%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU733^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n024.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:36:45 AM UTC 2026
% Result : Theorem 2.41s 1.47s
% Output : Refutation 2.41s
% Verified :
% SZS Type : Refutation
% Derivation depth : 17
% Number of leaves : 12
% Syntax : Number of formulae : 67 ( 17 unt; 0 typ; 2 def)
% Number of atoms : 4524 (1201 equ; 0 cnn)
% Maximal formula atoms : 254 ( 67 avg)
% Number of connectives : 2577 ( 97 ~; 66 |; 806 &; 564 @)
% ( 11 <=>;1033 =>; 0 <=; 0 <~>)
% Maximal formula depth : 263 ( 31 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 455 ( 452 usr; 266 con; 0-4 aty)
% Number of variables : 142 ( 0 ^; 97 !; 45 ?; 142 :)
% 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_301,type,
sP0: $i > $o > $i > $i > $o ).
thf(func_def_302,type,
sP1: $i > $i > $o ).
thf(func_def_303,type,
sP2: $i > $o ).
thf(func_def_304,type,
sP3: $i > $i > $o ).
thf(func_def_305,type,
sP4: $i > $i > $o ).
thf(func_def_308,type,
sP7: $i > $i > $i > $i > $o ).
thf(func_def_311,type,
sP10: $i > $i > $i > $o ).
thf(func_def_324,type,
sK23: $i > ( $i > $o ) > $i ).
thf(func_def_325,type,
sK24: $i > $o ).
thf(func_def_331,type,
sK30: ( $i > $i ) > $i > $i > $i ).
thf(func_def_334,type,
sK33: $i > $i ).
thf(func_def_337,type,
sK36: $i > $i > $i > $i > $i ).
thf(func_def_359,type,
sK58: $i > $o ).
thf(func_def_370,type,
sK69: $i > $o ).
thf(func_def_375,type,
sK74: $i > $o ).
thf(func_def_392,type,
sK91: $i > $o ).
thf(func_def_393,type,
sK92: $i > $i ).
thf(func_def_394,type,
sK93: ( $i > $o ) > $i ).
thf(func_def_403,type,
sK102: $i > $i ).
thf(func_def_404,type,
sK103: ( $i > $i ) > $i > $i > $i ).
thf(func_def_409,type,
sK108: $i > $o ).
thf(func_def_411,type,
sK110: $i > ( $i > $o ) > $i ).
thf(func_def_412,type,
sK111: $i > $o ).
thf(func_def_417,type,
sK116: $i > $o ).
thf(func_def_419,type,
sK118: ( $i > $o ) > $i > $i ).
thf(func_def_423,type,
sK122: $i > $i ).
thf(func_def_431,type,
sK130: $i > $o ).
thf(func_def_445,type,
sK144: $i > $i ).
thf(func_def_451,type,
sK150: $i > $o ).
thf(func_def_452,type,
sK151: ( $i > $o ) > $i > $i ).
thf(func_def_464,type,
sK163: $i > $i > $o ).
thf(func_def_472,type,
sK171: $i > $o ).
thf(func_def_473,type,
sK172: $i > $o ).
thf(func_def_474,type,
sK173: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_475,type,
sK174: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_501,type,
sK200: $i > $o ).
thf(func_def_503,type,
sK202: ( $i > $o ) > $i ).
thf(func_def_504,type,
sK203: ( $i > $o ) > $i ).
thf(func_def_511,type,
sK210: $i > $i > $o ).
thf(func_def_514,type,
sK213: $i > $i > $i ).
thf(func_def_518,type,
sK217: $i > $i > $i ).
thf(func_def_521,type,
sK220: $i > $i > $i ).
thf(func_def_522,type,
sK221: $i > $i > $i ).
thf(func_def_523,type,
sK222: $i > $i > $i ).
thf(func_def_524,type,
sK223: $i > $i > $i ).
thf(func_def_525,type,
sK224: $i > $i > $i > $i ).
thf(func_def_526,type,
sK225: $i > $i ).
thf(func_def_527,type,
sK226: $i > $i ).
thf(func_def_528,type,
sK227: $i > $i ).
thf(func_def_529,type,
sK228: $i > $i ).
thf(func_def_530,type,
sK229: $i > $i > $i > $i ).
thf(func_def_531,type,
sK230: $i > $i > $i > $i ).
thf(func_def_532,type,
sK231: $i > $i > $i ).
thf(func_def_533,type,
sK232: $i > $i > $i ).
thf(func_def_534,type,
sK233: $i > $i > $i ).
thf(func_def_536,type,
sK235: $i > $i ).
thf(func_def_537,type,
sK236: $i > $i ).
thf(func_def_538,type,
sK237: $i > $i ).
thf(func_def_539,type,
sK238: $i > $i > $i ).
thf(func_def_553,type,
sK252: ( $i > $i ) > $i > $i > $i ).
thf(func_def_556,type,
sK255: $i > $i ).
thf(func_def_563,type,
sK262: $i > $o ).
thf(func_def_566,type,
sK265: $i > ( $i > $o ) > $i ).
thf(func_def_567,type,
sK266: $i > ( $i > $o ) > $i ).
thf(func_def_568,type,
sK267: $i > $o ).
thf(func_def_571,type,
sK270: $i > ( $i > $o ) > $i ).
thf(func_def_576,type,
sK275: $i > $i > $o ).
thf(func_def_584,type,
sK283: $i > $i > $o ).
thf(func_def_597,type,
sK296: $i > $i > $i > $i > $i ).
thf(func_def_618,type,
sK317: ( $i > $o ) > $i ).
thf(func_def_619,type,
sK318: $i > $o ).
thf(func_def_648,type,
sK347: $i > $i > $i ).
thf(func_def_653,type,
sK352: $i > $i > $i ).
thf(func_def_654,type,
sK353: $i > $i > $i ).
thf(func_def_657,type,
sK356: $i > $i > $o ).
thf(func_def_671,type,
sK370: $i > $i > $i ).
thf(func_def_681,type,
sK380: $i > $i ).
thf(func_def_692,type,
sK391: $i > $o ).
thf(func_def_693,type,
sK392: ( $i > $o ) > $i ).
thf(func_def_694,type,
sK393: $i > $o ).
thf(func_def_695,type,
sK394: $i > $i ).
thf(func_def_699,type,
sK398: $i > $i > $i ).
thf(func_def_703,type,
sK402: $i > $o ).
thf(func_def_714,type,
sK413: $i > $o ).
thf(func_def_720,type,
sK419: $i > $i > $o ).
thf(func_def_737,type,
sK436: $i > $o ).
thf(func_def_763,type,
sK462: $i > $o ).
thf(func_def_777,type,
sK476: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_778,type,
sK477: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_781,type,
sK480: $i > $o ).
thf(func_def_782,type,
sK481: $i > $o ).
thf(func_def_785,type,
sK484: $i > $i ).
thf(func_def_786,type,
sK485: ( $i > $i ) > $i > $i > $i ).
thf(func_def_801,type,
sK500: $i > $i > $i ).
thf(func_def_815,type,
sK514: $i > $i > $i ).
thf(func_def_823,type,
sK522: $i > $i > $i > $i ).
thf(func_def_846,type,
sK545: $i > $i > $o ).
thf(func_def_853,type,
sK552: $i > $i > $i > $i > $i ).
thf(func_def_854,type,
sK553: $i > $i > $i > $i > $i ).
thf(func_def_855,type,
sK554: $i > $i > $i > $i > $i ).
thf(func_def_856,type,
sK555: $i > $i > $i > $i > $i ).
thf(func_def_861,type,
sK560: $i > $i > $i > $i > $i ).
thf(func_def_868,type,
sK567: $i > $i > $i > $i > $i ).
thf(func_def_869,type,
sK568: $i > $i > $i > $i > $i ).
thf(func_def_881,type,
sK580: $i > $o ).
thf(func_def_882,type,
sK581: ( $i > $o ) > $i > $i > $i ).
thf(func_def_885,type,
sK584: $i > $o ).
thf(func_def_886,type,
sK585: $i > $o ).
thf(func_def_887,type,
sK586: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_888,type,
sK587: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_890,type,
sK589: $i > $i ).
thf(func_def_901,type,
sK600: $i > $o ).
thf(func_def_904,type,
sK603: $i > ( $i > $o ) > $i ).
thf(func_def_905,type,
sK604: $i > ( $i > $o ) > $i ).
thf(func_def_909,type,
sK608: $i > $i ).
thf(func_def_912,type,
sK611: $i > $i > ( $i > $i ) > $i ).
thf(func_def_915,type,
sK614: $o > $i > $i > $i ).
thf(func_def_926,type,
sK625: $i > $i > $i > $i ).
thf(func_def_927,type,
sK626: $i > $i > $i > $i ).
thf(func_def_933,type,
sK632: $i > $i ).
thf(func_def_934,type,
sK633: $i > $i ).
thf(func_def_935,type,
sK634: $i > $i > $i > $i ).
thf(func_def_955,type,
sK654: $i > $i > $i ).
thf(func_def_957,type,
sK656: $i > $o ).
thf(func_def_959,type,
sK658: $i > ( $i > $o ) > $i ).
thf(func_def_976,type,
sK675: $i > $i > $i > $i ).
thf(func_def_977,type,
sK676: $i > $i > $i > $i ).
thf(func_def_990,type,
sK689: $i > $o ).
thf(func_def_1003,type,
sK702: $i > $i ).
thf(func_def_1032,type,
sK731: $i > $o ).
thf(func_def_1034,type,
sK733: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1035,type,
sK734: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1040,type,
sK739: $i > $o ).
thf(func_def_1052,type,
sK751: $i > $i ).
thf(func_def_1054,type,
sK753: ( $i > $i ) > $i > $i > $i ).
thf(func_def_1057,type,
sK756: $i > ( $i > $o ) > $i ).
thf(func_def_1058,type,
sK757: $i > $o ).
thf(func_def_1062,type,
sK761: $i > $o ).
thf(func_def_1065,type,
sK764: $i > ( $i > $o ) > $i ).
thf(func_def_1066,type,
sK765: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1067,type,
sK766: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1071,type,
sK770: $i > $o ).
thf(func_def_1074,type,
sK773: $i > $i > $o ).
thf(func_def_1075,type,
sK774: $i > $i ).
thf(func_def_1076,type,
sK775: $i > $i ).
thf(func_def_1077,type,
sK776: ( $i > $i > $o ) > $i > $i ).
thf(func_def_1078,type,
sK777: ( $i > $i > $o ) > $i > $i ).
thf(func_def_1079,type,
sK778: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_1081,type,
sK780: $i > $o ).
thf(func_def_1083,type,
sK782: $i > ( $i > $o ) > $i ).
thf(func_def_1084,type,
sK783: $i > $o ).
thf(func_def_1091,type,
ph790:
!>[X0: $tType] : X0 ).
thf(f4178,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f3892,f4122,f4141]) ).
thf(f4141,plain,
( spl789_7
| ~ spl789_14 ),
inference(avatar_split_clause,[status(thm)],[f4140,f3970,f3881]) ).
thf(f3881,definition,
( spl789_7
<=> ( ( in @ sK335 @ sK333 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl789_7])],[avatar_definition]) ).
thf(f3970,definition,
( spl789_14
<=> ( $true
= ( in @ sK335 @ ( binintersect @ sK333 @ sK334 ) ) ) ),
introduced(definition,[new_symbols(naming,[spl789_14])],[avatar_definition]) ).
thf(f4140,plain,
( ( ( in @ sK335 @ sK333 )
= $true )
| ~ spl789_14 ),
inference(trivial_inequality_removal,[status(thm)],[f4139]) ).
thf(f4139,plain,
( ( ( in @ sK335 @ sK333 )
= $true )
| ( $true != $true )
| ~ spl789_14 ),
inference(forward_demodulation,[status(thm)],[f4138,f2965]) ).
thf(f2965,plain,
binintersectEL = $true,
inference(cnf_transformation,[status(thm)],[f1613]) ).
thf(f1613,plain,
( ( setukpairinjR11 = $true )
& ( emptyinunitempty = $true )
& ( binintersectI = $true )
& ( quantDeMorgan2 = $true )
& ( omega__Cong = $true )
& ( nonemptyI = $true )
& ( cartprodpairin = $true )
& ( eqimpsubset2 = $true )
& ( exuE1 = $true )
& ( binintersectSubset4 = $true )
& ( dsetconstrER = $true )
& ( theprop = $true )
& ( sepSubset = $true )
& ( prop2set2propI = $true )
& ( binintersectSubset3 = $true )
& ( subPowSU = $true )
& ( symdiffE = $true )
& ( setukpairinjR2 = $true )
& ( lam2lamEq = $true )
& ( doubleComplementSub1 = $true )
& ( setadjoinSub = $true )
& ( singletonsubset = $true )
& ( subsetI1 = $true )
& ( complementT_lem = $true )
& ( lamp = $true )
& ( setminusELneg = $true )
& ( binintersectTELcontra = $true )
& ( notsubsetI = $true )
& ( powerset__Cong = $true )
& ( iffalse = $true )
& ( nonemptyImpWitness = $true )
& ( exuE3u = $true )
& ( emptysetE = $true )
& ( binunionE = $true )
& ( powersetTI1 = $true )
& ( binintersectEL = $true )
& ( cartprodpairsurjEq = $true )
& ( notequalI2 = $true )
& ( singletoninpowunion = $true )
& ( powersetI1 = $true )
& ( ex1E1 = $true )
& ( ksndpairEq = $true )
& ( beta2 = $true )
& ( upairsubunion = $true )
& ( omegaSAx = $true )
& ( in__Cong = $true )
& ( binintersectER = $true )
& ( setadjoin__Cong = $true )
& ( ifSingleton = $true )
& ( infuncsetfunc = $true )
& ( contrasubsetT1 = $true )
& ( emptyE1 = $true )
& ( wellorderingAx = $true )
& ( theeq = $true )
& ( inCongP = $true )
& ( disjointsetsI1 = $true )
& ( upairinpowunion = $true )
& ( lam2p = $true )
& ( funcGraphProp4 = $true )
& ( setukpairIR = $true )
& ( setminusSubset2 = $true )
& ( kpairp = $true )
& ( dpsetconstrERa = $true )
& ( cartprodfstpairEq = $true )
& ( dpsetconstrSub = $true )
& ( ksndsingleton = $true )
& ( omegaIndAx = $true )
& ( exuI3 = $true )
& ( sepInPowerset = $true )
& ( brelnall1 = $true )
& ( powersetTE1 = $true )
& ( complementTnotintersectT = $true )
& ( binunionIL = $true )
& ( exuI1 = $true )
& ( binintersectT_lem = $true )
& ( notinsingleton = $true )
& ( iffalseProp1 = $true )
& ( omega0Ax = $true )
& ( bs114d = $true )
& ( setukpairinjR1 = $true )
& ( exuE2 = $true )
& ( nonemptyI1 = $true )
& ( upairequniteq = $true )
& ( emptysetAx = $true )
& ( funcext2 = $true )
& ( setminusER = $true )
& ( setukpairIL = $true )
& ( setextT = $true )
& ( setminusLsub = $true )
& ( cartprodpairmemER = $true )
& ( setukpairinjL2 = $true )
& ( emptyinPowerset = $true )
& ( funcinfuncset = $true )
& ( singletonsuniq = $true )
& ( funcGraphProp1 = $true )
& ( setminusIRneg = $true )
& ( iftrueProp1 = $true )
& ( emptysetimpfalse = $true )
& ( setadjoinIR = $true )
& ( setukpairinjL1 = $true )
& ( dpsetconstrEL1 = $true )
& ( setadjoinAx = $true )
& ( emptysetsubset = $true )
& ( singletonprop = $true )
& ( binunionIR = $true )
& ( setunionI = $true )
& ( setunionsingleton2 = $true )
& ( eta1 = $true )
& ( ex1I = $true )
& ( subsetE = $true )
& ( binintersectTERcontra = $true )
& ( eqinunit = $true )
& ( notinemptyset = $true )
& ( $true
= ( in @ sK333 @ ( powerset @ sK332 ) ) )
& ( ( in @ sK334 @ ( powerset @ sK332 ) )
= $true )
& ( $true
= ( in @ sK335 @ ( setminus @ sK332 @ sK333 ) ) )
& ( ( in @ sK335 @ sK332 )
= $true )
& ( ( in @ sK335 @ ( setminus @ sK332 @ ( binintersect @ sK333 @ sK334 ) ) )
!= $true )
& ( ap2apEq2 = $true )
& ( setext = $true )
& ( notdexE = $true )
& ( symdiffIneg1 = $true )
& ( ap2apEq1 = $true )
& ( complementTI1 = $true )
& ( dsetconstrEL = $true )
& ( setunion__Cong = $true )
& ( setunionsingleton1 = $true )
& ( subsetE2 = $true )
& ( binintersectSubset5 = $true )
& ( quantDeMorgan3 = $true )
& ( powersetAx = $true )
& ( binintersectSubset2 = $true )
& ( setunionAx = $true )
& ( setminusEL = $true )
& ( powersetT_lem = $true )
& ( kpairiskpair = $true )
& ( kpairsurjEq = $true )
& ( setunionE = $true )
& ( setminusI = $true )
& ( binunionRsub = $true )
& ( lamProp = $true )
& ( cartprodmempair = $true )
& ( binintersectRsub = $true )
& ( descrp = $true )
& ( emptyset__Cong = $true )
& ( singletoninpowerset = $true )
& ( ifp = $true )
& ( binunionT_lem = $true )
& ( funcGraphProp2 = $true )
& ( emptyInPowerset = $true )
& ( setextAx = $true )
& ( subset2powerset = $true )
& ( cartprodpairmemEL = $true )
& ( setadjoinIL = $true )
& ( exu__Cong = $true )
& ( noeltsimpempty = $true )
& ( funcImageSingleton = $true )
& ( quantDeMorgan1 = $true )
& ( funcext = $true )
& ( binunionEcases = $true )
& ( setminusILneg = $true )
& ( ubforcartprodlem2 = $true )
& ( symdiffI2 = $true )
& ( powersetE = $true )
& ( kfstpairEq = $true )
& ( exuE3e = $true )
& ( upairsetIL = $true )
& ( contrasubsetT3 = $true )
& ( ap2p = $true )
& ( setoftrueEq = $true )
& ( binintersectSubset1 = $true )
& ( funcextLem = $true )
& ( setextsub = $true )
& ( vacuousDall = $true )
& ( app = $true )
& ( nonemptyE1 = $true )
& ( setukpairinjR12 = $true )
& ( setadjoinE = $true )
& ( eqbreln = $true )
& ( powersetE1 = $true )
& ( ex1I2 = $true )
& ( kfstsingleton = $true )
& ( contrasubsetT2 = $true )
& ( iftrue = $true )
& ( subsetRefl = $true )
& ( setadjoinSub2 = $true )
& ( cartprodmempair1 = $true )
& ( descr__Cong = $true )
& ( iftrueProp2 = $true )
& ( dsetconstrI = $true )
& ( inPowerset = $true )
& ( subbreln = $true )
& ( upairset2E = $true )
& ( emptyI = $true )
& ( apProp = $true )
& ( upairsetIR = $true )
& ( setadjoinOr = $true )
& ( doubleComplementEq = $true )
& ( setunionsingleton = $true )
& ( replAx = $true )
& ( dpsetconstrER = $true )
& ( subsetemptysetimpeq = $true )
& ( symdiffI1 = $true )
& ( contrasubsetT = $true )
& ( binunionLsub = $true )
& ( setminusERneg = $true )
& ( cartprodsndin = $true )
& ( exuEu = $true )
& ( doubleComplementE1 = $true )
& ( cartprodsndpairEq = $true )
& ( beta1 = $true )
& ( secondinupair = $true )
& ( ubforcartprodlem3 = $true )
& ( powersetI = $true )
& ( setunionE2 = $true )
& ( complementTE1 = $true )
& ( prop2setE = $true )
& ( upairsetE = $true )
& ( setbeta = $true )
& ( iffalseProp2 = $true )
& ( notequalI1 = $true )
& ( dpsetconstrI = $true )
& ( foundationAx = $true )
& ( iftrueorfalse = $true )
& ( setukpairinjL = $true )
& ( doubleComplementI1 = $true )
& ( exuI2 = $true )
& ( dpsetconstrEL2 = $true )
& ( upairset2IR = $true )
& ( prop2setI = $true )
& ( binintersectLsub = $true )
& ( powersetsubset = $true )
& ( setminusSubset1 = $true )
& ( singletonsswitch = $true )
& ( brelnall2 = $true )
& ( quantDeMorgan4 = $true )
& ( setminusT_lem = $true )
& ( notdallE = $true )
& ( cartprodmempaircEq = $true )
& ( setOfPairsIsBReln = $true )
& ( doubleComplementSub2 = $true )
& ( eta2 = $true )
& ( uniqinunit = $true )
& ( eqimpsubset1 = $true )
& ( ex1E2 = $true )
& ( setukpairinjR = $true )
& ( subsetI2 = $true )
& ( dsetconstr__Cong = $true )
& ( funcGraphProp3 = $true )
& ( cartprodfstin = $true )
& ( subsetTrans = $true )
& ( ubforcartprodlem1 = $true )
& ( subsetTI = $true )
& ( symdiffIneg2 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK332,sK333,sK334,sK335])],[f847,f1612,f1611,f1610]) ).
thf(f1610,plain,
( ? [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
& ? [X2: $i] :
( ( ( in @ X2 @ ( powerset @ X0 ) )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ ( setminus @ X0 @ X1 ) )
= $true )
& ( ( in @ X3 @ X0 )
= $true )
& ( ( in @ X3 @ ( setminus @ X0 @ ( binintersect @ X1 @ X2 ) ) )
!= $true ) ) ) )
=> ( ( $true
= ( in @ sK333 @ ( powerset @ sK332 ) ) )
& ? [X2: $i] :
( ( ( in @ X2 @ ( powerset @ sK332 ) )
= $true )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ ( setminus @ sK332 @ sK333 ) ) )
& ( ( in @ X3 @ sK332 )
= $true )
& ( ( in @ X3 @ ( setminus @ sK332 @ ( binintersect @ sK333 @ X2 ) ) )
!= $true ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1611,plain,
( ? [X2: $i] :
( ( ( in @ X2 @ ( powerset @ sK332 ) )
= $true )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ ( setminus @ sK332 @ sK333 ) ) )
& ( ( in @ X3 @ sK332 )
= $true )
& ( ( in @ X3 @ ( setminus @ sK332 @ ( binintersect @ sK333 @ X2 ) ) )
!= $true ) ) )
=> ( ( ( in @ sK334 @ ( powerset @ sK332 ) )
= $true )
& ? [X3: $i] :
( ( $true
= ( in @ X3 @ ( setminus @ sK332 @ sK333 ) ) )
& ( ( in @ X3 @ sK332 )
= $true )
& ( ( in @ X3 @ ( setminus @ sK332 @ ( binintersect @ sK333 @ sK334 ) ) )
!= $true ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1612,plain,
( ? [X3: $i] :
( ( $true
= ( in @ X3 @ ( setminus @ sK332 @ sK333 ) ) )
& ( ( in @ X3 @ sK332 )
= $true )
& ( ( in @ X3 @ ( setminus @ sK332 @ ( binintersect @ sK333 @ sK334 ) ) )
!= $true ) )
=> ( ( $true
= ( in @ sK335 @ ( setminus @ sK332 @ sK333 ) ) )
& ( ( in @ sK335 @ sK332 )
= $true )
& ( ( in @ sK335 @ ( setminus @ sK332 @ ( binintersect @ sK333 @ sK334 ) ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f847,plain,
( ( setukpairinjR11 = $true )
& ( emptyinunitempty = $true )
& ( binintersectI = $true )
& ( quantDeMorgan2 = $true )
& ( omega__Cong = $true )
& ( nonemptyI = $true )
& ( cartprodpairin = $true )
& ( eqimpsubset2 = $true )
& ( exuE1 = $true )
& ( binintersectSubset4 = $true )
& ( dsetconstrER = $true )
& ( theprop = $true )
& ( sepSubset = $true )
& ( prop2set2propI = $true )
& ( binintersectSubset3 = $true )
& ( subPowSU = $true )
& ( symdiffE = $true )
& ( setukpairinjR2 = $true )
& ( lam2lamEq = $true )
& ( doubleComplementSub1 = $true )
& ( setadjoinSub = $true )
& ( singletonsubset = $true )
& ( subsetI1 = $true )
& ( complementT_lem = $true )
& ( lamp = $true )
& ( setminusELneg = $true )
& ( binintersectTELcontra = $true )
& ( notsubsetI = $true )
& ( powerset__Cong = $true )
& ( iffalse = $true )
& ( nonemptyImpWitness = $true )
& ( exuE3u = $true )
& ( emptysetE = $true )
& ( binunionE = $true )
& ( powersetTI1 = $true )
& ( binintersectEL = $true )
& ( cartprodpairsurjEq = $true )
& ( notequalI2 = $true )
& ( singletoninpowunion = $true )
& ( powersetI1 = $true )
& ( ex1E1 = $true )
& ( ksndpairEq = $true )
& ( beta2 = $true )
& ( upairsubunion = $true )
& ( omegaSAx = $true )
& ( in__Cong = $true )
& ( binintersectER = $true )
& ( setadjoin__Cong = $true )
& ( ifSingleton = $true )
& ( infuncsetfunc = $true )
& ( contrasubsetT1 = $true )
& ( emptyE1 = $true )
& ( wellorderingAx = $true )
& ( theeq = $true )
& ( inCongP = $true )
& ( disjointsetsI1 = $true )
& ( upairinpowunion = $true )
& ( lam2p = $true )
& ( funcGraphProp4 = $true )
& ( setukpairIR = $true )
& ( setminusSubset2 = $true )
& ( kpairp = $true )
& ( dpsetconstrERa = $true )
& ( cartprodfstpairEq = $true )
& ( dpsetconstrSub = $true )
& ( ksndsingleton = $true )
& ( omegaIndAx = $true )
& ( exuI3 = $true )
& ( sepInPowerset = $true )
& ( brelnall1 = $true )
& ( powersetTE1 = $true )
& ( complementTnotintersectT = $true )
& ( binunionIL = $true )
& ( exuI1 = $true )
& ( binintersectT_lem = $true )
& ( notinsingleton = $true )
& ( iffalseProp1 = $true )
& ( omega0Ax = $true )
& ( bs114d = $true )
& ( setukpairinjR1 = $true )
& ( exuE2 = $true )
& ( nonemptyI1 = $true )
& ( upairequniteq = $true )
& ( emptysetAx = $true )
& ( funcext2 = $true )
& ( setminusER = $true )
& ( setukpairIL = $true )
& ( setextT = $true )
& ( setminusLsub = $true )
& ( cartprodpairmemER = $true )
& ( setukpairinjL2 = $true )
& ( emptyinPowerset = $true )
& ( funcinfuncset = $true )
& ( singletonsuniq = $true )
& ( funcGraphProp1 = $true )
& ( setminusIRneg = $true )
& ( iftrueProp1 = $true )
& ( emptysetimpfalse = $true )
& ( setadjoinIR = $true )
& ( setukpairinjL1 = $true )
& ( dpsetconstrEL1 = $true )
& ( setadjoinAx = $true )
& ( emptysetsubset = $true )
& ( singletonprop = $true )
& ( binunionIR = $true )
& ( setunionI = $true )
& ( setunionsingleton2 = $true )
& ( eta1 = $true )
& ( ex1I = $true )
& ( subsetE = $true )
& ( binintersectTERcontra = $true )
& ( eqinunit = $true )
& ( notinemptyset = $true )
& ? [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
& ? [X2: $i] :
( ( ( in @ X2 @ ( powerset @ X0 ) )
= $true )
& ? [X3: $i] :
( ( ( in @ X3 @ ( setminus @ X0 @ X1 ) )
= $true )
& ( ( in @ X3 @ X0 )
= $true )
& ( ( in @ X3 @ ( setminus @ X0 @ ( binintersect @ X1 @ X2 ) ) )
!= $true ) ) ) )
& ( ap2apEq2 = $true )
& ( setext = $true )
& ( notdexE = $true )
& ( symdiffIneg1 = $true )
& ( ap2apEq1 = $true )
& ( complementTI1 = $true )
& ( dsetconstrEL = $true )
& ( setunion__Cong = $true )
& ( setunionsingleton1 = $true )
& ( subsetE2 = $true )
& ( binintersectSubset5 = $true )
& ( quantDeMorgan3 = $true )
& ( powersetAx = $true )
& ( binintersectSubset2 = $true )
& ( setunionAx = $true )
& ( setminusEL = $true )
& ( powersetT_lem = $true )
& ( kpairiskpair = $true )
& ( kpairsurjEq = $true )
& ( setunionE = $true )
& ( setminusI = $true )
& ( binunionRsub = $true )
& ( lamProp = $true )
& ( cartprodmempair = $true )
& ( binintersectRsub = $true )
& ( descrp = $true )
& ( emptyset__Cong = $true )
& ( singletoninpowerset = $true )
& ( ifp = $true )
& ( binunionT_lem = $true )
& ( funcGraphProp2 = $true )
& ( emptyInPowerset = $true )
& ( setextAx = $true )
& ( subset2powerset = $true )
& ( cartprodpairmemEL = $true )
& ( setadjoinIL = $true )
& ( exu__Cong = $true )
& ( noeltsimpempty = $true )
& ( funcImageSingleton = $true )
& ( quantDeMorgan1 = $true )
& ( funcext = $true )
& ( binunionEcases = $true )
& ( setminusILneg = $true )
& ( ubforcartprodlem2 = $true )
& ( symdiffI2 = $true )
& ( powersetE = $true )
& ( kfstpairEq = $true )
& ( exuE3e = $true )
& ( upairsetIL = $true )
& ( contrasubsetT3 = $true )
& ( ap2p = $true )
& ( setoftrueEq = $true )
& ( binintersectSubset1 = $true )
& ( funcextLem = $true )
& ( setextsub = $true )
& ( vacuousDall = $true )
& ( app = $true )
& ( nonemptyE1 = $true )
& ( setukpairinjR12 = $true )
& ( setadjoinE = $true )
& ( eqbreln = $true )
& ( powersetE1 = $true )
& ( ex1I2 = $true )
& ( kfstsingleton = $true )
& ( contrasubsetT2 = $true )
& ( iftrue = $true )
& ( subsetRefl = $true )
& ( setadjoinSub2 = $true )
& ( cartprodmempair1 = $true )
& ( descr__Cong = $true )
& ( iftrueProp2 = $true )
& ( dsetconstrI = $true )
& ( inPowerset = $true )
& ( subbreln = $true )
& ( upairset2E = $true )
& ( emptyI = $true )
& ( apProp = $true )
& ( upairsetIR = $true )
& ( setadjoinOr = $true )
& ( doubleComplementEq = $true )
& ( setunionsingleton = $true )
& ( replAx = $true )
& ( dpsetconstrER = $true )
& ( subsetemptysetimpeq = $true )
& ( symdiffI1 = $true )
& ( contrasubsetT = $true )
& ( binunionLsub = $true )
& ( setminusERneg = $true )
& ( cartprodsndin = $true )
& ( exuEu = $true )
& ( doubleComplementE1 = $true )
& ( cartprodsndpairEq = $true )
& ( beta1 = $true )
& ( secondinupair = $true )
& ( ubforcartprodlem3 = $true )
& ( powersetI = $true )
& ( setunionE2 = $true )
& ( complementTE1 = $true )
& ( prop2setE = $true )
& ( upairsetE = $true )
& ( setbeta = $true )
& ( iffalseProp2 = $true )
& ( notequalI1 = $true )
& ( dpsetconstrI = $true )
& ( foundationAx = $true )
& ( iftrueorfalse = $true )
& ( setukpairinjL = $true )
& ( doubleComplementI1 = $true )
& ( exuI2 = $true )
& ( dpsetconstrEL2 = $true )
& ( upairset2IR = $true )
& ( prop2setI = $true )
& ( binintersectLsub = $true )
& ( powersetsubset = $true )
& ( setminusSubset1 = $true )
& ( singletonsswitch = $true )
& ( brelnall2 = $true )
& ( quantDeMorgan4 = $true )
& ( setminusT_lem = $true )
& ( notdallE = $true )
& ( cartprodmempaircEq = $true )
& ( setOfPairsIsBReln = $true )
& ( doubleComplementSub2 = $true )
& ( eta2 = $true )
& ( uniqinunit = $true )
& ( eqimpsubset1 = $true )
& ( ex1E2 = $true )
& ( setukpairinjR = $true )
& ( subsetI2 = $true )
& ( dsetconstr__Cong = $true )
& ( funcGraphProp3 = $true )
& ( cartprodfstin = $true )
& ( subsetTrans = $true )
& ( ubforcartprodlem1 = $true )
& ( subsetTI = $true )
& ( symdiffIneg2 = $true ) ),
inference(flattening,[status(thm)],[f846]) ).
thf(f846,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ( ( in @ X3 @ ( setminus @ X0 @ ( binintersect @ X1 @ X2 ) ) )
!= $true )
& ( ( in @ X3 @ ( setminus @ X0 @ X1 ) )
= $true )
& ( ( in @ X3 @ X0 )
= $true ) )
& ( ( in @ X2 @ ( powerset @ X0 ) )
= $true ) )
& ( ( in @ X1 @ ( powerset @ X0 ) )
= $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)],[f752]) ).
thf(f752,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 )
=> ! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
= $true )
=> ! [X2: $i] :
( ( ( in @ X2 @ ( powerset @ X0 ) )
= $true )
=> ! [X3: $i] :
( ( ( in @ X3 @ X0 )
= $true )
=> ( ( ( in @ X3 @ ( setminus @ X0 @ X1 ) )
= $true )
=> ( ( in @ X3 @ ( setminus @ X0 @ ( binintersect @ X1 @ X2 ) ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f751]) ).
thf(f751,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
=> ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X0 ) )
=> ! [X3: $i] :
( ( in @ X3 @ X0 )
=> ( ( in @ X3 @ ( setminus @ X0 @ X1 ) )
=> ( in @ X3 @ ( setminus @ X0 @ ( binintersect @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f255]) ).
thf(f255,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
=> ! [X3: $i,X11: $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 @ ( binintersect @ X11 @ X16 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f254]) ).
thf(f254,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
=> ! [X3: $i,X11: $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 @ ( binintersect @ X11 @ X16 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.V67yydYf3P/DTF2THF_3524.p',complementImpComplementIntersect) ).
thf(f4138,plain,
( ( binintersectEL != $true )
| ( ( in @ sK335 @ sK333 )
= $true )
| ~ spl789_14 ),
inference(trivial_inequality_removal,[status(thm)],[f4125]) ).
thf(f4125,plain,
( ( ( in @ sK335 @ sK333 )
= $true )
| ( binintersectEL != $true )
| ( $true != $true )
| ~ spl789_14 ),
inference(superposition,[status(thm)],[f2555,f3972]) ).
thf(f3972,plain,
( ( $true
= ( in @ sK335 @ ( binintersect @ sK333 @ sK334 ) ) )
| ~ spl789_14 ),
inference(avatar_component_clause,[status(thm)],[f3970]) ).
thf(f2555,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( ( in @ X0 @ ( binintersect @ X2 @ X1 ) )
!= $true )
| ( ( in @ X0 @ X2 )
= $true )
| ( binintersectEL != $true ) ),
inference(cnf_transformation,[status(thm)],[f1400]) ).
thf(f1400,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binintersect @ X2 @ X1 ) )
!= $true )
| ( ( in @ X0 @ X2 )
= $true ) )
| ( binintersectEL != $true ) )
& ( ( binintersectEL = $true )
| ( ( $true
= ( in @ sK192 @ ( binintersect @ sK194 @ sK193 ) ) )
& ( $true
!= ( in @ sK192 @ sK194 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK192,sK193,sK194])],[f1398,f1399]) ).
thf(f1399,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ ( binintersect @ X5 @ X4 ) )
= $true )
& ( $true
!= ( in @ X3 @ X5 ) ) )
=> ( ( $true
= ( in @ sK192 @ ( binintersect @ sK194 @ sK193 ) ) )
& ( $true
!= ( in @ sK192 @ sK194 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1398,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binintersect @ X2 @ X1 ) )
!= $true )
| ( ( in @ X0 @ X2 )
= $true ) )
| ( binintersectEL != $true ) )
& ( ( binintersectEL = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ ( binintersect @ X5 @ X4 ) )
= $true )
& ( $true
!= ( in @ X3 @ X5 ) ) ) ) ),
inference(rectify,[status(thm)],[f1397]) ).
thf(f1397,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binintersect @ X2 @ X1 ) )
!= $true )
| ( ( in @ X0 @ X2 )
= $true ) )
| ( binintersectEL != $true ) )
& ( ( binintersectEL = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binintersect @ X2 @ X1 ) )
= $true )
& ( ( in @ X0 @ X2 )
!= $true ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1085]) ).
thf(f1085,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binintersect @ X2 @ X1 ) )
!= $true )
| ( ( in @ X0 @ X2 )
= $true ) )
<=> ( binintersectEL = $true ) ),
inference(ennf_transformation,[status(thm)],[f559]) ).
thf(f559,plain,
( ( binintersectEL = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binintersect @ X2 @ X1 ) )
= $true )
=> ( ( in @ X0 @ X2 )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f558]) ).
thf(f558,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ ( binintersect @ X2 @ X1 ) )
=> ( in @ X0 @ X2 ) )
= binintersectEL ),
inference(rectify,[status(thm)],[f110]) ).
thf(f110,axiom,
! [X1: $i,X4: $i,X3: $i] :
( ( ( in @ X1 @ ( binintersect @ X3 @ X4 ) )
=> ( in @ X1 @ X3 ) )
= binintersectEL ),
file('/export/starexec/sandbox2/tmp/tmp.V67yydYf3P/DTF2THF_3524.p',binintersectEL) ).
thf(f4122,plain,
spl789_14,
inference(avatar_split_clause,[status(thm)],[f3954,f3970]) ).
thf(f3954,plain,
( $true
= ( in @ sK335 @ ( binintersect @ sK333 @ sK334 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3953]) ).
thf(f3953,plain,
( ( $true
= ( in @ sK335 @ ( binintersect @ sK333 @ sK334 ) ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3952,f2795]) ).
thf(f2795,plain,
setminusERneg = $true,
inference(cnf_transformation,[status(thm)],[f1613]) ).
thf(f3952,plain,
( ( setminusERneg != $true )
| ( $true
= ( in @ sK335 @ ( binintersect @ sK333 @ sK334 ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3951]) ).
thf(f3951,plain,
( ( setminusERneg != $true )
| ( $true
= ( in @ sK335 @ ( binintersect @ sK333 @ sK334 ) ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3937,f2884]) ).
thf(f2884,plain,
( ( in @ sK335 @ sK332 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1613]) ).
thf(f3937,plain,
( ( $true
= ( in @ sK335 @ ( binintersect @ sK333 @ sK334 ) ) )
| ( ( in @ sK335 @ sK332 )
!= $true )
| ( setminusERneg != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3925]) ).
thf(f3925,plain,
( ( $true
= ( in @ sK335 @ ( binintersect @ sK333 @ sK334 ) ) )
| ( ( in @ sK335 @ sK332 )
!= $true )
| ( $true != $true )
| ( setminusERneg != $true ) ),
inference(superposition,[status(thm)],[f2883,f3079]) ).
thf(f3079,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ ( setminus @ X4 @ X5 ) )
= $true )
| ( $true
= ( in @ X3 @ X5 ) )
| ( setminusERneg != $true )
| ( ( in @ X3 @ X4 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1713]) ).
thf(f1713,plain,
( ( ( setminusERneg = $true )
| ( ( ( in @ sK395 @ sK396 )
= $true )
& ( $true
!= ( in @ sK395 @ ( setminus @ sK396 @ sK397 ) ) )
& ( $true
!= ( in @ sK395 @ sK397 ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ X4 )
!= $true )
| ( ( in @ X3 @ ( setminus @ X4 @ X5 ) )
= $true )
| ( $true
= ( in @ X3 @ X5 ) ) )
| ( setminusERneg != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK395,sK396,sK397])],[f1711,f1712]) ).
thf(f1712,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
& ( $true
!= ( in @ X0 @ ( setminus @ X1 @ X2 ) ) )
& ( ( in @ X0 @ X2 )
!= $true ) )
=> ( ( ( in @ sK395 @ sK396 )
= $true )
& ( $true
!= ( in @ sK395 @ ( setminus @ sK396 @ sK397 ) ) )
& ( $true
!= ( in @ sK395 @ sK397 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1711,plain,
( ( ( setminusERneg = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
& ( $true
!= ( in @ X0 @ ( setminus @ X1 @ X2 ) ) )
& ( ( in @ X0 @ X2 )
!= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X3 @ X4 )
!= $true )
| ( ( in @ X3 @ ( setminus @ X4 @ X5 ) )
= $true )
| ( $true
= ( in @ X3 @ X5 ) ) )
| ( setminusERneg != $true ) ) ),
inference(rectify,[status(thm)],[f1710]) ).
thf(f1710,plain,
( ( ( setminusERneg = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X1 ) )
& ( $true
!= ( in @ X0 @ ( setminus @ X1 @ X2 ) ) )
& ( ( in @ X0 @ X2 )
!= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ ( setminus @ X1 @ X2 ) ) )
| ( ( in @ X0 @ X2 )
= $true ) )
| ( setminusERneg != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1049]) ).
thf(f1049,plain,
( ( setminusERneg = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ ( setminus @ X1 @ X2 ) ) )
| ( ( in @ X0 @ X2 )
= $true ) ) ),
inference(flattening,[status(thm)],[f1048]) ).
thf(f1048,plain,
( ( setminusERneg = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X0 @ X2 )
= $true )
| ( $true
!= ( in @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ ( setminus @ X1 @ X2 ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f798]) ).
thf(f798,plain,
( ( setminusERneg = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ ( setminus @ X1 @ X2 ) ) )
=> ( ( $true
= ( in @ X0 @ X1 ) )
=> ( ( in @ X0 @ X2 )
= $true ) ) ) ),
inference(flattening,[status(thm)],[f625]) ).
thf(f625,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ ( setminus @ X1 @ X2 ) ) )
=> ( ( $true
= ( in @ X0 @ X1 ) )
=> ( ( in @ X0 @ X2 )
= $true ) ) )
<=> ( setminusERneg = $true ) ),
inference(fool_elimination,[status(thm)],[f624]) ).
thf(f624,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ~ ( in @ X0 @ ( setminus @ X1 @ X2 ) )
=> ( ( in @ X0 @ X1 )
=> ( in @ X0 @ X2 ) ) )
= setminusERneg ),
inference(rectify,[status(thm)],[f124]) ).
thf(f124,axiom,
! [X1: $i,X3: $i,X4: $i] :
( ( ~ ( in @ X1 @ ( setminus @ X3 @ X4 ) )
=> ( ( in @ X1 @ X3 )
=> ( in @ X1 @ X4 ) ) )
= setminusERneg ),
file('/export/starexec/sandbox2/tmp/tmp.V67yydYf3P/DTF2THF_3524.p',setminusERneg) ).
thf(f2883,plain,
( ( in @ sK335 @ ( setminus @ sK332 @ ( binintersect @ sK333 @ sK334 ) ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1613]) ).
thf(f3892,plain,
~ spl789_7,
inference(avatar_split_clause,[status(thm)],[f3891,f3881]) ).
thf(f3891,plain,
( ( in @ sK335 @ sK333 )
!= $true ),
inference(trivial_inequality_removal,[status(thm)],[f3890]) ).
thf(f3890,plain,
( ( $true != $true )
| ( ( in @ sK335 @ sK333 )
!= $true ) ),
inference(forward_demodulation,[status(thm)],[f3868,f2915]) ).
thf(f2915,plain,
setminusER = $true,
inference(cnf_transformation,[status(thm)],[f1613]) ).
thf(f3868,plain,
( ( setminusER != $true )
| ( ( in @ sK335 @ sK333 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3861]) ).
thf(f3861,plain,
( ( $true != $true )
| ( ( in @ sK335 @ sK333 )
!= $true )
| ( setminusER != $true ) ),
inference(superposition,[status(thm)],[f3377,f2885]) ).
thf(f2885,plain,
( $true
= ( in @ sK335 @ ( setminus @ sK332 @ sK333 ) ) ),
inference(cnf_transformation,[status(thm)],[f1613]) ).
thf(f3377,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4 @ ( setminus @ X5 @ X3 ) ) )
| ( ( in @ X4 @ X3 )
!= $true )
| ( setminusER != $true ) ),
inference(cnf_transformation,[status(thm)],[f2051]) ).
thf(f2051,plain,
( ( ( setminusER = $true )
| ( ( $true
= ( in @ sK619 @ ( setminus @ sK620 @ sK618 ) ) )
& ( $true
= ( in @ sK619 @ sK618 ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4 @ ( setminus @ X5 @ X3 ) ) )
| ( ( in @ X4 @ X3 )
!= $true ) )
| ( setminusER != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK618,sK619,sK620])],[f2049,f2050]) ).
thf(f2050,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true )
& ( ( in @ X1 @ X0 )
= $true ) )
=> ( ( $true
= ( in @ sK619 @ ( setminus @ sK620 @ sK618 ) ) )
& ( $true
= ( in @ sK619 @ sK618 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2049,plain,
( ( ( setminusER = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true )
& ( ( in @ X1 @ X0 )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4 @ ( setminus @ X5 @ X3 ) ) )
| ( ( in @ X4 @ X3 )
!= $true ) )
| ( setminusER != $true ) ) ),
inference(rectify,[status(thm)],[f2048]) ).
thf(f2048,plain,
( ( ( setminusER = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true )
& ( ( in @ X1 @ X0 )
= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
!= $true )
| ( ( in @ X1 @ X0 )
!= $true ) )
| ( setminusER != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f920]) ).
thf(f920,plain,
( ( setminusER = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
!= $true )
| ( ( in @ X1 @ X0 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f773]) ).
thf(f773,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true )
=> ( ( in @ X1 @ X0 )
!= $true ) )
<=> ( setminusER = $true ) ),
inference(flattening,[status(thm)],[f396]) ).
thf(f396,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true )
=> ( ( in @ X1 @ X0 )
!= $true ) )
<=> ( setminusER = $true ) ),
inference(fool_elimination,[status(thm)],[f395]) ).
thf(f395,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
=> ~ ( in @ X1 @ X0 ) )
= setminusER ),
inference(rectify,[status(thm)],[f122]) ).
thf(f122,axiom,
! [X4: $i,X1: $i,X3: $i] :
( ( ( in @ X1 @ ( setminus @ X3 @ X4 ) )
=> ~ ( in @ X1 @ X4 ) )
= setminusER ),
file('/export/starexec/sandbox2/tmp/tmp.V67yydYf3P/DTF2THF_3524.p',setminusER) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : SEU733^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.07 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.07/0.25 % Computer : n024.cluster.edu
% 0.07/0.25 % Model : x86_64 x86_64
% 0.07/0.25 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.25 % Memory : 8042.1875MB
% 0.07/0.25 % OS : Linux 3.10.0-693.el7.x86_64
% 0.07/0.25 % CPULimit : 300
% 0.07/0.25 % WCLimit : 300
% 0.07/0.25 % DateTime : Tue Feb 24 07:32:10 EST 2026
% 0.07/0.25 % CPUTime :
% 0.07/0.25 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.07/0.31 ---- Original DTF file ---
% 0.07/0.31 thf(spec,logic,$$dhol).
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 % File : SEU733^1 : TPTP v9.2.1. Released v3.7.0.
% 0.07/0.31 % Domain : Set Theory
% 0.07/0.31 % Problem : Typed Set Theory - Laws for Typed Sets
% 0.07/0.31 % Version : Especial.
% 0.07/0.31 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.07/0.31 % (! x:i.in x A -> in x (setminus A X) ->
% 0.07/0.31 % in x (setminus A (binintersect X Y)))))
% 0.07/0.31
% 0.07/0.31 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.07/0.31 % Source : [Bro08]
% 0.07/0.31 % Names : ZFC235g [Bro08]
% 0.07/0.31
% 0.07/0.31 % Status : Theorem
% 0.07/0.31 % Rating : 0.22 v9.1.0, 0.38 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0, 0.29 v7.4.0, 0.33 v7.2.0, 0.25 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, 0.80 v4.1.0, 1.00 v3.7.0
% 0.07/0.31 % Syntax : Number of formulae : 541 ( 253 unt; 287 typ; 253 def)
% 0.07/0.31 % Number of atoms : 1455 ( 379 equ; 0 cnn)
% 0.07/0.31 % Maximal formula atoms : 254 ( 5 avg)
% 0.07/0.31 % Number of connectives : 2895 ( 65 ~; 12 |; 44 &;2055 @)
% 0.07/0.31 % ( 14 <=>; 705 =>; 0 <=; 0 <~>)
% 0.07/0.31 % Maximal formula depth : 262 ( 2 avg)
% 0.07/0.31 % Number of types : 2 ( 0 usr)
% 0.07/0.31 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.07/0.31 % Number of symbols : 290 ( 287 usr; 253 con; 0-4 aty)
% 0.07/0.31 % Number of variables : 850 ( 74 ^; 741 !; 35 ?; 850 :)
% 0.07/0.31 % SPC : TH0_THM_EQU_NAR
% 0.07/0.31
% 0.07/0.31 % Comments : http://mathgate.info/detsetitem.php?id=295
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 thf(in_type,type,
% 0.07/0.31 in: $i > $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(exu_type,type,
% 0.07/0.31 exu: ( $i > $o ) > $o ).
% 0.07/0.31
% 0.07/0.31 thf(exu,definition,
% 0.07/0.31 ( exu
% 0.07/0.31 = ( ^ [Xphi: $i > $o] :
% 0.07/0.31 ? [Xx: $i] :
% 0.07/0.31 ( ( Xphi @ Xx )
% 0.07/0.31 & ! [Xy: $i] :
% 0.07/0.31 ( ( Xphi @ Xy )
% 0.07/0.31 => ( Xx = Xy ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setextAx_type,type,
% 0.07/0.31 setextAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setextAx,definition,
% 0.07/0.31 ( setextAx
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 <=> ( in @ Xx @ B ) )
% 0.07/0.31 => ( A = B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(emptyset_type,type,
% 0.07/0.31 emptyset: $i ).
% 0.07/0.31
% 0.07/0.31 thf(emptysetAx_type,type,
% 0.07/0.31 emptysetAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(emptysetAx,definition,
% 0.07/0.31 ( emptysetAx
% 0.07/0.31 = ( ! [Xx: $i] :
% 0.07/0.31 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoin_type,type,
% 0.07/0.31 setadjoin: $i > $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinAx_type,type,
% 0.07/0.31 setadjoinAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setadjoinAx,definition,
% 0.07/0.31 ( setadjoinAx
% 0.07/0.31 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.07/0.31 <=> ( ( Xy = Xx )
% 0.07/0.31 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(powerset_type,type,
% 0.07/0.31 powerset: $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(powersetAx_type,type,
% 0.07/0.31 powersetAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(powersetAx,definition,
% 0.07/0.31 ( powersetAx
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ( in @ B @ ( powerset @ A ) )
% 0.07/0.31 <=> ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setunion_type,type,
% 0.07/0.31 setunion: $i > $i ).
% 0.07/0.31
% 0.07/0.31 thf(setunionAx_type,type,
% 0.07/0.31 setunionAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setunionAx,definition,
% 0.07/0.31 ( setunionAx
% 0.07/0.31 = ( ! [A: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ ( setunion @ A ) )
% 0.07/0.31 <=> ? [B: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 & ( in @ B @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(omega_type,type,
% 0.07/0.31 omega: $i ).
% 0.07/0.31
% 0.07/0.31 thf(omega0Ax_type,type,
% 0.07/0.31 omega0Ax: $o ).
% 0.07/0.31
% 0.07/0.31 thf(omega0Ax,definition,
% 0.07/0.31 ( omega0Ax
% 0.07/0.31 = ( in @ emptyset @ omega ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(omegaSAx_type,type,
% 0.07/0.31 omegaSAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(omegaSAx,definition,
% 0.07/0.31 ( omegaSAx
% 0.07/0.31 = ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ omega )
% 0.07/0.31 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(omegaIndAx_type,type,
% 0.07/0.31 omegaIndAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(omegaIndAx,definition,
% 0.07/0.31 ( omegaIndAx
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ( ( ( in @ emptyset @ A )
% 0.07/0.31 & ! [Xx: $i] :
% 0.07/0.31 ( ( ( in @ Xx @ omega )
% 0.07/0.31 & ( in @ Xx @ A ) )
% 0.07/0.31 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.07/0.31 => ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ omega )
% 0.07/0.31 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(replAx_type,type,
% 0.07/0.31 replAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(replAx,definition,
% 0.07/0.31 ( replAx
% 0.07/0.31 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( exu
% 0.07/0.31 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.07/0.31 => ? [B: $i] :
% 0.07/0.31 ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 <=> ? [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ A )
% 0.07/0.31 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(foundationAx_type,type,
% 0.07/0.31 foundationAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(foundationAx,definition,
% 0.07/0.31 ( foundationAx
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.07/0.31 => ? [B: $i] :
% 0.07/0.31 ( ( in @ B @ A )
% 0.07/0.31 & ~ ? [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(wellorderingAx_type,type,
% 0.07/0.31 wellorderingAx: $o ).
% 0.07/0.31
% 0.07/0.31 thf(wellorderingAx,definition,
% 0.07/0.31 ( wellorderingAx
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ? [B: $i] :
% 0.07/0.31 ( ! [C: $i] :
% 0.07/0.31 ( ( in @ C @ B )
% 0.07/0.31 => ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ C )
% 0.07/0.31 => ( in @ Xx @ A ) ) )
% 0.07/0.31 & ! [Xx: $i,Xy: $i] :
% 0.07/0.31 ( ( ( in @ Xx @ A )
% 0.07/0.31 & ( in @ Xy @ A ) )
% 0.07/0.31 => ( ! [C: $i] :
% 0.07/0.31 ( ( in @ C @ B )
% 0.07/0.31 => ( ( in @ Xx @ C )
% 0.07/0.31 <=> ( in @ Xy @ C ) ) )
% 0.07/0.31 => ( Xx = Xy ) ) )
% 0.07/0.31 & ! [C: $i,D: $i] :
% 0.07/0.31 ( ( ( in @ C @ B )
% 0.07/0.31 & ( in @ D @ B ) )
% 0.07/0.31 => ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ C )
% 0.07/0.31 => ( in @ Xx @ D ) )
% 0.07/0.31 | ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ D )
% 0.07/0.31 => ( in @ Xx @ C ) ) ) )
% 0.07/0.31 & ! [C: $i] :
% 0.07/0.31 ( ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ C )
% 0.07/0.31 => ( in @ Xx @ A ) )
% 0.07/0.31 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.07/0.31 => ? [D: $i,Xx: $i] :
% 0.07/0.31 ( ( in @ D @ B )
% 0.07/0.31 & ( in @ Xx @ C )
% 0.07/0.31 & ~ ? [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ D )
% 0.07/0.31 & ( in @ Xy @ C ) )
% 0.07/0.31 & ! [E: $i] :
% 0.07/0.31 ( ( in @ E @ B )
% 0.07/0.31 => ( ! [Xy: $i] :
% 0.07/0.31 ( ( in @ Xy @ E )
% 0.07/0.31 => ( in @ Xy @ D ) )
% 0.07/0.31 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(descr_type,type,
% 0.07/0.31 descr: ( $i > $o ) > $i ).
% 0.07/0.31
% 0.07/0.31 thf(descrp_type,type,
% 0.07/0.31 descrp: $o ).
% 0.07/0.31
% 0.07/0.31 thf(descrp,definition,
% 0.07/0.31 ( descrp
% 0.07/0.31 = ( ! [Xphi: $i > $o] :
% 0.07/0.31 ( ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 => ( Xphi
% 0.07/0.31 @ ( descr
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstr_type,type,
% 0.07/0.31 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrI_type,type,
% 0.07/0.31 dsetconstrI: $o ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrI,definition,
% 0.07/0.31 ( dsetconstrI
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( ( Xphi @ Xx )
% 0.07/0.31 => ( in @ Xx
% 0.07/0.31 @ ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrEL_type,type,
% 0.07/0.31 dsetconstrEL: $o ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrEL,definition,
% 0.07/0.31 ( dsetconstrEL
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx
% 0.07/0.31 @ ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.07/0.31 => ( in @ Xx @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrER_type,type,
% 0.07/0.31 dsetconstrER: $o ).
% 0.07/0.31
% 0.07/0.31 thf(dsetconstrER,definition,
% 0.07/0.31 ( dsetconstrER
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx
% 0.07/0.31 @ ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.07/0.31 => ( Xphi @ Xx ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(exuE1_type,type,
% 0.07/0.31 exuE1: $o ).
% 0.07/0.31
% 0.07/0.31 thf(exuE1,definition,
% 0.07/0.31 ( exuE1
% 0.07/0.31 = ( ! [Xphi: $i > $o] :
% 0.07/0.31 ( ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 => ? [Xx: $i] :
% 0.07/0.31 ( ( Xphi @ Xx )
% 0.07/0.31 & ! [Xy: $i] :
% 0.07/0.31 ( ( Xphi @ Xy )
% 0.07/0.31 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(prop2set_type,type,
% 0.07/0.31 prop2set: $o > $i ).
% 0.07/0.31
% 0.07/0.31 thf(prop2setE_type,type,
% 0.07/0.31 prop2setE: $o ).
% 0.07/0.31
% 0.07/0.31 thf(prop2setE,definition,
% 0.07/0.31 ( prop2setE
% 0.07/0.31 = ( ! [Xphi: $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.07/0.31 => Xphi ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(emptysetE_type,type,
% 0.07/0.31 emptysetE: $o ).
% 0.07/0.31
% 0.07/0.31 thf(emptysetE,definition,
% 0.07/0.31 ( emptysetE
% 0.07/0.31 = ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ emptyset )
% 0.07/0.31 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(emptysetimpfalse_type,type,
% 0.07/0.31 emptysetimpfalse: $o ).
% 0.07/0.31
% 0.07/0.31 thf(emptysetimpfalse,definition,
% 0.07/0.31 ( emptysetimpfalse
% 0.07/0.31 = ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ emptyset )
% 0.07/0.31 => $false ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(notinemptyset_type,type,
% 0.07/0.31 notinemptyset: $o ).
% 0.07/0.31
% 0.07/0.31 thf(notinemptyset,definition,
% 0.07/0.31 ( notinemptyset
% 0.07/0.31 = ( ! [Xx: $i] :
% 0.07/0.31 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(exuE3e_type,type,
% 0.07/0.31 exuE3e: $o ).
% 0.07/0.31
% 0.07/0.31 thf(exuE3e,definition,
% 0.07/0.31 ( exuE3e
% 0.07/0.31 = ( ! [Xphi: $i > $o] :
% 0.07/0.31 ( ( exu
% 0.07/0.31 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.31 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setext_type,type,
% 0.07/0.31 setext: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setext,definition,
% 0.07/0.31 ( setext
% 0.07/0.31 = ( ! [A: $i,B: $i] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( in @ Xx @ B ) )
% 0.07/0.31 => ( ! [Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ B )
% 0.07/0.31 => ( in @ Xx @ A ) )
% 0.07/0.31 => ( A = B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(emptyI_type,type,
% 0.07/0.31 emptyI: $o ).
% 0.07/0.31
% 0.07/0.31 thf(emptyI,definition,
% 0.07/0.31 ( emptyI
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ~ ( in @ Xx @ A )
% 0.07/0.31 => ( A = emptyset ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(noeltsimpempty_type,type,
% 0.07/0.31 noeltsimpempty: $o ).
% 0.07/0.31
% 0.07/0.31 thf(noeltsimpempty,definition,
% 0.07/0.31 ( noeltsimpempty
% 0.07/0.31 = ( ! [A: $i] :
% 0.07/0.31 ( ! [Xx: $i] :
% 0.07/0.31 ~ ( in @ Xx @ A )
% 0.07/0.31 => ( A = emptyset ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(setbeta_type,type,
% 0.07/0.31 setbeta: $o ).
% 0.07/0.31
% 0.07/0.31 thf(setbeta,definition,
% 0.07/0.31 ( setbeta
% 0.07/0.31 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.31 ( ( in @ Xx @ A )
% 0.07/0.31 => ( ( in @ Xx
% 0.07/0.31 @ ( dsetconstr @ A
% 0.07/0.31 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.07/0.31 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(nonempty_type,type,
% 0.07/0.31 nonempty: $i > $o ).
% 0.07/0.31
% 0.07/0.31 thf(nonempty,definition,
% 0.07/0.31 ( nonempty
% 0.07/0.31 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(nonemptyE1_type,type,
% 0.07/0.32 nonemptyE1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(nonemptyE1,definition,
% 0.07/0.32 ( nonemptyE1
% 0.07/0.32 = ( ! [A: $i] :
% 0.07/0.32 ( ( nonempty @ A )
% 0.07/0.32 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(nonemptyI_type,type,
% 0.07/0.32 nonemptyI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(nonemptyI,definition,
% 0.07/0.32 ( nonemptyI
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( Xphi @ Xx )
% 0.07/0.32 => ( nonempty
% 0.07/0.32 @ ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(nonemptyI1_type,type,
% 0.07/0.32 nonemptyI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(nonemptyI1,definition,
% 0.07/0.32 ( nonemptyI1
% 0.07/0.32 = ( ! [A: $i] :
% 0.07/0.32 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.07/0.32 => ( nonempty @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinIL_type,type,
% 0.07/0.32 setadjoinIL: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinIL,definition,
% 0.07/0.32 ( setadjoinIL
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(emptyinunitempty_type,type,
% 0.07/0.32 emptyinunitempty: $o ).
% 0.07/0.32
% 0.07/0.32 thf(emptyinunitempty,definition,
% 0.07/0.32 ( emptyinunitempty
% 0.07/0.32 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinIR_type,type,
% 0.07/0.32 setadjoinIR: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinIR,definition,
% 0.07/0.32 ( setadjoinIR
% 0.07/0.32 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinE_type,type,
% 0.07/0.32 setadjoinE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinE,definition,
% 0.07/0.32 ( setadjoinE
% 0.07/0.32 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.07/0.32 => ! [Xphi: $o] :
% 0.07/0.32 ( ( ( Xy = Xx )
% 0.07/0.32 => Xphi )
% 0.07/0.32 => ( ( ( in @ Xy @ A )
% 0.07/0.32 => Xphi )
% 0.07/0.32 => Xphi ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinOr_type,type,
% 0.07/0.32 setadjoinOr: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinOr,definition,
% 0.07/0.32 ( setadjoinOr
% 0.07/0.32 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.07/0.32 => ( ( Xy = Xx )
% 0.07/0.32 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setoftrueEq_type,type,
% 0.07/0.32 setoftrueEq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setoftrueEq,definition,
% 0.07/0.32 ( setoftrueEq
% 0.07/0.32 = ( ! [A: $i] :
% 0.07/0.32 ( ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xx: $i] : $true )
% 0.07/0.32 = A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powersetI_type,type,
% 0.07/0.32 powersetI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powersetI,definition,
% 0.07/0.32 ( powersetI
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ B )
% 0.07/0.32 => ( in @ Xx @ A ) )
% 0.07/0.32 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(emptyinPowerset_type,type,
% 0.07/0.32 emptyinPowerset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(emptyinPowerset,definition,
% 0.07/0.32 ( emptyinPowerset
% 0.07/0.32 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(emptyInPowerset_type,type,
% 0.07/0.32 emptyInPowerset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(emptyInPowerset,definition,
% 0.07/0.32 ( emptyInPowerset
% 0.07/0.32 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powersetE_type,type,
% 0.07/0.32 powersetE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powersetE,definition,
% 0.07/0.32 ( powersetE
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ B @ ( powerset @ A ) )
% 0.07/0.32 => ( ( in @ Xx @ B )
% 0.07/0.32 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setunionI_type,type,
% 0.07/0.32 setunionI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setunionI,definition,
% 0.07/0.32 ( setunionI
% 0.07/0.32 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.07/0.32 ( ( in @ Xx @ B )
% 0.07/0.32 => ( ( in @ B @ A )
% 0.07/0.32 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setunionE_type,type,
% 0.07/0.32 setunionE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setunionE,definition,
% 0.07/0.32 ( setunionE
% 0.07/0.32 = ( ! [A: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( setunion @ A ) )
% 0.07/0.32 => ! [Xphi: $o] :
% 0.07/0.32 ( ! [B: $i] :
% 0.07/0.32 ( ( in @ Xx @ B )
% 0.07/0.32 => ( ( in @ B @ A )
% 0.07/0.32 => Xphi ) )
% 0.07/0.32 => Xphi ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subPowSU_type,type,
% 0.07/0.32 subPowSU: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subPowSU,definition,
% 0.07/0.32 ( subPowSU
% 0.07/0.32 = ( ! [A: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(exuE2_type,type,
% 0.07/0.32 exuE2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(exuE2,definition,
% 0.07/0.32 ( exuE2
% 0.07/0.32 = ( ! [Xphi: $i > $o] :
% 0.07/0.32 ( ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 => ? [Xx: $i] :
% 0.07/0.32 ! [Xy: $i] :
% 0.07/0.32 ( ( Xphi @ Xy )
% 0.07/0.32 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(nonemptyImpWitness_type,type,
% 0.07/0.32 nonemptyImpWitness: $o ).
% 0.07/0.32
% 0.07/0.32 thf(nonemptyImpWitness,definition,
% 0.07/0.32 ( nonemptyImpWitness
% 0.07/0.32 = ( ! [A: $i] :
% 0.07/0.32 ( ( nonempty @ A )
% 0.07/0.32 => ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & $true ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(uniqinunit_type,type,
% 0.07/0.32 uniqinunit: $o ).
% 0.07/0.32
% 0.07/0.32 thf(uniqinunit,definition,
% 0.07/0.32 ( uniqinunit
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.07/0.32 => ( Xx = Xy ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(notinsingleton_type,type,
% 0.07/0.32 notinsingleton: $o ).
% 0.07/0.32
% 0.07/0.32 thf(notinsingleton,definition,
% 0.07/0.32 ( notinsingleton
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xx != Xy )
% 0.07/0.32 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(eqinunit_type,type,
% 0.07/0.32 eqinunit: $o ).
% 0.07/0.32
% 0.07/0.32 thf(eqinunit,definition,
% 0.07/0.32 ( eqinunit
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xx = Xy )
% 0.07/0.32 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(singletonsswitch_type,type,
% 0.07/0.32 singletonsswitch: $o ).
% 0.07/0.32
% 0.07/0.32 thf(singletonsswitch,definition,
% 0.07/0.32 ( singletonsswitch
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.07/0.32 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(upairsetE_type,type,
% 0.07/0.32 upairsetE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(upairsetE,definition,
% 0.07/0.32 ( upairsetE
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.07/0.32 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.07/0.32 => ( ( Xz = Xx )
% 0.07/0.32 | ( Xz = Xy ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(upairsetIL_type,type,
% 0.07/0.32 upairsetIL: $o ).
% 0.07/0.32
% 0.07/0.32 thf(upairsetIL,definition,
% 0.07/0.32 ( upairsetIL
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(upairsetIR_type,type,
% 0.07/0.32 upairsetIR: $o ).
% 0.07/0.32
% 0.07/0.32 thf(upairsetIR,definition,
% 0.07/0.32 ( upairsetIR
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(emptyE1_type,type,
% 0.07/0.32 emptyE1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(emptyE1,definition,
% 0.07/0.32 ( emptyE1
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ( Xphi @ Xx ) )
% 0.07/0.32 => ( ( ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 = emptyset )
% 0.07/0.32 => $false ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(vacuousDall_type,type,
% 0.07/0.32 vacuousDall: $o ).
% 0.07/0.32
% 0.07/0.32 thf(vacuousDall,definition,
% 0.07/0.32 ( vacuousDall
% 0.07/0.32 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ emptyset )
% 0.07/0.32 => ( Xphi @ Xx ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(quantDeMorgan1_type,type,
% 0.07/0.32 quantDeMorgan1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(quantDeMorgan1,definition,
% 0.07/0.32 ( quantDeMorgan1
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ~ ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( Xphi @ Xx ) )
% 0.07/0.32 => ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(quantDeMorgan2_type,type,
% 0.07/0.32 quantDeMorgan2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(quantDeMorgan2,definition,
% 0.07/0.32 ( quantDeMorgan2
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ~ ( Xphi @ Xx ) )
% 0.07/0.32 => ~ ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(quantDeMorgan3_type,type,
% 0.07/0.32 quantDeMorgan3: $o ).
% 0.07/0.32
% 0.07/0.32 thf(quantDeMorgan3,definition,
% 0.07/0.32 ( quantDeMorgan3
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ~ ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ( Xphi @ Xx ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(quantDeMorgan4_type,type,
% 0.07/0.32 quantDeMorgan4: $o ).
% 0.07/0.32
% 0.07/0.32 thf(quantDeMorgan4,definition,
% 0.07/0.32 ( quantDeMorgan4
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ~ ( Xphi @ Xx ) )
% 0.07/0.32 => ~ ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(prop2setI_type,type,
% 0.07/0.32 prop2setI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(prop2setI,definition,
% 0.07/0.32 ( prop2setI
% 0.07/0.32 = ( ! [Xphi: $o] :
% 0.07/0.32 ( Xphi
% 0.07/0.32 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(set2prop_type,type,
% 0.07/0.32 set2prop: $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(prop2set2propI_type,type,
% 0.07/0.32 prop2set2propI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(prop2set2propI,definition,
% 0.07/0.32 ( prop2set2propI
% 0.07/0.32 = ( ! [Xphi: $o] :
% 0.07/0.32 ( Xphi
% 0.07/0.32 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(notdexE_type,type,
% 0.07/0.32 notdexE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(notdexE,definition,
% 0.07/0.32 ( notdexE
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ~ ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ( Xphi @ Xx ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(notdallE_type,type,
% 0.07/0.32 notdallE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(notdallE,definition,
% 0.07/0.32 ( notdallE
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ~ ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( Xphi @ Xx ) )
% 0.07/0.32 => ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(exuI1_type,type,
% 0.07/0.32 exuI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(exuI1,definition,
% 0.07/0.32 ( exuI1
% 0.07/0.32 = ( ! [Xphi: $i > $o] :
% 0.07/0.32 ( ? [Xx: $i] :
% 0.07/0.32 ( ( Xphi @ Xx )
% 0.07/0.32 & ! [Xy: $i] :
% 0.07/0.32 ( ( Xphi @ Xy )
% 0.07/0.32 => ( Xx = Xy ) ) )
% 0.07/0.32 => ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(exuI3_type,type,
% 0.07/0.32 exuI3: $o ).
% 0.07/0.32
% 0.07/0.32 thf(exuI3,definition,
% 0.07/0.32 ( exuI3
% 0.07/0.32 = ( ! [Xphi: $i > $o] :
% 0.07/0.32 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.07/0.32 => ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xphi @ Xx )
% 0.07/0.32 => ( ( Xphi @ Xy )
% 0.07/0.32 => ( Xx = Xy ) ) )
% 0.07/0.32 => ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(exuI2_type,type,
% 0.07/0.32 exuI2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(exuI2,definition,
% 0.07/0.32 ( exuI2
% 0.07/0.32 = ( ! [Xphi: $i > $o] :
% 0.07/0.32 ( ? [Xx: $i] :
% 0.07/0.32 ! [Xy: $i] :
% 0.07/0.32 ( ( Xphi @ Xy )
% 0.07/0.32 <=> ( Xy = Xx ) )
% 0.07/0.32 => ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(inCongP_type,type,
% 0.07/0.32 inCongP: $o ).
% 0.07/0.32
% 0.07/0.32 thf(inCongP,definition,
% 0.07/0.32 ( inCongP
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( A = B )
% 0.07/0.32 => ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xx = Xy )
% 0.07/0.32 => ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(in__Cong_type,type,
% 0.07/0.32 in__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(in__Cong,definition,
% 0.07/0.32 ( in__Cong
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( A = B )
% 0.07/0.32 => ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xx = Xy )
% 0.07/0.32 => ( ( in @ Xx @ A )
% 0.07/0.32 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(exuE3u_type,type,
% 0.07/0.32 exuE3u: $o ).
% 0.07/0.32
% 0.07/0.32 thf(exuE3u,definition,
% 0.07/0.32 ( exuE3u
% 0.07/0.32 = ( ! [Xphi: $i > $o] :
% 0.07/0.32 ( ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 => ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xphi @ Xx )
% 0.07/0.32 => ( ( Xphi @ Xy )
% 0.07/0.32 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(exu__Cong_type,type,
% 0.07/0.32 exu__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(exu__Cong,definition,
% 0.07/0.32 ( exu__Cong
% 0.07/0.32 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.07/0.32 ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xx = Xy )
% 0.07/0.32 => ( ( Xphi @ Xx )
% 0.07/0.32 <=> ( Xpsi @ Xy ) ) )
% 0.07/0.32 => ( ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 <=> ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(emptyset__Cong_type,type,
% 0.07/0.32 emptyset__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(emptyset__Cong,definition,
% 0.07/0.32 ( emptyset__Cong
% 0.07/0.32 = ( emptyset = emptyset ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoin__Cong_type,type,
% 0.07/0.32 setadjoin__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoin__Cong,definition,
% 0.07/0.32 ( setadjoin__Cong
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xx = Xy )
% 0.07/0.32 => ! [Xz: $i,Xu: $i] :
% 0.07/0.32 ( ( Xz = Xu )
% 0.07/0.32 => ( ( setadjoin @ Xx @ Xz )
% 0.07/0.32 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powerset__Cong_type,type,
% 0.07/0.32 powerset__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powerset__Cong,definition,
% 0.07/0.32 ( powerset__Cong
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( A = B )
% 0.07/0.32 => ( ( powerset @ A )
% 0.07/0.32 = ( powerset @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setunion__Cong_type,type,
% 0.07/0.32 setunion__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setunion__Cong,definition,
% 0.07/0.32 ( setunion__Cong
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( A = B )
% 0.07/0.32 => ( ( setunion @ A )
% 0.07/0.32 = ( setunion @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(omega__Cong_type,type,
% 0.07/0.32 omega__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(omega__Cong,definition,
% 0.07/0.32 ( omega__Cong
% 0.07/0.32 = ( omega = omega ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(exuEu_type,type,
% 0.07/0.32 exuEu: $o ).
% 0.07/0.32
% 0.07/0.32 thf(exuEu,definition,
% 0.07/0.32 ( exuEu
% 0.07/0.32 = ( ! [Xphi: $i > $o] :
% 0.07/0.32 ( ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 => ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xphi @ Xx )
% 0.07/0.32 => ( ( Xphi @ Xy )
% 0.07/0.32 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(descr__Cong_type,type,
% 0.07/0.32 descr__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(descr__Cong,definition,
% 0.07/0.32 ( descr__Cong
% 0.07/0.32 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.07/0.32 ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xx = Xy )
% 0.07/0.32 => ( ( Xphi @ Xx )
% 0.07/0.32 <=> ( Xpsi @ Xy ) ) )
% 0.07/0.32 => ( ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 => ( ( exu
% 0.07/0.32 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.07/0.32 => ( ( descr
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 = ( descr
% 0.07/0.32 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(dsetconstr__Cong_type,type,
% 0.07/0.32 dsetconstr__Cong: $o ).
% 0.07/0.32
% 0.07/0.32 thf(dsetconstr__Cong,definition,
% 0.07/0.32 ( dsetconstr__Cong
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( A = B )
% 0.07/0.32 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( Xx = Xy )
% 0.07/0.32 => ( ( Xphi @ Xx )
% 0.07/0.32 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.07/0.32 => ( ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 = ( dsetconstr @ B
% 0.07/0.32 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subset_type,type,
% 0.07/0.32 subset: $i > $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(disjoint_type,type,
% 0.07/0.32 disjoint: $i > $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(setsmeet_type,type,
% 0.07/0.32 setsmeet: $i > $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetI1_type,type,
% 0.07/0.32 subsetI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetI1,definition,
% 0.07/0.32 ( subsetI1
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ Xx @ B ) )
% 0.07/0.32 => ( subset @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(eqimpsubset2_type,type,
% 0.07/0.32 eqimpsubset2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(eqimpsubset2,definition,
% 0.07/0.32 ( eqimpsubset2
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( A = B )
% 0.07/0.32 => ( subset @ B @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(eqimpsubset1_type,type,
% 0.07/0.32 eqimpsubset1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(eqimpsubset1,definition,
% 0.07/0.32 ( eqimpsubset1
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( A = B )
% 0.07/0.32 => ( subset @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subsetI2_type,type,
% 0.07/0.32 subsetI2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetI2,definition,
% 0.07/0.32 ( subsetI2
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ Xx @ B ) )
% 0.07/0.32 => ( subset @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(emptysetsubset_type,type,
% 0.07/0.32 emptysetsubset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(emptysetsubset,definition,
% 0.07/0.32 ( emptysetsubset
% 0.07/0.32 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subsetE_type,type,
% 0.07/0.32 subsetE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetE,definition,
% 0.07/0.32 ( subsetE
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subsetE2_type,type,
% 0.07/0.32 subsetE2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetE2,definition,
% 0.07/0.32 ( subsetE2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( ~ ( in @ Xx @ B )
% 0.07/0.32 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(notsubsetI_type,type,
% 0.07/0.32 notsubsetI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(notsubsetI,definition,
% 0.07/0.32 ( notsubsetI
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ B )
% 0.07/0.32 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(notequalI1_type,type,
% 0.07/0.32 notequalI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(notequalI1,definition,
% 0.07/0.32 ( notequalI1
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ~ ( subset @ A @ B )
% 0.07/0.32 => ( A != B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(notequalI2_type,type,
% 0.07/0.32 notequalI2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(notequalI2,definition,
% 0.07/0.32 ( notequalI2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ B )
% 0.07/0.32 => ( A != B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subsetRefl_type,type,
% 0.07/0.32 subsetRefl: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetRefl,definition,
% 0.07/0.32 ( subsetRefl
% 0.07/0.32 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subsetTrans_type,type,
% 0.07/0.32 subsetTrans: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetTrans,definition,
% 0.07/0.32 ( subsetTrans
% 0.07/0.32 = ( ! [A: $i,B: $i,C: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( ( subset @ B @ C )
% 0.07/0.32 => ( subset @ A @ C ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinSub_type,type,
% 0.07/0.32 setadjoinSub: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinSub,definition,
% 0.07/0.32 ( setadjoinSub
% 0.07/0.32 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinSub2_type,type,
% 0.07/0.32 setadjoinSub2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setadjoinSub2,definition,
% 0.07/0.32 ( setadjoinSub2
% 0.07/0.32 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subset2powerset_type,type,
% 0.07/0.32 subset2powerset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subset2powerset,definition,
% 0.07/0.32 ( subset2powerset
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setextsub_type,type,
% 0.07/0.32 setextsub: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setextsub,definition,
% 0.07/0.32 ( setextsub
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( ( subset @ B @ A )
% 0.07/0.32 => ( A = B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subsetemptysetimpeq_type,type,
% 0.07/0.32 subsetemptysetimpeq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetemptysetimpeq,definition,
% 0.07/0.32 ( subsetemptysetimpeq
% 0.07/0.32 = ( ! [A: $i] :
% 0.07/0.32 ( ( subset @ A @ emptyset )
% 0.07/0.32 => ( A = emptyset ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powersetI1_type,type,
% 0.07/0.32 powersetI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powersetI1,definition,
% 0.07/0.32 ( powersetI1
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( subset @ B @ A )
% 0.07/0.32 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powersetE1_type,type,
% 0.07/0.32 powersetE1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powersetE1,definition,
% 0.07/0.32 ( powersetE1
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( in @ B @ ( powerset @ A ) )
% 0.07/0.32 => ( subset @ B @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(inPowerset_type,type,
% 0.07/0.32 inPowerset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(inPowerset,definition,
% 0.07/0.32 ( inPowerset
% 0.07/0.32 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powersetsubset_type,type,
% 0.07/0.32 powersetsubset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powersetsubset,definition,
% 0.07/0.32 ( powersetsubset
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(sepInPowerset_type,type,
% 0.07/0.32 sepInPowerset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(sepInPowerset,definition,
% 0.07/0.32 ( sepInPowerset
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( in
% 0.07/0.32 @ ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 @ ( powerset @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(sepSubset_type,type,
% 0.07/0.32 sepSubset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(sepSubset,definition,
% 0.07/0.32 ( sepSubset
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( subset
% 0.07/0.32 @ ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunion_type,type,
% 0.07/0.32 binunion: $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(binunionIL_type,type,
% 0.07/0.32 binunionIL: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionIL,definition,
% 0.07/0.32 ( binunionIL
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(upairset2IR_type,type,
% 0.07/0.32 upairset2IR: $o ).
% 0.07/0.32
% 0.07/0.32 thf(upairset2IR,definition,
% 0.07/0.32 ( upairset2IR
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunionIR_type,type,
% 0.07/0.32 binunionIR: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionIR,definition,
% 0.07/0.32 ( binunionIR
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ B )
% 0.07/0.32 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunionEcases_type,type,
% 0.07/0.32 binunionEcases: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionEcases,definition,
% 0.07/0.32 ( binunionEcases
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.07/0.32 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.07/0.32 => ( ( ( in @ Xx @ A )
% 0.07/0.32 => Xphi )
% 0.07/0.32 => ( ( ( in @ Xx @ B )
% 0.07/0.32 => Xphi )
% 0.07/0.32 => Xphi ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunionE_type,type,
% 0.07/0.32 binunionE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionE,definition,
% 0.07/0.32 ( binunionE
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.07/0.32 => ( ( in @ Xx @ A )
% 0.07/0.32 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunionLsub_type,type,
% 0.07/0.32 binunionLsub: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionLsub,definition,
% 0.07/0.32 ( binunionLsub
% 0.07/0.32 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunionRsub_type,type,
% 0.07/0.32 binunionRsub: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionRsub,definition,
% 0.07/0.32 ( binunionRsub
% 0.07/0.32 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersect_type,type,
% 0.07/0.32 binintersect: $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectI_type,type,
% 0.07/0.32 binintersectI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectI,definition,
% 0.07/0.32 ( binintersectI
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ B )
% 0.07/0.32 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset5_type,type,
% 0.07/0.32 binintersectSubset5: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset5,definition,
% 0.07/0.32 ( binintersectSubset5
% 0.07/0.32 = ( ! [A: $i,B: $i,C: $i] :
% 0.07/0.32 ( ( subset @ C @ A )
% 0.07/0.32 => ( ( subset @ C @ B )
% 0.07/0.32 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectEL_type,type,
% 0.07/0.32 binintersectEL: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectEL,definition,
% 0.07/0.32 ( binintersectEL
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.07/0.32 => ( in @ Xx @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectLsub_type,type,
% 0.07/0.32 binintersectLsub: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectLsub,definition,
% 0.07/0.32 ( binintersectLsub
% 0.07/0.32 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset2_type,type,
% 0.07/0.32 binintersectSubset2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset2,definition,
% 0.07/0.32 ( binintersectSubset2
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( ( binintersect @ A @ B )
% 0.07/0.32 = A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset3_type,type,
% 0.07/0.32 binintersectSubset3: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset3,definition,
% 0.07/0.32 ( binintersectSubset3
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( ( binintersect @ A @ B )
% 0.07/0.32 = B )
% 0.07/0.32 => ( subset @ B @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectER_type,type,
% 0.07/0.32 binintersectER: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectER,definition,
% 0.07/0.32 ( binintersectER
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.07/0.32 => ( in @ Xx @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(disjointsetsI1_type,type,
% 0.07/0.32 disjointsetsI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(disjointsetsI1,definition,
% 0.07/0.32 ( disjointsetsI1
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ~ ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ( in @ Xx @ B ) )
% 0.07/0.32 => ( ( binintersect @ A @ B )
% 0.07/0.32 = emptyset ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectRsub_type,type,
% 0.07/0.32 binintersectRsub: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectRsub,definition,
% 0.07/0.32 ( binintersectRsub
% 0.07/0.32 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset4_type,type,
% 0.07/0.32 binintersectSubset4: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset4,definition,
% 0.07/0.32 ( binintersectSubset4
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( subset @ B @ A )
% 0.07/0.32 => ( ( binintersect @ A @ B )
% 0.07/0.32 = B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset1_type,type,
% 0.07/0.32 binintersectSubset1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectSubset1,definition,
% 0.07/0.32 ( binintersectSubset1
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( ( binintersect @ A @ B )
% 0.07/0.32 = A )
% 0.07/0.32 => ( subset @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(bs114d_type,type,
% 0.07/0.32 bs114d: $o ).
% 0.07/0.32
% 0.07/0.32 thf(bs114d,definition,
% 0.07/0.32 ( bs114d
% 0.07/0.32 = ( ! [A: $i,B: $i,C: $i] :
% 0.07/0.32 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.07/0.32 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(regular_type,type,
% 0.07/0.32 regular: $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminus_type,type,
% 0.07/0.32 setminus: $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(setminusI_type,type,
% 0.07/0.32 setminusI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusI,definition,
% 0.07/0.32 ( setminusI
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ B )
% 0.07/0.32 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusEL_type,type,
% 0.07/0.32 setminusEL: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusEL,definition,
% 0.07/0.32 ( setminusEL
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.07/0.32 => ( in @ Xx @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusER_type,type,
% 0.07/0.32 setminusER: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusER,definition,
% 0.07/0.32 ( setminusER
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.07/0.32 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusSubset2_type,type,
% 0.07/0.32 setminusSubset2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusSubset2,definition,
% 0.07/0.32 ( setminusSubset2
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( subset @ A @ B )
% 0.07/0.32 => ( ( setminus @ A @ B )
% 0.07/0.32 = emptyset ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusERneg_type,type,
% 0.07/0.32 setminusERneg: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusERneg,definition,
% 0.07/0.32 ( setminusERneg
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.07/0.32 => ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusELneg_type,type,
% 0.07/0.32 setminusELneg: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusELneg,definition,
% 0.07/0.32 ( setminusELneg
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.07/0.32 => ( ~ ( in @ Xx @ B )
% 0.07/0.32 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusILneg_type,type,
% 0.07/0.32 setminusILneg: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusILneg,definition,
% 0.07/0.32 ( setminusILneg
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ~ ( in @ Xx @ A )
% 0.07/0.32 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusIRneg_type,type,
% 0.07/0.32 setminusIRneg: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusIRneg,definition,
% 0.07/0.32 ( setminusIRneg
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ B )
% 0.07/0.32 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusLsub_type,type,
% 0.07/0.32 setminusLsub: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusLsub,definition,
% 0.07/0.32 ( setminusLsub
% 0.07/0.32 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusSubset1_type,type,
% 0.07/0.32 setminusSubset1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusSubset1,definition,
% 0.07/0.32 ( setminusSubset1
% 0.07/0.32 = ( ! [A: $i,B: $i] :
% 0.07/0.32 ( ( ( setminus @ A @ B )
% 0.07/0.32 = emptyset )
% 0.07/0.32 => ( subset @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(symdiff_type,type,
% 0.07/0.32 symdiff: $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffE_type,type,
% 0.07/0.32 symdiffE: $o ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffE,definition,
% 0.07/0.32 ( symdiffE
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.07/0.32 => ! [Xphi: $o] :
% 0.07/0.32 ( ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ B )
% 0.07/0.32 => Xphi ) )
% 0.07/0.32 => ( ( ~ ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ B )
% 0.07/0.32 => Xphi ) )
% 0.07/0.32 => Xphi ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffI1_type,type,
% 0.07/0.32 symdiffI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffI1,definition,
% 0.07/0.32 ( symdiffI1
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ B )
% 0.07/0.32 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffI2_type,type,
% 0.07/0.32 symdiffI2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffI2,definition,
% 0.07/0.32 ( symdiffI2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ~ ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ B )
% 0.07/0.32 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffIneg1_type,type,
% 0.07/0.32 symdiffIneg1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffIneg1,definition,
% 0.07/0.32 ( symdiffIneg1
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ B )
% 0.07/0.32 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffIneg2_type,type,
% 0.07/0.32 symdiffIneg2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(symdiffIneg2,definition,
% 0.07/0.32 ( symdiffIneg2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ~ ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ B )
% 0.07/0.32 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(iskpair_type,type,
% 0.07/0.32 iskpair: $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(secondinupair_type,type,
% 0.07/0.32 secondinupair: $o ).
% 0.07/0.32
% 0.07/0.32 thf(secondinupair,definition,
% 0.07/0.32 ( secondinupair
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairIL_type,type,
% 0.07/0.32 setukpairIL: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairIL,definition,
% 0.07/0.32 ( setukpairIL
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairIR_type,type,
% 0.07/0.32 setukpairIR: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairIR,definition,
% 0.07/0.32 ( setukpairIR
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(kpairiskpair_type,type,
% 0.07/0.32 kpairiskpair: $o ).
% 0.07/0.32
% 0.07/0.32 thf(kpairiskpair,definition,
% 0.07/0.32 ( kpairiskpair
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(kpair_type,type,
% 0.07/0.32 kpair: $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(kpairp_type,type,
% 0.07/0.32 kpairp: $o ).
% 0.07/0.32
% 0.07/0.32 thf(kpairp,definition,
% 0.07/0.32 ( kpairp
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(cartprod_type,type,
% 0.07/0.32 cartprod: $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(singletonsubset_type,type,
% 0.07/0.32 singletonsubset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(singletonsubset,definition,
% 0.07/0.32 ( singletonsubset
% 0.07/0.32 = ( ! [A: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(singletoninpowerset_type,type,
% 0.07/0.32 singletoninpowerset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(singletoninpowerset,definition,
% 0.07/0.32 ( singletoninpowerset
% 0.07/0.32 = ( ! [A: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(singletoninpowunion_type,type,
% 0.07/0.32 singletoninpowunion: $o ).
% 0.07/0.32
% 0.07/0.32 thf(singletoninpowunion,definition,
% 0.07/0.32 ( singletoninpowunion
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(upairset2E_type,type,
% 0.07/0.32 upairset2E: $o ).
% 0.07/0.32
% 0.07/0.32 thf(upairset2E,definition,
% 0.07/0.32 ( upairset2E
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.07/0.32 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.07/0.32 => ( ( Xz = Xx )
% 0.07/0.32 | ( Xz = Xy ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(upairsubunion_type,type,
% 0.07/0.32 upairsubunion: $o ).
% 0.07/0.32
% 0.07/0.32 thf(upairsubunion,definition,
% 0.07/0.32 ( upairsubunion
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(upairinpowunion_type,type,
% 0.07/0.32 upairinpowunion: $o ).
% 0.07/0.32
% 0.07/0.32 thf(upairinpowunion,definition,
% 0.07/0.32 ( upairinpowunion
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ubforcartprodlem1_type,type,
% 0.07/0.32 ubforcartprodlem1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ubforcartprodlem1,definition,
% 0.07/0.32 ( ubforcartprodlem1
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ubforcartprodlem2_type,type,
% 0.07/0.32 ubforcartprodlem2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ubforcartprodlem2,definition,
% 0.07/0.32 ( ubforcartprodlem2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ubforcartprodlem3_type,type,
% 0.07/0.32 ubforcartprodlem3: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ubforcartprodlem3,definition,
% 0.07/0.32 ( ubforcartprodlem3
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodpairin_type,type,
% 0.07/0.32 cartprodpairin: $o ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodpairin,definition,
% 0.07/0.32 ( cartprodpairin
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodmempair1_type,type,
% 0.07/0.32 cartprodmempair1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodmempair1,definition,
% 0.07/0.32 ( cartprodmempair1
% 0.07/0.32 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.07/0.32 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.07/0.32 => ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ? [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 & ( Xu
% 0.07/0.32 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodmempair_type,type,
% 0.07/0.32 cartprodmempair: $o ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodmempair,definition,
% 0.07/0.32 ( cartprodmempair
% 0.07/0.32 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.07/0.32 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.07/0.32 => ( iskpair @ Xu ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setunionE2_type,type,
% 0.07/0.32 setunionE2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setunionE2,definition,
% 0.07/0.32 ( setunionE2
% 0.07/0.32 = ( ! [A: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ ( setunion @ A ) )
% 0.07/0.32 => ? [X: $i] :
% 0.07/0.32 ( ( in @ X @ A )
% 0.07/0.32 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setunionsingleton1_type,type,
% 0.07/0.32 setunionsingleton1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setunionsingleton1,definition,
% 0.07/0.32 ( setunionsingleton1
% 0.07/0.32 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setunionsingleton2_type,type,
% 0.07/0.32 setunionsingleton2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setunionsingleton2,definition,
% 0.07/0.32 ( setunionsingleton2
% 0.07/0.32 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setunionsingleton_type,type,
% 0.07/0.32 setunionsingleton: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setunionsingleton,definition,
% 0.07/0.32 ( setunionsingleton
% 0.07/0.32 = ( ! [Xx: $i] :
% 0.07/0.32 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.07/0.32 = Xx ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(singleton_type,type,
% 0.07/0.32 singleton: $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(singletonprop_type,type,
% 0.07/0.32 singletonprop: $o ).
% 0.07/0.32
% 0.07/0.32 thf(singletonprop,definition,
% 0.07/0.32 ( singletonprop
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( ( Xphi @ Xx )
% 0.07/0.32 => ( ( Xphi @ Xy )
% 0.07/0.32 => ( Xx = Xy ) ) ) ) )
% 0.07/0.32 => ( ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ( Xphi @ Xx ) )
% 0.07/0.32 => ( singleton
% 0.07/0.32 @ ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ex1_type,type,
% 0.07/0.32 ex1: $i > ( $i > $o ) > $o ).
% 0.07/0.32
% 0.07/0.32 thf(ex1,definition,
% 0.07/0.32 ( ex1
% 0.07/0.32 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( singleton
% 0.07/0.32 @ ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ex1E1_type,type,
% 0.07/0.32 ex1E1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ex1E1,definition,
% 0.07/0.32 ( ex1E1
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ( ex1 @ A
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 => ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ex1I_type,type,
% 0.07/0.32 ex1I: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ex1I,definition,
% 0.07/0.32 ( ex1I
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( Xphi @ Xx )
% 0.07/0.32 => ( ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( ( Xphi @ Xy )
% 0.07/0.32 => ( Xy = Xx ) ) )
% 0.07/0.32 => ( ex1 @ A
% 0.07/0.32 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ex1I2_type,type,
% 0.07/0.32 ex1I2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ex1I2,definition,
% 0.07/0.32 ( ex1I2
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( ( Xphi @ Xx )
% 0.07/0.32 => ( ( Xphi @ Xy )
% 0.07/0.32 => ( Xx = Xy ) ) ) ) )
% 0.07/0.32 => ( ? [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 & ( Xphi @ Xx ) )
% 0.07/0.32 => ( ex1 @ A
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(singletonsuniq_type,type,
% 0.07/0.32 singletonsuniq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(singletonsuniq,definition,
% 0.07/0.32 ( singletonsuniq
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( ( setadjoin @ Xx @ emptyset )
% 0.07/0.32 = ( setadjoin @ Xy @ emptyset ) )
% 0.07/0.32 => ( Xx = Xy ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(atmost1p_type,type,
% 0.07/0.32 atmost1p: $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(atleast2p_type,type,
% 0.07/0.32 atleast2p: $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(atmost2p_type,type,
% 0.07/0.32 atmost2p: $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(upairsetp_type,type,
% 0.07/0.32 upairsetp: $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairinjL1_type,type,
% 0.07/0.32 setukpairinjL1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairinjL1,definition,
% 0.07/0.32 ( setukpairinjL1
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.07/0.32 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.07/0.32 => ( Xx = Xz ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(kfstsingleton_type,type,
% 0.07/0.32 kfstsingleton: $o ).
% 0.07/0.32
% 0.07/0.32 thf(kfstsingleton,definition,
% 0.07/0.32 ( kfstsingleton
% 0.07/0.32 = ( ! [Xu: $i] :
% 0.07/0.32 ( ( iskpair @ Xu )
% 0.07/0.32 => ( singleton
% 0.07/0.32 @ ( dsetconstr @ ( setunion @ Xu )
% 0.07/0.32 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(theprop_type,type,
% 0.07/0.32 theprop: $o ).
% 0.07/0.32
% 0.07/0.32 thf(theprop,definition,
% 0.07/0.32 ( theprop
% 0.07/0.32 = ( ! [X: $i] :
% 0.07/0.32 ( ( singleton @ X )
% 0.07/0.32 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(kfst_type,type,
% 0.07/0.32 kfst: $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(kfstpairEq_type,type,
% 0.07/0.32 kfstpairEq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(kfstpairEq,definition,
% 0.07/0.32 ( kfstpairEq
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.07/0.32 = Xx ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodfstin_type,type,
% 0.07/0.32 cartprodfstin: $o ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodfstin,definition,
% 0.07/0.32 ( cartprodfstin
% 0.07/0.32 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.07/0.32 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.07/0.32 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairinjL2_type,type,
% 0.07/0.32 setukpairinjL2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairinjL2,definition,
% 0.07/0.32 ( setukpairinjL2
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.07/0.32 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.07/0.32 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.07/0.32 => ( Xx = Xz ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairinjL_type,type,
% 0.07/0.32 setukpairinjL: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairinjL,definition,
% 0.07/0.32 ( setukpairinjL
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.07/0.32 ( ( ( kpair @ Xx @ Xy )
% 0.07/0.32 = ( kpair @ Xz @ Xu ) )
% 0.07/0.32 => ( Xx = Xz ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairinjR11_type,type,
% 0.07/0.32 setukpairinjR11: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairinjR11,definition,
% 0.07/0.32 ( setukpairinjR11
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xx = Xy )
% 0.07/0.32 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.07/0.32 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairinjR12_type,type,
% 0.07/0.32 setukpairinjR12: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairinjR12,definition,
% 0.07/0.32 ( setukpairinjR12
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xx = Xy )
% 0.07/0.32 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.07/0.32 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairinjR1_type,type,
% 0.07/0.32 setukpairinjR1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairinjR1,definition,
% 0.07/0.32 ( setukpairinjR1
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.07/0.32 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.07/0.32 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.07/0.32 => ( ( Xz = Xu )
% 0.07/0.32 => ( Xy = Xu ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(upairequniteq_type,type,
% 0.07/0.32 upairequniteq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(upairequniteq,definition,
% 0.07/0.32 ( upairequniteq
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.07/0.32 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.07/0.32 = ( setadjoin @ Xz @ emptyset ) )
% 0.07/0.32 => ( Xx = Xy ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairinjR2_type,type,
% 0.07/0.32 setukpairinjR2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairinjR2,definition,
% 0.07/0.32 ( setukpairinjR2
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.07/0.32 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.07/0.32 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.07/0.32 => ( Xy = Xu ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairinjR_type,type,
% 0.07/0.32 setukpairinjR: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setukpairinjR,definition,
% 0.07/0.32 ( setukpairinjR
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.07/0.32 ( ( ( kpair @ Xx @ Xy )
% 0.07/0.32 = ( kpair @ Xz @ Xu ) )
% 0.07/0.32 => ( Xy = Xu ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ksndsingleton_type,type,
% 0.07/0.32 ksndsingleton: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ksndsingleton,definition,
% 0.07/0.32 ( ksndsingleton
% 0.07/0.32 = ( ! [Xu: $i] :
% 0.07/0.32 ( ( iskpair @ Xu )
% 0.07/0.32 => ( singleton
% 0.07/0.32 @ ( dsetconstr @ ( setunion @ Xu )
% 0.07/0.32 @ ^ [Xx: $i] :
% 0.07/0.32 ( Xu
% 0.07/0.32 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ksnd_type,type,
% 0.07/0.32 ksnd: $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(ksndpairEq_type,type,
% 0.07/0.32 ksndpairEq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ksndpairEq,definition,
% 0.07/0.32 ( ksndpairEq
% 0.07/0.32 = ( ! [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.07/0.32 = Xy ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(kpairsurjEq_type,type,
% 0.07/0.32 kpairsurjEq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(kpairsurjEq,definition,
% 0.07/0.32 ( kpairsurjEq
% 0.07/0.32 = ( ! [Xu: $i] :
% 0.07/0.32 ( ( iskpair @ Xu )
% 0.07/0.32 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.07/0.32 = Xu ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodsndin_type,type,
% 0.07/0.32 cartprodsndin: $o ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodsndin,definition,
% 0.07/0.32 ( cartprodsndin
% 0.07/0.32 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.07/0.32 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.07/0.32 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodpairmemEL_type,type,
% 0.07/0.32 cartprodpairmemEL: $o ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodpairmemEL,definition,
% 0.07/0.32 ( cartprodpairmemEL
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.07/0.32 => ( in @ Xx @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodpairmemER_type,type,
% 0.07/0.32 cartprodpairmemER: $o ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodpairmemER,definition,
% 0.07/0.32 ( cartprodpairmemER
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.07/0.32 => ( in @ Xy @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodmempaircEq_type,type,
% 0.07/0.32 cartprodmempaircEq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodmempaircEq,definition,
% 0.07/0.32 ( cartprodmempaircEq
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( kpair @ Xx @ Xy )
% 0.07/0.32 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodfstpairEq_type,type,
% 0.07/0.32 cartprodfstpairEq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodfstpairEq,definition,
% 0.07/0.32 ( cartprodfstpairEq
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.07/0.32 = Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodsndpairEq_type,type,
% 0.07/0.32 cartprodsndpairEq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodsndpairEq,definition,
% 0.07/0.32 ( cartprodsndpairEq
% 0.07/0.32 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.07/0.32 = Xy ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodpairsurjEq_type,type,
% 0.07/0.32 cartprodpairsurjEq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(cartprodpairsurjEq,definition,
% 0.07/0.32 ( cartprodpairsurjEq
% 0.07/0.32 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.07/0.32 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.07/0.32 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.07/0.32 = Xu ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(breln_type,type,
% 0.07/0.32 breln: $i > $i > $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(breln,definition,
% 0.07/0.32 ( breln
% 0.07/0.32 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(dpsetconstr_type,type,
% 0.07/0.32 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.07/0.32
% 0.07/0.32 thf(dpsetconstrI_type,type,
% 0.07/0.32 dpsetconstrI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(dpsetconstrI,definition,
% 0.07/0.32 ( dpsetconstrI
% 0.07/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( Xphi @ Xx @ Xy )
% 0.07/0.32 => ( in @ ( kpair @ Xx @ Xy )
% 0.07/0.32 @ ( dpsetconstr @ A @ B
% 0.07/0.32 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(dpsetconstrSub_type,type,
% 0.07/0.32 dpsetconstrSub: $o ).
% 0.07/0.32
% 0.07/0.32 thf(dpsetconstrSub,definition,
% 0.07/0.32 ( dpsetconstrSub
% 0.07/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.07/0.32 ( subset
% 0.07/0.32 @ ( dpsetconstr @ A @ B
% 0.07/0.32 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.07/0.32 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setOfPairsIsBReln_type,type,
% 0.07/0.32 setOfPairsIsBReln: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setOfPairsIsBReln,definition,
% 0.07/0.32 ( setOfPairsIsBReln
% 0.07/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.07/0.32 ( breln @ A @ B
% 0.07/0.32 @ ( dpsetconstr @ A @ B
% 0.07/0.32 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(dpsetconstrERa_type,type,
% 0.07/0.32 dpsetconstrERa: $o ).
% 0.07/0.32
% 0.07/0.32 thf(dpsetconstrERa,definition,
% 0.07/0.32 ( dpsetconstrERa
% 0.07/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.07/0.32 @ ( dpsetconstr @ A @ B
% 0.07/0.32 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.07/0.32 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(dpsetconstrEL1_type,type,
% 0.07/0.32 dpsetconstrEL1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(dpsetconstrEL1,definition,
% 0.07/0.32 ( dpsetconstrEL1
% 0.07/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ ( kpair @ Xx @ Xy )
% 0.07/0.32 @ ( dpsetconstr @ A @ B
% 0.07/0.32 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.07/0.32 => ( in @ Xx @ A ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(dpsetconstrEL2_type,type,
% 0.07/0.32 dpsetconstrEL2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(dpsetconstrEL2,definition,
% 0.07/0.32 ( dpsetconstrEL2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ ( kpair @ Xx @ Xy )
% 0.07/0.32 @ ( dpsetconstr @ A @ B
% 0.07/0.32 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.07/0.32 => ( in @ Xy @ B ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(dpsetconstrER_type,type,
% 0.07/0.32 dpsetconstrER: $o ).
% 0.07/0.32
% 0.07/0.32 thf(dpsetconstrER,definition,
% 0.07/0.32 ( dpsetconstrER
% 0.07/0.32 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( in @ ( kpair @ Xx @ Xy )
% 0.07/0.32 @ ( dpsetconstr @ A @ B
% 0.07/0.32 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.07/0.32 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(func_type,type,
% 0.07/0.32 func: $i > $i > $i > $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcSet_type,type,
% 0.07/0.32 funcSet: $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(funcImageSingleton_type,type,
% 0.07/0.32 funcImageSingleton: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcImageSingleton,definition,
% 0.07/0.32 ( funcImageSingleton
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( singleton
% 0.07/0.32 @ ( dsetconstr @ B
% 0.07/0.32 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(apProp_type,type,
% 0.07/0.32 apProp: $o ).
% 0.07/0.32
% 0.07/0.32 thf(apProp,definition,
% 0.07/0.32 ( apProp
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in
% 0.07/0.32 @ ( setunion
% 0.07/0.32 @ ( dsetconstr @ B
% 0.07/0.32 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.07/0.32 @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ap_type,type,
% 0.07/0.32 ap: $i > $i > $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(app_type,type,
% 0.07/0.32 app: $o ).
% 0.07/0.32
% 0.07/0.32 thf(app,definition,
% 0.07/0.32 ( app
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(infuncsetfunc_type,type,
% 0.07/0.32 infuncsetfunc: $o ).
% 0.07/0.32
% 0.07/0.32 thf(infuncsetfunc,definition,
% 0.07/0.32 ( infuncsetfunc
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.07/0.32 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ap2p_type,type,
% 0.07/0.32 ap2p: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ap2p,definition,
% 0.07/0.32 ( ap2p
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(funcinfuncset_type,type,
% 0.07/0.32 funcinfuncset: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcinfuncset,definition,
% 0.07/0.32 ( funcinfuncset
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(lamProp_type,type,
% 0.07/0.32 lamProp: $o ).
% 0.07/0.32
% 0.07/0.32 thf(lamProp,definition,
% 0.07/0.32 ( lamProp
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.07/0.32 => ( func @ A @ B
% 0.07/0.32 @ ( dpsetconstr @ A @ B
% 0.07/0.32 @ ^ [Xx: $i,Xy: $i] :
% 0.07/0.32 ( ( Xf @ Xx )
% 0.07/0.32 = Xy ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(lam_type,type,
% 0.07/0.32 lam: $i > $i > ( $i > $i ) > $i ).
% 0.07/0.32
% 0.07/0.32 thf(lamp_type,type,
% 0.07/0.32 lamp: $o ).
% 0.07/0.32
% 0.07/0.32 thf(lamp,definition,
% 0.07/0.32 ( lamp
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.07/0.32 => ( func @ A @ B
% 0.07/0.32 @ ( lam @ A @ B
% 0.07/0.32 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(lam2p_type,type,
% 0.07/0.32 lam2p: $o ).
% 0.07/0.32
% 0.07/0.32 thf(lam2p,definition,
% 0.07/0.32 ( lam2p
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.07/0.32 => ( in
% 0.07/0.32 @ ( lam @ A @ B
% 0.07/0.32 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.07/0.32 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(brelnall1_type,type,
% 0.07/0.32 brelnall1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(brelnall1,definition,
% 0.07/0.32 ( brelnall1
% 0.07/0.32 = ( ! [A: $i,B: $i,R: $i] :
% 0.07/0.32 ( ( breln @ A @ B @ R )
% 0.07/0.32 => ! [Xphi: $i > $o] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.07/0.32 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ R )
% 0.07/0.32 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(brelnall2_type,type,
% 0.07/0.32 brelnall2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(brelnall2,definition,
% 0.07/0.32 ( brelnall2
% 0.07/0.32 = ( ! [A: $i,B: $i,R: $i] :
% 0.07/0.32 ( ( breln @ A @ B @ R )
% 0.07/0.32 => ! [Xphi: $i > $o] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.07/0.32 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ R )
% 0.07/0.32 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ex1E2_type,type,
% 0.07/0.32 ex1E2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ex1E2,definition,
% 0.07/0.32 ( ex1E2
% 0.07/0.32 = ( ! [A: $i,Xphi: $i > $o] :
% 0.07/0.32 ( ( ex1 @ A
% 0.07/0.32 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( ( Xphi @ Xx )
% 0.07/0.32 => ( ( Xphi @ Xy )
% 0.07/0.32 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(funcGraphProp1_type,type,
% 0.07/0.32 funcGraphProp1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcGraphProp1,definition,
% 0.07/0.32 ( funcGraphProp1
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(funcGraphProp3_type,type,
% 0.07/0.32 funcGraphProp3: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcGraphProp3,definition,
% 0.07/0.32 ( funcGraphProp3
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(funcGraphProp2_type,type,
% 0.07/0.32 funcGraphProp2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcGraphProp2,definition,
% 0.07/0.32 ( funcGraphProp2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.07/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.07/0.32 = Xy ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(funcextLem_type,type,
% 0.07/0.32 funcextLem: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcextLem,definition,
% 0.07/0.32 ( funcextLem
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ! [Xg: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xg )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.07/0.32 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.07/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(funcGraphProp4_type,type,
% 0.07/0.32 funcGraphProp4: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcGraphProp4,definition,
% 0.07/0.32 ( funcGraphProp4
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.07/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.07/0.32 = Xy ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subbreln_type,type,
% 0.07/0.32 subbreln: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subbreln,definition,
% 0.07/0.32 ( subbreln
% 0.07/0.32 = ( ! [A: $i,B: $i,R: $i] :
% 0.07/0.32 ( ( breln @ A @ B @ R )
% 0.07/0.32 => ! [S: $i] :
% 0.07/0.32 ( ( breln @ A @ B @ S )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.07/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.07/0.32 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(eqbreln_type,type,
% 0.07/0.32 eqbreln: $o ).
% 0.07/0.32
% 0.07/0.32 thf(eqbreln,definition,
% 0.07/0.32 ( eqbreln
% 0.07/0.32 = ( ! [A: $i,B: $i,R: $i] :
% 0.07/0.32 ( ( breln @ A @ B @ R )
% 0.07/0.32 => ! [S: $i] :
% 0.07/0.32 ( ( breln @ A @ B @ S )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.07/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ B )
% 0.07/0.32 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.07/0.32 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.07/0.32 => ( R = S ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(funcext_type,type,
% 0.07/0.32 funcext: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcext,definition,
% 0.07/0.32 ( funcext
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ! [Xg: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xg )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.07/0.32 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.07/0.32 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(funcext2_type,type,
% 0.07/0.32 funcext2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(funcext2,definition,
% 0.07/0.32 ( funcext2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.07/0.32 => ! [Xg: $i] :
% 0.07/0.32 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.07/0.32 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.07/0.32 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ap2apEq1_type,type,
% 0.07/0.32 ap2apEq1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ap2apEq1,definition,
% 0.07/0.32 ( ap2apEq1
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.07/0.32 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ap2apEq2_type,type,
% 0.07/0.32 ap2apEq2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ap2apEq2,definition,
% 0.07/0.32 ( ap2apEq2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.07/0.32 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(beta1_type,type,
% 0.07/0.32 beta1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(beta1,definition,
% 0.07/0.32 ( beta1
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( ap @ A @ B
% 0.07/0.32 @ ( lam @ A @ B
% 0.07/0.32 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.07/0.32 @ Xx )
% 0.07/0.32 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(eta1_type,type,
% 0.07/0.32 eta1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(eta1,definition,
% 0.07/0.32 ( eta1
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( func @ A @ B @ Xf )
% 0.07/0.32 => ( ( lam @ A @ B
% 0.07/0.32 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.07/0.32 = Xf ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(lam2lamEq_type,type,
% 0.07/0.32 lam2lamEq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(lam2lamEq,definition,
% 0.07/0.32 ( lam2lamEq
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.07/0.32 => ( ( lam @ A @ B
% 0.07/0.32 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.07/0.32 = ( lam @ A @ B
% 0.07/0.32 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(beta2_type,type,
% 0.07/0.32 beta2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(beta2,definition,
% 0.07/0.32 ( beta2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.07/0.32 ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( ap @ A @ B
% 0.07/0.32 @ ( lam @ A @ B
% 0.07/0.32 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.07/0.32 @ Xx )
% 0.07/0.32 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(eta2_type,type,
% 0.07/0.32 eta2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(eta2,definition,
% 0.07/0.32 ( eta2
% 0.07/0.32 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.07/0.32 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.07/0.32 => ( ( lam @ A @ B
% 0.07/0.32 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.07/0.32 = Xf ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(iffalseProp1_type,type,
% 0.07/0.32 iffalseProp1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(iffalseProp1,definition,
% 0.07/0.32 ( iffalseProp1
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( ~ Xphi
% 0.07/0.32 => ( in @ Xy
% 0.07/0.32 @ ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xz: $i] :
% 0.07/0.32 ( ( Xphi
% 0.07/0.32 & ( Xz = Xx ) )
% 0.07/0.32 | ( ~ Xphi
% 0.07/0.32 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(iffalseProp2_type,type,
% 0.07/0.32 iffalseProp2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(iffalseProp2,definition,
% 0.07/0.32 ( iffalseProp2
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( ~ Xphi
% 0.07/0.32 => ( ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xz: $i] :
% 0.07/0.32 ( ( Xphi
% 0.07/0.32 & ( Xz = Xx ) )
% 0.07/0.32 | ( ~ Xphi
% 0.07/0.32 & ( Xz = Xy ) ) ) )
% 0.07/0.32 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(iftrueProp1_type,type,
% 0.07/0.32 iftrueProp1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(iftrueProp1,definition,
% 0.07/0.32 ( iftrueProp1
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( Xphi
% 0.07/0.32 => ( in @ Xx
% 0.07/0.32 @ ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xz: $i] :
% 0.07/0.32 ( ( Xphi
% 0.07/0.32 & ( Xz = Xx ) )
% 0.07/0.32 | ( ~ Xphi
% 0.07/0.32 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(iftrueProp2_type,type,
% 0.07/0.32 iftrueProp2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(iftrueProp2,definition,
% 0.07/0.32 ( iftrueProp2
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( Xphi
% 0.07/0.32 => ( ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xz: $i] :
% 0.07/0.32 ( ( Xphi
% 0.07/0.32 & ( Xz = Xx ) )
% 0.07/0.32 | ( ~ Xphi
% 0.07/0.32 & ( Xz = Xy ) ) ) )
% 0.07/0.32 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(ifSingleton_type,type,
% 0.07/0.32 ifSingleton: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ifSingleton,definition,
% 0.07/0.32 ( ifSingleton
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( singleton
% 0.07/0.32 @ ( dsetconstr @ A
% 0.07/0.32 @ ^ [Xz: $i] :
% 0.07/0.32 ( ( Xphi
% 0.07/0.32 & ( Xz = Xx ) )
% 0.07/0.32 | ( ~ Xphi
% 0.07/0.32 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(if_type,type,
% 0.07/0.32 if: $i > $o > $i > $i > $i ).
% 0.07/0.32
% 0.07/0.32 thf(ifp_type,type,
% 0.07/0.32 ifp: $o ).
% 0.07/0.32
% 0.07/0.32 thf(ifp,definition,
% 0.07/0.32 ( ifp
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(theeq_type,type,
% 0.07/0.32 theeq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(theeq,definition,
% 0.07/0.32 ( theeq
% 0.07/0.32 = ( ! [X: $i] :
% 0.07/0.32 ( ( singleton @ X )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ X )
% 0.07/0.32 => ( ( setunion @ X )
% 0.07/0.32 = Xx ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(iftrue_type,type,
% 0.07/0.32 iftrue: $o ).
% 0.07/0.32
% 0.07/0.32 thf(iftrue,definition,
% 0.07/0.32 ( iftrue
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( Xphi
% 0.07/0.32 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.07/0.32 = Xx ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(iffalse_type,type,
% 0.07/0.32 iffalse: $o ).
% 0.07/0.32
% 0.07/0.32 thf(iffalse,definition,
% 0.07/0.32 ( iffalse
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( ~ Xphi
% 0.07/0.32 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.07/0.32 = Xy ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(iftrueorfalse_type,type,
% 0.07/0.32 iftrueorfalse: $o ).
% 0.07/0.32
% 0.07/0.32 thf(iftrueorfalse,definition,
% 0.07/0.32 ( iftrueorfalse
% 0.07/0.32 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ! [Xy: $i] :
% 0.07/0.32 ( ( in @ Xy @ A )
% 0.07/0.32 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectT_lem_type,type,
% 0.07/0.32 binintersectT_lem: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectT_lem,definition,
% 0.07/0.32 ( binintersectT_lem
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binunionT_lem_type,type,
% 0.07/0.32 binunionT_lem: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binunionT_lem,definition,
% 0.07/0.32 ( binunionT_lem
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powersetT_lem_type,type,
% 0.07/0.32 powersetT_lem: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powersetT_lem,definition,
% 0.07/0.32 ( powersetT_lem
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setminusT_lem_type,type,
% 0.07/0.32 setminusT_lem: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setminusT_lem,definition,
% 0.07/0.32 ( setminusT_lem
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(complementT_lem_type,type,
% 0.07/0.32 complementT_lem: $o ).
% 0.07/0.32
% 0.07/0.32 thf(complementT_lem,definition,
% 0.07/0.32 ( complementT_lem
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(setextT_type,type,
% 0.07/0.32 setextT: $o ).
% 0.07/0.32
% 0.07/0.32 thf(setextT,definition,
% 0.07/0.32 ( setextT
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ X )
% 0.07/0.32 => ( in @ Xx @ Y ) ) )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ Y )
% 0.07/0.32 => ( in @ Xx @ X ) ) )
% 0.07/0.32 => ( X = Y ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(subsetTI_type,type,
% 0.07/0.32 subsetTI: $o ).
% 0.07/0.32
% 0.07/0.32 thf(subsetTI,definition,
% 0.07/0.32 ( subsetTI
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ X )
% 0.07/0.32 => ( in @ Xx @ Y ) ) )
% 0.07/0.32 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powersetTI1_type,type,
% 0.07/0.32 powersetTI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powersetTI1,definition,
% 0.07/0.32 ( powersetTI1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ X )
% 0.07/0.32 => ( in @ Xx @ Y ) ) )
% 0.07/0.32 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(powersetTE1_type,type,
% 0.07/0.32 powersetTE1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(powersetTE1,definition,
% 0.07/0.32 ( powersetTE1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ X @ ( powerset @ Y ) )
% 0.07/0.32 => ( ( in @ Xx @ X )
% 0.07/0.32 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(complementTI1_type,type,
% 0.07/0.32 complementTI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(complementTI1,definition,
% 0.07/0.32 ( complementTI1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ X )
% 0.07/0.32 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(complementTE1_type,type,
% 0.07/0.32 complementTE1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(complementTE1,definition,
% 0.07/0.32 ( complementTE1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.07/0.32 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectTELcontra_type,type,
% 0.07/0.32 binintersectTELcontra: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectTELcontra,definition,
% 0.07/0.32 ( binintersectTELcontra
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ X )
% 0.07/0.32 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectTERcontra_type,type,
% 0.07/0.32 binintersectTERcontra: $o ).
% 0.07/0.32
% 0.07/0.32 thf(binintersectTERcontra,definition,
% 0.07/0.32 ( binintersectTERcontra
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ~ ( in @ Xx @ Y )
% 0.07/0.32 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(contrasubsetT_type,type,
% 0.07/0.32 contrasubsetT: $o ).
% 0.07/0.32
% 0.07/0.32 thf(contrasubsetT,definition,
% 0.07/0.32 ( contrasubsetT
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.07/0.32 => ( ( in @ Xx @ Y )
% 0.07/0.32 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(contrasubsetT1_type,type,
% 0.07/0.32 contrasubsetT1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(contrasubsetT1,definition,
% 0.07/0.32 ( contrasubsetT1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( subset @ X @ Y )
% 0.07/0.32 => ( ~ ( in @ Xx @ Y )
% 0.07/0.32 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(contrasubsetT2_type,type,
% 0.07/0.32 contrasubsetT2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(contrasubsetT2,definition,
% 0.07/0.32 ( contrasubsetT2
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( ( subset @ X @ Y )
% 0.07/0.32 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(contrasubsetT3_type,type,
% 0.07/0.32 contrasubsetT3: $o ).
% 0.07/0.32
% 0.07/0.32 thf(contrasubsetT3,definition,
% 0.07/0.32 ( contrasubsetT3
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.07/0.32 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementI1_type,type,
% 0.07/0.32 doubleComplementI1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementI1,definition,
% 0.07/0.32 ( doubleComplementI1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ X )
% 0.07/0.32 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementE1_type,type,
% 0.07/0.32 doubleComplementE1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementE1,definition,
% 0.07/0.32 ( doubleComplementE1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.07/0.32 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementSub1_type,type,
% 0.07/0.32 doubleComplementSub1: $o ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementSub1,definition,
% 0.07/0.32 ( doubleComplementSub1
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementSub2_type,type,
% 0.07/0.32 doubleComplementSub2: $o ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementSub2,definition,
% 0.07/0.32 ( doubleComplementSub2
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementEq_type,type,
% 0.07/0.32 doubleComplementEq: $o ).
% 0.07/0.32
% 0.07/0.32 thf(doubleComplementEq,definition,
% 0.07/0.32 ( doubleComplementEq
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ( X
% 0.07/0.32 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(complementTnotintersectT_type,type,
% 0.07/0.32 complementTnotintersectT: $o ).
% 0.07/0.32
% 0.07/0.32 thf(complementTnotintersectT,definition,
% 0.07/0.32 ( complementTnotintersectT
% 0.07/0.32 = ( ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.07/0.32 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(complementImpComplementIntersect,conjecture,
% 0.07/0.32 ( setextAx
% 0.07/0.32 => ( emptysetAx
% 0.07/0.32 => ( setadjoinAx
% 0.07/0.32 => ( powersetAx
% 0.07/0.32 => ( setunionAx
% 0.07/0.32 => ( omega0Ax
% 0.07/0.32 => ( omegaSAx
% 0.07/0.32 => ( omegaIndAx
% 0.07/0.32 => ( replAx
% 0.07/0.32 => ( foundationAx
% 0.07/0.32 => ( wellorderingAx
% 0.07/0.32 => ( descrp
% 0.07/0.32 => ( dsetconstrI
% 0.07/0.32 => ( dsetconstrEL
% 0.07/0.32 => ( dsetconstrER
% 0.07/0.32 => ( exuE1
% 0.07/0.32 => ( prop2setE
% 0.07/0.32 => ( emptysetE
% 0.07/0.32 => ( emptysetimpfalse
% 0.07/0.32 => ( notinemptyset
% 0.07/0.32 => ( exuE3e
% 0.07/0.32 => ( setext
% 0.07/0.32 => ( emptyI
% 0.07/0.32 => ( noeltsimpempty
% 0.07/0.32 => ( setbeta
% 0.07/0.32 => ( nonemptyE1
% 0.07/0.32 => ( nonemptyI
% 0.07/0.32 => ( nonemptyI1
% 0.07/0.32 => ( setadjoinIL
% 0.07/0.32 => ( emptyinunitempty
% 0.07/0.32 => ( setadjoinIR
% 0.07/0.32 => ( setadjoinE
% 0.07/0.32 => ( setadjoinOr
% 0.07/0.32 => ( setoftrueEq
% 0.07/0.32 => ( powersetI
% 0.07/0.32 => ( emptyinPowerset
% 0.07/0.32 => ( emptyInPowerset
% 0.07/0.32 => ( powersetE
% 0.07/0.32 => ( setunionI
% 0.07/0.32 => ( setunionE
% 0.07/0.32 => ( subPowSU
% 0.07/0.32 => ( exuE2
% 0.07/0.32 => ( nonemptyImpWitness
% 0.07/0.32 => ( uniqinunit
% 0.07/0.32 => ( notinsingleton
% 0.07/0.32 => ( eqinunit
% 0.07/0.32 => ( singletonsswitch
% 0.07/0.32 => ( upairsetE
% 0.07/0.32 => ( upairsetIL
% 0.07/0.32 => ( upairsetIR
% 0.07/0.32 => ( emptyE1
% 0.07/0.32 => ( vacuousDall
% 0.07/0.32 => ( quantDeMorgan1
% 0.07/0.32 => ( quantDeMorgan2
% 0.07/0.32 => ( quantDeMorgan3
% 0.07/0.32 => ( quantDeMorgan4
% 0.07/0.32 => ( prop2setI
% 0.07/0.32 => ( prop2set2propI
% 0.07/0.32 => ( notdexE
% 0.07/0.32 => ( notdallE
% 0.07/0.32 => ( exuI1
% 0.07/0.32 => ( exuI3
% 0.07/0.32 => ( exuI2
% 0.07/0.32 => ( inCongP
% 0.07/0.32 => ( in__Cong
% 0.07/0.32 => ( exuE3u
% 0.07/0.32 => ( exu__Cong
% 0.07/0.32 => ( emptyset__Cong
% 0.07/0.32 => ( setadjoin__Cong
% 0.07/0.32 => ( powerset__Cong
% 0.07/0.32 => ( setunion__Cong
% 0.07/0.32 => ( omega__Cong
% 0.07/0.32 => ( exuEu
% 0.07/0.32 => ( descr__Cong
% 0.07/0.32 => ( dsetconstr__Cong
% 0.07/0.32 => ( subsetI1
% 0.07/0.32 => ( eqimpsubset2
% 0.07/0.32 => ( eqimpsubset1
% 0.07/0.32 => ( subsetI2
% 0.07/0.32 => ( emptysetsubset
% 0.07/0.32 => ( subsetE
% 0.07/0.32 => ( subsetE2
% 0.07/0.32 => ( notsubsetI
% 0.07/0.32 => ( notequalI1
% 0.07/0.32 => ( notequalI2
% 0.07/0.32 => ( subsetRefl
% 0.07/0.32 => ( subsetTrans
% 0.07/0.32 => ( setadjoinSub
% 0.07/0.32 => ( setadjoinSub2
% 0.07/0.32 => ( subset2powerset
% 0.07/0.32 => ( setextsub
% 0.07/0.32 => ( subsetemptysetimpeq
% 0.07/0.32 => ( powersetI1
% 0.07/0.32 => ( powersetE1
% 0.07/0.32 => ( inPowerset
% 0.07/0.32 => ( powersetsubset
% 0.07/0.32 => ( sepInPowerset
% 0.07/0.32 => ( sepSubset
% 0.07/0.32 => ( binunionIL
% 0.07/0.32 => ( upairset2IR
% 0.07/0.32 => ( binunionIR
% 0.07/0.32 => ( binunionEcases
% 0.07/0.32 => ( binunionE
% 0.07/0.32 => ( binunionLsub
% 0.07/0.32 => ( binunionRsub
% 0.07/0.32 => ( binintersectI
% 0.07/0.32 => ( binintersectSubset5
% 0.07/0.32 => ( binintersectEL
% 0.07/0.32 => ( binintersectLsub
% 0.07/0.32 => ( binintersectSubset2
% 0.07/0.32 => ( binintersectSubset3
% 0.07/0.32 => ( binintersectER
% 0.07/0.32 => ( disjointsetsI1
% 0.07/0.32 => ( binintersectRsub
% 0.07/0.32 => ( binintersectSubset4
% 0.07/0.32 => ( binintersectSubset1
% 0.07/0.32 => ( bs114d
% 0.07/0.32 => ( setminusI
% 0.07/0.32 => ( setminusEL
% 0.07/0.32 => ( setminusER
% 0.07/0.32 => ( setminusSubset2
% 0.07/0.32 => ( setminusERneg
% 0.07/0.32 => ( setminusELneg
% 0.07/0.32 => ( setminusILneg
% 0.07/0.32 => ( setminusIRneg
% 0.07/0.32 => ( setminusLsub
% 0.07/0.32 => ( setminusSubset1
% 0.07/0.32 => ( symdiffE
% 0.07/0.32 => ( symdiffI1
% 0.07/0.32 => ( symdiffI2
% 0.07/0.32 => ( symdiffIneg1
% 0.07/0.32 => ( symdiffIneg2
% 0.07/0.32 => ( secondinupair
% 0.07/0.32 => ( setukpairIL
% 0.07/0.32 => ( setukpairIR
% 0.07/0.32 => ( kpairiskpair
% 0.07/0.32 => ( kpairp
% 0.07/0.32 => ( singletonsubset
% 0.07/0.32 => ( singletoninpowerset
% 0.07/0.32 => ( singletoninpowunion
% 0.07/0.32 => ( upairset2E
% 0.07/0.32 => ( upairsubunion
% 0.07/0.32 => ( upairinpowunion
% 0.07/0.32 => ( ubforcartprodlem1
% 0.07/0.32 => ( ubforcartprodlem2
% 0.07/0.32 => ( ubforcartprodlem3
% 0.07/0.32 => ( cartprodpairin
% 0.07/0.32 => ( cartprodmempair1
% 0.07/0.32 => ( cartprodmempair
% 0.07/0.32 => ( setunionE2
% 0.07/0.32 => ( setunionsingleton1
% 0.07/0.32 => ( setunionsingleton2
% 0.07/0.32 => ( setunionsingleton
% 0.07/0.32 => ( singletonprop
% 0.07/0.32 => ( ex1E1
% 0.07/0.32 => ( ex1I
% 0.07/0.32 => ( ex1I2
% 0.07/0.32 => ( singletonsuniq
% 0.07/0.32 => ( setukpairinjL1
% 0.07/0.32 => ( kfstsingleton
% 0.07/0.32 => ( theprop
% 0.07/0.32 => ( kfstpairEq
% 0.07/0.32 => ( cartprodfstin
% 0.07/0.32 => ( setukpairinjL2
% 0.07/0.32 => ( setukpairinjL
% 0.07/0.32 => ( setukpairinjR11
% 0.07/0.32 => ( setukpairinjR12
% 0.07/0.32 => ( setukpairinjR1
% 0.07/0.32 => ( upairequniteq
% 0.07/0.32 => ( setukpairinjR2
% 0.07/0.32 => ( setukpairinjR
% 0.07/0.32 => ( ksndsingleton
% 0.07/0.32 => ( ksndpairEq
% 0.07/0.32 => ( kpairsurjEq
% 0.07/0.32 => ( cartprodsndin
% 0.07/0.32 => ( cartprodpairmemEL
% 0.07/0.32 => ( cartprodpairmemER
% 0.07/0.32 => ( cartprodmempaircEq
% 0.07/0.32 => ( cartprodfstpairEq
% 0.07/0.32 => ( cartprodsndpairEq
% 0.07/0.32 => ( cartprodpairsurjEq
% 0.07/0.32 => ( dpsetconstrI
% 0.07/0.32 => ( dpsetconstrSub
% 0.07/0.32 => ( setOfPairsIsBReln
% 0.07/0.32 => ( dpsetconstrERa
% 0.07/0.32 => ( dpsetconstrEL1
% 0.07/0.32 => ( dpsetconstrEL2
% 0.07/0.32 => ( dpsetconstrER
% 0.07/0.32 => ( funcImageSingleton
% 0.07/0.32 => ( apProp
% 0.07/0.32 => ( app
% 0.07/0.32 => ( infuncsetfunc
% 0.07/0.32 => ( ap2p
% 0.07/0.32 => ( funcinfuncset
% 0.07/0.32 => ( lamProp
% 0.07/0.32 => ( lamp
% 0.07/0.32 => ( lam2p
% 0.07/0.32 => ( brelnall1
% 0.07/0.32 => ( brelnall2
% 0.07/0.32 => ( ex1E2
% 0.07/0.32 => ( funcGraphProp1
% 0.07/0.32 => ( funcGraphProp3
% 0.07/0.32 => ( funcGraphProp2
% 0.07/0.32 => ( funcextLem
% 0.07/0.32 => ( funcGraphProp4
% 0.07/0.32 => ( subbreln
% 0.07/0.32 => ( eqbreln
% 0.07/0.32 => ( funcext
% 0.07/0.32 => ( funcext2
% 0.07/0.32 => ( ap2apEq1
% 0.07/0.32 => ( ap2apEq2
% 0.07/0.32 => ( beta1
% 0.07/0.32 => ( eta1
% 0.07/0.32 => ( lam2lamEq
% 0.07/0.32 => ( beta2
% 0.07/0.32 => ( eta2
% 0.07/0.32 => ( iffalseProp1
% 0.07/0.32 => ( iffalseProp2
% 0.07/0.32 => ( iftrueProp1
% 0.07/0.32 => ( iftrueProp2
% 0.07/0.32 => ( ifSingleton
% 0.07/0.32 => ( ifp
% 0.07/0.32 => ( theeq
% 0.07/0.32 => ( iftrue
% 0.07/0.32 => ( iffalse
% 0.07/0.32 => ( iftrueorfalse
% 0.07/0.32 => ( binintersectT_lem
% 0.07/0.32 => ( binunionT_lem
% 0.07/0.32 => ( powersetT_lem
% 0.07/0.32 => ( setminusT_lem
% 0.07/0.32 => ( complementT_lem
% 0.07/0.32 => ( setextT
% 0.07/0.32 => ( subsetTI
% 0.07/0.32 => ( powersetTI1
% 0.07/0.32 => ( powersetTE1
% 0.07/0.32 => ( complementTI1
% 0.07/0.32 => ( complementTE1
% 0.07/0.32 => ( binintersectTELcontra
% 0.07/0.32 => ( binintersectTERcontra
% 0.07/0.32 => ( contrasubsetT
% 0.07/0.32 => ( contrasubsetT1
% 0.07/0.32 => ( contrasubsetT2
% 0.07/0.32 => ( contrasubsetT3
% 0.07/0.32 => ( doubleComplementI1
% 0.07/0.32 => ( doubleComplementE1
% 0.07/0.32 => ( doubleComplementSub1
% 0.07/0.32 => ( doubleComplementSub2
% 0.07/0.32 => ( doubleComplementEq
% 0.07/0.32 => ( complementTnotintersectT
% 0.07/0.32 => ! [A: $i,X: $i] :
% 0.07/0.32 ( ( in @ X @ ( powerset @ A ) )
% 0.07/0.32 => ! [Y: $i] :
% 0.07/0.32 ( ( in @ Y @ ( powerset @ A ) )
% 0.07/0.32 => ! [Xx: $i] :
% 0.07/0.32 ( ( in @ Xx @ A )
% 0.07/0.32 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.07/0.32 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 %------------------------------------------------------------------------------
% 0.07/0.32 ------------------------
% 2.35/1.15 ---- Embedded in THF ---
% 2.35/1.15 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.35/1.15 %%% Generated on Tue Feb 24 07:32:11 EST 2026
% 2.35/1.15 %%% using '$$dhol' embedding, version 1.3.0.
% 2.35/1.15 %%% Logic specification used:
% 2.35/1.15 %%% thf(spec, logic, $$dhol).
% 2.35/1.15
% 2.35/1.15 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.V67yydYf3P/DTF2DTF3524.p
% See solution above
% 2.35/1.15 ------------------------
% 2.35/1.18 ---- Cleaned THF ---
% 2.35/1.18 thf(in_type,type,
% 2.35/1.18 in: $i > $i > $o ).
% 2.35/1.18
% 2.35/1.18 thf(exu_type,type,
% 2.35/1.18 exu: ( $i > $o ) > $o ).
% 2.35/1.18
% 2.35/1.18 thf(exu,definition,
% 2.35/1.18 ( exu
% 2.35/1.18 = ( ^ [Xphi: $i > $o] :
% 2.35/1.18 ? [Xx: $i] :
% 2.35/1.18 ( ( Xphi @ Xx )
% 2.35/1.18 & ! [Xy: $i] :
% 2.35/1.18 ( ( Xphi @ Xy )
% 2.35/1.18 => ( Xx = Xy ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setextAx_type,type,
% 2.35/1.18 setextAx: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setextAx,definition,
% 2.35/1.18 ( setextAx
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 <=> ( in @ Xx @ B ) )
% 2.35/1.18 => ( A = B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(emptyset_type,type,
% 2.35/1.18 emptyset: $i ).
% 2.35/1.18
% 2.35/1.18 thf(emptysetAx_type,type,
% 2.35/1.18 emptysetAx: $o ).
% 2.35/1.18
% 2.35/1.18 thf(emptysetAx,definition,
% 2.35/1.18 ( emptysetAx
% 2.35/1.18 = ( ! [Xx: $i] :
% 2.35/1.18 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoin_type,type,
% 2.35/1.18 setadjoin: $i > $i > $i ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoinAx_type,type,
% 2.35/1.18 setadjoinAx: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoinAx,definition,
% 2.35/1.18 ( setadjoinAx
% 2.35/1.18 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.35/1.18 <=> ( ( Xy = Xx )
% 2.35/1.18 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(powerset_type,type,
% 2.35/1.18 powerset: $i > $i ).
% 2.35/1.18
% 2.35/1.18 thf(powersetAx_type,type,
% 2.35/1.18 powersetAx: $o ).
% 2.35/1.18
% 2.35/1.18 thf(powersetAx,definition,
% 2.35/1.18 ( powersetAx
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( in @ B @ ( powerset @ A ) )
% 2.35/1.18 <=> ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ B )
% 2.35/1.18 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setunion_type,type,
% 2.35/1.18 setunion: $i > $i ).
% 2.35/1.18
% 2.35/1.18 thf(setunionAx_type,type,
% 2.35/1.18 setunionAx: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setunionAx,definition,
% 2.35/1.18 ( setunionAx
% 2.35/1.18 = ( ! [A: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ ( setunion @ A ) )
% 2.35/1.18 <=> ? [B: $i] :
% 2.35/1.18 ( ( in @ Xx @ B )
% 2.35/1.18 & ( in @ B @ A ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(omega_type,type,
% 2.35/1.18 omega: $i ).
% 2.35/1.18
% 2.35/1.18 thf(omega0Ax_type,type,
% 2.35/1.18 omega0Ax: $o ).
% 2.35/1.18
% 2.35/1.18 thf(omega0Ax,definition,
% 2.35/1.18 ( omega0Ax
% 2.35/1.18 = ( in @ emptyset @ omega ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(omegaSAx_type,type,
% 2.35/1.18 omegaSAx: $o ).
% 2.35/1.18
% 2.35/1.18 thf(omegaSAx,definition,
% 2.35/1.18 ( omegaSAx
% 2.35/1.18 = ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ omega )
% 2.35/1.18 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(omegaIndAx_type,type,
% 2.35/1.18 omegaIndAx: $o ).
% 2.35/1.18
% 2.35/1.18 thf(omegaIndAx,definition,
% 2.35/1.18 ( omegaIndAx
% 2.35/1.18 = ( ! [A: $i] :
% 2.35/1.18 ( ( ( in @ emptyset @ A )
% 2.35/1.18 & ! [Xx: $i] :
% 2.35/1.18 ( ( ( in @ Xx @ omega )
% 2.35/1.18 & ( in @ Xx @ A ) )
% 2.35/1.18 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ omega )
% 2.35/1.18 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(replAx_type,type,
% 2.35/1.18 replAx: $o ).
% 2.35/1.18
% 2.35/1.18 thf(replAx,definition,
% 2.35/1.18 ( replAx
% 2.35/1.18 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( exu
% 2.35/1.18 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.35/1.18 => ? [B: $i] :
% 2.35/1.18 ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ B )
% 2.35/1.18 <=> ? [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ A )
% 2.35/1.18 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(foundationAx_type,type,
% 2.35/1.18 foundationAx: $o ).
% 2.35/1.18
% 2.35/1.18 thf(foundationAx,definition,
% 2.35/1.18 ( foundationAx
% 2.35/1.18 = ( ! [A: $i] :
% 2.35/1.18 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.35/1.18 => ? [B: $i] :
% 2.35/1.18 ( ( in @ B @ A )
% 2.35/1.18 & ~ ? [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ B )
% 2.35/1.18 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(wellorderingAx_type,type,
% 2.35/1.18 wellorderingAx: $o ).
% 2.35/1.18
% 2.35/1.18 thf(wellorderingAx,definition,
% 2.35/1.18 ( wellorderingAx
% 2.35/1.18 = ( ! [A: $i] :
% 2.35/1.18 ? [B: $i] :
% 2.35/1.18 ( ! [C: $i] :
% 2.35/1.18 ( ( in @ C @ B )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ C )
% 2.35/1.18 => ( in @ Xx @ A ) ) )
% 2.35/1.18 & ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( ( in @ Xx @ A )
% 2.35/1.18 & ( in @ Xy @ A ) )
% 2.35/1.18 => ( ! [C: $i] :
% 2.35/1.18 ( ( in @ C @ B )
% 2.35/1.18 => ( ( in @ Xx @ C )
% 2.35/1.18 <=> ( in @ Xy @ C ) ) )
% 2.35/1.18 => ( Xx = Xy ) ) )
% 2.35/1.18 & ! [C: $i,D: $i] :
% 2.35/1.18 ( ( ( in @ C @ B )
% 2.35/1.18 & ( in @ D @ B ) )
% 2.35/1.18 => ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ C )
% 2.35/1.18 => ( in @ Xx @ D ) )
% 2.35/1.18 | ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ D )
% 2.35/1.18 => ( in @ Xx @ C ) ) ) )
% 2.35/1.18 & ! [C: $i] :
% 2.35/1.18 ( ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ C )
% 2.35/1.18 => ( in @ Xx @ A ) )
% 2.35/1.18 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.35/1.18 => ? [D: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ D @ B )
% 2.35/1.18 & ( in @ Xx @ C )
% 2.35/1.18 & ~ ? [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ D )
% 2.35/1.18 & ( in @ Xy @ C ) )
% 2.35/1.18 & ! [E: $i] :
% 2.35/1.18 ( ( in @ E @ B )
% 2.35/1.18 => ( ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ E )
% 2.35/1.18 => ( in @ Xy @ D ) )
% 2.35/1.18 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(descr_type,type,
% 2.35/1.18 descr: ( $i > $o ) > $i ).
% 2.35/1.18
% 2.35/1.18 thf(descrp_type,type,
% 2.35/1.18 descrp: $o ).
% 2.35/1.18
% 2.35/1.18 thf(descrp,definition,
% 2.35/1.18 ( descrp
% 2.35/1.18 = ( ! [Xphi: $i > $o] :
% 2.35/1.18 ( ( exu
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.35/1.18 => ( Xphi
% 2.35/1.18 @ ( descr
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(dsetconstr_type,type,
% 2.35/1.18 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.35/1.18
% 2.35/1.18 thf(dsetconstrI_type,type,
% 2.35/1.18 dsetconstrI: $o ).
% 2.35/1.18
% 2.35/1.18 thf(dsetconstrI,definition,
% 2.35/1.18 ( dsetconstrI
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( Xphi @ Xx )
% 2.35/1.18 => ( in @ Xx
% 2.35/1.18 @ ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(dsetconstrEL_type,type,
% 2.35/1.18 dsetconstrEL: $o ).
% 2.35/1.18
% 2.35/1.18 thf(dsetconstrEL,definition,
% 2.35/1.18 ( dsetconstrEL
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx
% 2.35/1.18 @ ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.35/1.18 => ( in @ Xx @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(dsetconstrER_type,type,
% 2.35/1.18 dsetconstrER: $o ).
% 2.35/1.18
% 2.35/1.18 thf(dsetconstrER,definition,
% 2.35/1.18 ( dsetconstrER
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx
% 2.35/1.18 @ ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.35/1.18 => ( Xphi @ Xx ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(exuE1_type,type,
% 2.35/1.18 exuE1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(exuE1,definition,
% 2.35/1.18 ( exuE1
% 2.35/1.18 = ( ! [Xphi: $i > $o] :
% 2.35/1.18 ( ( exu
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.35/1.18 => ? [Xx: $i] :
% 2.35/1.18 ( ( Xphi @ Xx )
% 2.35/1.18 & ! [Xy: $i] :
% 2.35/1.18 ( ( Xphi @ Xy )
% 2.35/1.18 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(prop2set_type,type,
% 2.35/1.18 prop2set: $o > $i ).
% 2.35/1.18
% 2.35/1.18 thf(prop2setE_type,type,
% 2.35/1.18 prop2setE: $o ).
% 2.35/1.18
% 2.35/1.18 thf(prop2setE,definition,
% 2.35/1.18 ( prop2setE
% 2.35/1.18 = ( ! [Xphi: $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.35/1.18 => Xphi ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(emptysetE_type,type,
% 2.35/1.18 emptysetE: $o ).
% 2.35/1.18
% 2.35/1.18 thf(emptysetE,definition,
% 2.35/1.18 ( emptysetE
% 2.35/1.18 = ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ emptyset )
% 2.35/1.18 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(emptysetimpfalse_type,type,
% 2.35/1.18 emptysetimpfalse: $o ).
% 2.35/1.18
% 2.35/1.18 thf(emptysetimpfalse,definition,
% 2.35/1.18 ( emptysetimpfalse
% 2.35/1.18 = ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ emptyset )
% 2.35/1.18 => $false ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(notinemptyset_type,type,
% 2.35/1.18 notinemptyset: $o ).
% 2.35/1.18
% 2.35/1.18 thf(notinemptyset,definition,
% 2.35/1.18 ( notinemptyset
% 2.35/1.18 = ( ! [Xx: $i] :
% 2.35/1.18 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(exuE3e_type,type,
% 2.35/1.18 exuE3e: $o ).
% 2.35/1.18
% 2.35/1.18 thf(exuE3e,definition,
% 2.35/1.18 ( exuE3e
% 2.35/1.18 = ( ! [Xphi: $i > $o] :
% 2.35/1.18 ( ( exu
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.35/1.18 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setext_type,type,
% 2.35/1.18 setext: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setext,definition,
% 2.35/1.18 ( setext
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ Xx @ B ) )
% 2.35/1.18 => ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ B )
% 2.35/1.18 => ( in @ Xx @ A ) )
% 2.35/1.18 => ( A = B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(emptyI_type,type,
% 2.35/1.18 emptyI: $o ).
% 2.35/1.18
% 2.35/1.18 thf(emptyI,definition,
% 2.35/1.18 ( emptyI
% 2.35/1.18 = ( ! [A: $i] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ~ ( in @ Xx @ A )
% 2.35/1.18 => ( A = emptyset ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(noeltsimpempty_type,type,
% 2.35/1.18 noeltsimpempty: $o ).
% 2.35/1.18
% 2.35/1.18 thf(noeltsimpempty,definition,
% 2.35/1.18 ( noeltsimpempty
% 2.35/1.18 = ( ! [A: $i] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ~ ( in @ Xx @ A )
% 2.35/1.18 => ( A = emptyset ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setbeta_type,type,
% 2.35/1.18 setbeta: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setbeta,definition,
% 2.35/1.18 ( setbeta
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( in @ Xx
% 2.35/1.18 @ ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.35/1.18 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(nonempty_type,type,
% 2.35/1.18 nonempty: $i > $o ).
% 2.35/1.18
% 2.35/1.18 thf(nonempty,definition,
% 2.35/1.18 ( nonempty
% 2.35/1.18 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(nonemptyE1_type,type,
% 2.35/1.18 nonemptyE1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(nonemptyE1,definition,
% 2.35/1.18 ( nonemptyE1
% 2.35/1.18 = ( ! [A: $i] :
% 2.35/1.18 ( ( nonempty @ A )
% 2.35/1.18 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(nonemptyI_type,type,
% 2.35/1.18 nonemptyI: $o ).
% 2.35/1.18
% 2.35/1.18 thf(nonemptyI,definition,
% 2.35/1.18 ( nonemptyI
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( Xphi @ Xx )
% 2.35/1.18 => ( nonempty
% 2.35/1.18 @ ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(nonemptyI1_type,type,
% 2.35/1.18 nonemptyI1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(nonemptyI1,definition,
% 2.35/1.18 ( nonemptyI1
% 2.35/1.18 = ( ! [A: $i] :
% 2.35/1.18 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.35/1.18 => ( nonempty @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoinIL_type,type,
% 2.35/1.18 setadjoinIL: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoinIL,definition,
% 2.35/1.18 ( setadjoinIL
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(emptyinunitempty_type,type,
% 2.35/1.18 emptyinunitempty: $o ).
% 2.35/1.18
% 2.35/1.18 thf(emptyinunitempty,definition,
% 2.35/1.18 ( emptyinunitempty
% 2.35/1.18 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoinIR_type,type,
% 2.35/1.18 setadjoinIR: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoinIR,definition,
% 2.35/1.18 ( setadjoinIR
% 2.35/1.18 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ A )
% 2.35/1.18 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoinE_type,type,
% 2.35/1.18 setadjoinE: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoinE,definition,
% 2.35/1.18 ( setadjoinE
% 2.35/1.18 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.35/1.18 => ! [Xphi: $o] :
% 2.35/1.18 ( ( ( Xy = Xx )
% 2.35/1.18 => Xphi )
% 2.35/1.18 => ( ( ( in @ Xy @ A )
% 2.35/1.18 => Xphi )
% 2.35/1.18 => Xphi ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoinOr_type,type,
% 2.35/1.18 setadjoinOr: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoinOr,definition,
% 2.35/1.18 ( setadjoinOr
% 2.35/1.18 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.35/1.18 => ( ( Xy = Xx )
% 2.35/1.18 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setoftrueEq_type,type,
% 2.35/1.18 setoftrueEq: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setoftrueEq,definition,
% 2.35/1.18 ( setoftrueEq
% 2.35/1.18 = ( ! [A: $i] :
% 2.35/1.18 ( ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xx: $i] : $true )
% 2.35/1.18 = A ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(powersetI_type,type,
% 2.35/1.18 powersetI: $o ).
% 2.35/1.18
% 2.35/1.18 thf(powersetI,definition,
% 2.35/1.18 ( powersetI
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ B )
% 2.35/1.18 => ( in @ Xx @ A ) )
% 2.35/1.18 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(emptyinPowerset_type,type,
% 2.35/1.18 emptyinPowerset: $o ).
% 2.35/1.18
% 2.35/1.18 thf(emptyinPowerset,definition,
% 2.35/1.18 ( emptyinPowerset
% 2.35/1.18 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(emptyInPowerset_type,type,
% 2.35/1.18 emptyInPowerset: $o ).
% 2.35/1.18
% 2.35/1.18 thf(emptyInPowerset,definition,
% 2.35/1.18 ( emptyInPowerset
% 2.35/1.18 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(powersetE_type,type,
% 2.35/1.18 powersetE: $o ).
% 2.35/1.18
% 2.35/1.18 thf(powersetE,definition,
% 2.35/1.18 ( powersetE
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ B @ ( powerset @ A ) )
% 2.35/1.18 => ( ( in @ Xx @ B )
% 2.35/1.18 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setunionI_type,type,
% 2.35/1.18 setunionI: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setunionI,definition,
% 2.35/1.18 ( setunionI
% 2.35/1.18 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.35/1.18 ( ( in @ Xx @ B )
% 2.35/1.18 => ( ( in @ B @ A )
% 2.35/1.18 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setunionE_type,type,
% 2.35/1.18 setunionE: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setunionE,definition,
% 2.35/1.18 ( setunionE
% 2.35/1.18 = ( ! [A: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ ( setunion @ A ) )
% 2.35/1.18 => ! [Xphi: $o] :
% 2.35/1.18 ( ! [B: $i] :
% 2.35/1.18 ( ( in @ Xx @ B )
% 2.35/1.18 => ( ( in @ B @ A )
% 2.35/1.18 => Xphi ) )
% 2.35/1.18 => Xphi ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(subPowSU_type,type,
% 2.35/1.18 subPowSU: $o ).
% 2.35/1.18
% 2.35/1.18 thf(subPowSU,definition,
% 2.35/1.18 ( subPowSU
% 2.35/1.18 = ( ! [A: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(exuE2_type,type,
% 2.35/1.18 exuE2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(exuE2,definition,
% 2.35/1.18 ( exuE2
% 2.35/1.18 = ( ! [Xphi: $i > $o] :
% 2.35/1.18 ( ( exu
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.35/1.18 => ? [Xx: $i] :
% 2.35/1.18 ! [Xy: $i] :
% 2.35/1.18 ( ( Xphi @ Xy )
% 2.35/1.18 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(nonemptyImpWitness_type,type,
% 2.35/1.18 nonemptyImpWitness: $o ).
% 2.35/1.18
% 2.35/1.18 thf(nonemptyImpWitness,definition,
% 2.35/1.18 ( nonemptyImpWitness
% 2.35/1.18 = ( ! [A: $i] :
% 2.35/1.18 ( ( nonempty @ A )
% 2.35/1.18 => ? [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 & $true ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(uniqinunit_type,type,
% 2.35/1.18 uniqinunit: $o ).
% 2.35/1.18
% 2.35/1.18 thf(uniqinunit,definition,
% 2.35/1.18 ( uniqinunit
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.35/1.18 => ( Xx = Xy ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(notinsingleton_type,type,
% 2.35/1.18 notinsingleton: $o ).
% 2.35/1.18
% 2.35/1.18 thf(notinsingleton,definition,
% 2.35/1.18 ( notinsingleton
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( Xx != Xy )
% 2.35/1.18 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(eqinunit_type,type,
% 2.35/1.18 eqinunit: $o ).
% 2.35/1.18
% 2.35/1.18 thf(eqinunit,definition,
% 2.35/1.18 ( eqinunit
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( Xx = Xy )
% 2.35/1.18 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(singletonsswitch_type,type,
% 2.35/1.18 singletonsswitch: $o ).
% 2.35/1.18
% 2.35/1.18 thf(singletonsswitch,definition,
% 2.35/1.18 ( singletonsswitch
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.35/1.18 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(upairsetE_type,type,
% 2.35/1.18 upairsetE: $o ).
% 2.35/1.18
% 2.35/1.18 thf(upairsetE,definition,
% 2.35/1.18 ( upairsetE
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.35/1.18 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.35/1.18 => ( ( Xz = Xx )
% 2.35/1.18 | ( Xz = Xy ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(upairsetIL_type,type,
% 2.35/1.18 upairsetIL: $o ).
% 2.35/1.18
% 2.35/1.18 thf(upairsetIL,definition,
% 2.35/1.18 ( upairsetIL
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(upairsetIR_type,type,
% 2.35/1.18 upairsetIR: $o ).
% 2.35/1.18
% 2.35/1.18 thf(upairsetIR,definition,
% 2.35/1.18 ( upairsetIR
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(emptyE1_type,type,
% 2.35/1.18 emptyE1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(emptyE1,definition,
% 2.35/1.18 ( emptyE1
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o] :
% 2.35/1.18 ( ? [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 & ( Xphi @ Xx ) )
% 2.35/1.18 => ( ( ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.35/1.18 = emptyset )
% 2.35/1.18 => $false ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(vacuousDall_type,type,
% 2.35/1.18 vacuousDall: $o ).
% 2.35/1.18
% 2.35/1.18 thf(vacuousDall,definition,
% 2.35/1.18 ( vacuousDall
% 2.35/1.18 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ emptyset )
% 2.35/1.18 => ( Xphi @ Xx ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(quantDeMorgan1_type,type,
% 2.35/1.18 quantDeMorgan1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(quantDeMorgan1,definition,
% 2.35/1.18 ( quantDeMorgan1
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o] :
% 2.35/1.18 ( ~ ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( Xphi @ Xx ) )
% 2.35/1.18 => ? [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(quantDeMorgan2_type,type,
% 2.35/1.18 quantDeMorgan2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(quantDeMorgan2,definition,
% 2.35/1.18 ( quantDeMorgan2
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ~ ( Xphi @ Xx ) )
% 2.35/1.18 => ~ ? [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(quantDeMorgan3_type,type,
% 2.35/1.18 quantDeMorgan3: $o ).
% 2.35/1.18
% 2.35/1.18 thf(quantDeMorgan3,definition,
% 2.35/1.18 ( quantDeMorgan3
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o] :
% 2.35/1.18 ( ~ ? [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 & ( Xphi @ Xx ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(quantDeMorgan4_type,type,
% 2.35/1.18 quantDeMorgan4: $o ).
% 2.35/1.18
% 2.35/1.18 thf(quantDeMorgan4,definition,
% 2.35/1.18 ( quantDeMorgan4
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o] :
% 2.35/1.18 ( ? [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 & ~ ( Xphi @ Xx ) )
% 2.35/1.18 => ~ ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(prop2setI_type,type,
% 2.35/1.18 prop2setI: $o ).
% 2.35/1.18
% 2.35/1.18 thf(prop2setI,definition,
% 2.35/1.18 ( prop2setI
% 2.35/1.18 = ( ! [Xphi: $o] :
% 2.35/1.18 ( Xphi
% 2.35/1.18 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(set2prop_type,type,
% 2.35/1.18 set2prop: $i > $o ).
% 2.35/1.18
% 2.35/1.18 thf(prop2set2propI_type,type,
% 2.35/1.18 prop2set2propI: $o ).
% 2.35/1.18
% 2.35/1.18 thf(prop2set2propI,definition,
% 2.35/1.18 ( prop2set2propI
% 2.35/1.18 = ( ! [Xphi: $o] :
% 2.35/1.18 ( Xphi
% 2.35/1.18 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(notdexE_type,type,
% 2.35/1.18 notdexE: $o ).
% 2.35/1.18
% 2.35/1.18 thf(notdexE,definition,
% 2.35/1.18 ( notdexE
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o] :
% 2.35/1.18 ( ~ ? [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 & ( Xphi @ Xx ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(notdallE_type,type,
% 2.35/1.18 notdallE: $o ).
% 2.35/1.18
% 2.35/1.18 thf(notdallE,definition,
% 2.35/1.18 ( notdallE
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o] :
% 2.35/1.18 ( ~ ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( Xphi @ Xx ) )
% 2.35/1.18 => ? [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(exuI1_type,type,
% 2.35/1.18 exuI1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(exuI1,definition,
% 2.35/1.18 ( exuI1
% 2.35/1.18 = ( ! [Xphi: $i > $o] :
% 2.35/1.18 ( ? [Xx: $i] :
% 2.35/1.18 ( ( Xphi @ Xx )
% 2.35/1.18 & ! [Xy: $i] :
% 2.35/1.18 ( ( Xphi @ Xy )
% 2.35/1.18 => ( Xx = Xy ) ) )
% 2.35/1.18 => ( exu
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(exuI3_type,type,
% 2.35/1.18 exuI3: $o ).
% 2.35/1.18
% 2.35/1.18 thf(exuI3,definition,
% 2.35/1.18 ( exuI3
% 2.35/1.18 = ( ! [Xphi: $i > $o] :
% 2.35/1.18 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.35/1.18 => ( ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( Xphi @ Xx )
% 2.35/1.18 => ( ( Xphi @ Xy )
% 2.35/1.18 => ( Xx = Xy ) ) )
% 2.35/1.18 => ( exu
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(exuI2_type,type,
% 2.35/1.18 exuI2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(exuI2,definition,
% 2.35/1.18 ( exuI2
% 2.35/1.18 = ( ! [Xphi: $i > $o] :
% 2.35/1.18 ( ? [Xx: $i] :
% 2.35/1.18 ! [Xy: $i] :
% 2.35/1.18 ( ( Xphi @ Xy )
% 2.35/1.18 <=> ( Xy = Xx ) )
% 2.35/1.18 => ( exu
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(inCongP_type,type,
% 2.35/1.18 inCongP: $o ).
% 2.35/1.18
% 2.35/1.18 thf(inCongP,definition,
% 2.35/1.18 ( inCongP
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( A = B )
% 2.35/1.18 => ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( Xx = Xy )
% 2.35/1.18 => ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(in__Cong_type,type,
% 2.35/1.18 in__Cong: $o ).
% 2.35/1.18
% 2.35/1.18 thf(in__Cong,definition,
% 2.35/1.18 ( in__Cong
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( A = B )
% 2.35/1.18 => ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( Xx = Xy )
% 2.35/1.18 => ( ( in @ Xx @ A )
% 2.35/1.18 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(exuE3u_type,type,
% 2.35/1.18 exuE3u: $o ).
% 2.35/1.18
% 2.35/1.18 thf(exuE3u,definition,
% 2.35/1.18 ( exuE3u
% 2.35/1.18 = ( ! [Xphi: $i > $o] :
% 2.35/1.18 ( ( exu
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.35/1.18 => ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( Xphi @ Xx )
% 2.35/1.18 => ( ( Xphi @ Xy )
% 2.35/1.18 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(exu__Cong_type,type,
% 2.35/1.18 exu__Cong: $o ).
% 2.35/1.18
% 2.35/1.18 thf(exu__Cong,definition,
% 2.35/1.18 ( exu__Cong
% 2.35/1.18 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.35/1.18 ( ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( Xx = Xy )
% 2.35/1.18 => ( ( Xphi @ Xx )
% 2.35/1.18 <=> ( Xpsi @ Xy ) ) )
% 2.35/1.18 => ( ( exu
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.35/1.18 <=> ( exu
% 2.35/1.18 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(emptyset__Cong_type,type,
% 2.35/1.18 emptyset__Cong: $o ).
% 2.35/1.18
% 2.35/1.18 thf(emptyset__Cong,definition,
% 2.35/1.18 ( emptyset__Cong
% 2.35/1.18 = ( emptyset = emptyset ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoin__Cong_type,type,
% 2.35/1.18 setadjoin__Cong: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoin__Cong,definition,
% 2.35/1.18 ( setadjoin__Cong
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( Xx = Xy )
% 2.35/1.18 => ! [Xz: $i,Xu: $i] :
% 2.35/1.18 ( ( Xz = Xu )
% 2.35/1.18 => ( ( setadjoin @ Xx @ Xz )
% 2.35/1.18 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(powerset__Cong_type,type,
% 2.35/1.18 powerset__Cong: $o ).
% 2.35/1.18
% 2.35/1.18 thf(powerset__Cong,definition,
% 2.35/1.18 ( powerset__Cong
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( A = B )
% 2.35/1.18 => ( ( powerset @ A )
% 2.35/1.18 = ( powerset @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setunion__Cong_type,type,
% 2.35/1.18 setunion__Cong: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setunion__Cong,definition,
% 2.35/1.18 ( setunion__Cong
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( A = B )
% 2.35/1.18 => ( ( setunion @ A )
% 2.35/1.18 = ( setunion @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(omega__Cong_type,type,
% 2.35/1.18 omega__Cong: $o ).
% 2.35/1.18
% 2.35/1.18 thf(omega__Cong,definition,
% 2.35/1.18 ( omega__Cong
% 2.35/1.18 = ( omega = omega ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(exuEu_type,type,
% 2.35/1.18 exuEu: $o ).
% 2.35/1.18
% 2.35/1.18 thf(exuEu,definition,
% 2.35/1.18 ( exuEu
% 2.35/1.18 = ( ! [Xphi: $i > $o] :
% 2.35/1.18 ( ( exu
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.35/1.18 => ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( Xphi @ Xx )
% 2.35/1.18 => ( ( Xphi @ Xy )
% 2.35/1.18 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(descr__Cong_type,type,
% 2.35/1.18 descr__Cong: $o ).
% 2.35/1.18
% 2.35/1.18 thf(descr__Cong,definition,
% 2.35/1.18 ( descr__Cong
% 2.35/1.18 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.35/1.18 ( ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( Xx = Xy )
% 2.35/1.18 => ( ( Xphi @ Xx )
% 2.35/1.18 <=> ( Xpsi @ Xy ) ) )
% 2.35/1.18 => ( ( exu
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.35/1.18 => ( ( exu
% 2.35/1.18 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.35/1.18 => ( ( descr
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.35/1.18 = ( descr
% 2.35/1.18 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(dsetconstr__Cong_type,type,
% 2.35/1.18 dsetconstr__Cong: $o ).
% 2.35/1.18
% 2.35/1.18 thf(dsetconstr__Cong,definition,
% 2.35/1.18 ( dsetconstr__Cong
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( A = B )
% 2.35/1.18 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( ( Xx = Xy )
% 2.35/1.18 => ( ( Xphi @ Xx )
% 2.35/1.18 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.35/1.18 => ( ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.35/1.18 = ( dsetconstr @ B
% 2.35/1.18 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(subset_type,type,
% 2.35/1.18 subset: $i > $i > $o ).
% 2.35/1.18
% 2.35/1.18 thf(disjoint_type,type,
% 2.35/1.18 disjoint: $i > $i > $o ).
% 2.35/1.18
% 2.35/1.18 thf(setsmeet_type,type,
% 2.35/1.18 setsmeet: $i > $i > $o ).
% 2.35/1.18
% 2.35/1.18 thf(subsetI1_type,type,
% 2.35/1.18 subsetI1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(subsetI1,definition,
% 2.35/1.18 ( subsetI1
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ Xx @ B ) )
% 2.35/1.18 => ( subset @ A @ B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(eqimpsubset2_type,type,
% 2.35/1.18 eqimpsubset2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(eqimpsubset2,definition,
% 2.35/1.18 ( eqimpsubset2
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( A = B )
% 2.35/1.18 => ( subset @ B @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(eqimpsubset1_type,type,
% 2.35/1.18 eqimpsubset1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(eqimpsubset1,definition,
% 2.35/1.18 ( eqimpsubset1
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( A = B )
% 2.35/1.18 => ( subset @ A @ B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(subsetI2_type,type,
% 2.35/1.18 subsetI2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(subsetI2,definition,
% 2.35/1.18 ( subsetI2
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ Xx @ B ) )
% 2.35/1.18 => ( subset @ A @ B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(emptysetsubset_type,type,
% 2.35/1.18 emptysetsubset: $o ).
% 2.35/1.18
% 2.35/1.18 thf(emptysetsubset,definition,
% 2.35/1.18 ( emptysetsubset
% 2.35/1.18 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(subsetE_type,type,
% 2.35/1.18 subsetE: $o ).
% 2.35/1.18
% 2.35/1.18 thf(subsetE,definition,
% 2.35/1.18 ( subsetE
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( subset @ A @ B )
% 2.35/1.18 => ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(subsetE2_type,type,
% 2.35/1.18 subsetE2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(subsetE2,definition,
% 2.35/1.18 ( subsetE2
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( subset @ A @ B )
% 2.35/1.18 => ( ~ ( in @ Xx @ B )
% 2.35/1.18 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(notsubsetI_type,type,
% 2.35/1.18 notsubsetI: $o ).
% 2.35/1.18
% 2.35/1.18 thf(notsubsetI,definition,
% 2.35/1.18 ( notsubsetI
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ~ ( in @ Xx @ B )
% 2.35/1.18 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(notequalI1_type,type,
% 2.35/1.18 notequalI1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(notequalI1,definition,
% 2.35/1.18 ( notequalI1
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ~ ( subset @ A @ B )
% 2.35/1.18 => ( A != B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(notequalI2_type,type,
% 2.35/1.18 notequalI2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(notequalI2,definition,
% 2.35/1.18 ( notequalI2
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ~ ( in @ Xx @ B )
% 2.35/1.18 => ( A != B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(subsetRefl_type,type,
% 2.35/1.18 subsetRefl: $o ).
% 2.35/1.18
% 2.35/1.18 thf(subsetRefl,definition,
% 2.35/1.18 ( subsetRefl
% 2.35/1.18 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(subsetTrans_type,type,
% 2.35/1.18 subsetTrans: $o ).
% 2.35/1.18
% 2.35/1.18 thf(subsetTrans,definition,
% 2.35/1.18 ( subsetTrans
% 2.35/1.18 = ( ! [A: $i,B: $i,C: $i] :
% 2.35/1.18 ( ( subset @ A @ B )
% 2.35/1.18 => ( ( subset @ B @ C )
% 2.35/1.18 => ( subset @ A @ C ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoinSub_type,type,
% 2.35/1.18 setadjoinSub: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoinSub,definition,
% 2.35/1.18 ( setadjoinSub
% 2.35/1.18 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoinSub2_type,type,
% 2.35/1.18 setadjoinSub2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setadjoinSub2,definition,
% 2.35/1.18 ( setadjoinSub2
% 2.35/1.18 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.35/1.18 ( ( subset @ A @ B )
% 2.35/1.18 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(subset2powerset_type,type,
% 2.35/1.18 subset2powerset: $o ).
% 2.35/1.18
% 2.35/1.18 thf(subset2powerset,definition,
% 2.35/1.18 ( subset2powerset
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( subset @ A @ B )
% 2.35/1.18 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setextsub_type,type,
% 2.35/1.18 setextsub: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setextsub,definition,
% 2.35/1.18 ( setextsub
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( subset @ A @ B )
% 2.35/1.18 => ( ( subset @ B @ A )
% 2.35/1.18 => ( A = B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(subsetemptysetimpeq_type,type,
% 2.35/1.18 subsetemptysetimpeq: $o ).
% 2.35/1.18
% 2.35/1.18 thf(subsetemptysetimpeq,definition,
% 2.35/1.18 ( subsetemptysetimpeq
% 2.35/1.18 = ( ! [A: $i] :
% 2.35/1.18 ( ( subset @ A @ emptyset )
% 2.35/1.18 => ( A = emptyset ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(powersetI1_type,type,
% 2.35/1.18 powersetI1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(powersetI1,definition,
% 2.35/1.18 ( powersetI1
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( subset @ B @ A )
% 2.35/1.18 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(powersetE1_type,type,
% 2.35/1.18 powersetE1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(powersetE1,definition,
% 2.35/1.18 ( powersetE1
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( in @ B @ ( powerset @ A ) )
% 2.35/1.18 => ( subset @ B @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(inPowerset_type,type,
% 2.35/1.18 inPowerset: $o ).
% 2.35/1.18
% 2.35/1.18 thf(inPowerset,definition,
% 2.35/1.18 ( inPowerset
% 2.35/1.18 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(powersetsubset_type,type,
% 2.35/1.18 powersetsubset: $o ).
% 2.35/1.18
% 2.35/1.18 thf(powersetsubset,definition,
% 2.35/1.18 ( powersetsubset
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( subset @ A @ B )
% 2.35/1.18 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(sepInPowerset_type,type,
% 2.35/1.18 sepInPowerset: $o ).
% 2.35/1.18
% 2.35/1.18 thf(sepInPowerset,definition,
% 2.35/1.18 ( sepInPowerset
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o] :
% 2.35/1.18 ( in
% 2.35/1.18 @ ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.35/1.18 @ ( powerset @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(sepSubset_type,type,
% 2.35/1.18 sepSubset: $o ).
% 2.35/1.18
% 2.35/1.18 thf(sepSubset,definition,
% 2.35/1.18 ( sepSubset
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o] :
% 2.35/1.18 ( subset
% 2.35/1.18 @ ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.35/1.18 @ A ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binunion_type,type,
% 2.35/1.18 binunion: $i > $i > $i ).
% 2.35/1.18
% 2.35/1.18 thf(binunionIL_type,type,
% 2.35/1.18 binunionIL: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binunionIL,definition,
% 2.35/1.18 ( binunionIL
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(upairset2IR_type,type,
% 2.35/1.18 upairset2IR: $o ).
% 2.35/1.18
% 2.35/1.18 thf(upairset2IR,definition,
% 2.35/1.18 ( upairset2IR
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binunionIR_type,type,
% 2.35/1.18 binunionIR: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binunionIR,definition,
% 2.35/1.18 ( binunionIR
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ B )
% 2.35/1.18 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binunionEcases_type,type,
% 2.35/1.18 binunionEcases: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binunionEcases,definition,
% 2.35/1.18 ( binunionEcases
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.35/1.18 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.35/1.18 => ( ( ( in @ Xx @ A )
% 2.35/1.18 => Xphi )
% 2.35/1.18 => ( ( ( in @ Xx @ B )
% 2.35/1.18 => Xphi )
% 2.35/1.18 => Xphi ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binunionE_type,type,
% 2.35/1.18 binunionE: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binunionE,definition,
% 2.35/1.18 ( binunionE
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.35/1.18 => ( ( in @ Xx @ A )
% 2.35/1.18 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binunionLsub_type,type,
% 2.35/1.18 binunionLsub: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binunionLsub,definition,
% 2.35/1.18 ( binunionLsub
% 2.35/1.18 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binunionRsub_type,type,
% 2.35/1.18 binunionRsub: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binunionRsub,definition,
% 2.35/1.18 ( binunionRsub
% 2.35/1.18 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binintersect_type,type,
% 2.35/1.18 binintersect: $i > $i > $i ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectI_type,type,
% 2.35/1.18 binintersectI: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectI,definition,
% 2.35/1.18 ( binintersectI
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( in @ Xx @ B )
% 2.35/1.18 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectSubset5_type,type,
% 2.35/1.18 binintersectSubset5: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectSubset5,definition,
% 2.35/1.18 ( binintersectSubset5
% 2.35/1.18 = ( ! [A: $i,B: $i,C: $i] :
% 2.35/1.18 ( ( subset @ C @ A )
% 2.35/1.18 => ( ( subset @ C @ B )
% 2.35/1.18 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectEL_type,type,
% 2.35/1.18 binintersectEL: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectEL,definition,
% 2.35/1.18 ( binintersectEL
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.35/1.18 => ( in @ Xx @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectLsub_type,type,
% 2.35/1.18 binintersectLsub: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectLsub,definition,
% 2.35/1.18 ( binintersectLsub
% 2.35/1.18 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectSubset2_type,type,
% 2.35/1.18 binintersectSubset2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectSubset2,definition,
% 2.35/1.18 ( binintersectSubset2
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( subset @ A @ B )
% 2.35/1.18 => ( ( binintersect @ A @ B )
% 2.35/1.18 = A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectSubset3_type,type,
% 2.35/1.18 binintersectSubset3: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectSubset3,definition,
% 2.35/1.18 ( binintersectSubset3
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( ( binintersect @ A @ B )
% 2.35/1.18 = B )
% 2.35/1.18 => ( subset @ B @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectER_type,type,
% 2.35/1.18 binintersectER: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectER,definition,
% 2.35/1.18 ( binintersectER
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.35/1.18 => ( in @ Xx @ B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(disjointsetsI1_type,type,
% 2.35/1.18 disjointsetsI1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(disjointsetsI1,definition,
% 2.35/1.18 ( disjointsetsI1
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ~ ? [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 & ( in @ Xx @ B ) )
% 2.35/1.18 => ( ( binintersect @ A @ B )
% 2.35/1.18 = emptyset ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectRsub_type,type,
% 2.35/1.18 binintersectRsub: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectRsub,definition,
% 2.35/1.18 ( binintersectRsub
% 2.35/1.18 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectSubset4_type,type,
% 2.35/1.18 binintersectSubset4: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectSubset4,definition,
% 2.35/1.18 ( binintersectSubset4
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( subset @ B @ A )
% 2.35/1.18 => ( ( binintersect @ A @ B )
% 2.35/1.18 = B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectSubset1_type,type,
% 2.35/1.18 binintersectSubset1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectSubset1,definition,
% 2.35/1.18 ( binintersectSubset1
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( ( binintersect @ A @ B )
% 2.35/1.18 = A )
% 2.35/1.18 => ( subset @ A @ B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(bs114d_type,type,
% 2.35/1.18 bs114d: $o ).
% 2.35/1.18
% 2.35/1.18 thf(bs114d,definition,
% 2.35/1.18 ( bs114d
% 2.35/1.18 = ( ! [A: $i,B: $i,C: $i] :
% 2.35/1.18 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.35/1.18 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(regular_type,type,
% 2.35/1.18 regular: $i > $o ).
% 2.35/1.18
% 2.35/1.18 thf(setminus_type,type,
% 2.35/1.18 setminus: $i > $i > $i ).
% 2.35/1.18
% 2.35/1.18 thf(setminusI_type,type,
% 2.35/1.18 setminusI: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setminusI,definition,
% 2.35/1.18 ( setminusI
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ~ ( in @ Xx @ B )
% 2.35/1.18 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setminusEL_type,type,
% 2.35/1.18 setminusEL: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setminusEL,definition,
% 2.35/1.18 ( setminusEL
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.35/1.18 => ( in @ Xx @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setminusER_type,type,
% 2.35/1.18 setminusER: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setminusER,definition,
% 2.35/1.18 ( setminusER
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.35/1.18 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setminusSubset2_type,type,
% 2.35/1.18 setminusSubset2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setminusSubset2,definition,
% 2.35/1.18 ( setminusSubset2
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( subset @ A @ B )
% 2.35/1.18 => ( ( setminus @ A @ B )
% 2.35/1.18 = emptyset ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setminusERneg_type,type,
% 2.35/1.18 setminusERneg: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setminusERneg,definition,
% 2.35/1.18 ( setminusERneg
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.35/1.18 => ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setminusELneg_type,type,
% 2.35/1.18 setminusELneg: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setminusELneg,definition,
% 2.35/1.18 ( setminusELneg
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.35/1.18 => ( ~ ( in @ Xx @ B )
% 2.35/1.18 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setminusILneg_type,type,
% 2.35/1.18 setminusILneg: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setminusILneg,definition,
% 2.35/1.18 ( setminusILneg
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ~ ( in @ Xx @ A )
% 2.35/1.18 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setminusIRneg_type,type,
% 2.35/1.18 setminusIRneg: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setminusIRneg,definition,
% 2.35/1.18 ( setminusIRneg
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ B )
% 2.35/1.18 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setminusLsub_type,type,
% 2.35/1.18 setminusLsub: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setminusLsub,definition,
% 2.35/1.18 ( setminusLsub
% 2.35/1.18 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setminusSubset1_type,type,
% 2.35/1.18 setminusSubset1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setminusSubset1,definition,
% 2.35/1.18 ( setminusSubset1
% 2.35/1.18 = ( ! [A: $i,B: $i] :
% 2.35/1.18 ( ( ( setminus @ A @ B )
% 2.35/1.18 = emptyset )
% 2.35/1.18 => ( subset @ A @ B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(symdiff_type,type,
% 2.35/1.18 symdiff: $i > $i > $i ).
% 2.35/1.18
% 2.35/1.18 thf(symdiffE_type,type,
% 2.35/1.18 symdiffE: $o ).
% 2.35/1.18
% 2.35/1.18 thf(symdiffE,definition,
% 2.35/1.18 ( symdiffE
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.35/1.18 => ! [Xphi: $o] :
% 2.35/1.18 ( ( ( in @ Xx @ A )
% 2.35/1.18 => ( ~ ( in @ Xx @ B )
% 2.35/1.18 => Xphi ) )
% 2.35/1.18 => ( ( ~ ( in @ Xx @ A )
% 2.35/1.18 => ( ( in @ Xx @ B )
% 2.35/1.18 => Xphi ) )
% 2.35/1.18 => Xphi ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(symdiffI1_type,type,
% 2.35/1.18 symdiffI1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(symdiffI1,definition,
% 2.35/1.18 ( symdiffI1
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ~ ( in @ Xx @ B )
% 2.35/1.18 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(symdiffI2_type,type,
% 2.35/1.18 symdiffI2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(symdiffI2,definition,
% 2.35/1.18 ( symdiffI2
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ~ ( in @ Xx @ A )
% 2.35/1.18 => ( ( in @ Xx @ B )
% 2.35/1.18 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(symdiffIneg1_type,type,
% 2.35/1.18 symdiffIneg1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(symdiffIneg1,definition,
% 2.35/1.18 ( symdiffIneg1
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( in @ Xx @ B )
% 2.35/1.18 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(symdiffIneg2_type,type,
% 2.35/1.18 symdiffIneg2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(symdiffIneg2,definition,
% 2.35/1.18 ( symdiffIneg2
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ~ ( in @ Xx @ A )
% 2.35/1.18 => ( ~ ( in @ Xx @ B )
% 2.35/1.18 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(iskpair_type,type,
% 2.35/1.18 iskpair: $i > $o ).
% 2.35/1.18
% 2.35/1.18 thf(secondinupair_type,type,
% 2.35/1.18 secondinupair: $o ).
% 2.35/1.18
% 2.35/1.18 thf(secondinupair,definition,
% 2.35/1.18 ( secondinupair
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairIL_type,type,
% 2.35/1.18 setukpairIL: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairIL,definition,
% 2.35/1.18 ( setukpairIL
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairIR_type,type,
% 2.35/1.18 setukpairIR: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairIR,definition,
% 2.35/1.18 ( setukpairIR
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(kpairiskpair_type,type,
% 2.35/1.18 kpairiskpair: $o ).
% 2.35/1.18
% 2.35/1.18 thf(kpairiskpair,definition,
% 2.35/1.18 ( kpairiskpair
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(kpair_type,type,
% 2.35/1.18 kpair: $i > $i > $i ).
% 2.35/1.18
% 2.35/1.18 thf(kpairp_type,type,
% 2.35/1.18 kpairp: $o ).
% 2.35/1.18
% 2.35/1.18 thf(kpairp,definition,
% 2.35/1.18 ( kpairp
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(cartprod_type,type,
% 2.35/1.18 cartprod: $i > $i > $i ).
% 2.35/1.18
% 2.35/1.18 thf(singletonsubset_type,type,
% 2.35/1.18 singletonsubset: $o ).
% 2.35/1.18
% 2.35/1.18 thf(singletonsubset,definition,
% 2.35/1.18 ( singletonsubset
% 2.35/1.18 = ( ! [A: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(singletoninpowerset_type,type,
% 2.35/1.18 singletoninpowerset: $o ).
% 2.35/1.18
% 2.35/1.18 thf(singletoninpowerset,definition,
% 2.35/1.18 ( singletoninpowerset
% 2.35/1.18 = ( ! [A: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(singletoninpowunion_type,type,
% 2.35/1.18 singletoninpowunion: $o ).
% 2.35/1.18
% 2.35/1.18 thf(singletoninpowunion,definition,
% 2.35/1.18 ( singletoninpowunion
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(upairset2E_type,type,
% 2.35/1.18 upairset2E: $o ).
% 2.35/1.18
% 2.35/1.18 thf(upairset2E,definition,
% 2.35/1.18 ( upairset2E
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.35/1.18 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.35/1.18 => ( ( Xz = Xx )
% 2.35/1.18 | ( Xz = Xy ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(upairsubunion_type,type,
% 2.35/1.18 upairsubunion: $o ).
% 2.35/1.18
% 2.35/1.18 thf(upairsubunion,definition,
% 2.35/1.18 ( upairsubunion
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(upairinpowunion_type,type,
% 2.35/1.18 upairinpowunion: $o ).
% 2.35/1.18
% 2.35/1.18 thf(upairinpowunion,definition,
% 2.35/1.18 ( upairinpowunion
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(ubforcartprodlem1_type,type,
% 2.35/1.18 ubforcartprodlem1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(ubforcartprodlem1,definition,
% 2.35/1.18 ( ubforcartprodlem1
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(ubforcartprodlem2_type,type,
% 2.35/1.18 ubforcartprodlem2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(ubforcartprodlem2,definition,
% 2.35/1.18 ( ubforcartprodlem2
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(ubforcartprodlem3_type,type,
% 2.35/1.18 ubforcartprodlem3: $o ).
% 2.35/1.18
% 2.35/1.18 thf(ubforcartprodlem3,definition,
% 2.35/1.18 ( ubforcartprodlem3
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodpairin_type,type,
% 2.35/1.18 cartprodpairin: $o ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodpairin,definition,
% 2.35/1.18 ( cartprodpairin
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodmempair1_type,type,
% 2.35/1.18 cartprodmempair1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodmempair1,definition,
% 2.35/1.18 ( cartprodmempair1
% 2.35/1.18 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.35/1.18 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.35/1.18 => ? [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 & ? [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 & ( Xu
% 2.35/1.18 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodmempair_type,type,
% 2.35/1.18 cartprodmempair: $o ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodmempair,definition,
% 2.35/1.18 ( cartprodmempair
% 2.35/1.18 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.35/1.18 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.35/1.18 => ( iskpair @ Xu ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setunionE2_type,type,
% 2.35/1.18 setunionE2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setunionE2,definition,
% 2.35/1.18 ( setunionE2
% 2.35/1.18 = ( ! [A: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ ( setunion @ A ) )
% 2.35/1.18 => ? [X: $i] :
% 2.35/1.18 ( ( in @ X @ A )
% 2.35/1.18 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setunionsingleton1_type,type,
% 2.35/1.18 setunionsingleton1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setunionsingleton1,definition,
% 2.35/1.18 ( setunionsingleton1
% 2.35/1.18 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setunionsingleton2_type,type,
% 2.35/1.18 setunionsingleton2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setunionsingleton2,definition,
% 2.35/1.18 ( setunionsingleton2
% 2.35/1.18 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setunionsingleton_type,type,
% 2.35/1.18 setunionsingleton: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setunionsingleton,definition,
% 2.35/1.18 ( setunionsingleton
% 2.35/1.18 = ( ! [Xx: $i] :
% 2.35/1.18 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.35/1.18 = Xx ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(singleton_type,type,
% 2.35/1.18 singleton: $i > $o ).
% 2.35/1.18
% 2.35/1.18 thf(singletonprop_type,type,
% 2.35/1.18 singletonprop: $o ).
% 2.35/1.18
% 2.35/1.18 thf(singletonprop,definition,
% 2.35/1.18 ( singletonprop
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ A )
% 2.35/1.18 => ( ( Xphi @ Xx )
% 2.35/1.18 => ( ( Xphi @ Xy )
% 2.35/1.18 => ( Xx = Xy ) ) ) ) )
% 2.35/1.18 => ( ? [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 & ( Xphi @ Xx ) )
% 2.35/1.18 => ( singleton
% 2.35/1.18 @ ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(ex1_type,type,
% 2.35/1.18 ex1: $i > ( $i > $o ) > $o ).
% 2.35/1.18
% 2.35/1.18 thf(ex1,definition,
% 2.35/1.18 ( ex1
% 2.35/1.18 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.35/1.18 ( singleton
% 2.35/1.18 @ ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(ex1E1_type,type,
% 2.35/1.18 ex1E1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(ex1E1,definition,
% 2.35/1.18 ( ex1E1
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o] :
% 2.35/1.18 ( ( ex1 @ A
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.35/1.18 => ? [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(ex1I_type,type,
% 2.35/1.18 ex1I: $o ).
% 2.35/1.18
% 2.35/1.18 thf(ex1I,definition,
% 2.35/1.18 ( ex1I
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( Xphi @ Xx )
% 2.35/1.18 => ( ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ A )
% 2.35/1.18 => ( ( Xphi @ Xy )
% 2.35/1.18 => ( Xy = Xx ) ) )
% 2.35/1.18 => ( ex1 @ A
% 2.35/1.18 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(ex1I2_type,type,
% 2.35/1.18 ex1I2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(ex1I2,definition,
% 2.35/1.18 ( ex1I2
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ A )
% 2.35/1.18 => ( ( Xphi @ Xx )
% 2.35/1.18 => ( ( Xphi @ Xy )
% 2.35/1.18 => ( Xx = Xy ) ) ) ) )
% 2.35/1.18 => ( ? [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 & ( Xphi @ Xx ) )
% 2.35/1.18 => ( ex1 @ A
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(singletonsuniq_type,type,
% 2.35/1.18 singletonsuniq: $o ).
% 2.35/1.18
% 2.35/1.18 thf(singletonsuniq,definition,
% 2.35/1.18 ( singletonsuniq
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( ( setadjoin @ Xx @ emptyset )
% 2.35/1.18 = ( setadjoin @ Xy @ emptyset ) )
% 2.35/1.18 => ( Xx = Xy ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(atmost1p_type,type,
% 2.35/1.18 atmost1p: $i > $o ).
% 2.35/1.18
% 2.35/1.18 thf(atleast2p_type,type,
% 2.35/1.18 atleast2p: $i > $o ).
% 2.35/1.18
% 2.35/1.18 thf(atmost2p_type,type,
% 2.35/1.18 atmost2p: $i > $o ).
% 2.35/1.18
% 2.35/1.18 thf(upairsetp_type,type,
% 2.35/1.18 upairsetp: $i > $o ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairinjL1_type,type,
% 2.35/1.18 setukpairinjL1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairinjL1,definition,
% 2.35/1.18 ( setukpairinjL1
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.35/1.18 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.35/1.18 => ( Xx = Xz ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(kfstsingleton_type,type,
% 2.35/1.18 kfstsingleton: $o ).
% 2.35/1.18
% 2.35/1.18 thf(kfstsingleton,definition,
% 2.35/1.18 ( kfstsingleton
% 2.35/1.18 = ( ! [Xu: $i] :
% 2.35/1.18 ( ( iskpair @ Xu )
% 2.35/1.18 => ( singleton
% 2.35/1.18 @ ( dsetconstr @ ( setunion @ Xu )
% 2.35/1.18 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(theprop_type,type,
% 2.35/1.18 theprop: $o ).
% 2.35/1.18
% 2.35/1.18 thf(theprop,definition,
% 2.35/1.18 ( theprop
% 2.35/1.18 = ( ! [X: $i] :
% 2.35/1.18 ( ( singleton @ X )
% 2.35/1.18 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(kfst_type,type,
% 2.35/1.18 kfst: $i > $i ).
% 2.35/1.18
% 2.35/1.18 thf(kfstpairEq_type,type,
% 2.35/1.18 kfstpairEq: $o ).
% 2.35/1.18
% 2.35/1.18 thf(kfstpairEq,definition,
% 2.35/1.18 ( kfstpairEq
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.35/1.18 = Xx ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodfstin_type,type,
% 2.35/1.18 cartprodfstin: $o ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodfstin,definition,
% 2.35/1.18 ( cartprodfstin
% 2.35/1.18 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.35/1.18 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.35/1.18 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairinjL2_type,type,
% 2.35/1.18 setukpairinjL2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairinjL2,definition,
% 2.35/1.18 ( setukpairinjL2
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.35/1.18 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.35/1.18 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.35/1.18 => ( Xx = Xz ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairinjL_type,type,
% 2.35/1.18 setukpairinjL: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairinjL,definition,
% 2.35/1.18 ( setukpairinjL
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.35/1.18 ( ( ( kpair @ Xx @ Xy )
% 2.35/1.18 = ( kpair @ Xz @ Xu ) )
% 2.35/1.18 => ( Xx = Xz ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairinjR11_type,type,
% 2.35/1.18 setukpairinjR11: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairinjR11,definition,
% 2.35/1.18 ( setukpairinjR11
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( Xx = Xy )
% 2.35/1.18 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.35/1.18 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairinjR12_type,type,
% 2.35/1.18 setukpairinjR12: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairinjR12,definition,
% 2.35/1.18 ( setukpairinjR12
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( Xx = Xy )
% 2.35/1.18 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.35/1.18 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairinjR1_type,type,
% 2.35/1.18 setukpairinjR1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairinjR1,definition,
% 2.35/1.18 ( setukpairinjR1
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.35/1.18 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.35/1.18 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.35/1.18 => ( ( Xz = Xu )
% 2.35/1.18 => ( Xy = Xu ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(upairequniteq_type,type,
% 2.35/1.18 upairequniteq: $o ).
% 2.35/1.18
% 2.35/1.18 thf(upairequniteq,definition,
% 2.35/1.18 ( upairequniteq
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.35/1.18 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.35/1.18 = ( setadjoin @ Xz @ emptyset ) )
% 2.35/1.18 => ( Xx = Xy ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairinjR2_type,type,
% 2.35/1.18 setukpairinjR2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairinjR2,definition,
% 2.35/1.18 ( setukpairinjR2
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.35/1.18 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.35/1.18 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.35/1.18 => ( Xy = Xu ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairinjR_type,type,
% 2.35/1.18 setukpairinjR: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setukpairinjR,definition,
% 2.35/1.18 ( setukpairinjR
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.35/1.18 ( ( ( kpair @ Xx @ Xy )
% 2.35/1.18 = ( kpair @ Xz @ Xu ) )
% 2.35/1.18 => ( Xy = Xu ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(ksndsingleton_type,type,
% 2.35/1.18 ksndsingleton: $o ).
% 2.35/1.18
% 2.35/1.18 thf(ksndsingleton,definition,
% 2.35/1.18 ( ksndsingleton
% 2.35/1.18 = ( ! [Xu: $i] :
% 2.35/1.18 ( ( iskpair @ Xu )
% 2.35/1.18 => ( singleton
% 2.35/1.18 @ ( dsetconstr @ ( setunion @ Xu )
% 2.35/1.18 @ ^ [Xx: $i] :
% 2.35/1.18 ( Xu
% 2.35/1.18 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(ksnd_type,type,
% 2.35/1.18 ksnd: $i > $i ).
% 2.35/1.18
% 2.35/1.18 thf(ksndpairEq_type,type,
% 2.35/1.18 ksndpairEq: $o ).
% 2.35/1.18
% 2.35/1.18 thf(ksndpairEq,definition,
% 2.35/1.18 ( ksndpairEq
% 2.35/1.18 = ( ! [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.35/1.18 = Xy ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(kpairsurjEq_type,type,
% 2.35/1.18 kpairsurjEq: $o ).
% 2.35/1.18
% 2.35/1.18 thf(kpairsurjEq,definition,
% 2.35/1.18 ( kpairsurjEq
% 2.35/1.18 = ( ! [Xu: $i] :
% 2.35/1.18 ( ( iskpair @ Xu )
% 2.35/1.18 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.35/1.18 = Xu ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodsndin_type,type,
% 2.35/1.18 cartprodsndin: $o ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodsndin,definition,
% 2.35/1.18 ( cartprodsndin
% 2.35/1.18 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.35/1.18 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.35/1.18 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodpairmemEL_type,type,
% 2.35/1.18 cartprodpairmemEL: $o ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodpairmemEL,definition,
% 2.35/1.18 ( cartprodpairmemEL
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.35/1.18 => ( in @ Xx @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodpairmemER_type,type,
% 2.35/1.18 cartprodpairmemER: $o ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodpairmemER,definition,
% 2.35/1.18 ( cartprodpairmemER
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.35/1.18 => ( in @ Xy @ B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodmempaircEq_type,type,
% 2.35/1.18 cartprodmempaircEq: $o ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodmempaircEq,definition,
% 2.35/1.18 ( cartprodmempaircEq
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( ( kpair @ Xx @ Xy )
% 2.35/1.18 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodfstpairEq_type,type,
% 2.35/1.18 cartprodfstpairEq: $o ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodfstpairEq,definition,
% 2.35/1.18 ( cartprodfstpairEq
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.35/1.18 = Xx ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodsndpairEq_type,type,
% 2.35/1.18 cartprodsndpairEq: $o ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodsndpairEq,definition,
% 2.35/1.18 ( cartprodsndpairEq
% 2.35/1.18 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.35/1.18 = Xy ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodpairsurjEq_type,type,
% 2.35/1.18 cartprodpairsurjEq: $o ).
% 2.35/1.18
% 2.35/1.18 thf(cartprodpairsurjEq,definition,
% 2.35/1.18 ( cartprodpairsurjEq
% 2.35/1.18 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.35/1.18 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.35/1.18 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.35/1.18 = Xu ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(breln_type,type,
% 2.35/1.18 breln: $i > $i > $i > $o ).
% 2.35/1.18
% 2.35/1.18 thf(breln,definition,
% 2.35/1.18 ( breln
% 2.35/1.18 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(dpsetconstr_type,type,
% 2.35/1.18 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.35/1.18
% 2.35/1.18 thf(dpsetconstrI_type,type,
% 2.35/1.18 dpsetconstrI: $o ).
% 2.35/1.18
% 2.35/1.18 thf(dpsetconstrI,definition,
% 2.35/1.18 ( dpsetconstrI
% 2.35/1.18 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( ( Xphi @ Xx @ Xy )
% 2.35/1.18 => ( in @ ( kpair @ Xx @ Xy )
% 2.35/1.18 @ ( dpsetconstr @ A @ B
% 2.35/1.18 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(dpsetconstrSub_type,type,
% 2.35/1.18 dpsetconstrSub: $o ).
% 2.35/1.18
% 2.35/1.18 thf(dpsetconstrSub,definition,
% 2.35/1.18 ( dpsetconstrSub
% 2.35/1.18 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.35/1.18 ( subset
% 2.35/1.18 @ ( dpsetconstr @ A @ B
% 2.35/1.18 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.35/1.18 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setOfPairsIsBReln_type,type,
% 2.35/1.18 setOfPairsIsBReln: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setOfPairsIsBReln,definition,
% 2.35/1.18 ( setOfPairsIsBReln
% 2.35/1.18 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.35/1.18 ( breln @ A @ B
% 2.35/1.18 @ ( dpsetconstr @ A @ B
% 2.35/1.18 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(dpsetconstrERa_type,type,
% 2.35/1.18 dpsetconstrERa: $o ).
% 2.35/1.18
% 2.35/1.18 thf(dpsetconstrERa,definition,
% 2.35/1.18 ( dpsetconstrERa
% 2.35/1.18 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.35/1.18 @ ( dpsetconstr @ A @ B
% 2.35/1.18 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.35/1.18 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(dpsetconstrEL1_type,type,
% 2.35/1.18 dpsetconstrEL1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(dpsetconstrEL1,definition,
% 2.35/1.18 ( dpsetconstrEL1
% 2.35/1.18 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( in @ ( kpair @ Xx @ Xy )
% 2.35/1.18 @ ( dpsetconstr @ A @ B
% 2.35/1.18 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.35/1.18 => ( in @ Xx @ A ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(dpsetconstrEL2_type,type,
% 2.35/1.18 dpsetconstrEL2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(dpsetconstrEL2,definition,
% 2.35/1.18 ( dpsetconstrEL2
% 2.35/1.18 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( in @ ( kpair @ Xx @ Xy )
% 2.35/1.18 @ ( dpsetconstr @ A @ B
% 2.35/1.18 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.35/1.18 => ( in @ Xy @ B ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(dpsetconstrER_type,type,
% 2.35/1.18 dpsetconstrER: $o ).
% 2.35/1.18
% 2.35/1.18 thf(dpsetconstrER,definition,
% 2.35/1.18 ( dpsetconstrER
% 2.35/1.18 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( in @ ( kpair @ Xx @ Xy )
% 2.35/1.18 @ ( dpsetconstr @ A @ B
% 2.35/1.18 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.35/1.18 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(func_type,type,
% 2.35/1.18 func: $i > $i > $i > $o ).
% 2.35/1.18
% 2.35/1.18 thf(funcSet_type,type,
% 2.35/1.18 funcSet: $i > $i > $i ).
% 2.35/1.18
% 2.35/1.18 thf(funcImageSingleton_type,type,
% 2.35/1.18 funcImageSingleton: $o ).
% 2.35/1.18
% 2.35/1.18 thf(funcImageSingleton,definition,
% 2.35/1.18 ( funcImageSingleton
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( func @ A @ B @ Xf )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( singleton
% 2.35/1.18 @ ( dsetconstr @ B
% 2.35/1.18 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(apProp_type,type,
% 2.35/1.18 apProp: $o ).
% 2.35/1.18
% 2.35/1.18 thf(apProp,definition,
% 2.35/1.18 ( apProp
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( func @ A @ B @ Xf )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in
% 2.35/1.18 @ ( setunion
% 2.35/1.18 @ ( dsetconstr @ B
% 2.35/1.18 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.35/1.18 @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(ap_type,type,
% 2.35/1.18 ap: $i > $i > $i > $i > $i ).
% 2.35/1.18
% 2.35/1.18 thf(app_type,type,
% 2.35/1.18 app: $o ).
% 2.35/1.18
% 2.35/1.18 thf(app,definition,
% 2.35/1.18 ( app
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( func @ A @ B @ Xf )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(infuncsetfunc_type,type,
% 2.35/1.18 infuncsetfunc: $o ).
% 2.35/1.18
% 2.35/1.18 thf(infuncsetfunc,definition,
% 2.35/1.18 ( infuncsetfunc
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.35/1.18 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(ap2p_type,type,
% 2.35/1.18 ap2p: $o ).
% 2.35/1.18
% 2.35/1.18 thf(ap2p,definition,
% 2.35/1.18 ( ap2p
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(funcinfuncset_type,type,
% 2.35/1.18 funcinfuncset: $o ).
% 2.35/1.18
% 2.35/1.18 thf(funcinfuncset,definition,
% 2.35/1.18 ( funcinfuncset
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( func @ A @ B @ Xf )
% 2.35/1.18 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(lamProp_type,type,
% 2.35/1.18 lamProp: $o ).
% 2.35/1.18
% 2.35/1.18 thf(lamProp,definition,
% 2.35/1.18 ( lamProp
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.35/1.18 => ( func @ A @ B
% 2.35/1.18 @ ( dpsetconstr @ A @ B
% 2.35/1.18 @ ^ [Xx: $i,Xy: $i] :
% 2.35/1.18 ( ( Xf @ Xx )
% 2.35/1.18 = Xy ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(lam_type,type,
% 2.35/1.18 lam: $i > $i > ( $i > $i ) > $i ).
% 2.35/1.18
% 2.35/1.18 thf(lamp_type,type,
% 2.35/1.18 lamp: $o ).
% 2.35/1.18
% 2.35/1.18 thf(lamp,definition,
% 2.35/1.18 ( lamp
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.35/1.18 => ( func @ A @ B
% 2.35/1.18 @ ( lam @ A @ B
% 2.35/1.18 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(lam2p_type,type,
% 2.35/1.18 lam2p: $o ).
% 2.35/1.18
% 2.35/1.18 thf(lam2p,definition,
% 2.35/1.18 ( lam2p
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.35/1.18 => ( in
% 2.35/1.18 @ ( lam @ A @ B
% 2.35/1.18 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.35/1.18 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(brelnall1_type,type,
% 2.35/1.18 brelnall1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(brelnall1,definition,
% 2.35/1.18 ( brelnall1
% 2.35/1.18 = ( ! [A: $i,B: $i,R: $i] :
% 2.35/1.18 ( ( breln @ A @ B @ R )
% 2.35/1.18 => ! [Xphi: $i > $o] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.35/1.18 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ R )
% 2.35/1.18 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(brelnall2_type,type,
% 2.35/1.18 brelnall2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(brelnall2,definition,
% 2.35/1.18 ( brelnall2
% 2.35/1.18 = ( ! [A: $i,B: $i,R: $i] :
% 2.35/1.18 ( ( breln @ A @ B @ R )
% 2.35/1.18 => ! [Xphi: $i > $o] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.35/1.18 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ R )
% 2.35/1.18 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(ex1E2_type,type,
% 2.35/1.18 ex1E2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(ex1E2,definition,
% 2.35/1.18 ( ex1E2
% 2.35/1.18 = ( ! [A: $i,Xphi: $i > $o] :
% 2.35/1.18 ( ( ex1 @ A
% 2.35/1.18 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ A )
% 2.35/1.18 => ( ( Xphi @ Xx )
% 2.35/1.18 => ( ( Xphi @ Xy )
% 2.35/1.18 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(funcGraphProp1_type,type,
% 2.35/1.18 funcGraphProp1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(funcGraphProp1,definition,
% 2.35/1.18 ( funcGraphProp1
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( func @ A @ B @ Xf )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(funcGraphProp3_type,type,
% 2.35/1.18 funcGraphProp3: $o ).
% 2.35/1.18
% 2.35/1.18 thf(funcGraphProp3,definition,
% 2.35/1.18 ( funcGraphProp3
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(funcGraphProp2_type,type,
% 2.35/1.18 funcGraphProp2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(funcGraphProp2,definition,
% 2.35/1.18 ( funcGraphProp2
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( func @ A @ B @ Xf )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.35/1.18 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.35/1.18 = Xy ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(funcextLem_type,type,
% 2.35/1.18 funcextLem: $o ).
% 2.35/1.18
% 2.35/1.18 thf(funcextLem,definition,
% 2.35/1.18 ( funcextLem
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( func @ A @ B @ Xf )
% 2.35/1.18 => ! [Xg: $i] :
% 2.35/1.18 ( ( func @ A @ B @ Xg )
% 2.35/1.18 => ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.35/1.18 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.35/1.18 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(funcGraphProp4_type,type,
% 2.35/1.18 funcGraphProp4: $o ).
% 2.35/1.18
% 2.35/1.18 thf(funcGraphProp4,definition,
% 2.35/1.18 ( funcGraphProp4
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.35/1.18 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.35/1.18 = Xy ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(subbreln_type,type,
% 2.35/1.18 subbreln: $o ).
% 2.35/1.18
% 2.35/1.18 thf(subbreln,definition,
% 2.35/1.18 ( subbreln
% 2.35/1.18 = ( ! [A: $i,B: $i,R: $i] :
% 2.35/1.18 ( ( breln @ A @ B @ R )
% 2.35/1.18 => ! [S: $i] :
% 2.35/1.18 ( ( breln @ A @ B @ S )
% 2.35/1.18 => ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.35/1.18 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.35/1.18 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(eqbreln_type,type,
% 2.35/1.18 eqbreln: $o ).
% 2.35/1.18
% 2.35/1.18 thf(eqbreln,definition,
% 2.35/1.18 ( eqbreln
% 2.35/1.18 = ( ! [A: $i,B: $i,R: $i] :
% 2.35/1.18 ( ( breln @ A @ B @ R )
% 2.35/1.18 => ! [S: $i] :
% 2.35/1.18 ( ( breln @ A @ B @ S )
% 2.35/1.18 => ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.35/1.18 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.35/1.18 => ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ B )
% 2.35/1.18 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.35/1.18 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.35/1.18 => ( R = S ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(funcext_type,type,
% 2.35/1.18 funcext: $o ).
% 2.35/1.18
% 2.35/1.18 thf(funcext,definition,
% 2.35/1.18 ( funcext
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( func @ A @ B @ Xf )
% 2.35/1.18 => ! [Xg: $i] :
% 2.35/1.18 ( ( func @ A @ B @ Xg )
% 2.35/1.18 => ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.35/1.18 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.35/1.18 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(funcext2_type,type,
% 2.35/1.18 funcext2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(funcext2,definition,
% 2.35/1.18 ( funcext2
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.35/1.18 => ! [Xg: $i] :
% 2.35/1.18 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.35/1.18 => ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.35/1.18 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.35/1.18 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(ap2apEq1_type,type,
% 2.35/1.18 ap2apEq1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(ap2apEq1,definition,
% 2.35/1.18 ( ap2apEq1
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.35/1.18 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(ap2apEq2_type,type,
% 2.35/1.18 ap2apEq2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(ap2apEq2,definition,
% 2.35/1.18 ( ap2apEq2
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( func @ A @ B @ Xf )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.35/1.18 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(beta1_type,type,
% 2.35/1.18 beta1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(beta1,definition,
% 2.35/1.18 ( beta1
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( ap @ A @ B
% 2.35/1.18 @ ( lam @ A @ B
% 2.35/1.18 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.35/1.18 @ Xx )
% 2.35/1.18 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(eta1_type,type,
% 2.35/1.18 eta1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(eta1,definition,
% 2.35/1.18 ( eta1
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( func @ A @ B @ Xf )
% 2.35/1.18 => ( ( lam @ A @ B
% 2.35/1.18 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.35/1.18 = Xf ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(lam2lamEq_type,type,
% 2.35/1.18 lam2lamEq: $o ).
% 2.35/1.18
% 2.35/1.18 thf(lam2lamEq,definition,
% 2.35/1.18 ( lam2lamEq
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.35/1.18 => ( ( lam @ A @ B
% 2.35/1.18 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.35/1.18 = ( lam @ A @ B
% 2.35/1.18 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(beta2_type,type,
% 2.35/1.18 beta2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(beta2,definition,
% 2.35/1.18 ( beta2
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.35/1.18 ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( ap @ A @ B
% 2.35/1.18 @ ( lam @ A @ B
% 2.35/1.18 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.35/1.18 @ Xx )
% 2.35/1.18 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(eta2_type,type,
% 2.35/1.18 eta2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(eta2,definition,
% 2.35/1.18 ( eta2
% 2.35/1.18 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.35/1.18 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.35/1.18 => ( ( lam @ A @ B
% 2.35/1.18 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.35/1.18 = Xf ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(iffalseProp1_type,type,
% 2.35/1.18 iffalseProp1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(iffalseProp1,definition,
% 2.35/1.18 ( iffalseProp1
% 2.35/1.18 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ A )
% 2.35/1.18 => ( ~ Xphi
% 2.35/1.18 => ( in @ Xy
% 2.35/1.18 @ ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xz: $i] :
% 2.35/1.18 ( ( Xphi
% 2.35/1.18 & ( Xz = Xx ) )
% 2.35/1.18 | ( ~ Xphi
% 2.35/1.18 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(iffalseProp2_type,type,
% 2.35/1.18 iffalseProp2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(iffalseProp2,definition,
% 2.35/1.18 ( iffalseProp2
% 2.35/1.18 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ A )
% 2.35/1.18 => ( ~ Xphi
% 2.35/1.18 => ( ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xz: $i] :
% 2.35/1.18 ( ( Xphi
% 2.35/1.18 & ( Xz = Xx ) )
% 2.35/1.18 | ( ~ Xphi
% 2.35/1.18 & ( Xz = Xy ) ) ) )
% 2.35/1.18 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(iftrueProp1_type,type,
% 2.35/1.18 iftrueProp1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(iftrueProp1,definition,
% 2.35/1.18 ( iftrueProp1
% 2.35/1.18 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ A )
% 2.35/1.18 => ( Xphi
% 2.35/1.18 => ( in @ Xx
% 2.35/1.18 @ ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xz: $i] :
% 2.35/1.18 ( ( Xphi
% 2.35/1.18 & ( Xz = Xx ) )
% 2.35/1.18 | ( ~ Xphi
% 2.35/1.18 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(iftrueProp2_type,type,
% 2.35/1.18 iftrueProp2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(iftrueProp2,definition,
% 2.35/1.18 ( iftrueProp2
% 2.35/1.18 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ A )
% 2.35/1.18 => ( Xphi
% 2.35/1.18 => ( ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xz: $i] :
% 2.35/1.18 ( ( Xphi
% 2.35/1.18 & ( Xz = Xx ) )
% 2.35/1.18 | ( ~ Xphi
% 2.35/1.18 & ( Xz = Xy ) ) ) )
% 2.35/1.18 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(ifSingleton_type,type,
% 2.35/1.18 ifSingleton: $o ).
% 2.35/1.18
% 2.35/1.18 thf(ifSingleton,definition,
% 2.35/1.18 ( ifSingleton
% 2.35/1.18 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ A )
% 2.35/1.18 => ( singleton
% 2.35/1.18 @ ( dsetconstr @ A
% 2.35/1.18 @ ^ [Xz: $i] :
% 2.35/1.18 ( ( Xphi
% 2.35/1.18 & ( Xz = Xx ) )
% 2.35/1.18 | ( ~ Xphi
% 2.35/1.18 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(if_type,type,
% 2.35/1.18 if: $i > $o > $i > $i > $i ).
% 2.35/1.18
% 2.35/1.18 thf(ifp_type,type,
% 2.35/1.18 ifp: $o ).
% 2.35/1.18
% 2.35/1.18 thf(ifp,definition,
% 2.35/1.18 ( ifp
% 2.35/1.18 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ A )
% 2.35/1.18 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(theeq_type,type,
% 2.35/1.18 theeq: $o ).
% 2.35/1.18
% 2.35/1.18 thf(theeq,definition,
% 2.35/1.18 ( theeq
% 2.35/1.18 = ( ! [X: $i] :
% 2.35/1.18 ( ( singleton @ X )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ X )
% 2.35/1.18 => ( ( setunion @ X )
% 2.35/1.18 = Xx ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(iftrue_type,type,
% 2.35/1.18 iftrue: $o ).
% 2.35/1.18
% 2.35/1.18 thf(iftrue,definition,
% 2.35/1.18 ( iftrue
% 2.35/1.18 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ A )
% 2.35/1.18 => ( Xphi
% 2.35/1.18 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.35/1.18 = Xx ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(iffalse_type,type,
% 2.35/1.18 iffalse: $o ).
% 2.35/1.18
% 2.35/1.18 thf(iffalse,definition,
% 2.35/1.18 ( iffalse
% 2.35/1.18 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ A )
% 2.35/1.18 => ( ~ Xphi
% 2.35/1.18 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.35/1.18 = Xy ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(iftrueorfalse_type,type,
% 2.35/1.18 iftrueorfalse: $o ).
% 2.35/1.18
% 2.35/1.18 thf(iftrueorfalse,definition,
% 2.35/1.18 ( iftrueorfalse
% 2.35/1.18 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ! [Xy: $i] :
% 2.35/1.18 ( ( in @ Xy @ A )
% 2.35/1.18 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectT_lem_type,type,
% 2.35/1.18 binintersectT_lem: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectT_lem,definition,
% 2.35/1.18 ( binintersectT_lem
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Y: $i] :
% 2.35/1.18 ( ( in @ Y @ ( powerset @ A ) )
% 2.35/1.18 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binunionT_lem_type,type,
% 2.35/1.18 binunionT_lem: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binunionT_lem,definition,
% 2.35/1.18 ( binunionT_lem
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Y: $i] :
% 2.35/1.18 ( ( in @ Y @ ( powerset @ A ) )
% 2.35/1.18 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(powersetT_lem_type,type,
% 2.35/1.18 powersetT_lem: $o ).
% 2.35/1.18
% 2.35/1.18 thf(powersetT_lem,definition,
% 2.35/1.18 ( powersetT_lem
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setminusT_lem_type,type,
% 2.35/1.18 setminusT_lem: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setminusT_lem,definition,
% 2.35/1.18 ( setminusT_lem
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Y: $i] :
% 2.35/1.18 ( ( in @ Y @ ( powerset @ A ) )
% 2.35/1.18 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(complementT_lem_type,type,
% 2.35/1.18 complementT_lem: $o ).
% 2.35/1.18
% 2.35/1.18 thf(complementT_lem,definition,
% 2.35/1.18 ( complementT_lem
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(setextT_type,type,
% 2.35/1.18 setextT: $o ).
% 2.35/1.18
% 2.35/1.18 thf(setextT,definition,
% 2.35/1.18 ( setextT
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Y: $i] :
% 2.35/1.18 ( ( in @ Y @ ( powerset @ A ) )
% 2.35/1.18 => ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( in @ Xx @ X )
% 2.35/1.18 => ( in @ Xx @ Y ) ) )
% 2.35/1.18 => ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( in @ Xx @ Y )
% 2.35/1.18 => ( in @ Xx @ X ) ) )
% 2.35/1.18 => ( X = Y ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(subsetTI_type,type,
% 2.35/1.18 subsetTI: $o ).
% 2.35/1.18
% 2.35/1.18 thf(subsetTI,definition,
% 2.35/1.18 ( subsetTI
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Y: $i] :
% 2.35/1.18 ( ( in @ Y @ ( powerset @ A ) )
% 2.35/1.18 => ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( in @ Xx @ X )
% 2.35/1.18 => ( in @ Xx @ Y ) ) )
% 2.35/1.18 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(powersetTI1_type,type,
% 2.35/1.18 powersetTI1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(powersetTI1,definition,
% 2.35/1.18 ( powersetTI1
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Y: $i] :
% 2.35/1.18 ( ( in @ Y @ ( powerset @ A ) )
% 2.35/1.18 => ( ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( in @ Xx @ X )
% 2.35/1.18 => ( in @ Xx @ Y ) ) )
% 2.35/1.18 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(powersetTE1_type,type,
% 2.35/1.18 powersetTE1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(powersetTE1,definition,
% 2.35/1.18 ( powersetTE1
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Y: $i] :
% 2.35/1.18 ( ( in @ Y @ ( powerset @ A ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( in @ X @ ( powerset @ Y ) )
% 2.35/1.18 => ( ( in @ Xx @ X )
% 2.35/1.18 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(complementTI1_type,type,
% 2.35/1.18 complementTI1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(complementTI1,definition,
% 2.35/1.18 ( complementTI1
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( in @ Xx @ X )
% 2.35/1.18 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(complementTE1_type,type,
% 2.35/1.18 complementTE1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(complementTE1,definition,
% 2.35/1.18 ( complementTE1
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.35/1.18 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectTELcontra_type,type,
% 2.35/1.18 binintersectTELcontra: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectTELcontra,definition,
% 2.35/1.18 ( binintersectTELcontra
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Y: $i] :
% 2.35/1.18 ( ( in @ Y @ ( powerset @ A ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ~ ( in @ Xx @ X )
% 2.35/1.18 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectTERcontra_type,type,
% 2.35/1.18 binintersectTERcontra: $o ).
% 2.35/1.18
% 2.35/1.18 thf(binintersectTERcontra,definition,
% 2.35/1.18 ( binintersectTERcontra
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Y: $i] :
% 2.35/1.18 ( ( in @ Y @ ( powerset @ A ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ~ ( in @ Xx @ Y )
% 2.35/1.18 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(contrasubsetT_type,type,
% 2.35/1.18 contrasubsetT: $o ).
% 2.35/1.18
% 2.35/1.18 thf(contrasubsetT,definition,
% 2.35/1.18 ( contrasubsetT
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Y: $i] :
% 2.35/1.18 ( ( in @ Y @ ( powerset @ A ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.35/1.18 => ( ( in @ Xx @ Y )
% 2.35/1.18 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(contrasubsetT1_type,type,
% 2.35/1.18 contrasubsetT1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(contrasubsetT1,definition,
% 2.35/1.18 ( contrasubsetT1
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Y: $i] :
% 2.35/1.18 ( ( in @ Y @ ( powerset @ A ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( subset @ X @ Y )
% 2.35/1.18 => ( ~ ( in @ Xx @ Y )
% 2.35/1.18 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(contrasubsetT2_type,type,
% 2.35/1.18 contrasubsetT2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(contrasubsetT2,definition,
% 2.35/1.18 ( contrasubsetT2
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Y: $i] :
% 2.35/1.18 ( ( in @ Y @ ( powerset @ A ) )
% 2.35/1.18 => ( ( subset @ X @ Y )
% 2.35/1.18 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(contrasubsetT3_type,type,
% 2.35/1.18 contrasubsetT3: $o ).
% 2.35/1.18
% 2.35/1.18 thf(contrasubsetT3,definition,
% 2.35/1.18 ( contrasubsetT3
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Y: $i] :
% 2.35/1.18 ( ( in @ Y @ ( powerset @ A ) )
% 2.35/1.18 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.35/1.18 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(doubleComplementI1_type,type,
% 2.35/1.18 doubleComplementI1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(doubleComplementI1,definition,
% 2.35/1.18 ( doubleComplementI1
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( in @ Xx @ X )
% 2.35/1.18 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(doubleComplementE1_type,type,
% 2.35/1.18 doubleComplementE1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(doubleComplementE1,definition,
% 2.35/1.18 ( doubleComplementE1
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 2.35/1.18 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(doubleComplementSub1_type,type,
% 2.35/1.18 doubleComplementSub1: $o ).
% 2.35/1.18
% 2.35/1.18 thf(doubleComplementSub1,definition,
% 2.35/1.18 ( doubleComplementSub1
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(doubleComplementSub2_type,type,
% 2.35/1.18 doubleComplementSub2: $o ).
% 2.35/1.18
% 2.35/1.18 thf(doubleComplementSub2,definition,
% 2.35/1.18 ( doubleComplementSub2
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(doubleComplementEq_type,type,
% 2.35/1.18 doubleComplementEq: $o ).
% 2.35/1.18
% 2.35/1.18 thf(doubleComplementEq,definition,
% 2.35/1.18 ( doubleComplementEq
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ( X
% 2.35/1.18 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(complementTnotintersectT_type,type,
% 2.35/1.18 complementTnotintersectT: $o ).
% 2.35/1.18
% 2.35/1.18 thf(complementTnotintersectT,definition,
% 2.35/1.18 ( complementTnotintersectT
% 2.35/1.18 = ( ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Y: $i] :
% 2.35/1.18 ( ( in @ Y @ ( powerset @ A ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.35/1.18 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.35/1.18
% 2.35/1.18 thf(complementImpComplementIntersect,conjecture,
% 2.35/1.18 ( setextAx
% 2.35/1.18 => ( emptysetAx
% 2.35/1.18 => ( setadjoinAx
% 2.35/1.18 => ( powersetAx
% 2.35/1.18 => ( setunionAx
% 2.35/1.18 => ( omega0Ax
% 2.35/1.18 => ( omegaSAx
% 2.35/1.18 => ( omegaIndAx
% 2.35/1.18 => ( replAx
% 2.35/1.18 => ( foundationAx
% 2.35/1.18 => ( wellorderingAx
% 2.35/1.18 => ( descrp
% 2.35/1.18 => ( dsetconstrI
% 2.35/1.18 => ( dsetconstrEL
% 2.35/1.18 => ( dsetconstrER
% 2.35/1.18 => ( exuE1
% 2.35/1.18 => ( prop2setE
% 2.35/1.18 => ( emptysetE
% 2.35/1.18 => ( emptysetimpfalse
% 2.35/1.18 => ( notinemptyset
% 2.35/1.18 => ( exuE3e
% 2.35/1.18 => ( setext
% 2.35/1.18 => ( emptyI
% 2.35/1.18 => ( noeltsimpempty
% 2.35/1.18 => ( setbeta
% 2.35/1.18 => ( nonemptyE1
% 2.35/1.18 => ( nonemptyI
% 2.35/1.18 => ( nonemptyI1
% 2.35/1.18 => ( setadjoinIL
% 2.35/1.18 => ( emptyinunitempty
% 2.35/1.18 => ( setadjoinIR
% 2.35/1.18 => ( setadjoinE
% 2.35/1.18 => ( setadjoinOr
% 2.35/1.18 => ( setoftrueEq
% 2.35/1.18 => ( powersetI
% 2.35/1.18 => ( emptyinPowerset
% 2.35/1.18 => ( emptyInPowerset
% 2.35/1.18 => ( powersetE
% 2.35/1.18 => ( setunionI
% 2.35/1.18 => ( setunionE
% 2.35/1.18 => ( subPowSU
% 2.35/1.18 => ( exuE2
% 2.35/1.18 => ( nonemptyImpWitness
% 2.35/1.18 => ( uniqinunit
% 2.35/1.18 => ( notinsingleton
% 2.35/1.18 => ( eqinunit
% 2.35/1.18 => ( singletonsswitch
% 2.35/1.18 => ( upairsetE
% 2.35/1.18 => ( upairsetIL
% 2.35/1.18 => ( upairsetIR
% 2.35/1.18 => ( emptyE1
% 2.35/1.18 => ( vacuousDall
% 2.35/1.18 => ( quantDeMorgan1
% 2.35/1.18 => ( quantDeMorgan2
% 2.35/1.18 => ( quantDeMorgan3
% 2.35/1.18 => ( quantDeMorgan4
% 2.35/1.18 => ( prop2setI
% 2.35/1.18 => ( prop2set2propI
% 2.35/1.18 => ( notdexE
% 2.35/1.18 => ( notdallE
% 2.35/1.18 => ( exuI1
% 2.35/1.18 => ( exuI3
% 2.35/1.18 => ( exuI2
% 2.35/1.18 => ( inCongP
% 2.35/1.18 => ( in__Cong
% 2.35/1.18 => ( exuE3u
% 2.35/1.18 => ( exu__Cong
% 2.35/1.18 => ( emptyset__Cong
% 2.35/1.18 => ( setadjoin__Cong
% 2.35/1.18 => ( powerset__Cong
% 2.35/1.18 => ( setunion__Cong
% 2.35/1.18 => ( omega__Cong
% 2.35/1.18 => ( exuEu
% 2.35/1.18 => ( descr__Cong
% 2.35/1.18 => ( dsetconstr__Cong
% 2.35/1.18 => ( subsetI1
% 2.35/1.18 => ( eqimpsubset2
% 2.35/1.18 => ( eqimpsubset1
% 2.35/1.18 => ( subsetI2
% 2.35/1.18 => ( emptysetsubset
% 2.35/1.18 => ( subsetE
% 2.35/1.18 => ( subsetE2
% 2.35/1.18 => ( notsubsetI
% 2.35/1.18 => ( notequalI1
% 2.35/1.18 => ( notequalI2
% 2.35/1.18 => ( subsetRefl
% 2.35/1.18 => ( subsetTrans
% 2.35/1.18 => ( setadjoinSub
% 2.35/1.18 => ( setadjoinSub2
% 2.35/1.18 => ( subset2powerset
% 2.35/1.18 => ( setextsub
% 2.35/1.18 => ( subsetemptysetimpeq
% 2.35/1.18 => ( powersetI1
% 2.35/1.18 => ( powersetE1
% 2.35/1.18 => ( inPowerset
% 2.35/1.18 => ( powersetsubset
% 2.35/1.18 => ( sepInPowerset
% 2.35/1.18 => ( sepSubset
% 2.35/1.18 => ( binunionIL
% 2.35/1.18 => ( upairset2IR
% 2.35/1.18 => ( binunionIR
% 2.35/1.18 => ( binunionEcases
% 2.35/1.18 => ( binunionE
% 2.35/1.18 => ( binunionLsub
% 2.35/1.18 => ( binunionRsub
% 2.35/1.18 => ( binintersectI
% 2.35/1.18 => ( binintersectSubset5
% 2.35/1.18 => ( binintersectEL
% 2.35/1.18 => ( binintersectLsub
% 2.35/1.18 => ( binintersectSubset2
% 2.35/1.18 => ( binintersectSubset3
% 2.35/1.18 => ( binintersectER
% 2.35/1.18 => ( disjointsetsI1
% 2.35/1.18 => ( binintersectRsub
% 2.35/1.18 => ( binintersectSubset4
% 2.35/1.18 => ( binintersectSubset1
% 2.35/1.18 => ( bs114d
% 2.35/1.18 => ( setminusI
% 2.35/1.18 => ( setminusEL
% 2.35/1.18 => ( setminusER
% 2.35/1.18 => ( setminusSubset2
% 2.35/1.18 => ( setminusERneg
% 2.35/1.18 => ( setminusELneg
% 2.35/1.18 => ( setminusILneg
% 2.35/1.18 => ( setminusIRneg
% 2.35/1.18 => ( setminusLsub
% 2.35/1.18 => ( setminusSubset1
% 2.35/1.18 => ( symdiffE
% 2.35/1.18 => ( symdiffI1
% 2.35/1.18 => ( symdiffI2
% 2.35/1.18 => ( symdiffIneg1
% 2.35/1.18 => ( symdiffIneg2
% 2.35/1.18 => ( secondinupair
% 2.35/1.18 => ( setukpairIL
% 2.35/1.18 => ( setukpairIR
% 2.35/1.18 => ( kpairiskpair
% 2.35/1.18 => ( kpairp
% 2.35/1.18 => ( singletonsubset
% 2.35/1.18 => ( singletoninpowerset
% 2.35/1.18 => ( singletoninpowunion
% 2.35/1.18 => ( upairset2E
% 2.35/1.18 => ( upairsubunion
% 2.35/1.18 => ( upairinpowunion
% 2.35/1.18 => ( ubforcartprodlem1
% 2.35/1.18 => ( ubforcartprodlem2
% 2.35/1.18 => ( ubforcartprodlem3
% 2.35/1.18 => ( cartprodpairin
% 2.35/1.18 => ( cartprodmempair1
% 2.35/1.18 => ( cartprodmempair
% 2.35/1.18 => ( setunionE2
% 2.35/1.18 => ( setunionsingleton1
% 2.35/1.18 => ( setunionsingleton2
% 2.35/1.18 => ( setunionsingleton
% 2.35/1.18 => ( singletonprop
% 2.35/1.18 => ( ex1E1
% 2.35/1.18 => ( ex1I
% 2.35/1.18 => ( ex1I2
% 2.35/1.18 => ( singletonsuniq
% 2.35/1.18 => ( setukpairinjL1
% 2.35/1.18 => ( kfstsingleton
% 2.35/1.18 => ( theprop
% 2.35/1.18 => ( kfstpairEq
% 2.35/1.18 => ( cartprodfstin
% 2.35/1.18 => ( setukpairinjL2
% 2.35/1.18 => ( setukpairinjL
% 2.35/1.18 => ( setukpairinjR11
% 2.35/1.18 => ( setukpairinjR12
% 2.35/1.18 => ( setukpairinjR1
% 2.35/1.18 => ( upairequniteq
% 2.35/1.18 => ( setukpairinjR2
% 2.35/1.18 => ( setukpairinjR
% 2.35/1.18 => ( ksndsingleton
% 2.35/1.18 => ( ksndpairEq
% 2.35/1.18 => ( kpairsurjEq
% 2.35/1.18 => ( cartprodsndin
% 2.35/1.18 => ( cartprodpairmemEL
% 2.35/1.18 => ( cartprodpairmemER
% 2.35/1.18 => ( cartprodmempaircEq
% 2.35/1.18 => ( cartprodfstpairEq
% 2.35/1.18 => ( cartprodsndpairEq
% 2.35/1.18 => ( cartprodpairsurjEq
% 2.35/1.18 => ( dpsetconstrI
% 2.35/1.18 => ( dpsetconstrSub
% 2.35/1.18 => ( setOfPairsIsBReln
% 2.35/1.18 => ( dpsetconstrERa
% 2.35/1.18 => ( dpsetconstrEL1
% 2.35/1.18 => ( dpsetconstrEL2
% 2.35/1.18 => ( dpsetconstrER
% 2.35/1.18 => ( funcImageSingleton
% 2.35/1.18 => ( apProp
% 2.35/1.18 => ( app
% 2.35/1.18 => ( infuncsetfunc
% 2.35/1.18 => ( ap2p
% 2.35/1.18 => ( funcinfuncset
% 2.35/1.18 => ( lamProp
% 2.35/1.18 => ( lamp
% 2.35/1.18 => ( lam2p
% 2.35/1.18 => ( brelnall1
% 2.35/1.18 => ( brelnall2
% 2.35/1.18 => ( ex1E2
% 2.35/1.18 => ( funcGraphProp1
% 2.35/1.18 => ( funcGraphProp3
% 2.35/1.18 => ( funcGraphProp2
% 2.35/1.18 => ( funcextLem
% 2.35/1.18 => ( funcGraphProp4
% 2.35/1.18 => ( subbreln
% 2.35/1.18 => ( eqbreln
% 2.35/1.18 => ( funcext
% 2.35/1.18 => ( funcext2
% 2.35/1.18 => ( ap2apEq1
% 2.35/1.18 => ( ap2apEq2
% 2.35/1.18 => ( beta1
% 2.35/1.18 => ( eta1
% 2.35/1.18 => ( lam2lamEq
% 2.35/1.18 => ( beta2
% 2.35/1.18 => ( eta2
% 2.35/1.18 => ( iffalseProp1
% 2.35/1.18 => ( iffalseProp2
% 2.35/1.18 => ( iftrueProp1
% 2.35/1.18 => ( iftrueProp2
% 2.35/1.18 => ( ifSingleton
% 2.35/1.18 => ( ifp
% 2.35/1.18 => ( theeq
% 2.35/1.18 => ( iftrue
% 2.35/1.18 => ( iffalse
% 2.35/1.18 => ( iftrueorfalse
% 2.35/1.18 => ( binintersectT_lem
% 2.35/1.18 => ( binunionT_lem
% 2.35/1.18 => ( powersetT_lem
% 2.35/1.18 => ( setminusT_lem
% 2.35/1.18 => ( complementT_lem
% 2.35/1.18 => ( setextT
% 2.35/1.18 => ( subsetTI
% 2.35/1.18 => ( powersetTI1
% 2.35/1.18 => ( powersetTE1
% 2.35/1.18 => ( complementTI1
% 2.35/1.18 => ( complementTE1
% 2.35/1.18 => ( binintersectTELcontra
% 2.35/1.18 => ( binintersectTERcontra
% 2.35/1.18 => ( contrasubsetT
% 2.35/1.18 => ( contrasubsetT1
% 2.35/1.18 => ( contrasubsetT2
% 2.35/1.18 => ( contrasubsetT3
% 2.35/1.18 => ( doubleComplementI1
% 2.35/1.18 => ( doubleComplementE1
% 2.35/1.18 => ( doubleComplementSub1
% 2.35/1.18 => ( doubleComplementSub2
% 2.35/1.18 => ( doubleComplementEq
% 2.35/1.18 => ( complementTnotintersectT
% 2.35/1.18 => ! [A: $i,X: $i] :
% 2.35/1.18 ( ( in @ X @ ( powerset @ A ) )
% 2.35/1.18 => ! [Y: $i] :
% 2.35/1.18 ( ( in @ Y @ ( powerset @ A ) )
% 2.35/1.18 => ! [Xx: $i] :
% 2.35/1.18 ( ( in @ Xx @ A )
% 2.35/1.18 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.35/1.18 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.35/1.18 ------------------------
% 2.41/1.20 % (3690)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_3524 for (2999ds/183Mi)
% 2.41/1.21 % (3692)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_3524 for (2999ds/27Mi)
% 2.41/1.21 % (3694)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_3524 for (2999ds/2Mi)
% 2.41/1.21 % (3693)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_3524 for (2999ds/2Mi)
% 2.41/1.21 % (3691)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_3524 for (2999ds/4Mi)
% 2.41/1.21 % (3696)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_3524 for (2999ds/18Mi)
% 2.41/1.21 % (3695)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_3524 for (2999ds/275Mi)
% 2.41/1.21 % (3693)Instruction limit reached!
% 2.41/1.21 % (3693)------------------------------
% 2.41/1.21 % (3693)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.21 % (3693)Termination reason: Unknown
% 2.41/1.21 % (3694)Instruction limit reached!
% 2.41/1.21 % (3694)------------------------------
% 2.41/1.21 % (3694)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.21 % (3694)Termination reason: Unknown
% 2.41/1.21 % (3694)Termination phase: shuffling
% 2.41/1.21
% 2.41/1.21 % (3694)Memory used [KB]: 1535
% 2.41/1.21 % (3694)Time elapsed: 0.003 s
% 2.41/1.21 % (3694)Instructions burned: 3 (million)
% 2.41/1.21 % (3694)------------------------------
% 2.41/1.21 % (3694)------------------------------
% 2.41/1.21 % (3693)Termination phase: shuffling
% 2.41/1.21
% 2.41/1.21 % (3693)Memory used [KB]: 1535
% 2.41/1.21 % (3693)Time elapsed: 0.003 s
% 2.41/1.21 % (3693)Instructions burned: 3 (million)
% 2.41/1.21 % (3693)------------------------------
% 2.41/1.21 % (3693)------------------------------
% 2.41/1.21 % (3691)Instruction limit reached!
% 2.41/1.21 % (3691)------------------------------
% 2.41/1.21 % (3691)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.21 % (3691)Termination reason: Unknown
% 2.41/1.21 % (3691)Termination phase: shuffling
% 2.41/1.21
% 2.41/1.21 % (3691)Memory used [KB]: 1535
% 2.41/1.21 % (3691)Time elapsed: 0.004 s
% 2.41/1.21 % (3691)Instructions burned: 5 (million)
% 2.41/1.21 % (3691)------------------------------
% 2.41/1.21 % (3691)------------------------------
% 2.41/1.21 % (3692)Instruction limit reached!
% 2.41/1.21 % (3692)------------------------------
% 2.41/1.21 % (3692)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.21 % (3692)Termination reason: Unknown
% 2.41/1.21 % (3692)Termination phase: shuffling
% 2.41/1.21
% 2.41/1.21 % (3692)Memory used [KB]: 1918
% 2.41/1.21 % (3692)Time elapsed: 0.009 s
% 2.41/1.21 % (3692)Instructions burned: 27 (million)
% 2.41/1.21 % (3692)------------------------------
% 2.41/1.21 % (3692)------------------------------
% 2.41/1.22 % (3696)Instruction limit reached!
% 2.41/1.22 % (3696)------------------------------
% 2.41/1.22 % (3696)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.22 % (3696)Termination reason: Unknown
% 2.41/1.22 % (3696)Termination phase: shuffling
% 2.41/1.22
% 2.41/1.22 % (3696)Memory used [KB]: 1791
% 2.41/1.22 % (3696)Time elapsed: 0.011 s
% 2.41/1.22 % (3696)Instructions burned: 19 (million)
% 2.41/1.22 % (3696)------------------------------
% 2.41/1.22 % (3696)------------------------------
% 2.41/1.22 % (3704)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_3524 for (2999ds/15Mi)
% 2.41/1.22 % (3703)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_3524 for (2999ds/37Mi)
% 2.41/1.23 % (3702)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_3524 for (2999ds/3Mi)
% 2.41/1.23 % (3705)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_3524 for (2999ds/3Mi)
% 2.41/1.23 % (3704)Instruction limit reached!
% 2.41/1.23 % (3704)------------------------------
% 2.41/1.23 % (3704)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.23 % (3704)Termination reason: Unknown
% 2.41/1.23 % (3704)Termination phase: shuffling
% 2.41/1.23
% 2.41/1.23 % (3704)Memory used [KB]: 1663
% 2.41/1.23 % (3704)Time elapsed: 0.008 s
% 2.41/1.23 % (3704)Instructions burned: 16 (million)
% 2.41/1.23 % (3704)------------------------------
% 2.41/1.23 % (3704)------------------------------
% 2.41/1.23 % (3705)Instruction limit reached!
% 2.41/1.23 % (3705)------------------------------
% 2.41/1.23 % (3705)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.23 % (3705)Termination reason: Unknown
% 2.41/1.23 % (3705)Termination phase: shuffling
% 2.41/1.23
% 2.41/1.23 % (3705)Memory used [KB]: 1407
% 2.41/1.23 % (3705)Time elapsed: 0.003 s
% 2.41/1.23 % (3705)Instructions burned: 3 (million)
% 2.41/1.23 % (3705)------------------------------
% 2.41/1.23 % (3705)------------------------------
% 2.41/1.23 % (3702)Instruction limit reached!
% 2.41/1.23 % (3702)------------------------------
% 2.41/1.23 % (3702)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.23 % (3702)Termination reason: Unknown
% 2.41/1.23 % (3702)Termination phase: shuffling
% 2.41/1.23
% 2.41/1.23 % (3702)Memory used [KB]: 1535
% 2.41/1.23 % (3702)Time elapsed: 0.004 s
% 2.41/1.23 % (3702)Instructions burned: 5 (million)
% 2.41/1.23 % (3702)------------------------------
% 2.41/1.23 % (3702)------------------------------
% 2.41/1.23 % (3703)Instruction limit reached!
% 2.41/1.23 % (3703)------------------------------
% 2.41/1.23 % (3703)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.23 % (3703)Termination reason: Unknown
% 2.41/1.23 % (3703)Termination phase: shuffling
% 2.41/1.23
% 2.41/1.23 % (3703)Memory used [KB]: 2302
% 2.41/1.23 % (3703)Time elapsed: 0.012 s
% 2.41/1.23 % (3703)Instructions burned: 39 (million)
% 2.41/1.23 % (3703)------------------------------
% 2.41/1.23 % (3703)------------------------------
% 2.41/1.24 % (3707)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_3524 for (2999ds/1041Mi)
% 2.41/1.24 % (3713)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_3524 for (2999ds/3Mi)
% 2.41/1.24 % (3713)Instruction limit reached!
% 2.41/1.24 % (3713)------------------------------
% 2.41/1.24 % (3713)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.24 % (3713)Termination reason: Unknown
% 2.41/1.24 % (3713)Termination phase: shuffling
% 2.41/1.24
% 2.41/1.24 % (3713)Memory used [KB]: 1535
% 2.41/1.24 % (3713)Time elapsed: 0.003 s
% 2.41/1.24 % (3713)Instructions burned: 6 (million)
% 2.41/1.24 % (3713)------------------------------
% 2.41/1.24 % (3713)------------------------------
% 2.41/1.24 % (3710)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_3524 for (2999ds/7Mi)
% 2.41/1.25 % (3712)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_3524 for (2999ds/16Mi)
% 2.41/1.25 % (3710)Instruction limit reached!
% 2.41/1.25 % (3710)------------------------------
% 2.41/1.25 % (3710)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.25 % (3710)Termination reason: Unknown
% 2.41/1.25 % (3710)Termination phase: shuffling
% 2.41/1.25
% 2.41/1.25 % (3710)Memory used [KB]: 1535
% 2.41/1.25 % (3710)Time elapsed: 0.005 s
% 2.41/1.25 % (3710)Instructions burned: 7 (million)
% 2.41/1.25 % (3710)------------------------------
% 2.41/1.25 % (3710)------------------------------
% 2.41/1.25 % (3712)Instruction limit reached!
% 2.41/1.25 % (3712)------------------------------
% 2.41/1.25 % (3712)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.25 % (3712)Termination reason: Unknown
% 2.41/1.25 % (3712)Termination phase: shuffling
% 2.41/1.25
% 2.41/1.25 % (3712)Memory used [KB]: 1791
% 2.41/1.25 % (3712)Time elapsed: 0.009 s
% 2.41/1.25 % (3712)Instructions burned: 17 (million)
% 2.41/1.25 % (3712)------------------------------
% 2.41/1.25 % (3712)------------------------------
% 2.41/1.26 % (3714)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_3524 for (2999ds/3Mi)
% 2.41/1.26 % (3714)Instruction limit reached!
% 2.41/1.26 % (3714)------------------------------
% 2.41/1.26 % (3714)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.26 % (3714)Termination reason: Unknown
% 2.41/1.26 % (3714)Termination phase: shuffling
% 2.41/1.26
% 2.41/1.26 % (3714)Memory used [KB]: 1535
% 2.41/1.26 % (3714)Time elapsed: 0.003 s
% 2.41/1.26 % (3714)Instructions burned: 3 (million)
% 2.41/1.26 % (3714)------------------------------
% 2.41/1.26 % (3714)------------------------------
% 2.41/1.26 % (3716)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_3524 for (2999ds/7Mi)
% 2.41/1.26 % (3716)Instruction limit reached!
% 2.41/1.26 % (3716)------------------------------
% 2.41/1.26 % (3716)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.26 % (3716)Termination reason: Unknown
% 2.41/1.26 % (3716)Termination phase: shuffling
% 2.41/1.26
% 2.41/1.26 % (3716)Memory used [KB]: 1535
% 2.41/1.26 % (3716)Time elapsed: 0.005 s
% 2.41/1.26 % (3716)Instructions burned: 7 (million)
% 2.41/1.26 % (3716)------------------------------
% 2.41/1.26 % (3716)------------------------------
% 2.41/1.27 % (3719)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_3524 for (2999ds/3Mi)
% 2.41/1.27 % (3720)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_3524 for (2999ds/4Mi)
% 2.41/1.27 % (3719)Instruction limit reached!
% 2.41/1.27 % (3719)------------------------------
% 2.41/1.27 % (3719)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.27 % (3719)Termination reason: Unknown
% 2.41/1.27 % (3719)Termination phase: shuffling
% 2.41/1.27
% 2.41/1.27 % (3719)Memory used [KB]: 1535
% 2.41/1.27 % (3719)Time elapsed: 0.003 s
% 2.41/1.27 % (3719)Instructions burned: 4 (million)
% 2.41/1.27 % (3719)------------------------------
% 2.41/1.27 % (3719)------------------------------
% 2.41/1.27 % (3720)Instruction limit reached!
% 2.41/1.27 % (3720)------------------------------
% 2.41/1.27 % (3720)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.27 % (3720)Termination reason: Unknown
% 2.41/1.27 % (3720)Termination phase: shuffling
% 2.41/1.27
% 2.41/1.27 % (3720)Memory used [KB]: 1535
% 2.41/1.27 % (3720)Time elapsed: 0.004 s
% 2.41/1.27 % (3720)Instructions burned: 5 (million)
% 2.41/1.27 % (3720)------------------------------
% 2.41/1.27 % (3720)------------------------------
% 2.41/1.28 % (3723)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_3524 for (2999ds/18Mi)
% 2.41/1.28 % (3690)Instruction limit reached!
% 2.41/1.28 % (3690)------------------------------
% 2.41/1.28 % (3690)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.28 % (3690)Termination reason: Unknown
% 2.41/1.28 % (3690)Termination phase: Saturation
% 2.41/1.28
% 2.41/1.28 % (3690)Memory used [KB]: 7675
% 2.41/1.28 % (3690)Time elapsed: 0.077 s
% 2.41/1.28 % (3690)Instructions burned: 184 (million)
% 2.41/1.28 % (3690)------------------------------
% 2.41/1.28 % (3690)------------------------------
% 2.41/1.29 % (3723)Instruction limit reached!
% 2.41/1.29 % (3723)------------------------------
% 2.41/1.29 % (3723)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.29 % (3723)Termination reason: Unknown
% 2.41/1.29 % (3723)Termination phase: shuffling
% 2.41/1.29
% 2.41/1.29 % (3723)Memory used [KB]: 1791
% 2.41/1.29 % (3723)Time elapsed: 0.010 s
% 2.41/1.29 % (3723)Instructions burned: 18 (million)
% 2.41/1.29 % (3723)------------------------------
% 2.41/1.29 % (3723)------------------------------
% 2.41/1.29 % (3727)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_3524 for (2999ds/6Mi)
% 2.41/1.29 % (3729)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_3524 for (2999ds/902Mi)
% 2.41/1.29 % (3727)Instruction limit reached!
% 2.41/1.29 % (3727)------------------------------
% 2.41/1.29 % (3727)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.29 % (3727)Termination reason: Unknown
% 2.41/1.29 % (3727)Termination phase: shuffling
% 2.41/1.29
% 2.41/1.29 % (3727)Memory used [KB]: 1535
% 2.41/1.29 % (3727)Time elapsed: 0.005 s
% 2.41/1.29 % (3727)Instructions burned: 7 (million)
% 2.41/1.29 % (3727)------------------------------
% 2.41/1.29 % (3727)------------------------------
% 2.41/1.30 % (3725)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_3524 for (2999ds/710Mi)
% 2.41/1.30 % (3735)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_3524 for (2999ds/6Mi)
% 2.41/1.30 % (3735)Instruction limit reached!
% 2.41/1.30 % (3735)------------------------------
% 2.41/1.30 % (3735)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.30 % (3735)Termination reason: Unknown
% 2.41/1.30 % (3735)Termination phase: shuffling
% 2.41/1.30
% 2.41/1.30 % (3735)Memory used [KB]: 1535
% 2.41/1.30 % (3735)Time elapsed: 0.004 s
% 2.41/1.30 % (3735)Instructions burned: 8 (million)
% 2.41/1.30 % (3735)------------------------------
% 2.41/1.30 % (3735)------------------------------
% 2.41/1.30 % (3732)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_3524 for (2999ds/21Mi)
% 2.41/1.31 % (3733)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_3524 for (2999ds/5Mi)
% 2.41/1.31 % (3733)Instruction limit reached!
% 2.41/1.31 % (3733)------------------------------
% 2.41/1.31 % (3733)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.31 % (3733)Termination reason: Unknown
% 2.41/1.31 % (3733)Termination phase: shuffling
% 2.41/1.31
% 2.41/1.31 % (3733)Memory used [KB]: 1535
% 2.41/1.31 % (3733)Time elapsed: 0.004 s
% 2.41/1.31 % (3733)Instructions burned: 5 (million)
% 2.41/1.31 % (3733)------------------------------
% 2.41/1.31 % (3733)------------------------------
% 2.41/1.31 % (3732)Instruction limit reached!
% 2.41/1.31 % (3732)------------------------------
% 2.41/1.31 % (3732)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.31 % (3732)Termination reason: Unknown
% 2.41/1.31 % (3732)Termination phase: shuffling
% 2.41/1.31
% 2.41/1.31 % (3732)Memory used [KB]: 1918
% 2.41/1.31 % (3732)Time elapsed: 0.012 s
% 2.41/1.31 % (3732)Instructions burned: 23 (million)
% 2.41/1.31 % (3732)------------------------------
% 2.41/1.31 % (3732)------------------------------
% 2.41/1.31 % (3737)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_3524 for (2998ds/377Mi)
% 2.41/1.32 % (3695)Instruction limit reached!
% 2.41/1.32 % (3695)------------------------------
% 2.41/1.32 % (3695)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.32 % (3695)Termination reason: Unknown
% 2.41/1.32 % (3695)Termination phase: Saturation
% 2.41/1.32
% 2.41/1.32 % (3695)Memory used [KB]: 9594
% 2.41/1.32 % (3695)Time elapsed: 0.114 s
% 2.41/1.32 % (3695)Instructions burned: 275 (million)
% 2.41/1.32 % (3695)------------------------------
% 2.41/1.32 % (3695)------------------------------
% 2.41/1.33 % (3741)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_3524 for (2998ds/19Mi)
% 2.41/1.33 % (3740)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_3524 for (2998ds/779Mi)
% 2.41/1.33 % (3742)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_3524 for (2998ds/879Mi)
% 2.41/1.34 % (3741)Instruction limit reached!
% 2.41/1.34 % (3741)------------------------------
% 2.41/1.34 % (3741)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.34 % (3741)Termination reason: Unknown
% 2.41/1.34 % (3741)Termination phase: shuffling
% 2.41/1.34
% 2.41/1.34 % (3741)Memory used [KB]: 1791
% 2.41/1.34 % (3741)Time elapsed: 0.010 s
% 2.41/1.34 % (3741)Instructions burned: 19 (million)
% 2.41/1.34 % (3741)------------------------------
% 2.41/1.34 % (3741)------------------------------
% 2.41/1.36 % (3744)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_3524 for (2998ds/17Mi)
% 2.41/1.36 % (3744)Instruction limit reached!
% 2.41/1.36 % (3744)------------------------------
% 2.41/1.36 % (3744)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.36 % (3744)Termination reason: Unknown
% 2.41/1.36 % (3744)Termination phase: shuffling
% 2.41/1.36
% 2.41/1.36 % (3744)Memory used [KB]: 1791
% 2.41/1.36 % (3744)Time elapsed: 0.010 s
% 2.41/1.36 % (3744)Instructions burned: 18 (million)
% 2.41/1.36 % (3744)------------------------------
% 2.41/1.36 % (3744)------------------------------
% 2.41/1.38 % (3746)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_3524 for (2998ds/3Mi)
% 2.41/1.38 % (3746)Instruction limit reached!
% 2.41/1.38 % (3746)------------------------------
% 2.41/1.38 % (3746)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.38 % (3746)Termination reason: Unknown
% 2.41/1.38 % (3746)Termination phase: shuffling
% 2.41/1.38
% 2.41/1.38 % (3746)Memory used [KB]: 1535
% 2.41/1.38 % (3746)Time elapsed: 0.004 s
% 2.41/1.38 % (3746)Instructions burned: 5 (million)
% 2.41/1.38 % (3746)------------------------------
% 2.41/1.38 % (3746)------------------------------
% 2.41/1.40 % (3748)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_3524 for (2998ds/30Mi)
% 2.41/1.42 % (3748)Instruction limit reached!
% 2.41/1.42 % (3748)------------------------------
% 2.41/1.42 % (3748)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.42 % (3748)Termination reason: Unknown
% 2.41/1.42 % (3748)Termination phase: shuffling
% 2.41/1.42
% 2.41/1.42 % (3748)Memory used [KB]: 2046
% 2.41/1.42 % (3748)Time elapsed: 0.017 s
% 2.41/1.42 % (3748)Instructions burned: 32 (million)
% 2.41/1.42 % (3748)------------------------------
% 2.41/1.42 % (3748)------------------------------
% 2.41/1.43 % (3750)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_3524 for (2997ds/127Mi)
% 2.41/1.46 % (3729)First to succeed.
% 2.41/1.47 % (3729)Refutation found. Thanks to Tanya!
% 2.41/1.47 % SZS status Theorem for DTF2THF_3524
% 2.41/1.47 % SZS output start Proof for DTF2THF_3524
% 2.41/1.47 thf(func_def_0, type, in: $i > $i > $o).
% 2.41/1.47 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.41/1.47 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.41/1.47 thf(func_def_8, type, powerset: $i > $i).
% 2.41/1.47 thf(func_def_10, type, setunion: $i > $i).
% 2.41/1.47 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.41/1.47 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.41/1.47 thf(func_def_26, type, prop2set: $o > $i).
% 2.41/1.47 thf(func_def_36, type, nonempty: $i > $o).
% 2.41/1.47 thf(func_def_69, type, set2prop: $i > $o).
% 2.41/1.47 thf(func_def_88, type, subset: $i > $i > $o).
% 2.41/1.47 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.41/1.47 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.41/1.47 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.41/1.47 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.41/1.47 thf(func_def_135, type, regular: $i > $o).
% 2.41/1.47 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.41/1.47 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.41/1.47 thf(func_def_153, type, iskpair: $i > $o).
% 2.41/1.47 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.41/1.47 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.41/1.47 thf(func_def_177, type, singleton: $i > $o).
% 2.41/1.47 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.41/1.47 thf(func_def_184, type, atmost1p: $i > $o).
% 2.41/1.47 thf(func_def_185, type, atleast2p: $i > $o).
% 2.41/1.47 thf(func_def_186, type, atmost2p: $i > $o).
% 2.41/1.47 thf(func_def_187, type, upairsetp: $i > $o).
% 2.41/1.47 thf(func_def_191, type, kfst: $i > $i).
% 2.41/1.47 thf(func_def_203, type, ksnd: $i > $i).
% 2.41/1.47 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.41/1.47 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.41/1.47 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.41/1.47 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.41/1.47 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.41/1.47 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.41/1.47 thf(func_def_301, type, sP0: $i > $o > $i > $i > $o).
% 2.41/1.47 thf(func_def_302, type, sP1: $i > $i > $o).
% 2.41/1.47 thf(func_def_303, type, sP2: $i > $o).
% 2.41/1.47 thf(func_def_304, type, sP3: $i > $i > $o).
% 2.41/1.47 thf(func_def_305, type, sP4: $i > $i > $o).
% 2.41/1.47 thf(func_def_308, type, sP7: $i > $i > $i > $i > $o).
% 2.41/1.47 thf(func_def_311, type, sP10: $i > $i > $i > $o).
% 2.41/1.47 thf(func_def_324, type, sK23: $i > ($i > $o) > $i).
% 2.41/1.47 thf(func_def_325, type, sK24: $i > $o).
% 2.41/1.47 thf(func_def_331, type, sK30: ($i > $i) > $i > $i > $i).
% 2.41/1.47 thf(func_def_334, type, sK33: $i > $i).
% 2.41/1.47 thf(func_def_337, type, sK36: $i > $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_359, type, sK58: $i > $o).
% 2.41/1.47 thf(func_def_370, type, sK69: $i > $o).
% 2.41/1.47 thf(func_def_375, type, sK74: $i > $o).
% 2.41/1.47 thf(func_def_392, type, sK91: $i > $o).
% 2.41/1.47 thf(func_def_393, type, sK92: $i > $i).
% 2.41/1.47 thf(func_def_394, type, sK93: ($i > $o) > $i).
% 2.41/1.47 thf(func_def_403, type, sK102: $i > $i).
% 2.41/1.47 thf(func_def_404, type, sK103: ($i > $i) > $i > $i > $i).
% 2.41/1.47 thf(func_def_409, type, sK108: $i > $o).
% 2.41/1.47 thf(func_def_411, type, sK110: $i > ($i > $o) > $i).
% 2.41/1.47 thf(func_def_412, type, sK111: $i > $o).
% 2.41/1.47 thf(func_def_417, type, sK116: $i > $o).
% 2.41/1.47 thf(func_def_419, type, sK118: ($i > $o) > $i > $i).
% 2.41/1.47 thf(func_def_423, type, sK122: $i > $i).
% 2.41/1.47 thf(func_def_431, type, sK130: $i > $o).
% 2.41/1.47 thf(func_def_445, type, sK144: $i > $i).
% 2.41/1.47 thf(func_def_451, type, sK150: $i > $o).
% 2.41/1.47 thf(func_def_452, type, sK151: ($i > $o) > $i > $i).
% 2.41/1.47 thf(func_def_464, type, sK163: $i > $i > $o).
% 2.41/1.47 thf(func_def_472, type, sK171: $i > $o).
% 2.41/1.47 thf(func_def_473, type, sK172: $i > $o).
% 2.41/1.47 thf(func_def_474, type, sK173: ($i > $o) > ($i > $o) > $i).
% 2.41/1.47 thf(func_def_475, type, sK174: ($i > $o) > ($i > $o) > $i).
% 2.41/1.47 thf(func_def_501, type, sK200: $i > $o).
% 2.41/1.47 thf(func_def_503, type, sK202: ($i > $o) > $i).
% 2.41/1.47 thf(func_def_504, type, sK203: ($i > $o) > $i).
% 2.41/1.47 thf(func_def_511, type, sK210: $i > $i > $o).
% 2.41/1.47 thf(func_def_514, type, sK213: $i > $i > $i).
% 2.41/1.47 thf(func_def_518, type, sK217: $i > $i > $i).
% 2.41/1.47 thf(func_def_521, type, sK220: $i > $i > $i).
% 2.41/1.47 thf(func_def_522, type, sK221: $i > $i > $i).
% 2.41/1.47 thf(func_def_523, type, sK222: $i > $i > $i).
% 2.41/1.47 thf(func_def_524, type, sK223: $i > $i > $i).
% 2.41/1.47 thf(func_def_525, type, sK224: $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_526, type, sK225: $i > $i).
% 2.41/1.47 thf(func_def_527, type, sK226: $i > $i).
% 2.41/1.47 thf(func_def_528, type, sK227: $i > $i).
% 2.41/1.47 thf(func_def_529, type, sK228: $i > $i).
% 2.41/1.47 thf(func_def_530, type, sK229: $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_531, type, sK230: $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_532, type, sK231: $i > $i > $i).
% 2.41/1.47 thf(func_def_533, type, sK232: $i > $i > $i).
% 2.41/1.47 thf(func_def_534, type, sK233: $i > $i > $i).
% 2.41/1.47 thf(func_def_536, type, sK235: $i > $i).
% 2.41/1.47 thf(func_def_537, type, sK236: $i > $i).
% 2.41/1.47 thf(func_def_538, type, sK237: $i > $i).
% 2.41/1.47 thf(func_def_539, type, sK238: $i > $i > $i).
% 2.41/1.47 thf(func_def_553, type, sK252: ($i > $i) > $i > $i > $i).
% 2.41/1.47 thf(func_def_556, type, sK255: $i > $i).
% 2.41/1.47 thf(func_def_563, type, sK262: $i > $o).
% 2.41/1.47 thf(func_def_566, type, sK265: $i > ($i > $o) > $i).
% 2.41/1.47 thf(func_def_567, type, sK266: $i > ($i > $o) > $i).
% 2.41/1.47 thf(func_def_568, type, sK267: $i > $o).
% 2.41/1.47 thf(func_def_571, type, sK270: $i > ($i > $o) > $i).
% 2.41/1.47 thf(func_def_576, type, sK275: $i > $i > $o).
% 2.41/1.47 thf(func_def_584, type, sK283: $i > $i > $o).
% 2.41/1.47 thf(func_def_597, type, sK296: $i > $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_618, type, sK317: ($i > $o) > $i).
% 2.41/1.47 thf(func_def_619, type, sK318: $i > $o).
% 2.41/1.47 thf(func_def_648, type, sK347: $i > $i > $i).
% 2.41/1.47 thf(func_def_653, type, sK352: $i > $i > $i).
% 2.41/1.47 thf(func_def_654, type, sK353: $i > $i > $i).
% 2.41/1.47 thf(func_def_657, type, sK356: $i > $i > $o).
% 2.41/1.47 thf(func_def_671, type, sK370: $i > $i > $i).
% 2.41/1.47 thf(func_def_681, type, sK380: $i > $i).
% 2.41/1.47 thf(func_def_692, type, sK391: $i > $o).
% 2.41/1.47 thf(func_def_693, type, sK392: ($i > $o) > $i).
% 2.41/1.47 thf(func_def_694, type, sK393: $i > $o).
% 2.41/1.47 thf(func_def_695, type, sK394: $i > $i).
% 2.41/1.47 thf(func_def_699, type, sK398: $i > $i > $i).
% 2.41/1.47 thf(func_def_703, type, sK402: $i > $o).
% 2.41/1.47 thf(func_def_714, type, sK413: $i > $o).
% 2.41/1.47 thf(func_def_720, type, sK419: $i > $i > $o).
% 2.41/1.47 thf(func_def_737, type, sK436: $i > $o).
% 2.41/1.47 thf(func_def_763, type, sK462: $i > $o).
% 2.41/1.47 thf(func_def_777, type, sK476: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.41/1.47 thf(func_def_778, type, sK477: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.41/1.47 thf(func_def_781, type, sK480: $i > $o).
% 2.41/1.47 thf(func_def_782, type, sK481: $i > $o).
% 2.41/1.47 thf(func_def_785, type, sK484: $i > $i).
% 2.41/1.47 thf(func_def_786, type, sK485: ($i > $i) > $i > $i > $i).
% 2.41/1.47 thf(func_def_801, type, sK500: $i > $i > $i).
% 2.41/1.47 thf(func_def_815, type, sK514: $i > $i > $i).
% 2.41/1.47 thf(func_def_823, type, sK522: $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_846, type, sK545: $i > $i > $o).
% 2.41/1.47 thf(func_def_853, type, sK552: $i > $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_854, type, sK553: $i > $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_855, type, sK554: $i > $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_856, type, sK555: $i > $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_861, type, sK560: $i > $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_868, type, sK567: $i > $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_869, type, sK568: $i > $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_881, type, sK580: $i > $o).
% 2.41/1.47 thf(func_def_882, type, sK581: ($i > $o) > $i > $i > $i).
% 2.41/1.47 thf(func_def_885, type, sK584: $i > $o).
% 2.41/1.47 thf(func_def_886, type, sK585: $i > $o).
% 2.41/1.47 thf(func_def_887, type, sK586: ($i > $o) > ($i > $o) > $i).
% 2.41/1.47 thf(func_def_888, type, sK587: ($i > $o) > ($i > $o) > $i).
% 2.41/1.47 thf(func_def_890, type, sK589: $i > $i).
% 2.41/1.47 thf(func_def_901, type, sK600: $i > $o).
% 2.41/1.47 thf(func_def_904, type, sK603: $i > ($i > $o) > $i).
% 2.41/1.47 thf(func_def_905, type, sK604: $i > ($i > $o) > $i).
% 2.41/1.47 thf(func_def_909, type, sK608: $i > $i).
% 2.41/1.47 thf(func_def_912, type, sK611: $i > $i > ($i > $i) > $i).
% 2.41/1.47 thf(func_def_915, type, sK614: $o > $i > $i > $i).
% 2.41/1.47 thf(func_def_926, type, sK625: $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_927, type, sK626: $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_933, type, sK632: $i > $i).
% 2.41/1.47 thf(func_def_934, type, sK633: $i > $i).
% 2.41/1.47 thf(func_def_935, type, sK634: $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_955, type, sK654: $i > $i > $i).
% 2.41/1.47 thf(func_def_957, type, sK656: $i > $o).
% 2.41/1.47 thf(func_def_959, type, sK658: $i > ($i > $o) > $i).
% 2.41/1.47 thf(func_def_976, type, sK675: $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_977, type, sK676: $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_990, type, sK689: $i > $o).
% 2.41/1.47 thf(func_def_1003, type, sK702: $i > $i).
% 2.41/1.47 thf(func_def_1032, type, sK731: $i > $o).
% 2.41/1.47 thf(func_def_1034, type, sK733: ($i > $o) > $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_1035, type, sK734: ($i > $o) > $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_1040, type, sK739: $i > $o).
% 2.41/1.47 thf(func_def_1052, type, sK751: $i > $i).
% 2.41/1.47 thf(func_def_1054, type, sK753: ($i > $i) > $i > $i > $i).
% 2.41/1.47 thf(func_def_1057, type, sK756: $i > ($i > $o) > $i).
% 2.41/1.47 thf(func_def_1058, type, sK757: $i > $o).
% 2.41/1.47 thf(func_def_1062, type, sK761: $i > $o).
% 2.41/1.47 thf(func_def_1065, type, sK764: $i > ($i > $o) > $i).
% 2.41/1.47 thf(func_def_1066, type, sK765: ($i > $o) > $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_1067, type, sK766: ($i > $o) > $i > $i > $i > $i).
% 2.41/1.47 thf(func_def_1071, type, sK770: $i > $o).
% 2.41/1.47 thf(func_def_1074, type, sK773: $i > $i > $o).
% 2.41/1.47 thf(func_def_1075, type, sK774: $i > $i).
% 2.41/1.47 thf(func_def_1076, type, sK775: $i > $i).
% 2.41/1.47 thf(func_def_1077, type, sK776: ($i > $i > $o) > $i > $i).
% 2.41/1.47 thf(func_def_1078, type, sK777: ($i > $i > $o) > $i > $i).
% 2.41/1.47 thf(func_def_1079, type, sK778: $i > ($i > $i > $o) > $i > $i).
% 2.41/1.47 thf(func_def_1081, type, sK780: $i > $o).
% 2.41/1.47 thf(func_def_1083, type, sK782: $i > ($i > $o) > $i).
% 2.41/1.47 thf(func_def_1084, type, sK783: $i > $o).
% 2.41/1.47 thf(func_def_1091, type, ph790: !>[X0: $tType]:(X0)).
% 2.41/1.47 thf(f4178,plain,(
% 2.41/1.47 $false),
% 2.41/1.47 inference(avatar_sat_refutation,[status(thm)],[f3892,f4122,f4141])).
% 2.41/1.47 thf(f4141,plain,(
% 2.41/1.47 spl789_7 | ~spl789_14),
% 2.41/1.47 inference(avatar_split_clause,[status(thm)],[f4140,f3970,f3881])).
% 2.41/1.47 thf(f3881,definition,(
% 2.41/1.47 spl789_7 <=> ((in @ sK335 @ sK333) = $true)),
% 2.41/1.47 introduced(definition,[new_symbols(naming,[spl789_7])],[avatar_definition])).
% 2.41/1.47 thf(f3970,definition,(
% 2.41/1.47 spl789_14 <=> ($true = (in @ sK335 @ (binintersect @ sK333 @ sK334)))),
% 2.41/1.47 introduced(definition,[new_symbols(naming,[spl789_14])],[avatar_definition])).
% 2.41/1.47 thf(f4140,plain,(
% 2.41/1.47 ((in @ sK335 @ sK333) = $true) | ~spl789_14),
% 2.41/1.47 inference(trivial_inequality_removal,[status(thm)],[f4139])).
% 2.41/1.47 thf(f4139,plain,(
% 2.41/1.47 ((in @ sK335 @ sK333) = $true) | ($true != $true) | ~spl789_14),
% 2.41/1.47 inference(forward_demodulation,[status(thm)],[f4138,f2965])).
% 2.41/1.47 thf(f2965,plain,(
% 2.41/1.47 (binintersectEL = $true)),
% 2.41/1.47 inference(cnf_transformation,[status(thm)],[f1613])).
% 2.41/1.47 thf(f1613,plain,(
% 2.41/1.47 (setukpairinjR11 = $true) & (emptyinunitempty = $true) & (binintersectI = $true) & (quantDeMorgan2 = $true) & (omega__Cong = $true) & (nonemptyI = $true) & (cartprodpairin = $true) & (eqimpsubset2 = $true) & (exuE1 = $true) & (binintersectSubset4 = $true) & (dsetconstrER = $true) & (theprop = $true) & (sepSubset = $true) & (prop2set2propI = $true) & (binintersectSubset3 = $true) & (subPowSU = $true) & (symdiffE = $true) & (setukpairinjR2 = $true) & (lam2lamEq = $true) & (doubleComplementSub1 = $true) & (setadjoinSub = $true) & (singletonsubset = $true) & (subsetI1 = $true) & (complementT_lem = $true) & (lamp = $true) & (setminusELneg = $true) & (binintersectTELcontra = $true) & (notsubsetI = $true) & (powerset__Cong = $true) & (iffalse = $true) & (nonemptyImpWitness = $true) & (exuE3u = $true) & (emptysetE = $true) & (binunionE = $true) & (powersetTI1 = $true) & (binintersectEL = $true) & (cartprodpairsurjEq = $true) & (notequalI2 = $true) & (singletoninpowunion = $true) & (powersetI1 = $true) & (ex1E1 = $true) & (ksndpairEq = $true) & (beta2 = $true) & (upairsubunion = $true) & (omegaSAx = $true) & (in__Cong = $true) & (binintersectER = $true) & (setadjoin__Cong = $true) & (ifSingleton = $true) & (infuncsetfunc = $true) & (contrasubsetT1 = $true) & (emptyE1 = $true) & (wellorderingAx = $true) & (theeq = $true) & (inCongP = $true) & (disjointsetsI1 = $true) & (upairinpowunion = $true) & (lam2p = $true) & (funcGraphProp4 = $true) & (setukpairIR = $true) & (setminusSubset2 = $true) & (kpairp = $true) & (dpsetconstrERa = $true) & (cartprodfstpairEq = $true) & (dpsetconstrSub = $true) & (ksndsingleton = $true) & (omegaIndAx = $true) & (exuI3 = $true) & (sepInPowerset = $true) & (brelnall1 = $true) & (powersetTE1 = $true) & (complementTnotintersectT = $true) & (binunionIL = $true) & (exuI1 = $true) & (binintersectT_lem = $true) & (notinsingleton = $true) & (iffalseProp1 = $true) & (omega0Ax = $true) & (bs114d = $true) & (setukpairinjR1 = $true) & (exuE2 = $true) & (nonemptyI1 = $true) & (upairequniteq = $true) & (emptysetAx = $true) & (funcext2 = $true) & (setminusER = $true) & (setukpairIL = $true) & (setextT = $true) & (setminusLsub = $true) & (cartprodpairmemER = $true) & (setukpairinjL2 = $true) & (emptyinPowerset = $true) & (funcinfuncset = $true) & (singletonsuniq = $true) & (funcGraphProp1 = $true) & (setminusIRneg = $true) & (iftrueProp1 = $true) & (emptysetimpfalse = $true) & (setadjoinIR = $true) & (setukpairinjL1 = $true) & (dpsetconstrEL1 = $true) & (setadjoinAx = $true) & (emptysetsubset = $true) & (singletonprop = $true) & (binunionIR = $true) & (setunionI = $true) & (setunionsingleton2 = $true) & (eta1 = $true) & (ex1I = $true) & (subsetE = $true) & (binintersectTERcontra = $true) & (eqinunit = $true) & (notinemptyset = $true) & (($true = (in @ sK333 @ (powerset @ sK332))) & (((in @ sK334 @ (powerset @ sK332)) = $true) & (($true = (in @ sK335 @ (setminus @ sK332 @ sK333))) & ((in @ sK335 @ sK332) = $true) & ((in @ sK335 @ (setminus @ sK332 @ (binintersect @ sK333 @ sK334))) != $true)))) & (ap2apEq2 = $true) & (setext = $true) & (notdexE = $true) & (symdiffIneg1 = $true) & (ap2apEq1 = $true) & (complementTI1 = $true) & (dsetconstrEL = $true) & (setunion__Cong = $true) & (setunionsingleton1 = $true) & (subsetE2 = $true) & (binintersectSubset5 = $true) & (quantDeMorgan3 = $true) & (powersetAx = $true) & (binintersectSubset2 = $true) & (setunionAx = $true) & (setminusEL = $true) & (powersetT_lem = $true) & (kpairiskpair = $true) & (kpairsurjEq = $true) & (setunionE = $true) & (setminusI = $true) & (binunionRsub = $true) & (lamProp = $true) & (cartprodmempair = $true) & (binintersectRsub = $true) & (descrp = $true) & (emptyset__Cong = $true) & (singletoninpowerset = $true) & (ifp = $true) & (binunionT_lem = $true) & (funcGraphProp2 = $true) & (emptyInPowerset = $true) & (setextAx = $true) & (subset2powerset = $true) & (cartprodpairmemEL = $true) & (setadjoinIL = $true) & (exu__Cong = $true) & (noeltsimpempty = $true) & (funcImageSingleton = $true) & (quantDeMorgan1 = $true) & (funcext = $true) & (binunionEcases = $true) & (setminusILneg = $true) & (ubforcartprodlem2 = $true) & (symdiffI2 = $true) & (powersetE = $true) & (kfstpairEq = $true) & (exuE3e = $true) & (upairsetIL = $true) & (contrasubsetT3 = $true) & (ap2p = $true) & (setoftrueEq = $true) & (binintersectSubset1 = $true) & (funcextLem = $true) & (setextsub = $true) & (vacuousDall = $true) & (app = $true) & (nonemptyE1 = $true) & (setukpairinjR12 = $true) & (setadjoinE = $true) & (eqbreln = $true) & (powersetE1 = $true) & (ex1I2 = $true) & (kfstsingleton = $true) & (contrasubsetT2 = $true) & (iftrue = $true) & (subsetRefl = $true) & (setadjoinSub2 = $true) & (cartprodmempair1 = $true) & (descr__Cong = $true) & (iftrueProp2 = $true) & (dsetconstrI = $true) & (inPowerset = $true) & (subbreln = $true) & (upairset2E = $true) & (emptyI = $true) & (apProp = $true) & (upairsetIR = $true) & (setadjoinOr = $true) & (doubleComplementEq = $true) & (setunionsingleton = $true) & (replAx = $true) & (dpsetconstrER = $true) & (subsetemptysetimpeq = $true) & (symdiffI1 = $true) & (contrasubsetT = $true) & (binunionLsub = $true) & (setminusERneg = $true) & (cartprodsndin = $true) & (exuEu = $true) & (doubleComplementE1 = $true) & (cartprodsndpairEq = $true) & (beta1 = $true) & (secondinupair = $true) & (ubforcartprodlem3 = $true) & (powersetI = $true) & (setunionE2 = $true) & (complementTE1 = $true) & (prop2setE = $true) & (upairsetE = $true) & (setbeta = $true) & (iffalseProp2 = $true) & (notequalI1 = $true) & (dpsetconstrI = $true) & (foundationAx = $true) & (iftrueorfalse = $true) & (setukpairinjL = $true) & (doubleComplementI1 = $true) & (exuI2 = $true) & (dpsetconstrEL2 = $true) & (upairset2IR = $true) & (prop2setI = $true) & (binintersectLsub = $true) & (powersetsubset = $true) & (setminusSubset1 = $true) & (singletonsswitch = $true) & (brelnall2 = $true) & (quantDeMorgan4 = $true) & (setminusT_lem = $true) & (notdallE = $true) & (cartprodmempaircEq = $true) & (setOfPairsIsBReln = $true) & (doubleComplementSub2 = $true) & (eta2 = $true) & (uniqinunit = $true) & (eqimpsubset1 = $true) & (ex1E2 = $true) & (setukpairinjR = $true) & (subsetI2 = $true) & (dsetconstr__Cong = $true) & (funcGraphProp3 = $true) & (cartprodfstin = $true) & (subsetTrans = $true) & (ubforcartprodlem1 = $true) & (subsetTI = $true) & (symdiffIneg2 = $true)),
% 2.41/1.47 inference(skolemisation,[status(esa),new_symbols(skolem,[sK332,sK333,sK334,sK335])],[f847,f1612,f1611,f1610])).
% 2.41/1.47 thf(f1610,plain,(
% 2.41/1.47 ? [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) & ? [X2 : $i] : (((in @ X2 @ (powerset @ X0)) = $true) & ? [X3 : $i] : (((in @ X3 @ (setminus @ X0 @ X1)) = $true) & ((in @ X3 @ X0) = $true) & ((in @ X3 @ (setminus @ X0 @ (binintersect @ X1 @ X2))) != $true)))) => (($true = (in @ sK333 @ (powerset @ sK332))) & ? [X2 : $i] : (((in @ X2 @ (powerset @ sK332)) = $true) & ? [X3 : $i] : (($true = (in @ X3 @ (setminus @ sK332 @ sK333))) & ((in @ X3 @ sK332) = $true) & ((in @ X3 @ (setminus @ sK332 @ (binintersect @ sK333 @ X2))) != $true))))),
% 2.41/1.47 introduced(definition,[],[choice_axiom])).
% 2.41/1.47 thf(f1611,plain,(
% 2.41/1.47 ? [X2 : $i] : (((in @ X2 @ (powerset @ sK332)) = $true) & ? [X3 : $i] : (($true = (in @ X3 @ (setminus @ sK332 @ sK333))) & ((in @ X3 @ sK332) = $true) & ((in @ X3 @ (setminus @ sK332 @ (binintersect @ sK333 @ X2))) != $true))) => (((in @ sK334 @ (powerset @ sK332)) = $true) & ? [X3 : $i] : (($true = (in @ X3 @ (setminus @ sK332 @ sK333))) & ((in @ X3 @ sK332) = $true) & ((in @ X3 @ (setminus @ sK332 @ (binintersect @ sK333 @ sK334))) != $true)))),
% 2.41/1.47 introduced(definition,[],[choice_axiom])).
% 2.41/1.47 thf(f1612,plain,(
% 2.41/1.47 ? [X3 : $i] : (($true = (in @ X3 @ (setminus @ sK332 @ sK333))) & ((in @ X3 @ sK332) = $true) & ((in @ X3 @ (setminus @ sK332 @ (binintersect @ sK333 @ sK334))) != $true)) => (($true = (in @ sK335 @ (setminus @ sK332 @ sK333))) & ((in @ sK335 @ sK332) = $true) & ((in @ sK335 @ (setminus @ sK332 @ (binintersect @ sK333 @ sK334))) != $true))),
% 2.41/1.47 introduced(definition,[],[choice_axiom])).
% 2.41/1.47 thf(f847,plain,(
% 2.41/1.47 (setukpairinjR11 = $true) & (emptyinunitempty = $true) & (binintersectI = $true) & (quantDeMorgan2 = $true) & (omega__Cong = $true) & (nonemptyI = $true) & (cartprodpairin = $true) & (eqimpsubset2 = $true) & (exuE1 = $true) & (binintersectSubset4 = $true) & (dsetconstrER = $true) & (theprop = $true) & (sepSubset = $true) & (prop2set2propI = $true) & (binintersectSubset3 = $true) & (subPowSU = $true) & (symdiffE = $true) & (setukpairinjR2 = $true) & (lam2lamEq = $true) & (doubleComplementSub1 = $true) & (setadjoinSub = $true) & (singletonsubset = $true) & (subsetI1 = $true) & (complementT_lem = $true) & (lamp = $true) & (setminusELneg = $true) & (binintersectTELcontra = $true) & (notsubsetI = $true) & (powerset__Cong = $true) & (iffalse = $true) & (nonemptyImpWitness = $true) & (exuE3u = $true) & (emptysetE = $true) & (binunionE = $true) & (powersetTI1 = $true) & (binintersectEL = $true) & (cartprodpairsurjEq = $true) & (notequalI2 = $true) & (singletoninpowunion = $true) & (powersetI1 = $true) & (ex1E1 = $true) & (ksndpairEq = $true) & (beta2 = $true) & (upairsubunion = $true) & (omegaSAx = $true) & (in__Cong = $true) & (binintersectER = $true) & (setadjoin__Cong = $true) & (ifSingleton = $true) & (infuncsetfunc = $true) & (contrasubsetT1 = $true) & (emptyE1 = $true) & (wellorderingAx = $true) & (theeq = $true) & (inCongP = $true) & (disjointsetsI1 = $true) & (upairinpowunion = $true) & (lam2p = $true) & (funcGraphProp4 = $true) & (setukpairIR = $true) & (setminusSubset2 = $true) & (kpairp = $true) & (dpsetconstrERa = $true) & (cartprodfstpairEq = $true) & (dpsetconstrSub = $true) & (ksndsingleton = $true) & (omegaIndAx = $true) & (exuI3 = $true) & (sepInPowerset = $true) & (brelnall1 = $true) & (powersetTE1 = $true) & (complementTnotintersectT = $true) & (binunionIL = $true) & (exuI1 = $true) & (binintersectT_lem = $true) & (notinsingleton = $true) & (iffalseProp1 = $true) & (omega0Ax = $true) & (bs114d = $true) & (setukpairinjR1 = $true) & (exuE2 = $true) & (nonemptyI1 = $true) & (upairequniteq = $true) & (emptysetAx = $true) & (funcext2 = $true) & (setminusER = $true) & (setukpairIL = $true) & (setextT = $true) & (setminusLsub = $true) & (cartprodpairmemER = $true) & (setukpairinjL2 = $true) & (emptyinPowerset = $true) & (funcinfuncset = $true) & (singletonsuniq = $true) & (funcGraphProp1 = $true) & (setminusIRneg = $true) & (iftrueProp1 = $true) & (emptysetimpfalse = $true) & (setadjoinIR = $true) & (setukpairinjL1 = $true) & (dpsetconstrEL1 = $true) & (setadjoinAx = $true) & (emptysetsubset = $true) & (singletonprop = $true) & (binunionIR = $true) & (setunionI = $true) & (setunionsingleton2 = $true) & (eta1 = $true) & (ex1I = $true) & (subsetE = $true) & (binintersectTERcontra = $true) & (eqinunit = $true) & (notinemptyset = $true) & ? [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) & ? [X2 : $i] : (((in @ X2 @ (powerset @ X0)) = $true) & ? [X3 : $i] : (((in @ X3 @ (setminus @ X0 @ X1)) = $true) & ((in @ X3 @ X0) = $true) & ((in @ X3 @ (setminus @ X0 @ (binintersect @ X1 @ X2))) != $true)))) & (ap2apEq2 = $true) & (setext = $true) & (notdexE = $true) & (symdiffIneg1 = $true) & (ap2apEq1 = $true) & (complementTI1 = $true) & (dsetconstrEL = $true) & (setunion__Cong = $true) & (setunionsingleton1 = $true) & (subsetE2 = $true) & (binintersectSubset5 = $true) & (quantDeMorgan3 = $true) & (powersetAx = $true) & (binintersectSubset2 = $true) & (setunionAx = $true) & (setminusEL = $true) & (powersetT_lem = $true) & (kpairiskpair = $true) & (kpairsurjEq = $true) & (setunionE = $true) & (setminusI = $true) & (binunionRsub = $true) & (lamProp = $true) & (cartprodmempair = $true) & (binintersectRsub = $true) & (descrp = $true) & (emptyset__Cong = $true) & (singletoninpowerset = $true) & (ifp = $true) & (binunionT_lem = $true) & (funcGraphProp2 = $true) & (emptyInPowerset = $true) & (setextAx = $true) & (subset2powerset = $true) & (cartprodpairmemEL = $true) & (setadjoinIL = $true) & (exu__Cong = $true) & (noeltsimpempty = $true) & (funcImageSingleton = $true) & (quantDeMorgan1 = $true) & (funcext = $true) & (binunionEcases = $true) & (setminusILneg = $true) & (ubforcartprodlem2 = $true) & (symdiffI2 = $true) & (powersetE = $true) & (kfstpairEq = $true) & (exuE3e = $true) & (upairsetIL = $true) & (contrasubsetT3 = $true) & (ap2p = $true) & (setoftrueEq = $true) & (binintersectSubset1 = $true) & (funcextLem = $true) & (setextsub = $true) & (vacuousDall = $true) & (app = $true) & (nonemptyE1 = $true) & (setukpairinjR12 = $true) & (setadjoinE = $true) & (eqbreln = $true) & (powersetE1 = $true) & (ex1I2 = $true) & (kfstsingleton = $true) & (contrasubsetT2 = $true) & (iftrue = $true) & (subsetRefl = $true) & (setadjoinSub2 = $true) & (cartprodmempair1 = $true) & (descr__Cong = $true) & (iftrueProp2 = $true) & (dsetconstrI = $true) & (inPowerset = $true) & (subbreln = $true) & (upairset2E = $true) & (emptyI = $true) & (apProp = $true) & (upairsetIR = $true) & (setadjoinOr = $true) & (doubleComplementEq = $true) & (setunionsingleton = $true) & (replAx = $true) & (dpsetconstrER = $true) & (subsetemptysetimpeq = $true) & (symdiffI1 = $true) & (contrasubsetT = $true) & (binunionLsub = $true) & (setminusERneg = $true) & (cartprodsndin = $true) & (exuEu = $true) & (doubleComplementE1 = $true) & (cartprodsndpairEq = $true) & (beta1 = $true) & (secondinupair = $true) & (ubforcartprodlem3 = $true) & (powersetI = $true) & (setunionE2 = $true) & (complementTE1 = $true) & (prop2setE = $true) & (upairsetE = $true) & (setbeta = $true) & (iffalseProp2 = $true) & (notequalI1 = $true) & (dpsetconstrI = $true) & (foundationAx = $true) & (iftrueorfalse = $true) & (setukpairinjL = $true) & (doubleComplementI1 = $true) & (exuI2 = $true) & (dpsetconstrEL2 = $true) & (upairset2IR = $true) & (prop2setI = $true) & (binintersectLsub = $true) & (powersetsubset = $true) & (setminusSubset1 = $true) & (singletonsswitch = $true) & (brelnall2 = $true) & (quantDeMorgan4 = $true) & (setminusT_lem = $true) & (notdallE = $true) & (cartprodmempaircEq = $true) & (setOfPairsIsBReln = $true) & (doubleComplementSub2 = $true) & (eta2 = $true) & (uniqinunit = $true) & (eqimpsubset1 = $true) & (ex1E2 = $true) & (setukpairinjR = $true) & (subsetI2 = $true) & (dsetconstr__Cong = $true) & (funcGraphProp3 = $true) & (cartprodfstin = $true) & (subsetTrans = $true) & (ubforcartprodlem1 = $true) & (subsetTI = $true) & (symdiffIneg2 = $true)),
% 2.41/1.47 inference(flattening,[status(thm)],[f846])).
% 2.41/1.47 thf(f846,plain,(
% 2.41/1.47 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : ((((in @ X3 @ (setminus @ X0 @ (binintersect @ X1 @ X2))) != $true) & ((in @ X3 @ (setminus @ X0 @ X1)) = $true)) & ((in @ X3 @ X0) = $true)) & ((in @ X2 @ (powerset @ X0)) = $true)) & ((in @ X1 @ (powerset @ X0)) = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.41/1.47 inference(ennf_transformation,[status(thm)],[f752])).
% 2.41/1.47 thf(f752,plain,(
% 2.41/1.47 ~((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) => ! [X0 : $i,X1 : $i] : (((in @ X1 @ (powerset @ X0)) = $true) => ! [X2 : $i] : (((in @ X2 @ (powerset @ X0)) = $true) => ! [X3 : $i] : (((in @ X3 @ X0) = $true) => (((in @ X3 @ (setminus @ X0 @ X1)) = $true) => ((in @ X3 @ (setminus @ X0 @ (binintersect @ X1 @ X2))) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.41/1.47 inference(fool_elimination,[status(thm)],[f751])).
% 2.41/1.47 thf(f751,plain,(
% 2.41/1.47 ~(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 => ! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X0)) => ! [X3 : $i] : ((in @ X3 @ X0) => ((in @ X3 @ (setminus @ X0 @ X1)) => (in @ X3 @ (setminus @ X0 @ (binintersect @ X1 @ X2))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.41/1.47 inference(rectify,[status(thm)],[f255])).
% 2.41/1.47 thf(f255,negated_conjecture,(
% 2.41/1.47 ~(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 => ! [X3 : $i,X11 : $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 @ (binintersect @ X11 @ X16))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.41/1.47 inference(negated_conjecture,[status(cth)],[f254])).
% 2.41/1.47 thf(f254,conjecture,(
% 2.41/1.47 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 => ! [X3 : $i,X11 : $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 @ (binintersect @ X11 @ X16)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.41/1.47 file('/export/starexec/sandbox2/tmp/tmp.V67yydYf3P/DTF2THF_3524.p',complementImpComplementIntersect)).
% 2.41/1.47 thf(f4138,plain,(
% 2.41/1.47 (binintersectEL != $true) | ((in @ sK335 @ sK333) = $true) | ~spl789_14),
% 2.41/1.47 inference(trivial_inequality_removal,[status(thm)],[f4125])).
% 2.41/1.47 thf(f4125,plain,(
% 2.41/1.47 ((in @ sK335 @ sK333) = $true) | (binintersectEL != $true) | ($true != $true) | ~spl789_14),
% 2.41/1.47 inference(superposition,[status(thm)],[f2555,f3972])).
% 2.41/1.47 thf(f3972,plain,(
% 2.41/1.47 ($true = (in @ sK335 @ (binintersect @ sK333 @ sK334))) | ~spl789_14),
% 2.41/1.47 inference(avatar_component_clause,[status(thm)],[f3970])).
% 2.41/1.47 thf(f2555,plain,(
% 2.41/1.47 ( ! [X2 : $i,X0 : $i,X1 : $i] : (((in @ X0 @ (binintersect @ X2 @ X1)) != $true) | ((in @ X0 @ X2) = $true) | (binintersectEL != $true)) )),
% 2.41/1.47 inference(cnf_transformation,[status(thm)],[f1400])).
% 2.41/1.47 thf(f1400,plain,(
% 2.41/1.47 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binintersect @ X2 @ X1)) != $true) | ((in @ X0 @ X2) = $true)) | (binintersectEL != $true)) & ((binintersectEL = $true) | (($true = (in @ sK192 @ (binintersect @ sK194 @ sK193))) & ($true != (in @ sK192 @ sK194))))),
% 2.41/1.47 inference(skolemisation,[status(esa),new_symbols(skolem,[sK192,sK193,sK194])],[f1398,f1399])).
% 2.41/1.47 thf(f1399,plain,(
% 2.41/1.47 ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ (binintersect @ X5 @ X4)) = $true) & ($true != (in @ X3 @ X5))) => (($true = (in @ sK192 @ (binintersect @ sK194 @ sK193))) & ($true != (in @ sK192 @ sK194)))),
% 2.41/1.47 introduced(definition,[],[choice_axiom])).
% 2.41/1.47 thf(f1398,plain,(
% 2.41/1.47 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binintersect @ X2 @ X1)) != $true) | ((in @ X0 @ X2) = $true)) | (binintersectEL != $true)) & ((binintersectEL = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ (binintersect @ X5 @ X4)) = $true) & ($true != (in @ X3 @ X5))))),
% 2.41/1.47 inference(rectify,[status(thm)],[f1397])).
% 2.41/1.47 thf(f1397,plain,(
% 2.41/1.47 (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binintersect @ X2 @ X1)) != $true) | ((in @ X0 @ X2) = $true)) | (binintersectEL != $true)) & ((binintersectEL = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binintersect @ X2 @ X1)) = $true) & ((in @ X0 @ X2) != $true)))),
% 2.41/1.47 inference(nnf_transformation,[status(thm)],[f1085])).
% 2.41/1.47 thf(f1085,plain,(
% 2.41/1.47 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binintersect @ X2 @ X1)) != $true) | ((in @ X0 @ X2) = $true)) <=> (binintersectEL = $true)),
% 2.41/1.47 inference(ennf_transformation,[status(thm)],[f559])).
% 2.41/1.47 thf(f559,plain,(
% 2.41/1.47 (binintersectEL = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X0 @ (binintersect @ X2 @ X1)) = $true) => ((in @ X0 @ X2) = $true))),
% 2.41/1.47 inference(fool_elimination,[status(thm)],[f558])).
% 2.41/1.47 thf(f558,plain,(
% 2.41/1.47 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ (binintersect @ X2 @ X1)) => (in @ X0 @ X2)) = binintersectEL)),
% 2.41/1.47 inference(rectify,[status(thm)],[f110])).
% 2.41/1.47 thf(f110,axiom,(
% 2.41/1.47 (! [X1 : $i,X4 : $i,X3 : $i] : ((in @ X1 @ (binintersect @ X3 @ X4)) => (in @ X1 @ X3)) = binintersectEL)),
% 2.41/1.47 file('/export/starexec/sandbox2/tmp/tmp.V67yydYf3P/DTF2THF_3524.p',binintersectEL)).
% 2.41/1.47 thf(f4122,plain,(
% 2.41/1.47 spl789_14),
% 2.41/1.47 inference(avatar_split_clause,[status(thm)],[f3954,f3970])).
% 2.41/1.47 thf(f3954,plain,(
% 2.41/1.47 ($true = (in @ sK335 @ (binintersect @ sK333 @ sK334)))),
% 2.41/1.47 inference(trivial_inequality_removal,[status(thm)],[f3953])).
% 2.41/1.47 thf(f3953,plain,(
% 2.41/1.47 ($true = (in @ sK335 @ (binintersect @ sK333 @ sK334))) | ($true != $true)),
% 2.41/1.47 inference(forward_demodulation,[status(thm)],[f3952,f2795])).
% 2.41/1.47 thf(f2795,plain,(
% 2.41/1.47 (setminusERneg = $true)),
% 2.41/1.47 inference(cnf_transformation,[status(thm)],[f1613])).
% 2.41/1.47 thf(f3952,plain,(
% 2.41/1.47 (setminusERneg != $true) | ($true = (in @ sK335 @ (binintersect @ sK333 @ sK334)))),
% 2.41/1.47 inference(trivial_inequality_removal,[status(thm)],[f3951])).
% 2.41/1.47 thf(f3951,plain,(
% 2.41/1.47 (setminusERneg != $true) | ($true = (in @ sK335 @ (binintersect @ sK333 @ sK334))) | ($true != $true)),
% 2.41/1.47 inference(forward_demodulation,[status(thm)],[f3937,f2884])).
% 2.41/1.47 thf(f2884,plain,(
% 2.41/1.47 ((in @ sK335 @ sK332) = $true)),
% 2.41/1.47 inference(cnf_transformation,[status(thm)],[f1613])).
% 2.41/1.47 thf(f3937,plain,(
% 2.41/1.47 ($true = (in @ sK335 @ (binintersect @ sK333 @ sK334))) | ((in @ sK335 @ sK332) != $true) | (setminusERneg != $true)),
% 2.41/1.47 inference(trivial_inequality_removal,[status(thm)],[f3925])).
% 2.41/1.47 thf(f3925,plain,(
% 2.41/1.47 ($true = (in @ sK335 @ (binintersect @ sK333 @ sK334))) | ((in @ sK335 @ sK332) != $true) | ($true != $true) | (setminusERneg != $true)),
% 2.41/1.47 inference(superposition,[status(thm)],[f2883,f3079])).
% 2.41/1.47 thf(f3079,plain,(
% 2.41/1.47 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ (setminus @ X4 @ X5)) = $true) | ($true = (in @ X3 @ X5)) | (setminusERneg != $true) | ((in @ X3 @ X4) != $true)) )),
% 2.41/1.47 inference(cnf_transformation,[status(thm)],[f1713])).
% 2.41/1.47 thf(f1713,plain,(
% 2.41/1.47 ((setminusERneg = $true) | (((in @ sK395 @ sK396) = $true) & ($true != (in @ sK395 @ (setminus @ sK396 @ sK397))) & ($true != (in @ sK395 @ sK397)))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ X4) != $true) | ((in @ X3 @ (setminus @ X4 @ X5)) = $true) | ($true = (in @ X3 @ X5))) | (setminusERneg != $true))),
% 2.41/1.47 inference(skolemisation,[status(esa),new_symbols(skolem,[sK395,sK396,sK397])],[f1711,f1712])).
% 2.41/1.47 thf(f1712,plain,(
% 2.41/1.47 ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X1)) & ($true != (in @ X0 @ (setminus @ X1 @ X2))) & ((in @ X0 @ X2) != $true)) => (((in @ sK395 @ sK396) = $true) & ($true != (in @ sK395 @ (setminus @ sK396 @ sK397))) & ($true != (in @ sK395 @ sK397)))),
% 2.41/1.47 introduced(definition,[],[choice_axiom])).
% 2.41/1.47 thf(f1711,plain,(
% 2.41/1.47 ((setminusERneg = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X1)) & ($true != (in @ X0 @ (setminus @ X1 @ X2))) & ((in @ X0 @ X2) != $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X3 @ X4) != $true) | ((in @ X3 @ (setminus @ X4 @ X5)) = $true) | ($true = (in @ X3 @ X5))) | (setminusERneg != $true))),
% 2.41/1.47 inference(rectify,[status(thm)],[f1710])).
% 2.41/1.47 thf(f1710,plain,(
% 2.41/1.47 ((setminusERneg = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X1)) & ($true != (in @ X0 @ (setminus @ X1 @ X2))) & ((in @ X0 @ X2) != $true))) & (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ X1)) | ($true = (in @ X0 @ (setminus @ X1 @ X2))) | ((in @ X0 @ X2) = $true)) | (setminusERneg != $true))),
% 2.41/1.47 inference(nnf_transformation,[status(thm)],[f1049])).
% 2.41/1.47 thf(f1049,plain,(
% 2.41/1.47 (setminusERneg = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ X1)) | ($true = (in @ X0 @ (setminus @ X1 @ X2))) | ((in @ X0 @ X2) = $true))),
% 2.41/1.47 inference(flattening,[status(thm)],[f1048])).
% 2.41/1.47 thf(f1048,plain,(
% 2.41/1.47 (setminusERneg = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : ((((in @ X0 @ X2) = $true) | ($true != (in @ X0 @ X1))) | ($true = (in @ X0 @ (setminus @ X1 @ X2))))),
% 2.41/1.47 inference(ennf_transformation,[status(thm)],[f798])).
% 2.41/1.47 thf(f798,plain,(
% 2.41/1.47 (setminusERneg = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ (setminus @ X1 @ X2))) => (($true = (in @ X0 @ X1)) => ((in @ X0 @ X2) = $true)))),
% 2.41/1.47 inference(flattening,[status(thm)],[f625])).
% 2.41/1.47 thf(f625,plain,(
% 2.41/1.47 ! [X0 : $i,X1 : $i,X2 : $i] : (~($true = (in @ X0 @ (setminus @ X1 @ X2))) => (($true = (in @ X0 @ X1)) => ((in @ X0 @ X2) = $true))) <=> (setminusERneg = $true)),
% 2.41/1.47 inference(fool_elimination,[status(thm)],[f624])).
% 2.41/1.47 thf(f624,plain,(
% 2.41/1.47 (! [X0 : $i,X1 : $i,X2 : $i] : (~(in @ X0 @ (setminus @ X1 @ X2)) => ((in @ X0 @ X1) => (in @ X0 @ X2))) = setminusERneg)),
% 2.41/1.47 inference(rectify,[status(thm)],[f124])).
% 2.41/1.47 thf(f124,axiom,(
% 2.41/1.47 (! [X1 : $i,X3 : $i,X4 : $i] : (~(in @ X1 @ (setminus @ X3 @ X4)) => ((in @ X1 @ X3) => (in @ X1 @ X4))) = setminusERneg)),
% 2.41/1.47 file('/export/starexec/sandbox2/tmp/tmp.V67yydYf3P/DTF2THF_3524.p',setminusERneg)).
% 2.41/1.47 thf(f2883,plain,(
% 2.41/1.47 ((in @ sK335 @ (setminus @ sK332 @ (binintersect @ sK333 @ sK334))) != $true)),
% 2.41/1.47 inference(cnf_transformation,[status(thm)],[f1613])).
% 2.41/1.47 thf(f3892,plain,(
% 2.41/1.47 ~spl789_7),
% 2.41/1.47 inference(avatar_split_clause,[status(thm)],[f3891,f3881])).
% 2.41/1.47 thf(f3891,plain,(
% 2.41/1.47 ((in @ sK335 @ sK333) != $true)),
% 2.41/1.47 inference(trivial_inequality_removal,[status(thm)],[f3890])).
% 2.41/1.47 thf(f3890,plain,(
% 2.41/1.47 ($true != $true) | ((in @ sK335 @ sK333) != $true)),
% 2.41/1.47 inference(forward_demodulation,[status(thm)],[f3868,f2915])).
% 2.41/1.47 thf(f2915,plain,(
% 2.41/1.47 (setminusER = $true)),
% 2.41/1.47 inference(cnf_transformation,[status(thm)],[f1613])).
% 2.41/1.47 thf(f3868,plain,(
% 2.41/1.47 (setminusER != $true) | ((in @ sK335 @ sK333) != $true)),
% 2.41/1.47 inference(trivial_inequality_removal,[status(thm)],[f3861])).
% 2.41/1.47 thf(f3861,plain,(
% 2.41/1.47 ($true != $true) | ((in @ sK335 @ sK333) != $true) | (setminusER != $true)),
% 2.41/1.47 inference(superposition,[status(thm)],[f3377,f2885])).
% 2.41/1.47 thf(f2885,plain,(
% 2.41/1.47 ($true = (in @ sK335 @ (setminus @ sK332 @ sK333)))),
% 2.41/1.47 inference(cnf_transformation,[status(thm)],[f1613])).
% 2.41/1.47 thf(f3377,plain,(
% 2.41/1.47 ( ! [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X4 @ (setminus @ X5 @ X3))) | ((in @ X4 @ X3) != $true) | (setminusER != $true)) )),
% 2.41/1.47 inference(cnf_transformation,[status(thm)],[f2051])).
% 2.41/1.47 thf(f2051,plain,(
% 2.41/1.47 ((setminusER = $true) | (($true = (in @ sK619 @ (setminus @ sK620 @ sK618))) & ($true = (in @ sK619 @ sK618)))) & (! [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X4 @ (setminus @ X5 @ X3))) | ((in @ X4 @ X3) != $true)) | (setminusER != $true))),
% 2.41/1.47 inference(skolemisation,[status(esa),new_symbols(skolem,[sK618,sK619,sK620])],[f2049,f2050])).
% 2.41/1.47 thf(f2050,plain,(
% 2.41/1.47 ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (setminus @ X2 @ X0)) = $true) & ((in @ X1 @ X0) = $true)) => (($true = (in @ sK619 @ (setminus @ sK620 @ sK618))) & ($true = (in @ sK619 @ sK618)))),
% 2.41/1.47 introduced(definition,[],[choice_axiom])).
% 2.41/1.47 thf(f2049,plain,(
% 2.41/1.47 ((setminusER = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (setminus @ X2 @ X0)) = $true) & ((in @ X1 @ X0) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X4 @ (setminus @ X5 @ X3))) | ((in @ X4 @ X3) != $true)) | (setminusER != $true))),
% 2.41/1.47 inference(rectify,[status(thm)],[f2048])).
% 2.41/1.47 thf(f2048,plain,(
% 2.41/1.47 ((setminusER = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (setminus @ X2 @ X0)) = $true) & ((in @ X1 @ X0) = $true))) & (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (setminus @ X2 @ X0)) != $true) | ((in @ X1 @ X0) != $true)) | (setminusER != $true))),
% 2.41/1.47 inference(nnf_transformation,[status(thm)],[f920])).
% 2.41/1.47 thf(f920,plain,(
% 2.41/1.47 (setminusER = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (setminus @ X2 @ X0)) != $true) | ((in @ X1 @ X0) != $true))),
% 2.41/1.47 inference(ennf_transformation,[status(thm)],[f773])).
% 2.41/1.47 thf(f773,plain,(
% 2.41/1.47 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (setminus @ X2 @ X0)) = $true) => ((in @ X1 @ X0) != $true)) <=> (setminusER = $true)),
% 2.41/1.47 inference(flattening,[status(thm)],[f396])).
% 2.41/1.47 thf(f396,plain,(
% 2.41/1.47 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (setminus @ X2 @ X0)) = $true) => ~((in @ X1 @ X0) = $true)) <=> (setminusER = $true)),
% 2.41/1.47 inference(fool_elimination,[status(thm)],[f395])).
% 2.41/1.47 thf(f395,plain,(
% 2.41/1.47 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ (setminus @ X2 @ X0)) => ~(in @ X1 @ X0)) = setminusER)),
% 2.41/1.47 inference(rectify,[status(thm)],[f122])).
% 2.41/1.47 thf(f122,axiom,(
% 2.41/1.47 (! [X4 : $i,X1 : $i,X3 : $i] : ((in @ X1 @ (setminus @ X3 @ X4)) => ~(in @ X1 @ X4)) = setminusER)),
% 2.41/1.47 file('/export/starexec/sandbox2/tmp/tmp.V67yydYf3P/DTF2THF_3524.p',setminusER)).
% 2.41/1.47 % SZS output end Proof for DTF2THF_3524
% 2.41/1.47 % (3729)------------------------------
% 2.41/1.47 % (3729)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.41/1.47 % (3729)Termination reason: Refutation
% 2.41/1.47
% 2.41/1.47 % (3729)Memory used [KB]: 10106
% 2.41/1.47 % (3729)Time elapsed: 0.178 s
% 2.41/1.47 % (3729)Instructions burned: 351 (million)
% 2.41/1.47 % (3729)------------------------------
% 2.41/1.47 % (3729)------------------------------
% 2.41/1.47 % (3682)Success in time 0.269 s
%------------------------------------------------------------------------------