3 @ sK277 )
= $true )
& ( ( in @ X3 @ ( setminus @ sK277 @ sK278 ) )
= $true )
& ( ( in @ X3 @ ( setminus @ sK277 @ sK276 ) )
= $true ) )
=> ( ( $true
!= ( in @ sK279 @ ( setminus @ sK277 @ ( binunion @ sK276 @ sK278 ) ) ) )
& ( $true
= ( in @ sK279 @ sK277 ) )
& ( ( in @ sK279 @ ( setminus @ sK277 @ sK278 ) )
= $true )
& ( ( in @ sK279 @ ( setminus @ sK277 @ sK276 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1107,plain,
( ( setunionI = $true )
& ( ap2apEq1 = $true )
& ( intersectInPowersetIntersectUnions = $true )
& ( subsetTI = $true )
& ( subsetE2 = $true )
& ( binunionE = $true )
& ( demorgan2a = $true )
& ( inIntersectImpInIntersectUnions = $true )
& ( powersetE1 = $true )
& ( binintersectTERcontra = $true )
& ( setOfPairsIsBReln = $true )
& ( iftrue = $true )
& ( ksndpairEq = $true )
& ( powersetTI1 = $true )
& ( app = $true )
& ( notequalI1 = $true )
& ( singletoninpowunion = $true )
& ( binintersectSubset3 = $true )
& ( binunionTILcontra = $true )
& ( funcGraphProp3 = $true )
& ( upairsetIR = $true )
& ( dpsetconstrERa = $true )
& ( binunionEcases = $true )
& ( wellorderingAx = $true )
& ( lamp = $true )
& ( theprop = $true )
& ( binintersectTELcontra = $true )
& ( prop2setI = $true )
& ( setunion__Cong = $true )
& ( iftrueProp2 = $true )
& ( emptysetsubset = $true )
& ( eqinunit = $true )
& ( quantDeMorgan2 = $true )
& ( notdexE = $true )
& ( binunionRsub = $true )
& ( symdiffI2 = $true )
& ( singletonsubset = $true )
& ( exuE2 = $true )
& ( replAx = $true )
& ( kfstpairEq = $true )
& ( contraSubsetComplement = $true )
& ( secondinupair = $true )
& ( beta1 = $true )
& ( setadjoinSub2 = $true )
& ( singletonsswitch = $true )
& ( doubleComplementEq = $true )
& ( complementTI1 = $true )
& ( setminusIRneg = $true )
& ( symdiffIneg1 = $true )
& ( setadjoinIR = $true )
& ( nonemptyI1 = $true )
& ( notinemptyset = $true )
& ( subPowSU = $true )
& ( ex1E1 = $true )
& ( emptyI = $true )
& ( cartprodfstin = $true )
& ( disjointsetsI1 = $true )
& ( emptyinunitempty = $true )
& ( funcext2 = $true )
& ( binunionIL = $true )
& ( powerset__Cong = $true )
& ( contrasubsetT3 = $true )
& ( binunionTE = $true )
& ( binintersectSubset1 = $true )
& ( emptysetE = $true )
& ( setadjoinAx = $true )
& ( binintersectEL = $true )
& ( iftrueProp1 = $true )
& ( ubforcartprodlem2 = $true )
& ( lamProp = $true )
& ( emptysetimpfalse = $true )
& ( setminusERneg = $true )
& ( upairset2IR = $true )
& ( binintersectSubset4 = $true )
& ( kfstsingleton = $true )
& ( complementImpComplementIntersect = $true )
& ( inPowerset = $true )
& ( cartprodsndpairEq = $true )
& ( exuI2 = $true )
& ( symdiffI1 = $true )
& ( powersetI1 = $true )
& ( cartprodpairmemEL = $true )
& ( binunionTIRcontra = $true )
& ( setadjoinE = $true )
& ( powersetAx = $true )
& ( contrasubsetT = $true )
& ( powersetT_lem = $true )
& ( inCongP = $true )
& ( subbreln = $true )
& ( demorgan1b = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( exuEu = $true )
& ( singletoninpowerset = $true )
& ( apProp = $true )
& ( demorgan2a2 = $true )
& ( dsetconstr__Cong = $true )
& ( funcext = $true )
& ( binintersectSubset5 = $true )
& ( upairsubunion = $true )
& ( notinsingleton = $true )
& ( setminusSubset1 = $true )
& ( beta2 = $true )
& ( setadjoinSub = $true )
& ( powersetE = $true )
& ( setunionAx = $true )
& ( demorgan1a = $true )
& ( nonemptyE1 = $true )
& ( funcGraphProp4 = $true )
& ( prop2setE = $true )
& ( emptyInPowerset = $true )
& ( complementUnionInPowersetComplement = $true )
& ( complementTnotintersectT = $true )
& ( setukpairIR = $true )
& ( binintersectER = $true )
& ( exu__Cong = $true )
& ( doubleComplementI1 = $true )
& ( notsubsetI = $true )
& ( setminusI = $true )
& ( ksndsingleton = $true )
& ( powersetTE1 = $true )
& ( doubleComplementSub2 = $true )
& ( exuI3 = $true )
& ( setminusER = $true )
& ( emptyset__Cong = $true )
& ( cartprodpairin = $true )
& ( powersetI = $true )
& ( complementTE1 = $true )
& ( eqimpsubset1 = $true )
& ( symdiffIneg2 = $true )
& ( setminusLsub = $true )
& ( setunionsingleton = $true )
& ( demorgan2a1 = $true )
& ( cartprodmempair1 = $true )
& ( iffalseProp2 = $true )
& ( eqbreln = $true )
& ( emptyE1 = $true )
& ( setextsub = $true )
& ( cartprodsndin = $true )
& ( kpairiskpair = $true )
& ( setbeta = $true )
& ( complementSubsetComplementIntersect = $true )
& ( descrp = $true )
& ( inComplementUnionImpNotIn1 = $true )
& ( dpsetconstrI = $true )
& ( iftrueorfalse = $true )
& ? [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
& ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ ( setminus @ X1 @ ( binunion @ X0 @ X2 ) ) )
!= $true )
& ( ( in @ X3 @ X1 )
= $true )
& ( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
= $true )
& ( $true
= ( in @ X3 @ ( setminus @ X1 @ X0 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) ) ) )
& ( brelnall1 = $true )
& ( setukpairIL = $true )
& ( binintersectRsub = $true )
& ( setunionE2 = $true )
& ( subsetI1 = $true )
& ( dsetconstrER = $true )
& ( upairequniteq = $true )
& ( binunionTEcontra = $true )
& ( upairsetIL = $true )
& ( eqimpsubset2 = $true )
& ( doubleComplementSub1 = $true )
& ( setext = $true )
& ( brelnall2 = $true )
& ( binunionLsub = $true )
& ( noeltsimpempty = $true )
& ( ap2p = $true )
& ( ubforcartprodlem3 = $true )
& ( kpairsurjEq = $true )
& ( nonemptyI = $true )
& ( cartprodpairmemER = $true )
& ( exuE1 = $true )
& ( nonemptyImpWitness = $true )
& ( exuE3e = $true )
& ( uniqinunit = $true )
& ( quantDeMorgan4 = $true )
& ( ex1I = $true )
& ( upairinpowunion = $true )
& ( binintersectLsub = $true )
& ( bs114d = $true )
& ( dpsetconstrEL1 = $true )
& ( complementT_lem = $true )
& ( inIntersectImpInUnion = $true )
& ( inComplementUnionImpInComplement1 = $true )
& ( dpsetconstrER = $true )
& ( setukpairinjL2 = $true )
& ( exuE3u = $true )
& ( sepInPowerset = $true )
& ( dsetconstrI = $true )
& ( complementTcontraSubset = $true )
& ( ifSingleton = $true )
& ( eta2 = $true )
& ( setunionE = $true )
& ( dsetconstrEL = $true )
& ( setminusT_lem = $true )
& ( binunionT_lem = $true )
& ( setextAx = $true )
& ( cartprodpairsurjEq = $true )
& ( subsetRefl = $true )
& ( dpsetconstrSub = $true )
& ( emptysetAx = $true )
& ( ap2apEq2 = $true )
& ( omegaSAx = $true )
& ( descr__Cong = $true )
& ( setukpairinjL1 = $true )
& ( ex1I2 = $true )
& ( iffalse = $true )
& ( contrasubsetT2 = $true )
& ( theeq = $true )
& ( singletonsuniq = $true )
& ( ifp = $true )
& ( kpairp = $true )
& ( powersetsubset = $true )
& ( setunionsingleton2 = $true )
& ( lam2lamEq = $true )
& ( lam2p = $true )
& ( exuI1 = $true )
& ( subset2powerset = $true )
& ( doubleComplementE1 = $true )
& ( binintersectI = $true )
& ( funcinfuncset = $true )
& ( emptyinPowerset = $true )
& ( singletonprop = $true )
& ( cartprodmempair = $true )
& ( sepSubset = $true )
& ( setukpairinjR = $true )
& ( vacuousDall = $true )
& ( binintersectSubset2 = $true )
& ( symdiffE = $true )
& ( setminusILneg = $true )
& ( ex1E2 = $true )
& ( ubforcartprodlem1 = $true )
& ( funcGraphProp1 = $true )
& ( omegaIndAx = $true )
& ( notequalI2 = $true )
& ( omega__Cong = $true )
& ( setukpairinjR2 = $true )
& ( demorgan1 = $true )
& ( quantDeMorgan1 = $true )
& ( setadjoin__Cong = $true )
& ( upairsetE = $true )
& ( setukpairinjL = $true )
& ( setukpairinjR1 = $true )
& ( prop2set2propI = $true )
& ( subsetI2 = $true )
& ( subsetTrans = $true )
& ( in__Cong = $true )
& ( setminusEL = $true )
& ( setadjoinOr = $true )
& ( setextT = $true )
& ( setukpairinjR12 = $true )
& ( inIntersectImpInUnion2 = $true )
& ( setadjoinIL = $true )
& ( setoftrueEq = $true )
& ( quantDeMorgan3 = $true )
& ( setukpairinjR11 = $true )
& ( upairset2E = $true )
& ( dpsetconstrEL2 = $true )
& ( binunionIR = $true )
& ( omega0Ax = $true )
& ( notdallE = $true )
& ( subsetemptysetimpeq = $true )
& ( setunionsingleton1 = $true )
& ( binintersectT_lem = $true )
& ( funcextLem = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodfstpairEq = $true )
& ( setminusSubset2 = $true )
& ( setminusELneg = $true )
& ( eta1 = $true )
& ( subsetE = $true )
& ( contrasubsetT1 = $true )
& ( infuncsetfunc = $true )
& ( funcGraphProp2 = $true )
& ( iffalseProp1 = $true )
& ( foundationAx = $true )
& ( funcImageSingleton = $true ) ),
inference(flattening,[status(thm)],[f1106]) ).
thf(f1106,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ ( setminus @ X1 @ ( binunion @ X0 @ X2 ) ) )
!= $true )
& ( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
= $true )
& ( $true
= ( in @ X3 @ ( setminus @ X1 @ X0 ) ) )
& ( ( in @ X3 @ X1 )
= $true ) )
& ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) ) )
& ( ( in @ X0 @ ( powerset @ X1 ) )
= $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)],[f768]) ).
thf(f768,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 )
=> ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X1 ) ) )
=> ! [X3: $i] :
( ( ( in @ X3 @ X1 )
= $true )
=> ( ( $true
= ( in @ X3 @ ( setminus @ X1 @ X0 ) ) )
=> ( ( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
= $true )
=> ( ( in @ X3 @ ( setminus @ X1 @ ( binunion @ X0 @ X2 ) ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f767]) ).
thf(f767,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
=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( powerset @ X1 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X1 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X1 )
=> ( ( in @ X3 @ ( setminus @ X1 @ X0 ) )
=> ( ( in @ X3 @ ( setminus @ X1 @ X2 ) )
=> ( in @ X3 @ ( setminus @ X1 @ ( binunion @ X0 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f277]) ).
thf(f277,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
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( in @ X1 @ ( setminus @ X3 @ X11 ) )
=> ( ( in @ X1 @ ( setminus @ X3 @ X16 ) )
=> ( in @ X1 @ ( setminus @ X3 @ ( binunion @ X11 @ X16 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f276]) ).
thf(f276,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
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( in @ X1 @ ( setminus @ X3 @ X11 ) )
=> ( ( in @ X1 @ ( setminus @ X3 @ X16 ) )
=> ( in @ X1 @ ( setminus @ X3 @ ( binunion @ X11 @ X16 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.UUdmaiX54M/DTF2THF_26421.p',demorgan2b2) ).
thf(f4633,plain,
( ( $true
= ( in @ sK279 @ sK276 ) )
| ( $true
= ( in @ sK279 @ sK278 ) )
| ( binunionE != $true )
| ~ spl877_18 ),
inference(trivial_inequality_removal,[status(thm)],[f4623]) ).
thf(f4623,plain,
( ( $true != $true )
| ( $true
= ( in @ sK279 @ sK276 ) )
| ( $true
= ( in @ sK279 @ sK278 ) )
| ( binunionE != $true )
| ~ spl877_18 ),
inference(superposition,[status(thm)],[f3425,f4440]) ).
thf(f4440,plain,
( ( ( in @ sK279 @ ( binunion @ sK276 @ sK278 ) )
= $true )
| ~ spl877_18 ),
inference(avatar_component_clause,[status(thm)],[f4438]) ).
thf(f3425,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4 @ ( binunion @ X3 @ X5 ) ) )
| ( $true
= ( in @ X4 @ X5 ) )
| ( ( in @ X4 @ X3 )
= $true )
| ( binunionE != $true ) ),
inference(cnf_transformation,[status(thm)],[f1912]) ).
thf(f1912,plain,
( ( ( binunionE = $true )
| ( ( $true
!= ( in @ sK455 @ sK454 ) )
& ( $true
!= ( in @ sK455 @ sK456 ) )
& ( ( in @ sK455 @ ( binunion @ sK454 @ sK456 ) )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ X3 )
= $true )
| ( $true
= ( in @ X4 @ X5 ) )
| ( $true
!= ( in @ X4 @ ( binunion @ X3 @ X5 ) ) ) )
| ( binunionE != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK454,sK455,sK456])],[f1910,f1911]) ).
thf(f1911,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ X0 ) )
& ( ( in @ X1 @ X2 )
!= $true )
& ( $true
= ( in @ X1 @ ( binunion @ X0 @ X2 ) ) ) )
=> ( ( $true
!= ( in @ sK455 @ sK454 ) )
& ( $true
!= ( in @ sK455 @ sK456 ) )
& ( ( in @ sK455 @ ( binunion @ sK454 @ sK456 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1910,plain,
( ( ( binunionE = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ X0 ) )
& ( ( in @ X1 @ X2 )
!= $true )
& ( $true
= ( in @ X1 @ ( binunion @ X0 @ X2 ) ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ X3 )
= $true )
| ( $true
= ( in @ X4 @ X5 ) )
| ( $true
!= ( in @ X4 @ ( binunion @ X3 @ X5 ) ) ) )
| ( binunionE != $true ) ) ),
inference(rectify,[status(thm)],[f1909]) ).
thf(f1909,plain,
( ( ( binunionE = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ X0 ) )
& ( ( in @ X1 @ X2 )
!= $true )
& ( $true
= ( in @ X1 @ ( binunion @ X0 @ X2 ) ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
| ( ( in @ X1 @ X2 )
= $true )
| ( $true
!= ( in @ X1 @ ( binunion @ X0 @ X2 ) ) ) )
| ( binunionE != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1196]) ).
thf(f1196,plain,
( ( binunionE = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
| ( ( in @ X1 @ X2 )
= $true )
| ( $true
!= ( in @ X1 @ ( binunion @ X0 @ X2 ) ) ) ) ),
inference(flattening,[status(thm)],[f1195]) ).
thf(f1195,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
| ( $true
= ( in @ X1 @ X0 ) )
| ( $true
!= ( in @ X1 @ ( binunion @ X0 @ X2 ) ) ) )
<=> ( binunionE = $true ) ),
inference(ennf_transformation,[status(thm)],[f584]) ).
thf(f584,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ ( binunion @ X0 @ X2 ) ) )
=> ( ( ( in @ X1 @ X2 )
= $true )
| ( $true
= ( in @ X1 @ X0 ) ) ) )
<=> ( binunionE = $true ) ),
inference(fool_elimination,[status(thm)],[f583]) ).
thf(f583,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( binunion @ X0 @ X2 ) )
=> ( ( in @ X1 @ X0 )
| ( in @ X1 @ X2 ) ) )
= binunionE ),
inference(rectify,[status(thm)],[f105]) ).
thf(f105,axiom,
! [X3: $i,X1: $i,X4: $i] :
( ( ( in @ X1 @ ( binunion @ X3 @ X4 ) )
=> ( ( in @ X1 @ X3 )
| ( in @ X1 @ X4 ) ) )
= binunionE ),
file('/export/starexec/sandbox/tmp/tmp.UUdmaiX54M/DTF2THF_26421.p',binunionE) ).
thf(f4622,plain,
spl877_18,
inference(avatar_split_clause,[status(thm)],[f4419,f4438]) ).
thf(f4419,plain,
( ( in @ sK279 @ ( binunion @ sK276 @ sK278 ) )
= $true ),
inference(trivial_inequality_removal,[status(thm)],[f4418]) ).
thf(f4418,plain,
( ( $true != $true )
| ( ( in @ sK279 @ ( binunion @ sK276 @ sK278 ) )
= $true ) ),
inference(forward_demodulation,[status(thm)],[f4417,f3040]) ).
thf(f3040,plain,
( $true
= ( in @ sK279 @ sK277 ) ),
inference(cnf_transformation,[status(thm)],[f1628]) ).
thf(f4417,plain,
( ( ( in @ sK279 @ ( binunion @ sK276 @ sK278 ) )
= $true )
| ( $true
!= ( in @ sK279 @ sK277 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f4416]) ).
thf(f4416,plain,
( ( ( in @ sK279 @ ( binunion @ sK276 @ sK278 ) )
= $true )
| ( $true
!= ( in @ sK279 @ sK277 ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f4415,f2919]) ).
thf(f2919,plain,
setminusELneg = $true,
inference(cnf_transformation,[status(thm)],[f1628]) ).
thf(f4415,plain,
( ( setminusELneg != $true )
| ( $true
!= ( in @ sK279 @ sK277 ) )
| ( ( in @ sK279 @ ( binunion @ sK276 @ sK278 ) )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4398]) ).
thf(f4398,plain,
( ( $true != $true )
| ( setminusELneg != $true )
| ( $true
!= ( in @ sK279 @ sK277 ) )
| ( ( in @ sK279 @ ( binunion @ sK276 @ sK278 ) )
= $true ) ),
inference(superposition,[status(thm)],[f3041,f3746]) ).
thf(f3746,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X2 @ ( setminus @ X0 @ X1 ) )
= $true )
| ( $true
= ( in @ X2 @ X1 ) )
| ( setminusELneg != $true )
| ( $true
!= ( in @ X2 @ X0 ) ) ),
inference(cnf_transformation,[status(thm)],[f2275]) ).
thf(f2275,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
| ( $true
= ( in @ X2 @ X1 ) )
| ( ( in @ X2 @ ( setminus @ X0 @ X1 ) )
= $true ) )
| ( setminusELneg != $true ) )
& ( ( setminusELneg = $true )
| ( ( $true
= ( in @ sK688 @ sK686 ) )
& ( $true
!= ( in @ sK688 @ sK687 ) )
& ( $true
!= ( in @ sK688 @ ( setminus @ sK686 @ sK687 ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK686,sK687,sK688])],[f2273,f2274]) ).
thf(f2274,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X5 @ X3 ) )
& ( ( in @ X5 @ X4 )
!= $true )
& ( $true
!= ( in @ X5 @ ( setminus @ X3 @ X4 ) ) ) )
=> ( ( $true
= ( in @ sK688 @ sK686 ) )
& ( $true
!= ( in @ sK688 @ sK687 ) )
& ( $true
!= ( in @ sK688 @ ( setminus @ sK686 @ sK687 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2273,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
| ( $true
= ( in @ X2 @ X1 ) )
| ( ( in @ X2 @ ( setminus @ X0 @ X1 ) )
= $true ) )
| ( setminusELneg != $true ) )
& ( ( setminusELneg = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X5 @ X3 ) )
& ( ( in @ X5 @ X4 )
!= $true )
& ( $true
!= ( in @ X5 @ ( setminus @ X3 @ X4 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f2272]) ).
thf(f2272,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
| ( $true
= ( in @ X2 @ X1 ) )
| ( ( in @ X2 @ ( setminus @ X0 @ X1 ) )
= $true ) )
| ( setminusELneg != $true ) )
& ( ( setminusELneg = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
& ( $true
!= ( in @ X2 @ X1 ) )
& ( ( in @ X2 @ ( setminus @ X0 @ X1 ) )
!= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1081]) ).
thf(f1081,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
| ( $true
= ( in @ X2 @ X1 ) )
| ( ( in @ X2 @ ( setminus @ X0 @ X1 ) )
= $true ) )
<=> ( setminusELneg = $true ) ),
inference(flattening,[status(thm)],[f1080]) ).
thf(f1080,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
| ( $true
= ( in @ X2 @ X1 ) )
| ( ( in @ X2 @ ( setminus @ X0 @ X1 ) )
= $true ) )
<=> ( setminusELneg = $true ) ),
inference(ennf_transformation,[status(thm)],[f844]) ).
thf(f844,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( setminus @ X0 @ X1 ) )
!= $true )
=> ( ( $true
!= ( in @ X2 @ X1 ) )
=> ( $true
!= ( in @ X2 @ X0 ) ) ) )
<=> ( setminusELneg = $true ) ),
inference(flattening,[status(thm)],[f498]) ).
thf(f498,plain,
( ( setminusELneg = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X2 @ ( setminus @ X0 @ X1 ) )
!= $true )
=> ( ( $true
!= ( in @ X2 @ X1 ) )
=> ( $true
!= ( in @ X2 @ X0 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f497]) ).
thf(f497,plain,
( setminusELneg
= ( ! [X0: $i,X1: $i,X2: $i] :
( ~ ( in @ X2 @ ( setminus @ X0 @ X1 ) )
=> ( ~ ( in @ X2 @ X1 )
=> ~ ( in @ X2 @ X0 ) ) ) ) ),
inference(rectify,[status(thm)],[f125]) ).
thf(f125,axiom,
( setminusELneg
= ( ! [X3: $i,X4: $i,X1: $i] :
( ~ ( in @ X1 @ ( setminus @ X3 @ X4 ) )
=> ( ~ ( in @ X1 @ X4 )
=> ~ ( in @ X1 @ X3 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.UUdmaiX54M/DTF2THF_26421.p',setminusELneg) ).
thf(f3041,plain,
( $true
!= ( in @ sK279 @ ( setminus @ sK277 @ ( binunion @ sK276 @ sK278 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f1628]) ).
thf(f4394,plain,
~ spl877_13,
inference(avatar_split_clause,[status(thm)],[f4393,f4365]) ).
thf(f4393,plain,
( $true
!= ( in @ sK279 @ sK278 ) ),
inference(trivial_inequality_removal,[status(thm)],[f4392]) ).
thf(f4392,plain,
( ( $true
!= ( in @ sK279 @ sK278 ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f4338,f3065]) ).
thf(f3065,plain,
setminusER = $true,
inference(cnf_transformation,[status(thm)],[f1628]) ).
thf(f4338,plain,
( ( setminusER != $true )
| ( $true
!= ( in @ sK279 @ sK278 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f4333]) ).
thf(f4333,plain,
( ( $true != $true )
| ( $true
!= ( in @ sK279 @ sK278 ) )
| ( setminusER != $true ) ),
inference(superposition,[status(thm)],[f3318,f3039]) ).
thf(f3039,plain,
( ( in @ sK279 @ ( setminus @ sK277 @ sK278 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1628]) ).
thf(f3318,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ ( setminus @ X5 @ X4 ) )
!= $true )
| ( setminusER != $true )
| ( ( in @ X3 @ X4 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1786]) ).
thf(f1786,plain,
( ( ( setminusER = $true )
| ( ( $true
= ( in @ sK376 @ sK377 ) )
& ( ( in @ sK376 @ ( setminus @ sK378 @ sK377 ) )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ X4 )
!= $true )
| ( ( in @ X3 @ ( setminus @ X5 @ X4 ) )
!= $true ) )
| ( setminusER != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK376,sK377,sK378])],[f1784,f1785]) ).
thf(f1785,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
= $true )
& ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
= $true ) )
=> ( ( $true
= ( in @ sK376 @ sK377 ) )
& ( ( in @ sK376 @ ( setminus @ sK378 @ sK377 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1784,plain,
( ( ( setminusER = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
= $true )
& ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ X4 )
!= $true )
| ( ( in @ X3 @ ( setminus @ X5 @ X4 ) )
!= $true ) )
| ( setminusER != $true ) ) ),
inference(rectify,[status(thm)],[f1783]) ).
thf(f1783,plain,
( ( ( setminusER = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
= $true )
& ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
!= $true )
| ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
!= $true ) )
| ( setminusER != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f958]) ).
thf(f958,plain,
( ( setminusER = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X1 )
!= $true )
| ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f875]) ).
thf(f875,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
= $true )
=> ( ( in @ X0 @ X1 )
!= $true ) )
<=> ( setminusER = $true ) ),
inference(flattening,[status(thm)],[f804]) ).
thf(f804,plain,
( ( setminusER = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
= $true )
=> ( ( in @ X0 @ X1 )
!= $true ) ) ),
inference(fool_elimination,[status(thm)],[f803]) ).
thf(f803,plain,
( setminusER
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ ( setminus @ X2 @ X1 ) )
=> ~ ( in @ X0 @ X1 ) ) ) ),
inference(rectify,[status(thm)],[f122]) ).
thf(f122,axiom,
( setminusER
= ( ! [X1: $i,X4: $i,X3: $i] :
( ( in @ X1 @ ( setminus @ X3 @ X4 ) )
=> ~ ( in @ X1 @ X4 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.UUdmaiX54M/DTF2THF_26421.p',setminusER) ).
thf(f4328,plain,
~ spl877_6,
inference(avatar_split_clause,[status(thm)],[f4327,f4283]) ).
thf(f4327,plain,
( $true
!= ( in @ sK279 @ sK276 ) ),
inference(trivial_inequality_removal,[status(thm)],[f4326]) ).
thf(f4326,plain,
( ( $true
!= ( in @ sK279 @ sK276 ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f4268,f3140]) ).
thf(f3140,plain,
setminusIRneg = $true,
inference(cnf_transformation,[status(thm)],[f1628]) ).
thf(f4268,plain,
( ( $true
!= ( in @ sK279 @ sK276 ) )
| ( setminusIRneg != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4261]) ).
thf(f4261,plain,
( ( setminusIRneg != $true )
| ( $true != $true )
| ( $true
!= ( in @ sK279 @ sK276 ) ) ),
inference(superposition,[status(thm)],[f2769,f3038]) ).
thf(f3038,plain,
( ( in @ sK279 @ ( setminus @ sK277 @ sK276 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f1628]) ).
thf(f2769,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X1 @ ( setminus @ X0 @ X2 ) )
!= $true )
| ( ( in @ X1 @ X2 )
!= $true )
| ( setminusIRneg != $true ) ),
inference(cnf_transformation,[status(thm)],[f1471]) ).
thf(f1471,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
!= $true )
| ( ( in @ X1 @ ( setminus @ X0 @ X2 ) )
!= $true ) )
| ( setminusIRneg != $true ) )
& ( ( setminusIRneg = $true )
| ( ( ( in @ sK177 @ sK178 )
= $true )
& ( ( in @ sK177 @ ( setminus @ sK176 @ sK178 ) )
= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK176,sK177,sK178])],[f1469,f1470]) ).
thf(f1470,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X4 @ X5 ) )
& ( ( in @ X4 @ ( setminus @ X3 @ X5 ) )
= $true ) )
=> ( ( ( in @ sK177 @ sK178 )
= $true )
& ( ( in @ sK177 @ ( setminus @ sK176 @ sK178 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1469,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
!= $true )
| ( ( in @ X1 @ ( setminus @ X0 @ X2 ) )
!= $true ) )
| ( setminusIRneg != $true ) )
& ( ( setminusIRneg = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( in @ X4 @ X5 ) )
& ( ( in @ X4 @ ( setminus @ X3 @ X5 ) )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f1468]) ).
thf(f1468,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
!= $true )
| ( ( in @ X1 @ ( setminus @ X0 @ X2 ) )
!= $true ) )
| ( setminusIRneg != $true ) )
& ( ( setminusIRneg = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
& ( ( in @ X1 @ ( setminus @ X0 @ X2 ) )
= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f974]) ).
thf(f974,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
!= $true )
| ( ( in @ X1 @ ( setminus @ X0 @ X2 ) )
!= $true ) )
<=> ( setminusIRneg = $true ) ),
inference(ennf_transformation,[status(thm)],[f838]) ).
thf(f838,plain,
( ( setminusIRneg = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
=> ( ( in @ X1 @ ( setminus @ X0 @ X2 ) )
!= $true ) ) ),
inference(flattening,[status(thm)],[f435]) ).
thf(f435,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
=> ( ( in @ X1 @ ( setminus @ X0 @ X2 ) )
!= $true ) )
<=> ( setminusIRneg = $true ) ),
inference(fool_elimination,[status(thm)],[f434]) ).
thf(f434,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
=> ~ ( in @ X1 @ ( setminus @ X0 @ X2 ) ) )
= setminusIRneg ),
inference(rectify,[status(thm)],[f127]) ).
thf(f127,axiom,
! [X3: $i,X1: $i,X4: $i] :
( ( ( in @ X1 @ X4 )
=> ~ ( in @ X1 @ ( setminus @ X3 @ X4 ) ) )
= setminusIRneg ),
file('/export/starexec/sandbox/tmp/tmp.UUdmaiX54M/DTF2THF_26421.p',setminusIRneg) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU755^1 : TPTP v9.2.1. Released v3.7.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n027.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 12:33:22 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.47 ---- Original DTF file ---
% 0.19/0.47 thf(spec,logic,$$dhol).
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 % File : SEU755^1 : TPTP v9.2.1. Released v3.7.0.
% 0.19/0.47 % Domain : Set Theory
% 0.19/0.47 % Problem : Typed Set Theory - Laws for Typed Sets - DeMorgan Laws
% 0.19/0.47 % Version : Especial.
% 0.19/0.47 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.19/0.47 % (! x:i.in x A -> in x (setminus A X) -> in x (setminus A Y) ->
% 0.19/0.47 % in x (setminus A (binunion X Y)))))
% 0.19/0.47
% 0.19/0.47 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.19/0.47 % Source : [Bro08]
% 0.19/0.47 % Names : ZFC257g [Bro08]
% 0.19/0.47
% 0.19/0.47 % Status : Theorem
% 0.19/0.47 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0, 0.29 v7.4.0, 0.44 v7.2.0, 0.38 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.60 v6.2.0, 0.43 v6.1.0, 0.57 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.19/0.47 % Syntax : Number of formulae : 585 ( 275 unt; 309 typ; 275 def)
% 0.19/0.47 % Number of atoms : 1628 ( 402 equ; 0 cnn)
% 0.19/0.47 % Maximal formula atoms : 277 ( 5 avg)
% 0.19/0.47 % Number of connectives : 3404 ( 73 ~; 12 |; 44 &;2446 @)
% 0.19/0.47 % ( 14 <=>; 815 =>; 0 <=; 0 <~>)
% 0.19/0.47 % Maximal formula depth : 285 ( 2 avg)
% 0.19/0.47 % Number of types : 2 ( 0 usr)
% 0.19/0.47 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.19/0.47 % Number of symbols : 312 ( 309 usr; 275 con; 0-4 aty)
% 0.19/0.47 % Number of variables : 937 ( 74 ^; 828 !; 35 ?; 937 :)
% 0.19/0.47 % SPC : TH0_THM_EQU_NAR
% 0.19/0.47
% 0.19/0.47 % Comments : http://mathgate.info/detsetitem.php?id=316
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 thf(in_type,type,
% 0.19/0.47 in: $i > $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(exu_type,type,
% 0.19/0.47 exu: ( $i > $o ) > $o ).
% 0.19/0.47
% 0.19/0.47 thf(exu,definition,
% 0.19/0.47 ( exu
% 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
% 0.19/0.47 thf(setextAx_type,type,
% 0.19/0.47 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 ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoin__Cong,definition,
% 0.19/0.47 ( setadjoin__Cong
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.47 ( ( Xx = Xy )
% 0.19/0.47 => ! [Xz: $i,Xu: $i] :
% 0.19/0.47 ( ( Xz = Xu )
% 0.19/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.19/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(powerset__Cong_type,type,
% 0.19/0.47 powerset__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(powerset__Cong,definition,
% 0.19/0.47 ( powerset__Cong
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ( ( powerset @ A )
% 0.19/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setunion__Cong_type,type,
% 0.19/0.47 setunion__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setunion__Cong,definition,
% 0.19/0.47 ( setunion__Cong
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ( ( setunion @ A )
% 0.19/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(omega__Cong_type,type,
% 0.19/0.47 omega__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(omega__Cong,definition,
% 0.19/0.47 ( omega__Cong
% 0.19/0.47 = ( omega = omega ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(exuEu_type,type,
% 0.19/0.47 exuEu: $o ).
% 0.19/0.47
% 0.19/0.47 thf(exuEu,definition,
% 0.19/0.47 ( exuEu
% 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(descr__Cong_type,type,
% 0.19/0.47 descr__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(descr__Cong,definition,
% 0.19/0.47 ( descr__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 => ( ( descr
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 = ( descr
% 0.19/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstr__Cong_type,type,
% 0.19/0.47 dsetconstr__Cong: $o ).
% 0.19/0.47
% 0.19/0.47 thf(dsetconstr__Cong,definition,
% 0.19/0.47 ( dsetconstr__Cong
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.19/0.47 ( ! [Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( ( Xx = Xy )
% 0.19/0.47 => ( ( Xphi @ Xx )
% 0.19/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.19/0.47 => ( ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 = ( dsetconstr @ B
% 0.19/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subset_type,type,
% 0.19/0.47 subset: $i > $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(disjoint_type,type,
% 0.19/0.47 disjoint: $i > $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(setsmeet_type,type,
% 0.19/0.47 setsmeet: $i > $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(subsetI1_type,type,
% 0.19/0.47 subsetI1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subsetI1,definition,
% 0.19/0.47 ( subsetI1
% 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 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(eqimpsubset2_type,type,
% 0.19/0.47 eqimpsubset2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(eqimpsubset2,definition,
% 0.19/0.47 ( eqimpsubset2
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(eqimpsubset1_type,type,
% 0.19/0.47 eqimpsubset1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(eqimpsubset1,definition,
% 0.19/0.47 ( eqimpsubset1
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( A = B )
% 0.19/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subsetI2_type,type,
% 0.19/0.47 subsetI2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subsetI2,definition,
% 0.19/0.47 ( subsetI2
% 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 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetsubset_type,type,
% 0.19/0.47 emptysetsubset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(emptysetsubset,definition,
% 0.19/0.47 ( emptysetsubset
% 0.19/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subsetE_type,type,
% 0.19/0.47 subsetE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subsetE,definition,
% 0.19/0.47 ( subsetE
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subsetE2_type,type,
% 0.19/0.47 subsetE2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subsetE2,definition,
% 0.19/0.47 ( subsetE2
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( ~ ( in @ Xx @ B )
% 0.19/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notsubsetI_type,type,
% 0.19/0.47 notsubsetI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notsubsetI,definition,
% 0.19/0.47 ( notsubsetI
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ~ ( in @ Xx @ B )
% 0.19/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notequalI1_type,type,
% 0.19/0.47 notequalI1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notequalI1,definition,
% 0.19/0.47 ( notequalI1
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ~ ( subset @ A @ B )
% 0.19/0.47 => ( A != B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(notequalI2_type,type,
% 0.19/0.47 notequalI2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(notequalI2,definition,
% 0.19/0.47 ( notequalI2
% 0.19/0.47 = ( ! [A: $i,B: $i,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(subsetRefl_type,type,
% 0.19/0.47 subsetRefl: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subsetRefl,definition,
% 0.19/0.47 ( subsetRefl
% 0.19/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subsetTrans_type,type,
% 0.19/0.47 subsetTrans: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subsetTrans,definition,
% 0.19/0.47 ( subsetTrans
% 0.19/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( ( subset @ B @ C )
% 0.19/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinSub_type,type,
% 0.19/0.47 setadjoinSub: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinSub,definition,
% 0.19/0.47 ( setadjoinSub
% 0.19/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinSub2_type,type,
% 0.19/0.47 setadjoinSub2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setadjoinSub2,definition,
% 0.19/0.47 ( setadjoinSub2
% 0.19/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subset2powerset_type,type,
% 0.19/0.47 subset2powerset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subset2powerset,definition,
% 0.19/0.47 ( subset2powerset
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setextsub_type,type,
% 0.19/0.47 setextsub: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setextsub,definition,
% 0.19/0.47 ( setextsub
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( ( subset @ B @ A )
% 0.19/0.47 => ( A = B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(subsetemptysetimpeq_type,type,
% 0.19/0.47 subsetemptysetimpeq: $o ).
% 0.19/0.47
% 0.19/0.47 thf(subsetemptysetimpeq,definition,
% 0.19/0.47 ( subsetemptysetimpeq
% 0.19/0.47 = ( ! [A: $i] :
% 0.19/0.47 ( ( subset @ A @ emptyset )
% 0.19/0.47 => ( A = emptyset ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(powersetI1_type,type,
% 0.19/0.47 powersetI1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(powersetI1,definition,
% 0.19/0.47 ( powersetI1
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( subset @ B @ A )
% 0.19/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(powersetE1_type,type,
% 0.19/0.47 powersetE1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(powersetE1,definition,
% 0.19/0.47 ( powersetE1
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.19/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(inPowerset_type,type,
% 0.19/0.47 inPowerset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(inPowerset,definition,
% 0.19/0.47 ( inPowerset
% 0.19/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(powersetsubset_type,type,
% 0.19/0.47 powersetsubset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(powersetsubset,definition,
% 0.19/0.47 ( powersetsubset
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sepInPowerset_type,type,
% 0.19/0.47 sepInPowerset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(sepInPowerset,definition,
% 0.19/0.47 ( sepInPowerset
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( in
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(sepSubset_type,type,
% 0.19/0.47 sepSubset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(sepSubset,definition,
% 0.19/0.47 ( sepSubset
% 0.19/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.47 ( subset
% 0.19/0.47 @ ( dsetconstr @ A
% 0.19/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.47 @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binunion_type,type,
% 0.19/0.47 binunion: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(binunionIL_type,type,
% 0.19/0.47 binunionIL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binunionIL,definition,
% 0.19/0.47 ( binunionIL
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairset2IR_type,type,
% 0.19/0.47 upairset2IR: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairset2IR,definition,
% 0.19/0.47 ( upairset2IR
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binunionIR_type,type,
% 0.19/0.47 binunionIR: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binunionIR,definition,
% 0.19/0.47 ( binunionIR
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binunionEcases_type,type,
% 0.19/0.47 binunionEcases: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binunionEcases,definition,
% 0.19/0.47 ( binunionEcases
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.19/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.47 => ( ( ( in @ Xx @ A )
% 0.19/0.47 => Xphi )
% 0.19/0.47 => ( ( ( in @ Xx @ B )
% 0.19/0.47 => Xphi )
% 0.19/0.47 => Xphi ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binunionE_type,type,
% 0.19/0.47 binunionE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binunionE,definition,
% 0.19/0.47 ( binunionE
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.19/0.47 => ( ( in @ Xx @ A )
% 0.19/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binunionLsub_type,type,
% 0.19/0.47 binunionLsub: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binunionLsub,definition,
% 0.19/0.47 ( binunionLsub
% 0.19/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binunionRsub_type,type,
% 0.19/0.47 binunionRsub: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binunionRsub,definition,
% 0.19/0.47 ( binunionRsub
% 0.19/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersect_type,type,
% 0.19/0.47 binintersect: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectI_type,type,
% 0.19/0.47 binintersectI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectI,definition,
% 0.19/0.47 ( binintersectI
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset5_type,type,
% 0.19/0.47 binintersectSubset5: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset5,definition,
% 0.19/0.47 ( binintersectSubset5
% 0.19/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.47 ( ( subset @ C @ A )
% 0.19/0.47 => ( ( subset @ C @ B )
% 0.19/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectEL_type,type,
% 0.19/0.47 binintersectEL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectEL,definition,
% 0.19/0.47 ( binintersectEL
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectLsub_type,type,
% 0.19/0.47 binintersectLsub: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectLsub,definition,
% 0.19/0.47 ( binintersectLsub
% 0.19/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset2_type,type,
% 0.19/0.47 binintersectSubset2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset2,definition,
% 0.19/0.47 ( binintersectSubset2
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( ( binintersect @ A @ B )
% 0.19/0.47 = A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset3_type,type,
% 0.19/0.47 binintersectSubset3: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset3,definition,
% 0.19/0.47 ( binintersectSubset3
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( ( binintersect @ A @ B )
% 0.19/0.47 = B )
% 0.19/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectER_type,type,
% 0.19/0.47 binintersectER: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectER,definition,
% 0.19/0.47 ( binintersectER
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.19/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(disjointsetsI1_type,type,
% 0.19/0.47 disjointsetsI1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(disjointsetsI1,definition,
% 0.19/0.47 ( disjointsetsI1
% 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 => ( ( binintersect @ A @ B )
% 0.19/0.47 = emptyset ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectRsub_type,type,
% 0.19/0.47 binintersectRsub: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectRsub,definition,
% 0.19/0.47 ( binintersectRsub
% 0.19/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset4_type,type,
% 0.19/0.47 binintersectSubset4: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset4,definition,
% 0.19/0.47 ( binintersectSubset4
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( subset @ B @ A )
% 0.19/0.47 => ( ( binintersect @ A @ B )
% 0.19/0.47 = B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset1_type,type,
% 0.19/0.47 binintersectSubset1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(binintersectSubset1,definition,
% 0.19/0.47 ( binintersectSubset1
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( ( binintersect @ A @ B )
% 0.19/0.47 = A )
% 0.19/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(bs114d_type,type,
% 0.19/0.47 bs114d: $o ).
% 0.19/0.47
% 0.19/0.47 thf(bs114d,definition,
% 0.19/0.47 ( bs114d
% 0.19/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.19/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.19/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(regular_type,type,
% 0.19/0.47 regular: $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminus_type,type,
% 0.19/0.47 setminus: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(setminusI_type,type,
% 0.19/0.47 setminusI: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusI,definition,
% 0.19/0.47 ( setminusI
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ~ ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusEL_type,type,
% 0.19/0.47 setminusEL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusEL,definition,
% 0.19/0.47 ( setminusEL
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusER_type,type,
% 0.19/0.47 setminusER: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusER,definition,
% 0.19/0.47 ( setminusER
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusSubset2_type,type,
% 0.19/0.47 setminusSubset2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusSubset2,definition,
% 0.19/0.47 ( setminusSubset2
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( subset @ A @ B )
% 0.19/0.47 => ( ( setminus @ A @ B )
% 0.19/0.47 = emptyset ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusERneg_type,type,
% 0.19/0.47 setminusERneg: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusERneg,definition,
% 0.19/0.47 ( setminusERneg
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.47 => ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusELneg_type,type,
% 0.19/0.47 setminusELneg: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusELneg,definition,
% 0.19/0.47 ( setminusELneg
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.19/0.47 => ( ~ ( in @ Xx @ B )
% 0.19/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusILneg_type,type,
% 0.19/0.47 setminusILneg: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusILneg,definition,
% 0.19/0.47 ( setminusILneg
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ~ ( in @ Xx @ A )
% 0.19/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusIRneg_type,type,
% 0.19/0.47 setminusIRneg: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusIRneg,definition,
% 0.19/0.47 ( setminusIRneg
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ B )
% 0.19/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusLsub_type,type,
% 0.19/0.47 setminusLsub: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusLsub,definition,
% 0.19/0.47 ( setminusLsub
% 0.19/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setminusSubset1_type,type,
% 0.19/0.47 setminusSubset1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setminusSubset1,definition,
% 0.19/0.47 ( setminusSubset1
% 0.19/0.47 = ( ! [A: $i,B: $i] :
% 0.19/0.47 ( ( ( setminus @ A @ B )
% 0.19/0.47 = emptyset )
% 0.19/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(symdiff_type,type,
% 0.19/0.47 symdiff: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffE_type,type,
% 0.19/0.47 symdiffE: $o ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffE,definition,
% 0.19/0.47 ( symdiffE
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.19/0.47 => ! [Xphi: $o] :
% 0.19/0.47 ( ( ( in @ Xx @ A )
% 0.19/0.47 => ( ~ ( in @ Xx @ B )
% 0.19/0.47 => Xphi ) )
% 0.19/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.19/0.47 => ( ( in @ Xx @ B )
% 0.19/0.47 => Xphi ) )
% 0.19/0.47 => Xphi ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffI1_type,type,
% 0.19/0.47 symdiffI1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffI1,definition,
% 0.19/0.47 ( symdiffI1
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ~ ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffI2_type,type,
% 0.19/0.47 symdiffI2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffI2,definition,
% 0.19/0.47 ( symdiffI2
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ~ ( in @ Xx @ A )
% 0.19/0.47 => ( ( in @ Xx @ B )
% 0.19/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffIneg1_type,type,
% 0.19/0.47 symdiffIneg1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffIneg1,definition,
% 0.19/0.47 ( symdiffIneg1
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( ( in @ Xx @ B )
% 0.19/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffIneg2_type,type,
% 0.19/0.47 symdiffIneg2: $o ).
% 0.19/0.47
% 0.19/0.47 thf(symdiffIneg2,definition,
% 0.19/0.47 ( symdiffIneg2
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ~ ( in @ Xx @ A )
% 0.19/0.47 => ( ~ ( in @ Xx @ B )
% 0.19/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(iskpair_type,type,
% 0.19/0.47 iskpair: $i > $o ).
% 0.19/0.47
% 0.19/0.47 thf(secondinupair_type,type,
% 0.19/0.47 secondinupair: $o ).
% 0.19/0.47
% 0.19/0.47 thf(secondinupair,definition,
% 0.19/0.47 ( secondinupair
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairIL_type,type,
% 0.19/0.47 setukpairIL: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairIL,definition,
% 0.19/0.47 ( setukpairIL
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairIR_type,type,
% 0.19/0.47 setukpairIR: $o ).
% 0.19/0.47
% 0.19/0.47 thf(setukpairIR,definition,
% 0.19/0.47 ( setukpairIR
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(kpairiskpair_type,type,
% 0.19/0.47 kpairiskpair: $o ).
% 0.19/0.47
% 0.19/0.47 thf(kpairiskpair,definition,
% 0.19/0.47 ( kpairiskpair
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(kpair_type,type,
% 0.19/0.47 kpair: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(kpairp_type,type,
% 0.19/0.47 kpairp: $o ).
% 0.19/0.47
% 0.19/0.47 thf(kpairp,definition,
% 0.19/0.47 ( kpairp
% 0.19/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(cartprod_type,type,
% 0.19/0.47 cartprod: $i > $i > $i ).
% 0.19/0.47
% 0.19/0.47 thf(singletonsubset_type,type,
% 0.19/0.47 singletonsubset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(singletonsubset,definition,
% 0.19/0.47 ( singletonsubset
% 0.19/0.47 = ( ! [A: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(singletoninpowerset_type,type,
% 0.19/0.47 singletoninpowerset: $o ).
% 0.19/0.47
% 0.19/0.47 thf(singletoninpowerset,definition,
% 0.19/0.47 ( singletoninpowerset
% 0.19/0.47 = ( ! [A: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(singletoninpowunion_type,type,
% 0.19/0.47 singletoninpowunion: $o ).
% 0.19/0.47
% 0.19/0.47 thf(singletoninpowunion,definition,
% 0.19/0.47 ( singletoninpowunion
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairset2E_type,type,
% 0.19/0.47 upairset2E: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairset2E,definition,
% 0.19/0.47 ( upairset2E
% 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(upairsubunion_type,type,
% 0.19/0.47 upairsubunion: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairsubunion,definition,
% 0.19/0.47 ( upairsubunion
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(upairinpowunion_type,type,
% 0.19/0.47 upairinpowunion: $o ).
% 0.19/0.47
% 0.19/0.47 thf(upairinpowunion,definition,
% 0.19/0.47 ( upairinpowunion
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(ubforcartprodlem1_type,type,
% 0.19/0.47 ubforcartprodlem1: $o ).
% 0.19/0.47
% 0.19/0.47 thf(ubforcartprodlem1,definition,
% 0.19/0.47 ( ubforcartprodlem1
% 0.19/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.47 ( ( in @ Xx @ A )
% 0.19/0.47 => ! [Xy: $i] :
% 0.19/0.47 ( ( in @ Xy @ B )
% 0.19/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(ubforcartprodlem2_type,type,
% 0.19/0.48 ubforcartprodlem2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(ubforcartprodlem2,definition,
% 0.19/0.48 ( ubforcartprodlem2
% 0.19/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(ubforcartprodlem3_type,type,
% 0.19/0.48 ubforcartprodlem3: $o ).
% 0.19/0.48
% 0.19/0.48 thf(ubforcartprodlem3,definition,
% 0.19/0.48 ( ubforcartprodlem3
% 0.19/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodpairin_type,type,
% 0.19/0.48 cartprodpairin: $o ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodpairin,definition,
% 0.19/0.48 ( cartprodpairin
% 0.19/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodmempair1_type,type,
% 0.19/0.48 cartprodmempair1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodmempair1,definition,
% 0.19/0.48 ( cartprodmempair1
% 0.19/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.48 => ? [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 & ? [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 & ( Xu
% 0.19/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodmempair_type,type,
% 0.19/0.48 cartprodmempair: $o ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodmempair,definition,
% 0.19/0.48 ( cartprodmempair
% 0.19/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.48 => ( iskpair @ Xu ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setunionE2_type,type,
% 0.19/0.48 setunionE2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setunionE2,definition,
% 0.19/0.48 ( setunionE2
% 0.19/0.48 = ( ! [A: $i,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ ( setunion @ A ) )
% 0.19/0.48 => ? [X: $i] :
% 0.19/0.48 ( ( in @ X @ A )
% 0.19/0.48 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setunionsingleton1_type,type,
% 0.19/0.48 setunionsingleton1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setunionsingleton1,definition,
% 0.19/0.48 ( setunionsingleton1
% 0.19/0.48 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setunionsingleton2_type,type,
% 0.19/0.48 setunionsingleton2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setunionsingleton2,definition,
% 0.19/0.48 ( setunionsingleton2
% 0.19/0.48 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setunionsingleton_type,type,
% 0.19/0.48 setunionsingleton: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setunionsingleton,definition,
% 0.19/0.48 ( setunionsingleton
% 0.19/0.48 = ( ! [Xx: $i] :
% 0.19/0.48 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.19/0.48 = Xx ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(singleton_type,type,
% 0.19/0.48 singleton: $i > $o ).
% 0.19/0.48
% 0.19/0.48 thf(singletonprop_type,type,
% 0.19/0.48 singletonprop: $o ).
% 0.19/0.48
% 0.19/0.48 thf(singletonprop,definition,
% 0.19/0.48 ( singletonprop
% 0.19/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ A )
% 0.19/0.48 => ( ( Xphi @ Xx )
% 0.19/0.48 => ( ( Xphi @ Xy )
% 0.19/0.48 => ( Xx = Xy ) ) ) ) )
% 0.19/0.48 => ( ? [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 & ( Xphi @ Xx ) )
% 0.19/0.48 => ( singleton
% 0.19/0.48 @ ( dsetconstr @ A
% 0.19/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(ex1_type,type,
% 0.19/0.48 ex1: $i > ( $i > $o ) > $o ).
% 0.19/0.48
% 0.19/0.48 thf(ex1,definition,
% 0.19/0.48 ( ex1
% 0.19/0.48 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.19/0.48 ( singleton
% 0.19/0.48 @ ( dsetconstr @ A
% 0.19/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(ex1E1_type,type,
% 0.19/0.48 ex1E1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(ex1E1,definition,
% 0.19/0.48 ( ex1E1
% 0.19/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.48 ( ( ex1 @ A
% 0.19/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.48 => ? [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(ex1I_type,type,
% 0.19/0.48 ex1I: $o ).
% 0.19/0.48
% 0.19/0.48 thf(ex1I,definition,
% 0.19/0.48 ( ex1I
% 0.19/0.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( Xphi @ Xx )
% 0.19/0.48 => ( ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ A )
% 0.19/0.48 => ( ( Xphi @ Xy )
% 0.19/0.48 => ( Xy = Xx ) ) )
% 0.19/0.48 => ( ex1 @ A
% 0.19/0.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(ex1I2_type,type,
% 0.19/0.48 ex1I2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(ex1I2,definition,
% 0.19/0.48 ( ex1I2
% 0.19/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ A )
% 0.19/0.48 => ( ( Xphi @ Xx )
% 0.19/0.48 => ( ( Xphi @ Xy )
% 0.19/0.48 => ( Xx = Xy ) ) ) ) )
% 0.19/0.48 => ( ? [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 & ( Xphi @ Xx ) )
% 0.19/0.48 => ( ex1 @ A
% 0.19/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(singletonsuniq_type,type,
% 0.19/0.48 singletonsuniq: $o ).
% 0.19/0.48
% 0.19/0.48 thf(singletonsuniq,definition,
% 0.19/0.48 ( singletonsuniq
% 0.19/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.48 ( ( ( setadjoin @ Xx @ emptyset )
% 0.19/0.48 = ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(atmost1p_type,type,
% 0.19/0.48 atmost1p: $i > $o ).
% 0.19/0.48
% 0.19/0.48 thf(atleast2p_type,type,
% 0.19/0.48 atleast2p: $i > $o ).
% 0.19/0.48
% 0.19/0.48 thf(atmost2p_type,type,
% 0.19/0.48 atmost2p: $i > $o ).
% 0.19/0.48
% 0.19/0.48 thf(upairsetp_type,type,
% 0.19/0.48 upairsetp: $i > $o ).
% 0.19/0.48
% 0.19/0.48 thf(setukpairinjL1_type,type,
% 0.19/0.48 setukpairinjL1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setukpairinjL1,definition,
% 0.19/0.48 ( setukpairinjL1
% 0.19/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.48 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(kfstsingleton_type,type,
% 0.19/0.48 kfstsingleton: $o ).
% 0.19/0.48
% 0.19/0.48 thf(kfstsingleton,definition,
% 0.19/0.48 ( kfstsingleton
% 0.19/0.48 = ( ! [Xu: $i] :
% 0.19/0.48 ( ( iskpair @ Xu )
% 0.19/0.48 => ( singleton
% 0.19/0.48 @ ( dsetconstr @ ( setunion @ Xu )
% 0.19/0.48 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(theprop_type,type,
% 0.19/0.48 theprop: $o ).
% 0.19/0.48
% 0.19/0.48 thf(theprop,definition,
% 0.19/0.48 ( theprop
% 0.19/0.48 = ( ! [X: $i] :
% 0.19/0.48 ( ( singleton @ X )
% 0.19/0.48 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(kfst_type,type,
% 0.19/0.48 kfst: $i > $i ).
% 0.19/0.48
% 0.19/0.48 thf(kfstpairEq_type,type,
% 0.19/0.48 kfstpairEq: $o ).
% 0.19/0.48
% 0.19/0.48 thf(kfstpairEq,definition,
% 0.19/0.48 ( kfstpairEq
% 0.19/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.48 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.19/0.48 = Xx ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodfstin_type,type,
% 0.19/0.48 cartprodfstin: $o ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodfstin,definition,
% 0.19/0.48 ( cartprodfstin
% 0.19/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.48 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setukpairinjL2_type,type,
% 0.19/0.48 setukpairinjL2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setukpairinjL2,definition,
% 0.19/0.48 ( setukpairinjL2
% 0.19/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setukpairinjL_type,type,
% 0.19/0.48 setukpairinjL: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setukpairinjL,definition,
% 0.19/0.48 ( setukpairinjL
% 0.19/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.48 ( ( ( kpair @ Xx @ Xy )
% 0.19/0.48 = ( kpair @ Xz @ Xu ) )
% 0.19/0.48 => ( Xx = Xz ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setukpairinjR11_type,type,
% 0.19/0.48 setukpairinjR11: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setukpairinjR11,definition,
% 0.19/0.48 ( setukpairinjR11
% 0.19/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.48 ( ( Xx = Xy )
% 0.19/0.48 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setukpairinjR12_type,type,
% 0.19/0.48 setukpairinjR12: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setukpairinjR12,definition,
% 0.19/0.48 ( setukpairinjR12
% 0.19/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.48 ( ( Xx = Xy )
% 0.19/0.48 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.48 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setukpairinjR1_type,type,
% 0.19/0.48 setukpairinjR1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setukpairinjR1,definition,
% 0.19/0.48 ( setukpairinjR1
% 0.19/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.48 => ( ( Xz = Xu )
% 0.19/0.48 => ( Xy = Xu ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(upairequniteq_type,type,
% 0.19/0.48 upairequniteq: $o ).
% 0.19/0.48
% 0.19/0.48 thf(upairequniteq,definition,
% 0.19/0.48 ( upairequniteq
% 0.19/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.19/0.48 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.19/0.48 = ( setadjoin @ Xz @ emptyset ) )
% 0.19/0.48 => ( Xx = Xy ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setukpairinjR2_type,type,
% 0.19/0.48 setukpairinjR2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setukpairinjR2,definition,
% 0.19/0.48 ( setukpairinjR2
% 0.19/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.19/0.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.19/0.48 => ( Xy = Xu ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setukpairinjR_type,type,
% 0.19/0.48 setukpairinjR: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setukpairinjR,definition,
% 0.19/0.48 ( setukpairinjR
% 0.19/0.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.19/0.48 ( ( ( kpair @ Xx @ Xy )
% 0.19/0.48 = ( kpair @ Xz @ Xu ) )
% 0.19/0.48 => ( Xy = Xu ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(ksndsingleton_type,type,
% 0.19/0.48 ksndsingleton: $o ).
% 0.19/0.48
% 0.19/0.48 thf(ksndsingleton,definition,
% 0.19/0.48 ( ksndsingleton
% 0.19/0.48 = ( ! [Xu: $i] :
% 0.19/0.48 ( ( iskpair @ Xu )
% 0.19/0.48 => ( singleton
% 0.19/0.48 @ ( dsetconstr @ ( setunion @ Xu )
% 0.19/0.48 @ ^ [Xx: $i] :
% 0.19/0.48 ( Xu
% 0.19/0.48 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(ksnd_type,type,
% 0.19/0.48 ksnd: $i > $i ).
% 0.19/0.48
% 0.19/0.48 thf(ksndpairEq_type,type,
% 0.19/0.48 ksndpairEq: $o ).
% 0.19/0.48
% 0.19/0.48 thf(ksndpairEq,definition,
% 0.19/0.48 ( ksndpairEq
% 0.19/0.48 = ( ! [Xx: $i,Xy: $i] :
% 0.19/0.48 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.19/0.48 = Xy ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(kpairsurjEq_type,type,
% 0.19/0.48 kpairsurjEq: $o ).
% 0.19/0.48
% 0.19/0.48 thf(kpairsurjEq,definition,
% 0.19/0.48 ( kpairsurjEq
% 0.19/0.48 = ( ! [Xu: $i] :
% 0.19/0.48 ( ( iskpair @ Xu )
% 0.19/0.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.19/0.48 = Xu ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodsndin_type,type,
% 0.19/0.48 cartprodsndin: $o ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodsndin,definition,
% 0.19/0.48 ( cartprodsndin
% 0.19/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.48 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodpairmemEL_type,type,
% 0.19/0.48 cartprodpairmemEL: $o ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodpairmemEL,definition,
% 0.19/0.48 ( cartprodpairmemEL
% 0.19/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.19/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.19/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodpairmemER_type,type,
% 0.19/0.48 cartprodpairmemER: $o ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodpairmemER,definition,
% 0.19/0.48 ( cartprodpairmemER
% 0.19/0.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.19/0.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.19/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodmempaircEq_type,type,
% 0.19/0.48 cartprodmempaircEq: $o ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodmempaircEq,definition,
% 0.19/0.48 ( cartprodmempaircEq
% 0.19/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 => ( ( kpair @ Xx @ Xy )
% 0.19/0.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodfstpairEq_type,type,
% 0.19/0.48 cartprodfstpairEq: $o ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodfstpairEq,definition,
% 0.19/0.48 ( cartprodfstpairEq
% 0.19/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.19/0.48 = Xx ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodsndpairEq_type,type,
% 0.19/0.48 cartprodsndpairEq: $o ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodsndpairEq,definition,
% 0.19/0.48 ( cartprodsndpairEq
% 0.19/0.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.19/0.48 = Xy ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodpairsurjEq_type,type,
% 0.19/0.48 cartprodpairsurjEq: $o ).
% 0.19/0.48
% 0.19/0.48 thf(cartprodpairsurjEq,definition,
% 0.19/0.48 ( cartprodpairsurjEq
% 0.19/0.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.19/0.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.19/0.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.19/0.48 = Xu ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(breln_type,type,
% 0.19/0.48 breln: $i > $i > $i > $o ).
% 0.19/0.48
% 0.19/0.48 thf(breln,definition,
% 0.19/0.48 ( breln
% 0.19/0.48 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(dpsetconstr_type,type,
% 0.19/0.48 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.19/0.48
% 0.19/0.48 thf(dpsetconstrI_type,type,
% 0.19/0.48 dpsetconstrI: $o ).
% 0.19/0.48
% 0.19/0.48 thf(dpsetconstrI,definition,
% 0.19/0.48 ( dpsetconstrI
% 0.19/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 => ( ( Xphi @ Xx @ Xy )
% 0.19/0.48 => ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.48 @ ( dpsetconstr @ A @ B
% 0.19/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(dpsetconstrSub_type,type,
% 0.19/0.48 dpsetconstrSub: $o ).
% 0.19/0.48
% 0.19/0.48 thf(dpsetconstrSub,definition,
% 0.19/0.48 ( dpsetconstrSub
% 0.19/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.19/0.48 ( subset
% 0.19/0.48 @ ( dpsetconstr @ A @ B
% 0.19/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.19/0.48 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setOfPairsIsBReln_type,type,
% 0.19/0.48 setOfPairsIsBReln: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setOfPairsIsBReln,definition,
% 0.19/0.48 ( setOfPairsIsBReln
% 0.19/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.19/0.48 ( breln @ A @ B
% 0.19/0.48 @ ( dpsetconstr @ A @ B
% 0.19/0.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(dpsetconstrERa_type,type,
% 0.19/0.48 dpsetconstrERa: $o ).
% 0.19/0.48
% 0.19/0.48 thf(dpsetconstrERa,definition,
% 0.19/0.48 ( dpsetconstrERa
% 0.19/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.48 @ ( dpsetconstr @ A @ B
% 0.19/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.48 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(dpsetconstrEL1_type,type,
% 0.19/0.48 dpsetconstrEL1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(dpsetconstrEL1,definition,
% 0.19/0.48 ( dpsetconstrEL1
% 0.19/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.48 @ ( dpsetconstr @ A @ B
% 0.19/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.48 => ( in @ Xx @ A ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(dpsetconstrEL2_type,type,
% 0.19/0.48 dpsetconstrEL2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(dpsetconstrEL2,definition,
% 0.19/0.48 ( dpsetconstrEL2
% 0.19/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.48 @ ( dpsetconstr @ A @ B
% 0.19/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.48 => ( in @ Xy @ B ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(dpsetconstrER_type,type,
% 0.19/0.48 dpsetconstrER: $o ).
% 0.19/0.48
% 0.19/0.48 thf(dpsetconstrER,definition,
% 0.19/0.48 ( dpsetconstrER
% 0.19/0.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.19/0.48 ( ( in @ ( kpair @ Xx @ Xy )
% 0.19/0.48 @ ( dpsetconstr @ A @ B
% 0.19/0.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.19/0.48 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(func_type,type,
% 0.19/0.48 func: $i > $i > $i > $o ).
% 0.19/0.48
% 0.19/0.48 thf(funcSet_type,type,
% 0.19/0.48 funcSet: $i > $i > $i ).
% 0.19/0.48
% 0.19/0.48 thf(funcImageSingleton_type,type,
% 0.19/0.48 funcImageSingleton: $o ).
% 0.19/0.48
% 0.19/0.48 thf(funcImageSingleton,definition,
% 0.19/0.48 ( funcImageSingleton
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( func @ A @ B @ Xf )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( singleton
% 0.19/0.48 @ ( dsetconstr @ B
% 0.19/0.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(apProp_type,type,
% 0.19/0.48 apProp: $o ).
% 0.19/0.48
% 0.19/0.48 thf(apProp,definition,
% 0.19/0.48 ( apProp
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( func @ A @ B @ Xf )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( in
% 0.19/0.48 @ ( setunion
% 0.19/0.48 @ ( dsetconstr @ B
% 0.19/0.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.19/0.48 @ B ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(ap_type,type,
% 0.19/0.48 ap: $i > $i > $i > $i > $i ).
% 0.19/0.48
% 0.19/0.48 thf(app_type,type,
% 0.19/0.48 app: $o ).
% 0.19/0.48
% 0.19/0.48 thf(app,definition,
% 0.19/0.48 ( app
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( func @ A @ B @ Xf )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(infuncsetfunc_type,type,
% 0.19/0.48 infuncsetfunc: $o ).
% 0.19/0.48
% 0.19/0.48 thf(infuncsetfunc,definition,
% 0.19/0.48 ( infuncsetfunc
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.48 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(ap2p_type,type,
% 0.19/0.48 ap2p: $o ).
% 0.19/0.48
% 0.19/0.48 thf(ap2p,definition,
% 0.19/0.48 ( ap2p
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(funcinfuncset_type,type,
% 0.19/0.48 funcinfuncset: $o ).
% 0.19/0.48
% 0.19/0.48 thf(funcinfuncset,definition,
% 0.19/0.48 ( funcinfuncset
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( func @ A @ B @ Xf )
% 0.19/0.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(lamProp_type,type,
% 0.19/0.48 lamProp: $o ).
% 0.19/0.48
% 0.19/0.48 thf(lamProp,definition,
% 0.19/0.48 ( lamProp
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.48 => ( func @ A @ B
% 0.19/0.48 @ ( dpsetconstr @ A @ B
% 0.19/0.48 @ ^ [Xx: $i,Xy: $i] :
% 0.19/0.48 ( ( Xf @ Xx )
% 0.19/0.48 = Xy ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(lam_type,type,
% 0.19/0.48 lam: $i > $i > ( $i > $i ) > $i ).
% 0.19/0.48
% 0.19/0.48 thf(lamp_type,type,
% 0.19/0.48 lamp: $o ).
% 0.19/0.48
% 0.19/0.48 thf(lamp,definition,
% 0.19/0.48 ( lamp
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.48 => ( func @ A @ B
% 0.19/0.48 @ ( lam @ A @ B
% 0.19/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(lam2p_type,type,
% 0.19/0.48 lam2p: $o ).
% 0.19/0.48
% 0.19/0.48 thf(lam2p,definition,
% 0.19/0.48 ( lam2p
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.48 => ( in
% 0.19/0.48 @ ( lam @ A @ B
% 0.19/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.19/0.48 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(brelnall1_type,type,
% 0.19/0.48 brelnall1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(brelnall1,definition,
% 0.19/0.48 ( brelnall1
% 0.19/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.48 ( ( breln @ A @ B @ R )
% 0.19/0.48 => ! [Xphi: $i > $o] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ R )
% 0.19/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(brelnall2_type,type,
% 0.19/0.48 brelnall2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(brelnall2,definition,
% 0.19/0.48 ( brelnall2
% 0.19/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.48 ( ( breln @ A @ B @ R )
% 0.19/0.48 => ! [Xphi: $i > $o] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ R )
% 0.19/0.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(ex1E2_type,type,
% 0.19/0.48 ex1E2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(ex1E2,definition,
% 0.19/0.48 ( ex1E2
% 0.19/0.48 = ( ! [A: $i,Xphi: $i > $o] :
% 0.19/0.48 ( ( ex1 @ A
% 0.19/0.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ A )
% 0.19/0.48 => ( ( Xphi @ Xx )
% 0.19/0.48 => ( ( Xphi @ Xy )
% 0.19/0.48 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(funcGraphProp1_type,type,
% 0.19/0.48 funcGraphProp1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(funcGraphProp1,definition,
% 0.19/0.48 ( funcGraphProp1
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( func @ A @ B @ Xf )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(funcGraphProp3_type,type,
% 0.19/0.48 funcGraphProp3: $o ).
% 0.19/0.48
% 0.19/0.48 thf(funcGraphProp3,definition,
% 0.19/0.48 ( funcGraphProp3
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(funcGraphProp2_type,type,
% 0.19/0.48 funcGraphProp2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(funcGraphProp2,definition,
% 0.19/0.48 ( funcGraphProp2
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( func @ A @ B @ Xf )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.19/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.48 = Xy ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(funcextLem_type,type,
% 0.19/0.48 funcextLem: $o ).
% 0.19/0.48
% 0.19/0.48 thf(funcextLem,definition,
% 0.19/0.48 ( funcextLem
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( func @ A @ B @ Xf )
% 0.19/0.48 => ! [Xg: $i] :
% 0.19/0.48 ( ( func @ A @ B @ Xg )
% 0.19/0.48 => ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.19/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(funcGraphProp4_type,type,
% 0.19/0.48 funcGraphProp4: $o ).
% 0.19/0.48
% 0.19/0.48 thf(funcGraphProp4,definition,
% 0.19/0.48 ( funcGraphProp4
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.19/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.48 = Xy ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(subbreln_type,type,
% 0.19/0.48 subbreln: $o ).
% 0.19/0.48
% 0.19/0.48 thf(subbreln,definition,
% 0.19/0.48 ( subbreln
% 0.19/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.48 ( ( breln @ A @ B @ R )
% 0.19/0.48 => ! [S: $i] :
% 0.19/0.48 ( ( breln @ A @ B @ S )
% 0.19/0.48 => ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.19/0.48 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(eqbreln_type,type,
% 0.19/0.48 eqbreln: $o ).
% 0.19/0.48
% 0.19/0.48 thf(eqbreln,definition,
% 0.19/0.48 ( eqbreln
% 0.19/0.48 = ( ! [A: $i,B: $i,R: $i] :
% 0.19/0.48 ( ( breln @ A @ B @ R )
% 0.19/0.48 => ! [S: $i] :
% 0.19/0.48 ( ( breln @ A @ B @ S )
% 0.19/0.48 => ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.19/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.19/0.48 => ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ B )
% 0.19/0.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.19/0.48 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.19/0.48 => ( R = S ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(funcext_type,type,
% 0.19/0.48 funcext: $o ).
% 0.19/0.48
% 0.19/0.48 thf(funcext,definition,
% 0.19/0.48 ( funcext
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( func @ A @ B @ Xf )
% 0.19/0.48 => ! [Xg: $i] :
% 0.19/0.48 ( ( func @ A @ B @ Xg )
% 0.19/0.48 => ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.19/0.48 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(funcext2_type,type,
% 0.19/0.48 funcext2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(funcext2,definition,
% 0.19/0.48 ( funcext2
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.48 => ! [Xg: $i] :
% 0.19/0.48 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.19/0.48 => ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.19/0.48 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(ap2apEq1_type,type,
% 0.19/0.48 ap2apEq1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(ap2apEq1,definition,
% 0.19/0.48 ( ap2apEq1
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.48 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(ap2apEq2_type,type,
% 0.19/0.48 ap2apEq2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(ap2apEq2,definition,
% 0.19/0.48 ( ap2apEq2
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( func @ A @ B @ Xf )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.19/0.48 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(beta1_type,type,
% 0.19/0.48 beta1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(beta1,definition,
% 0.19/0.48 ( beta1
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( ap @ A @ B
% 0.19/0.48 @ ( lam @ A @ B
% 0.19/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.19/0.48 @ Xx )
% 0.19/0.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(eta1_type,type,
% 0.19/0.48 eta1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(eta1,definition,
% 0.19/0.48 ( eta1
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( func @ A @ B @ Xf )
% 0.19/0.48 => ( ( lam @ A @ B
% 0.19/0.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.19/0.48 = Xf ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(lam2lamEq_type,type,
% 0.19/0.48 lam2lamEq: $o ).
% 0.19/0.48
% 0.19/0.48 thf(lam2lamEq,definition,
% 0.19/0.48 ( lam2lamEq
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.48 => ( ( lam @ A @ B
% 0.19/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.19/0.48 = ( lam @ A @ B
% 0.19/0.48 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(beta2_type,type,
% 0.19/0.48 beta2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(beta2,definition,
% 0.19/0.48 ( beta2
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.19/0.48 ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( ap @ A @ B
% 0.19/0.48 @ ( lam @ A @ B
% 0.19/0.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.19/0.48 @ Xx )
% 0.19/0.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(eta2_type,type,
% 0.19/0.48 eta2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(eta2,definition,
% 0.19/0.48 ( eta2
% 0.19/0.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.19/0.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.19/0.48 => ( ( lam @ A @ B
% 0.19/0.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.19/0.48 = Xf ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(iffalseProp1_type,type,
% 0.19/0.48 iffalseProp1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(iffalseProp1,definition,
% 0.19/0.48 ( iffalseProp1
% 0.19/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ A )
% 0.19/0.48 => ( ~ Xphi
% 0.19/0.48 => ( in @ Xy
% 0.19/0.48 @ ( dsetconstr @ A
% 0.19/0.48 @ ^ [Xz: $i] :
% 0.19/0.48 ( ( Xphi
% 0.19/0.48 & ( Xz = Xx ) )
% 0.19/0.48 | ( ~ Xphi
% 0.19/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(iffalseProp2_type,type,
% 0.19/0.48 iffalseProp2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(iffalseProp2,definition,
% 0.19/0.48 ( iffalseProp2
% 0.19/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ A )
% 0.19/0.48 => ( ~ Xphi
% 0.19/0.48 => ( ( dsetconstr @ A
% 0.19/0.48 @ ^ [Xz: $i] :
% 0.19/0.48 ( ( Xphi
% 0.19/0.48 & ( Xz = Xx ) )
% 0.19/0.48 | ( ~ Xphi
% 0.19/0.48 & ( Xz = Xy ) ) ) )
% 0.19/0.48 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(iftrueProp1_type,type,
% 0.19/0.48 iftrueProp1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(iftrueProp1,definition,
% 0.19/0.48 ( iftrueProp1
% 0.19/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ A )
% 0.19/0.48 => ( Xphi
% 0.19/0.48 => ( in @ Xx
% 0.19/0.48 @ ( dsetconstr @ A
% 0.19/0.48 @ ^ [Xz: $i] :
% 0.19/0.48 ( ( Xphi
% 0.19/0.48 & ( Xz = Xx ) )
% 0.19/0.48 | ( ~ Xphi
% 0.19/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(iftrueProp2_type,type,
% 0.19/0.48 iftrueProp2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(iftrueProp2,definition,
% 0.19/0.48 ( iftrueProp2
% 0.19/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ A )
% 0.19/0.48 => ( Xphi
% 0.19/0.48 => ( ( dsetconstr @ A
% 0.19/0.48 @ ^ [Xz: $i] :
% 0.19/0.48 ( ( Xphi
% 0.19/0.48 & ( Xz = Xx ) )
% 0.19/0.48 | ( ~ Xphi
% 0.19/0.48 & ( Xz = Xy ) ) ) )
% 0.19/0.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(ifSingleton_type,type,
% 0.19/0.48 ifSingleton: $o ).
% 0.19/0.48
% 0.19/0.48 thf(ifSingleton,definition,
% 0.19/0.48 ( ifSingleton
% 0.19/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ A )
% 0.19/0.48 => ( singleton
% 0.19/0.48 @ ( dsetconstr @ A
% 0.19/0.48 @ ^ [Xz: $i] :
% 0.19/0.48 ( ( Xphi
% 0.19/0.48 & ( Xz = Xx ) )
% 0.19/0.48 | ( ~ Xphi
% 0.19/0.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(if_type,type,
% 0.19/0.48 if: $i > $o > $i > $i > $i ).
% 0.19/0.48
% 0.19/0.48 thf(ifp_type,type,
% 0.19/0.48 ifp: $o ).
% 0.19/0.48
% 0.19/0.48 thf(ifp,definition,
% 0.19/0.48 ( ifp
% 0.19/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ A )
% 0.19/0.48 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(theeq_type,type,
% 0.19/0.48 theeq: $o ).
% 0.19/0.48
% 0.19/0.48 thf(theeq,definition,
% 0.19/0.48 ( theeq
% 0.19/0.48 = ( ! [X: $i] :
% 0.19/0.48 ( ( singleton @ X )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ X )
% 0.19/0.48 => ( ( setunion @ X )
% 0.19/0.48 = Xx ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(iftrue_type,type,
% 0.19/0.48 iftrue: $o ).
% 0.19/0.48
% 0.19/0.48 thf(iftrue,definition,
% 0.19/0.48 ( iftrue
% 0.19/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ A )
% 0.19/0.48 => ( Xphi
% 0.19/0.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.19/0.48 = Xx ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(iffalse_type,type,
% 0.19/0.48 iffalse: $o ).
% 0.19/0.48
% 0.19/0.48 thf(iffalse,definition,
% 0.19/0.48 ( iffalse
% 0.19/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ A )
% 0.19/0.48 => ( ~ Xphi
% 0.19/0.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.19/0.48 = Xy ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(iftrueorfalse_type,type,
% 0.19/0.48 iftrueorfalse: $o ).
% 0.19/0.48
% 0.19/0.48 thf(iftrueorfalse,definition,
% 0.19/0.48 ( iftrueorfalse
% 0.19/0.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ! [Xy: $i] :
% 0.19/0.48 ( ( in @ Xy @ A )
% 0.19/0.48 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(binintersectT_lem_type,type,
% 0.19/0.48 binintersectT_lem: $o ).
% 0.19/0.48
% 0.19/0.48 thf(binintersectT_lem,definition,
% 0.19/0.48 ( binintersectT_lem
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(binunionT_lem_type,type,
% 0.19/0.48 binunionT_lem: $o ).
% 0.19/0.48
% 0.19/0.48 thf(binunionT_lem,definition,
% 0.19/0.48 ( binunionT_lem
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(powersetT_lem_type,type,
% 0.19/0.48 powersetT_lem: $o ).
% 0.19/0.48
% 0.19/0.48 thf(powersetT_lem,definition,
% 0.19/0.48 ( powersetT_lem
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setminusT_lem_type,type,
% 0.19/0.48 setminusT_lem: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setminusT_lem,definition,
% 0.19/0.48 ( setminusT_lem
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(complementT_lem_type,type,
% 0.19/0.48 complementT_lem: $o ).
% 0.19/0.48
% 0.19/0.48 thf(complementT_lem,definition,
% 0.19/0.48 ( complementT_lem
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(setextT_type,type,
% 0.19/0.48 setextT: $o ).
% 0.19/0.48
% 0.19/0.48 thf(setextT,definition,
% 0.19/0.48 ( setextT
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( in @ Xx @ X )
% 0.19/0.48 => ( in @ Xx @ Y ) ) )
% 0.19/0.48 => ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( in @ Xx @ Y )
% 0.19/0.48 => ( in @ Xx @ X ) ) )
% 0.19/0.48 => ( X = Y ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(subsetTI_type,type,
% 0.19/0.48 subsetTI: $o ).
% 0.19/0.48
% 0.19/0.48 thf(subsetTI,definition,
% 0.19/0.48 ( subsetTI
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( in @ Xx @ X )
% 0.19/0.48 => ( in @ Xx @ Y ) ) )
% 0.19/0.48 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(powersetTI1_type,type,
% 0.19/0.48 powersetTI1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(powersetTI1,definition,
% 0.19/0.48 ( powersetTI1
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ( ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( in @ Xx @ X )
% 0.19/0.48 => ( in @ Xx @ Y ) ) )
% 0.19/0.48 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(powersetTE1_type,type,
% 0.19/0.48 powersetTE1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(powersetTE1,definition,
% 0.19/0.48 ( powersetTE1
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( in @ X @ ( powerset @ Y ) )
% 0.19/0.48 => ( ( in @ Xx @ X )
% 0.19/0.48 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(complementTI1_type,type,
% 0.19/0.48 complementTI1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(complementTI1,definition,
% 0.19/0.48 ( complementTI1
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( in @ Xx @ X )
% 0.19/0.48 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(complementTE1_type,type,
% 0.19/0.48 complementTE1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(complementTE1,definition,
% 0.19/0.48 ( complementTE1
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.19/0.48 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(binintersectTELcontra_type,type,
% 0.19/0.48 binintersectTELcontra: $o ).
% 0.19/0.48
% 0.19/0.48 thf(binintersectTELcontra,definition,
% 0.19/0.48 ( binintersectTELcontra
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ~ ( in @ Xx @ X )
% 0.19/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(binintersectTERcontra_type,type,
% 0.19/0.48 binintersectTERcontra: $o ).
% 0.19/0.48
% 0.19/0.48 thf(binintersectTERcontra,definition,
% 0.19/0.48 ( binintersectTERcontra
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ~ ( in @ Xx @ Y )
% 0.19/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(contrasubsetT_type,type,
% 0.19/0.48 contrasubsetT: $o ).
% 0.19/0.48
% 0.19/0.48 thf(contrasubsetT,definition,
% 0.19/0.48 ( contrasubsetT
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.19/0.48 => ( ( in @ Xx @ Y )
% 0.19/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(contrasubsetT1_type,type,
% 0.19/0.48 contrasubsetT1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(contrasubsetT1,definition,
% 0.19/0.48 ( contrasubsetT1
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( subset @ X @ Y )
% 0.19/0.48 => ( ~ ( in @ Xx @ Y )
% 0.19/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(contrasubsetT2_type,type,
% 0.19/0.48 contrasubsetT2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(contrasubsetT2,definition,
% 0.19/0.48 ( contrasubsetT2
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ( ( subset @ X @ Y )
% 0.19/0.48 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(contrasubsetT3_type,type,
% 0.19/0.48 contrasubsetT3: $o ).
% 0.19/0.48
% 0.19/0.48 thf(contrasubsetT3,definition,
% 0.19/0.48 ( contrasubsetT3
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.19/0.48 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(doubleComplementI1_type,type,
% 0.19/0.48 doubleComplementI1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(doubleComplementI1,definition,
% 0.19/0.48 ( doubleComplementI1
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( in @ Xx @ X )
% 0.19/0.48 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(doubleComplementE1_type,type,
% 0.19/0.48 doubleComplementE1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(doubleComplementE1,definition,
% 0.19/0.48 ( doubleComplementE1
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.19/0.48 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(doubleComplementSub1_type,type,
% 0.19/0.48 doubleComplementSub1: $o ).
% 0.19/0.48
% 0.19/0.48 thf(doubleComplementSub1,definition,
% 0.19/0.48 ( doubleComplementSub1
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(doubleComplementSub2_type,type,
% 0.19/0.48 doubleComplementSub2: $o ).
% 0.19/0.48
% 0.19/0.48 thf(doubleComplementSub2,definition,
% 0.19/0.48 ( doubleComplementSub2
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(doubleComplementEq_type,type,
% 0.19/0.48 doubleComplementEq: $o ).
% 0.19/0.48
% 0.19/0.48 thf(doubleComplementEq,definition,
% 0.19/0.48 ( doubleComplementEq
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ( X
% 0.19/0.48 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(complementTnotintersectT_type,type,
% 0.19/0.48 complementTnotintersectT: $o ).
% 0.19/0.48
% 0.19/0.48 thf(complementTnotintersectT,definition,
% 0.19/0.48 ( complementTnotintersectT
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.19/0.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(complementImpComplementIntersect_type,type,
% 0.19/0.48 complementImpComplementIntersect: $o ).
% 0.19/0.48
% 0.19/0.48 thf(complementImpComplementIntersect,definition,
% 0.19/0.48 ( complementImpComplementIntersect
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.19/0.48 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(complementSubsetComplementIntersect_type,type,
% 0.19/0.48 complementSubsetComplementIntersect: $o ).
% 0.19/0.48
% 0.19/0.48 thf(complementSubsetComplementIntersect,definition,
% 0.19/0.48 ( complementSubsetComplementIntersect
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(complementInPowersetComplementIntersect_type,type,
% 0.19/0.48 complementInPowersetComplementIntersect: $o ).
% 0.19/0.48
% 0.19/0.48 thf(complementInPowersetComplementIntersect,definition,
% 0.19/0.48 ( complementInPowersetComplementIntersect
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(contraSubsetComplement_type,type,
% 0.19/0.48 contraSubsetComplement: $o ).
% 0.19/0.48
% 0.19/0.48 thf(contraSubsetComplement,definition,
% 0.19/0.48 ( contraSubsetComplement
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( in @ Xx @ Y )
% 0.19/0.48 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(complementTcontraSubset_type,type,
% 0.19/0.48 complementTcontraSubset: $o ).
% 0.19/0.48
% 0.19/0.48 thf(complementTcontraSubset,definition,
% 0.19/0.48 ( complementTcontraSubset
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.19/0.48 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(binunionTILcontra_type,type,
% 0.19/0.48 binunionTILcontra: $o ).
% 0.19/0.48
% 0.19/0.48 thf(binunionTILcontra,definition,
% 0.19/0.48 ( binunionTILcontra
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.19/0.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(binunionTIRcontra_type,type,
% 0.19/0.48 binunionTIRcontra: $o ).
% 0.19/0.48
% 0.19/0.48 thf(binunionTIRcontra,definition,
% 0.19/0.48 ( binunionTIRcontra
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.19/0.48 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.19/0.48
% 0.19/0.48 thf(inIntersectImpInUnion_type,type,
% 0.19/0.48 inIntersectImpInUnion: $o ).
% 0.19/0.48
% 0.19/0.48 thf(inIntersectImpInUnion,definition,
% 0.19/0.48 ( inIntersectImpInUnion
% 0.19/0.48 = ( ! [A: $i,X: $i] :
% 0.19/0.48 ( ( in @ X @ ( powerset @ A ) )
% 0.19/0.48 => ! [Y: $i] :
% 0.19/0.48 ( ( in @ Y @ ( powerset @ A ) )
% 0.19/0.48 => ! [Z: $i] :
% 0.19/0.48 ( ( in @ Z @ ( powerset @ A ) )
% 0.19/0.48 => ! [Xx: $i] :
% 0.19/0.48 ( ( in @ Xx @ A )
% 0.19/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.31/0.48 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(inIntersectImpInUnion2_type,type,
% 0.31/0.48 inIntersectImpInUnion2: $o ).
% 0.31/0.48
% 0.31/0.48 thf(inIntersectImpInUnion2,definition,
% 0.31/0.48 ( inIntersectImpInUnion2
% 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 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.31/0.48 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(inIntersectImpInIntersectUnions_type,type,
% 0.31/0.48 inIntersectImpInIntersectUnions: $o ).
% 0.31/0.48
% 0.31/0.48 thf(inIntersectImpInIntersectUnions,definition,
% 0.31/0.48 ( inIntersectImpInIntersectUnions
% 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 => ! [Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.31/0.48 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(intersectInPowersetIntersectUnions_type,type,
% 0.31/0.48 intersectInPowersetIntersectUnions: $o ).
% 0.31/0.48
% 0.31/0.48 thf(intersectInPowersetIntersectUnions,definition,
% 0.31/0.48 ( intersectInPowersetIntersectUnions
% 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 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(inComplementUnionImpNotIn1_type,type,
% 0.31/0.48 inComplementUnionImpNotIn1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(inComplementUnionImpNotIn1,definition,
% 0.31/0.48 ( inComplementUnionImpNotIn1
% 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 @ X ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(inComplementUnionImpInComplement1_type,type,
% 0.31/0.48 inComplementUnionImpInComplement1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(inComplementUnionImpInComplement1,definition,
% 0.31/0.48 ( inComplementUnionImpInComplement1
% 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 @ X ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binunionTE_type,type,
% 0.31/0.48 binunionTE: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binunionTE,definition,
% 0.31/0.48 ( binunionTE
% 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 => ! [Xphi: $o,Xx: $i] :
% 0.31/0.48 ( ( in @ Xx @ A )
% 0.31/0.48 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.31/0.48 => ( ( ( in @ Xx @ X )
% 0.31/0.48 => Xphi )
% 0.31/0.48 => ( ( ( in @ Xx @ Y )
% 0.31/0.48 => Xphi )
% 0.31/0.48 => Xphi ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(binunionTEcontra_type,type,
% 0.31/0.48 binunionTEcontra: $o ).
% 0.31/0.48
% 0.31/0.48 thf(binunionTEcontra,definition,
% 0.31/0.48 ( binunionTEcontra
% 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 @ X )
% 0.31/0.48 => ( ~ ( in @ Xx @ Y )
% 0.31/0.48 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan2a1_type,type,
% 0.31/0.48 demorgan2a1: $o ).
% 0.31/0.48
% 0.31/0.48 thf(demorgan2a1,definition,
% 0.31/0.48 ( demorgan2a1
% 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 @ 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,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 => ! [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 %------------------------------------------------------------------------------
% 0.31/0.48 ------------------------
% 0.31/1.80 ---- Embedded in THF ---
% 0.31/1.80 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.31/1.80 %%% Generated on Tue Feb 24 12:33:23 EST 2026
% 0.31/1.80 %%% using '$$dhol' embedding, version 1.3.0.
% 0.31/1.80 %%% Logic specification used:
% 0.31/1.80 %%% thf(spec, logic, $$dhol).
% 0.31/1.80
% 0.31/1.80 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.UUdmaiX54M/DTF2DTF26421.p
% See solution above
% 0.31/1.81 ------------------------
% 0.31/1.86 ---- Cleaned THF ---
% 0.31/1.86 thf(in_type,type,
% 0.31/1.86 in: $i > $i > $o ).
% 0.31/1.86
% 0.31/1.86 thf(exu_type,type,
% 0.31/1.86 exu: ( $i > $o ) > $o ).
% 0.31/1.86
% 0.31/1.86 thf(exu,definition,
% 0.31/1.86 ( exu
% 0.31/1.86 = ( ^ [Xphi: $i > $o] :
% 0.31/1.86 ? [Xx: $i] :
% 0.31/1.86 ( ( Xphi @ Xx )
% 0.31/1.86 & ! [Xy: $i] :
% 0.31/1.86 ( ( Xphi @ Xy )
% 0.31/1.86 => ( Xx = Xy ) ) ) ) ) ).
% 0.31/1.86
% 0.31/1.86 thf(setextAx_type,type,
% 0.31/1.86 setextAx: $o ).
% 0.31/1.86
% 0.31/1.86 thf(setextAx,definition,
% 0.31/1.86 ( setextAx
% 0.31/1.86 = ( ! [A: $i,B: $i] :
% 0.31/1.86 ( ! [Xx: $i] :
% 0.31/1.86 ( ( in @ Xx @ A )
% 0.31/1.86 <=> ( in @ Xx @ B ) )
% 0.31/1.86 => ( A = B ) ) ) ) ).
% 0.31/1.86
% 0.31/1.86 thf(emptyset_type,type,
% 0.31/1.86 emptyset: $i ).
% 0.31/1.86
% 0.31/1.86 thf(emptysetAx_type,type,
% 0.31/1.86 emptysetAx: $o ).
% 0.31/1.86
% 0.31/1.86 thf(emptysetAx,definition,
% 0.31/1.86 ( emptysetAx
% 0.31/1.86 = ( ! [Xx: $i] :
% 0.31/1.86 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/1.86
% 0.31/1.86 thf(setadjoin_type,type,
% 0.31/1.86 setadjoin: $i > $i > $i ).
% 0.31/1.86
% 0.31/1.86 thf(setadjoinAx_type,type,
% 0.31/1.86 setadjoinAx: $o ).
% 0.31/1.86
% 0.31/1.86 thf(setadjoinAx,definition,
% 0.31/1.86 ( setadjoinAx
% 0.31/1.86 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/1.86 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/1.86 <=> ( ( Xy = Xx )
% 0.31/1.86 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/1.86
% 0.31/1.86 thf(powerset_type,type,
% 0.31/1.86 powerset: $i > $i ).
% 0.31/1.86
% 0.31/1.86 thf(powersetAx_type,type,
% 0.31/1.86 powersetAx: $o ).
% 0.31/1.86
% 0.31/1.86 thf(powersetAx,definition,
% 0.31/1.86 ( powersetAx
% 0.31/1.86 = ( ! [A: $i,B: $i] :
% 0.31/1.86 ( ( in @ B @ ( powerset @ A ) )
% 0.31/1.86 <=> ! [Xx: $i] :
% 0.31/1.86 ( ( in @ Xx @ B )
% 0.31/1.86 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/1.86
% 0.31/1.86 thf(setunion_type,type,
% 0.31/1.86 setunion: $i > $i ).
% 0.31/1.86
% 0.31/1.86 thf(setunionAx_type,type,
% 0.31/1.86 setunionAx: $o ).
% 0.31/1.86
% 0.31/1.86 thf(setunionAx,definition,
% 0.31/1.86 ( setunionAx
% 0.31/1.86 = ( ! [A: $i,Xx: $i] :
% 0.31/1.86 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/1.86 <=> ? [B: $i] :
% 0.31/1.86 ( ( in @ Xx @ B )
% 0.31/1.86 & ( in @ B @ A ) ) ) ) ) ).
% 0.31/1.86
% 0.31/1.86 thf(omega_type,type,
% 0.31/1.86 omega: $i ).
% 0.31/1.86
% 0.31/1.86 thf(omega0Ax_type,type,
% 0.31/1.86 omega0Ax: $o ).
% 0.31/1.86
% 0.31/1.86 thf(omega0Ax,definition,
% 0.31/1.86 ( omega0Ax
% 0.31/1.86 = ( in @ emptyset @ omega ) ) ).
% 0.31/1.86
% 0.31/1.86 thf(omegaSAx_type,type,
% 0.31/1.86 omegaSAx: $o ).
% 0.31/1.86
% 0.31/1.86 thf(omegaSAx,definition,
% 0.31/1.86 ( omegaSAx
% 0.31/1.86 = ( ! [Xx: $i] :
% 0.31/1.86 ( ( in @ Xx @ omega )
% 0.31/1.86 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.31/1.86
% 0.31/1.86 thf(omegaIndAx_type,type,
% 0.31/1.86 omegaIndAx: $o ).
% 0.31/1.86
% 0.31/1.86 thf(omegaIndAx,definition,
% 0.31/1.86 ( omegaIndAx
% 0.31/1.86 = ( ! [A: $i] :
% 0.31/1.86 ( ( ( in @ emptyset @ A )
% 0.31/1.86 & ! [Xx: $i] :
% 0.31/1.86 ( ( ( in @ Xx @ omega )
% 0.31/1.86 & ( in @ Xx @ A ) )
% 0.31/1.86 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.31/1.86 => ! [Xx: $i] :
% 0.31/1.86 ( ( in @ Xx @ omega )
% 0.31/1.86 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/1.86
% 0.31/1.86 thf(replAx_type,type,
% 0.31/1.86 replAx: $o ).
% 0.31/1.86
% 0.31/1.86 thf(replAx,definition,
% 0.31/1.86 ( replAx
% 0.31/1.86 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.31/1.86 ( ! [Xx: $i] :
% 0.31/1.86 ( ( in @ Xx @ A )
% 0.31/1.86 => ( exu
% 0.31/1.86 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.31/1.86 => ? [B: $i] :
% 0.31/1.86 ! [Xx: $i] :
% 0.31/1.86 ( ( in @ Xx @ B )
% 0.31/1.86 <=> ? [Xy: $i] :
% 0.31/1.86 ( ( in @ Xy @ A )
% 0.31/1.86 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.31/1.86
% 0.31/1.86 thf(foundationAx_type,type,
% 0.31/1.86 foundationAx: $o ).
% 0.31/1.86
% 0.31/1.86 thf(foundationAx,definition,
% 0.31/1.86 ( foundationAx
% 0.31/1.86 = ( ! [A: $i] :
% 0.31/1.86 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/1.86 => ? [B: $i] :
% 0.31/1.86 ( ( in @ B @ A )
% 0.31/1.86 & ~ ? [Xx: $i] :
% 0.31/1.86 ( ( in @ Xx @ B )
% 0.31/1.86 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.31/1.86
% 0.31/1.86 thf(wellorderingAx_type,type,
% 0.31/1.86 wellorderingAx: $o ).
% 0.31/1.86
% 0.31/1.86 thf(wellorderingAx,definition,
% 0.31/1.86 ( wellorderingAx
% 0.31/1.86 = ( ! [A: $i] :
% 0.31/1.86 ? [B: $i] :
% 0.31/1.86 ( ! [C: $i] :
% 0.31/1.86 ( ( in @ C @ B )
% 0.31/1.86 => ! [Xx: $i] :
% 0.31/1.86 ( ( in @ Xx @ C )
% 0.31/1.86 => ( in @ Xx @ A ) ) )
% 0.31/1.86 & ! [Xx: $i,Xy: $i] :
% 0.31/1.86 ( ( ( in @ Xx @ A )
% 0.31/1.86 & ( in @ Xy @ A ) )
% 0.31/1.86 => ( ! [C: $i] :
% 0.31/1.86 ( ( in @ C @ B )
% 0.31/1.86 => ( ( in @ Xx @ C )
% 0.31/1.86 <=> ( in @ Xy @ C ) ) )
% 0.31/1.86 => ( Xx = Xy ) ) )
% 0.31/1.86 & ! [C: $i,D: $i] :
% 0.31/1.86 ( ( ( in @ C @ B )
% 0.31/1.86 & ( in @ D @ B ) )
% 0.31/1.86 => ( ! [Xx: $i] :
% 0.31/1.86 ( ( in @ Xx @ C )
% 0.31/1.86 => ( in @ Xx @ D ) )
% 0.31/1.86 | ! [Xx: $i] :
% 0.31/1.86 ( ( in @ Xx @ D )
% 0.31/1.86 => ( in @ Xx @ C ) ) ) )
% 0.31/1.86 & ! [C: $i] :
% 0.31/1.86 ( ( ! [Xx: $i] :
% 0.31/1.86 ( ( in @ Xx @ C )
% 0.31/1.86 => ( in @ Xx @ A ) )
% 0.31/1.86 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.31/1.86 => ? [D: $i,Xx: $i] :
% 0.31/1.86 ( ( in @ D @ B )
% 0.31/1.86 & ( in @ Xx @ C )
% 0.31/1.86 & ~ ? [Xy: $i] :
% 0.31/1.86 ( ( in @ Xy @ D )
% 0.31/1.86 & ( in @ Xy @ C ) )
% 0.31/1.86 & ! [E: $i] :
% 0.31/1.86 ( ( in @ E @ B )
% 0.31/1.86 => ( ! [Xy: $i] :
% 0.31/1.86 ( ( in @ Xy @ E )
% 0.31/1.86 => ( in @ Xy @ D ) )
% 0.31/1.86 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.31/1.86
% 0.31/1.86 thf(descr_type,type,
% 0.31/1.86 descr: ( $i > $o ) > $i ).
% 0.31/1.86
% 0.31/1.86 thf(descrp_type,type,
% 0.31/1.86 descrp: $o ).
% 0.31/1.86
% 0.31/1.86 thf(descrp,definition,
% 0.31/1.86 ( descrp
% 0.31/1.86 = ( ! [Xphi: $i > $o] :
% 0.31/1.86 ( ( exu
% 0.31/1.86 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.86 => ( Xphi
% 0.31/1.86 @ ( descr
% 0.31/1.86 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(dsetconstr_type,type,
% 0.31/1.87 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.31/1.87
% 0.31/1.87 thf(dsetconstrI_type,type,
% 0.31/1.87 dsetconstrI: $o ).
% 0.31/1.87
% 0.31/1.87 thf(dsetconstrI,definition,
% 0.31/1.87 ( dsetconstrI
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( Xphi @ Xx )
% 0.31/1.87 => ( in @ Xx
% 0.31/1.87 @ ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(dsetconstrEL_type,type,
% 0.31/1.87 dsetconstrEL: $o ).
% 0.31/1.87
% 0.31/1.87 thf(dsetconstrEL,definition,
% 0.31/1.87 ( dsetconstrEL
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx
% 0.31/1.87 @ ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/1.87 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(dsetconstrER_type,type,
% 0.31/1.87 dsetconstrER: $o ).
% 0.31/1.87
% 0.31/1.87 thf(dsetconstrER,definition,
% 0.31/1.87 ( dsetconstrER
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx
% 0.31/1.87 @ ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/1.87 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(exuE1_type,type,
% 0.31/1.87 exuE1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(exuE1,definition,
% 0.31/1.87 ( exuE1
% 0.31/1.87 = ( ! [Xphi: $i > $o] :
% 0.31/1.87 ( ( exu
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.87 => ? [Xx: $i] :
% 0.31/1.87 ( ( Xphi @ Xx )
% 0.31/1.87 & ! [Xy: $i] :
% 0.31/1.87 ( ( Xphi @ Xy )
% 0.31/1.87 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(prop2set_type,type,
% 0.31/1.87 prop2set: $o > $i ).
% 0.31/1.87
% 0.31/1.87 thf(prop2setE_type,type,
% 0.31/1.87 prop2setE: $o ).
% 0.31/1.87
% 0.31/1.87 thf(prop2setE,definition,
% 0.31/1.87 ( prop2setE
% 0.31/1.87 = ( ! [Xphi: $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.31/1.87 => Xphi ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(emptysetE_type,type,
% 0.31/1.87 emptysetE: $o ).
% 0.31/1.87
% 0.31/1.87 thf(emptysetE,definition,
% 0.31/1.87 ( emptysetE
% 0.31/1.87 = ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ emptyset )
% 0.31/1.87 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(emptysetimpfalse_type,type,
% 0.31/1.87 emptysetimpfalse: $o ).
% 0.31/1.87
% 0.31/1.87 thf(emptysetimpfalse,definition,
% 0.31/1.87 ( emptysetimpfalse
% 0.31/1.87 = ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ emptyset )
% 0.31/1.87 => $false ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(notinemptyset_type,type,
% 0.31/1.87 notinemptyset: $o ).
% 0.31/1.87
% 0.31/1.87 thf(notinemptyset,definition,
% 0.31/1.87 ( notinemptyset
% 0.31/1.87 = ( ! [Xx: $i] :
% 0.31/1.87 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(exuE3e_type,type,
% 0.31/1.87 exuE3e: $o ).
% 0.31/1.87
% 0.31/1.87 thf(exuE3e,definition,
% 0.31/1.87 ( exuE3e
% 0.31/1.87 = ( ! [Xphi: $i > $o] :
% 0.31/1.87 ( ( exu
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.87 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setext_type,type,
% 0.31/1.87 setext: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setext,definition,
% 0.31/1.87 ( setext
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ Xx @ B ) )
% 0.31/1.87 => ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ B )
% 0.31/1.87 => ( in @ Xx @ A ) )
% 0.31/1.87 => ( A = B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(emptyI_type,type,
% 0.31/1.87 emptyI: $o ).
% 0.31/1.87
% 0.31/1.87 thf(emptyI,definition,
% 0.31/1.87 ( emptyI
% 0.31/1.87 = ( ! [A: $i] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ~ ( in @ Xx @ A )
% 0.31/1.87 => ( A = emptyset ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(noeltsimpempty_type,type,
% 0.31/1.87 noeltsimpempty: $o ).
% 0.31/1.87
% 0.31/1.87 thf(noeltsimpempty,definition,
% 0.31/1.87 ( noeltsimpempty
% 0.31/1.87 = ( ! [A: $i] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ~ ( in @ Xx @ A )
% 0.31/1.87 => ( A = emptyset ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setbeta_type,type,
% 0.31/1.87 setbeta: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setbeta,definition,
% 0.31/1.87 ( setbeta
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx
% 0.31/1.87 @ ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.31/1.87 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(nonempty_type,type,
% 0.31/1.87 nonempty: $i > $o ).
% 0.31/1.87
% 0.31/1.87 thf(nonempty,definition,
% 0.31/1.87 ( nonempty
% 0.31/1.87 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(nonemptyE1_type,type,
% 0.31/1.87 nonemptyE1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(nonemptyE1,definition,
% 0.31/1.87 ( nonemptyE1
% 0.31/1.87 = ( ! [A: $i] :
% 0.31/1.87 ( ( nonempty @ A )
% 0.31/1.87 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(nonemptyI_type,type,
% 0.31/1.87 nonemptyI: $o ).
% 0.31/1.87
% 0.31/1.87 thf(nonemptyI,definition,
% 0.31/1.87 ( nonemptyI
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( Xphi @ Xx )
% 0.31/1.87 => ( nonempty
% 0.31/1.87 @ ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(nonemptyI1_type,type,
% 0.31/1.87 nonemptyI1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(nonemptyI1,definition,
% 0.31/1.87 ( nonemptyI1
% 0.31/1.87 = ( ! [A: $i] :
% 0.31/1.87 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.31/1.87 => ( nonempty @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setadjoinIL_type,type,
% 0.31/1.87 setadjoinIL: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setadjoinIL,definition,
% 0.31/1.87 ( setadjoinIL
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(emptyinunitempty_type,type,
% 0.31/1.87 emptyinunitempty: $o ).
% 0.31/1.87
% 0.31/1.87 thf(emptyinunitempty,definition,
% 0.31/1.87 ( emptyinunitempty
% 0.31/1.87 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setadjoinIR_type,type,
% 0.31/1.87 setadjoinIR: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setadjoinIR,definition,
% 0.31/1.87 ( setadjoinIR
% 0.31/1.87 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ A )
% 0.31/1.87 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setadjoinE_type,type,
% 0.31/1.87 setadjoinE: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setadjoinE,definition,
% 0.31/1.87 ( setadjoinE
% 0.31/1.87 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/1.87 => ! [Xphi: $o] :
% 0.31/1.87 ( ( ( Xy = Xx )
% 0.31/1.87 => Xphi )
% 0.31/1.87 => ( ( ( in @ Xy @ A )
% 0.31/1.87 => Xphi )
% 0.31/1.87 => Xphi ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setadjoinOr_type,type,
% 0.31/1.87 setadjoinOr: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setadjoinOr,definition,
% 0.31/1.87 ( setadjoinOr
% 0.31/1.87 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.31/1.87 => ( ( Xy = Xx )
% 0.31/1.87 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setoftrueEq_type,type,
% 0.31/1.87 setoftrueEq: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setoftrueEq,definition,
% 0.31/1.87 ( setoftrueEq
% 0.31/1.87 = ( ! [A: $i] :
% 0.31/1.87 ( ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xx: $i] : $true )
% 0.31/1.87 = A ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(powersetI_type,type,
% 0.31/1.87 powersetI: $o ).
% 0.31/1.87
% 0.31/1.87 thf(powersetI,definition,
% 0.31/1.87 ( powersetI
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ B )
% 0.31/1.87 => ( in @ Xx @ A ) )
% 0.31/1.87 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(emptyinPowerset_type,type,
% 0.31/1.87 emptyinPowerset: $o ).
% 0.31/1.87
% 0.31/1.87 thf(emptyinPowerset,definition,
% 0.31/1.87 ( emptyinPowerset
% 0.31/1.87 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(emptyInPowerset_type,type,
% 0.31/1.87 emptyInPowerset: $o ).
% 0.31/1.87
% 0.31/1.87 thf(emptyInPowerset,definition,
% 0.31/1.87 ( emptyInPowerset
% 0.31/1.87 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(powersetE_type,type,
% 0.31/1.87 powersetE: $o ).
% 0.31/1.87
% 0.31/1.87 thf(powersetE,definition,
% 0.31/1.87 ( powersetE
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ B @ ( powerset @ A ) )
% 0.31/1.87 => ( ( in @ Xx @ B )
% 0.31/1.87 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setunionI_type,type,
% 0.31/1.87 setunionI: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setunionI,definition,
% 0.31/1.87 ( setunionI
% 0.31/1.87 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/1.87 ( ( in @ Xx @ B )
% 0.31/1.87 => ( ( in @ B @ A )
% 0.31/1.87 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setunionE_type,type,
% 0.31/1.87 setunionE: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setunionE,definition,
% 0.31/1.87 ( setunionE
% 0.31/1.87 = ( ! [A: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/1.87 => ! [Xphi: $o] :
% 0.31/1.87 ( ! [B: $i] :
% 0.31/1.87 ( ( in @ Xx @ B )
% 0.31/1.87 => ( ( in @ B @ A )
% 0.31/1.87 => Xphi ) )
% 0.31/1.87 => Xphi ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(subPowSU_type,type,
% 0.31/1.87 subPowSU: $o ).
% 0.31/1.87
% 0.31/1.87 thf(subPowSU,definition,
% 0.31/1.87 ( subPowSU
% 0.31/1.87 = ( ! [A: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(exuE2_type,type,
% 0.31/1.87 exuE2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(exuE2,definition,
% 0.31/1.87 ( exuE2
% 0.31/1.87 = ( ! [Xphi: $i > $o] :
% 0.31/1.87 ( ( exu
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.87 => ? [Xx: $i] :
% 0.31/1.87 ! [Xy: $i] :
% 0.31/1.87 ( ( Xphi @ Xy )
% 0.31/1.87 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(nonemptyImpWitness_type,type,
% 0.31/1.87 nonemptyImpWitness: $o ).
% 0.31/1.87
% 0.31/1.87 thf(nonemptyImpWitness,definition,
% 0.31/1.87 ( nonemptyImpWitness
% 0.31/1.87 = ( ! [A: $i] :
% 0.31/1.87 ( ( nonempty @ A )
% 0.31/1.87 => ? [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 & $true ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(uniqinunit_type,type,
% 0.31/1.87 uniqinunit: $o ).
% 0.31/1.87
% 0.31/1.87 thf(uniqinunit,definition,
% 0.31/1.87 ( uniqinunit
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/1.87 => ( Xx = Xy ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(notinsingleton_type,type,
% 0.31/1.87 notinsingleton: $o ).
% 0.31/1.87
% 0.31/1.87 thf(notinsingleton,definition,
% 0.31/1.87 ( notinsingleton
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( Xx != Xy )
% 0.31/1.87 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(eqinunit_type,type,
% 0.31/1.87 eqinunit: $o ).
% 0.31/1.87
% 0.31/1.87 thf(eqinunit,definition,
% 0.31/1.87 ( eqinunit
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( Xx = Xy )
% 0.31/1.87 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(singletonsswitch_type,type,
% 0.31/1.87 singletonsswitch: $o ).
% 0.31/1.87
% 0.31/1.87 thf(singletonsswitch,definition,
% 0.31/1.87 ( singletonsswitch
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/1.87 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(upairsetE_type,type,
% 0.31/1.87 upairsetE: $o ).
% 0.31/1.87
% 0.31/1.87 thf(upairsetE,definition,
% 0.31/1.87 ( upairsetE
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/1.87 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/1.87 => ( ( Xz = Xx )
% 0.31/1.87 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(upairsetIL_type,type,
% 0.31/1.87 upairsetIL: $o ).
% 0.31/1.87
% 0.31/1.87 thf(upairsetIL,definition,
% 0.31/1.87 ( upairsetIL
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(upairsetIR_type,type,
% 0.31/1.87 upairsetIR: $o ).
% 0.31/1.87
% 0.31/1.87 thf(upairsetIR,definition,
% 0.31/1.87 ( upairsetIR
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(emptyE1_type,type,
% 0.31/1.87 emptyE1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(emptyE1,definition,
% 0.31/1.87 ( emptyE1
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.87 ( ? [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 & ( Xphi @ Xx ) )
% 0.31/1.87 => ( ( ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.87 = emptyset )
% 0.31/1.87 => $false ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(vacuousDall_type,type,
% 0.31/1.87 vacuousDall: $o ).
% 0.31/1.87
% 0.31/1.87 thf(vacuousDall,definition,
% 0.31/1.87 ( vacuousDall
% 0.31/1.87 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ emptyset )
% 0.31/1.87 => ( Xphi @ Xx ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(quantDeMorgan1_type,type,
% 0.31/1.87 quantDeMorgan1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(quantDeMorgan1,definition,
% 0.31/1.87 ( quantDeMorgan1
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.87 ( ~ ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( Xphi @ Xx ) )
% 0.31/1.87 => ? [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(quantDeMorgan2_type,type,
% 0.31/1.87 quantDeMorgan2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(quantDeMorgan2,definition,
% 0.31/1.87 ( quantDeMorgan2
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ~ ( Xphi @ Xx ) )
% 0.31/1.87 => ~ ? [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(quantDeMorgan3_type,type,
% 0.31/1.87 quantDeMorgan3: $o ).
% 0.31/1.87
% 0.31/1.87 thf(quantDeMorgan3,definition,
% 0.31/1.87 ( quantDeMorgan3
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.87 ( ~ ? [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 & ( Xphi @ Xx ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(quantDeMorgan4_type,type,
% 0.31/1.87 quantDeMorgan4: $o ).
% 0.31/1.87
% 0.31/1.87 thf(quantDeMorgan4,definition,
% 0.31/1.87 ( quantDeMorgan4
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.87 ( ? [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 & ~ ( Xphi @ Xx ) )
% 0.31/1.87 => ~ ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(prop2setI_type,type,
% 0.31/1.87 prop2setI: $o ).
% 0.31/1.87
% 0.31/1.87 thf(prop2setI,definition,
% 0.31/1.87 ( prop2setI
% 0.31/1.87 = ( ! [Xphi: $o] :
% 0.31/1.87 ( Xphi
% 0.31/1.87 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(set2prop_type,type,
% 0.31/1.87 set2prop: $i > $o ).
% 0.31/1.87
% 0.31/1.87 thf(prop2set2propI_type,type,
% 0.31/1.87 prop2set2propI: $o ).
% 0.31/1.87
% 0.31/1.87 thf(prop2set2propI,definition,
% 0.31/1.87 ( prop2set2propI
% 0.31/1.87 = ( ! [Xphi: $o] :
% 0.31/1.87 ( Xphi
% 0.31/1.87 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(notdexE_type,type,
% 0.31/1.87 notdexE: $o ).
% 0.31/1.87
% 0.31/1.87 thf(notdexE,definition,
% 0.31/1.87 ( notdexE
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.87 ( ~ ? [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 & ( Xphi @ Xx ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(notdallE_type,type,
% 0.31/1.87 notdallE: $o ).
% 0.31/1.87
% 0.31/1.87 thf(notdallE,definition,
% 0.31/1.87 ( notdallE
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.87 ( ~ ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( Xphi @ Xx ) )
% 0.31/1.87 => ? [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(exuI1_type,type,
% 0.31/1.87 exuI1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(exuI1,definition,
% 0.31/1.87 ( exuI1
% 0.31/1.87 = ( ! [Xphi: $i > $o] :
% 0.31/1.87 ( ? [Xx: $i] :
% 0.31/1.87 ( ( Xphi @ Xx )
% 0.31/1.87 & ! [Xy: $i] :
% 0.31/1.87 ( ( Xphi @ Xy )
% 0.31/1.87 => ( Xx = Xy ) ) )
% 0.31/1.87 => ( exu
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(exuI3_type,type,
% 0.31/1.87 exuI3: $o ).
% 0.31/1.87
% 0.31/1.87 thf(exuI3,definition,
% 0.31/1.87 ( exuI3
% 0.31/1.87 = ( ! [Xphi: $i > $o] :
% 0.31/1.87 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.31/1.87 => ( ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( Xphi @ Xx )
% 0.31/1.87 => ( ( Xphi @ Xy )
% 0.31/1.87 => ( Xx = Xy ) ) )
% 0.31/1.87 => ( exu
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(exuI2_type,type,
% 0.31/1.87 exuI2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(exuI2,definition,
% 0.31/1.87 ( exuI2
% 0.31/1.87 = ( ! [Xphi: $i > $o] :
% 0.31/1.87 ( ? [Xx: $i] :
% 0.31/1.87 ! [Xy: $i] :
% 0.31/1.87 ( ( Xphi @ Xy )
% 0.31/1.87 <=> ( Xy = Xx ) )
% 0.31/1.87 => ( exu
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(inCongP_type,type,
% 0.31/1.87 inCongP: $o ).
% 0.31/1.87
% 0.31/1.87 thf(inCongP,definition,
% 0.31/1.87 ( inCongP
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( A = B )
% 0.31/1.87 => ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( Xx = Xy )
% 0.31/1.87 => ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(in__Cong_type,type,
% 0.31/1.87 in__Cong: $o ).
% 0.31/1.87
% 0.31/1.87 thf(in__Cong,definition,
% 0.31/1.87 ( in__Cong
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( A = B )
% 0.31/1.87 => ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( Xx = Xy )
% 0.31/1.87 => ( ( in @ Xx @ A )
% 0.31/1.87 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(exuE3u_type,type,
% 0.31/1.87 exuE3u: $o ).
% 0.31/1.87
% 0.31/1.87 thf(exuE3u,definition,
% 0.31/1.87 ( exuE3u
% 0.31/1.87 = ( ! [Xphi: $i > $o] :
% 0.31/1.87 ( ( exu
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.87 => ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( Xphi @ Xx )
% 0.31/1.87 => ( ( Xphi @ Xy )
% 0.31/1.87 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(exu__Cong_type,type,
% 0.31/1.87 exu__Cong: $o ).
% 0.31/1.87
% 0.31/1.87 thf(exu__Cong,definition,
% 0.31/1.87 ( exu__Cong
% 0.31/1.87 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/1.87 ( ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( Xx = Xy )
% 0.31/1.87 => ( ( Xphi @ Xx )
% 0.31/1.87 <=> ( Xpsi @ Xy ) ) )
% 0.31/1.87 => ( ( exu
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.87 <=> ( exu
% 0.31/1.87 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(emptyset__Cong_type,type,
% 0.31/1.87 emptyset__Cong: $o ).
% 0.31/1.87
% 0.31/1.87 thf(emptyset__Cong,definition,
% 0.31/1.87 ( emptyset__Cong
% 0.31/1.87 = ( emptyset = emptyset ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setadjoin__Cong_type,type,
% 0.31/1.87 setadjoin__Cong: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setadjoin__Cong,definition,
% 0.31/1.87 ( setadjoin__Cong
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( Xx = Xy )
% 0.31/1.87 => ! [Xz: $i,Xu: $i] :
% 0.31/1.87 ( ( Xz = Xu )
% 0.31/1.87 => ( ( setadjoin @ Xx @ Xz )
% 0.31/1.87 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(powerset__Cong_type,type,
% 0.31/1.87 powerset__Cong: $o ).
% 0.31/1.87
% 0.31/1.87 thf(powerset__Cong,definition,
% 0.31/1.87 ( powerset__Cong
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( A = B )
% 0.31/1.87 => ( ( powerset @ A )
% 0.31/1.87 = ( powerset @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setunion__Cong_type,type,
% 0.31/1.87 setunion__Cong: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setunion__Cong,definition,
% 0.31/1.87 ( setunion__Cong
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( A = B )
% 0.31/1.87 => ( ( setunion @ A )
% 0.31/1.87 = ( setunion @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(omega__Cong_type,type,
% 0.31/1.87 omega__Cong: $o ).
% 0.31/1.87
% 0.31/1.87 thf(omega__Cong,definition,
% 0.31/1.87 ( omega__Cong
% 0.31/1.87 = ( omega = omega ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(exuEu_type,type,
% 0.31/1.87 exuEu: $o ).
% 0.31/1.87
% 0.31/1.87 thf(exuEu,definition,
% 0.31/1.87 ( exuEu
% 0.31/1.87 = ( ! [Xphi: $i > $o] :
% 0.31/1.87 ( ( exu
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.87 => ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( Xphi @ Xx )
% 0.31/1.87 => ( ( Xphi @ Xy )
% 0.31/1.87 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(descr__Cong_type,type,
% 0.31/1.87 descr__Cong: $o ).
% 0.31/1.87
% 0.31/1.87 thf(descr__Cong,definition,
% 0.31/1.87 ( descr__Cong
% 0.31/1.87 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/1.87 ( ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( Xx = Xy )
% 0.31/1.87 => ( ( Xphi @ Xx )
% 0.31/1.87 <=> ( Xpsi @ Xy ) ) )
% 0.31/1.87 => ( ( exu
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.87 => ( ( exu
% 0.31/1.87 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.31/1.87 => ( ( descr
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.87 = ( descr
% 0.31/1.87 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(dsetconstr__Cong_type,type,
% 0.31/1.87 dsetconstr__Cong: $o ).
% 0.31/1.87
% 0.31/1.87 thf(dsetconstr__Cong,definition,
% 0.31/1.87 ( dsetconstr__Cong
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( A = B )
% 0.31/1.87 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( ( Xx = Xy )
% 0.31/1.87 => ( ( Xphi @ Xx )
% 0.31/1.87 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.31/1.87 => ( ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.87 = ( dsetconstr @ B
% 0.31/1.87 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(subset_type,type,
% 0.31/1.87 subset: $i > $i > $o ).
% 0.31/1.87
% 0.31/1.87 thf(disjoint_type,type,
% 0.31/1.87 disjoint: $i > $i > $o ).
% 0.31/1.87
% 0.31/1.87 thf(setsmeet_type,type,
% 0.31/1.87 setsmeet: $i > $i > $o ).
% 0.31/1.87
% 0.31/1.87 thf(subsetI1_type,type,
% 0.31/1.87 subsetI1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(subsetI1,definition,
% 0.31/1.87 ( subsetI1
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ Xx @ B ) )
% 0.31/1.87 => ( subset @ A @ B ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(eqimpsubset2_type,type,
% 0.31/1.87 eqimpsubset2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(eqimpsubset2,definition,
% 0.31/1.87 ( eqimpsubset2
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( A = B )
% 0.31/1.87 => ( subset @ B @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(eqimpsubset1_type,type,
% 0.31/1.87 eqimpsubset1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(eqimpsubset1,definition,
% 0.31/1.87 ( eqimpsubset1
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( A = B )
% 0.31/1.87 => ( subset @ A @ B ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(subsetI2_type,type,
% 0.31/1.87 subsetI2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(subsetI2,definition,
% 0.31/1.87 ( subsetI2
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ Xx @ B ) )
% 0.31/1.87 => ( subset @ A @ B ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(emptysetsubset_type,type,
% 0.31/1.87 emptysetsubset: $o ).
% 0.31/1.87
% 0.31/1.87 thf(emptysetsubset,definition,
% 0.31/1.87 ( emptysetsubset
% 0.31/1.87 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(subsetE_type,type,
% 0.31/1.87 subsetE: $o ).
% 0.31/1.87
% 0.31/1.87 thf(subsetE,definition,
% 0.31/1.87 ( subsetE
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( subset @ A @ B )
% 0.31/1.87 => ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(subsetE2_type,type,
% 0.31/1.87 subsetE2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(subsetE2,definition,
% 0.31/1.87 ( subsetE2
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( subset @ A @ B )
% 0.31/1.87 => ( ~ ( in @ Xx @ B )
% 0.31/1.87 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(notsubsetI_type,type,
% 0.31/1.87 notsubsetI: $o ).
% 0.31/1.87
% 0.31/1.87 thf(notsubsetI,definition,
% 0.31/1.87 ( notsubsetI
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ~ ( in @ Xx @ B )
% 0.31/1.87 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(notequalI1_type,type,
% 0.31/1.87 notequalI1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(notequalI1,definition,
% 0.31/1.87 ( notequalI1
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ~ ( subset @ A @ B )
% 0.31/1.87 => ( A != B ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(notequalI2_type,type,
% 0.31/1.87 notequalI2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(notequalI2,definition,
% 0.31/1.87 ( notequalI2
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ~ ( in @ Xx @ B )
% 0.31/1.87 => ( A != B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(subsetRefl_type,type,
% 0.31/1.87 subsetRefl: $o ).
% 0.31/1.87
% 0.31/1.87 thf(subsetRefl,definition,
% 0.31/1.87 ( subsetRefl
% 0.31/1.87 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(subsetTrans_type,type,
% 0.31/1.87 subsetTrans: $o ).
% 0.31/1.87
% 0.31/1.87 thf(subsetTrans,definition,
% 0.31/1.87 ( subsetTrans
% 0.31/1.87 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/1.87 ( ( subset @ A @ B )
% 0.31/1.87 => ( ( subset @ B @ C )
% 0.31/1.87 => ( subset @ A @ C ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setadjoinSub_type,type,
% 0.31/1.87 setadjoinSub: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setadjoinSub,definition,
% 0.31/1.87 ( setadjoinSub
% 0.31/1.87 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setadjoinSub2_type,type,
% 0.31/1.87 setadjoinSub2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setadjoinSub2,definition,
% 0.31/1.87 ( setadjoinSub2
% 0.31/1.87 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.31/1.87 ( ( subset @ A @ B )
% 0.31/1.87 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(subset2powerset_type,type,
% 0.31/1.87 subset2powerset: $o ).
% 0.31/1.87
% 0.31/1.87 thf(subset2powerset,definition,
% 0.31/1.87 ( subset2powerset
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( subset @ A @ B )
% 0.31/1.87 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setextsub_type,type,
% 0.31/1.87 setextsub: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setextsub,definition,
% 0.31/1.87 ( setextsub
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( subset @ A @ B )
% 0.31/1.87 => ( ( subset @ B @ A )
% 0.31/1.87 => ( A = B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(subsetemptysetimpeq_type,type,
% 0.31/1.87 subsetemptysetimpeq: $o ).
% 0.31/1.87
% 0.31/1.87 thf(subsetemptysetimpeq,definition,
% 0.31/1.87 ( subsetemptysetimpeq
% 0.31/1.87 = ( ! [A: $i] :
% 0.31/1.87 ( ( subset @ A @ emptyset )
% 0.31/1.87 => ( A = emptyset ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(powersetI1_type,type,
% 0.31/1.87 powersetI1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(powersetI1,definition,
% 0.31/1.87 ( powersetI1
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( subset @ B @ A )
% 0.31/1.87 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(powersetE1_type,type,
% 0.31/1.87 powersetE1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(powersetE1,definition,
% 0.31/1.87 ( powersetE1
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( in @ B @ ( powerset @ A ) )
% 0.31/1.87 => ( subset @ B @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(inPowerset_type,type,
% 0.31/1.87 inPowerset: $o ).
% 0.31/1.87
% 0.31/1.87 thf(inPowerset,definition,
% 0.31/1.87 ( inPowerset
% 0.31/1.87 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(powersetsubset_type,type,
% 0.31/1.87 powersetsubset: $o ).
% 0.31/1.87
% 0.31/1.87 thf(powersetsubset,definition,
% 0.31/1.87 ( powersetsubset
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( subset @ A @ B )
% 0.31/1.87 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(sepInPowerset_type,type,
% 0.31/1.87 sepInPowerset: $o ).
% 0.31/1.87
% 0.31/1.87 thf(sepInPowerset,definition,
% 0.31/1.87 ( sepInPowerset
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.87 ( in
% 0.31/1.87 @ ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.87 @ ( powerset @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(sepSubset_type,type,
% 0.31/1.87 sepSubset: $o ).
% 0.31/1.87
% 0.31/1.87 thf(sepSubset,definition,
% 0.31/1.87 ( sepSubset
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.87 ( subset
% 0.31/1.87 @ ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.87 @ A ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binunion_type,type,
% 0.31/1.87 binunion: $i > $i > $i ).
% 0.31/1.87
% 0.31/1.87 thf(binunionIL_type,type,
% 0.31/1.87 binunionIL: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binunionIL,definition,
% 0.31/1.87 ( binunionIL
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(upairset2IR_type,type,
% 0.31/1.87 upairset2IR: $o ).
% 0.31/1.87
% 0.31/1.87 thf(upairset2IR,definition,
% 0.31/1.87 ( upairset2IR
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binunionIR_type,type,
% 0.31/1.87 binunionIR: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binunionIR,definition,
% 0.31/1.87 ( binunionIR
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ B )
% 0.31/1.87 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binunionEcases_type,type,
% 0.31/1.87 binunionEcases: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binunionEcases,definition,
% 0.31/1.87 ( binunionEcases
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.31/1.87 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/1.87 => ( ( ( in @ Xx @ A )
% 0.31/1.87 => Xphi )
% 0.31/1.87 => ( ( ( in @ Xx @ B )
% 0.31/1.87 => Xphi )
% 0.31/1.87 => Xphi ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binunionE_type,type,
% 0.31/1.87 binunionE: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binunionE,definition,
% 0.31/1.87 ( binunionE
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.31/1.87 => ( ( in @ Xx @ A )
% 0.31/1.87 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binunionLsub_type,type,
% 0.31/1.87 binunionLsub: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binunionLsub,definition,
% 0.31/1.87 ( binunionLsub
% 0.31/1.87 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binunionRsub_type,type,
% 0.31/1.87 binunionRsub: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binunionRsub,definition,
% 0.31/1.87 ( binunionRsub
% 0.31/1.87 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binintersect_type,type,
% 0.31/1.87 binintersect: $i > $i > $i ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectI_type,type,
% 0.31/1.87 binintersectI: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectI,definition,
% 0.31/1.87 ( binintersectI
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ B )
% 0.31/1.87 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectSubset5_type,type,
% 0.31/1.87 binintersectSubset5: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectSubset5,definition,
% 0.31/1.87 ( binintersectSubset5
% 0.31/1.87 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/1.87 ( ( subset @ C @ A )
% 0.31/1.87 => ( ( subset @ C @ B )
% 0.31/1.87 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectEL_type,type,
% 0.31/1.87 binintersectEL: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectEL,definition,
% 0.31/1.87 ( binintersectEL
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/1.87 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectLsub_type,type,
% 0.31/1.87 binintersectLsub: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectLsub,definition,
% 0.31/1.87 ( binintersectLsub
% 0.31/1.87 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectSubset2_type,type,
% 0.31/1.87 binintersectSubset2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectSubset2,definition,
% 0.31/1.87 ( binintersectSubset2
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( subset @ A @ B )
% 0.31/1.87 => ( ( binintersect @ A @ B )
% 0.31/1.87 = A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectSubset3_type,type,
% 0.31/1.87 binintersectSubset3: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectSubset3,definition,
% 0.31/1.87 ( binintersectSubset3
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( ( binintersect @ A @ B )
% 0.31/1.87 = B )
% 0.31/1.87 => ( subset @ B @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectER_type,type,
% 0.31/1.87 binintersectER: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectER,definition,
% 0.31/1.87 ( binintersectER
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.31/1.87 => ( in @ Xx @ B ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(disjointsetsI1_type,type,
% 0.31/1.87 disjointsetsI1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(disjointsetsI1,definition,
% 0.31/1.87 ( disjointsetsI1
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ~ ? [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 & ( in @ Xx @ B ) )
% 0.31/1.87 => ( ( binintersect @ A @ B )
% 0.31/1.87 = emptyset ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectRsub_type,type,
% 0.31/1.87 binintersectRsub: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectRsub,definition,
% 0.31/1.87 ( binintersectRsub
% 0.31/1.87 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectSubset4_type,type,
% 0.31/1.87 binintersectSubset4: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectSubset4,definition,
% 0.31/1.87 ( binintersectSubset4
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( subset @ B @ A )
% 0.31/1.87 => ( ( binintersect @ A @ B )
% 0.31/1.87 = B ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectSubset1_type,type,
% 0.31/1.87 binintersectSubset1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectSubset1,definition,
% 0.31/1.87 ( binintersectSubset1
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( ( binintersect @ A @ B )
% 0.31/1.87 = A )
% 0.31/1.87 => ( subset @ A @ B ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(bs114d_type,type,
% 0.31/1.87 bs114d: $o ).
% 0.31/1.87
% 0.31/1.87 thf(bs114d,definition,
% 0.31/1.87 ( bs114d
% 0.31/1.87 = ( ! [A: $i,B: $i,C: $i] :
% 0.31/1.87 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.31/1.87 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(regular_type,type,
% 0.31/1.87 regular: $i > $o ).
% 0.31/1.87
% 0.31/1.87 thf(setminus_type,type,
% 0.31/1.87 setminus: $i > $i > $i ).
% 0.31/1.87
% 0.31/1.87 thf(setminusI_type,type,
% 0.31/1.87 setminusI: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setminusI,definition,
% 0.31/1.87 ( setminusI
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ~ ( in @ Xx @ B )
% 0.31/1.87 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setminusEL_type,type,
% 0.31/1.87 setminusEL: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setminusEL,definition,
% 0.31/1.87 ( setminusEL
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/1.87 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setminusER_type,type,
% 0.31/1.87 setminusER: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setminusER,definition,
% 0.31/1.87 ( setminusER
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/1.87 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setminusSubset2_type,type,
% 0.31/1.87 setminusSubset2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setminusSubset2,definition,
% 0.31/1.87 ( setminusSubset2
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( subset @ A @ B )
% 0.31/1.87 => ( ( setminus @ A @ B )
% 0.31/1.87 = emptyset ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setminusERneg_type,type,
% 0.31/1.87 setminusERneg: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setminusERneg,definition,
% 0.31/1.87 ( setminusERneg
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/1.87 => ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setminusELneg_type,type,
% 0.31/1.87 setminusELneg: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setminusELneg,definition,
% 0.31/1.87 ( setminusELneg
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.31/1.87 => ( ~ ( in @ Xx @ B )
% 0.31/1.87 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setminusILneg_type,type,
% 0.31/1.87 setminusILneg: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setminusILneg,definition,
% 0.31/1.87 ( setminusILneg
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ~ ( in @ Xx @ A )
% 0.31/1.87 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setminusIRneg_type,type,
% 0.31/1.87 setminusIRneg: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setminusIRneg,definition,
% 0.31/1.87 ( setminusIRneg
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ B )
% 0.31/1.87 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setminusLsub_type,type,
% 0.31/1.87 setminusLsub: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setminusLsub,definition,
% 0.31/1.87 ( setminusLsub
% 0.31/1.87 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setminusSubset1_type,type,
% 0.31/1.87 setminusSubset1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setminusSubset1,definition,
% 0.31/1.87 ( setminusSubset1
% 0.31/1.87 = ( ! [A: $i,B: $i] :
% 0.31/1.87 ( ( ( setminus @ A @ B )
% 0.31/1.87 = emptyset )
% 0.31/1.87 => ( subset @ A @ B ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(symdiff_type,type,
% 0.31/1.87 symdiff: $i > $i > $i ).
% 0.31/1.87
% 0.31/1.87 thf(symdiffE_type,type,
% 0.31/1.87 symdiffE: $o ).
% 0.31/1.87
% 0.31/1.87 thf(symdiffE,definition,
% 0.31/1.87 ( symdiffE
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.31/1.87 => ! [Xphi: $o] :
% 0.31/1.87 ( ( ( in @ Xx @ A )
% 0.31/1.87 => ( ~ ( in @ Xx @ B )
% 0.31/1.87 => Xphi ) )
% 0.31/1.87 => ( ( ~ ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ B )
% 0.31/1.87 => Xphi ) )
% 0.31/1.87 => Xphi ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(symdiffI1_type,type,
% 0.31/1.87 symdiffI1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(symdiffI1,definition,
% 0.31/1.87 ( symdiffI1
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ~ ( in @ Xx @ B )
% 0.31/1.87 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(symdiffI2_type,type,
% 0.31/1.87 symdiffI2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(symdiffI2,definition,
% 0.31/1.87 ( symdiffI2
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ~ ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ B )
% 0.31/1.87 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(symdiffIneg1_type,type,
% 0.31/1.87 symdiffIneg1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(symdiffIneg1,definition,
% 0.31/1.87 ( symdiffIneg1
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ B )
% 0.31/1.87 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(symdiffIneg2_type,type,
% 0.31/1.87 symdiffIneg2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(symdiffIneg2,definition,
% 0.31/1.87 ( symdiffIneg2
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ~ ( in @ Xx @ A )
% 0.31/1.87 => ( ~ ( in @ Xx @ B )
% 0.31/1.87 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(iskpair_type,type,
% 0.31/1.87 iskpair: $i > $o ).
% 0.31/1.87
% 0.31/1.87 thf(secondinupair_type,type,
% 0.31/1.87 secondinupair: $o ).
% 0.31/1.87
% 0.31/1.87 thf(secondinupair,definition,
% 0.31/1.87 ( secondinupair
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairIL_type,type,
% 0.31/1.87 setukpairIL: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairIL,definition,
% 0.31/1.87 ( setukpairIL
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairIR_type,type,
% 0.31/1.87 setukpairIR: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairIR,definition,
% 0.31/1.87 ( setukpairIR
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(kpairiskpair_type,type,
% 0.31/1.87 kpairiskpair: $o ).
% 0.31/1.87
% 0.31/1.87 thf(kpairiskpair,definition,
% 0.31/1.87 ( kpairiskpair
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(kpair_type,type,
% 0.31/1.87 kpair: $i > $i > $i ).
% 0.31/1.87
% 0.31/1.87 thf(kpairp_type,type,
% 0.31/1.87 kpairp: $o ).
% 0.31/1.87
% 0.31/1.87 thf(kpairp,definition,
% 0.31/1.87 ( kpairp
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(cartprod_type,type,
% 0.31/1.87 cartprod: $i > $i > $i ).
% 0.31/1.87
% 0.31/1.87 thf(singletonsubset_type,type,
% 0.31/1.87 singletonsubset: $o ).
% 0.31/1.87
% 0.31/1.87 thf(singletonsubset,definition,
% 0.31/1.87 ( singletonsubset
% 0.31/1.87 = ( ! [A: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(singletoninpowerset_type,type,
% 0.31/1.87 singletoninpowerset: $o ).
% 0.31/1.87
% 0.31/1.87 thf(singletoninpowerset,definition,
% 0.31/1.87 ( singletoninpowerset
% 0.31/1.87 = ( ! [A: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(singletoninpowunion_type,type,
% 0.31/1.87 singletoninpowunion: $o ).
% 0.31/1.87
% 0.31/1.87 thf(singletoninpowunion,definition,
% 0.31/1.87 ( singletoninpowunion
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(upairset2E_type,type,
% 0.31/1.87 upairset2E: $o ).
% 0.31/1.87
% 0.31/1.87 thf(upairset2E,definition,
% 0.31/1.87 ( upairset2E
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/1.87 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.31/1.87 => ( ( Xz = Xx )
% 0.31/1.87 | ( Xz = Xy ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(upairsubunion_type,type,
% 0.31/1.87 upairsubunion: $o ).
% 0.31/1.87
% 0.31/1.87 thf(upairsubunion,definition,
% 0.31/1.87 ( upairsubunion
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(upairinpowunion_type,type,
% 0.31/1.87 upairinpowunion: $o ).
% 0.31/1.87
% 0.31/1.87 thf(upairinpowunion,definition,
% 0.31/1.87 ( upairinpowunion
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(ubforcartprodlem1_type,type,
% 0.31/1.87 ubforcartprodlem1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(ubforcartprodlem1,definition,
% 0.31/1.87 ( ubforcartprodlem1
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(ubforcartprodlem2_type,type,
% 0.31/1.87 ubforcartprodlem2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(ubforcartprodlem2,definition,
% 0.31/1.87 ( ubforcartprodlem2
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(ubforcartprodlem3_type,type,
% 0.31/1.87 ubforcartprodlem3: $o ).
% 0.31/1.87
% 0.31/1.87 thf(ubforcartprodlem3,definition,
% 0.31/1.87 ( ubforcartprodlem3
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodpairin_type,type,
% 0.31/1.87 cartprodpairin: $o ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodpairin,definition,
% 0.31/1.87 ( cartprodpairin
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodmempair1_type,type,
% 0.31/1.87 cartprodmempair1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodmempair1,definition,
% 0.31/1.87 ( cartprodmempair1
% 0.31/1.87 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/1.87 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/1.87 => ? [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 & ? [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 & ( Xu
% 0.31/1.87 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodmempair_type,type,
% 0.31/1.87 cartprodmempair: $o ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodmempair,definition,
% 0.31/1.87 ( cartprodmempair
% 0.31/1.87 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/1.87 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/1.87 => ( iskpair @ Xu ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setunionE2_type,type,
% 0.31/1.87 setunionE2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setunionE2,definition,
% 0.31/1.87 ( setunionE2
% 0.31/1.87 = ( ! [A: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ ( setunion @ A ) )
% 0.31/1.87 => ? [X: $i] :
% 0.31/1.87 ( ( in @ X @ A )
% 0.31/1.87 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setunionsingleton1_type,type,
% 0.31/1.87 setunionsingleton1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setunionsingleton1,definition,
% 0.31/1.87 ( setunionsingleton1
% 0.31/1.87 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setunionsingleton2_type,type,
% 0.31/1.87 setunionsingleton2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setunionsingleton2,definition,
% 0.31/1.87 ( setunionsingleton2
% 0.31/1.87 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setunionsingleton_type,type,
% 0.31/1.87 setunionsingleton: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setunionsingleton,definition,
% 0.31/1.87 ( setunionsingleton
% 0.31/1.87 = ( ! [Xx: $i] :
% 0.31/1.87 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.31/1.87 = Xx ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(singleton_type,type,
% 0.31/1.87 singleton: $i > $o ).
% 0.31/1.87
% 0.31/1.87 thf(singletonprop_type,type,
% 0.31/1.87 singletonprop: $o ).
% 0.31/1.87
% 0.31/1.87 thf(singletonprop,definition,
% 0.31/1.87 ( singletonprop
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ A )
% 0.31/1.87 => ( ( Xphi @ Xx )
% 0.31/1.87 => ( ( Xphi @ Xy )
% 0.31/1.87 => ( Xx = Xy ) ) ) ) )
% 0.31/1.87 => ( ? [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 & ( Xphi @ Xx ) )
% 0.31/1.87 => ( singleton
% 0.31/1.87 @ ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(ex1_type,type,
% 0.31/1.87 ex1: $i > ( $i > $o ) > $o ).
% 0.31/1.87
% 0.31/1.87 thf(ex1,definition,
% 0.31/1.87 ( ex1
% 0.31/1.87 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.31/1.87 ( singleton
% 0.31/1.87 @ ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(ex1E1_type,type,
% 0.31/1.87 ex1E1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(ex1E1,definition,
% 0.31/1.87 ( ex1E1
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.87 ( ( ex1 @ A
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.87 => ? [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(ex1I_type,type,
% 0.31/1.87 ex1I: $o ).
% 0.31/1.87
% 0.31/1.87 thf(ex1I,definition,
% 0.31/1.87 ( ex1I
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( Xphi @ Xx )
% 0.31/1.87 => ( ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ A )
% 0.31/1.87 => ( ( Xphi @ Xy )
% 0.31/1.87 => ( Xy = Xx ) ) )
% 0.31/1.87 => ( ex1 @ A
% 0.31/1.87 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(ex1I2_type,type,
% 0.31/1.87 ex1I2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(ex1I2,definition,
% 0.31/1.87 ( ex1I2
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ A )
% 0.31/1.87 => ( ( Xphi @ Xx )
% 0.31/1.87 => ( ( Xphi @ Xy )
% 0.31/1.87 => ( Xx = Xy ) ) ) ) )
% 0.31/1.87 => ( ? [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 & ( Xphi @ Xx ) )
% 0.31/1.87 => ( ex1 @ A
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(singletonsuniq_type,type,
% 0.31/1.87 singletonsuniq: $o ).
% 0.31/1.87
% 0.31/1.87 thf(singletonsuniq,definition,
% 0.31/1.87 ( singletonsuniq
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( ( setadjoin @ Xx @ emptyset )
% 0.31/1.87 = ( setadjoin @ Xy @ emptyset ) )
% 0.31/1.87 => ( Xx = Xy ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(atmost1p_type,type,
% 0.31/1.87 atmost1p: $i > $o ).
% 0.31/1.87
% 0.31/1.87 thf(atleast2p_type,type,
% 0.31/1.87 atleast2p: $i > $o ).
% 0.31/1.87
% 0.31/1.87 thf(atmost2p_type,type,
% 0.31/1.87 atmost2p: $i > $o ).
% 0.31/1.87
% 0.31/1.87 thf(upairsetp_type,type,
% 0.31/1.87 upairsetp: $i > $o ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairinjL1_type,type,
% 0.31/1.87 setukpairinjL1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairinjL1,definition,
% 0.31/1.87 ( setukpairinjL1
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/1.87 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.31/1.87 => ( Xx = Xz ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(kfstsingleton_type,type,
% 0.31/1.87 kfstsingleton: $o ).
% 0.31/1.87
% 0.31/1.87 thf(kfstsingleton,definition,
% 0.31/1.87 ( kfstsingleton
% 0.31/1.87 = ( ! [Xu: $i] :
% 0.31/1.87 ( ( iskpair @ Xu )
% 0.31/1.87 => ( singleton
% 0.31/1.87 @ ( dsetconstr @ ( setunion @ Xu )
% 0.31/1.87 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(theprop_type,type,
% 0.31/1.87 theprop: $o ).
% 0.31/1.87
% 0.31/1.87 thf(theprop,definition,
% 0.31/1.87 ( theprop
% 0.31/1.87 = ( ! [X: $i] :
% 0.31/1.87 ( ( singleton @ X )
% 0.31/1.87 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(kfst_type,type,
% 0.31/1.87 kfst: $i > $i ).
% 0.31/1.87
% 0.31/1.87 thf(kfstpairEq_type,type,
% 0.31/1.87 kfstpairEq: $o ).
% 0.31/1.87
% 0.31/1.87 thf(kfstpairEq,definition,
% 0.31/1.87 ( kfstpairEq
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.31/1.87 = Xx ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodfstin_type,type,
% 0.31/1.87 cartprodfstin: $o ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodfstin,definition,
% 0.31/1.87 ( cartprodfstin
% 0.31/1.87 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/1.87 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/1.87 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairinjL2_type,type,
% 0.31/1.87 setukpairinjL2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairinjL2,definition,
% 0.31/1.87 ( setukpairinjL2
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/1.87 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/1.87 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/1.87 => ( Xx = Xz ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairinjL_type,type,
% 0.31/1.87 setukpairinjL: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairinjL,definition,
% 0.31/1.87 ( setukpairinjL
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/1.87 ( ( ( kpair @ Xx @ Xy )
% 0.31/1.87 = ( kpair @ Xz @ Xu ) )
% 0.31/1.87 => ( Xx = Xz ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairinjR11_type,type,
% 0.31/1.87 setukpairinjR11: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairinjR11,definition,
% 0.31/1.87 ( setukpairinjR11
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( Xx = Xy )
% 0.31/1.87 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/1.87 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairinjR12_type,type,
% 0.31/1.87 setukpairinjR12: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairinjR12,definition,
% 0.31/1.87 ( setukpairinjR12
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( Xx = Xy )
% 0.31/1.87 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/1.87 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairinjR1_type,type,
% 0.31/1.87 setukpairinjR1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairinjR1,definition,
% 0.31/1.87 ( setukpairinjR1
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/1.87 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/1.87 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/1.87 => ( ( Xz = Xu )
% 0.31/1.87 => ( Xy = Xu ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(upairequniteq_type,type,
% 0.31/1.87 upairequniteq: $o ).
% 0.31/1.87
% 0.31/1.87 thf(upairequniteq,definition,
% 0.31/1.87 ( upairequniteq
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.31/1.87 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.31/1.87 = ( setadjoin @ Xz @ emptyset ) )
% 0.31/1.87 => ( Xx = Xy ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairinjR2_type,type,
% 0.31/1.87 setukpairinjR2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairinjR2,definition,
% 0.31/1.87 ( setukpairinjR2
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/1.87 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.31/1.87 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.31/1.87 => ( Xy = Xu ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairinjR_type,type,
% 0.31/1.87 setukpairinjR: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setukpairinjR,definition,
% 0.31/1.87 ( setukpairinjR
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.31/1.87 ( ( ( kpair @ Xx @ Xy )
% 0.31/1.87 = ( kpair @ Xz @ Xu ) )
% 0.31/1.87 => ( Xy = Xu ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(ksndsingleton_type,type,
% 0.31/1.87 ksndsingleton: $o ).
% 0.31/1.87
% 0.31/1.87 thf(ksndsingleton,definition,
% 0.31/1.87 ( ksndsingleton
% 0.31/1.87 = ( ! [Xu: $i] :
% 0.31/1.87 ( ( iskpair @ Xu )
% 0.31/1.87 => ( singleton
% 0.31/1.87 @ ( dsetconstr @ ( setunion @ Xu )
% 0.31/1.87 @ ^ [Xx: $i] :
% 0.31/1.87 ( Xu
% 0.31/1.87 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(ksnd_type,type,
% 0.31/1.87 ksnd: $i > $i ).
% 0.31/1.87
% 0.31/1.87 thf(ksndpairEq_type,type,
% 0.31/1.87 ksndpairEq: $o ).
% 0.31/1.87
% 0.31/1.87 thf(ksndpairEq,definition,
% 0.31/1.87 ( ksndpairEq
% 0.31/1.87 = ( ! [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.31/1.87 = Xy ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(kpairsurjEq_type,type,
% 0.31/1.87 kpairsurjEq: $o ).
% 0.31/1.87
% 0.31/1.87 thf(kpairsurjEq,definition,
% 0.31/1.87 ( kpairsurjEq
% 0.31/1.87 = ( ! [Xu: $i] :
% 0.31/1.87 ( ( iskpair @ Xu )
% 0.31/1.87 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.31/1.87 = Xu ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodsndin_type,type,
% 0.31/1.87 cartprodsndin: $o ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodsndin,definition,
% 0.31/1.87 ( cartprodsndin
% 0.31/1.87 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/1.87 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/1.87 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodpairmemEL_type,type,
% 0.31/1.87 cartprodpairmemEL: $o ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodpairmemEL,definition,
% 0.31/1.87 ( cartprodpairmemEL
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.31/1.87 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodpairmemER_type,type,
% 0.31/1.87 cartprodpairmemER: $o ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodpairmemER,definition,
% 0.31/1.87 ( cartprodpairmemER
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.31/1.87 => ( in @ Xy @ B ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodmempaircEq_type,type,
% 0.31/1.87 cartprodmempaircEq: $o ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodmempaircEq,definition,
% 0.31/1.87 ( cartprodmempaircEq
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( ( kpair @ Xx @ Xy )
% 0.31/1.87 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodfstpairEq_type,type,
% 0.31/1.87 cartprodfstpairEq: $o ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodfstpairEq,definition,
% 0.31/1.87 ( cartprodfstpairEq
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.31/1.87 = Xx ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodsndpairEq_type,type,
% 0.31/1.87 cartprodsndpairEq: $o ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodsndpairEq,definition,
% 0.31/1.87 ( cartprodsndpairEq
% 0.31/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.31/1.87 = Xy ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodpairsurjEq_type,type,
% 0.31/1.87 cartprodpairsurjEq: $o ).
% 0.31/1.87
% 0.31/1.87 thf(cartprodpairsurjEq,definition,
% 0.31/1.87 ( cartprodpairsurjEq
% 0.31/1.87 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.31/1.87 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.31/1.87 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.31/1.87 = Xu ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(breln_type,type,
% 0.31/1.87 breln: $i > $i > $i > $o ).
% 0.31/1.87
% 0.31/1.87 thf(breln,definition,
% 0.31/1.87 ( breln
% 0.31/1.87 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(dpsetconstr_type,type,
% 0.31/1.87 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.31/1.87
% 0.31/1.87 thf(dpsetconstrI_type,type,
% 0.31/1.87 dpsetconstrI: $o ).
% 0.31/1.87
% 0.31/1.87 thf(dpsetconstrI,definition,
% 0.31/1.87 ( dpsetconstrI
% 0.31/1.87 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( ( Xphi @ Xx @ Xy )
% 0.31/1.87 => ( in @ ( kpair @ Xx @ Xy )
% 0.31/1.87 @ ( dpsetconstr @ A @ B
% 0.31/1.87 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(dpsetconstrSub_type,type,
% 0.31/1.87 dpsetconstrSub: $o ).
% 0.31/1.87
% 0.31/1.87 thf(dpsetconstrSub,definition,
% 0.31/1.87 ( dpsetconstrSub
% 0.31/1.87 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/1.87 ( subset
% 0.31/1.87 @ ( dpsetconstr @ A @ B
% 0.31/1.87 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.31/1.87 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setOfPairsIsBReln_type,type,
% 0.31/1.87 setOfPairsIsBReln: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setOfPairsIsBReln,definition,
% 0.31/1.87 ( setOfPairsIsBReln
% 0.31/1.87 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.31/1.87 ( breln @ A @ B
% 0.31/1.87 @ ( dpsetconstr @ A @ B
% 0.31/1.87 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(dpsetconstrERa_type,type,
% 0.31/1.87 dpsetconstrERa: $o ).
% 0.31/1.87
% 0.31/1.87 thf(dpsetconstrERa,definition,
% 0.31/1.87 ( dpsetconstrERa
% 0.31/1.87 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/1.87 @ ( dpsetconstr @ A @ B
% 0.31/1.87 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/1.87 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(dpsetconstrEL1_type,type,
% 0.31/1.87 dpsetconstrEL1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(dpsetconstrEL1,definition,
% 0.31/1.87 ( dpsetconstrEL1
% 0.31/1.87 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/1.87 @ ( dpsetconstr @ A @ B
% 0.31/1.87 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/1.87 => ( in @ Xx @ A ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(dpsetconstrEL2_type,type,
% 0.31/1.87 dpsetconstrEL2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(dpsetconstrEL2,definition,
% 0.31/1.87 ( dpsetconstrEL2
% 0.31/1.87 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/1.87 @ ( dpsetconstr @ A @ B
% 0.31/1.87 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/1.87 => ( in @ Xy @ B ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(dpsetconstrER_type,type,
% 0.31/1.87 dpsetconstrER: $o ).
% 0.31/1.87
% 0.31/1.87 thf(dpsetconstrER,definition,
% 0.31/1.87 ( dpsetconstrER
% 0.31/1.87 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( in @ ( kpair @ Xx @ Xy )
% 0.31/1.87 @ ( dpsetconstr @ A @ B
% 0.31/1.87 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.31/1.87 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(func_type,type,
% 0.31/1.87 func: $i > $i > $i > $o ).
% 0.31/1.87
% 0.31/1.87 thf(funcSet_type,type,
% 0.31/1.87 funcSet: $i > $i > $i ).
% 0.31/1.87
% 0.31/1.87 thf(funcImageSingleton_type,type,
% 0.31/1.87 funcImageSingleton: $o ).
% 0.31/1.87
% 0.31/1.87 thf(funcImageSingleton,definition,
% 0.31/1.87 ( funcImageSingleton
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( func @ A @ B @ Xf )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( singleton
% 0.31/1.87 @ ( dsetconstr @ B
% 0.31/1.87 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(apProp_type,type,
% 0.31/1.87 apProp: $o ).
% 0.31/1.87
% 0.31/1.87 thf(apProp,definition,
% 0.31/1.87 ( apProp
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( func @ A @ B @ Xf )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in
% 0.31/1.87 @ ( setunion
% 0.31/1.87 @ ( dsetconstr @ B
% 0.31/1.87 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.31/1.87 @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(ap_type,type,
% 0.31/1.87 ap: $i > $i > $i > $i > $i ).
% 0.31/1.87
% 0.31/1.87 thf(app_type,type,
% 0.31/1.87 app: $o ).
% 0.31/1.87
% 0.31/1.87 thf(app,definition,
% 0.31/1.87 ( app
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( func @ A @ B @ Xf )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(infuncsetfunc_type,type,
% 0.31/1.87 infuncsetfunc: $o ).
% 0.31/1.87
% 0.31/1.87 thf(infuncsetfunc,definition,
% 0.31/1.87 ( infuncsetfunc
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/1.87 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(ap2p_type,type,
% 0.31/1.87 ap2p: $o ).
% 0.31/1.87
% 0.31/1.87 thf(ap2p,definition,
% 0.31/1.87 ( ap2p
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(funcinfuncset_type,type,
% 0.31/1.87 funcinfuncset: $o ).
% 0.31/1.87
% 0.31/1.87 thf(funcinfuncset,definition,
% 0.31/1.87 ( funcinfuncset
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( func @ A @ B @ Xf )
% 0.31/1.87 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(lamProp_type,type,
% 0.31/1.87 lamProp: $o ).
% 0.31/1.87
% 0.31/1.87 thf(lamProp,definition,
% 0.31/1.87 ( lamProp
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/1.87 => ( func @ A @ B
% 0.31/1.87 @ ( dpsetconstr @ A @ B
% 0.31/1.87 @ ^ [Xx: $i,Xy: $i] :
% 0.31/1.87 ( ( Xf @ Xx )
% 0.31/1.87 = Xy ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(lam_type,type,
% 0.31/1.87 lam: $i > $i > ( $i > $i ) > $i ).
% 0.31/1.87
% 0.31/1.87 thf(lamp_type,type,
% 0.31/1.87 lamp: $o ).
% 0.31/1.87
% 0.31/1.87 thf(lamp,definition,
% 0.31/1.87 ( lamp
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/1.87 => ( func @ A @ B
% 0.31/1.87 @ ( lam @ A @ B
% 0.31/1.87 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(lam2p_type,type,
% 0.31/1.87 lam2p: $o ).
% 0.31/1.87
% 0.31/1.87 thf(lam2p,definition,
% 0.31/1.87 ( lam2p
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/1.87 => ( in
% 0.31/1.87 @ ( lam @ A @ B
% 0.31/1.87 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.31/1.87 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(brelnall1_type,type,
% 0.31/1.87 brelnall1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(brelnall1,definition,
% 0.31/1.87 ( brelnall1
% 0.31/1.87 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/1.87 ( ( breln @ A @ B @ R )
% 0.31/1.87 => ! [Xphi: $i > $o] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/1.87 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ R )
% 0.31/1.87 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(brelnall2_type,type,
% 0.31/1.87 brelnall2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(brelnall2,definition,
% 0.31/1.87 ( brelnall2
% 0.31/1.87 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/1.87 ( ( breln @ A @ B @ R )
% 0.31/1.87 => ! [Xphi: $i > $o] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/1.87 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ R )
% 0.31/1.87 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(ex1E2_type,type,
% 0.31/1.87 ex1E2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(ex1E2,definition,
% 0.31/1.87 ( ex1E2
% 0.31/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 0.31/1.87 ( ( ex1 @ A
% 0.31/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ A )
% 0.31/1.87 => ( ( Xphi @ Xx )
% 0.31/1.87 => ( ( Xphi @ Xy )
% 0.31/1.87 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(funcGraphProp1_type,type,
% 0.31/1.87 funcGraphProp1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(funcGraphProp1,definition,
% 0.31/1.87 ( funcGraphProp1
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( func @ A @ B @ Xf )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(funcGraphProp3_type,type,
% 0.31/1.87 funcGraphProp3: $o ).
% 0.31/1.87
% 0.31/1.87 thf(funcGraphProp3,definition,
% 0.31/1.87 ( funcGraphProp3
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(funcGraphProp2_type,type,
% 0.31/1.87 funcGraphProp2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(funcGraphProp2,definition,
% 0.31/1.87 ( funcGraphProp2
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( func @ A @ B @ Xf )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.31/1.87 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/1.87 = Xy ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(funcextLem_type,type,
% 0.31/1.87 funcextLem: $o ).
% 0.31/1.87
% 0.31/1.87 thf(funcextLem,definition,
% 0.31/1.87 ( funcextLem
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( func @ A @ B @ Xf )
% 0.31/1.87 => ! [Xg: $i] :
% 0.31/1.87 ( ( func @ A @ B @ Xg )
% 0.31/1.87 => ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/1.87 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.31/1.87 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(funcGraphProp4_type,type,
% 0.31/1.87 funcGraphProp4: $o ).
% 0.31/1.87
% 0.31/1.87 thf(funcGraphProp4,definition,
% 0.31/1.87 ( funcGraphProp4
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.31/1.87 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/1.87 = Xy ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(subbreln_type,type,
% 0.31/1.87 subbreln: $o ).
% 0.31/1.87
% 0.31/1.87 thf(subbreln,definition,
% 0.31/1.87 ( subbreln
% 0.31/1.87 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/1.87 ( ( breln @ A @ B @ R )
% 0.31/1.87 => ! [S: $i] :
% 0.31/1.87 ( ( breln @ A @ B @ S )
% 0.31/1.87 => ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/1.87 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.31/1.87 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(eqbreln_type,type,
% 0.31/1.87 eqbreln: $o ).
% 0.31/1.87
% 0.31/1.87 thf(eqbreln,definition,
% 0.31/1.87 ( eqbreln
% 0.31/1.87 = ( ! [A: $i,B: $i,R: $i] :
% 0.31/1.87 ( ( breln @ A @ B @ R )
% 0.31/1.87 => ! [S: $i] :
% 0.31/1.87 ( ( breln @ A @ B @ S )
% 0.31/1.87 => ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.31/1.87 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.31/1.87 => ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ B )
% 0.31/1.87 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.31/1.87 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.31/1.87 => ( R = S ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(funcext_type,type,
% 0.31/1.87 funcext: $o ).
% 0.31/1.87
% 0.31/1.87 thf(funcext,definition,
% 0.31/1.87 ( funcext
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( func @ A @ B @ Xf )
% 0.31/1.87 => ! [Xg: $i] :
% 0.31/1.87 ( ( func @ A @ B @ Xg )
% 0.31/1.87 => ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/1.87 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/1.87 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(funcext2_type,type,
% 0.31/1.87 funcext2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(funcext2,definition,
% 0.31/1.87 ( funcext2
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/1.87 => ! [Xg: $i] :
% 0.31/1.87 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.31/1.87 => ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/1.87 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.31/1.87 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(ap2apEq1_type,type,
% 0.31/1.87 ap2apEq1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(ap2apEq1,definition,
% 0.31/1.87 ( ap2apEq1
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/1.87 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(ap2apEq2_type,type,
% 0.31/1.87 ap2apEq2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(ap2apEq2,definition,
% 0.31/1.87 ( ap2apEq2
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( func @ A @ B @ Xf )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.31/1.87 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(beta1_type,type,
% 0.31/1.87 beta1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(beta1,definition,
% 0.31/1.87 ( beta1
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( ap @ A @ B
% 0.31/1.87 @ ( lam @ A @ B
% 0.31/1.87 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.31/1.87 @ Xx )
% 0.31/1.87 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(eta1_type,type,
% 0.31/1.87 eta1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(eta1,definition,
% 0.31/1.87 ( eta1
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( func @ A @ B @ Xf )
% 0.31/1.87 => ( ( lam @ A @ B
% 0.31/1.87 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.31/1.87 = Xf ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(lam2lamEq_type,type,
% 0.31/1.87 lam2lamEq: $o ).
% 0.31/1.87
% 0.31/1.87 thf(lam2lamEq,definition,
% 0.31/1.87 ( lam2lamEq
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/1.87 => ( ( lam @ A @ B
% 0.31/1.87 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.31/1.87 = ( lam @ A @ B
% 0.31/1.87 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(beta2_type,type,
% 0.31/1.87 beta2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(beta2,definition,
% 0.31/1.87 ( beta2
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.31/1.87 ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( ap @ A @ B
% 0.31/1.87 @ ( lam @ A @ B
% 0.31/1.87 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.31/1.87 @ Xx )
% 0.31/1.87 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(eta2_type,type,
% 0.31/1.87 eta2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(eta2,definition,
% 0.31/1.87 ( eta2
% 0.31/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.31/1.87 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.31/1.87 => ( ( lam @ A @ B
% 0.31/1.87 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.31/1.87 = Xf ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(iffalseProp1_type,type,
% 0.31/1.87 iffalseProp1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(iffalseProp1,definition,
% 0.31/1.87 ( iffalseProp1
% 0.31/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ A )
% 0.31/1.87 => ( ~ Xphi
% 0.31/1.87 => ( in @ Xy
% 0.31/1.87 @ ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xz: $i] :
% 0.31/1.87 ( ( Xphi
% 0.31/1.87 & ( Xz = Xx ) )
% 0.31/1.87 | ( ~ Xphi
% 0.31/1.87 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(iffalseProp2_type,type,
% 0.31/1.87 iffalseProp2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(iffalseProp2,definition,
% 0.31/1.87 ( iffalseProp2
% 0.31/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ A )
% 0.31/1.87 => ( ~ Xphi
% 0.31/1.87 => ( ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xz: $i] :
% 0.31/1.87 ( ( Xphi
% 0.31/1.87 & ( Xz = Xx ) )
% 0.31/1.87 | ( ~ Xphi
% 0.31/1.87 & ( Xz = Xy ) ) ) )
% 0.31/1.87 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(iftrueProp1_type,type,
% 0.31/1.87 iftrueProp1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(iftrueProp1,definition,
% 0.31/1.87 ( iftrueProp1
% 0.31/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ A )
% 0.31/1.87 => ( Xphi
% 0.31/1.87 => ( in @ Xx
% 0.31/1.87 @ ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xz: $i] :
% 0.31/1.87 ( ( Xphi
% 0.31/1.87 & ( Xz = Xx ) )
% 0.31/1.87 | ( ~ Xphi
% 0.31/1.87 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(iftrueProp2_type,type,
% 0.31/1.87 iftrueProp2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(iftrueProp2,definition,
% 0.31/1.87 ( iftrueProp2
% 0.31/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ A )
% 0.31/1.87 => ( Xphi
% 0.31/1.87 => ( ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xz: $i] :
% 0.31/1.87 ( ( Xphi
% 0.31/1.87 & ( Xz = Xx ) )
% 0.31/1.87 | ( ~ Xphi
% 0.31/1.87 & ( Xz = Xy ) ) ) )
% 0.31/1.87 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(ifSingleton_type,type,
% 0.31/1.87 ifSingleton: $o ).
% 0.31/1.87
% 0.31/1.87 thf(ifSingleton,definition,
% 0.31/1.87 ( ifSingleton
% 0.31/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ A )
% 0.31/1.87 => ( singleton
% 0.31/1.87 @ ( dsetconstr @ A
% 0.31/1.87 @ ^ [Xz: $i] :
% 0.31/1.87 ( ( Xphi
% 0.31/1.87 & ( Xz = Xx ) )
% 0.31/1.87 | ( ~ Xphi
% 0.31/1.87 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(if_type,type,
% 0.31/1.87 if: $i > $o > $i > $i > $i ).
% 0.31/1.87
% 0.31/1.87 thf(ifp_type,type,
% 0.31/1.87 ifp: $o ).
% 0.31/1.87
% 0.31/1.87 thf(ifp,definition,
% 0.31/1.87 ( ifp
% 0.31/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ A )
% 0.31/1.87 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(theeq_type,type,
% 0.31/1.87 theeq: $o ).
% 0.31/1.87
% 0.31/1.87 thf(theeq,definition,
% 0.31/1.87 ( theeq
% 0.31/1.87 = ( ! [X: $i] :
% 0.31/1.87 ( ( singleton @ X )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ X )
% 0.31/1.87 => ( ( setunion @ X )
% 0.31/1.87 = Xx ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(iftrue_type,type,
% 0.31/1.87 iftrue: $o ).
% 0.31/1.87
% 0.31/1.87 thf(iftrue,definition,
% 0.31/1.87 ( iftrue
% 0.31/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ A )
% 0.31/1.87 => ( Xphi
% 0.31/1.87 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.31/1.87 = Xx ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(iffalse_type,type,
% 0.31/1.87 iffalse: $o ).
% 0.31/1.87
% 0.31/1.87 thf(iffalse,definition,
% 0.31/1.87 ( iffalse
% 0.31/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ A )
% 0.31/1.87 => ( ~ Xphi
% 0.31/1.87 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.31/1.87 = Xy ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(iftrueorfalse_type,type,
% 0.31/1.87 iftrueorfalse: $o ).
% 0.31/1.87
% 0.31/1.87 thf(iftrueorfalse,definition,
% 0.31/1.87 ( iftrueorfalse
% 0.31/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ! [Xy: $i] :
% 0.31/1.87 ( ( in @ Xy @ A )
% 0.31/1.87 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectT_lem_type,type,
% 0.31/1.87 binintersectT_lem: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectT_lem,definition,
% 0.31/1.87 ( binintersectT_lem
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binunionT_lem_type,type,
% 0.31/1.87 binunionT_lem: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binunionT_lem,definition,
% 0.31/1.87 ( binunionT_lem
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(powersetT_lem_type,type,
% 0.31/1.87 powersetT_lem: $o ).
% 0.31/1.87
% 0.31/1.87 thf(powersetT_lem,definition,
% 0.31/1.87 ( powersetT_lem
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setminusT_lem_type,type,
% 0.31/1.87 setminusT_lem: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setminusT_lem,definition,
% 0.31/1.87 ( setminusT_lem
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(complementT_lem_type,type,
% 0.31/1.87 complementT_lem: $o ).
% 0.31/1.87
% 0.31/1.87 thf(complementT_lem,definition,
% 0.31/1.87 ( complementT_lem
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(setextT_type,type,
% 0.31/1.87 setextT: $o ).
% 0.31/1.87
% 0.31/1.87 thf(setextT,definition,
% 0.31/1.87 ( setextT
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ X )
% 0.31/1.87 => ( in @ Xx @ Y ) ) )
% 0.31/1.87 => ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ Y )
% 0.31/1.87 => ( in @ Xx @ X ) ) )
% 0.31/1.87 => ( X = Y ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(subsetTI_type,type,
% 0.31/1.87 subsetTI: $o ).
% 0.31/1.87
% 0.31/1.87 thf(subsetTI,definition,
% 0.31/1.87 ( subsetTI
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ X )
% 0.31/1.87 => ( in @ Xx @ Y ) ) )
% 0.31/1.87 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(powersetTI1_type,type,
% 0.31/1.87 powersetTI1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(powersetTI1,definition,
% 0.31/1.87 ( powersetTI1
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ( ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ X )
% 0.31/1.87 => ( in @ Xx @ Y ) ) )
% 0.31/1.87 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(powersetTE1_type,type,
% 0.31/1.87 powersetTE1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(powersetTE1,definition,
% 0.31/1.87 ( powersetTE1
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ X @ ( powerset @ Y ) )
% 0.31/1.87 => ( ( in @ Xx @ X )
% 0.31/1.87 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(complementTI1_type,type,
% 0.31/1.87 complementTI1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(complementTI1,definition,
% 0.31/1.87 ( complementTI1
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ X )
% 0.31/1.87 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(complementTE1_type,type,
% 0.31/1.87 complementTE1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(complementTE1,definition,
% 0.31/1.87 ( complementTE1
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.31/1.87 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectTELcontra_type,type,
% 0.31/1.87 binintersectTELcontra: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectTELcontra,definition,
% 0.31/1.87 ( binintersectTELcontra
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ~ ( in @ Xx @ X )
% 0.31/1.87 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectTERcontra_type,type,
% 0.31/1.87 binintersectTERcontra: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binintersectTERcontra,definition,
% 0.31/1.87 ( binintersectTERcontra
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ~ ( in @ Xx @ Y )
% 0.31/1.87 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(contrasubsetT_type,type,
% 0.31/1.87 contrasubsetT: $o ).
% 0.31/1.87
% 0.31/1.87 thf(contrasubsetT,definition,
% 0.31/1.87 ( contrasubsetT
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.31/1.87 => ( ( in @ Xx @ Y )
% 0.31/1.87 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(contrasubsetT1_type,type,
% 0.31/1.87 contrasubsetT1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(contrasubsetT1,definition,
% 0.31/1.87 ( contrasubsetT1
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( subset @ X @ Y )
% 0.31/1.87 => ( ~ ( in @ Xx @ Y )
% 0.31/1.87 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(contrasubsetT2_type,type,
% 0.31/1.87 contrasubsetT2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(contrasubsetT2,definition,
% 0.31/1.87 ( contrasubsetT2
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ( ( subset @ X @ Y )
% 0.31/1.87 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(contrasubsetT3_type,type,
% 0.31/1.87 contrasubsetT3: $o ).
% 0.31/1.87
% 0.31/1.87 thf(contrasubsetT3,definition,
% 0.31/1.87 ( contrasubsetT3
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.31/1.87 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(doubleComplementI1_type,type,
% 0.31/1.87 doubleComplementI1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(doubleComplementI1,definition,
% 0.31/1.87 ( doubleComplementI1
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ X )
% 0.31/1.87 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(doubleComplementE1_type,type,
% 0.31/1.87 doubleComplementE1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(doubleComplementE1,definition,
% 0.31/1.87 ( doubleComplementE1
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.31/1.87 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(doubleComplementSub1_type,type,
% 0.31/1.87 doubleComplementSub1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(doubleComplementSub1,definition,
% 0.31/1.87 ( doubleComplementSub1
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(doubleComplementSub2_type,type,
% 0.31/1.87 doubleComplementSub2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(doubleComplementSub2,definition,
% 0.31/1.87 ( doubleComplementSub2
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(doubleComplementEq_type,type,
% 0.31/1.87 doubleComplementEq: $o ).
% 0.31/1.87
% 0.31/1.87 thf(doubleComplementEq,definition,
% 0.31/1.87 ( doubleComplementEq
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ( X
% 0.31/1.87 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(complementTnotintersectT_type,type,
% 0.31/1.87 complementTnotintersectT: $o ).
% 0.31/1.87
% 0.31/1.87 thf(complementTnotintersectT,definition,
% 0.31/1.87 ( complementTnotintersectT
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.31/1.87 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(complementImpComplementIntersect_type,type,
% 0.31/1.87 complementImpComplementIntersect: $o ).
% 0.31/1.87
% 0.31/1.87 thf(complementImpComplementIntersect,definition,
% 0.31/1.87 ( complementImpComplementIntersect
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.31/1.87 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(complementSubsetComplementIntersect_type,type,
% 0.31/1.87 complementSubsetComplementIntersect: $o ).
% 0.31/1.87
% 0.31/1.87 thf(complementSubsetComplementIntersect,definition,
% 0.31/1.87 ( complementSubsetComplementIntersect
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(complementInPowersetComplementIntersect_type,type,
% 0.31/1.87 complementInPowersetComplementIntersect: $o ).
% 0.31/1.87
% 0.31/1.87 thf(complementInPowersetComplementIntersect,definition,
% 0.31/1.87 ( complementInPowersetComplementIntersect
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(contraSubsetComplement_type,type,
% 0.31/1.87 contraSubsetComplement: $o ).
% 0.31/1.87
% 0.31/1.87 thf(contraSubsetComplement,definition,
% 0.31/1.87 ( contraSubsetComplement
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ Y )
% 0.31/1.87 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(complementTcontraSubset_type,type,
% 0.31/1.87 complementTcontraSubset: $o ).
% 0.31/1.87
% 0.31/1.87 thf(complementTcontraSubset,definition,
% 0.31/1.87 ( complementTcontraSubset
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.31/1.87 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binunionTILcontra_type,type,
% 0.31/1.87 binunionTILcontra: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binunionTILcontra,definition,
% 0.31/1.87 ( binunionTILcontra
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.31/1.87 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binunionTIRcontra_type,type,
% 0.31/1.87 binunionTIRcontra: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binunionTIRcontra,definition,
% 0.31/1.87 ( binunionTIRcontra
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.31/1.87 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(inIntersectImpInUnion_type,type,
% 0.31/1.87 inIntersectImpInUnion: $o ).
% 0.31/1.87
% 0.31/1.87 thf(inIntersectImpInUnion,definition,
% 0.31/1.87 ( inIntersectImpInUnion
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Z: $i] :
% 0.31/1.87 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.31/1.87 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(inIntersectImpInUnion2_type,type,
% 0.31/1.87 inIntersectImpInUnion2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(inIntersectImpInUnion2,definition,
% 0.31/1.87 ( inIntersectImpInUnion2
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Z: $i] :
% 0.31/1.87 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.31/1.87 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(inIntersectImpInIntersectUnions_type,type,
% 0.31/1.87 inIntersectImpInIntersectUnions: $o ).
% 0.31/1.87
% 0.31/1.87 thf(inIntersectImpInIntersectUnions,definition,
% 0.31/1.87 ( inIntersectImpInIntersectUnions
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Z: $i] :
% 0.31/1.87 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.31/1.87 => ( in @ Xx @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(intersectInPowersetIntersectUnions_type,type,
% 0.31/1.87 intersectInPowersetIntersectUnions: $o ).
% 0.31/1.87
% 0.31/1.87 thf(intersectInPowersetIntersectUnions,definition,
% 0.31/1.87 ( intersectInPowersetIntersectUnions
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Z: $i] :
% 0.31/1.87 ( ( in @ Z @ ( powerset @ A ) )
% 0.31/1.87 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ ( binintersect @ ( binunion @ X @ Z ) @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(inComplementUnionImpNotIn1_type,type,
% 0.31/1.87 inComplementUnionImpNotIn1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(inComplementUnionImpNotIn1,definition,
% 0.31/1.87 ( inComplementUnionImpNotIn1
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.31/1.87 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(inComplementUnionImpInComplement1_type,type,
% 0.31/1.87 inComplementUnionImpInComplement1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(inComplementUnionImpInComplement1,definition,
% 0.31/1.87 ( inComplementUnionImpInComplement1
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.31/1.87 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binunionTE_type,type,
% 0.31/1.87 binunionTE: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binunionTE,definition,
% 0.31/1.87 ( binunionTE
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xphi: $o,Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.31/1.87 => ( ( ( in @ Xx @ X )
% 0.31/1.87 => Xphi )
% 0.31/1.87 => ( ( ( in @ Xx @ Y )
% 0.31/1.87 => Xphi )
% 0.31/1.87 => Xphi ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(binunionTEcontra_type,type,
% 0.31/1.87 binunionTEcontra: $o ).
% 0.31/1.87
% 0.31/1.87 thf(binunionTEcontra,definition,
% 0.31/1.87 ( binunionTEcontra
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ~ ( in @ Xx @ X )
% 0.31/1.87 => ( ~ ( in @ Xx @ Y )
% 0.31/1.87 => ~ ( in @ Xx @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(demorgan2a1_type,type,
% 0.31/1.87 demorgan2a1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(demorgan2a1,definition,
% 0.31/1.87 ( demorgan2a1
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.31/1.87 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(complementUnionInPowersetComplement_type,type,
% 0.31/1.87 complementUnionInPowersetComplement: $o ).
% 0.31/1.87
% 0.31/1.87 thf(complementUnionInPowersetComplement,definition,
% 0.31/1.87 ( complementUnionInPowersetComplement
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ( in @ ( setminus @ A @ ( binunion @ X @ Y ) ) @ ( powerset @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(demorgan2a2_type,type,
% 0.31/1.87 demorgan2a2: $o ).
% 0.31/1.87
% 0.31/1.87 thf(demorgan2a2,definition,
% 0.31/1.87 ( demorgan2a2
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.31/1.87 => ( in @ Xx @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(demorgan1a_type,type,
% 0.31/1.87 demorgan1a: $o ).
% 0.31/1.87
% 0.31/1.87 thf(demorgan1a,definition,
% 0.31/1.87 ( demorgan1a
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) )
% 0.31/1.87 => ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(demorgan1b_type,type,
% 0.31/1.87 demorgan1b: $o ).
% 0.31/1.87
% 0.31/1.87 thf(demorgan1b,definition,
% 0.31/1.87 ( demorgan1b
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) )
% 0.31/1.87 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(demorgan1_type,type,
% 0.31/1.87 demorgan1: $o ).
% 0.31/1.87
% 0.31/1.87 thf(demorgan1,definition,
% 0.31/1.87 ( demorgan1
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ( ( setminus @ A @ ( binintersect @ X @ Y ) )
% 0.31/1.87 = ( binunion @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(demorgan2a_type,type,
% 0.31/1.87 demorgan2a: $o ).
% 0.31/1.87
% 0.31/1.87 thf(demorgan2a,definition,
% 0.31/1.87 ( demorgan2a
% 0.31/1.87 = ( ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) )
% 0.31/1.87 => ( in @ Xx @ ( binintersect @ ( setminus @ A @ X ) @ ( setminus @ A @ Y ) ) ) ) ) ) ) ) ) ).
% 0.31/1.87
% 0.31/1.87 thf(demorgan2b2,conjecture,
% 0.31/1.87 ( setextAx
% 0.31/1.87 => ( emptysetAx
% 0.31/1.87 => ( setadjoinAx
% 0.31/1.87 => ( powersetAx
% 0.31/1.87 => ( setunionAx
% 0.31/1.87 => ( omega0Ax
% 0.31/1.87 => ( omegaSAx
% 0.31/1.87 => ( omegaIndAx
% 0.31/1.87 => ( replAx
% 0.31/1.87 => ( foundationAx
% 0.31/1.87 => ( wellorderingAx
% 0.31/1.87 => ( descrp
% 0.31/1.87 => ( dsetconstrI
% 0.31/1.87 => ( dsetconstrEL
% 0.31/1.87 => ( dsetconstrER
% 0.31/1.87 => ( exuE1
% 0.31/1.87 => ( prop2setE
% 0.31/1.87 => ( emptysetE
% 0.31/1.87 => ( emptysetimpfalse
% 0.31/1.87 => ( notinemptyset
% 0.31/1.87 => ( exuE3e
% 0.31/1.87 => ( setext
% 0.31/1.87 => ( emptyI
% 0.31/1.87 => ( noeltsimpempty
% 0.31/1.87 => ( setbeta
% 0.31/1.87 => ( nonemptyE1
% 0.31/1.87 => ( nonemptyI
% 0.31/1.87 => ( nonemptyI1
% 0.31/1.87 => ( setadjoinIL
% 0.31/1.87 => ( emptyinunitempty
% 0.31/1.87 => ( setadjoinIR
% 0.31/1.87 => ( setadjoinE
% 0.31/1.87 => ( setadjoinOr
% 0.31/1.87 => ( setoftrueEq
% 0.31/1.87 => ( powersetI
% 0.31/1.87 => ( emptyinPowerset
% 0.31/1.87 => ( emptyInPowerset
% 0.31/1.87 => ( powersetE
% 0.31/1.87 => ( setunionI
% 0.31/1.87 => ( setunionE
% 0.31/1.87 => ( subPowSU
% 0.31/1.87 => ( exuE2
% 0.31/1.87 => ( nonemptyImpWitness
% 0.31/1.87 => ( uniqinunit
% 0.31/1.87 => ( notinsingleton
% 0.31/1.87 => ( eqinunit
% 0.31/1.87 => ( singletonsswitch
% 0.31/1.87 => ( upairsetE
% 0.31/1.87 => ( upairsetIL
% 0.31/1.87 => ( upairsetIR
% 0.31/1.87 => ( emptyE1
% 0.31/1.87 => ( vacuousDall
% 0.31/1.87 => ( quantDeMorgan1
% 0.31/1.87 => ( quantDeMorgan2
% 0.31/1.87 => ( quantDeMorgan3
% 0.31/1.87 => ( quantDeMorgan4
% 0.31/1.87 => ( prop2setI
% 0.31/1.87 => ( prop2set2propI
% 0.31/1.87 => ( notdexE
% 0.31/1.87 => ( notdallE
% 0.31/1.87 => ( exuI1
% 0.31/1.87 => ( exuI3
% 0.31/1.87 => ( exuI2
% 0.31/1.87 => ( inCongP
% 0.31/1.87 => ( in__Cong
% 0.31/1.87 => ( exuE3u
% 0.31/1.87 => ( exu__Cong
% 0.31/1.87 => ( emptyset__Cong
% 0.31/1.87 => ( setadjoin__Cong
% 0.31/1.87 => ( powerset__Cong
% 0.31/1.87 => ( setunion__Cong
% 0.31/1.87 => ( omega__Cong
% 0.31/1.87 => ( exuEu
% 0.31/1.87 => ( descr__Cong
% 0.31/1.87 => ( dsetconstr__Cong
% 0.31/1.87 => ( subsetI1
% 0.31/1.87 => ( eqimpsubset2
% 0.31/1.87 => ( eqimpsubset1
% 0.31/1.87 => ( subsetI2
% 0.31/1.87 => ( emptysetsubset
% 0.31/1.87 => ( subsetE
% 0.31/1.87 => ( subsetE2
% 0.31/1.87 => ( notsubsetI
% 0.31/1.87 => ( notequalI1
% 0.31/1.87 => ( notequalI2
% 0.31/1.87 => ( subsetRefl
% 0.31/1.87 => ( subsetTrans
% 0.31/1.87 => ( setadjoinSub
% 0.31/1.87 => ( setadjoinSub2
% 0.31/1.87 => ( subset2powerset
% 0.31/1.87 => ( setextsub
% 0.31/1.87 => ( subsetemptysetimpeq
% 0.31/1.87 => ( powersetI1
% 0.31/1.87 => ( powersetE1
% 0.31/1.87 => ( inPowerset
% 0.31/1.87 => ( powersetsubset
% 0.31/1.87 => ( sepInPowerset
% 0.31/1.87 => ( sepSubset
% 0.31/1.87 => ( binunionIL
% 0.31/1.87 => ( upairset2IR
% 0.31/1.87 => ( binunionIR
% 0.31/1.87 => ( binunionEcases
% 0.31/1.87 => ( binunionE
% 0.31/1.87 => ( binunionLsub
% 0.31/1.87 => ( binunionRsub
% 0.31/1.87 => ( binintersectI
% 0.31/1.87 => ( binintersectSubset5
% 0.31/1.87 => ( binintersectEL
% 0.31/1.87 => ( binintersectLsub
% 0.31/1.87 => ( binintersectSubset2
% 0.31/1.87 => ( binintersectSubset3
% 0.31/1.87 => ( binintersectER
% 0.31/1.87 => ( disjointsetsI1
% 0.31/1.87 => ( binintersectRsub
% 0.31/1.87 => ( binintersectSubset4
% 0.31/1.87 => ( binintersectSubset1
% 0.31/1.87 => ( bs114d
% 0.31/1.87 => ( setminusI
% 0.31/1.87 => ( setminusEL
% 0.31/1.87 => ( setminusER
% 0.31/1.87 => ( setminusSubset2
% 0.31/1.87 => ( setminusERneg
% 0.31/1.87 => ( setminusELneg
% 0.31/1.87 => ( setminusILneg
% 0.31/1.87 => ( setminusIRneg
% 0.31/1.87 => ( setminusLsub
% 0.31/1.87 => ( setminusSubset1
% 0.31/1.87 => ( symdiffE
% 0.31/1.87 => ( symdiffI1
% 0.31/1.87 => ( symdiffI2
% 0.31/1.87 => ( symdiffIneg1
% 0.31/1.87 => ( symdiffIneg2
% 0.31/1.87 => ( secondinupair
% 0.31/1.87 => ( setukpairIL
% 0.31/1.87 => ( setukpairIR
% 0.31/1.87 => ( kpairiskpair
% 0.31/1.87 => ( kpairp
% 0.31/1.87 => ( singletonsubset
% 0.31/1.87 => ( singletoninpowerset
% 0.31/1.87 => ( singletoninpowunion
% 0.31/1.87 => ( upairset2E
% 0.31/1.87 => ( upairsubunion
% 0.31/1.87 => ( upairinpowunion
% 0.31/1.87 => ( ubforcartprodlem1
% 0.31/1.87 => ( ubforcartprodlem2
% 0.31/1.87 => ( ubforcartprodlem3
% 0.31/1.87 => ( cartprodpairin
% 0.31/1.87 => ( cartprodmempair1
% 0.31/1.87 => ( cartprodmempair
% 0.31/1.87 => ( setunionE2
% 0.31/1.87 => ( setunionsingleton1
% 0.31/1.87 => ( setunionsingleton2
% 0.31/1.87 => ( setunionsingleton
% 0.31/1.87 => ( singletonprop
% 0.31/1.87 => ( ex1E1
% 0.31/1.87 => ( ex1I
% 0.31/1.87 => ( ex1I2
% 0.31/1.87 => ( singletonsuniq
% 0.31/1.87 => ( setukpairinjL1
% 0.31/1.87 => ( kfstsingleton
% 0.31/1.87 => ( theprop
% 0.31/1.87 => ( kfstpairEq
% 0.31/1.87 => ( cartprodfstin
% 0.31/1.87 => ( setukpairinjL2
% 0.31/1.87 => ( setukpairinjL
% 0.31/1.87 => ( setukpairinjR11
% 0.31/1.87 => ( setukpairinjR12
% 0.31/1.87 => ( setukpairinjR1
% 0.31/1.87 => ( upairequniteq
% 0.31/1.87 => ( setukpairinjR2
% 0.31/1.87 => ( setukpairinjR
% 0.31/1.87 => ( ksndsingleton
% 0.31/1.87 => ( ksndpairEq
% 0.31/1.87 => ( kpairsurjEq
% 0.31/1.87 => ( cartprodsndin
% 0.31/1.87 => ( cartprodpairmemEL
% 0.31/1.87 => ( cartprodpairmemER
% 0.31/1.87 => ( cartprodmempaircEq
% 0.31/1.87 => ( cartprodfstpairEq
% 0.31/1.87 => ( cartprodsndpairEq
% 0.31/1.87 => ( cartprodpairsurjEq
% 0.31/1.87 => ( dpsetconstrI
% 0.31/1.87 => ( dpsetconstrSub
% 0.31/1.87 => ( setOfPairsIsBReln
% 0.31/1.87 => ( dpsetconstrERa
% 0.31/1.87 => ( dpsetconstrEL1
% 0.31/1.87 => ( dpsetconstrEL2
% 0.31/1.87 => ( dpsetconstrER
% 0.31/1.87 => ( funcImageSingleton
% 0.31/1.87 => ( apProp
% 0.31/1.87 => ( app
% 0.31/1.87 => ( infuncsetfunc
% 0.31/1.87 => ( ap2p
% 0.31/1.87 => ( funcinfuncset
% 0.31/1.87 => ( lamProp
% 0.31/1.87 => ( lamp
% 0.31/1.87 => ( lam2p
% 0.31/1.87 => ( brelnall1
% 0.31/1.87 => ( brelnall2
% 0.31/1.87 => ( ex1E2
% 0.31/1.87 => ( funcGraphProp1
% 0.31/1.87 => ( funcGraphProp3
% 0.31/1.87 => ( funcGraphProp2
% 0.31/1.87 => ( funcextLem
% 0.31/1.87 => ( funcGraphProp4
% 0.31/1.87 => ( subbreln
% 0.31/1.87 => ( eqbreln
% 0.31/1.87 => ( funcext
% 0.31/1.87 => ( funcext2
% 0.31/1.87 => ( ap2apEq1
% 0.31/1.87 => ( ap2apEq2
% 0.31/1.87 => ( beta1
% 0.31/1.87 => ( eta1
% 0.31/1.87 => ( lam2lamEq
% 0.31/1.87 => ( beta2
% 0.31/1.87 => ( eta2
% 0.31/1.87 => ( iffalseProp1
% 0.31/1.87 => ( iffalseProp2
% 0.31/1.87 => ( iftrueProp1
% 0.31/1.87 => ( iftrueProp2
% 0.31/1.87 => ( ifSingleton
% 0.31/1.87 => ( ifp
% 0.31/1.87 => ( theeq
% 0.31/1.87 => ( iftrue
% 0.31/1.87 => ( iffalse
% 0.31/1.87 => ( iftrueorfalse
% 0.31/1.87 => ( binintersectT_lem
% 0.31/1.87 => ( binunionT_lem
% 0.31/1.87 => ( powersetT_lem
% 0.31/1.87 => ( setminusT_lem
% 0.31/1.87 => ( complementT_lem
% 0.31/1.87 => ( setextT
% 0.31/1.87 => ( subsetTI
% 0.31/1.87 => ( powersetTI1
% 0.31/1.87 => ( powersetTE1
% 0.31/1.87 => ( complementTI1
% 0.31/1.87 => ( complementTE1
% 0.31/1.87 => ( binintersectTELcontra
% 0.31/1.87 => ( binintersectTERcontra
% 0.31/1.87 => ( contrasubsetT
% 0.31/1.87 => ( contrasubsetT1
% 0.31/1.87 => ( contrasubsetT2
% 0.31/1.87 => ( contrasubsetT3
% 0.31/1.87 => ( doubleComplementI1
% 0.31/1.87 => ( doubleComplementE1
% 0.31/1.87 => ( doubleComplementSub1
% 0.31/1.87 => ( doubleComplementSub2
% 0.31/1.87 => ( doubleComplementEq
% 0.31/1.87 => ( complementTnotintersectT
% 0.31/1.87 => ( complementImpComplementIntersect
% 0.31/1.87 => ( complementSubsetComplementIntersect
% 0.31/1.87 => ( complementInPowersetComplementIntersect
% 0.31/1.87 => ( contraSubsetComplement
% 0.31/1.87 => ( complementTcontraSubset
% 0.31/1.87 => ( binunionTILcontra
% 0.31/1.87 => ( binunionTIRcontra
% 0.31/1.87 => ( inIntersectImpInUnion
% 0.31/1.87 => ( inIntersectImpInUnion2
% 0.31/1.87 => ( inIntersectImpInIntersectUnions
% 0.31/1.87 => ( intersectInPowersetIntersectUnions
% 0.31/1.87 => ( inComplementUnionImpNotIn1
% 0.31/1.87 => ( inComplementUnionImpInComplement1
% 0.31/1.87 => ( binunionTE
% 0.31/1.87 => ( binunionTEcontra
% 0.31/1.87 => ( demorgan2a1
% 0.31/1.87 => ( complementUnionInPowersetComplement
% 0.31/1.87 => ( demorgan2a2
% 0.31/1.87 => ( demorgan1a
% 0.31/1.87 => ( demorgan1b
% 0.31/1.87 => ( demorgan1
% 0.31/1.87 => ( demorgan2a
% 0.31/1.87 => ! [A: $i,X: $i] :
% 0.31/1.87 ( ( in @ X @ ( powerset @ A ) )
% 0.31/1.87 => ! [Y: $i] :
% 0.31/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 0.31/1.87 => ! [Xx: $i] :
% 0.31/1.87 ( ( in @ Xx @ A )
% 0.31/1.87 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.31/1.87 => ( ( in @ Xx @ ( setminus @ A @ Y ) )
% 0.31/1.87 => ( in @ Xx @ ( setminus @ A @ ( binunion @ X @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.31/1.87 ------------------------
% 3.46/1.92 % (26525)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_26421 for (2999ds/2Mi)
% 3.46/1.92 % (26526)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_26421 for (2999ds/275Mi)
% 3.46/1.92 % (26523)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_26421 for (2999ds/27Mi)
% 3.46/1.92 % (26521)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_26421 for (2999ds/183Mi)
% 3.46/1.92 % (26524)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_26421 for (2999ds/2Mi)
% 3.46/1.92 % (26522)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_26421 for (2999ds/4Mi)
% 3.46/1.92 % (26527)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_26421 for (2999ds/18Mi)
% 3.46/1.92 % (26524)Instruction limit reached!
% 3.46/1.92 % (26524)------------------------------
% 3.46/1.92 % (26524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.92 % (26524)Termination reason: Unknown
% 3.46/1.92 % (26524)Termination phase: shuffling
% 3.46/1.92
% 3.46/1.92 % (26524)Memory used [KB]: 1535
% 3.46/1.92 % (26524)Time elapsed: 0.003 s
% 3.46/1.92 % (26524)Instructions burned: 2 (million)
% 3.46/1.92 % (26524)------------------------------
% 3.46/1.92 % (26524)------------------------------
% 3.46/1.92 % (26525)Instruction limit reached!
% 3.46/1.92 % (26525)------------------------------
% 3.46/1.92 % (26525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.92 % (26525)Termination reason: Unknown
% 3.46/1.92 % (26525)Termination phase: shuffling
% 3.46/1.92
% 3.46/1.92 % (26525)Memory used [KB]: 1535
% 3.46/1.92 % (26525)Time elapsed: 0.003 s
% 3.46/1.92 % (26525)Instructions burned: 2 (million)
% 3.46/1.92 % (26525)------------------------------
% 3.46/1.92 % (26525)------------------------------
% 3.46/1.92 % (26522)Instruction limit reached!
% 3.46/1.92 % (26522)------------------------------
% 3.46/1.92 % (26522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.92 % (26522)Termination reason: Unknown
% 3.46/1.92 % (26522)Termination phase: shuffling
% 3.46/1.92
% 3.46/1.92 % (26522)Memory used [KB]: 1535
% 3.46/1.92 % (26522)Time elapsed: 0.004 s
% 3.46/1.92 % (26522)Instructions burned: 6 (million)
% 3.46/1.92 % (26522)------------------------------
% 3.46/1.92 % (26522)------------------------------
% 3.46/1.93 % (26527)Instruction limit reached!
% 3.46/1.93 % (26527)------------------------------
% 3.46/1.93 % (26527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.93 % (26527)Termination reason: Unknown
% 3.46/1.93 % (26527)Termination phase: shuffling
% 3.46/1.93
% 3.46/1.93 % (26527)Memory used [KB]: 1791
% 3.46/1.93 % (26527)Time elapsed: 0.011 s
% 3.46/1.93 % (26527)Instructions burned: 18 (million)
% 3.46/1.93 % (26527)------------------------------
% 3.46/1.93 % (26527)------------------------------
% 3.46/1.93 % (26523)Instruction limit reached!
% 3.46/1.93 % (26523)------------------------------
% 3.46/1.93 % (26523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.93 % (26523)Termination reason: Unknown
% 3.46/1.93 % (26523)Termination phase: shuffling
% 3.46/1.93
% 3.46/1.93 % (26523)Memory used [KB]: 1918
% 3.46/1.93 % (26523)Time elapsed: 0.015 s
% 3.46/1.93 % (26523)Instructions burned: 28 (million)
% 3.46/1.93 % (26523)------------------------------
% 3.46/1.93 % (26523)------------------------------
% 3.46/1.93 % (26529)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_26421 for (2999ds/37Mi)
% 3.46/1.93 % (26528)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_26421 for (2999ds/3Mi)
% 3.46/1.93 % (26528)Instruction limit reached!
% 3.46/1.93 % (26528)------------------------------
% 3.46/1.93 % (26528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.94 % (26528)Termination reason: Unknown
% 3.46/1.94 % (26528)Termination phase: shuffling
% 3.46/1.94
% 3.46/1.94 % (26528)Memory used [KB]: 1535
% 3.46/1.94 % (26528)Time elapsed: 0.003 s
% 3.46/1.94 % (26528)Instructions burned: 3 (million)
% 3.46/1.94 % (26528)------------------------------
% 3.46/1.94 % (26528)------------------------------
% 3.46/1.94 % (26530)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_26421 for (2999ds/15Mi)
% 3.46/1.94 % (26531)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_26421 for (2999ds/3Mi)
% 3.46/1.94 % (26530)Instruction limit reached!
% 3.46/1.94 % (26530)------------------------------
% 3.46/1.94 % (26530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.94 % (26530)Termination reason: Unknown
% 3.46/1.94 % (26530)Termination phase: shuffling
% 3.46/1.94
% 3.46/1.94 % (26530)Memory used [KB]: 1791
% 3.46/1.94 % (26530)Time elapsed: 0.009 s
% 3.46/1.94 % (26530)Instructions burned: 15 (million)
% 3.46/1.94 % (26530)------------------------------
% 3.46/1.94 % (26530)------------------------------
% 3.46/1.94 % (26531)Instruction limit reached!
% 3.46/1.94 % (26531)------------------------------
% 3.46/1.94 % (26531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.94 % (26531)Termination reason: Unknown
% 3.46/1.94 % (26531)Termination phase: shuffling
% 3.46/1.94
% 3.46/1.94 % (26531)Memory used [KB]: 1535
% 3.46/1.94 % (26531)Time elapsed: 0.003 s
% 3.46/1.94 % (26531)Instructions burned: 3 (million)
% 3.46/1.94 % (26531)------------------------------
% 3.46/1.94 % (26531)------------------------------
% 3.46/1.95 % (26532)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_26421 for (2999ds/1041Mi)
% 3.46/1.95 % (26529)Instruction limit reached!
% 3.46/1.95 % (26529)------------------------------
% 3.46/1.95 % (26529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.95 % (26529)Termination reason: Unknown
% 3.46/1.95 % (26529)Termination phase: shuffling
% 3.46/1.95
% 3.46/1.95 % (26529)Memory used [KB]: 2302
% 3.46/1.95 % (26529)Time elapsed: 0.019 s
% 3.46/1.95 % (26529)Instructions burned: 38 (million)
% 3.46/1.95 % (26529)------------------------------
% 3.46/1.95 % (26529)------------------------------
% 3.46/1.95 % (26533)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_26421 for (2999ds/7Mi)
% 3.46/1.95 % (26533)Instruction limit reached!
% 3.46/1.95 % (26533)------------------------------
% 3.46/1.95 % (26533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.95 % (26533)Termination reason: Unknown
% 3.46/1.95 % (26533)Termination phase: shuffling
% 3.46/1.95
% 3.46/1.95 % (26533)Memory used [KB]: 1535
% 3.46/1.95 % (26533)Time elapsed: 0.005 s
% 3.46/1.95 % (26533)Instructions burned: 7 (million)
% 3.46/1.95 % (26533)------------------------------
% 3.46/1.95 % (26533)------------------------------
% 3.46/1.96 % (26534)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_26421 for (2999ds/16Mi)
% 3.46/1.96 % (26535)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_26421 for (2999ds/3Mi)
% 3.46/1.96 % (26535)Instruction limit reached!
% 3.46/1.96 % (26535)------------------------------
% 3.46/1.96 % (26535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.96 % (26535)Termination reason: Unknown
% 3.46/1.96 % (26535)Termination phase: shuffling
% 3.46/1.96
% 3.46/1.96 % (26535)Memory used [KB]: 1535
% 3.46/1.96 % (26535)Time elapsed: 0.003 s
% 3.46/1.96 % (26535)Instructions burned: 3 (million)
% 3.46/1.96 % (26535)------------------------------
% 3.46/1.96 % (26535)------------------------------
% 3.46/1.97 % (26534)Instruction limit reached!
% 3.46/1.97 % (26534)------------------------------
% 3.46/1.97 % (26534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.97 % (26534)Termination reason: Unknown
% 3.46/1.97 % (26534)Termination phase: shuffling
% 3.46/1.97
% 3.46/1.97 % (26534)Memory used [KB]: 1791
% 3.46/1.97 % (26534)Time elapsed: 0.009 s
% 3.46/1.97 % (26534)Instructions burned: 16 (million)
% 3.46/1.97 % (26534)------------------------------
% 3.46/1.97 % (26534)------------------------------
% 3.46/1.97 % (26536)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_26421 for (2999ds/3Mi)
% 3.46/1.97 % (26536)Instruction limit reached!
% 3.46/1.97 % (26536)------------------------------
% 3.46/1.97 % (26536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.97 % (26536)Termination reason: Unknown
% 3.46/1.97 % (26536)Termination phase: shuffling
% 3.46/1.97
% 3.46/1.97 % (26536)Memory used [KB]: 1535
% 3.46/1.97 % (26536)Time elapsed: 0.003 s
% 3.46/1.97 % (26536)Instructions burned: 3 (million)
% 3.46/1.97 % (26536)------------------------------
% 3.46/1.97 % (26536)------------------------------
% 3.46/1.97 % (26537)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_26421 for (2999ds/7Mi)
% 3.46/1.97 % (26537)Instruction limit reached!
% 3.46/1.97 % (26537)------------------------------
% 3.46/1.97 % (26537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.97 % (26537)Termination reason: Unknown
% 3.46/1.97 % (26537)Termination phase: shuffling
% 3.46/1.97
% 3.46/1.97 % (26537)Memory used [KB]: 1535
% 3.46/1.97 % (26537)Time elapsed: 0.005 s
% 3.46/1.97 % (26537)Instructions burned: 7 (million)
% 3.46/1.97 % (26537)------------------------------
% 3.46/1.97 % (26537)------------------------------
% 3.46/1.98 % (26538)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_26421 for (2999ds/3Mi)
% 3.46/1.98 % (26538)Instruction limit reached!
% 3.46/1.98 % (26538)------------------------------
% 3.46/1.98 % (26538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.98 % (26538)Termination reason: Unknown
% 3.46/1.98 % (26538)Termination phase: shuffling
% 3.46/1.98
% 3.46/1.98 % (26538)Memory used [KB]: 1535
% 3.46/1.98 % (26538)Time elapsed: 0.003 s
% 3.46/1.98 % (26538)Instructions burned: 3 (million)
% 3.46/1.98 % (26538)------------------------------
% 3.46/1.98 % (26538)------------------------------
% 3.46/1.98 % (26539)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_26421 for (2999ds/4Mi)
% 3.46/1.98 % (26539)Instruction limit reached!
% 3.46/1.98 % (26539)------------------------------
% 3.46/1.98 % (26539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.98 % (26539)Termination reason: Unknown
% 3.46/1.98 % (26539)Termination phase: shuffling
% 3.46/1.98
% 3.46/1.98 % (26540)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_26421 for (2999ds/18Mi)
% 3.46/1.98 % (26539)Memory used [KB]: 1535
% 3.46/1.98 % (26539)Time elapsed: 0.004 s
% 3.46/1.98 % (26539)Instructions burned: 5 (million)
% 3.46/1.98 % (26539)------------------------------
% 3.46/1.98 % (26539)------------------------------
% 3.46/1.99 % (26541)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_26421 for (2999ds/710Mi)
% 3.46/1.99 % (26540)Instruction limit reached!
% 3.46/1.99 % (26540)------------------------------
% 3.46/1.99 % (26540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/1.99 % (26540)Termination reason: Unknown
% 3.46/1.99 % (26540)Termination phase: shuffling
% 3.46/1.99
% 3.46/1.99 % (26540)Memory used [KB]: 1918
% 3.46/1.99 % (26540)Time elapsed: 0.010 s
% 3.46/1.99 % (26540)Instructions burned: 18 (million)
% 3.46/1.99 % (26540)------------------------------
% 3.46/1.99 % (26540)------------------------------
% 3.46/1.99 % (26542)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_26421 for (2999ds/6Mi)
% 3.46/2.00 % (26542)Instruction limit reached!
% 3.46/2.00 % (26542)------------------------------
% 3.46/2.00 % (26542)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/2.00 % (26542)Termination reason: Unknown
% 3.46/2.00 % (26542)Termination phase: shuffling
% 3.46/2.00
% 3.46/2.00 % (26542)Memory used [KB]: 1663
% 3.46/2.00 % (26542)Time elapsed: 0.005 s
% 3.46/2.00 % (26542)Instructions burned: 7 (million)
% 3.46/2.00 % (26542)------------------------------
% 3.46/2.00 % (26542)------------------------------
% 3.46/2.00 % (26521)Instruction limit reached!
% 3.46/2.00 % (26521)------------------------------
% 3.46/2.00 % (26521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/2.00 % (26521)Termination reason: Unknown
% 3.46/2.00 % (26521)Termination phase: Saturation
% 3.46/2.00
% 3.46/2.00 % (26521)Memory used [KB]: 7547
% 3.46/2.00 % (26521)Time elapsed: 0.085 s
% 3.46/2.00 % (26521)Instructions burned: 184 (million)
% 3.46/2.00 % (26521)------------------------------
% 3.46/2.00 % (26521)------------------------------
% 3.46/2.00 % (26543)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_26421 for (2999ds/902Mi)
% 3.46/2.01 % (26544)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_26421 for (2998ds/21Mi)
% 3.46/2.01 % (26545)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_26421 for (2998ds/5Mi)
% 3.46/2.02 % (26545)Instruction limit reached!
% 3.46/2.02 % (26545)------------------------------
% 3.46/2.02 % (26545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/2.02 % (26545)Termination reason: Unknown
% 3.46/2.02 % (26545)Termination phase: shuffling
% 3.46/2.02
% 3.46/2.02 % (26545)Memory used [KB]: 1535
% 3.46/2.02 % (26545)Time elapsed: 0.004 s
% 3.46/2.02 % (26545)Instructions burned: 5 (million)
% 3.46/2.02 % (26545)------------------------------
% 3.46/2.02 % (26545)------------------------------
% 3.46/2.02 % (26546)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_26421 for (2998ds/6Mi)
% 3.46/2.02 % (26546)Instruction limit reached!
% 3.46/2.02 % (26546)------------------------------
% 3.46/2.02 % (26546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/2.02 % (26544)Instruction limit reached!
% 3.46/2.02 % (26544)------------------------------
% 3.46/2.02 % (26544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.46/2.02 % (26544)Termination reason: Unknown
% 3.46/2.02 % (26544)Termination phase: shuffling
% 3.46/2.02
% 3.46/2.02 % (26544)Memory used [KB]: 1918
% 3.46/2.02 % (26544)Time elapsed: 0.012 s
% 3.46/2.02 % (26544)Instructions burned: 22 (million)
% 3.46/2.02 % (26544)------------------------------
% 3.46/2.02 % (26544)------------------------------
% 3.46/2.02 % (26546)Termination reason: Unknown
% 3.46/2.02 % (26546)Termination phase: shuffling
% 3.46/2.02
% 3.46/2.02 % (26546)Memory used [KB]: 1535
% 3.46/2.02 % (26546)Time elapsed: 0.005 s
% 3.46/2.02 % (26546)Instructions burned: 6 (million)
% 3.46/2.02 % (26546)------------------------------
% 3.46/2.02 % (26546)------------------------------
% 4.28/2.03 % (26547)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_26421 for (2998ds/377Mi)
% 4.28/2.04 % (26548)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_26421 for (2998ds/779Mi)
% 4.28/2.04 % (26549)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_26421 for (2998ds/19Mi)
% 4.28/2.05 % (26549)Instruction limit reached!
% 4.28/2.05 % (26549)------------------------------
% 4.28/2.05 % (26549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.28/2.05 % (26549)Termination reason: Unknown
% 4.28/2.05 % (26549)Termination phase: shuffling
% 4.28/2.05
% 4.28/2.05 % (26549)Memory used [KB]: 1791
% 4.28/2.05 % (26549)Time elapsed: 0.011 s
% 4.28/2.05 % (26549)Instructions burned: 19 (million)
% 4.28/2.05 % (26549)------------------------------
% 4.28/2.05 % (26549)------------------------------
% 4.28/2.05 % (26526)Instruction limit reached!
% 4.28/2.05 % (26526)------------------------------
% 4.28/2.05 % (26526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.28/2.05 % (26526)Termination reason: Unknown
% 4.28/2.05 % (26526)Termination phase: Saturation
% 4.28/2.05
% 4.28/2.05 % (26526)Memory used [KB]: 9978
% 4.28/2.05 % (26526)Time elapsed: 0.133 s
% 4.28/2.05 % (26526)Instructions burned: 276 (million)
% 4.28/2.05 % (26526)------------------------------
% 4.28/2.05 % (26526)------------------------------
% 4.28/2.06 % (26550)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_26421 for (2998ds/879Mi)
% 4.28/2.06 % (26551)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_26421 for (2998ds/17Mi)
% 4.28/2.07 % (26551)Instruction limit reached!
% 4.28/2.07 % (26551)------------------------------
% 4.28/2.07 % (26551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.28/2.07 % (26551)Termination reason: Unknown
% 4.28/2.07 % (26551)Termination phase: shuffling
% 4.28/2.07
% 4.28/2.07 % (26551)Memory used [KB]: 1918
% 4.28/2.07 % (26551)Time elapsed: 0.010 s
% 4.28/2.07 % (26551)Instructions burned: 18 (million)
% 4.28/2.07 % (26551)------------------------------
% 4.28/2.07 % (26551)------------------------------
% 4.28/2.09 % (26552)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_26421 for (2998ds/3Mi)
% 4.28/2.09 % (26552)Instruction limit reached!
% 4.28/2.09 % (26552)------------------------------
% 4.28/2.09 % (26552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.28/2.09 % (26552)Termination reason: Unknown
% 4.28/2.09 % (26552)Termination phase: shuffling
% 4.28/2.09
% 4.28/2.09 % (26552)Memory used [KB]: 1535
% 4.28/2.09 % (26552)Time elapsed: 0.004 s
% 4.28/2.09 % (26552)Instructions burned: 4 (million)
% 4.28/2.09 % (26552)------------------------------
% 4.28/2.09 % (26552)------------------------------
% 4.28/2.11 % (26553)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_26421 for (2997ds/30Mi)
% 4.28/2.12 % (26553)Instruction limit reached!
% 4.28/2.12 % (26553)------------------------------
% 4.28/2.12 % (26553)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.28/2.12 % (26553)Termination reason: Unknown
% 4.28/2.12 % (26553)Termination phase: shuffling
% 4.28/2.12
% 4.28/2.12 % (26553)Memory used [KB]: 2046
% 4.28/2.12 % (26553)Time elapsed: 0.016 s
% 4.28/2.12 % (26553)Instructions burned: 30 (million)
% 4.28/2.12 % (26553)------------------------------
% 4.28/2.12 % (26553)------------------------------
% 4.71/2.14 % (26554)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_26421 for (2997ds/127Mi)
% 4.71/2.19 % (26554)Instruction limit reached!
% 4.71/2.19 % (26554)------------------------------
% 4.71/2.19 % (26554)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.71/2.19 % (26554)Termination reason: Unknown
% 4.71/2.19 % (26554)Termination phase: Property scanning
% 4.71/2.19
% 4.71/2.19 % (26554)Memory used [KB]: 2686
% 4.71/2.19 % (26554)Time elapsed: 0.058 s
% 4.71/2.19 % (26554)Instructions burned: 128 (million)
% 4.71/2.19 % (26554)------------------------------
% 4.71/2.19 % (26554)------------------------------
% 4.99/2.20 % (26543)First to succeed.
% 4.99/2.20 % (26547)Instruction limit reached!
% 4.99/2.20 % (26547)------------------------------
% 4.99/2.20 % (26547)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.99/2.21 % (26547)Termination reason: Unknown
% 4.99/2.21 % (26547)Termination phase: Saturation
% 4.99/2.21
% 4.99/2.21 % (26547)Memory used [KB]: 9466
% 4.99/2.21 % (26547)Time elapsed: 0.174 s
% 4.99/2.21 % (26547)Instructions burned: 378 (million)
% 4.99/2.21 % (26547)------------------------------
% 4.99/2.21 % (26547)------------------------------
% 4.99/2.21 % (26543)Refutation found. Thanks to Tanya!
% 4.99/2.21 % SZS status Theorem for DTF2THF_26421
% 4.99/2.21 % SZS output start Proof for DTF2THF_26421
% 4.99/2.21 thf(func_def_0, type, in: $i > $i > $o).
% 4.99/2.21 thf(func_def_1, type, exu: ($i > $o) > $o).
% 4.99/2.21 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 4.99/2.21 thf(func_def_8, type, powerset: $i > $i).
% 4.99/2.21 thf(func_def_10, type, setunion: $i > $i).
% 4.99/2.21 thf(func_def_19, type, descr: ($i > $o) > $i).
% 4.99/2.21 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 4.99/2.21 thf(func_def_26, type, prop2set: $o > $i).
% 4.99/2.21 thf(func_def_36, type, nonempty: $i > $o).
% 4.99/2.21 thf(func_def_69, type, set2prop: $i > $o).
% 4.99/2.21 thf(func_def_88, type, subset: $i > $i > $o).
% 4.99/2.21 thf(func_def_89, type, disjoint: $i > $i > $o).
% 4.99/2.21 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 4.99/2.21 thf(func_def_114, type, binunion: $i > $i > $i).
% 4.99/2.21 thf(func_def_122, type, binintersect: $i > $i > $i).
% 4.99/2.21 thf(func_def_135, type, regular: $i > $o).
% 4.99/2.21 thf(func_def_136, type, setminus: $i > $i > $i).
% 4.99/2.21 thf(func_def_147, type, symdiff: $i > $i > $i).
% 4.99/2.21 thf(func_def_153, type, iskpair: $i > $o).
% 4.99/2.21 thf(func_def_158, type, kpair: $i > $i > $i).
% 4.99/2.21 thf(func_def_160, type, cartprod: $i > $i > $i).
% 4.99/2.21 thf(func_def_177, type, singleton: $i > $o).
% 4.99/2.21 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 4.99/2.21 thf(func_def_184, type, atmost1p: $i > $o).
% 4.99/2.21 thf(func_def_185, type, atleast2p: $i > $o).
% 4.99/2.21 thf(func_def_186, type, atmost2p: $i > $o).
% 4.99/2.21 thf(func_def_187, type, upairsetp: $i > $o).
% 4.99/2.21 thf(func_def_191, type, kfst: $i > $i).
% 4.99/2.21 thf(func_def_203, type, ksnd: $i > $i).
% 4.99/2.21 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 4.99/2.21 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 4.99/2.21 thf(func_def_222, type, func: $i > $i > $i > $o).
% 4.99/2.21 thf(func_def_223, type, funcSet: $i > $i > $i).
% 4.99/2.21 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 4.99/2.21 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 4.99/2.21 thf(func_def_323, type, sP0: $i > $i > $i > $o).
% 4.99/2.21 thf(func_def_324, type, sP1: $i > $i > $o).
% 4.99/2.21 thf(func_def_325, type, sP2: $i > $o).
% 4.99/2.21 thf(func_def_326, type, sP3: $i > $i > $o).
% 4.99/2.21 thf(func_def_327, type, sP4: $i > $i > $o).
% 4.99/2.21 thf(func_def_329, type, sP6: $i > $i > $i > $i > $o).
% 4.99/2.21 thf(func_def_335, type, sP12: $o > $i > $i > $i > $o).
% 4.99/2.21 thf(func_def_339, type, sK16: $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_357, type, sK34: $i > $o).
% 4.99/2.21 thf(func_def_362, type, sK39: $i > $o).
% 4.99/2.21 thf(func_def_363, type, sK40: $i > $o).
% 4.99/2.21 thf(func_def_364, type, sK41: ($i > $o) > ($i > $o) > $i).
% 4.99/2.21 thf(func_def_365, type, sK42: ($i > $o) > ($i > $o) > $i).
% 4.99/2.21 thf(func_def_374, type, sK51: $i > $o).
% 4.99/2.21 thf(func_def_381, type, sK58: ($i > $o) > ($i > $o) > $i).
% 4.99/2.21 thf(func_def_382, type, sK59: ($i > $o) > ($i > $o) > $i).
% 4.99/2.21 thf(func_def_383, type, sK60: $i > $o).
% 4.99/2.21 thf(func_def_384, type, sK61: $i > $o).
% 4.99/2.21 thf(func_def_393, type, sK70: $i > ($i > $o) > $i).
% 4.99/2.21 thf(func_def_394, type, sK71: $i > $o).
% 4.99/2.21 thf(func_def_405, type, sK82: ($i > $o) > $i).
% 4.99/2.21 thf(func_def_406, type, sK83: $i > $o).
% 4.99/2.21 thf(func_def_407, type, sK84: $i > $i).
% 4.99/2.21 thf(func_def_451, type, sK128: $i > $o).
% 4.99/2.21 thf(func_def_469, type, sK146: $i > $o).
% 4.99/2.21 thf(func_def_483, type, sK160: $i > $o).
% 4.99/2.21 thf(func_def_485, type, sK162: ($i > $o) > $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_486, type, sK163: ($i > $o) > $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_489, type, sK166: $i > $i > $o).
% 4.99/2.21 thf(func_def_492, type, sK169: $i > $i).
% 4.99/2.21 thf(func_def_494, type, sK171: $i > $i).
% 4.99/2.21 thf(func_def_505, type, sK182: $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_509, type, sK186: $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_522, type, sK199: $i > $o).
% 4.99/2.21 thf(func_def_527, type, sK204: $i > $i > $o).
% 4.99/2.21 thf(func_def_533, type, sK210: $i > $o).
% 4.99/2.21 thf(func_def_549, type, sK226: $i > $o).
% 4.99/2.21 thf(func_def_558, type, sK235: $i > $o).
% 4.99/2.21 thf(func_def_560, type, sK237: $i > $i > $i).
% 4.99/2.21 thf(func_def_561, type, sK238: $i > $i > $i).
% 4.99/2.21 thf(func_def_562, type, sK239: $i > $i > $i).
% 4.99/2.21 thf(func_def_563, type, sK240: $i > $i > $i).
% 4.99/2.21 thf(func_def_564, type, sK241: $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_565, type, sK242: $i > $i).
% 4.99/2.21 thf(func_def_566, type, sK243: $i > $i).
% 4.99/2.21 thf(func_def_567, type, sK244: $i > $i).
% 4.99/2.21 thf(func_def_568, type, sK245: $i > $i).
% 4.99/2.21 thf(func_def_569, type, sK246: $i > $i > $i).
% 4.99/2.21 thf(func_def_570, type, sK247: $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_571, type, sK248: $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_572, type, sK249: $i > $i > $i).
% 4.99/2.21 thf(func_def_573, type, sK250: $i > $i > $i).
% 4.99/2.21 thf(func_def_574, type, sK251: $i > $i).
% 4.99/2.21 thf(func_def_575, type, sK252: $i > $i > $i).
% 4.99/2.21 thf(func_def_577, type, sK254: $i > $i).
% 4.99/2.21 thf(func_def_578, type, sK255: $i > $i).
% 4.99/2.21 thf(func_def_579, type, sK256: $i > $i > $i).
% 4.99/2.21 thf(func_def_591, type, sK268: $i > $i > $i).
% 4.99/2.21 thf(func_def_606, type, sK283: $i > $i > $o).
% 4.99/2.21 thf(func_def_617, type, sK294: $i > $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_648, type, sK325: $i > $o).
% 4.99/2.21 thf(func_def_649, type, sK326: ($i > $o) > $i > $i).
% 4.99/2.21 thf(func_def_666, type, sK343: $o > $i > $i > $i).
% 4.99/2.21 thf(func_def_686, type, sK363: $i > $i).
% 4.99/2.21 thf(func_def_694, type, sK371: $i > $i > $o).
% 4.99/2.21 thf(func_def_695, type, sK372: ($i > $o) > $i).
% 4.99/2.21 thf(func_def_696, type, sK373: ($i > $o) > $i).
% 4.99/2.21 thf(func_def_697, type, sK374: $i > $o).
% 4.99/2.21 thf(func_def_709, type, sK386: $i > $i > $i).
% 4.99/2.21 thf(func_def_719, type, sK396: $i > $i > $i).
% 4.99/2.21 thf(func_def_724, type, sK401: $i > $i > ($i > $i) > $i).
% 4.99/2.21 thf(func_def_725, type, sK402: $i > $i).
% 4.99/2.21 thf(func_def_747, type, sK424: ($i > $o) > $i).
% 4.99/2.21 thf(func_def_748, type, sK425: $i > $o).
% 4.99/2.21 thf(func_def_767, type, sK444: $i > $i > $o).
% 4.99/2.21 thf(func_def_769, type, sK446: $i > ($i > $o) > $i > $i).
% 4.99/2.21 thf(func_def_771, type, sK448: $i > $o).
% 4.99/2.21 thf(func_def_776, type, sK453: $i > $o).
% 4.99/2.21 thf(func_def_783, type, sK460: $i > $i).
% 4.99/2.21 thf(func_def_793, type, sK470: $i > $i > ($i > $i) > $i).
% 4.99/2.21 thf(func_def_794, type, sK471: $i > $i).
% 4.99/2.21 thf(func_def_801, type, sK478: $i > $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_802, type, sK479: $i > $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_803, type, sK480: $i > $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_804, type, sK481: $i > $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_811, type, sK488: $i > $i > $o).
% 4.99/2.21 thf(func_def_812, type, sK489: $i > $i).
% 4.99/2.21 thf(func_def_813, type, sK490: $i > $i).
% 4.99/2.21 thf(func_def_814, type, sK491: ($i > $i > $o) > $i > $i).
% 4.99/2.21 thf(func_def_815, type, sK492: $i > ($i > $i > $o) > $i > $i).
% 4.99/2.21 thf(func_def_816, type, sK493: ($i > $i > $o) > $i > $i).
% 4.99/2.21 thf(func_def_817, type, sK494: ($i > $o) > $i > $i).
% 4.99/2.21 thf(func_def_819, type, sK496: $i > $o).
% 4.99/2.21 thf(func_def_821, type, sK498: $i > $o).
% 4.99/2.21 thf(func_def_824, type, sK501: $i > $i > $i).
% 4.99/2.21 thf(func_def_832, type, sK509: $i > $o).
% 4.99/2.21 thf(func_def_834, type, sK511: $i > ($i > $o) > $i).
% 4.99/2.21 thf(func_def_835, type, sK512: ($i > $i) > $i > $i > $i).
% 4.99/2.21 thf(func_def_838, type, sK515: $i > $i).
% 4.99/2.21 thf(func_def_844, type, sK521: $i > ($i > $o) > $i).
% 4.99/2.21 thf(func_def_845, type, sK522: $i > $o).
% 4.99/2.21 thf(func_def_849, type, sK526: $i > $i).
% 4.99/2.21 thf(func_def_850, type, sK527: ($i > $o) > $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_851, type, sK528: ($i > $o) > $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_855, type, sK532: $i > $o).
% 4.99/2.21 thf(func_def_868, type, sK545: $i > $i).
% 4.99/2.21 thf(func_def_885, type, sK562: $i > $o).
% 4.99/2.21 thf(func_def_888, type, sK565: $i > ($i > $o) > $i).
% 4.99/2.21 thf(func_def_898, type, sK575: $i > $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_904, type, sK581: $i > ($i > $i) > $i > $i).
% 4.99/2.21 thf(func_def_906, type, sK583: $i > $i).
% 4.99/2.21 thf(func_def_909, type, sK586: $i > $i > $i).
% 4.99/2.21 thf(func_def_930, type, sK607: $i > $i).
% 4.99/2.21 thf(func_def_934, type, sK611: $i > $o).
% 4.99/2.21 thf(func_def_936, type, sK613: $i > $i > $i).
% 4.99/2.21 thf(func_def_937, type, sK614: $i > $i > $i).
% 4.99/2.21 thf(func_def_953, type, sK630: $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_962, type, sK639: $i > $i).
% 4.99/2.21 thf(func_def_963, type, sK640: ($i > $i) > $i > $i > $i).
% 4.99/2.21 thf(func_def_981, type, sK658: $i > $o).
% 4.99/2.21 thf(func_def_984, type, sK661: $i > ($i > $o) > $i).
% 4.99/2.21 thf(func_def_996, type, sK673: ($i > $o) > $i).
% 4.99/2.21 thf(func_def_997, type, sK674: $i > $o).
% 4.99/2.21 thf(func_def_998, type, sK675: $i > $i).
% 4.99/2.21 thf(func_def_1021, type, sK698: ($i > $i) > $i > $i > $i).
% 4.99/2.21 thf(func_def_1024, type, sK701: $i > $i).
% 4.99/2.21 thf(func_def_1029, type, sK706: $i > $o).
% 4.99/2.21 thf(func_def_1032, type, sK709: $i > ($i > $o) > $i).
% 4.99/2.21 thf(func_def_1033, type, sK710: $i > $o).
% 4.99/2.21 thf(func_def_1050, type, sK727: $i > $o).
% 4.99/2.21 thf(func_def_1070, type, sK747: $i > $i > $o).
% 4.99/2.21 thf(func_def_1077, type, sK754: $i > $o).
% 4.99/2.21 thf(func_def_1080, type, sK757: $i > ($i > $o) > $i).
% 4.99/2.21 thf(func_def_1083, type, sK760: $i > $i > $i).
% 4.99/2.21 thf(func_def_1097, type, sK774: $i > $o).
% 4.99/2.21 thf(func_def_1100, type, sK777: $i > ($i > $o) > $i).
% 4.99/2.21 thf(func_def_1101, type, sK778: $i > ($i > $o) > $i).
% 4.99/2.21 thf(func_def_1106, type, sK783: $i > $o).
% 4.99/2.21 thf(func_def_1109, type, sK786: $i > ($i > $o) > $i).
% 4.99/2.21 thf(func_def_1110, type, sK787: $i > ($i > $o) > $i).
% 4.99/2.21 thf(func_def_1132, type, sK809: ($i > $o) > ($i > $o) > $i > $i > $i).
% 4.99/2.21 thf(func_def_1133, type, sK810: ($i > $o) > ($i > $o) > $i > $i > $i).
% 4.99/2.21 thf(func_def_1136, type, sK813: $i > $o).
% 4.99/2.21 thf(func_def_1137, type, sK814: $i > $o).
% 4.99/2.21 thf(func_def_1151, type, sK828: $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_1152, type, sK829: $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_1163, type, sK840: $i > $i > $o).
% 4.99/2.21 thf(func_def_1175, type, sK852: $i > $i > $i).
% 4.99/2.21 thf(func_def_1181, type, sK858: $i > $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_1182, type, sK859: $i > $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_1191, type, sK868: $i > $i > $i > $i > $i).
% 4.99/2.21 thf(func_def_1201, type, ph878: !>[X0: $tType]:(X0)).
% 4.99/2.21 thf(f4668,plain,(
% 4.99/2.21 $false),
% 4.99/2.21 inference(avatar_sat_refutation,[status(thm)],[f4328,f4394,f4622,f4643])).
% 4.99/2.21 thf(f4643,plain,(
% 4.99/2.21 spl877_13 | spl877_6 | ~spl877_18),
% 4.99/2.21 inference(avatar_split_clause,[status(thm)],[f4642,f4438,f4283,f4365])).
% 4.99/2.21 thf(f4365,definition,(
% 4.99/2.21 spl877_13 <=> ($true = (in @ sK279 @ sK278))),
% 4.99/2.21 introduced(definition,[new_symbols(naming,[spl877_13])],[avatar_definition])).
% 4.99/2.21 thf(f4283,definition,(
% 4.99/2.21 spl877_6 <=> ($true = (in @ sK279 @ sK276))),
% 4.99/2.21 introduced(definition,[new_symbols(naming,[spl877_6])],[avatar_definition])).
% 4.99/2.21 thf(f4438,definition,(
% 4.99/2.21 spl877_18 <=> ((in @ sK279 @ (binunion @ sK276 @ sK278)) = $true)),
% 4.99/2.21 introduced(definition,[new_symbols(naming,[spl877_18])],[avatar_definition])).
% 4.99/2.21 thf(f4642,plain,(
% 4.99/2.21 ($true = (in @ sK279 @ sK278)) | ($true = (in @ sK279 @ sK276)) | ~spl877_18),
% 4.99/2.21 inference(trivial_inequality_removal,[status(thm)],[f4641])).
% 4.99/2.21 thf(f4641,plain,(
% 4.99/2.21 ($true = (in @ sK279 @ sK278)) | ($true != $true) | ($true = (in @ sK279 @ sK276)) | ~spl877_18),
% 4.99/2.21 inference(forward_demodulation,[status(thm)],[f4633,f3182])).
% 4.99/2.21 thf(f3182,plain,(
% 4.99/2.21 (binunionE = $true)),
% 4.99/2.21 inference(cnf_transformation,[status(thm)],[f1628])).
% 4.99/2.21 thf(f1628,plain,(
% 4.99/2.21 (setunionI = $true) & (ap2apEq1 = $true) & (intersectInPowersetIntersectUnions = $true) & (subsetTI = $true) & (subsetE2 = $true) & (binunionE = $true) & (demorgan2a = $true) & (inIntersectImpInIntersectUnions = $true) & (powersetE1 = $true) & (binintersectTERcontra = $true) & (setOfPairsIsBReln = $true) & (iftrue = $true) & (ksndpairEq = $true) & (powersetTI1 = $true) & (app = $true) & (notequalI1 = $true) & (singletoninpowunion = $true) & (binintersectSubset3 = $true) & (binunionTILcontra = $true) & (funcGraphProp3 = $true) & (upairsetIR = $true) & (dpsetconstrERa = $true) & (binunionEcases = $true) & (wellorderingAx = $true) & (lamp = $true) & (theprop = $true) & (binintersectTELcontra = $true) & (prop2setI = $true) & (setunion__Cong = $true) & (iftrueProp2 = $true) & (emptysetsubset = $true) & (eqinunit = $true) & (quantDeMorgan2 = $true) & (notdexE = $true) & (binunionRsub = $true) & (symdiffI2 = $true) & (singletonsubset = $true) & (exuE2 = $true) & (replAx = $true) & (kfstpairEq = $true) & (contraSubsetComplement = $true) & (secondinupair = $true) & (beta1 = $true) & (setadjoinSub2 = $true) & (singletonsswitch = $true) & (doubleComplementEq = $true) & (complementTI1 = $true) & (setminusIRneg = $true) & (symdiffIneg1 = $true) & (setadjoinIR = $true) & (nonemptyI1 = $true) & (notinemptyset = $true) & (subPowSU = $true) & (ex1E1 = $true) & (emptyI = $true) & (cartprodfstin = $true) & (disjointsetsI1 = $true) & (emptyinunitempty = $true) & (funcext2 = $true) & (binunionIL = $true) & (powerset__Cong = $true) & (contrasubsetT3 = $true) & (binunionTE = $true) & (binintersectSubset1 = $true) & (emptysetE = $true) & (setadjoinAx = $true) & (binintersectEL = $true) & (iftrueProp1 = $true) & (ubforcartprodlem2 = $true) & (lamProp = $true) & (emptysetimpfalse = $true) & (setminusERneg = $true) & (upairset2IR = $true) & (binintersectSubset4 = $true) & (kfstsingleton = $true) & (complementImpComplementIntersect = $true) & (inPowerset = $true) & (cartprodsndpairEq = $true) & (exuI2 = $true) & (symdiffI1 = $true) & (powersetI1 = $true) & (cartprodpairmemEL = $true) & (binunionTIRcontra = $true) & (setadjoinE = $true) & (powersetAx = $true) & (contrasubsetT = $true) & (powersetT_lem = $true) & (inCongP = $true) & (subbreln = $true) & (demorgan1b = $true) & (complementInPowersetComplementIntersect = $true) & (exuEu = $true) & (singletoninpowerset = $true) & (apProp = $true) & (demorgan2a2 = $true) & (dsetconstr__Cong = $true) & (funcext = $true) & (binintersectSubset5 = $true) & (upairsubunion = $true) & (notinsingleton = $true) & (setminusSubset1 = $true) & (beta2 = $true) & (setadjoinSub = $true) & (powersetE = $true) & (setunionAx = $true) & (demorgan1a = $true) & (nonemptyE1 = $true) & (funcGraphProp4 = $true) & (prop2setE = $true) & (emptyInPowerset = $true) & (complementUnionInPowersetComplement = $true) & (complementTnotintersectT = $true) & (setukpairIR = $true) & (binintersectER = $true) & (exu__Cong = $true) & (doubleComplementI1 = $true) & (notsubsetI = $true) & (setminusI = $true) & (ksndsingleton = $true) & (powersetTE1 = $true) & (doubleComplementSub2 = $true) & (exuI3 = $true) & (setminusER = $true) & (emptyset__Cong = $true) & (cartprodpairin = $true) & (powersetI = $true) & (complementTE1 = $true) & (eqimpsubset1 = $true) & (symdiffIneg2 = $true) & (setminusLsub = $true) & (setunionsingleton = $true) & (demorgan2a1 = $true) & (cartprodmempair1 = $true) & (iffalseProp2 = $true) & (eqbreln = $true) & (emptyE1 = $true) & (setextsub = $true) & (cartprodsndin = $true) & (kpairiskpair = $true) & (setbeta = $true) & (complementSubsetComplementIntersect = $true) & (descrp = $true) & (inComplementUnionImpNotIn1 = $true) & (dpsetconstrI = $true) & (iftrueorfalse = $true) & (($true = (in @ sK276 @ (powerset @ sK277))) & ((($true != (in @ sK279 @ (setminus @ sK277 @ (binunion @ sK276 @ sK278)))) & ($true = (in @ sK279 @ sK277)) & ((in @ sK279 @ (setminus @ sK277 @ sK278)) = $true) & ((in @ sK279 @ (setminus @ sK277 @ sK276)) = $true)) & ((in @ sK278 @ (powerset @ sK277)) = $true))) & (brelnall1 = $true) & (setukpairIL = $true) & (binintersectRsub = $true) & (setunionE2 = $true) & (subsetI1 = $true) & (dsetconstrER = $true) & (upairequniteq = $true) & (binunionTEcontra = $true) & (upairsetIL = $true) & (eqimpsubset2 = $true) & (doubleComplementSub1 = $true) & (setext = $true) & (brelnall2 = $true) & (binunionLsub = $true) & (noeltsimpempty = $true) & (ap2p = $true) & (ubforcartprodlem3 = $true) & (kpairsurjEq = $true) & (nonemptyI = $true) & (cartprodpairmemER = $true) & (exuE1 = $true) & (nonemptyImpWitness = $true) & (exuE3e = $true) & (uniqinunit = $true) & (quantDeMorgan4 = $true) & (ex1I = $true) & (upairinpowunion = $true) & (binintersectLsub = $true) & (bs114d = $true) & (dpsetconstrEL1 = $true) & (complementT_lem = $true) & (inIntersectImpInUnion = $true) & (inComplementUnionImpInComplement1 = $true) & (dpsetconstrER = $true) & (setukpairinjL2 = $true) & (exuE3u = $true) & (sepInPowerset = $true) & (dsetconstrI = $true) & (complementTcontraSubset = $true) & (ifSingleton = $true) & (eta2 = $true) & (setunionE = $true) & (dsetconstrEL = $true) & (setminusT_lem = $true) & (binunionT_lem = $true) & (setextAx = $true) & (cartprodpairsurjEq = $true) & (subsetRefl = $true) & (dpsetconstrSub = $true) & (emptysetAx = $true) & (ap2apEq2 = $true) & (omegaSAx = $true) & (descr__Cong = $true) & (setukpairinjL1 = $true) & (ex1I2 = $true) & (iffalse = $true) & (contrasubsetT2 = $true) & (theeq = $true) & (singletonsuniq = $true) & (ifp = $true) & (kpairp = $true) & (powersetsubset = $true) & (setunionsingleton2 = $true) & (lam2lamEq = $true) & (lam2p = $true) & (exuI1 = $true) & (subset2powerset = $true) & (doubleComplementE1 = $true) & (binintersectI = $true) & (funcinfuncset = $true) & (emptyinPowerset = $true) & (singletonprop = $true) & (cartprodmempair = $true) & (sepSubset = $true) & (setukpairinjR = $true) & (vacuousDall = $true) & (binintersectSubset2 = $true) & (symdiffE = $true) & (setminusILneg = $true) & (ex1E2 = $true) & (ubforcartprodlem1 = $true) & (funcGraphProp1 = $true) & (omegaIndAx = $true) & (notequalI2 = $true) & (omega__Cong = $true) & (setukpairinjR2 = $true) & (demorgan1 = $true) & (quantDeMorgan1 = $true) & (setadjoin__Cong = $true) & (upairsetE = $true) & (setukpairinjL = $true) & (setukpairinjR1 = $true) & (prop2set2propI = $true) & (subsetI2 = $true) & (subsetTrans = $true) & (in__Cong = $true) & (setminusEL = $true) & (setadjoinOr = $true) & (setextT = $true) & (setukpairinjR12 = $true) & (inIntersectImpInUnion2 = $true) & (setadjoinIL = $true) & (setoftrueEq = $true) & (quantDeMorgan3 = $true) & (setukpairinjR11 = $true) & (upairset2E = $true) & (dpsetconstrEL2 = $true) & (binunionIR = $true) & (omega0Ax = $true) & (notdallE = $true) & (subsetemptysetimpeq = $true) & (setunionsingleton1 = $true) & (binintersectT_lem = $true) & (funcextLem = $true) & (cartprodmempaircEq = $true) & (cartprodfstpairEq = $true) & (setminusSubset2 = $true) & (setminusELneg = $true) & (eta1 = $true) & (subsetE = $true) & (contrasubsetT1 = $true) & (infuncsetfunc = $true) & (funcGraphProp2 = $true) & (iffalseProp1 = $true) & (foundationAx = $true) & (funcImageSingleton = $true)),
% 4.99/2.21 inference(skolemisation,[status(esa),new_symbols(skolem,[sK276,sK277,sK278,sK279])],[f1107,f1627,f1626,f1625])).
% 4.99/2.21 thf(f1625,plain,(
% 4.99/2.21 ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ (setminus @ X1 @ (binunion @ X0 @ X2))) != $true) & ((in @ X3 @ X1) = $true) & ((in @ X3 @ (setminus @ X1 @ X2)) = $true) & ($true = (in @ X3 @ (setminus @ X1 @ X0)))) & ($true = (in @ X2 @ (powerset @ X1))))) => (($true = (in @ sK276 @ (powerset @ sK277))) & ? [X2 : $i] : (? [X3 : $i] : (($true != (in @ X3 @ (setminus @ sK277 @ (binunion @ sK276 @ X2)))) & ((in @ X3 @ sK277) = $true) & ($true = (in @ X3 @ (setminus @ sK277 @ X2))) & ((in @ X3 @ (setminus @ sK277 @ sK276)) = $true)) & ($true = (in @ X2 @ (powerset @ sK277)))))),
% 4.99/2.21 introduced(definition,[],[choice_axiom])).
% 4.99/2.21 thf(f1626,plain,(
% 4.99/2.21 ? [X2 : $i] : (? [X3 : $i] : (($true != (in @ X3 @ (setminus @ sK277 @ (binunion @ sK276 @ X2)))) & ((in @ X3 @ sK277) = $true) & ($true = (in @ X3 @ (setminus @ sK277 @ X2))) & ((in @ X3 @ (setminus @ sK277 @ sK276)) = $true)) & ($true = (in @ X2 @ (powerset @ sK277)))) => (? [X3 : $i] : (($true != (in @ X3 @ (setminus @ sK277 @ (binunion @ sK276 @ sK278)))) & ((in @ X3 @ sK277) = $true) & ((in @ X3 @ (setminus @ sK277 @ sK278)) = $true) & ((in @ X3 @ (setminus @ sK277 @ sK276)) = $true)) & ((in @ sK278 @ (powerset @ sK277)) = $true))),
% 4.99/2.21 introduced(definition,[],[choice_axiom])).
% 4.99/2.21 thf(f1627,plain,(
% 4.99/2.21 ? [X3 : $i] : (($true != (in @ X3 @ (setminus @ sK277 @ (binunion @ sK276 @ sK278)))) & ((in @ X3 @ sK277) = $true) & ((in @ X3 @ (setminus @ sK277 @ sK278)) = $true) & ((in @ X3 @ (setminus @ sK277 @ sK276)) = $true)) => (($true != (in @ sK279 @ (setminus @ sK277 @ (binunion @ sK276 @ sK278)))) & ($true = (in @ sK279 @ sK277)) & ((in @ sK279 @ (setminus @ sK277 @ sK278)) = $true) & ((in @ sK279 @ (setminus @ sK277 @ sK276)) = $true))),
% 4.99/2.21 introduced(definition,[],[choice_axiom])).
% 4.99/2.21 thf(f1107,plain,(
% 4.99/2.21 (setunionI = $true) & (ap2apEq1 = $true) & (intersectInPowersetIntersectUnions = $true) & (subsetTI = $true) & (subsetE2 = $true) & (binunionE = $true) & (demorgan2a = $true) & (inIntersectImpInIntersectUnions = $true) & (powersetE1 = $true) & (binintersectTERcontra = $true) & (setOfPairsIsBReln = $true) & (iftrue = $true) & (ksndpairEq = $true) & (powersetTI1 = $true) & (app = $true) & (notequalI1 = $true) & (singletoninpowunion = $true) & (binintersectSubset3 = $true) & (binunionTILcontra = $true) & (funcGraphProp3 = $true) & (upairsetIR = $true) & (dpsetconstrERa = $true) & (binunionEcases = $true) & (wellorderingAx = $true) & (lamp = $true) & (theprop = $true) & (binintersectTELcontra = $true) & (prop2setI = $true) & (setunion__Cong = $true) & (iftrueProp2 = $true) & (emptysetsubset = $true) & (eqinunit = $true) & (quantDeMorgan2 = $true) & (notdexE = $true) & (binunionRsub = $true) & (symdiffI2 = $true) & (singletonsubset = $true) & (exuE2 = $true) & (replAx = $true) & (kfstpairEq = $true) & (contraSubsetComplement = $true) & (secondinupair = $true) & (beta1 = $true) & (setadjoinSub2 = $true) & (singletonsswitch = $true) & (doubleComplementEq = $true) & (complementTI1 = $true) & (setminusIRneg = $true) & (symdiffIneg1 = $true) & (setadjoinIR = $true) & (nonemptyI1 = $true) & (notinemptyset = $true) & (subPowSU = $true) & (ex1E1 = $true) & (emptyI = $true) & (cartprodfstin = $true) & (disjointsetsI1 = $true) & (emptyinunitempty = $true) & (funcext2 = $true) & (binunionIL = $true) & (powerset__Cong = $true) & (contrasubsetT3 = $true) & (binunionTE = $true) & (binintersectSubset1 = $true) & (emptysetE = $true) & (setadjoinAx = $true) & (binintersectEL = $true) & (iftrueProp1 = $true) & (ubforcartprodlem2 = $true) & (lamProp = $true) & (emptysetimpfalse = $true) & (setminusERneg = $true) & (upairset2IR = $true) & (binintersectSubset4 = $true) & (kfstsingleton = $true) & (complementImpComplementIntersect = $true) & (inPowerset = $true) & (cartprodsndpairEq = $true) & (exuI2 = $true) & (symdiffI1 = $true) & (powersetI1 = $true) & (cartprodpairmemEL = $true) & (binunionTIRcontra = $true) & (setadjoinE = $true) & (powersetAx = $true) & (contrasubsetT = $true) & (powersetT_lem = $true) & (inCongP = $true) & (subbreln = $true) & (demorgan1b = $true) & (complementInPowersetComplementIntersect = $true) & (exuEu = $true) & (singletoninpowerset = $true) & (apProp = $true) & (demorgan2a2 = $true) & (dsetconstr__Cong = $true) & (funcext = $true) & (binintersectSubset5 = $true) & (upairsubunion = $true) & (notinsingleton = $true) & (setminusSubset1 = $true) & (beta2 = $true) & (setadjoinSub = $true) & (powersetE = $true) & (setunionAx = $true) & (demorgan1a = $true) & (nonemptyE1 = $true) & (funcGraphProp4 = $true) & (prop2setE = $true) & (emptyInPowerset = $true) & (complementUnionInPowersetComplement = $true) & (complementTnotintersectT = $true) & (setukpairIR = $true) & (binintersectER = $true) & (exu__Cong = $true) & (doubleComplementI1 = $true) & (notsubsetI = $true) & (setminusI = $true) & (ksndsingleton = $true) & (powersetTE1 = $true) & (doubleComplementSub2 = $true) & (exuI3 = $true) & (setminusER = $true) & (emptyset__Cong = $true) & (cartprodpairin = $true) & (powersetI = $true) & (complementTE1 = $true) & (eqimpsubset1 = $true) & (symdiffIneg2 = $true) & (setminusLsub = $true) & (setunionsingleton = $true) & (demorgan2a1 = $true) & (cartprodmempair1 = $true) & (iffalseProp2 = $true) & (eqbreln = $true) & (emptyE1 = $true) & (setextsub = $true) & (cartprodsndin = $true) & (kpairiskpair = $true) & (setbeta = $true) & (complementSubsetComplementIntersect = $true) & (descrp = $true) & (inComplementUnionImpNotIn1 = $true) & (dpsetconstrI = $true) & (iftrueorfalse = $true) & ? [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) & ? [X2 : $i] : (? [X3 : $i] : (((in @ X3 @ (setminus @ X1 @ (binunion @ X0 @ X2))) != $true) & ((in @ X3 @ X1) = $true) & ((in @ X3 @ (setminus @ X1 @ X2)) = $true) & ($true = (in @ X3 @ (setminus @ X1 @ X0)))) & ($true = (in @ X2 @ (powerset @ X1))))) & (brelnall1 = $true) & (setukpairIL = $true) & (binintersectRsub = $true) & (setunionE2 = $true) & (subsetI1 = $true) & (dsetconstrER = $true) & (upairequniteq = $true) & (binunionTEcontra = $true) & (upairsetIL = $true) & (eqimpsubset2 = $true) & (doubleComplementSub1 = $true) & (setext = $true) & (brelnall2 = $true) & (binunionLsub = $true) & (noeltsimpempty = $true) & (ap2p = $true) & (ubforcartprodlem3 = $true) & (kpairsurjEq = $true) & (nonemptyI = $true) & (cartprodpairmemER = $true) & (exuE1 = $true) & (nonemptyImpWitness = $true) & (exuE3e = $true) & (uniqinunit = $true) & (quantDeMorgan4 = $true) & (ex1I = $true) & (upairinpowunion = $true) & (binintersectLsub = $true) & (bs114d = $true) & (dpsetconstrEL1 = $true) & (complementT_lem = $true) & (inIntersectImpInUnion = $true) & (inComplementUnionImpInComplement1 = $true) & (dpsetconstrER = $true) & (setukpairinjL2 = $true) & (exuE3u = $true) & (sepInPowerset = $true) & (dsetconstrI = $true) & (complementTcontraSubset = $true) & (ifSingleton = $true) & (eta2 = $true) & (setunionE = $true) & (dsetconstrEL = $true) & (setminusT_lem = $true) & (binunionT_lem = $true) & (setextAx = $true) & (cartprodpairsurjEq = $true) & (subsetRefl = $true) & (dpsetconstrSub = $true) & (emptysetAx = $true) & (ap2apEq2 = $true) & (omegaSAx = $true) & (descr__Cong = $true) & (setukpairinjL1 = $true) & (ex1I2 = $true) & (iffalse = $true) & (contrasubsetT2 = $true) & (theeq = $true) & (singletonsuniq = $true) & (ifp = $true) & (kpairp = $true) & (powersetsubset = $true) & (setunionsingleton2 = $true) & (lam2lamEq = $true) & (lam2p = $true) & (exuI1 = $true) & (subset2powerset = $true) & (doubleComplementE1 = $true) & (binintersectI = $true) & (funcinfuncset = $true) & (emptyinPowerset = $true) & (singletonprop = $true) & (cartprodmempair = $true) & (sepSubset = $true) & (setukpairinjR = $true) & (vacuousDall = $true) & (binintersectSubset2 = $true) & (symdiffE = $true) & (setminusILneg = $true) & (ex1E2 = $true) & (ubforcartprodlem1 = $true) & (funcGraphProp1 = $true) & (omegaIndAx = $true) & (notequalI2 = $true) & (omega__Cong = $true) & (setukpairinjR2 = $true) & (demorgan1 = $true) & (quantDeMorgan1 = $true) & (setadjoin__Cong = $true) & (upairsetE = $true) & (setukpairinjL = $true) & (setukpairinjR1 = $true) & (prop2set2propI = $true) & (subsetI2 = $true) & (subsetTrans = $true) & (in__Cong = $true) & (setminusEL = $true) & (setadjoinOr = $true) & (setextT = $true) & (setukpairinjR12 = $true) & (inIntersectImpInUnion2 = $true) & (setadjoinIL = $true) & (setoftrueEq = $true) & (quantDeMorgan3 = $true) & (setukpairinjR11 = $true) & (upairset2E = $true) & (dpsetconstrEL2 = $true) & (binunionIR = $true) & (omega0Ax = $true) & (notdallE = $true) & (subsetemptysetimpeq = $true) & (setunionsingleton1 = $true) & (binintersectT_lem = $true) & (funcextLem = $true) & (cartprodmempaircEq = $true) & (cartprodfstpairEq = $true) & (setminusSubset2 = $true) & (setminusELneg = $true) & (eta1 = $true) & (subsetE = $true) & (contrasubsetT1 = $true) & (infuncsetfunc = $true) & (funcGraphProp2 = $true) & (iffalseProp1 = $true) & (foundationAx = $true) & (funcImageSingleton = $true)),
% 4.99/2.21 inference(flattening,[status(thm)],[f1106])).
% 4.99/2.21 thf(f1106,plain,(
% 4.99/2.21 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (((((in @ X3 @ (setminus @ X1 @ (binunion @ X0 @ X2))) != $true) & ((in @ X3 @ (setminus @ X1 @ X2)) = $true)) & ($true = (in @ X3 @ (setminus @ X1 @ X0)))) & ((in @ X3 @ X1) = $true)) & ($true = (in @ X2 @ (powerset @ X1)))) & ((in @ X0 @ (powerset @ X1)) = $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)),
% 4.99/2.21 inference(ennf_transformation,[status(thm)],[f768])).
% 4.99/2.21 thf(f768,plain,(
% 4.99/2.21 ~((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) => ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X1))) => ! [X3 : $i] : (((in @ X3 @ X1) = $true) => (($true = (in @ X3 @ (setminus @ X1 @ X0))) => (((in @ X3 @ (setminus @ X1 @ X2)) = $true) => ((in @ X3 @ (setminus @ X1 @ (binunion @ X0 @ X2))) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 4.99/2.21 inference(fool_elimination,[status(thm)],[f767])).
% 4.99/2.21 thf(f767,plain,(
% 4.99/2.21 ~(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 => ! [X0 : $i,X1 : $i] : ((in @ X0 @ (powerset @ X1)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X1)) => ! [X3 : $i] : ((in @ X3 @ X1) => ((in @ X3 @ (setminus @ X1 @ X0)) => ((in @ X3 @ (setminus @ X1 @ X2)) => (in @ X3 @ (setminus @ X1 @ (binunion @ X0 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 4.99/2.21 inference(rectify,[status(thm)],[f277])).
% 4.99/2.21 thf(f277,negated_conjecture,(
% 4.99/2.21 ~(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 => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((in @ X1 @ (setminus @ X3 @ X11)) => ((in @ X1 @ (setminus @ X3 @ X16)) => (in @ X1 @ (setminus @ X3 @ (binunion @ X11 @ X16)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 4.99/2.21 inference(negated_conjecture,[status(cth)],[f276])).
% 4.99/2.21 thf(f276,conjecture,(
% 4.99/2.21 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 => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((in @ X1 @ (setminus @ X3 @ X11)) => ((in @ X1 @ (setminus @ X3 @ X16)) => (in @ X1 @ (setminus @ X3 @ (binunion @ X11 @ X16))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 4.99/2.21 file('/export/starexec/sandbox/tmp/tmp.UUdmaiX54M/DTF2THF_26421.p',demorgan2b2)).
% 4.99/2.21 thf(f4633,plain,(
% 4.99/2.21 ($true = (in @ sK279 @ sK276)) | ($true = (in @ sK279 @ sK278)) | (binunionE != $true) | ~spl877_18),
% 4.99/2.21 inference(trivial_inequality_removal,[status(thm)],[f4623])).
% 4.99/2.21 thf(f4623,plain,(
% 4.99/2.21 ($true != $true) | ($true = (in @ sK279 @ sK276)) | ($true = (in @ sK279 @ sK278)) | (binunionE != $true) | ~spl877_18),
% 4.99/2.21 inference(superposition,[status(thm)],[f3425,f4440])).
% 4.99/2.21 thf(f4440,plain,(
% 4.99/2.21 ((in @ sK279 @ (binunion @ sK276 @ sK278)) = $true) | ~spl877_18),
% 4.99/2.21 inference(avatar_component_clause,[status(thm)],[f4438])).
% 4.99/2.21 thf(f3425,plain,(
% 4.99/2.21 ( ! [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X4 @ (binunion @ X3 @ X5))) | ($true = (in @ X4 @ X5)) | ((in @ X4 @ X3) = $true) | (binunionE != $true)) )),
% 4.99/2.21 inference(cnf_transformation,[status(thm)],[f1912])).
% 4.99/2.21 thf(f1912,plain,(
% 4.99/2.21 ((binunionE = $true) | (($true != (in @ sK455 @ sK454)) & ($true != (in @ sK455 @ sK456)) & ((in @ sK455 @ (binunion @ sK454 @ sK456)) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ X3) = $true) | ($true = (in @ X4 @ X5)) | ($true != (in @ X4 @ (binunion @ X3 @ X5)))) | (binunionE != $true))),
% 4.99/2.21 inference(skolemisation,[status(esa),new_symbols(skolem,[sK454,sK455,sK456])],[f1910,f1911])).
% 4.99/2.21 thf(f1911,plain,(
% 4.99/2.21 ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ X0)) & ((in @ X1 @ X2) != $true) & ($true = (in @ X1 @ (binunion @ X0 @ X2)))) => (($true != (in @ sK455 @ sK454)) & ($true != (in @ sK455 @ sK456)) & ((in @ sK455 @ (binunion @ sK454 @ sK456)) = $true))),
% 4.99/2.21 introduced(definition,[],[choice_axiom])).
% 4.99/2.21 thf(f1910,plain,(
% 4.99/2.21 ((binunionE = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ X0)) & ((in @ X1 @ X2) != $true) & ($true = (in @ X1 @ (binunion @ X0 @ X2))))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ X3) = $true) | ($true = (in @ X4 @ X5)) | ($true != (in @ X4 @ (binunion @ X3 @ X5)))) | (binunionE != $true))),
% 4.99/2.21 inference(rectify,[status(thm)],[f1909])).
% 4.99/2.21 thf(f1909,plain,(
% 4.99/2.21 ((binunionE = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ X0)) & ((in @ X1 @ X2) != $true) & ($true = (in @ X1 @ (binunion @ X0 @ X2))))) & (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ X0)) | ((in @ X1 @ X2) = $true) | ($true != (in @ X1 @ (binunion @ X0 @ X2)))) | (binunionE != $true))),
% 4.99/2.21 inference(nnf_transformation,[status(thm)],[f1196])).
% 4.99/2.21 thf(f1196,plain,(
% 4.99/2.21 (binunionE = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ X0)) | ((in @ X1 @ X2) = $true) | ($true != (in @ X1 @ (binunion @ X0 @ X2))))),
% 4.99/2.21 inference(flattening,[status(thm)],[f1195])).
% 4.99/2.21 thf(f1195,plain,(
% 4.99/2.21 ! [X0 : $i,X1 : $i,X2 : $i] : ((((in @ X1 @ X2) = $true) | ($true = (in @ X1 @ X0))) | ($true != (in @ X1 @ (binunion @ X0 @ X2)))) <=> (binunionE = $true)),
% 4.99/2.21 inference(ennf_transformation,[status(thm)],[f584])).
% 4.99/2.21 thf(f584,plain,(
% 4.99/2.21 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ (binunion @ X0 @ X2))) => (((in @ X1 @ X2) = $true) | ($true = (in @ X1 @ X0)))) <=> (binunionE = $true)),
% 4.99/2.21 inference(fool_elimination,[status(thm)],[f583])).
% 4.99/2.21 thf(f583,plain,(
% 4.99/2.21 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ (binunion @ X0 @ X2)) => ((in @ X1 @ X0) | (in @ X1 @ X2))) = binunionE)),
% 4.99/2.21 inference(rectify,[status(thm)],[f105])).
% 4.99/2.21 thf(f105,axiom,(
% 4.99/2.21 (! [X3 : $i,X1 : $i,X4 : $i] : ((in @ X1 @ (binunion @ X3 @ X4)) => ((in @ X1 @ X3) | (in @ X1 @ X4))) = binunionE)),
% 4.99/2.21 file('/export/starexec/sandbox/tmp/tmp.UUdmaiX54M/DTF2THF_26421.p',binunionE)).
% 4.99/2.21 thf(f4622,plain,(
% 4.99/2.21 spl877_18),
% 4.99/2.21 inference(avatar_split_clause,[status(thm)],[f4419,f4438])).
% 4.99/2.21 thf(f4419,plain,(
% 4.99/2.21 ((in @ sK279 @ (binunion @ sK276 @ sK278)) = $true)),
% 4.99/2.21 inference(trivial_inequality_removal,[status(thm)],[f4418])).
% 4.99/2.21 thf(f4418,plain,(
% 4.99/2.21 ($true != $true) | ((in @ sK279 @ (binunion @ sK276 @ sK278)) = $true)),
% 4.99/2.21 inference(forward_demodulation,[status(thm)],[f4417,f3040])).
% 4.99/2.21 thf(f3040,plain,(
% 4.99/2.21 ($true = (in @ sK279 @ sK277))),
% 4.99/2.21 inference(cnf_transformation,[status(thm)],[f1628])).
% 4.99/2.21 thf(f4417,plain,(
% 4.99/2.21 ((in @ sK279 @ (binunion @ sK276 @ sK278)) = $true) | ($true != (in @ sK279 @ sK277))),
% 4.99/2.21 inference(trivial_inequality_removal,[status(thm)],[f4416])).
% 4.99/2.21 thf(f4416,plain,(
% 4.99/2.21 ((in @ sK279 @ (binunion @ sK276 @ sK278)) = $true) | ($true != (in @ sK279 @ sK277)) | ($true != $true)),
% 4.99/2.21 inference(forward_demodulation,[status(thm)],[f4415,f2919])).
% 4.99/2.21 thf(f2919,plain,(
% 4.99/2.21 (setminusELneg = $true)),
% 4.99/2.21 inference(cnf_transformation,[status(thm)],[f1628])).
% 4.99/2.21 thf(f4415,plain,(
% 4.99/2.21 (setminusELneg != $true) | ($true != (in @ sK279 @ sK277)) | ((in @ sK279 @ (binunion @ sK276 @ sK278)) = $true)),
% 4.99/2.21 inference(trivial_inequality_removal,[status(thm)],[f4398])).
% 4.99/2.21 thf(f4398,plain,(
% 4.99/2.21 ($true != $true) | (setminusELneg != $true) | ($true != (in @ sK279 @ sK277)) | ((in @ sK279 @ (binunion @ sK276 @ sK278)) = $true)),
% 4.99/2.21 inference(superposition,[status(thm)],[f3041,f3746])).
% 4.99/2.21 thf(f3746,plain,(
% 4.99/2.21 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X2 @ (setminus @ X0 @ X1)) = $true) | ($true = (in @ X2 @ X1)) | (setminusELneg != $true) | ($true != (in @ X2 @ X0))) )),
% 4.99/2.21 inference(cnf_transformation,[status(thm)],[f2275])).
% 4.99/2.21 thf(f2275,plain,(
% 4.99/2.21 (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X2 @ X0)) | ($true = (in @ X2 @ X1)) | ((in @ X2 @ (setminus @ X0 @ X1)) = $true)) | (setminusELneg != $true)) & ((setminusELneg = $true) | (($true = (in @ sK688 @ sK686)) & ($true != (in @ sK688 @ sK687)) & ($true != (in @ sK688 @ (setminus @ sK686 @ sK687)))))),
% 4.99/2.21 inference(skolemisation,[status(esa),new_symbols(skolem,[sK686,sK687,sK688])],[f2273,f2274])).
% 4.99/2.21 thf(f2274,plain,(
% 4.99/2.21 ? [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X5 @ X3)) & ((in @ X5 @ X4) != $true) & ($true != (in @ X5 @ (setminus @ X3 @ X4)))) => (($true = (in @ sK688 @ sK686)) & ($true != (in @ sK688 @ sK687)) & ($true != (in @ sK688 @ (setminus @ sK686 @ sK687))))),
% 4.99/2.21 introduced(definition,[],[choice_axiom])).
% 4.99/2.21 thf(f2273,plain,(
% 4.99/2.21 (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X2 @ X0)) | ($true = (in @ X2 @ X1)) | ((in @ X2 @ (setminus @ X0 @ X1)) = $true)) | (setminusELneg != $true)) & ((setminusELneg = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X5 @ X3)) & ((in @ X5 @ X4) != $true) & ($true != (in @ X5 @ (setminus @ X3 @ X4)))))),
% 4.99/2.21 inference(rectify,[status(thm)],[f2272])).
% 4.99/2.21 thf(f2272,plain,(
% 4.99/2.21 (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X2 @ X0)) | ($true = (in @ X2 @ X1)) | ((in @ X2 @ (setminus @ X0 @ X1)) = $true)) | (setminusELneg != $true)) & ((setminusELneg = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X2 @ X0)) & ($true != (in @ X2 @ X1)) & ((in @ X2 @ (setminus @ X0 @ X1)) != $true)))),
% 4.99/2.21 inference(nnf_transformation,[status(thm)],[f1081])).
% 4.99/2.21 thf(f1081,plain,(
% 4.99/2.21 ! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X2 @ X0)) | ($true = (in @ X2 @ X1)) | ((in @ X2 @ (setminus @ X0 @ X1)) = $true)) <=> (setminusELneg = $true)),
% 4.99/2.21 inference(flattening,[status(thm)],[f1080])).
% 4.99/2.21 thf(f1080,plain,(
% 4.99/2.21 ! [X0 : $i,X1 : $i,X2 : $i] : ((($true != (in @ X2 @ X0)) | ($true = (in @ X2 @ X1))) | ((in @ X2 @ (setminus @ X0 @ X1)) = $true)) <=> (setminusELneg = $true)),
% 4.99/2.21 inference(ennf_transformation,[status(thm)],[f844])).
% 4.99/2.21 thf(f844,plain,(
% 4.99/2.21 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X2 @ (setminus @ X0 @ X1)) != $true) => (($true != (in @ X2 @ X1)) => ($true != (in @ X2 @ X0)))) <=> (setminusELneg = $true)),
% 4.99/2.21 inference(flattening,[status(thm)],[f498])).
% 4.99/2.21 thf(f498,plain,(
% 4.99/2.21 (setminusELneg = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (~((in @ X2 @ (setminus @ X0 @ X1)) = $true) => (~($true = (in @ X2 @ X1)) => ~($true = (in @ X2 @ X0))))),
% 4.99/2.21 inference(fool_elimination,[status(thm)],[f497])).
% 4.99/2.21 thf(f497,plain,(
% 4.99/2.21 (setminusELneg = ! [X0 : $i,X1 : $i,X2 : $i] : (~(in @ X2 @ (setminus @ X0 @ X1)) => (~(in @ X2 @ X1) => ~(in @ X2 @ X0))))),
% 4.99/2.21 inference(rectify,[status(thm)],[f125])).
% 4.99/2.21 thf(f125,axiom,(
% 4.99/2.21 (setminusELneg = ! [X3 : $i,X4 : $i,X1 : $i] : (~(in @ X1 @ (setminus @ X3 @ X4)) => (~(in @ X1 @ X4) => ~(in @ X1 @ X3))))),
% 4.99/2.21 file('/export/starexec/sandbox/tmp/tmp.UUdmaiX54M/DTF2THF_26421.p',setminusELneg)).
% 4.99/2.21 thf(f3041,plain,(
% 4.99/2.21 ($true != (in @ sK279 @ (setminus @ sK277 @ (binunion @ sK276 @ sK278))))),
% 4.99/2.21 inference(cnf_transformation,[status(thm)],[f1628])).
% 4.99/2.21 thf(f4394,plain,(
% 4.99/2.21 ~spl877_13),
% 4.99/2.21 inference(avatar_split_clause,[status(thm)],[f4393,f4365])).
% 4.99/2.21 thf(f4393,plain,(
% 4.99/2.21 ($true != (in @ sK279 @ sK278))),
% 4.99/2.21 inference(trivial_inequality_removal,[status(thm)],[f4392])).
% 4.99/2.21 thf(f4392,plain,(
% 4.99/2.21 ($true != (in @ sK279 @ sK278)) | ($true != $true)),
% 4.99/2.21 inference(forward_demodulation,[status(thm)],[f4338,f3065])).
% 4.99/2.21 thf(f3065,plain,(
% 4.99/2.21 (setminusER = $true)),
% 4.99/2.21 inference(cnf_transformation,[status(thm)],[f1628])).
% 4.99/2.21 thf(f4338,plain,(
% 4.99/2.21 (setminusER != $true) | ($true != (in @ sK279 @ sK278))),
% 4.99/2.21 inference(trivial_inequality_removal,[status(thm)],[f4333])).
% 4.99/2.21 thf(f4333,plain,(
% 4.99/2.21 ($true != $true) | ($true != (in @ sK279 @ sK278)) | (setminusER != $true)),
% 4.99/2.21 inference(superposition,[status(thm)],[f3318,f3039])).
% 4.99/2.21 thf(f3039,plain,(
% 4.99/2.21 ((in @ sK279 @ (setminus @ sK277 @ sK278)) = $true)),
% 4.99/2.21 inference(cnf_transformation,[status(thm)],[f1628])).
% 4.99/2.21 thf(f3318,plain,(
% 4.99/2.21 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ (setminus @ X5 @ X4)) != $true) | (setminusER != $true) | ((in @ X3 @ X4) != $true)) )),
% 4.99/2.21 inference(cnf_transformation,[status(thm)],[f1786])).
% 4.99/2.21 thf(f1786,plain,(
% 4.99/2.21 ((setminusER = $true) | (($true = (in @ sK376 @ sK377)) & ((in @ sK376 @ (setminus @ sK378 @ sK377)) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ X4) != $true) | ((in @ X3 @ (setminus @ X5 @ X4)) != $true)) | (setminusER != $true))),
% 4.99/2.21 inference(skolemisation,[status(esa),new_symbols(skolem,[sK376,sK377,sK378])],[f1784,f1785])).
% 4.99/2.21 thf(f1785,plain,(
% 4.99/2.21 ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) = $true) & ((in @ X0 @ (setminus @ X2 @ X1)) = $true)) => (($true = (in @ sK376 @ sK377)) & ((in @ sK376 @ (setminus @ sK378 @ sK377)) = $true))),
% 4.99/2.21 introduced(definition,[],[choice_axiom])).
% 4.99/2.21 thf(f1784,plain,(
% 4.99/2.21 ((setminusER = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) = $true) & ((in @ X0 @ (setminus @ X2 @ X1)) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ X4) != $true) | ((in @ X3 @ (setminus @ X5 @ X4)) != $true)) | (setminusER != $true))),
% 4.99/2.21 inference(rectify,[status(thm)],[f1783])).
% 4.99/2.21 thf(f1783,plain,(
% 4.99/2.21 ((setminusER = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) = $true) & ((in @ X0 @ (setminus @ X2 @ X1)) = $true))) & (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) != $true) | ((in @ X0 @ (setminus @ X2 @ X1)) != $true)) | (setminusER != $true))),
% 4.99/2.21 inference(nnf_transformation,[status(thm)],[f958])).
% 4.99/2.21 thf(f958,plain,(
% 4.99/2.21 (setminusER = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ X1) != $true) | ((in @ X0 @ (setminus @ X2 @ X1)) != $true))),
% 4.99/2.21 inference(ennf_transformation,[status(thm)],[f875])).
% 4.99/2.21 thf(f875,plain,(
% 4.99/2.21 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (setminus @ X2 @ X1)) = $true) => ((in @ X0 @ X1) != $true)) <=> (setminusER = $true)),
% 4.99/2.21 inference(flattening,[status(thm)],[f804])).
% 4.99/2.21 thf(f804,plain,(
% 4.99/2.21 (setminusER = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (setminus @ X2 @ X1)) = $true) => ~((in @ X0 @ X1) = $true))),
% 4.99/2.21 inference(fool_elimination,[status(thm)],[f803])).
% 4.99/2.21 thf(f803,plain,(
% 4.99/2.21 (setminusER = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ (setminus @ X2 @ X1)) => ~(in @ X0 @ X1)))),
% 4.99/2.21 inference(rectify,[status(thm)],[f122])).
% 4.99/2.21 thf(f122,axiom,(
% 4.99/2.21 (setminusER = ! [X1 : $i,X4 : $i,X3 : $i] : ((in @ X1 @ (setminus @ X3 @ X4)) => ~(in @ X1 @ X4)))),
% 4.99/2.21 file('/export/starexec/sandbox/tmp/tmp.UUdmaiX54M/DTF2THF_26421.p',setminusER)).
% 4.99/2.21 thf(f4328,plain,(
% 4.99/2.21 ~spl877_6),
% 4.99/2.21 inference(avatar_split_clause,[status(thm)],[f4327,f4283])).
% 4.99/2.21 thf(f4327,plain,(
% 4.99/2.21 ($true != (in @ sK279 @ sK276))),
% 4.99/2.21 inference(trivial_inequality_removal,[status(thm)],[f4326])).
% 4.99/2.21 thf(f4326,plain,(
% 4.99/2.21 ($true != (in @ sK279 @ sK276)) | ($true != $true)),
% 4.99/2.21 inference(forward_demodulation,[status(thm)],[f4268,f3140])).
% 4.99/2.21 thf(f3140,plain,(
% 4.99/2.21 (setminusIRneg = $true)),
% 4.99/2.21 inference(cnf_transformation,[status(thm)],[f1628])).
% 4.99/2.21 thf(f4268,plain,(
% 4.99/2.21 ($true != (in @ sK279 @ sK276)) | (setminusIRneg != $true)),
% 4.99/2.21 inference(trivial_inequality_removal,[status(thm)],[f4261])).
% 4.99/2.21 thf(f4261,plain,(
% 4.99/2.21 (setminusIRneg != $true) | ($true != $true) | ($true != (in @ sK279 @ sK276))),
% 4.99/2.21 inference(superposition,[status(thm)],[f2769,f3038])).
% 4.99/2.21 thf(f3038,plain,(
% 4.99/2.21 ((in @ sK279 @ (setminus @ sK277 @ sK276)) = $true)),
% 4.99/2.21 inference(cnf_transformation,[status(thm)],[f1628])).
% 4.99/2.21 thf(f2769,plain,(
% 4.99/2.21 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X1 @ (setminus @ X0 @ X2)) != $true) | ((in @ X1 @ X2) != $true) | (setminusIRneg != $true)) )),
% 4.99/2.21 inference(cnf_transformation,[status(thm)],[f1471])).
% 4.99/2.21 thf(f1471,plain,(
% 4.99/2.21 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X2) != $true) | ((in @ X1 @ (setminus @ X0 @ X2)) != $true)) | (setminusIRneg != $true)) & ((setminusIRneg = $true) | (((in @ sK177 @ sK178) = $true) & ((in @ sK177 @ (setminus @ sK176 @ sK178)) = $true)))),
% 4.99/2.21 inference(skolemisation,[status(esa),new_symbols(skolem,[sK176,sK177,sK178])],[f1469,f1470])).
% 4.99/2.21 thf(f1470,plain,(
% 4.99/2.21 ? [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X4 @ X5)) & ((in @ X4 @ (setminus @ X3 @ X5)) = $true)) => (((in @ sK177 @ sK178) = $true) & ((in @ sK177 @ (setminus @ sK176 @ sK178)) = $true))),
% 4.99/2.21 introduced(definition,[],[choice_axiom])).
% 4.99/2.21 thf(f1469,plain,(
% 4.99/2.21 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X2) != $true) | ((in @ X1 @ (setminus @ X0 @ X2)) != $true)) | (setminusIRneg != $true)) & ((setminusIRneg = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (($true = (in @ X4 @ X5)) & ((in @ X4 @ (setminus @ X3 @ X5)) = $true)))),
% 4.99/2.21 inference(rectify,[status(thm)],[f1468])).
% 4.99/2.21 thf(f1468,plain,(
% 4.99/2.21 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X2) != $true) | ((in @ X1 @ (setminus @ X0 @ X2)) != $true)) | (setminusIRneg != $true)) & ((setminusIRneg = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) & ((in @ X1 @ (setminus @ X0 @ X2)) = $true)))),
% 4.99/2.21 inference(nnf_transformation,[status(thm)],[f974])).
% 4.99/2.21 thf(f974,plain,(
% 4.99/2.21 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X2) != $true) | ((in @ X1 @ (setminus @ X0 @ X2)) != $true)) <=> (setminusIRneg = $true)),
% 4.99/2.21 inference(ennf_transformation,[status(thm)],[f838])).
% 4.99/2.21 thf(f838,plain,(
% 4.99/2.21 (setminusIRneg = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) => ((in @ X1 @ (setminus @ X0 @ X2)) != $true))),
% 4.99/2.21 inference(flattening,[status(thm)],[f435])).
% 4.99/2.21 thf(f435,plain,(
% 4.99/2.21 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) => ~((in @ X1 @ (setminus @ X0 @ X2)) = $true)) <=> (setminusIRneg = $true)),
% 4.99/2.21 inference(fool_elimination,[status(thm)],[f434])).
% 4.99/2.21 thf(f434,plain,(
% 4.99/2.21 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ X2) => ~(in @ X1 @ (setminus @ X0 @ X2))) = setminusIRneg)),
% 4.99/2.21 inference(rectify,[status(thm)],[f127])).
% 4.99/2.21 thf(f127,axiom,(
% 4.99/2.21 (! [X3 : $i,X1 : $i,X4 : $i] : ((in @ X1 @ X4) => ~(in @ X1 @ (setminus @ X3 @ X4))) = setminusIRneg)),
% 4.99/2.21 file('/export/starexec/sandbox/tmp/tmp.UUdmaiX54M/DTF2THF_26421.p',setminusIRneg)).
% 4.99/2.21 % SZS output end Proof for DTF2THF_26421
% 4.99/2.21 % (26543)------------------------------
% 4.99/2.21 % (26543)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.99/2.21 % (26543)Termination reason: Refutation
% 4.99/2.21
% 4.99/2.21 % (26543)Memory used [KB]: 10746
% 4.99/2.21 % (26543)Time elapsed: 0.207 s
% 4.99/2.21 % (26543)Instructions burned: 403 (million)
% 4.99/2.21 % (26543)------------------------------
% 4.99/2.21 % (26543)------------------------------
% 4.99/2.21 % (26520)Success in time 0.316 s
%------------------------------------------------------------------------------