%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU713^1 : 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 : n019.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:36:35 AM UTC 2026
% Result : Theorem 2.97s 1.98s
% Output : Refutation 2.97s
% Verified :
% SZS Type : Refutation
% Derivation depth : 22
% Number of leaves : 17
% Syntax : Number of formulae : 94 ( 23 unt; 0 typ; 1 def)
% Number of atoms : 4574 (1237 equ; 0 cnn)
% Maximal formula atoms : 232 ( 48 avg)
% Number of connectives : 2713 ( 163 ~; 130 |; 760 &; 693 @)
% ( 15 <=>; 952 =>; 0 <=; 0 <~>)
% Maximal formula depth : 240 ( 22 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 431 ( 428 usr; 247 con; 0-4 aty)
% Number of variables : 196 ( 0 ^; 140 !; 56 ?; 196 :)
% 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_284,type,
sP3: $i > $i > $i > $o > $o ).
thf(func_def_285,type,
sP4: $i > $i > $i > $i > $o ).
thf(func_def_287,type,
sP6: $i > $i > $o ).
thf(func_def_288,type,
sP7: $i > $o ).
thf(func_def_289,type,
sP8: $i > $i > $o ).
thf(func_def_290,type,
sP9: $i > $i > $o ).
thf(func_def_292,type,
sK11: $i > $i > ( $i > $o ) > $i ).
thf(func_def_293,type,
sK12: $i > $o ).
thf(func_def_303,type,
sK22: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_304,type,
sK23: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_305,type,
sK24: $i > $o ).
thf(func_def_306,type,
sK25: $i > $o ).
thf(func_def_309,type,
sK28: $i > $i ).
thf(func_def_310,type,
sK29: ( $i > $i ) > $i > $i > $i ).
thf(func_def_318,type,
sK37: $i > $i > $o ).
thf(func_def_325,type,
sK44: $i > $i > $o ).
thf(func_def_327,type,
sK46: $i > $i ).
thf(func_def_328,type,
sK47: $i > $i ).
thf(func_def_329,type,
sK48: $i > ( $i > $i > $o ) > $i ).
thf(func_def_330,type,
sK49: $i > ( $i > $i > $o ) > $i ).
thf(func_def_331,type,
sK50: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_336,type,
sK55: $i > ( $i > $o ) > $i ).
thf(func_def_337,type,
sK56: $i > $o ).
thf(func_def_346,type,
sK65: $i > ( $i > $o ) > $i ).
thf(func_def_347,type,
sK66: $i > $o ).
thf(func_def_357,type,
sK76: $i > $i ).
thf(func_def_359,type,
sK78: $i > $o ).
thf(func_def_368,type,
sK87: $i > $i > $i ).
thf(func_def_376,type,
sK95: ( $i > $o ) > $i ).
thf(func_def_377,type,
sK96: ( $i > $o ) > $i ).
thf(func_def_378,type,
sK97: $i > $o ).
thf(func_def_403,type,
sK122: $i > $o ).
thf(func_def_408,type,
sK127: $i > $o ).
thf(func_def_410,type,
sK129: ( $i > $o ) > $i > $i ).
thf(func_def_415,type,
sK134: $i > $i > $i ).
thf(func_def_416,type,
sK135: $i > $i ).
thf(func_def_419,type,
sK138: $i > $i > ( $i > $i ) > $i ).
thf(func_def_425,type,
sK144: $i > $i > $o ).
thf(func_def_434,type,
sK153: $i > $i > $i > $i > $i ).
thf(func_def_447,type,
sK166: $i > $o ).
thf(func_def_448,type,
sK167: $i > $o ).
thf(func_def_449,type,
sK168: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_450,type,
sK169: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_454,type,
sK173: $i > $i > $o ).
thf(func_def_465,type,
sK184: $i > $o ).
thf(func_def_471,type,
sK190: ( $i > $o ) > $i ).
thf(func_def_472,type,
sK191: $i > $o ).
thf(func_def_473,type,
sK192: $i > $i ).
thf(func_def_480,type,
sK199: $i > $i > $i ).
thf(func_def_499,type,
sK218: $i > $i > $i > $i > $i ).
thf(func_def_507,type,
sK226: $i > $i > $i > $i ).
thf(func_def_508,type,
sK227: $i > $i > $i > $i ).
thf(func_def_512,type,
sK231: $i > $o ).
thf(func_def_533,type,
sK252: $i > $o ).
thf(func_def_535,type,
sK254: ( $i > $o ) > $i > $i ).
thf(func_def_536,type,
sK255: ( $i > $o ) > $i > $i ).
thf(func_def_540,type,
sK259: $i > $i > $i ).
thf(func_def_550,type,
sK269: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_551,type,
sK270: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_554,type,
sK273: $i > $o ).
thf(func_def_555,type,
sK274: $i > $o ).
thf(func_def_561,type,
sK280: $i > $i ).
thf(func_def_563,type,
sK282: $o > $i > $i > $i ).
thf(func_def_577,type,
sK296: $i > $i > $i > $i > $i ).
thf(func_def_578,type,
sK297: $i > $i > $i > $i > $i ).
thf(func_def_584,type,
sK303: $i > $i > $i > $i > $i ).
thf(func_def_598,type,
sK317: $i > $i > $o ).
thf(func_def_609,type,
sK328: ( $i > $o ) > $i ).
thf(func_def_610,type,
sK329: $i > $o ).
thf(func_def_611,type,
sK330: $i > $i ).
thf(func_def_613,type,
sK332: $i > $o ).
thf(func_def_621,type,
sK340: $i > $o ).
thf(func_def_640,type,
sK359: $i > $i > $i ).
thf(func_def_643,type,
sK362: $i > $o ).
thf(func_def_646,type,
sK365: $i > $i > $o ).
thf(func_def_654,type,
sK373: $i > $o ).
thf(func_def_657,type,
sK376: $i > ( $i > $o ) > $i ).
thf(func_def_658,type,
sK377: $i > $o ).
thf(func_def_687,type,
sK406: $i > $i ).
thf(func_def_688,type,
sK407: ( $i > $i ) > $i > $i > $i ).
thf(func_def_693,type,
sK412: $i > $i ).
thf(func_def_694,type,
sK413: $i > $i > $i ).
thf(func_def_704,type,
sK423: $i > $o ).
thf(func_def_707,type,
sK426: ( $i > $o ) > $i > $i ).
thf(func_def_709,type,
sK428: $i > $o ).
thf(func_def_714,type,
sK433: $i > ( $i > $o ) > $i ).
thf(func_def_715,type,
sK434: $i > $o ).
thf(func_def_718,type,
sK437: $i > $i > $i ).
thf(func_def_726,type,
sK445: ( $i > $o ) > $i ).
thf(func_def_727,type,
sK446: $i > $o ).
thf(func_def_735,type,
sK454: $i > $i > ( $i > $i ) > $i ).
thf(func_def_736,type,
sK455: $i > $i ).
thf(func_def_761,type,
sK480: $i > $i ).
thf(func_def_769,type,
sK488: $i > $i ).
thf(func_def_778,type,
sK497: $i > $i > $o ).
thf(func_def_787,type,
sK506: $i > $i > $i ).
thf(func_def_788,type,
sK507: $i > $i > $i ).
thf(func_def_794,type,
sK513: $i > $o ).
thf(func_def_796,type,
sK515: ( $i > $o ) > $i > $i ).
thf(func_def_797,type,
sK516: $i > $o ).
thf(func_def_801,type,
sK520: $i > $o ).
thf(func_def_810,type,
sK529: $i > $i > $i ).
thf(func_def_816,type,
sK535: $i > $o ).
thf(func_def_818,type,
sK537: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_819,type,
sK538: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_836,type,
sK555: $i > $o ).
thf(func_def_837,type,
sK556: $i > $o ).
thf(func_def_850,type,
sK569: $i > ( $i > $o ) > $i ).
thf(func_def_851,type,
sK570: $i > $o ).
thf(func_def_853,type,
sK572: $i > $i ).
thf(func_def_856,type,
sK575: $i > $i ).
thf(func_def_858,type,
sK577: $i > $i ).
thf(func_def_861,type,
sK580: $i > ( $i > $i ) > $i > $i ).
thf(func_def_873,type,
sK592: $i > $i > $o ).
thf(func_def_884,type,
sK603: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_885,type,
sK604: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_889,type,
sK608: $i > $o ).
thf(func_def_895,type,
sK614: $i > $i > $i > $i > $i ).
thf(func_def_896,type,
sK615: $i > $i > $i > $i > $i ).
thf(func_def_897,type,
sK616: $i > $i > $i > $i > $i ).
thf(func_def_898,type,
sK617: $i > $i > $i > $i > $i ).
thf(func_def_899,type,
sK618: $i > $i > $i ).
thf(func_def_900,type,
sK619: $i > $i > $i ).
thf(func_def_901,type,
sK620: $i > $i > $i ).
thf(func_def_902,type,
sK621: $i > $i > $i ).
thf(func_def_903,type,
sK622: $i > $i > $i ).
thf(func_def_904,type,
sK623: $i > $i > $i > $i ).
thf(func_def_905,type,
sK624: $i > $i ).
thf(func_def_906,type,
sK625: $i > $i ).
thf(func_def_907,type,
sK626: $i > $i ).
thf(func_def_908,type,
sK627: $i > $i ).
thf(func_def_909,type,
sK628: $i > $i > $i > $i ).
thf(func_def_910,type,
sK629: $i > $i > $i > $i ).
thf(func_def_911,type,
sK630: $i > $i > $i ).
thf(func_def_912,type,
sK631: $i > $i > $i ).
thf(func_def_913,type,
sK632: $i > $i > $i ).
thf(func_def_915,type,
sK634: $i > $i ).
thf(func_def_916,type,
sK635: $i > $i ).
thf(func_def_917,type,
sK636: $i > $i ).
thf(func_def_934,type,
sK653: $i > $i > ( $i > $i ) > $i ).
thf(func_def_935,type,
sK654: $i > $i ).
thf(func_def_942,type,
sK661: ( $i > $o ) > $i > $i ).
thf(func_def_944,type,
sK663: $i > $o ).
thf(func_def_962,type,
sK681: $i > $o ).
thf(func_def_964,type,
sK683: ( $i > $o ) > $i > $i ).
thf(func_def_965,type,
sK684: ( $i > $o ) > $i > $i ).
thf(func_def_1003,type,
ph722:
!>[X0: $tType] : X0 ).
thf(f9119,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f7069,f9118]) ).
thf(f9118,plain,
~ spl721_127,
inference(avatar_split_clause,[status(thm)],[f6759,f6940]) ).
thf(f6940,definition,
( spl721_127
<=> ( $true
= ( subset @ sK74 @ sK73 ) ) ),
introduced(definition,[new_symbols(naming,[spl721_127])],[avatar_definition]) ).
thf(f6759,plain,
( $true
!= ( subset @ sK74 @ sK73 ) ),
inference(trivial_inequality_removal,[status(thm)],[f6758]) ).
thf(f6758,plain,
( ( $true
!= ( subset @ sK74 @ sK73 ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f6622,f2311]) ).
thf(f2311,plain,
setminusLsub = $true,
inference(cnf_transformation,[status(thm)],[f1106]) ).
thf(f1106,plain,
( ( eta2 = $true )
& ( brelnall1 = $true )
& ( setbeta = $true )
& ( upairequniteq = $true )
& ( notequalI1 = $true )
& ( exuE3e = $true )
& ( ubforcartprodlem1 = $true )
& ( cartprodfstin = $true )
& ( binintersectSubset2 = $true )
& ( powersetI = $true )
& ( setadjoinSub = $true )
& ( setadjoinE = $true )
& ( omegaSAx = $true )
& ( upairinpowunion = $true )
& ( symdiffI2 = $true )
& ( emptysetsubset = $true )
& ( cartprodpairmemER = $true )
& ( setukpairinjR12 = $true )
& ( binintersectSubset1 = $true )
& ( emptyinPowerset = $true )
& ( subsetTrans = $true )
& ( notdallE = $true )
& ( descrp = $true )
& ( kpairiskpair = $true )
& ( funcext = $true )
& ( lam2lamEq = $true )
& ( beta1 = $true )
& ( setunion__Cong = $true )
& ( sepInPowerset = $true )
& ( prop2set2propI = $true )
& ( dsetconstrEL = $true )
& ( prop2setI = $true )
& ( cartprodmempair1 = $true )
& ( brelnall2 = $true )
& ( setminusIRneg = $true )
& ( binintersectRsub = $true )
& ( setunionsingleton2 = $true )
& ( omega__Cong = $true )
& ( setukpairinjR11 = $true )
& ( emptyInPowerset = $true )
& ( iftrueorfalse = $true )
& ( setminusSubset2 = $true )
& ( powersetE = $true )
& ( nonemptyI1 = $true )
& ( setextAx = $true )
& ( inCongP = $true )
& ( setunionE = $true )
& ( emptyE1 = $true )
& ( setminusELneg = $true )
& ( binunionIL = $true )
& ( upairsetIL = $true )
& ( cartprodpairmemEL = $true )
& ( vacuousDall = $true )
& ( ifp = $true )
& ( ap2apEq1 = $true )
& ( funcGraphProp4 = $true )
& ( dsetconstrI = $true )
& ( eqinunit = $true )
& ( binunionEcases = $true )
& ( replAx = $true )
& ( setoftrueEq = $true )
& ( bs114d = $true )
& ( symdiffI1 = $true )
& ( ex1I2 = $true )
& ( secondinupair = $true )
& ( ubforcartprodlem2 = $true )
& ( subsetE2 = $true )
& ( lam2p = $true )
& ( setukpairinjR = $true )
& ( ap2p = $true )
& ( subset2powerset = $true )
& ( ex1E1 = $true )
& ( setOfPairsIsBReln = $true )
& ( binintersectT_lem = $true )
& ( setadjoinSub2 = $true )
& ( setminusER = $true )
& ( eqbreln = $true )
& ( singletoninpowunion = $true )
& ( kfstpairEq = $true )
& ( quantDeMorgan4 = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodmempair = $true )
& ( subsetE = $true )
& ( singletonsswitch = $true )
& ( setunionI = $true )
& ( descr__Cong = $true )
& ( lamProp = $true )
& ( symdiffE = $true )
& ( cartprodfstpairEq = $true )
& ( setminusSubset1 = $true )
& ( dsetconstr__Cong = $true )
& ( subsetRefl = $true )
& ( ksndpairEq = $true )
& ( funcGraphProp2 = $true )
& ( setminusEL = $true )
& ( subsetI1 = $true )
& ( ksndsingleton = $true )
& ( in__Cong = $true )
& ( setadjoin__Cong = $true )
& ( subsetemptysetimpeq = $true )
& ( setminusLsub = $true )
& ( inPowerset = $true )
& ( funcGraphProp1 = $true )
& ( singletonsuniq = $true )
& ( dpsetconstrEL1 = $true )
& ( eqimpsubset1 = $true )
& ( setadjoinAx = $true )
& ( iftrue = $true )
& ( upairset2IR = $true )
& ( apProp = $true )
& ( funcext2 = $true )
& ( cartprodpairsurjEq = $true )
& ( binintersectER = $true )
& ( cartprodsndpairEq = $true )
& ( setunionsingleton1 = $true )
& ( exuE2 = $true )
& ( symdiffIneg2 = $true )
& ( quantDeMorgan2 = $true )
& ( emptyI = $true )
& ( binintersectSubset4 = $true )
& ( setminusI = $true )
& ( setunionAx = $true )
& ( powersetI1 = $true )
& ( $true
= ( in @ sK74 @ ( powerset @ sK73 ) ) )
& ( $true
= ( in @ sK75 @ ( powerset @ sK73 ) ) )
& ( ( in @ ( setminus @ sK74 @ sK75 ) @ ( powerset @ sK73 ) )
!= $true )
& ( dpsetconstrEL2 = $true )
& ( quantDeMorgan3 = $true )
& ( nonemptyImpWitness = $true )
& ( dpsetconstrER = $true )
& ( binintersectI = $true )
& ( binintersectLsub = $true )
& ( powerset__Cong = $true )
& ( omega0Ax = $true )
& ( iffalse = $true )
& ( dpsetconstrERa = $true )
& ( infuncsetfunc = $true )
& ( binintersectSubset3 = $true )
& ( omegaIndAx = $true )
& ( notequalI2 = $true )
& ( emptysetE = $true )
& ( funcextLem = $true )
& ( dpsetconstrI = $true )
& ( subsetI2 = $true )
& ( funcinfuncset = $true )
& ( setext = $true )
& ( quantDeMorgan1 = $true )
& ( notsubsetI = $true )
& ( setadjoinIL = $true )
& ( binunionLsub = $true )
& ( exuE3u = $true )
& ( subbreln = $true )
& ( nonemptyE1 = $true )
& ( emptyset__Cong = $true )
& ( ubforcartprodlem3 = $true )
& ( cartprodpairin = $true )
& ( iftrueProp1 = $true )
& ( disjointsetsI1 = $true )
& ( eta1 = $true )
& ( funcImageSingleton = $true )
& ( kpairsurjEq = $true )
& ( setukpairinjR1 = $true )
& ( ex1E2 = $true )
& ( cartprodsndin = $true )
& ( lamp = $true )
& ( setminusILneg = $true )
& ( iffalseProp2 = $true )
& ( binunionRsub = $true )
& ( setukpairIR = $true )
& ( singletonprop = $true )
& ( notinsingleton = $true )
& ( ex1I = $true )
& ( wellorderingAx = $true )
& ( theeq = $true )
& ( theprop = $true )
& ( setukpairinjL = $true )
& ( iftrueProp2 = $true )
& ( setukpairinjR2 = $true )
& ( exuE1 = $true )
& ( setunionsingleton = $true )
& ( binunionIR = $true )
& ( prop2setE = $true )
& ( setadjoinOr = $true )
& ( setukpairinjL2 = $true )
& ( ifSingleton = $true )
& ( eqimpsubset2 = $true )
& ( notinemptyset = $true )
& ( nonemptyI = $true )
& ( exu__Cong = $true )
& ( emptysetAx = $true )
& ( noeltsimpempty = $true )
& ( dsetconstrER = $true )
& ( powersetsubset = $true )
& ( sepSubset = $true )
& ( setextsub = $true )
& ( iffalseProp1 = $true )
& ( exuI2 = $true )
& ( powersetE1 = $true )
& ( binintersectSubset5 = $true )
& ( setadjoinIR = $true )
& ( powersetAx = $true )
& ( upairsetIR = $true )
& ( notdexE = $true )
& ( setukpairIL = $true )
& ( subPowSU = $true )
& ( setminusERneg = $true )
& ( binunionT_lem = $true )
& ( exuI3 = $true )
& ( symdiffIneg1 = $true )
& ( dpsetconstrSub = $true )
& ( singletoninpowerset = $true )
& ( setunionE2 = $true )
& ( kpairp = $true )
& ( binunionE = $true )
& ( singletonsubset = $true )
& ( emptysetimpfalse = $true )
& ( uniqinunit = $true )
& ( kfstsingleton = $true )
& ( foundationAx = $true )
& ( ap2apEq2 = $true )
& ( app = $true )
& ( emptyinunitempty = $true )
& ( upairsetE = $true )
& ( exuI1 = $true )
& ( setukpairinjL1 = $true )
& ( binintersectEL = $true )
& ( funcGraphProp3 = $true )
& ( upairset2E = $true )
& ( upairsubunion = $true )
& ( exuEu = $true )
& ( powersetT_lem = $true )
& ( beta2 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK73,sK74,sK75])],[f831,f1105,f1104]) ).
thf(f1104,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
& ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
& ( $true
!= ( in @ ( setminus @ X1 @ X2 ) @ ( powerset @ X0 ) ) ) ) )
=> ( ( $true
= ( in @ sK74 @ ( powerset @ sK73 ) ) )
& ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ sK73 ) ) )
& ( $true
!= ( in @ ( setminus @ sK74 @ X2 ) @ ( powerset @ sK73 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1105,plain,
( ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ sK73 ) ) )
& ( $true
!= ( in @ ( setminus @ sK74 @ X2 ) @ ( powerset @ sK73 ) ) ) )
=> ( ( $true
= ( in @ sK75 @ ( powerset @ sK73 ) ) )
& ( ( in @ ( setminus @ sK74 @ sK75 ) @ ( powerset @ sK73 ) )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f831,plain,
( ( eta2 = $true )
& ( brelnall1 = $true )
& ( setbeta = $true )
& ( upairequniteq = $true )
& ( notequalI1 = $true )
& ( exuE3e = $true )
& ( ubforcartprodlem1 = $true )
& ( cartprodfstin = $true )
& ( binintersectSubset2 = $true )
& ( powersetI = $true )
& ( setadjoinSub = $true )
& ( setadjoinE = $true )
& ( omegaSAx = $true )
& ( upairinpowunion = $true )
& ( symdiffI2 = $true )
& ( emptysetsubset = $true )
& ( cartprodpairmemER = $true )
& ( setukpairinjR12 = $true )
& ( binintersectSubset1 = $true )
& ( emptyinPowerset = $true )
& ( subsetTrans = $true )
& ( notdallE = $true )
& ( descrp = $true )
& ( kpairiskpair = $true )
& ( funcext = $true )
& ( lam2lamEq = $true )
& ( beta1 = $true )
& ( setunion__Cong = $true )
& ( sepInPowerset = $true )
& ( prop2set2propI = $true )
& ( dsetconstrEL = $true )
& ( prop2setI = $true )
& ( cartprodmempair1 = $true )
& ( brelnall2 = $true )
& ( setminusIRneg = $true )
& ( binintersectRsub = $true )
& ( setunionsingleton2 = $true )
& ( omega__Cong = $true )
& ( setukpairinjR11 = $true )
& ( emptyInPowerset = $true )
& ( iftrueorfalse = $true )
& ( setminusSubset2 = $true )
& ( powersetE = $true )
& ( nonemptyI1 = $true )
& ( setextAx = $true )
& ( inCongP = $true )
& ( setunionE = $true )
& ( emptyE1 = $true )
& ( setminusELneg = $true )
& ( binunionIL = $true )
& ( upairsetIL = $true )
& ( cartprodpairmemEL = $true )
& ( vacuousDall = $true )
& ( ifp = $true )
& ( ap2apEq1 = $true )
& ( funcGraphProp4 = $true )
& ( dsetconstrI = $true )
& ( eqinunit = $true )
& ( binunionEcases = $true )
& ( replAx = $true )
& ( setoftrueEq = $true )
& ( bs114d = $true )
& ( symdiffI1 = $true )
& ( ex1I2 = $true )
& ( secondinupair = $true )
& ( ubforcartprodlem2 = $true )
& ( subsetE2 = $true )
& ( lam2p = $true )
& ( setukpairinjR = $true )
& ( ap2p = $true )
& ( subset2powerset = $true )
& ( ex1E1 = $true )
& ( setOfPairsIsBReln = $true )
& ( binintersectT_lem = $true )
& ( setadjoinSub2 = $true )
& ( setminusER = $true )
& ( eqbreln = $true )
& ( singletoninpowunion = $true )
& ( kfstpairEq = $true )
& ( quantDeMorgan4 = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodmempair = $true )
& ( subsetE = $true )
& ( singletonsswitch = $true )
& ( setunionI = $true )
& ( descr__Cong = $true )
& ( lamProp = $true )
& ( symdiffE = $true )
& ( cartprodfstpairEq = $true )
& ( setminusSubset1 = $true )
& ( dsetconstr__Cong = $true )
& ( subsetRefl = $true )
& ( ksndpairEq = $true )
& ( funcGraphProp2 = $true )
& ( setminusEL = $true )
& ( subsetI1 = $true )
& ( ksndsingleton = $true )
& ( in__Cong = $true )
& ( setadjoin__Cong = $true )
& ( subsetemptysetimpeq = $true )
& ( setminusLsub = $true )
& ( inPowerset = $true )
& ( funcGraphProp1 = $true )
& ( singletonsuniq = $true )
& ( dpsetconstrEL1 = $true )
& ( eqimpsubset1 = $true )
& ( setadjoinAx = $true )
& ( iftrue = $true )
& ( upairset2IR = $true )
& ( apProp = $true )
& ( funcext2 = $true )
& ( cartprodpairsurjEq = $true )
& ( binintersectER = $true )
& ( cartprodsndpairEq = $true )
& ( setunionsingleton1 = $true )
& ( exuE2 = $true )
& ( symdiffIneg2 = $true )
& ( quantDeMorgan2 = $true )
& ( emptyI = $true )
& ( binintersectSubset4 = $true )
& ( setminusI = $true )
& ( setunionAx = $true )
& ( powersetI1 = $true )
& ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
& ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
& ( $true
!= ( in @ ( setminus @ X1 @ X2 ) @ ( powerset @ X0 ) ) ) ) )
& ( dpsetconstrEL2 = $true )
& ( quantDeMorgan3 = $true )
& ( nonemptyImpWitness = $true )
& ( dpsetconstrER = $true )
& ( binintersectI = $true )
& ( binintersectLsub = $true )
& ( powerset__Cong = $true )
& ( omega0Ax = $true )
& ( iffalse = $true )
& ( dpsetconstrERa = $true )
& ( infuncsetfunc = $true )
& ( binintersectSubset3 = $true )
& ( omegaIndAx = $true )
& ( notequalI2 = $true )
& ( emptysetE = $true )
& ( funcextLem = $true )
& ( dpsetconstrI = $true )
& ( subsetI2 = $true )
& ( funcinfuncset = $true )
& ( setext = $true )
& ( quantDeMorgan1 = $true )
& ( notsubsetI = $true )
& ( setadjoinIL = $true )
& ( binunionLsub = $true )
& ( exuE3u = $true )
& ( subbreln = $true )
& ( nonemptyE1 = $true )
& ( emptyset__Cong = $true )
& ( ubforcartprodlem3 = $true )
& ( cartprodpairin = $true )
& ( iftrueProp1 = $true )
& ( disjointsetsI1 = $true )
& ( eta1 = $true )
& ( funcImageSingleton = $true )
& ( kpairsurjEq = $true )
& ( setukpairinjR1 = $true )
& ( ex1E2 = $true )
& ( cartprodsndin = $true )
& ( lamp = $true )
& ( setminusILneg = $true )
& ( iffalseProp2 = $true )
& ( binunionRsub = $true )
& ( setukpairIR = $true )
& ( singletonprop = $true )
& ( notinsingleton = $true )
& ( ex1I = $true )
& ( wellorderingAx = $true )
& ( theeq = $true )
& ( theprop = $true )
& ( setukpairinjL = $true )
& ( iftrueProp2 = $true )
& ( setukpairinjR2 = $true )
& ( exuE1 = $true )
& ( setunionsingleton = $true )
& ( binunionIR = $true )
& ( prop2setE = $true )
& ( setadjoinOr = $true )
& ( setukpairinjL2 = $true )
& ( ifSingleton = $true )
& ( eqimpsubset2 = $true )
& ( notinemptyset = $true )
& ( nonemptyI = $true )
& ( exu__Cong = $true )
& ( emptysetAx = $true )
& ( noeltsimpempty = $true )
& ( dsetconstrER = $true )
& ( powersetsubset = $true )
& ( sepSubset = $true )
& ( setextsub = $true )
& ( iffalseProp1 = $true )
& ( exuI2 = $true )
& ( powersetE1 = $true )
& ( binintersectSubset5 = $true )
& ( setadjoinIR = $true )
& ( powersetAx = $true )
& ( upairsetIR = $true )
& ( notdexE = $true )
& ( setukpairIL = $true )
& ( subPowSU = $true )
& ( setminusERneg = $true )
& ( binunionT_lem = $true )
& ( exuI3 = $true )
& ( symdiffIneg1 = $true )
& ( dpsetconstrSub = $true )
& ( singletoninpowerset = $true )
& ( setunionE2 = $true )
& ( kpairp = $true )
& ( binunionE = $true )
& ( singletonsubset = $true )
& ( emptysetimpfalse = $true )
& ( uniqinunit = $true )
& ( kfstsingleton = $true )
& ( foundationAx = $true )
& ( ap2apEq2 = $true )
& ( app = $true )
& ( emptyinunitempty = $true )
& ( upairsetE = $true )
& ( exuI1 = $true )
& ( setukpairinjL1 = $true )
& ( binintersectEL = $true )
& ( funcGraphProp3 = $true )
& ( upairset2E = $true )
& ( upairsubunion = $true )
& ( exuEu = $true )
& ( powersetT_lem = $true )
& ( beta2 = $true ) ),
inference(flattening,[status(thm)],[f830]) ).
thf(f830,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
& ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
& ( $true
!= ( in @ ( setminus @ X1 @ X2 ) @ ( powerset @ X0 ) ) ) ) )
& ( 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)],[f316]) ).
thf(f316,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 )
=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
=> ( $true
= ( in @ ( setminus @ X1 @ X2 ) @ ( powerset @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f315]) ).
thf(f315,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
=> ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X0 ) )
=> ( in @ ( setminus @ X1 @ X2 ) @ ( powerset @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f235]) ).
thf(f235,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
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( in @ ( setminus @ X11 @ X16 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f234]) ).
thf(f234,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
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( in @ ( setminus @ X11 @ X16 ) @ ( powerset @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.9EhP1wCwEA/DTF2THF_22036.p',setminusT_lem) ).
thf(f6622,plain,
( ( setminusLsub != $true )
| ( $true
!= ( subset @ sK74 @ sK73 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f6591]) ).
thf(f6591,plain,
( ( setminusLsub != $true )
| ( $true != $true )
| ( $true
!= ( subset @ sK74 @ sK73 ) ) ),
inference(superposition,[status(thm)],[f3574,f2124]) ).
thf(f2124,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( subset @ ( setminus @ X1 @ X0 ) @ X1 ) )
| ( setminusLsub != $true ) ),
inference(cnf_transformation,[status(thm)],[f1039]) ).
thf(f1039,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( subset @ ( setminus @ X1 @ X0 ) @ X1 ) )
| ( setminusLsub != $true ) )
& ( ( setminusLsub = $true )
| ( $true
!= ( subset @ ( setminus @ sK32 @ sK31 ) @ sK32 ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK31,sK32])],[f1037,f1038]) ).
thf(f1038,plain,
( ? [X2: $i,X3: $i] :
( $true
!= ( subset @ ( setminus @ X3 @ X2 ) @ X3 ) )
=> ( $true
!= ( subset @ ( setminus @ sK32 @ sK31 ) @ sK32 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1037,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( subset @ ( setminus @ X1 @ X0 ) @ X1 ) )
| ( setminusLsub != $true ) )
& ( ( setminusLsub = $true )
| ? [X2: $i,X3: $i] :
( $true
!= ( subset @ ( setminus @ X3 @ X2 ) @ X3 ) ) ) ),
inference(rectify,[status(thm)],[f1036]) ).
thf(f1036,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( subset @ ( setminus @ X1 @ X0 ) @ X1 ) )
| ( setminusLsub != $true ) )
& ( ( setminusLsub = $true )
| ? [X0: $i,X1: $i] :
( $true
!= ( subset @ ( setminus @ X1 @ X0 ) @ X1 ) ) ) ),
inference(nnf_transformation,[status(thm)],[f475]) ).
thf(f475,plain,
( ! [X0: $i,X1: $i] :
( $true
= ( subset @ ( setminus @ X1 @ X0 ) @ X1 ) )
<=> ( setminusLsub = $true ) ),
inference(fool_elimination,[status(thm)],[f474]) ).
thf(f474,plain,
( setminusLsub
= ( ! [X0: $i,X1: $i] : ( subset @ ( setminus @ X1 @ X0 ) @ X1 ) ) ),
inference(rectify,[status(thm)],[f128]) ).
thf(f128,axiom,
( setminusLsub
= ( ! [X4: $i,X3: $i] : ( subset @ ( setminus @ X3 @ X4 ) @ X3 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.9EhP1wCwEA/DTF2THF_22036.p',setminusLsub) ).
thf(f3574,plain,
! [X0: $i] :
( ( $true
!= ( subset @ ( setminus @ sK74 @ sK75 ) @ X0 ) )
| ( $true
!= ( subset @ X0 @ sK73 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3573]) ).
thf(f3573,plain,
! [X0: $i] :
( ( $true
!= ( subset @ ( setminus @ sK74 @ sK75 ) @ X0 ) )
| ( $true != $true )
| ( $true
!= ( subset @ X0 @ sK73 ) ) ),
inference(forward_demodulation,[status(thm)],[f3559,f2391]) ).
thf(f2391,plain,
subsetTrans = $true,
inference(cnf_transformation,[status(thm)],[f1106]) ).
thf(f3559,plain,
! [X0: $i] :
( ( subsetTrans != $true )
| ( $true
!= ( subset @ ( setminus @ sK74 @ sK75 ) @ X0 ) )
| ( $true
!= ( subset @ X0 @ sK73 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3554]) ).
thf(f3554,plain,
! [X0: $i] :
( ( subsetTrans != $true )
| ( $true
!= ( subset @ ( setminus @ sK74 @ sK75 ) @ X0 ) )
| ( $true
!= ( subset @ X0 @ sK73 ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f3506,f2951]) ).
thf(f2951,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
| ( subsetTrans != $true )
| ( $true
!= ( subset @ X2 @ X1 ) )
| ( $true
!= ( subset @ X0 @ X2 ) ) ),
inference(cnf_transformation,[status(thm)],[f1764]) ).
thf(f1764,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( subset @ X2 @ X1 ) )
| ( $true
!= ( subset @ X0 @ X2 ) )
| ( $true
= ( subset @ X0 @ X1 ) ) )
| ( subsetTrans != $true ) )
& ( ( subsetTrans = $true )
| ( ( $true
= ( subset @ sK502 @ sK501 ) )
& ( $true
= ( subset @ sK500 @ sK502 ) )
& ( ( subset @ sK500 @ sK501 )
!= $true ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK500,sK501,sK502])],[f1762,f1763]) ).
thf(f1763,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X5 @ X4 )
= $true )
& ( ( subset @ X3 @ X5 )
= $true )
& ( ( subset @ X3 @ X4 )
!= $true ) )
=> ( ( $true
= ( subset @ sK502 @ sK501 ) )
& ( $true
= ( subset @ sK500 @ sK502 ) )
& ( ( subset @ sK500 @ sK501 )
!= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1762,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( subset @ X2 @ X1 ) )
| ( $true
!= ( subset @ X0 @ X2 ) )
| ( $true
= ( subset @ X0 @ X1 ) ) )
| ( subsetTrans != $true ) )
& ( ( subsetTrans = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( ( subset @ X5 @ X4 )
= $true )
& ( ( subset @ X3 @ X5 )
= $true )
& ( ( subset @ X3 @ X4 )
!= $true ) ) ) ),
inference(rectify,[status(thm)],[f1761]) ).
thf(f1761,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( subset @ X2 @ X1 ) )
| ( $true
!= ( subset @ X0 @ X2 ) )
| ( $true
= ( subset @ X0 @ X1 ) ) )
| ( subsetTrans != $true ) )
& ( ( subsetTrans = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( subset @ X2 @ X1 ) )
& ( $true
= ( subset @ X0 @ X2 ) )
& ( $true
!= ( subset @ X0 @ X1 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f891]) ).
thf(f891,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( subset @ X2 @ X1 ) )
| ( $true
!= ( subset @ X0 @ X2 ) )
| ( $true
= ( subset @ X0 @ X1 ) ) )
<=> ( subsetTrans = $true ) ),
inference(flattening,[status(thm)],[f890]) ).
thf(f890,plain,
( ( subsetTrans = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
| ( $true
!= ( subset @ X2 @ X1 ) )
| ( $true
!= ( subset @ X0 @ X2 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f574]) ).
thf(f574,plain,
( ( subsetTrans = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( subset @ X0 @ X2 ) )
=> ( ( $true
= ( subset @ X2 @ X1 ) )
=> ( $true
= ( subset @ X0 @ X1 ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f573]) ).
thf(f573,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( subset @ X0 @ X2 )
=> ( ( subset @ X2 @ X1 )
=> ( subset @ X0 @ X1 ) ) )
= subsetTrans ),
inference(rectify,[status(thm)],[f89]) ).
thf(f89,axiom,
! [X3: $i,X5: $i,X4: $i] :
( ( ( subset @ X3 @ X4 )
=> ( ( subset @ X4 @ X5 )
=> ( subset @ X3 @ X5 ) ) )
= subsetTrans ),
file('/export/starexec/sandbox2/tmp/tmp.9EhP1wCwEA/DTF2THF_22036.p',subsetTrans) ).
thf(f3506,plain,
( $true
!= ( subset @ ( setminus @ sK74 @ sK75 ) @ sK73 ) ),
inference(trivial_inequality_removal,[status(thm)],[f3505]) ).
thf(f3505,plain,
( ( $true != $true )
| ( $true
!= ( subset @ ( setminus @ sK74 @ sK75 ) @ sK73 ) ) ),
inference(forward_demodulation,[status(thm)],[f3487,f2341]) ).
thf(f2341,plain,
subset2powerset = $true,
inference(cnf_transformation,[status(thm)],[f1106]) ).
thf(f3487,plain,
( ( subset2powerset != $true )
| ( $true
!= ( subset @ ( setminus @ sK74 @ sK75 ) @ sK73 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3479]) ).
thf(f3479,plain,
( ( $true != $true )
| ( $true
!= ( subset @ ( setminus @ sK74 @ sK75 ) @ sK73 ) )
| ( subset2powerset != $true ) ),
inference(superposition,[status(thm)],[f2286,f2581]) ).
thf(f2581,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) )
| ( subset2powerset != $true )
| ( $true
!= ( subset @ X0 @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f1322]) ).
thf(f1322,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) )
| ( subset2powerset != $true ) )
& ( ( subset2powerset = $true )
| ( ( $true
= ( subset @ sK208 @ sK209 ) )
& ( $true
!= ( in @ sK208 @ ( powerset @ sK209 ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK208,sK209])],[f1320,f1321]) ).
thf(f1321,plain,
( ? [X2: $i,X3: $i] :
( ( $true
= ( subset @ X2 @ X3 ) )
& ( ( in @ X2 @ ( powerset @ X3 ) )
!= $true ) )
=> ( ( $true
= ( subset @ sK208 @ sK209 ) )
& ( $true
!= ( in @ sK208 @ ( powerset @ sK209 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1320,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) )
| ( subset2powerset != $true ) )
& ( ( subset2powerset = $true )
| ? [X2: $i,X3: $i] :
( ( $true
= ( subset @ X2 @ X3 ) )
& ( ( in @ X2 @ ( powerset @ X3 ) )
!= $true ) ) ) ),
inference(rectify,[status(thm)],[f1319]) ).
thf(f1319,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) )
| ( subset2powerset != $true ) )
& ( ( subset2powerset = $true )
| ? [X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
& ( $true
!= ( in @ X0 @ ( powerset @ X1 ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f917]) ).
thf(f917,plain,
( ! [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
| ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) )
<=> ( subset2powerset = $true ) ),
inference(ennf_transformation,[status(thm)],[f377]) ).
thf(f377,plain,
( ! [X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
=> ( $true
= ( in @ X0 @ ( powerset @ X1 ) ) ) )
<=> ( subset2powerset = $true ) ),
inference(fool_elimination,[status(thm)],[f376]) ).
thf(f376,plain,
! [X0: $i,X1: $i] :
( ( ( subset @ X0 @ X1 )
=> ( in @ X0 @ ( powerset @ X1 ) ) )
= subset2powerset ),
inference(rectify,[status(thm)],[f92]) ).
thf(f92,axiom,
! [X3: $i,X4: $i] :
( ( ( subset @ X3 @ X4 )
=> ( in @ X3 @ ( powerset @ X4 ) ) )
= subset2powerset ),
file('/export/starexec/sandbox2/tmp/tmp.9EhP1wCwEA/DTF2THF_22036.p',subset2powerset) ).
thf(f2286,plain,
( ( in @ ( setminus @ sK74 @ sK75 ) @ ( powerset @ sK73 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1106]) ).
thf(f7069,plain,
spl721_127,
inference(avatar_split_clause,[status(thm)],[f7068,f6940]) ).
thf(f7068,plain,
( $true
= ( subset @ sK74 @ sK73 ) ),
inference(trivial_inequality_removal,[status(thm)],[f7067]) ).
thf(f7067,plain,
( ( $true != $true )
| ( $true
= ( subset @ sK74 @ sK73 ) ) ),
inference(forward_demodulation,[status(thm)],[f7063,f2268]) ).
thf(f2268,plain,
subsetI2 = $true,
inference(cnf_transformation,[status(thm)],[f1106]) ).
thf(f7063,plain,
( ( subsetI2 != $true )
| ( $true
= ( subset @ sK74 @ sK73 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f7062]) ).
thf(f7062,plain,
( ( $true
= ( subset @ sK74 @ sK73 ) )
| ( $true != $true )
| ( subsetI2 != $true ) ),
inference(duplicate_literal_removal,[status(thm)],[f7054]) ).
thf(f7054,plain,
( ( $true
= ( subset @ sK74 @ sK73 ) )
| ( $true
= ( subset @ sK74 @ sK73 ) )
| ( $true != $true )
| ( subsetI2 != $true ) ),
inference(superposition,[status(thm)],[f3877,f2771]) ).
thf(f2771,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( in @ ( sK359 @ X1 @ X0 ) @ X0 ) )
| ( subsetI2 != $true )
| ( $true
= ( subset @ X0 @ X1 ) ) ),
inference(cnf_transformation,[status(thm)],[f1546]) ).
thf(f1546,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
| ( ( $true
= ( in @ ( sK359 @ X1 @ X0 ) @ X0 ) )
& ( $true
!= ( in @ ( sK359 @ X1 @ X0 ) @ X1 ) ) ) )
| ( subsetI2 != $true ) )
& ( ( subsetI2 = $true )
| ( ( $true
!= ( subset @ sK360 @ sK361 ) )
& ! [X5: $i] :
( ( $true
!= ( in @ X5 @ sK360 ) )
| ( $true
= ( in @ X5 @ sK361 ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK359,sK360,sK361])],[f1543,f1545,f1544]) ).
thf(f1544,plain,
! [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
& ( $true
!= ( in @ X2 @ X1 ) ) )
=> ( ( $true
= ( in @ ( sK359 @ X1 @ X0 ) @ X0 ) )
& ( $true
!= ( in @ ( sK359 @ X1 @ X0 ) @ X1 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1545,plain,
( ? [X3: $i,X4: $i] :
( ( ( subset @ X3 @ X4 )
!= $true )
& ! [X5: $i] :
( ( $true
!= ( in @ X5 @ X3 ) )
| ( ( in @ X5 @ X4 )
= $true ) ) )
=> ( ( $true
!= ( subset @ sK360 @ sK361 ) )
& ! [X5: $i] :
( ( $true
!= ( in @ X5 @ sK360 ) )
| ( $true
= ( in @ X5 @ sK361 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1543,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
| ? [X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
& ( $true
!= ( in @ X2 @ X1 ) ) ) )
| ( subsetI2 != $true ) )
& ( ( subsetI2 = $true )
| ? [X3: $i,X4: $i] :
( ( ( subset @ X3 @ X4 )
!= $true )
& ! [X5: $i] :
( ( $true
!= ( in @ X5 @ X3 ) )
| ( ( in @ X5 @ X4 )
= $true ) ) ) ) ),
inference(rectify,[status(thm)],[f1542]) ).
thf(f1542,plain,
( ( ! [X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
| ? [X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
& ( $true
!= ( in @ X2 @ X1 ) ) ) )
| ( subsetI2 != $true ) )
& ( ( subsetI2 = $true )
| ? [X0: $i,X1: $i] :
( ( $true
!= ( subset @ X0 @ X1 ) )
& ! [X2: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
| ( $true
= ( in @ X2 @ X1 ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f801]) ).
thf(f801,plain,
( ! [X0: $i,X1: $i] :
( ( $true
= ( subset @ X0 @ X1 ) )
| ? [X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
& ( $true
!= ( in @ X2 @ X1 ) ) ) )
<=> ( subsetI2 = $true ) ),
inference(ennf_transformation,[status(thm)],[f264]) ).
thf(f264,plain,
( ( subsetI2 = $true )
<=> ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
=> ( $true
= ( in @ X2 @ X1 ) ) )
=> ( $true
= ( subset @ X0 @ X1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f263]) ).
thf(f263,plain,
! [X0: $i,X1: $i] :
( ( ! [X2: $i] :
( ( in @ X2 @ X0 )
=> ( in @ X2 @ X1 ) )
=> ( subset @ X0 @ X1 ) )
= subsetI2 ),
inference(rectify,[status(thm)],[f81]) ).
thf(f81,axiom,
! [X3: $i,X4: $i] :
( ( ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( in @ X1 @ X4 ) )
=> ( subset @ X3 @ X4 ) )
= subsetI2 ),
file('/export/starexec/sandbox2/tmp/tmp.9EhP1wCwEA/DTF2THF_22036.p',subsetI2) ).
thf(f3877,plain,
! [X0: $i] :
( ( $true
!= ( in @ ( sK359 @ sK73 @ X0 ) @ sK74 ) )
| ( $true
= ( subset @ X0 @ sK73 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3876]) ).
thf(f3876,plain,
! [X0: $i] :
( ( $true
!= ( in @ ( sK359 @ sK73 @ X0 ) @ sK74 ) )
| ( $true != $true )
| ( $true
= ( subset @ X0 @ sK73 ) ) ),
inference(forward_demodulation,[status(thm)],[f3851,f2268]) ).
thf(f3851,plain,
! [X0: $i] :
( ( $true
!= ( in @ ( sK359 @ sK73 @ X0 ) @ sK74 ) )
| ( subsetI2 != $true )
| ( $true
= ( subset @ X0 @ sK73 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3828]) ).
thf(f3828,plain,
! [X0: $i] :
( ( $true != $true )
| ( $true
!= ( in @ ( sK359 @ sK73 @ X0 ) @ sK74 ) )
| ( $true
= ( subset @ X0 @ sK73 ) )
| ( subsetI2 != $true ) ),
inference(superposition,[status(thm)],[f2770,f3476]) ).
thf(f3476,plain,
! [X0: $i] :
( ( $true
= ( in @ X0 @ sK73 ) )
| ( $true
!= ( in @ X0 @ sK74 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3475]) ).
thf(f3475,plain,
! [X0: $i] :
( ( $true
= ( in @ X0 @ sK73 ) )
| ( $true != $true )
| ( $true
!= ( in @ X0 @ sK74 ) ) ),
inference(forward_demodulation,[status(thm)],[f3457,f2211]) ).
thf(f2211,plain,
powersetAx = $true,
inference(cnf_transformation,[status(thm)],[f1106]) ).
thf(f3457,plain,
! [X0: $i] :
( ( powersetAx != $true )
| ( $true
!= ( in @ X0 @ sK74 ) )
| ( $true
= ( in @ X0 @ sK73 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3453]) ).
thf(f3453,plain,
! [X0: $i] :
( ( $true != $true )
| ( $true
= ( in @ X0 @ sK73 ) )
| ( $true
!= ( in @ X0 @ sK74 ) )
| ( powersetAx != $true ) ),
inference(superposition,[status(thm)],[f2485,f2288]) ).
thf(f2288,plain,
( $true
= ( in @ sK74 @ ( powerset @ sK73 ) ) ),
inference(cnf_transformation,[status(thm)],[f1106]) ).
thf(f2485,plain,
! [X7: $i,X4: $i,X5: $i] :
( ( ( in @ X5 @ ( powerset @ X4 ) )
!= $true )
| ( powersetAx != $true )
| ( ( in @ X7 @ X4 )
= $true )
| ( $true
!= ( in @ X7 @ X5 ) ) ),
inference(cnf_transformation,[status(thm)],[f1209]) ).
thf(f1209,plain,
( ( ( powersetAx = $true )
| ( ( ( ( $true
!= ( in @ sK133 @ sK131 ) )
& ( $true
= ( in @ sK133 @ sK132 ) ) )
| ( $true
!= ( in @ sK132 @ ( powerset @ sK131 ) ) ) )
& ( ! [X3: $i] :
( ( $true
= ( in @ X3 @ sK131 ) )
| ( $true
!= ( in @ X3 @ sK132 ) ) )
| ( $true
= ( in @ sK132 @ ( powerset @ sK131 ) ) ) ) ) )
& ( ! [X4: $i,X5: $i] :
( ( ( ( in @ X5 @ ( powerset @ X4 ) )
= $true )
| ( ( $true
!= ( in @ ( sK134 @ X5 @ X4 ) @ X4 ) )
& ( $true
= ( in @ ( sK134 @ X5 @ X4 ) @ X5 ) ) ) )
& ( ! [X7: $i] :
( ( ( in @ X7 @ X4 )
= $true )
| ( $true
!= ( in @ X7 @ X5 ) ) )
| ( ( in @ X5 @ ( powerset @ X4 ) )
!= $true ) ) )
| ( powersetAx != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK131,sK132,sK133,sK134])],[f1205,f1208,f1207,f1206]) ).
thf(f1206,plain,
( ? [X0: $i,X1: $i] :
( ( ? [X2: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
& ( $true
= ( in @ X2 @ X1 ) ) )
| ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) )
& ( ! [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
| ( $true
!= ( in @ X3 @ X1 ) ) )
| ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) ) )
=> ( ( ? [X2: $i] :
( ( $true
!= ( in @ X2 @ sK131 ) )
& ( $true
= ( in @ X2 @ sK132 ) ) )
| ( $true
!= ( in @ sK132 @ ( powerset @ sK131 ) ) ) )
& ( ! [X3: $i] :
( ( $true
= ( in @ X3 @ sK131 ) )
| ( $true
!= ( in @ X3 @ sK132 ) ) )
| ( $true
= ( in @ sK132 @ ( powerset @ sK131 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1207,plain,
( ? [X2: $i] :
( ( $true
!= ( in @ X2 @ sK131 ) )
& ( $true
= ( in @ X2 @ sK132 ) ) )
=> ( ( $true
!= ( in @ sK133 @ sK131 ) )
& ( $true
= ( in @ sK133 @ sK132 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1208,plain,
! [X4: $i,X5: $i] :
( ? [X6: $i] :
( ( ( in @ X6 @ X4 )
!= $true )
& ( $true
= ( in @ X6 @ X5 ) ) )
=> ( ( $true
!= ( in @ ( sK134 @ X5 @ X4 ) @ X4 ) )
& ( $true
= ( in @ ( sK134 @ X5 @ X4 ) @ X5 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1205,plain,
( ( ( powersetAx = $true )
| ? [X0: $i,X1: $i] :
( ( ? [X2: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
& ( $true
= ( in @ X2 @ X1 ) ) )
| ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) )
& ( ! [X3: $i] :
( ( $true
= ( in @ X3 @ X0 ) )
| ( $true
!= ( in @ X3 @ X1 ) ) )
| ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) ) ) )
& ( ! [X4: $i,X5: $i] :
( ( ( ( in @ X5 @ ( powerset @ X4 ) )
= $true )
| ? [X6: $i] :
( ( ( in @ X6 @ X4 )
!= $true )
& ( $true
= ( in @ X6 @ X5 ) ) ) )
& ( ! [X7: $i] :
( ( ( in @ X7 @ X4 )
= $true )
| ( $true
!= ( in @ X7 @ X5 ) ) )
| ( ( in @ X5 @ ( powerset @ X4 ) )
!= $true ) ) )
| ( powersetAx != $true ) ) ),
inference(rectify,[status(thm)],[f1204]) ).
thf(f1204,plain,
( ( ( powersetAx = $true )
| ? [X0: $i,X1: $i] :
( ( ? [X2: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
& ( $true
= ( in @ X2 @ X1 ) ) )
| ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) )
& ( ! [X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
| ( $true
!= ( in @ X2 @ X1 ) ) )
| ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) ) ) )
& ( ! [X0: $i,X1: $i] :
( ( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
| ? [X2: $i] :
( ( $true
!= ( in @ X2 @ X0 ) )
& ( $true
= ( in @ X2 @ X1 ) ) ) )
& ( ! [X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
| ( $true
!= ( in @ X2 @ X1 ) ) )
| ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) ) )
| ( powersetAx != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f918]) ).
thf(f918,plain,
( ( powersetAx = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
<=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ X0 ) )
| ( $true
!= ( in @ X2 @ X1 ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f545]) ).
thf(f545,plain,
( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
= ( in @ X2 @ X1 ) )
=> ( $true
= ( in @ X2 @ X0 ) ) )
<=> ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) )
<=> ( powersetAx = $true ) ),
inference(fool_elimination,[status(thm)],[f544]) ).
thf(f544,plain,
( powersetAx
= ( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( in @ X2 @ X0 ) )
<=> ( in @ X1 @ ( powerset @ X0 ) ) ) ) ),
inference(rectify,[status(thm)],[f5]) ).
thf(f5,axiom,
( powersetAx
= ( ! [X3: $i,X4: $i] :
( ! [X1: $i] :
( ( in @ X1 @ X4 )
=> ( in @ X1 @ X3 ) )
<=> ( in @ X4 @ ( powerset @ X3 ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.9EhP1wCwEA/DTF2THF_22036.p',powersetAx) ).
thf(f2770,plain,
! [X0: $i,X1: $i] :
( ( $true
!= ( in @ ( sK359 @ X1 @ X0 ) @ X1 ) )
| ( $true
= ( subset @ X0 @ X1 ) )
| ( subsetI2 != $true ) ),
inference(cnf_transformation,[status(thm)],[f1546]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU713^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.34 % Computer : n019.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 12:31:54 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.28/0.46 ---- Original DTF file ---
% 0.28/0.46 thf(spec,logic,$$dhol).
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % File : SEU713^1 : TPTP v9.2.1. Released v3.7.0.
% 0.28/0.46 % Domain : Set Theory
% 0.28/0.46 % Problem : Typed Set Theory - Types of Set Operators
% 0.28/0.46 % Version : Especial.
% 0.28/0.46 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.28/0.46 % in (setminus X Y) (powerset A)))
% 0.28/0.46
% 0.28/0.46 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.28/0.46 % Source : [Bro08]
% 0.28/0.46 % Names : ZFC215g [Bro08]
% 0.28/0.46
% 0.28/0.46 % Status : Theorem
% 0.28/0.46 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.27 v7.5.0, 0.29 v7.4.0, 0.44 v7.2.0, 0.38 v7.1.0, 0.50 v7.0.0, 0.43 v6.4.0, 0.50 v6.3.0, 0.80 v6.2.0, 0.57 v5.5.0, 0.67 v5.4.0, 0.80 v5.2.0, 1.00 v3.7.0
% 0.28/0.46 % Syntax : Number of formulae : 501 ( 233 unt; 267 typ; 233 def)
% 0.28/0.46 % Number of atoms : 1304 ( 357 equ; 0 cnn)
% 0.28/0.46 % Maximal formula atoms : 232 ( 5 avg)
% 0.28/0.46 % Number of connectives : 2531 ( 55 ~; 12 |; 44 &;1792 @)
% 0.28/0.46 % ( 14 <=>; 614 =>; 0 <=; 0 <~>)
% 0.28/0.46 % Maximal formula depth : 239 ( 2 avg)
% 0.28/0.46 % Number of types : 2 ( 0 usr)
% 0.28/0.46 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.28/0.46 % Number of symbols : 270 ( 267 usr; 233 con; 0-4 aty)
% 0.28/0.46 % Number of variables : 783 ( 74 ^; 674 !; 35 ?; 783 :)
% 0.28/0.46 % SPC : TH0_THM_EQU_NAR
% 0.28/0.46
% 0.28/0.46 % Comments : http://mathgate.info/detsetitem.php?id=271
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 thf(in_type,type,
% 0.28/0.46 in: $i > $i > $o ).
% 0.28/0.46
% 0.28/0.46 thf(exu_type,type,
% 0.28/0.46 exu: ( $i > $o ) > $o ).
% 0.28/0.46
% 0.28/0.46 thf(exu,definition,
% 0.28/0.46 ( exu
% 0.28/0.46 = ( ^ [Xphi: $i > $o] :
% 0.28/0.46 ? [Xx: $i] :
% 0.28/0.46 ( ( Xphi @ Xx )
% 0.28/0.46 & ! [Xy: $i] :
% 0.28/0.46 ( ( Xphi @ Xy )
% 0.28/0.46 => ( Xx = Xy ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setextAx_type,type,
% 0.28/0.46 setextAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setextAx,definition,
% 0.28/0.46 ( setextAx
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 <=> ( in @ Xx @ B ) )
% 0.28/0.46 => ( A = B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptyset_type,type,
% 0.28/0.46 emptyset: $i ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetAx_type,type,
% 0.28/0.46 emptysetAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetAx,definition,
% 0.28/0.46 ( emptysetAx
% 0.28/0.46 = ( ! [Xx: $i] :
% 0.28/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoin_type,type,
% 0.28/0.46 setadjoin: $i > $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinAx_type,type,
% 0.28/0.46 setadjoinAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setadjoinAx,definition,
% 0.28/0.46 ( setadjoinAx
% 0.28/0.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.28/0.46 <=> ( ( Xy = Xx )
% 0.28/0.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(powerset_type,type,
% 0.28/0.46 powerset: $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(powersetAx_type,type,
% 0.28/0.46 powersetAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(powersetAx,definition,
% 0.28/0.46 ( powersetAx
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ( in @ B @ ( powerset @ A ) )
% 0.28/0.46 <=> ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setunion_type,type,
% 0.28/0.46 setunion: $i > $i ).
% 0.28/0.46
% 0.28/0.46 thf(setunionAx_type,type,
% 0.28/0.46 setunionAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setunionAx,definition,
% 0.28/0.46 ( setunionAx
% 0.28/0.46 = ( ! [A: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( setunion @ A ) )
% 0.28/0.46 <=> ? [B: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 & ( in @ B @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(omega_type,type,
% 0.28/0.46 omega: $i ).
% 0.28/0.46
% 0.28/0.46 thf(omega0Ax_type,type,
% 0.28/0.46 omega0Ax: $o ).
% 0.28/0.46
% 0.28/0.46 thf(omega0Ax,definition,
% 0.28/0.46 ( omega0Ax
% 0.28/0.46 = ( in @ emptyset @ omega ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(omegaSAx_type,type,
% 0.28/0.46 omegaSAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(omegaSAx,definition,
% 0.28/0.46 ( omegaSAx
% 0.28/0.46 = ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ omega )
% 0.28/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(omegaIndAx_type,type,
% 0.28/0.46 omegaIndAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(omegaIndAx,definition,
% 0.28/0.46 ( omegaIndAx
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ( ( in @ emptyset @ A )
% 0.28/0.46 & ! [Xx: $i] :
% 0.28/0.46 ( ( ( in @ Xx @ omega )
% 0.28/0.46 & ( in @ Xx @ A ) )
% 0.28/0.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.28/0.46 => ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ omega )
% 0.28/0.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(replAx_type,type,
% 0.28/0.46 replAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(replAx,definition,
% 0.28/0.46 ( replAx
% 0.28/0.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( exu
% 0.28/0.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.28/0.46 => ? [B: $i] :
% 0.28/0.46 ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 <=> ? [Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ A )
% 0.28/0.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(foundationAx_type,type,
% 0.28/0.46 foundationAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(foundationAx,definition,
% 0.28/0.46 ( foundationAx
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.28/0.46 => ? [B: $i] :
% 0.28/0.46 ( ( in @ B @ A )
% 0.28/0.46 & ~ ? [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(wellorderingAx_type,type,
% 0.28/0.46 wellorderingAx: $o ).
% 0.28/0.46
% 0.28/0.46 thf(wellorderingAx,definition,
% 0.28/0.46 ( wellorderingAx
% 0.28/0.46 = ( ! [A: $i] :
% 0.28/0.46 ? [B: $i] :
% 0.28/0.46 ( ! [C: $i] :
% 0.28/0.46 ( ( in @ C @ B )
% 0.28/0.46 => ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ C )
% 0.28/0.46 => ( in @ Xx @ A ) ) )
% 0.28/0.46 & ! [Xx: $i,Xy: $i] :
% 0.28/0.46 ( ( ( in @ Xx @ A )
% 0.28/0.46 & ( in @ Xy @ A ) )
% 0.28/0.46 => ( ! [C: $i] :
% 0.28/0.46 ( ( in @ C @ B )
% 0.28/0.46 => ( ( in @ Xx @ C )
% 0.28/0.46 <=> ( in @ Xy @ C ) ) )
% 0.28/0.46 => ( Xx = Xy ) ) )
% 0.28/0.46 & ! [C: $i,D: $i] :
% 0.28/0.46 ( ( ( in @ C @ B )
% 0.28/0.46 & ( in @ D @ B ) )
% 0.28/0.46 => ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ C )
% 0.28/0.46 => ( in @ Xx @ D ) )
% 0.28/0.46 | ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ D )
% 0.28/0.46 => ( in @ Xx @ C ) ) ) )
% 0.28/0.46 & ! [C: $i] :
% 0.28/0.46 ( ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ C )
% 0.28/0.46 => ( in @ Xx @ A ) )
% 0.28/0.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.28/0.46 => ? [D: $i,Xx: $i] :
% 0.28/0.46 ( ( in @ D @ B )
% 0.28/0.46 & ( in @ Xx @ C )
% 0.28/0.46 & ~ ? [Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ D )
% 0.28/0.46 & ( in @ Xy @ C ) )
% 0.28/0.46 & ! [E: $i] :
% 0.28/0.46 ( ( in @ E @ B )
% 0.28/0.46 => ( ! [Xy: $i] :
% 0.28/0.46 ( ( in @ Xy @ E )
% 0.28/0.46 => ( in @ Xy @ D ) )
% 0.28/0.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(descr_type,type,
% 0.28/0.46 descr: ( $i > $o ) > $i ).
% 0.28/0.46
% 0.28/0.46 thf(descrp_type,type,
% 0.28/0.46 descrp: $o ).
% 0.28/0.46
% 0.28/0.46 thf(descrp,definition,
% 0.28/0.46 ( descrp
% 0.28/0.46 = ( ! [Xphi: $i > $o] :
% 0.28/0.46 ( ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 => ( Xphi
% 0.28/0.46 @ ( descr
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstr_type,type,
% 0.28/0.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrI_type,type,
% 0.28/0.46 dsetconstrI: $o ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrI,definition,
% 0.28/0.46 ( dsetconstrI
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( ( Xphi @ Xx )
% 0.28/0.46 => ( in @ Xx
% 0.28/0.46 @ ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrEL_type,type,
% 0.28/0.46 dsetconstrEL: $o ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrEL,definition,
% 0.28/0.46 ( dsetconstrEL
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx
% 0.28/0.46 @ ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.28/0.46 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrER_type,type,
% 0.28/0.46 dsetconstrER: $o ).
% 0.28/0.46
% 0.28/0.46 thf(dsetconstrER,definition,
% 0.28/0.46 ( dsetconstrER
% 0.28/0.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx
% 0.28/0.46 @ ( dsetconstr @ A
% 0.28/0.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.28/0.46 => ( Xphi @ Xx ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(exuE1_type,type,
% 0.28/0.46 exuE1: $o ).
% 0.28/0.46
% 0.28/0.46 thf(exuE1,definition,
% 0.28/0.46 ( exuE1
% 0.28/0.46 = ( ! [Xphi: $i > $o] :
% 0.28/0.46 ( ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 => ? [Xx: $i] :
% 0.28/0.46 ( ( Xphi @ Xx )
% 0.28/0.46 & ! [Xy: $i] :
% 0.28/0.46 ( ( Xphi @ Xy )
% 0.28/0.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(prop2set_type,type,
% 0.28/0.46 prop2set: $o > $i ).
% 0.28/0.46
% 0.28/0.46 thf(prop2setE_type,type,
% 0.28/0.46 prop2setE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(prop2setE,definition,
% 0.28/0.46 ( prop2setE
% 0.28/0.46 = ( ! [Xphi: $o,Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.28/0.46 => Xphi ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetE_type,type,
% 0.28/0.46 emptysetE: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetE,definition,
% 0.28/0.46 ( emptysetE
% 0.28/0.46 = ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ emptyset )
% 0.28/0.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetimpfalse_type,type,
% 0.28/0.46 emptysetimpfalse: $o ).
% 0.28/0.46
% 0.28/0.46 thf(emptysetimpfalse,definition,
% 0.28/0.46 ( emptysetimpfalse
% 0.28/0.46 = ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ emptyset )
% 0.28/0.46 => $false ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(notinemptyset_type,type,
% 0.28/0.46 notinemptyset: $o ).
% 0.28/0.46
% 0.28/0.46 thf(notinemptyset,definition,
% 0.28/0.46 ( notinemptyset
% 0.28/0.46 = ( ! [Xx: $i] :
% 0.28/0.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(exuE3e_type,type,
% 0.28/0.46 exuE3e: $o ).
% 0.28/0.46
% 0.28/0.46 thf(exuE3e,definition,
% 0.28/0.46 ( exuE3e
% 0.28/0.46 = ( ! [Xphi: $i > $o] :
% 0.28/0.46 ( ( exu
% 0.28/0.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(setext_type,type,
% 0.28/0.46 setext: $o ).
% 0.28/0.46
% 0.28/0.46 thf(setext,definition,
% 0.28/0.46 ( setext
% 0.28/0.46 = ( ! [A: $i,B: $i] :
% 0.28/0.46 ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ A )
% 0.28/0.46 => ( in @ Xx @ B ) )
% 0.28/0.46 => ( ! [Xx: $i] :
% 0.28/0.46 ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ A ) )
% 0.28/0.47 => ( A = B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyI_type,type,
% 0.28/0.47 emptyI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyI,definition,
% 0.28/0.47 ( emptyI
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ~ ( in @ Xx @ A )
% 0.28/0.47 => ( A = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(noeltsimpempty_type,type,
% 0.28/0.47 noeltsimpempty: $o ).
% 0.28/0.47
% 0.28/0.47 thf(noeltsimpempty,definition,
% 0.28/0.47 ( noeltsimpempty
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ~ ( in @ Xx @ A )
% 0.28/0.47 => ( A = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setbeta_type,type,
% 0.28/0.47 setbeta: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setbeta,definition,
% 0.28/0.47 ( setbeta
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.28/0.47 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(nonempty_type,type,
% 0.28/0.47 nonempty: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(nonempty,definition,
% 0.28/0.47 ( nonempty
% 0.28/0.47 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyE1_type,type,
% 0.28/0.47 nonemptyE1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyE1,definition,
% 0.28/0.47 ( nonemptyE1
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ( nonempty @ A )
% 0.28/0.47 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyI_type,type,
% 0.28/0.47 nonemptyI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyI,definition,
% 0.28/0.47 ( nonemptyI
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( nonempty
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyI1_type,type,
% 0.28/0.47 nonemptyI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyI1,definition,
% 0.28/0.47 ( nonemptyI1
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.28/0.47 => ( nonempty @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinIL_type,type,
% 0.28/0.47 setadjoinIL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinIL,definition,
% 0.28/0.47 ( setadjoinIL
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyinunitempty_type,type,
% 0.28/0.47 emptyinunitempty: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyinunitempty,definition,
% 0.28/0.47 ( emptyinunitempty
% 0.28/0.47 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinIR_type,type,
% 0.28/0.47 setadjoinIR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinIR,definition,
% 0.28/0.47 ( setadjoinIR
% 0.28/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinE_type,type,
% 0.28/0.47 setadjoinE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinE,definition,
% 0.28/0.47 ( setadjoinE
% 0.28/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.28/0.47 => ! [Xphi: $o] :
% 0.28/0.47 ( ( ( Xy = Xx )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => ( ( ( in @ Xy @ A )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinOr_type,type,
% 0.28/0.47 setadjoinOr: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinOr,definition,
% 0.28/0.47 ( setadjoinOr
% 0.28/0.47 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.28/0.47 => ( ( Xy = Xx )
% 0.28/0.47 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setoftrueEq_type,type,
% 0.28/0.47 setoftrueEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setoftrueEq,definition,
% 0.28/0.47 ( setoftrueEq
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : $true )
% 0.28/0.47 = A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetI_type,type,
% 0.28/0.47 powersetI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetI,definition,
% 0.28/0.47 ( powersetI
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ A ) )
% 0.28/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyinPowerset_type,type,
% 0.28/0.47 emptyinPowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyinPowerset,definition,
% 0.28/0.47 ( emptyinPowerset
% 0.28/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyInPowerset_type,type,
% 0.28/0.47 emptyInPowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyInPowerset,definition,
% 0.28/0.47 ( emptyInPowerset
% 0.28/0.47 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetE_type,type,
% 0.28/0.47 powersetE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetE,definition,
% 0.28/0.47 ( powersetE
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionI_type,type,
% 0.28/0.47 setunionI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionI,definition,
% 0.28/0.47 ( setunionI
% 0.28/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( ( in @ B @ A )
% 0.28/0.47 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionE_type,type,
% 0.28/0.47 setunionE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionE,definition,
% 0.28/0.47 ( setunionE
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.28/0.47 => ! [Xphi: $o] :
% 0.28/0.47 ( ! [B: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( ( in @ B @ A )
% 0.28/0.47 => Xphi ) )
% 0.28/0.47 => Xphi ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subPowSU_type,type,
% 0.28/0.47 subPowSU: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subPowSU,definition,
% 0.28/0.47 ( subPowSU
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuE2_type,type,
% 0.28/0.47 exuE2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuE2,definition,
% 0.28/0.47 ( exuE2
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ! [Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xy )
% 0.28/0.47 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyImpWitness_type,type,
% 0.28/0.47 nonemptyImpWitness: $o ).
% 0.28/0.47
% 0.28/0.47 thf(nonemptyImpWitness,definition,
% 0.28/0.47 ( nonemptyImpWitness
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ( nonempty @ A )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & $true ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(uniqinunit_type,type,
% 0.28/0.47 uniqinunit: $o ).
% 0.28/0.47
% 0.28/0.47 thf(uniqinunit,definition,
% 0.28/0.47 ( uniqinunit
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notinsingleton_type,type,
% 0.28/0.47 notinsingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notinsingleton,definition,
% 0.28/0.47 ( notinsingleton
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx != Xy )
% 0.28/0.47 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eqinunit_type,type,
% 0.28/0.47 eqinunit: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eqinunit,definition,
% 0.28/0.47 ( eqinunit
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsswitch_type,type,
% 0.28/0.47 singletonsswitch: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsswitch,definition,
% 0.28/0.47 ( singletonsswitch
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.47 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetE_type,type,
% 0.28/0.47 upairsetE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetE,definition,
% 0.28/0.47 ( upairsetE
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.28/0.47 => ( ( Xz = Xx )
% 0.28/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetIL_type,type,
% 0.28/0.47 upairsetIL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetIL,definition,
% 0.28/0.47 ( upairsetIL
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetIR_type,type,
% 0.28/0.47 upairsetIR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetIR,definition,
% 0.28/0.47 ( upairsetIR
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyE1_type,type,
% 0.28/0.47 emptyE1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyE1,definition,
% 0.28/0.47 ( emptyE1
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ( ( ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 = emptyset )
% 0.28/0.47 => $false ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(vacuousDall_type,type,
% 0.28/0.47 vacuousDall: $o ).
% 0.28/0.47
% 0.28/0.47 thf(vacuousDall,definition,
% 0.28/0.47 ( vacuousDall
% 0.28/0.47 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ emptyset )
% 0.28/0.47 => ( Xphi @ Xx ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan1_type,type,
% 0.28/0.47 quantDeMorgan1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan1,definition,
% 0.28/0.47 ( quantDeMorgan1
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ~ ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan2_type,type,
% 0.28/0.47 quantDeMorgan2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan2,definition,
% 0.28/0.47 ( quantDeMorgan2
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ~ ( Xphi @ Xx ) )
% 0.28/0.47 => ~ ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan3_type,type,
% 0.28/0.47 quantDeMorgan3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan3,definition,
% 0.28/0.47 ( quantDeMorgan3
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ~ ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan4_type,type,
% 0.28/0.47 quantDeMorgan4: $o ).
% 0.28/0.47
% 0.28/0.47 thf(quantDeMorgan4,definition,
% 0.28/0.47 ( quantDeMorgan4
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ~ ( Xphi @ Xx ) )
% 0.28/0.47 => ~ ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(prop2setI_type,type,
% 0.28/0.47 prop2setI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(prop2setI,definition,
% 0.28/0.47 ( prop2setI
% 0.28/0.47 = ( ! [Xphi: $o] :
% 0.28/0.47 ( Xphi
% 0.28/0.47 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(set2prop_type,type,
% 0.28/0.47 set2prop: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(prop2set2propI_type,type,
% 0.28/0.47 prop2set2propI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(prop2set2propI,definition,
% 0.28/0.47 ( prop2set2propI
% 0.28/0.47 = ( ! [Xphi: $o] :
% 0.28/0.47 ( Xphi
% 0.28/0.47 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notdexE_type,type,
% 0.28/0.47 notdexE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notdexE,definition,
% 0.28/0.47 ( notdexE
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ~ ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notdallE_type,type,
% 0.28/0.47 notdallE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notdallE,definition,
% 0.28/0.47 ( notdallE
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ~ ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuI1_type,type,
% 0.28/0.47 exuI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuI1,definition,
% 0.28/0.47 ( exuI1
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 & ! [Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) )
% 0.28/0.47 => ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuI3_type,type,
% 0.28/0.47 exuI3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuI3,definition,
% 0.28/0.47 ( exuI3
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.28/0.47 => ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) )
% 0.28/0.47 => ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuI2_type,type,
% 0.28/0.47 exuI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuI2,definition,
% 0.28/0.47 ( exuI2
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ? [Xx: $i] :
% 0.28/0.47 ! [Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xy )
% 0.28/0.47 <=> ( Xy = Xx ) )
% 0.28/0.47 => ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(inCongP_type,type,
% 0.28/0.47 inCongP: $o ).
% 0.28/0.47
% 0.28/0.47 thf(inCongP,definition,
% 0.28/0.47 ( inCongP
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(in__Cong_type,type,
% 0.28/0.47 in__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(in__Cong,definition,
% 0.28/0.47 ( in__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuE3u_type,type,
% 0.28/0.47 exuE3u: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuE3u,definition,
% 0.28/0.47 ( exuE3u
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exu__Cong_type,type,
% 0.28/0.47 exu__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exu__Cong,definition,
% 0.28/0.47 ( exu__Cong
% 0.28/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.28/0.47 => ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 <=> ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptyset__Cong_type,type,
% 0.28/0.47 emptyset__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptyset__Cong,definition,
% 0.28/0.47 ( emptyset__Cong
% 0.28/0.47 = ( emptyset = emptyset ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoin__Cong_type,type,
% 0.28/0.47 setadjoin__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoin__Cong,definition,
% 0.28/0.47 ( setadjoin__Cong
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ! [Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( Xz = Xu )
% 0.28/0.47 => ( ( setadjoin @ Xx @ Xz )
% 0.28/0.47 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powerset__Cong_type,type,
% 0.28/0.47 powerset__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powerset__Cong,definition,
% 0.28/0.47 ( powerset__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ( ( powerset @ A )
% 0.28/0.47 = ( powerset @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunion__Cong_type,type,
% 0.28/0.47 setunion__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunion__Cong,definition,
% 0.28/0.47 ( setunion__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ( ( setunion @ A )
% 0.28/0.47 = ( setunion @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(omega__Cong_type,type,
% 0.28/0.47 omega__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(omega__Cong,definition,
% 0.28/0.47 ( omega__Cong
% 0.28/0.47 = ( omega = omega ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(exuEu_type,type,
% 0.28/0.47 exuEu: $o ).
% 0.28/0.47
% 0.28/0.47 thf(exuEu,definition,
% 0.28/0.47 ( exuEu
% 0.28/0.47 = ( ! [Xphi: $i > $o] :
% 0.28/0.47 ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(descr__Cong_type,type,
% 0.28/0.47 descr__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(descr__Cong,definition,
% 0.28/0.47 ( descr__Cong
% 0.28/0.47 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 <=> ( Xpsi @ Xy ) ) )
% 0.28/0.47 => ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ( ( exu
% 0.28/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.28/0.47 => ( ( descr
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 = ( descr
% 0.28/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstr__Cong_type,type,
% 0.28/0.47 dsetconstr__Cong: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dsetconstr__Cong,definition,
% 0.28/0.47 ( dsetconstr__Cong
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( Xx = Xy )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.28/0.47 => ( ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 = ( dsetconstr @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subset_type,type,
% 0.28/0.47 subset: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(disjoint_type,type,
% 0.28/0.47 disjoint: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(setsmeet_type,type,
% 0.28/0.47 setsmeet: $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI1_type,type,
% 0.28/0.47 subsetI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI1,definition,
% 0.28/0.47 ( subsetI1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eqimpsubset2_type,type,
% 0.28/0.47 eqimpsubset2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eqimpsubset2,definition,
% 0.28/0.47 ( eqimpsubset2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eqimpsubset1_type,type,
% 0.28/0.47 eqimpsubset1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eqimpsubset1,definition,
% 0.28/0.47 ( eqimpsubset1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( A = B )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI2_type,type,
% 0.28/0.47 subsetI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetI2,definition,
% 0.28/0.47 ( subsetI2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetsubset_type,type,
% 0.28/0.47 emptysetsubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(emptysetsubset,definition,
% 0.28/0.47 ( emptysetsubset
% 0.28/0.47 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetE_type,type,
% 0.28/0.47 subsetE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetE,definition,
% 0.28/0.47 ( subsetE
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetE2_type,type,
% 0.28/0.47 subsetE2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetE2,definition,
% 0.28/0.47 ( subsetE2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notsubsetI_type,type,
% 0.28/0.47 notsubsetI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notsubsetI,definition,
% 0.28/0.47 ( notsubsetI
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notequalI1_type,type,
% 0.28/0.47 notequalI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notequalI1,definition,
% 0.28/0.47 ( notequalI1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ~ ( subset @ A @ B )
% 0.28/0.47 => ( A != B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(notequalI2_type,type,
% 0.28/0.47 notequalI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(notequalI2,definition,
% 0.28/0.47 ( notequalI2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ( A != B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetRefl_type,type,
% 0.28/0.47 subsetRefl: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetRefl,definition,
% 0.28/0.47 ( subsetRefl
% 0.28/0.47 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetTrans_type,type,
% 0.28/0.47 subsetTrans: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetTrans,definition,
% 0.28/0.47 ( subsetTrans
% 0.28/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( subset @ B @ C )
% 0.28/0.47 => ( subset @ A @ C ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinSub_type,type,
% 0.28/0.47 setadjoinSub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinSub,definition,
% 0.28/0.47 ( setadjoinSub
% 0.28/0.47 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinSub2_type,type,
% 0.28/0.47 setadjoinSub2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setadjoinSub2,definition,
% 0.28/0.47 ( setadjoinSub2
% 0.28/0.47 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subset2powerset_type,type,
% 0.28/0.47 subset2powerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subset2powerset,definition,
% 0.28/0.47 ( subset2powerset
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setextsub_type,type,
% 0.28/0.47 setextsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setextsub,definition,
% 0.28/0.47 ( setextsub
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( subset @ B @ A )
% 0.28/0.47 => ( A = B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subsetemptysetimpeq_type,type,
% 0.28/0.47 subsetemptysetimpeq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subsetemptysetimpeq,definition,
% 0.28/0.47 ( subsetemptysetimpeq
% 0.28/0.47 = ( ! [A: $i] :
% 0.28/0.47 ( ( subset @ A @ emptyset )
% 0.28/0.47 => ( A = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetI1_type,type,
% 0.28/0.47 powersetI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetI1,definition,
% 0.28/0.47 ( powersetI1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ B @ A )
% 0.28/0.47 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetE1_type,type,
% 0.28/0.47 powersetE1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetE1,definition,
% 0.28/0.47 ( powersetE1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( in @ B @ ( powerset @ A ) )
% 0.28/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(inPowerset_type,type,
% 0.28/0.47 inPowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(inPowerset,definition,
% 0.28/0.47 ( inPowerset
% 0.28/0.47 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetsubset_type,type,
% 0.28/0.47 powersetsubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetsubset,definition,
% 0.28/0.47 ( powersetsubset
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sepInPowerset_type,type,
% 0.28/0.47 sepInPowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(sepInPowerset,definition,
% 0.28/0.47 ( sepInPowerset
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( in
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 @ ( powerset @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(sepSubset_type,type,
% 0.28/0.47 sepSubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(sepSubset,definition,
% 0.28/0.47 ( sepSubset
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( subset
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunion_type,type,
% 0.28/0.47 binunion: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(binunionIL_type,type,
% 0.28/0.47 binunionIL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionIL,definition,
% 0.28/0.47 ( binunionIL
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairset2IR_type,type,
% 0.28/0.47 upairset2IR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairset2IR,definition,
% 0.28/0.47 ( upairset2IR
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionIR_type,type,
% 0.28/0.47 binunionIR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionIR,definition,
% 0.28/0.47 ( binunionIR
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionEcases_type,type,
% 0.28/0.47 binunionEcases: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionEcases,definition,
% 0.28/0.47 ( binunionEcases
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.28/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.28/0.47 => ( ( ( in @ Xx @ A )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => ( ( ( in @ Xx @ B )
% 0.28/0.47 => Xphi )
% 0.28/0.47 => Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionE_type,type,
% 0.28/0.47 binunionE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionE,definition,
% 0.28/0.47 ( binunionE
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionLsub_type,type,
% 0.28/0.47 binunionLsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionLsub,definition,
% 0.28/0.47 ( binunionLsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionRsub_type,type,
% 0.28/0.47 binunionRsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionRsub,definition,
% 0.28/0.47 ( binunionRsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersect_type,type,
% 0.28/0.47 binintersect: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectI_type,type,
% 0.28/0.47 binintersectI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectI,definition,
% 0.28/0.47 ( binintersectI
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset5_type,type,
% 0.28/0.47 binintersectSubset5: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset5,definition,
% 0.28/0.47 ( binintersectSubset5
% 0.28/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.28/0.47 ( ( subset @ C @ A )
% 0.28/0.47 => ( ( subset @ C @ B )
% 0.28/0.47 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectEL_type,type,
% 0.28/0.47 binintersectEL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectEL,definition,
% 0.28/0.47 ( binintersectEL
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectLsub_type,type,
% 0.28/0.47 binintersectLsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectLsub,definition,
% 0.28/0.47 ( binintersectLsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset2_type,type,
% 0.28/0.47 binintersectSubset2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset2,definition,
% 0.28/0.47 ( binintersectSubset2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( binintersect @ A @ B )
% 0.28/0.47 = A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset3_type,type,
% 0.28/0.47 binintersectSubset3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset3,definition,
% 0.28/0.47 ( binintersectSubset3
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( ( binintersect @ A @ B )
% 0.28/0.47 = B )
% 0.28/0.47 => ( subset @ B @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectER_type,type,
% 0.28/0.47 binintersectER: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectER,definition,
% 0.28/0.47 ( binintersectER
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.28/0.47 => ( in @ Xx @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(disjointsetsI1_type,type,
% 0.28/0.47 disjointsetsI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(disjointsetsI1,definition,
% 0.28/0.47 ( disjointsetsI1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ~ ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( in @ Xx @ B ) )
% 0.28/0.47 => ( ( binintersect @ A @ B )
% 0.28/0.47 = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectRsub_type,type,
% 0.28/0.47 binintersectRsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectRsub,definition,
% 0.28/0.47 ( binintersectRsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset4_type,type,
% 0.28/0.47 binintersectSubset4: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset4,definition,
% 0.28/0.47 ( binintersectSubset4
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ B @ A )
% 0.28/0.47 => ( ( binintersect @ A @ B )
% 0.28/0.47 = B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset1_type,type,
% 0.28/0.47 binintersectSubset1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectSubset1,definition,
% 0.28/0.47 ( binintersectSubset1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( ( binintersect @ A @ B )
% 0.28/0.47 = A )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(bs114d_type,type,
% 0.28/0.47 bs114d: $o ).
% 0.28/0.47
% 0.28/0.47 thf(bs114d,definition,
% 0.28/0.47 ( bs114d
% 0.28/0.47 = ( ! [A: $i,B: $i,C: $i] :
% 0.28/0.47 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.28/0.47 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(regular_type,type,
% 0.28/0.47 regular: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminus_type,type,
% 0.28/0.47 setminus: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(setminusI_type,type,
% 0.28/0.47 setminusI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusI,definition,
% 0.28/0.47 ( setminusI
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusEL_type,type,
% 0.28/0.47 setminusEL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusEL,definition,
% 0.28/0.47 ( setminusEL
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusER_type,type,
% 0.28/0.47 setminusER: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusER,definition,
% 0.28/0.47 ( setminusER
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.47 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset2_type,type,
% 0.28/0.47 setminusSubset2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset2,definition,
% 0.28/0.47 ( setminusSubset2
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( subset @ A @ B )
% 0.28/0.47 => ( ( setminus @ A @ B )
% 0.28/0.47 = emptyset ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusERneg_type,type,
% 0.28/0.47 setminusERneg: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusERneg,definition,
% 0.28/0.47 ( setminusERneg
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.47 => ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusELneg_type,type,
% 0.28/0.47 setminusELneg: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusELneg,definition,
% 0.28/0.47 ( setminusELneg
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusILneg_type,type,
% 0.28/0.47 setminusILneg: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusILneg,definition,
% 0.28/0.47 ( setminusILneg
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ A )
% 0.28/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusIRneg_type,type,
% 0.28/0.47 setminusIRneg: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusIRneg,definition,
% 0.28/0.47 ( setminusIRneg
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusLsub_type,type,
% 0.28/0.47 setminusLsub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusLsub,definition,
% 0.28/0.47 ( setminusLsub
% 0.28/0.47 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset1_type,type,
% 0.28/0.47 setminusSubset1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setminusSubset1,definition,
% 0.28/0.47 ( setminusSubset1
% 0.28/0.47 = ( ! [A: $i,B: $i] :
% 0.28/0.47 ( ( ( setminus @ A @ B )
% 0.28/0.47 = emptyset )
% 0.28/0.47 => ( subset @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiff_type,type,
% 0.28/0.47 symdiff: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffE_type,type,
% 0.28/0.47 symdiffE: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffE,definition,
% 0.28/0.47 ( symdiffE
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.28/0.47 => ! [Xphi: $o] :
% 0.28/0.47 ( ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => Xphi ) )
% 0.28/0.47 => ( ( ~ ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => Xphi ) )
% 0.28/0.47 => Xphi ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffI1_type,type,
% 0.28/0.47 symdiffI1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffI1,definition,
% 0.28/0.47 ( symdiffI1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffI2_type,type,
% 0.28/0.47 symdiffI2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffI2,definition,
% 0.28/0.47 ( symdiffI2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffIneg1_type,type,
% 0.28/0.47 symdiffIneg1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffIneg1,definition,
% 0.28/0.47 ( symdiffIneg1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffIneg2_type,type,
% 0.28/0.47 symdiffIneg2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(symdiffIneg2,definition,
% 0.28/0.47 ( symdiffIneg2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ~ ( in @ Xx @ A )
% 0.28/0.47 => ( ~ ( in @ Xx @ B )
% 0.28/0.47 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iskpair_type,type,
% 0.28/0.47 iskpair: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(secondinupair_type,type,
% 0.28/0.47 secondinupair: $o ).
% 0.28/0.47
% 0.28/0.47 thf(secondinupair,definition,
% 0.28/0.47 ( secondinupair
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairIL_type,type,
% 0.28/0.47 setukpairIL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairIL,definition,
% 0.28/0.47 ( setukpairIL
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairIR_type,type,
% 0.28/0.47 setukpairIR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairIR,definition,
% 0.28/0.47 ( setukpairIR
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kpairiskpair_type,type,
% 0.28/0.47 kpairiskpair: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kpairiskpair,definition,
% 0.28/0.47 ( kpairiskpair
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kpair_type,type,
% 0.28/0.47 kpair: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(kpairp_type,type,
% 0.28/0.47 kpairp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kpairp,definition,
% 0.28/0.47 ( kpairp
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprod_type,type,
% 0.28/0.47 cartprod: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsubset_type,type,
% 0.28/0.47 singletonsubset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsubset,definition,
% 0.28/0.47 ( singletonsubset
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singletoninpowerset_type,type,
% 0.28/0.47 singletoninpowerset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletoninpowerset,definition,
% 0.28/0.47 ( singletoninpowerset
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singletoninpowunion_type,type,
% 0.28/0.47 singletoninpowunion: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletoninpowunion,definition,
% 0.28/0.47 ( singletoninpowunion
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairset2E_type,type,
% 0.28/0.47 upairset2E: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairset2E,definition,
% 0.28/0.47 ( upairset2E
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.47 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.28/0.47 => ( ( Xz = Xx )
% 0.28/0.47 | ( Xz = Xy ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairsubunion_type,type,
% 0.28/0.47 upairsubunion: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsubunion,definition,
% 0.28/0.47 ( upairsubunion
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairinpowunion_type,type,
% 0.28/0.47 upairinpowunion: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairinpowunion,definition,
% 0.28/0.47 ( upairinpowunion
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem1_type,type,
% 0.28/0.47 ubforcartprodlem1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem1,definition,
% 0.28/0.47 ( ubforcartprodlem1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem2_type,type,
% 0.28/0.47 ubforcartprodlem2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem2,definition,
% 0.28/0.47 ( ubforcartprodlem2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem3_type,type,
% 0.28/0.47 ubforcartprodlem3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ubforcartprodlem3,definition,
% 0.28/0.47 ( ubforcartprodlem3
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairin_type,type,
% 0.28/0.47 cartprodpairin: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairin,definition,
% 0.28/0.47 ( cartprodpairin
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempair1_type,type,
% 0.28/0.47 cartprodmempair1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempair1,definition,
% 0.28/0.47 ( cartprodmempair1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ? [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 & ( Xu
% 0.28/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempair_type,type,
% 0.28/0.47 cartprodmempair: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempair,definition,
% 0.28/0.47 ( cartprodmempair
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( iskpair @ Xu ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionE2_type,type,
% 0.28/0.47 setunionE2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionE2,definition,
% 0.28/0.47 ( setunionE2
% 0.28/0.47 = ( ! [A: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ ( setunion @ A ) )
% 0.28/0.47 => ? [X: $i] :
% 0.28/0.47 ( ( in @ X @ A )
% 0.28/0.47 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton1_type,type,
% 0.28/0.47 setunionsingleton1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton1,definition,
% 0.28/0.47 ( setunionsingleton1
% 0.28/0.47 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton2_type,type,
% 0.28/0.47 setunionsingleton2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton2,definition,
% 0.28/0.47 ( setunionsingleton2
% 0.28/0.47 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton_type,type,
% 0.28/0.47 setunionsingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setunionsingleton,definition,
% 0.28/0.47 ( setunionsingleton
% 0.28/0.47 = ( ! [Xx: $i] :
% 0.28/0.47 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.28/0.47 = Xx ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singleton_type,type,
% 0.28/0.47 singleton: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonprop_type,type,
% 0.28/0.47 singletonprop: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonprop,definition,
% 0.28/0.47 ( singletonprop
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) )
% 0.28/0.47 => ( ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1_type,type,
% 0.28/0.47 ex1: $i > ( $i > $o ) > $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1,definition,
% 0.28/0.47 ( ex1
% 0.28/0.47 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( singleton
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1E1_type,type,
% 0.28/0.47 ex1E1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1E1,definition,
% 0.28/0.47 ( ex1E1
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ( ex1 @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1I_type,type,
% 0.28/0.47 ex1I: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1I,definition,
% 0.28/0.47 ( ex1I
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xy = Xx ) ) )
% 0.28/0.47 => ( ex1 @ A
% 0.28/0.47 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1I2_type,type,
% 0.28/0.47 ex1I2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1I2,definition,
% 0.28/0.47 ( ex1I2
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) )
% 0.28/0.47 => ( ? [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 & ( Xphi @ Xx ) )
% 0.28/0.47 => ( ex1 @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsuniq_type,type,
% 0.28/0.47 singletonsuniq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(singletonsuniq,definition,
% 0.28/0.47 ( singletonsuniq
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( ( setadjoin @ Xx @ emptyset )
% 0.28/0.47 = ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(atmost1p_type,type,
% 0.28/0.47 atmost1p: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(atleast2p_type,type,
% 0.28/0.47 atleast2p: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(atmost2p_type,type,
% 0.28/0.47 atmost2p: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairsetp_type,type,
% 0.28/0.47 upairsetp: $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL1_type,type,
% 0.28/0.47 setukpairinjL1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL1,definition,
% 0.28/0.47 ( setukpairinjL1
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.47 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.28/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kfstsingleton_type,type,
% 0.28/0.47 kfstsingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kfstsingleton,definition,
% 0.28/0.47 ( kfstsingleton
% 0.28/0.47 = ( ! [Xu: $i] :
% 0.28/0.47 ( ( iskpair @ Xu )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.28/0.47 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(theprop_type,type,
% 0.28/0.47 theprop: $o ).
% 0.28/0.47
% 0.28/0.47 thf(theprop,definition,
% 0.28/0.47 ( theprop
% 0.28/0.47 = ( ! [X: $i] :
% 0.28/0.47 ( ( singleton @ X )
% 0.28/0.47 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kfst_type,type,
% 0.28/0.47 kfst: $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(kfstpairEq_type,type,
% 0.28/0.47 kfstpairEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kfstpairEq,definition,
% 0.28/0.47 ( kfstpairEq
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.28/0.47 = Xx ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodfstin_type,type,
% 0.28/0.47 cartprodfstin: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodfstin,definition,
% 0.28/0.47 ( cartprodfstin
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL2_type,type,
% 0.28/0.47 setukpairinjL2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL2,definition,
% 0.28/0.47 ( setukpairinjL2
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.28/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL_type,type,
% 0.28/0.47 setukpairinjL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjL,definition,
% 0.28/0.47 ( setukpairinjL
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.28/0.47 = ( kpair @ Xz @ Xu ) )
% 0.28/0.47 => ( Xx = Xz ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR11_type,type,
% 0.28/0.47 setukpairinjR11: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR11,definition,
% 0.28/0.47 ( setukpairinjR11
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR12_type,type,
% 0.28/0.47 setukpairinjR12: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR12,definition,
% 0.28/0.47 ( setukpairinjR12
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xx = Xy )
% 0.28/0.47 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR1_type,type,
% 0.28/0.47 setukpairinjR1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR1,definition,
% 0.28/0.47 ( setukpairinjR1
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.28/0.47 => ( ( Xz = Xu )
% 0.28/0.47 => ( Xy = Xu ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(upairequniteq_type,type,
% 0.28/0.47 upairequniteq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(upairequniteq,definition,
% 0.28/0.47 ( upairequniteq
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.28/0.47 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ Xz @ emptyset ) )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR2_type,type,
% 0.28/0.47 setukpairinjR2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR2,definition,
% 0.28/0.47 ( setukpairinjR2
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.28/0.47 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.28/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR_type,type,
% 0.28/0.47 setukpairinjR: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setukpairinjR,definition,
% 0.28/0.47 ( setukpairinjR
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.28/0.47 ( ( ( kpair @ Xx @ Xy )
% 0.28/0.47 = ( kpair @ Xz @ Xu ) )
% 0.28/0.47 => ( Xy = Xu ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ksndsingleton_type,type,
% 0.28/0.47 ksndsingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ksndsingleton,definition,
% 0.28/0.47 ( ksndsingleton
% 0.28/0.47 = ( ! [Xu: $i] :
% 0.28/0.47 ( ( iskpair @ Xu )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ ( setunion @ Xu )
% 0.28/0.47 @ ^ [Xx: $i] :
% 0.28/0.47 ( Xu
% 0.28/0.47 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ksnd_type,type,
% 0.28/0.47 ksnd: $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(ksndpairEq_type,type,
% 0.28/0.47 ksndpairEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ksndpairEq,definition,
% 0.28/0.47 ( ksndpairEq
% 0.28/0.47 = ( ! [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.28/0.47 = Xy ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(kpairsurjEq_type,type,
% 0.28/0.47 kpairsurjEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(kpairsurjEq,definition,
% 0.28/0.47 ( kpairsurjEq
% 0.28/0.47 = ( ! [Xu: $i] :
% 0.28/0.47 ( ( iskpair @ Xu )
% 0.28/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.28/0.47 = Xu ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodsndin_type,type,
% 0.28/0.47 cartprodsndin: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodsndin,definition,
% 0.28/0.47 ( cartprodsndin
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairmemEL_type,type,
% 0.28/0.47 cartprodpairmemEL: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairmemEL,definition,
% 0.28/0.47 ( cartprodpairmemEL
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairmemER_type,type,
% 0.28/0.47 cartprodpairmemER: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairmemER,definition,
% 0.28/0.47 ( cartprodpairmemER
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempaircEq_type,type,
% 0.28/0.47 cartprodmempaircEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodmempaircEq,definition,
% 0.28/0.47 ( cartprodmempaircEq
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( kpair @ Xx @ Xy )
% 0.28/0.47 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodfstpairEq_type,type,
% 0.28/0.47 cartprodfstpairEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodfstpairEq,definition,
% 0.28/0.47 ( cartprodfstpairEq
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.28/0.47 = Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodsndpairEq_type,type,
% 0.28/0.47 cartprodsndpairEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodsndpairEq,definition,
% 0.28/0.47 ( cartprodsndpairEq
% 0.28/0.47 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.28/0.47 = Xy ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairsurjEq_type,type,
% 0.28/0.47 cartprodpairsurjEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(cartprodpairsurjEq,definition,
% 0.28/0.47 ( cartprodpairsurjEq
% 0.28/0.47 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.28/0.47 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.28/0.47 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.28/0.47 = Xu ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(breln_type,type,
% 0.28/0.47 breln: $i > $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(breln,definition,
% 0.28/0.47 ( breln
% 0.28/0.47 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstr_type,type,
% 0.28/0.47 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrI_type,type,
% 0.28/0.47 dpsetconstrI: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrI,definition,
% 0.28/0.47 ( dpsetconstrI
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( Xphi @ Xx @ Xy )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrSub_type,type,
% 0.28/0.47 dpsetconstrSub: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrSub,definition,
% 0.28/0.47 ( dpsetconstrSub
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.28/0.47 ( subset
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.28/0.47 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setOfPairsIsBReln_type,type,
% 0.28/0.47 setOfPairsIsBReln: $o ).
% 0.28/0.47
% 0.28/0.47 thf(setOfPairsIsBReln,definition,
% 0.28/0.47 ( setOfPairsIsBReln
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.28/0.47 ( breln @ A @ B
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrERa_type,type,
% 0.28/0.47 dpsetconstrERa: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrERa,definition,
% 0.28/0.47 ( dpsetconstrERa
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.28/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrEL1_type,type,
% 0.28/0.47 dpsetconstrEL1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrEL1,definition,
% 0.28/0.47 ( dpsetconstrEL1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.28/0.47 => ( in @ Xx @ A ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrEL2_type,type,
% 0.28/0.47 dpsetconstrEL2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrEL2,definition,
% 0.28/0.47 ( dpsetconstrEL2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.28/0.47 => ( in @ Xy @ B ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrER_type,type,
% 0.28/0.47 dpsetconstrER: $o ).
% 0.28/0.47
% 0.28/0.47 thf(dpsetconstrER,definition,
% 0.28/0.47 ( dpsetconstrER
% 0.28/0.47 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( in @ ( kpair @ Xx @ Xy )
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.28/0.47 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(func_type,type,
% 0.28/0.47 func: $i > $i > $i > $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcSet_type,type,
% 0.28/0.47 funcSet: $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(funcImageSingleton_type,type,
% 0.28/0.47 funcImageSingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcImageSingleton,definition,
% 0.28/0.47 ( funcImageSingleton
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ B
% 0.28/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(apProp_type,type,
% 0.28/0.47 apProp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(apProp,definition,
% 0.28/0.47 ( apProp
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in
% 0.28/0.47 @ ( setunion
% 0.28/0.47 @ ( dsetconstr @ B
% 0.28/0.47 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.28/0.47 @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ap_type,type,
% 0.28/0.47 ap: $i > $i > $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(app_type,type,
% 0.28/0.47 app: $o ).
% 0.28/0.47
% 0.28/0.47 thf(app,definition,
% 0.28/0.47 ( app
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(infuncsetfunc_type,type,
% 0.28/0.47 infuncsetfunc: $o ).
% 0.28/0.47
% 0.28/0.47 thf(infuncsetfunc,definition,
% 0.28/0.47 ( infuncsetfunc
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ap2p_type,type,
% 0.28/0.47 ap2p: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ap2p,definition,
% 0.28/0.47 ( ap2p
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcinfuncset_type,type,
% 0.28/0.47 funcinfuncset: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcinfuncset,definition,
% 0.28/0.47 ( funcinfuncset
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(lamProp_type,type,
% 0.28/0.47 lamProp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(lamProp,definition,
% 0.28/0.47 ( lamProp
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.47 => ( func @ A @ B
% 0.28/0.47 @ ( dpsetconstr @ A @ B
% 0.28/0.47 @ ^ [Xx: $i,Xy: $i] :
% 0.28/0.47 ( ( Xf @ Xx )
% 0.28/0.47 = Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(lam_type,type,
% 0.28/0.47 lam: $i > $i > ( $i > $i ) > $i ).
% 0.28/0.47
% 0.28/0.47 thf(lamp_type,type,
% 0.28/0.47 lamp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(lamp,definition,
% 0.28/0.47 ( lamp
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.47 => ( func @ A @ B
% 0.28/0.47 @ ( lam @ A @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(lam2p_type,type,
% 0.28/0.47 lam2p: $o ).
% 0.28/0.47
% 0.28/0.47 thf(lam2p,definition,
% 0.28/0.47 ( lam2p
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.47 => ( in
% 0.28/0.47 @ ( lam @ A @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.28/0.47 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(brelnall1_type,type,
% 0.28/0.47 brelnall1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(brelnall1,definition,
% 0.28/0.47 ( brelnall1
% 0.28/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ R )
% 0.28/0.47 => ! [Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ R )
% 0.28/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(brelnall2_type,type,
% 0.28/0.47 brelnall2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(brelnall2,definition,
% 0.28/0.47 ( brelnall2
% 0.28/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ R )
% 0.28/0.47 => ! [Xphi: $i > $o] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ R )
% 0.28/0.47 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ex1E2_type,type,
% 0.28/0.47 ex1E2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ex1E2,definition,
% 0.28/0.47 ( ex1E2
% 0.28/0.47 = ( ! [A: $i,Xphi: $i > $o] :
% 0.28/0.47 ( ( ex1 @ A
% 0.28/0.47 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ( Xphi @ Xx )
% 0.28/0.47 => ( ( Xphi @ Xy )
% 0.28/0.47 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp1_type,type,
% 0.28/0.47 funcGraphProp1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp1,definition,
% 0.28/0.47 ( funcGraphProp1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp3_type,type,
% 0.28/0.47 funcGraphProp3: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp3,definition,
% 0.28/0.47 ( funcGraphProp3
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp2_type,type,
% 0.28/0.47 funcGraphProp2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp2,definition,
% 0.28/0.47 ( funcGraphProp2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcextLem_type,type,
% 0.28/0.47 funcextLem: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcextLem,definition,
% 0.28/0.47 ( funcextLem
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xg: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xg )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp4_type,type,
% 0.28/0.47 funcGraphProp4: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcGraphProp4,definition,
% 0.28/0.47 ( funcGraphProp4
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = Xy ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(subbreln_type,type,
% 0.28/0.47 subbreln: $o ).
% 0.28/0.47
% 0.28/0.47 thf(subbreln,definition,
% 0.28/0.47 ( subbreln
% 0.28/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ S )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.28/0.47 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eqbreln_type,type,
% 0.28/0.47 eqbreln: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eqbreln,definition,
% 0.28/0.47 ( eqbreln
% 0.28/0.47 = ( ! [A: $i,B: $i,R: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ R )
% 0.28/0.47 => ! [S: $i] :
% 0.28/0.47 ( ( breln @ A @ B @ S )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ B )
% 0.28/0.47 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.28/0.47 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.28/0.47 => ( R = S ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcext_type,type,
% 0.28/0.47 funcext: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcext,definition,
% 0.28/0.47 ( funcext
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xg: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xg )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.28/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(funcext2_type,type,
% 0.28/0.47 funcext2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(funcext2,definition,
% 0.28/0.47 ( funcext2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ! [Xg: $i] :
% 0.28/0.47 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.28/0.47 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ap2apEq1_type,type,
% 0.28/0.47 ap2apEq1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ap2apEq1,definition,
% 0.28/0.47 ( ap2apEq1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ap2apEq2_type,type,
% 0.28/0.47 ap2apEq2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ap2apEq2,definition,
% 0.28/0.47 ( ap2apEq2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.28/0.47 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(beta1_type,type,
% 0.28/0.47 beta1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(beta1,definition,
% 0.28/0.47 ( beta1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B
% 0.28/0.47 @ ( lam @ A @ B
% 0.28/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.28/0.47 @ Xx )
% 0.28/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eta1_type,type,
% 0.28/0.47 eta1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eta1,definition,
% 0.28/0.47 ( eta1
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( func @ A @ B @ Xf )
% 0.28/0.47 => ( ( lam @ A @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.28/0.47 = Xf ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(lam2lamEq_type,type,
% 0.28/0.47 lam2lamEq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(lam2lamEq,definition,
% 0.28/0.47 ( lam2lamEq
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.47 => ( ( lam @ A @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.28/0.47 = ( lam @ A @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(beta2_type,type,
% 0.28/0.47 beta2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(beta2,definition,
% 0.28/0.47 ( beta2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.28/0.47 ( ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ( ( ap @ A @ B
% 0.28/0.47 @ ( lam @ A @ B
% 0.28/0.47 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.28/0.47 @ Xx )
% 0.28/0.47 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(eta2_type,type,
% 0.28/0.47 eta2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(eta2,definition,
% 0.28/0.47 ( eta2
% 0.28/0.47 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.28/0.47 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.28/0.47 => ( ( lam @ A @ B
% 0.28/0.47 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.28/0.47 = Xf ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iffalseProp1_type,type,
% 0.28/0.47 iffalseProp1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(iffalseProp1,definition,
% 0.28/0.47 ( iffalseProp1
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ~ Xphi
% 0.28/0.47 => ( in @ Xy
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xz: $i] :
% 0.28/0.47 ( ( Xphi
% 0.28/0.47 & ( Xz = Xx ) )
% 0.28/0.47 | ( ~ Xphi
% 0.28/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iffalseProp2_type,type,
% 0.28/0.47 iffalseProp2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(iffalseProp2,definition,
% 0.28/0.47 ( iffalseProp2
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ~ Xphi
% 0.28/0.47 => ( ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xz: $i] :
% 0.28/0.47 ( ( Xphi
% 0.28/0.47 & ( Xz = Xx ) )
% 0.28/0.47 | ( ~ Xphi
% 0.28/0.47 & ( Xz = Xy ) ) ) )
% 0.28/0.47 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iftrueProp1_type,type,
% 0.28/0.47 iftrueProp1: $o ).
% 0.28/0.47
% 0.28/0.47 thf(iftrueProp1,definition,
% 0.28/0.47 ( iftrueProp1
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( Xphi
% 0.28/0.47 => ( in @ Xx
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xz: $i] :
% 0.28/0.47 ( ( Xphi
% 0.28/0.47 & ( Xz = Xx ) )
% 0.28/0.47 | ( ~ Xphi
% 0.28/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iftrueProp2_type,type,
% 0.28/0.47 iftrueProp2: $o ).
% 0.28/0.47
% 0.28/0.47 thf(iftrueProp2,definition,
% 0.28/0.47 ( iftrueProp2
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( Xphi
% 0.28/0.47 => ( ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xz: $i] :
% 0.28/0.47 ( ( Xphi
% 0.28/0.47 & ( Xz = Xx ) )
% 0.28/0.47 | ( ~ Xphi
% 0.28/0.47 & ( Xz = Xy ) ) ) )
% 0.28/0.47 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(ifSingleton_type,type,
% 0.28/0.47 ifSingleton: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ifSingleton,definition,
% 0.28/0.47 ( ifSingleton
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( singleton
% 0.28/0.47 @ ( dsetconstr @ A
% 0.28/0.47 @ ^ [Xz: $i] :
% 0.28/0.47 ( ( Xphi
% 0.28/0.47 & ( Xz = Xx ) )
% 0.28/0.47 | ( ~ Xphi
% 0.28/0.47 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(if_type,type,
% 0.28/0.47 if: $i > $o > $i > $i > $i ).
% 0.28/0.47
% 0.28/0.47 thf(ifp_type,type,
% 0.28/0.47 ifp: $o ).
% 0.28/0.47
% 0.28/0.47 thf(ifp,definition,
% 0.28/0.47 ( ifp
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(theeq_type,type,
% 0.28/0.47 theeq: $o ).
% 0.28/0.47
% 0.28/0.47 thf(theeq,definition,
% 0.28/0.47 ( theeq
% 0.28/0.47 = ( ! [X: $i] :
% 0.28/0.47 ( ( singleton @ X )
% 0.28/0.47 => ! [Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ X )
% 0.28/0.47 => ( ( setunion @ X )
% 0.28/0.47 = Xx ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iftrue_type,type,
% 0.28/0.47 iftrue: $o ).
% 0.28/0.47
% 0.28/0.47 thf(iftrue,definition,
% 0.28/0.47 ( iftrue
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( Xphi
% 0.28/0.47 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.28/0.47 = Xx ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iffalse_type,type,
% 0.28/0.47 iffalse: $o ).
% 0.28/0.47
% 0.28/0.47 thf(iffalse,definition,
% 0.28/0.47 ( iffalse
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( ~ Xphi
% 0.28/0.47 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.28/0.47 = Xy ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(iftrueorfalse_type,type,
% 0.28/0.47 iftrueorfalse: $o ).
% 0.28/0.47
% 0.28/0.47 thf(iftrueorfalse,definition,
% 0.28/0.47 ( iftrueorfalse
% 0.28/0.47 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.28/0.47 ( ( in @ Xx @ A )
% 0.28/0.47 => ! [Xy: $i] :
% 0.28/0.47 ( ( in @ Xy @ A )
% 0.28/0.47 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectT_lem_type,type,
% 0.28/0.47 binintersectT_lem: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binintersectT_lem,definition,
% 0.28/0.47 ( binintersectT_lem
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(binunionT_lem_type,type,
% 0.28/0.47 binunionT_lem: $o ).
% 0.28/0.47
% 0.28/0.47 thf(binunionT_lem,definition,
% 0.28/0.47 ( binunionT_lem
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(powersetT_lem_type,type,
% 0.28/0.47 powersetT_lem: $o ).
% 0.28/0.47
% 0.28/0.47 thf(powersetT_lem,definition,
% 0.28/0.47 ( powersetT_lem
% 0.28/0.47 = ( ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 thf(setminusT_lem,conjecture,
% 0.28/0.47 ( setextAx
% 0.28/0.47 => ( emptysetAx
% 0.28/0.47 => ( setadjoinAx
% 0.28/0.47 => ( powersetAx
% 0.28/0.47 => ( setunionAx
% 0.28/0.47 => ( omega0Ax
% 0.28/0.47 => ( omegaSAx
% 0.28/0.47 => ( omegaIndAx
% 0.28/0.47 => ( replAx
% 0.28/0.47 => ( foundationAx
% 0.28/0.47 => ( wellorderingAx
% 0.28/0.47 => ( descrp
% 0.28/0.47 => ( dsetconstrI
% 0.28/0.47 => ( dsetconstrEL
% 0.28/0.47 => ( dsetconstrER
% 0.28/0.47 => ( exuE1
% 0.28/0.47 => ( prop2setE
% 0.28/0.47 => ( emptysetE
% 0.28/0.47 => ( emptysetimpfalse
% 0.28/0.47 => ( notinemptyset
% 0.28/0.47 => ( exuE3e
% 0.28/0.47 => ( setext
% 0.28/0.47 => ( emptyI
% 0.28/0.47 => ( noeltsimpempty
% 0.28/0.47 => ( setbeta
% 0.28/0.47 => ( nonemptyE1
% 0.28/0.47 => ( nonemptyI
% 0.28/0.47 => ( nonemptyI1
% 0.28/0.47 => ( setadjoinIL
% 0.28/0.47 => ( emptyinunitempty
% 0.28/0.47 => ( setadjoinIR
% 0.28/0.47 => ( setadjoinE
% 0.28/0.47 => ( setadjoinOr
% 0.28/0.47 => ( setoftrueEq
% 0.28/0.47 => ( powersetI
% 0.28/0.47 => ( emptyinPowerset
% 0.28/0.47 => ( emptyInPowerset
% 0.28/0.47 => ( powersetE
% 0.28/0.47 => ( setunionI
% 0.28/0.47 => ( setunionE
% 0.28/0.47 => ( subPowSU
% 0.28/0.47 => ( exuE2
% 0.28/0.47 => ( nonemptyImpWitness
% 0.28/0.47 => ( uniqinunit
% 0.28/0.47 => ( notinsingleton
% 0.28/0.47 => ( eqinunit
% 0.28/0.47 => ( singletonsswitch
% 0.28/0.47 => ( upairsetE
% 0.28/0.47 => ( upairsetIL
% 0.28/0.47 => ( upairsetIR
% 0.28/0.47 => ( emptyE1
% 0.28/0.47 => ( vacuousDall
% 0.28/0.47 => ( quantDeMorgan1
% 0.28/0.47 => ( quantDeMorgan2
% 0.28/0.47 => ( quantDeMorgan3
% 0.28/0.47 => ( quantDeMorgan4
% 0.28/0.47 => ( prop2setI
% 0.28/0.47 => ( prop2set2propI
% 0.28/0.47 => ( notdexE
% 0.28/0.47 => ( notdallE
% 0.28/0.47 => ( exuI1
% 0.28/0.47 => ( exuI3
% 0.28/0.47 => ( exuI2
% 0.28/0.47 => ( inCongP
% 0.28/0.47 => ( in__Cong
% 0.28/0.47 => ( exuE3u
% 0.28/0.47 => ( exu__Cong
% 0.28/0.47 => ( emptyset__Cong
% 0.28/0.47 => ( setadjoin__Cong
% 0.28/0.47 => ( powerset__Cong
% 0.28/0.47 => ( setunion__Cong
% 0.28/0.47 => ( omega__Cong
% 0.28/0.47 => ( exuEu
% 0.28/0.47 => ( descr__Cong
% 0.28/0.47 => ( dsetconstr__Cong
% 0.28/0.47 => ( subsetI1
% 0.28/0.47 => ( eqimpsubset2
% 0.28/0.47 => ( eqimpsubset1
% 0.28/0.47 => ( subsetI2
% 0.28/0.47 => ( emptysetsubset
% 0.28/0.47 => ( subsetE
% 0.28/0.47 => ( subsetE2
% 0.28/0.47 => ( notsubsetI
% 0.28/0.47 => ( notequalI1
% 0.28/0.47 => ( notequalI2
% 0.28/0.47 => ( subsetRefl
% 0.28/0.47 => ( subsetTrans
% 0.28/0.47 => ( setadjoinSub
% 0.28/0.47 => ( setadjoinSub2
% 0.28/0.47 => ( subset2powerset
% 0.28/0.47 => ( setextsub
% 0.28/0.47 => ( subsetemptysetimpeq
% 0.28/0.47 => ( powersetI1
% 0.28/0.47 => ( powersetE1
% 0.28/0.47 => ( inPowerset
% 0.28/0.47 => ( powersetsubset
% 0.28/0.47 => ( sepInPowerset
% 0.28/0.47 => ( sepSubset
% 0.28/0.47 => ( binunionIL
% 0.28/0.47 => ( upairset2IR
% 0.28/0.47 => ( binunionIR
% 0.28/0.47 => ( binunionEcases
% 0.28/0.47 => ( binunionE
% 0.28/0.47 => ( binunionLsub
% 0.28/0.47 => ( binunionRsub
% 0.28/0.47 => ( binintersectI
% 0.28/0.47 => ( binintersectSubset5
% 0.28/0.47 => ( binintersectEL
% 0.28/0.47 => ( binintersectLsub
% 0.28/0.47 => ( binintersectSubset2
% 0.28/0.47 => ( binintersectSubset3
% 0.28/0.47 => ( binintersectER
% 0.28/0.47 => ( disjointsetsI1
% 0.28/0.47 => ( binintersectRsub
% 0.28/0.47 => ( binintersectSubset4
% 0.28/0.47 => ( binintersectSubset1
% 0.28/0.47 => ( bs114d
% 0.28/0.47 => ( setminusI
% 0.28/0.47 => ( setminusEL
% 0.28/0.47 => ( setminusER
% 0.28/0.47 => ( setminusSubset2
% 0.28/0.47 => ( setminusERneg
% 0.28/0.47 => ( setminusELneg
% 0.28/0.47 => ( setminusILneg
% 0.28/0.47 => ( setminusIRneg
% 0.28/0.47 => ( setminusLsub
% 0.28/0.47 => ( setminusSubset1
% 0.28/0.47 => ( symdiffE
% 0.28/0.47 => ( symdiffI1
% 0.28/0.47 => ( symdiffI2
% 0.28/0.47 => ( symdiffIneg1
% 0.28/0.47 => ( symdiffIneg2
% 0.28/0.47 => ( secondinupair
% 0.28/0.47 => ( setukpairIL
% 0.28/0.47 => ( setukpairIR
% 0.28/0.47 => ( kpairiskpair
% 0.28/0.47 => ( kpairp
% 0.28/0.47 => ( singletonsubset
% 0.28/0.47 => ( singletoninpowerset
% 0.28/0.47 => ( singletoninpowunion
% 0.28/0.47 => ( upairset2E
% 0.28/0.47 => ( upairsubunion
% 0.28/0.47 => ( upairinpowunion
% 0.28/0.47 => ( ubforcartprodlem1
% 0.28/0.47 => ( ubforcartprodlem2
% 0.28/0.47 => ( ubforcartprodlem3
% 0.28/0.47 => ( cartprodpairin
% 0.28/0.47 => ( cartprodmempair1
% 0.28/0.47 => ( cartprodmempair
% 0.28/0.47 => ( setunionE2
% 0.28/0.47 => ( setunionsingleton1
% 0.28/0.47 => ( setunionsingleton2
% 0.28/0.47 => ( setunionsingleton
% 0.28/0.47 => ( singletonprop
% 0.28/0.47 => ( ex1E1
% 0.28/0.47 => ( ex1I
% 0.28/0.47 => ( ex1I2
% 0.28/0.47 => ( singletonsuniq
% 0.28/0.47 => ( setukpairinjL1
% 0.28/0.47 => ( kfstsingleton
% 0.28/0.47 => ( theprop
% 0.28/0.47 => ( kfstpairEq
% 0.28/0.47 => ( cartprodfstin
% 0.28/0.47 => ( setukpairinjL2
% 0.28/0.47 => ( setukpairinjL
% 0.28/0.47 => ( setukpairinjR11
% 0.28/0.47 => ( setukpairinjR12
% 0.28/0.47 => ( setukpairinjR1
% 0.28/0.47 => ( upairequniteq
% 0.28/0.47 => ( setukpairinjR2
% 0.28/0.47 => ( setukpairinjR
% 0.28/0.47 => ( ksndsingleton
% 0.28/0.47 => ( ksndpairEq
% 0.28/0.47 => ( kpairsurjEq
% 0.28/0.47 => ( cartprodsndin
% 0.28/0.47 => ( cartprodpairmemEL
% 0.28/0.47 => ( cartprodpairmemER
% 0.28/0.47 => ( cartprodmempaircEq
% 0.28/0.47 => ( cartprodfstpairEq
% 0.28/0.47 => ( cartprodsndpairEq
% 0.28/0.47 => ( cartprodpairsurjEq
% 0.28/0.47 => ( dpsetconstrI
% 0.28/0.47 => ( dpsetconstrSub
% 0.28/0.47 => ( setOfPairsIsBReln
% 0.28/0.47 => ( dpsetconstrERa
% 0.28/0.47 => ( dpsetconstrEL1
% 0.28/0.47 => ( dpsetconstrEL2
% 0.28/0.47 => ( dpsetconstrER
% 0.28/0.47 => ( funcImageSingleton
% 0.28/0.47 => ( apProp
% 0.28/0.47 => ( app
% 0.28/0.47 => ( infuncsetfunc
% 0.28/0.47 => ( ap2p
% 0.28/0.47 => ( funcinfuncset
% 0.28/0.47 => ( lamProp
% 0.28/0.47 => ( lamp
% 0.28/0.47 => ( lam2p
% 0.28/0.47 => ( brelnall1
% 0.28/0.47 => ( brelnall2
% 0.28/0.47 => ( ex1E2
% 0.28/0.47 => ( funcGraphProp1
% 0.28/0.47 => ( funcGraphProp3
% 0.28/0.47 => ( funcGraphProp2
% 0.28/0.47 => ( funcextLem
% 0.28/0.47 => ( funcGraphProp4
% 0.28/0.47 => ( subbreln
% 0.28/0.47 => ( eqbreln
% 0.28/0.47 => ( funcext
% 0.28/0.47 => ( funcext2
% 0.28/0.47 => ( ap2apEq1
% 0.28/0.47 => ( ap2apEq2
% 0.28/0.47 => ( beta1
% 0.28/0.47 => ( eta1
% 0.28/0.47 => ( lam2lamEq
% 0.28/0.47 => ( beta2
% 0.28/0.47 => ( eta2
% 0.28/0.47 => ( iffalseProp1
% 0.28/0.47 => ( iffalseProp2
% 0.28/0.47 => ( iftrueProp1
% 0.28/0.47 => ( iftrueProp2
% 0.28/0.47 => ( ifSingleton
% 0.28/0.47 => ( ifp
% 0.28/0.47 => ( theeq
% 0.28/0.47 => ( iftrue
% 0.28/0.47 => ( iffalse
% 0.28/0.47 => ( iftrueorfalse
% 0.28/0.47 => ( binintersectT_lem
% 0.28/0.47 => ( binunionT_lem
% 0.28/0.47 => ( powersetT_lem
% 0.28/0.47 => ! [A: $i,X: $i] :
% 0.28/0.47 ( ( in @ X @ ( powerset @ A ) )
% 0.28/0.47 => ! [Y: $i] :
% 0.28/0.47 ( ( in @ Y @ ( powerset @ A ) )
% 0.28/0.47 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.28/0.47
% 0.28/0.47 %------------------------------------------------------------------------------
% 0.28/0.47 ------------------------
% 2.70/1.59 ---- Embedded in THF ---
% 2.70/1.59 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.70/1.59 %%% Generated on Tue Feb 24 12:31:55 EST 2026
% 2.70/1.59 %%% using '$$dhol' embedding, version 1.3.0.
% 2.70/1.59 %%% Logic specification used:
% 2.70/1.59 %%% thf(spec, logic, $$dhol).
% 2.70/1.59
% 2.70/1.59 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.9EhP1wCwEA/DTF2DTF22036.p
% See solution above
% 2.70/1.59 ------------------------
% 2.94/1.62 ---- Cleaned THF ---
% 2.94/1.62 thf(in_type,type,
% 2.94/1.62 in: $i > $i > $o ).
% 2.94/1.62
% 2.94/1.62 thf(exu_type,type,
% 2.94/1.62 exu: ( $i > $o ) > $o ).
% 2.94/1.62
% 2.94/1.62 thf(exu,definition,
% 2.94/1.62 ( exu
% 2.94/1.62 = ( ^ [Xphi: $i > $o] :
% 2.94/1.62 ? [Xx: $i] :
% 2.94/1.62 ( ( Xphi @ Xx )
% 2.94/1.62 & ! [Xy: $i] :
% 2.94/1.62 ( ( Xphi @ Xy )
% 2.94/1.62 => ( Xx = Xy ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(setextAx_type,type,
% 2.94/1.62 setextAx: $o ).
% 2.94/1.62
% 2.94/1.62 thf(setextAx,definition,
% 2.94/1.62 ( setextAx
% 2.94/1.62 = ( ! [A: $i,B: $i] :
% 2.94/1.62 ( ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 <=> ( in @ Xx @ B ) )
% 2.94/1.62 => ( A = B ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(emptyset_type,type,
% 2.94/1.62 emptyset: $i ).
% 2.94/1.62
% 2.94/1.62 thf(emptysetAx_type,type,
% 2.94/1.62 emptysetAx: $o ).
% 2.94/1.62
% 2.94/1.62 thf(emptysetAx,definition,
% 2.94/1.62 ( emptysetAx
% 2.94/1.62 = ( ! [Xx: $i] :
% 2.94/1.62 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(setadjoin_type,type,
% 2.94/1.62 setadjoin: $i > $i > $i ).
% 2.94/1.62
% 2.94/1.62 thf(setadjoinAx_type,type,
% 2.94/1.62 setadjoinAx: $o ).
% 2.94/1.62
% 2.94/1.62 thf(setadjoinAx,definition,
% 2.94/1.62 ( setadjoinAx
% 2.94/1.62 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.94/1.62 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.94/1.62 <=> ( ( Xy = Xx )
% 2.94/1.62 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(powerset_type,type,
% 2.94/1.62 powerset: $i > $i ).
% 2.94/1.62
% 2.94/1.62 thf(powersetAx_type,type,
% 2.94/1.62 powersetAx: $o ).
% 2.94/1.62
% 2.94/1.62 thf(powersetAx,definition,
% 2.94/1.62 ( powersetAx
% 2.94/1.62 = ( ! [A: $i,B: $i] :
% 2.94/1.62 ( ( in @ B @ ( powerset @ A ) )
% 2.94/1.62 <=> ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ B )
% 2.94/1.62 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(setunion_type,type,
% 2.94/1.62 setunion: $i > $i ).
% 2.94/1.62
% 2.94/1.62 thf(setunionAx_type,type,
% 2.94/1.62 setunionAx: $o ).
% 2.94/1.62
% 2.94/1.62 thf(setunionAx,definition,
% 2.94/1.62 ( setunionAx
% 2.94/1.62 = ( ! [A: $i,Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ ( setunion @ A ) )
% 2.94/1.62 <=> ? [B: $i] :
% 2.94/1.62 ( ( in @ Xx @ B )
% 2.94/1.62 & ( in @ B @ A ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(omega_type,type,
% 2.94/1.62 omega: $i ).
% 2.94/1.62
% 2.94/1.62 thf(omega0Ax_type,type,
% 2.94/1.62 omega0Ax: $o ).
% 2.94/1.62
% 2.94/1.62 thf(omega0Ax,definition,
% 2.94/1.62 ( omega0Ax
% 2.94/1.62 = ( in @ emptyset @ omega ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(omegaSAx_type,type,
% 2.94/1.62 omegaSAx: $o ).
% 2.94/1.62
% 2.94/1.62 thf(omegaSAx,definition,
% 2.94/1.62 ( omegaSAx
% 2.94/1.62 = ( ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ omega )
% 2.94/1.62 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(omegaIndAx_type,type,
% 2.94/1.62 omegaIndAx: $o ).
% 2.94/1.62
% 2.94/1.62 thf(omegaIndAx,definition,
% 2.94/1.62 ( omegaIndAx
% 2.94/1.62 = ( ! [A: $i] :
% 2.94/1.62 ( ( ( in @ emptyset @ A )
% 2.94/1.62 & ! [Xx: $i] :
% 2.94/1.62 ( ( ( in @ Xx @ omega )
% 2.94/1.62 & ( in @ Xx @ A ) )
% 2.94/1.62 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.94/1.62 => ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ omega )
% 2.94/1.62 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(replAx_type,type,
% 2.94/1.62 replAx: $o ).
% 2.94/1.62
% 2.94/1.62 thf(replAx,definition,
% 2.94/1.62 ( replAx
% 2.94/1.62 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.94/1.62 ( ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 => ( exu
% 2.94/1.62 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.94/1.62 => ? [B: $i] :
% 2.94/1.62 ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ B )
% 2.94/1.62 <=> ? [Xy: $i] :
% 2.94/1.62 ( ( in @ Xy @ A )
% 2.94/1.62 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(foundationAx_type,type,
% 2.94/1.62 foundationAx: $o ).
% 2.94/1.62
% 2.94/1.62 thf(foundationAx,definition,
% 2.94/1.62 ( foundationAx
% 2.94/1.62 = ( ! [A: $i] :
% 2.94/1.62 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.94/1.62 => ? [B: $i] :
% 2.94/1.62 ( ( in @ B @ A )
% 2.94/1.62 & ~ ? [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ B )
% 2.94/1.62 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(wellorderingAx_type,type,
% 2.94/1.62 wellorderingAx: $o ).
% 2.94/1.62
% 2.94/1.62 thf(wellorderingAx,definition,
% 2.94/1.62 ( wellorderingAx
% 2.94/1.62 = ( ! [A: $i] :
% 2.94/1.62 ? [B: $i] :
% 2.94/1.62 ( ! [C: $i] :
% 2.94/1.62 ( ( in @ C @ B )
% 2.94/1.62 => ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ C )
% 2.94/1.62 => ( in @ Xx @ A ) ) )
% 2.94/1.62 & ! [Xx: $i,Xy: $i] :
% 2.94/1.62 ( ( ( in @ Xx @ A )
% 2.94/1.62 & ( in @ Xy @ A ) )
% 2.94/1.62 => ( ! [C: $i] :
% 2.94/1.62 ( ( in @ C @ B )
% 2.94/1.62 => ( ( in @ Xx @ C )
% 2.94/1.62 <=> ( in @ Xy @ C ) ) )
% 2.94/1.62 => ( Xx = Xy ) ) )
% 2.94/1.62 & ! [C: $i,D: $i] :
% 2.94/1.62 ( ( ( in @ C @ B )
% 2.94/1.62 & ( in @ D @ B ) )
% 2.94/1.62 => ( ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ C )
% 2.94/1.62 => ( in @ Xx @ D ) )
% 2.94/1.62 | ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ D )
% 2.94/1.62 => ( in @ Xx @ C ) ) ) )
% 2.94/1.62 & ! [C: $i] :
% 2.94/1.62 ( ( ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ C )
% 2.94/1.62 => ( in @ Xx @ A ) )
% 2.94/1.62 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.94/1.62 => ? [D: $i,Xx: $i] :
% 2.94/1.62 ( ( in @ D @ B )
% 2.94/1.62 & ( in @ Xx @ C )
% 2.94/1.62 & ~ ? [Xy: $i] :
% 2.94/1.62 ( ( in @ Xy @ D )
% 2.94/1.62 & ( in @ Xy @ C ) )
% 2.94/1.62 & ! [E: $i] :
% 2.94/1.62 ( ( in @ E @ B )
% 2.94/1.62 => ( ! [Xy: $i] :
% 2.94/1.62 ( ( in @ Xy @ E )
% 2.94/1.62 => ( in @ Xy @ D ) )
% 2.94/1.62 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(descr_type,type,
% 2.94/1.62 descr: ( $i > $o ) > $i ).
% 2.94/1.62
% 2.94/1.62 thf(descrp_type,type,
% 2.94/1.62 descrp: $o ).
% 2.94/1.62
% 2.94/1.62 thf(descrp,definition,
% 2.94/1.62 ( descrp
% 2.94/1.62 = ( ! [Xphi: $i > $o] :
% 2.94/1.62 ( ( exu
% 2.94/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.94/1.62 => ( Xphi
% 2.94/1.62 @ ( descr
% 2.94/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(dsetconstr_type,type,
% 2.94/1.62 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.94/1.62
% 2.94/1.62 thf(dsetconstrI_type,type,
% 2.94/1.62 dsetconstrI: $o ).
% 2.94/1.62
% 2.94/1.62 thf(dsetconstrI,definition,
% 2.94/1.62 ( dsetconstrI
% 2.94/1.62 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 => ( ( Xphi @ Xx )
% 2.94/1.62 => ( in @ Xx
% 2.94/1.62 @ ( dsetconstr @ A
% 2.94/1.62 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(dsetconstrEL_type,type,
% 2.94/1.62 dsetconstrEL: $o ).
% 2.94/1.62
% 2.94/1.62 thf(dsetconstrEL,definition,
% 2.94/1.62 ( dsetconstrEL
% 2.94/1.62 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.94/1.62 ( ( in @ Xx
% 2.94/1.62 @ ( dsetconstr @ A
% 2.94/1.62 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.94/1.62 => ( in @ Xx @ A ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(dsetconstrER_type,type,
% 2.94/1.62 dsetconstrER: $o ).
% 2.94/1.62
% 2.94/1.62 thf(dsetconstrER,definition,
% 2.94/1.62 ( dsetconstrER
% 2.94/1.62 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.94/1.62 ( ( in @ Xx
% 2.94/1.62 @ ( dsetconstr @ A
% 2.94/1.62 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.94/1.62 => ( Xphi @ Xx ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(exuE1_type,type,
% 2.94/1.62 exuE1: $o ).
% 2.94/1.62
% 2.94/1.62 thf(exuE1,definition,
% 2.94/1.62 ( exuE1
% 2.94/1.62 = ( ! [Xphi: $i > $o] :
% 2.94/1.62 ( ( exu
% 2.94/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.94/1.62 => ? [Xx: $i] :
% 2.94/1.62 ( ( Xphi @ Xx )
% 2.94/1.62 & ! [Xy: $i] :
% 2.94/1.62 ( ( Xphi @ Xy )
% 2.94/1.62 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(prop2set_type,type,
% 2.94/1.62 prop2set: $o > $i ).
% 2.94/1.62
% 2.94/1.62 thf(prop2setE_type,type,
% 2.94/1.62 prop2setE: $o ).
% 2.94/1.62
% 2.94/1.62 thf(prop2setE,definition,
% 2.94/1.62 ( prop2setE
% 2.94/1.62 = ( ! [Xphi: $o,Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.94/1.62 => Xphi ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(emptysetE_type,type,
% 2.94/1.62 emptysetE: $o ).
% 2.94/1.62
% 2.94/1.62 thf(emptysetE,definition,
% 2.94/1.62 ( emptysetE
% 2.94/1.62 = ( ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ emptyset )
% 2.94/1.62 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(emptysetimpfalse_type,type,
% 2.94/1.62 emptysetimpfalse: $o ).
% 2.94/1.62
% 2.94/1.62 thf(emptysetimpfalse,definition,
% 2.94/1.62 ( emptysetimpfalse
% 2.94/1.62 = ( ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ emptyset )
% 2.94/1.62 => $false ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(notinemptyset_type,type,
% 2.94/1.62 notinemptyset: $o ).
% 2.94/1.62
% 2.94/1.62 thf(notinemptyset,definition,
% 2.94/1.62 ( notinemptyset
% 2.94/1.62 = ( ! [Xx: $i] :
% 2.94/1.62 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(exuE3e_type,type,
% 2.94/1.62 exuE3e: $o ).
% 2.94/1.62
% 2.94/1.62 thf(exuE3e,definition,
% 2.94/1.62 ( exuE3e
% 2.94/1.62 = ( ! [Xphi: $i > $o] :
% 2.94/1.62 ( ( exu
% 2.94/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.94/1.62 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(setext_type,type,
% 2.94/1.62 setext: $o ).
% 2.94/1.62
% 2.94/1.62 thf(setext,definition,
% 2.94/1.62 ( setext
% 2.94/1.62 = ( ! [A: $i,B: $i] :
% 2.94/1.62 ( ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 => ( in @ Xx @ B ) )
% 2.94/1.62 => ( ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ B )
% 2.94/1.62 => ( in @ Xx @ A ) )
% 2.94/1.62 => ( A = B ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(emptyI_type,type,
% 2.94/1.62 emptyI: $o ).
% 2.94/1.62
% 2.94/1.62 thf(emptyI,definition,
% 2.94/1.62 ( emptyI
% 2.94/1.62 = ( ! [A: $i] :
% 2.94/1.62 ( ! [Xx: $i] :
% 2.94/1.62 ~ ( in @ Xx @ A )
% 2.94/1.62 => ( A = emptyset ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(noeltsimpempty_type,type,
% 2.94/1.62 noeltsimpempty: $o ).
% 2.94/1.62
% 2.94/1.62 thf(noeltsimpempty,definition,
% 2.94/1.62 ( noeltsimpempty
% 2.94/1.62 = ( ! [A: $i] :
% 2.94/1.62 ( ! [Xx: $i] :
% 2.94/1.62 ~ ( in @ Xx @ A )
% 2.94/1.62 => ( A = emptyset ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(setbeta_type,type,
% 2.94/1.62 setbeta: $o ).
% 2.94/1.62
% 2.94/1.62 thf(setbeta,definition,
% 2.94/1.62 ( setbeta
% 2.94/1.62 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 => ( ( in @ Xx
% 2.94/1.62 @ ( dsetconstr @ A
% 2.94/1.62 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.94/1.62 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(nonempty_type,type,
% 2.94/1.62 nonempty: $i > $o ).
% 2.94/1.62
% 2.94/1.62 thf(nonempty,definition,
% 2.94/1.62 ( nonempty
% 2.94/1.62 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(nonemptyE1_type,type,
% 2.94/1.62 nonemptyE1: $o ).
% 2.94/1.62
% 2.94/1.62 thf(nonemptyE1,definition,
% 2.94/1.62 ( nonemptyE1
% 2.94/1.62 = ( ! [A: $i] :
% 2.94/1.62 ( ( nonempty @ A )
% 2.94/1.62 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(nonemptyI_type,type,
% 2.94/1.62 nonemptyI: $o ).
% 2.94/1.62
% 2.94/1.62 thf(nonemptyI,definition,
% 2.94/1.62 ( nonemptyI
% 2.94/1.62 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 => ( ( Xphi @ Xx )
% 2.94/1.62 => ( nonempty
% 2.94/1.62 @ ( dsetconstr @ A
% 2.94/1.62 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(nonemptyI1_type,type,
% 2.94/1.62 nonemptyI1: $o ).
% 2.94/1.62
% 2.94/1.62 thf(nonemptyI1,definition,
% 2.94/1.62 ( nonemptyI1
% 2.94/1.62 = ( ! [A: $i] :
% 2.94/1.62 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.94/1.62 => ( nonempty @ A ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(setadjoinIL_type,type,
% 2.94/1.62 setadjoinIL: $o ).
% 2.94/1.62
% 2.94/1.62 thf(setadjoinIL,definition,
% 2.94/1.62 ( setadjoinIL
% 2.94/1.62 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(emptyinunitempty_type,type,
% 2.94/1.62 emptyinunitempty: $o ).
% 2.94/1.62
% 2.94/1.62 thf(emptyinunitempty,definition,
% 2.94/1.62 ( emptyinunitempty
% 2.94/1.62 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(setadjoinIR_type,type,
% 2.94/1.62 setadjoinIR: $o ).
% 2.94/1.62
% 2.94/1.62 thf(setadjoinIR,definition,
% 2.94/1.62 ( setadjoinIR
% 2.94/1.62 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.94/1.62 ( ( in @ Xy @ A )
% 2.94/1.62 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(setadjoinE_type,type,
% 2.94/1.62 setadjoinE: $o ).
% 2.94/1.62
% 2.94/1.62 thf(setadjoinE,definition,
% 2.94/1.62 ( setadjoinE
% 2.94/1.62 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.94/1.62 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.94/1.62 => ! [Xphi: $o] :
% 2.94/1.62 ( ( ( Xy = Xx )
% 2.94/1.62 => Xphi )
% 2.94/1.62 => ( ( ( in @ Xy @ A )
% 2.94/1.62 => Xphi )
% 2.94/1.62 => Xphi ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(setadjoinOr_type,type,
% 2.94/1.62 setadjoinOr: $o ).
% 2.94/1.62
% 2.94/1.62 thf(setadjoinOr,definition,
% 2.94/1.62 ( setadjoinOr
% 2.94/1.62 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.94/1.62 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.94/1.62 => ( ( Xy = Xx )
% 2.94/1.62 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(setoftrueEq_type,type,
% 2.94/1.62 setoftrueEq: $o ).
% 2.94/1.62
% 2.94/1.62 thf(setoftrueEq,definition,
% 2.94/1.62 ( setoftrueEq
% 2.94/1.62 = ( ! [A: $i] :
% 2.94/1.62 ( ( dsetconstr @ A
% 2.94/1.62 @ ^ [Xx: $i] : $true )
% 2.94/1.62 = A ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(powersetI_type,type,
% 2.94/1.62 powersetI: $o ).
% 2.94/1.62
% 2.94/1.62 thf(powersetI,definition,
% 2.94/1.62 ( powersetI
% 2.94/1.62 = ( ! [A: $i,B: $i] :
% 2.94/1.62 ( ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ B )
% 2.94/1.62 => ( in @ Xx @ A ) )
% 2.94/1.62 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(emptyinPowerset_type,type,
% 2.94/1.62 emptyinPowerset: $o ).
% 2.94/1.62
% 2.94/1.62 thf(emptyinPowerset,definition,
% 2.94/1.62 ( emptyinPowerset
% 2.94/1.62 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(emptyInPowerset_type,type,
% 2.94/1.62 emptyInPowerset: $o ).
% 2.94/1.62
% 2.94/1.62 thf(emptyInPowerset,definition,
% 2.94/1.62 ( emptyInPowerset
% 2.94/1.62 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(powersetE_type,type,
% 2.94/1.62 powersetE: $o ).
% 2.94/1.62
% 2.94/1.62 thf(powersetE,definition,
% 2.94/1.62 ( powersetE
% 2.94/1.62 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.62 ( ( in @ B @ ( powerset @ A ) )
% 2.94/1.62 => ( ( in @ Xx @ B )
% 2.94/1.62 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(setunionI_type,type,
% 2.94/1.62 setunionI: $o ).
% 2.94/1.62
% 2.94/1.62 thf(setunionI,definition,
% 2.94/1.62 ( setunionI
% 2.94/1.62 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.94/1.62 ( ( in @ Xx @ B )
% 2.94/1.62 => ( ( in @ B @ A )
% 2.94/1.62 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(setunionE_type,type,
% 2.94/1.62 setunionE: $o ).
% 2.94/1.62
% 2.94/1.62 thf(setunionE,definition,
% 2.94/1.62 ( setunionE
% 2.94/1.62 = ( ! [A: $i,Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ ( setunion @ A ) )
% 2.94/1.62 => ! [Xphi: $o] :
% 2.94/1.62 ( ! [B: $i] :
% 2.94/1.62 ( ( in @ Xx @ B )
% 2.94/1.62 => ( ( in @ B @ A )
% 2.94/1.62 => Xphi ) )
% 2.94/1.62 => Xphi ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(subPowSU_type,type,
% 2.94/1.62 subPowSU: $o ).
% 2.94/1.62
% 2.94/1.62 thf(subPowSU,definition,
% 2.94/1.62 ( subPowSU
% 2.94/1.62 = ( ! [A: $i,Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(exuE2_type,type,
% 2.94/1.62 exuE2: $o ).
% 2.94/1.62
% 2.94/1.62 thf(exuE2,definition,
% 2.94/1.62 ( exuE2
% 2.94/1.62 = ( ! [Xphi: $i > $o] :
% 2.94/1.62 ( ( exu
% 2.94/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.94/1.62 => ? [Xx: $i] :
% 2.94/1.62 ! [Xy: $i] :
% 2.94/1.62 ( ( Xphi @ Xy )
% 2.94/1.62 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(nonemptyImpWitness_type,type,
% 2.94/1.62 nonemptyImpWitness: $o ).
% 2.94/1.62
% 2.94/1.62 thf(nonemptyImpWitness,definition,
% 2.94/1.62 ( nonemptyImpWitness
% 2.94/1.62 = ( ! [A: $i] :
% 2.94/1.62 ( ( nonempty @ A )
% 2.94/1.62 => ? [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 & $true ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(uniqinunit_type,type,
% 2.94/1.62 uniqinunit: $o ).
% 2.94/1.62
% 2.94/1.62 thf(uniqinunit,definition,
% 2.94/1.62 ( uniqinunit
% 2.94/1.62 = ( ! [Xx: $i,Xy: $i] :
% 2.94/1.62 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.94/1.62 => ( Xx = Xy ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(notinsingleton_type,type,
% 2.94/1.62 notinsingleton: $o ).
% 2.94/1.62
% 2.94/1.62 thf(notinsingleton,definition,
% 2.94/1.62 ( notinsingleton
% 2.94/1.62 = ( ! [Xx: $i,Xy: $i] :
% 2.94/1.62 ( ( Xx != Xy )
% 2.94/1.62 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(eqinunit_type,type,
% 2.94/1.62 eqinunit: $o ).
% 2.94/1.62
% 2.94/1.62 thf(eqinunit,definition,
% 2.94/1.62 ( eqinunit
% 2.94/1.62 = ( ! [Xx: $i,Xy: $i] :
% 2.94/1.62 ( ( Xx = Xy )
% 2.94/1.62 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(singletonsswitch_type,type,
% 2.94/1.62 singletonsswitch: $o ).
% 2.94/1.62
% 2.94/1.62 thf(singletonsswitch,definition,
% 2.94/1.62 ( singletonsswitch
% 2.94/1.62 = ( ! [Xx: $i,Xy: $i] :
% 2.94/1.62 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.94/1.62 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(upairsetE_type,type,
% 2.94/1.62 upairsetE: $o ).
% 2.94/1.62
% 2.94/1.62 thf(upairsetE,definition,
% 2.94/1.62 ( upairsetE
% 2.94/1.62 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.94/1.62 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.94/1.62 => ( ( Xz = Xx )
% 2.94/1.62 | ( Xz = Xy ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(upairsetIL_type,type,
% 2.94/1.62 upairsetIL: $o ).
% 2.94/1.62
% 2.94/1.62 thf(upairsetIL,definition,
% 2.94/1.62 ( upairsetIL
% 2.94/1.62 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(upairsetIR_type,type,
% 2.94/1.62 upairsetIR: $o ).
% 2.94/1.62
% 2.94/1.62 thf(upairsetIR,definition,
% 2.94/1.62 ( upairsetIR
% 2.94/1.62 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(emptyE1_type,type,
% 2.94/1.62 emptyE1: $o ).
% 2.94/1.62
% 2.94/1.62 thf(emptyE1,definition,
% 2.94/1.62 ( emptyE1
% 2.94/1.62 = ( ! [A: $i,Xphi: $i > $o] :
% 2.94/1.62 ( ? [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 & ( Xphi @ Xx ) )
% 2.94/1.62 => ( ( ( dsetconstr @ A
% 2.94/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.94/1.62 = emptyset )
% 2.94/1.62 => $false ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(vacuousDall_type,type,
% 2.94/1.62 vacuousDall: $o ).
% 2.94/1.62
% 2.94/1.62 thf(vacuousDall,definition,
% 2.94/1.62 ( vacuousDall
% 2.94/1.62 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ emptyset )
% 2.94/1.62 => ( Xphi @ Xx ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(quantDeMorgan1_type,type,
% 2.94/1.62 quantDeMorgan1: $o ).
% 2.94/1.62
% 2.94/1.62 thf(quantDeMorgan1,definition,
% 2.94/1.62 ( quantDeMorgan1
% 2.94/1.62 = ( ! [A: $i,Xphi: $i > $o] :
% 2.94/1.62 ( ~ ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 => ( Xphi @ Xx ) )
% 2.94/1.62 => ? [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(quantDeMorgan2_type,type,
% 2.94/1.62 quantDeMorgan2: $o ).
% 2.94/1.62
% 2.94/1.62 thf(quantDeMorgan2,definition,
% 2.94/1.62 ( quantDeMorgan2
% 2.94/1.62 = ( ! [A: $i,Xphi: $i > $o] :
% 2.94/1.62 ( ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 => ~ ( Xphi @ Xx ) )
% 2.94/1.62 => ~ ? [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(quantDeMorgan3_type,type,
% 2.94/1.62 quantDeMorgan3: $o ).
% 2.94/1.62
% 2.94/1.62 thf(quantDeMorgan3,definition,
% 2.94/1.62 ( quantDeMorgan3
% 2.94/1.62 = ( ! [A: $i,Xphi: $i > $o] :
% 2.94/1.62 ( ~ ? [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 & ( Xphi @ Xx ) )
% 2.94/1.62 => ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(quantDeMorgan4_type,type,
% 2.94/1.62 quantDeMorgan4: $o ).
% 2.94/1.62
% 2.94/1.62 thf(quantDeMorgan4,definition,
% 2.94/1.62 ( quantDeMorgan4
% 2.94/1.62 = ( ! [A: $i,Xphi: $i > $o] :
% 2.94/1.62 ( ? [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 & ~ ( Xphi @ Xx ) )
% 2.94/1.62 => ~ ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(prop2setI_type,type,
% 2.94/1.62 prop2setI: $o ).
% 2.94/1.62
% 2.94/1.62 thf(prop2setI,definition,
% 2.94/1.62 ( prop2setI
% 2.94/1.62 = ( ! [Xphi: $o] :
% 2.94/1.62 ( Xphi
% 2.94/1.62 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(set2prop_type,type,
% 2.94/1.62 set2prop: $i > $o ).
% 2.94/1.62
% 2.94/1.62 thf(prop2set2propI_type,type,
% 2.94/1.62 prop2set2propI: $o ).
% 2.94/1.62
% 2.94/1.62 thf(prop2set2propI,definition,
% 2.94/1.62 ( prop2set2propI
% 2.94/1.62 = ( ! [Xphi: $o] :
% 2.94/1.62 ( Xphi
% 2.94/1.62 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(notdexE_type,type,
% 2.94/1.62 notdexE: $o ).
% 2.94/1.62
% 2.94/1.62 thf(notdexE,definition,
% 2.94/1.62 ( notdexE
% 2.94/1.62 = ( ! [A: $i,Xphi: $i > $o] :
% 2.94/1.62 ( ~ ? [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 & ( Xphi @ Xx ) )
% 2.94/1.62 => ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(notdallE_type,type,
% 2.94/1.62 notdallE: $o ).
% 2.94/1.62
% 2.94/1.62 thf(notdallE,definition,
% 2.94/1.62 ( notdallE
% 2.94/1.62 = ( ! [A: $i,Xphi: $i > $o] :
% 2.94/1.62 ( ~ ! [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 => ( Xphi @ Xx ) )
% 2.94/1.62 => ? [Xx: $i] :
% 2.94/1.62 ( ( in @ Xx @ A )
% 2.94/1.62 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(exuI1_type,type,
% 2.94/1.62 exuI1: $o ).
% 2.94/1.62
% 2.94/1.62 thf(exuI1,definition,
% 2.94/1.62 ( exuI1
% 2.94/1.62 = ( ! [Xphi: $i > $o] :
% 2.94/1.62 ( ? [Xx: $i] :
% 2.94/1.62 ( ( Xphi @ Xx )
% 2.94/1.62 & ! [Xy: $i] :
% 2.94/1.62 ( ( Xphi @ Xy )
% 2.94/1.62 => ( Xx = Xy ) ) )
% 2.94/1.62 => ( exu
% 2.94/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(exuI3_type,type,
% 2.94/1.62 exuI3: $o ).
% 2.94/1.62
% 2.94/1.62 thf(exuI3,definition,
% 2.94/1.62 ( exuI3
% 2.94/1.62 = ( ! [Xphi: $i > $o] :
% 2.94/1.62 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.94/1.62 => ( ! [Xx: $i,Xy: $i] :
% 2.94/1.62 ( ( Xphi @ Xx )
% 2.94/1.62 => ( ( Xphi @ Xy )
% 2.94/1.62 => ( Xx = Xy ) ) )
% 2.94/1.62 => ( exu
% 2.94/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(exuI2_type,type,
% 2.94/1.62 exuI2: $o ).
% 2.94/1.62
% 2.94/1.62 thf(exuI2,definition,
% 2.94/1.62 ( exuI2
% 2.94/1.62 = ( ! [Xphi: $i > $o] :
% 2.94/1.62 ( ? [Xx: $i] :
% 2.94/1.62 ! [Xy: $i] :
% 2.94/1.62 ( ( Xphi @ Xy )
% 2.94/1.62 <=> ( Xy = Xx ) )
% 2.94/1.62 => ( exu
% 2.94/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(inCongP_type,type,
% 2.94/1.62 inCongP: $o ).
% 2.94/1.62
% 2.94/1.62 thf(inCongP,definition,
% 2.94/1.62 ( inCongP
% 2.94/1.62 = ( ! [A: $i,B: $i] :
% 2.94/1.62 ( ( A = B )
% 2.94/1.62 => ! [Xx: $i,Xy: $i] :
% 2.94/1.62 ( ( Xx = Xy )
% 2.94/1.62 => ( ( in @ Xx @ A )
% 2.94/1.62 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(in__Cong_type,type,
% 2.94/1.62 in__Cong: $o ).
% 2.94/1.62
% 2.94/1.62 thf(in__Cong,definition,
% 2.94/1.62 ( in__Cong
% 2.94/1.62 = ( ! [A: $i,B: $i] :
% 2.94/1.62 ( ( A = B )
% 2.94/1.62 => ! [Xx: $i,Xy: $i] :
% 2.94/1.62 ( ( Xx = Xy )
% 2.94/1.62 => ( ( in @ Xx @ A )
% 2.94/1.62 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(exuE3u_type,type,
% 2.94/1.62 exuE3u: $o ).
% 2.94/1.62
% 2.94/1.62 thf(exuE3u,definition,
% 2.94/1.62 ( exuE3u
% 2.94/1.62 = ( ! [Xphi: $i > $o] :
% 2.94/1.62 ( ( exu
% 2.94/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.94/1.62 => ! [Xx: $i,Xy: $i] :
% 2.94/1.62 ( ( Xphi @ Xx )
% 2.94/1.62 => ( ( Xphi @ Xy )
% 2.94/1.62 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(exu__Cong_type,type,
% 2.94/1.62 exu__Cong: $o ).
% 2.94/1.62
% 2.94/1.62 thf(exu__Cong,definition,
% 2.94/1.62 ( exu__Cong
% 2.94/1.62 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.94/1.62 ( ! [Xx: $i,Xy: $i] :
% 2.94/1.62 ( ( Xx = Xy )
% 2.94/1.62 => ( ( Xphi @ Xx )
% 2.94/1.62 <=> ( Xpsi @ Xy ) ) )
% 2.94/1.62 => ( ( exu
% 2.94/1.62 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.94/1.62 <=> ( exu
% 2.94/1.62 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(emptyset__Cong_type,type,
% 2.94/1.62 emptyset__Cong: $o ).
% 2.94/1.62
% 2.94/1.62 thf(emptyset__Cong,definition,
% 2.94/1.62 ( emptyset__Cong
% 2.94/1.62 = ( emptyset = emptyset ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(setadjoin__Cong_type,type,
% 2.94/1.62 setadjoin__Cong: $o ).
% 2.94/1.62
% 2.94/1.62 thf(setadjoin__Cong,definition,
% 2.94/1.62 ( setadjoin__Cong
% 2.94/1.62 = ( ! [Xx: $i,Xy: $i] :
% 2.94/1.62 ( ( Xx = Xy )
% 2.94/1.62 => ! [Xz: $i,Xu: $i] :
% 2.94/1.62 ( ( Xz = Xu )
% 2.94/1.62 => ( ( setadjoin @ Xx @ Xz )
% 2.94/1.62 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(powerset__Cong_type,type,
% 2.94/1.62 powerset__Cong: $o ).
% 2.94/1.62
% 2.94/1.62 thf(powerset__Cong,definition,
% 2.94/1.62 ( powerset__Cong
% 2.94/1.62 = ( ! [A: $i,B: $i] :
% 2.94/1.62 ( ( A = B )
% 2.94/1.62 => ( ( powerset @ A )
% 2.94/1.62 = ( powerset @ B ) ) ) ) ) ).
% 2.94/1.62
% 2.94/1.62 thf(setunion__Cong_type,type,
% 2.94/1.62 setunion__Cong: $o ).
% 2.94/1.62
% 2.94/1.62 thf(setunion__Cong,definition,
% 2.94/1.62 ( setunion__Cong
% 2.94/1.62 = ( ! [A: $i,B: $i] :
% 2.94/1.62 ( ( A = B )
% 2.94/1.62 => ( ( setunion @ A )
% 2.94/1.63 = ( setunion @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(omega__Cong_type,type,
% 2.94/1.63 omega__Cong: $o ).
% 2.94/1.63
% 2.94/1.63 thf(omega__Cong,definition,
% 2.94/1.63 ( omega__Cong
% 2.94/1.63 = ( omega = omega ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(exuEu_type,type,
% 2.94/1.63 exuEu: $o ).
% 2.94/1.63
% 2.94/1.63 thf(exuEu,definition,
% 2.94/1.63 ( exuEu
% 2.94/1.63 = ( ! [Xphi: $i > $o] :
% 2.94/1.63 ( ( exu
% 2.94/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.94/1.63 => ! [Xx: $i,Xy: $i] :
% 2.94/1.63 ( ( Xphi @ Xx )
% 2.94/1.63 => ( ( Xphi @ Xy )
% 2.94/1.63 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(descr__Cong_type,type,
% 2.94/1.63 descr__Cong: $o ).
% 2.94/1.63
% 2.94/1.63 thf(descr__Cong,definition,
% 2.94/1.63 ( descr__Cong
% 2.94/1.63 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.94/1.63 ( ! [Xx: $i,Xy: $i] :
% 2.94/1.63 ( ( Xx = Xy )
% 2.94/1.63 => ( ( Xphi @ Xx )
% 2.94/1.63 <=> ( Xpsi @ Xy ) ) )
% 2.94/1.63 => ( ( exu
% 2.94/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.94/1.63 => ( ( exu
% 2.94/1.63 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.94/1.63 => ( ( descr
% 2.94/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.94/1.63 = ( descr
% 2.94/1.63 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(dsetconstr__Cong_type,type,
% 2.94/1.63 dsetconstr__Cong: $o ).
% 2.94/1.63
% 2.94/1.63 thf(dsetconstr__Cong,definition,
% 2.94/1.63 ( dsetconstr__Cong
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ( A = B )
% 2.94/1.63 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.94/1.63 ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( ( Xx = Xy )
% 2.94/1.63 => ( ( Xphi @ Xx )
% 2.94/1.63 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.94/1.63 => ( ( dsetconstr @ A
% 2.94/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.94/1.63 = ( dsetconstr @ B
% 2.94/1.63 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(subset_type,type,
% 2.94/1.63 subset: $i > $i > $o ).
% 2.94/1.63
% 2.94/1.63 thf(disjoint_type,type,
% 2.94/1.63 disjoint: $i > $i > $o ).
% 2.94/1.63
% 2.94/1.63 thf(setsmeet_type,type,
% 2.94/1.63 setsmeet: $i > $i > $o ).
% 2.94/1.63
% 2.94/1.63 thf(subsetI1_type,type,
% 2.94/1.63 subsetI1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(subsetI1,definition,
% 2.94/1.63 ( subsetI1
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ Xx @ B ) )
% 2.94/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(eqimpsubset2_type,type,
% 2.94/1.63 eqimpsubset2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(eqimpsubset2,definition,
% 2.94/1.63 ( eqimpsubset2
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ( A = B )
% 2.94/1.63 => ( subset @ B @ A ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(eqimpsubset1_type,type,
% 2.94/1.63 eqimpsubset1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(eqimpsubset1,definition,
% 2.94/1.63 ( eqimpsubset1
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ( A = B )
% 2.94/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(subsetI2_type,type,
% 2.94/1.63 subsetI2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(subsetI2,definition,
% 2.94/1.63 ( subsetI2
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ Xx @ B ) )
% 2.94/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(emptysetsubset_type,type,
% 2.94/1.63 emptysetsubset: $o ).
% 2.94/1.63
% 2.94/1.63 thf(emptysetsubset,definition,
% 2.94/1.63 ( emptysetsubset
% 2.94/1.63 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(subsetE_type,type,
% 2.94/1.63 subsetE: $o ).
% 2.94/1.63
% 2.94/1.63 thf(subsetE,definition,
% 2.94/1.63 ( subsetE
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( subset @ A @ B )
% 2.94/1.63 => ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(subsetE2_type,type,
% 2.94/1.63 subsetE2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(subsetE2,definition,
% 2.94/1.63 ( subsetE2
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( subset @ A @ B )
% 2.94/1.63 => ( ~ ( in @ Xx @ B )
% 2.94/1.63 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(notsubsetI_type,type,
% 2.94/1.63 notsubsetI: $o ).
% 2.94/1.63
% 2.94/1.63 thf(notsubsetI,definition,
% 2.94/1.63 ( notsubsetI
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( ~ ( in @ Xx @ B )
% 2.94/1.63 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(notequalI1_type,type,
% 2.94/1.63 notequalI1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(notequalI1,definition,
% 2.94/1.63 ( notequalI1
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ~ ( subset @ A @ B )
% 2.94/1.63 => ( A != B ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(notequalI2_type,type,
% 2.94/1.63 notequalI2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(notequalI2,definition,
% 2.94/1.63 ( notequalI2
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( ~ ( in @ Xx @ B )
% 2.94/1.63 => ( A != B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(subsetRefl_type,type,
% 2.94/1.63 subsetRefl: $o ).
% 2.94/1.63
% 2.94/1.63 thf(subsetRefl,definition,
% 2.94/1.63 ( subsetRefl
% 2.94/1.63 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(subsetTrans_type,type,
% 2.94/1.63 subsetTrans: $o ).
% 2.94/1.63
% 2.94/1.63 thf(subsetTrans,definition,
% 2.94/1.63 ( subsetTrans
% 2.94/1.63 = ( ! [A: $i,B: $i,C: $i] :
% 2.94/1.63 ( ( subset @ A @ B )
% 2.94/1.63 => ( ( subset @ B @ C )
% 2.94/1.63 => ( subset @ A @ C ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setadjoinSub_type,type,
% 2.94/1.63 setadjoinSub: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setadjoinSub,definition,
% 2.94/1.63 ( setadjoinSub
% 2.94/1.63 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setadjoinSub2_type,type,
% 2.94/1.63 setadjoinSub2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setadjoinSub2,definition,
% 2.94/1.63 ( setadjoinSub2
% 2.94/1.63 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.94/1.63 ( ( subset @ A @ B )
% 2.94/1.63 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(subset2powerset_type,type,
% 2.94/1.63 subset2powerset: $o ).
% 2.94/1.63
% 2.94/1.63 thf(subset2powerset,definition,
% 2.94/1.63 ( subset2powerset
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ( subset @ A @ B )
% 2.94/1.63 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setextsub_type,type,
% 2.94/1.63 setextsub: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setextsub,definition,
% 2.94/1.63 ( setextsub
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ( subset @ A @ B )
% 2.94/1.63 => ( ( subset @ B @ A )
% 2.94/1.63 => ( A = B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(subsetemptysetimpeq_type,type,
% 2.94/1.63 subsetemptysetimpeq: $o ).
% 2.94/1.63
% 2.94/1.63 thf(subsetemptysetimpeq,definition,
% 2.94/1.63 ( subsetemptysetimpeq
% 2.94/1.63 = ( ! [A: $i] :
% 2.94/1.63 ( ( subset @ A @ emptyset )
% 2.94/1.63 => ( A = emptyset ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(powersetI1_type,type,
% 2.94/1.63 powersetI1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(powersetI1,definition,
% 2.94/1.63 ( powersetI1
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ( subset @ B @ A )
% 2.94/1.63 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(powersetE1_type,type,
% 2.94/1.63 powersetE1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(powersetE1,definition,
% 2.94/1.63 ( powersetE1
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ( in @ B @ ( powerset @ A ) )
% 2.94/1.63 => ( subset @ B @ A ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(inPowerset_type,type,
% 2.94/1.63 inPowerset: $o ).
% 2.94/1.63
% 2.94/1.63 thf(inPowerset,definition,
% 2.94/1.63 ( inPowerset
% 2.94/1.63 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(powersetsubset_type,type,
% 2.94/1.63 powersetsubset: $o ).
% 2.94/1.63
% 2.94/1.63 thf(powersetsubset,definition,
% 2.94/1.63 ( powersetsubset
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ( subset @ A @ B )
% 2.94/1.63 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(sepInPowerset_type,type,
% 2.94/1.63 sepInPowerset: $o ).
% 2.94/1.63
% 2.94/1.63 thf(sepInPowerset,definition,
% 2.94/1.63 ( sepInPowerset
% 2.94/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.94/1.63 ( in
% 2.94/1.63 @ ( dsetconstr @ A
% 2.94/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.94/1.63 @ ( powerset @ A ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(sepSubset_type,type,
% 2.94/1.63 sepSubset: $o ).
% 2.94/1.63
% 2.94/1.63 thf(sepSubset,definition,
% 2.94/1.63 ( sepSubset
% 2.94/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.94/1.63 ( subset
% 2.94/1.63 @ ( dsetconstr @ A
% 2.94/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.94/1.63 @ A ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binunion_type,type,
% 2.94/1.63 binunion: $i > $i > $i ).
% 2.94/1.63
% 2.94/1.63 thf(binunionIL_type,type,
% 2.94/1.63 binunionIL: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binunionIL,definition,
% 2.94/1.63 ( binunionIL
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(upairset2IR_type,type,
% 2.94/1.63 upairset2IR: $o ).
% 2.94/1.63
% 2.94/1.63 thf(upairset2IR,definition,
% 2.94/1.63 ( upairset2IR
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binunionIR_type,type,
% 2.94/1.63 binunionIR: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binunionIR,definition,
% 2.94/1.63 ( binunionIR
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ B )
% 2.94/1.63 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binunionEcases_type,type,
% 2.94/1.63 binunionEcases: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binunionEcases,definition,
% 2.94/1.63 ( binunionEcases
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.94/1.63 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.94/1.63 => ( ( ( in @ Xx @ A )
% 2.94/1.63 => Xphi )
% 2.94/1.63 => ( ( ( in @ Xx @ B )
% 2.94/1.63 => Xphi )
% 2.94/1.63 => Xphi ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binunionE_type,type,
% 2.94/1.63 binunionE: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binunionE,definition,
% 2.94/1.63 ( binunionE
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.94/1.63 => ( ( in @ Xx @ A )
% 2.94/1.63 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binunionLsub_type,type,
% 2.94/1.63 binunionLsub: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binunionLsub,definition,
% 2.94/1.63 ( binunionLsub
% 2.94/1.63 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binunionRsub_type,type,
% 2.94/1.63 binunionRsub: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binunionRsub,definition,
% 2.94/1.63 ( binunionRsub
% 2.94/1.63 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binintersect_type,type,
% 2.94/1.63 binintersect: $i > $i > $i ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectI_type,type,
% 2.94/1.63 binintersectI: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectI,definition,
% 2.94/1.63 ( binintersectI
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( ( in @ Xx @ B )
% 2.94/1.63 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectSubset5_type,type,
% 2.94/1.63 binintersectSubset5: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectSubset5,definition,
% 2.94/1.63 ( binintersectSubset5
% 2.94/1.63 = ( ! [A: $i,B: $i,C: $i] :
% 2.94/1.63 ( ( subset @ C @ A )
% 2.94/1.63 => ( ( subset @ C @ B )
% 2.94/1.63 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectEL_type,type,
% 2.94/1.63 binintersectEL: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectEL,definition,
% 2.94/1.63 ( binintersectEL
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.94/1.63 => ( in @ Xx @ A ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectLsub_type,type,
% 2.94/1.63 binintersectLsub: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectLsub,definition,
% 2.94/1.63 ( binintersectLsub
% 2.94/1.63 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectSubset2_type,type,
% 2.94/1.63 binintersectSubset2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectSubset2,definition,
% 2.94/1.63 ( binintersectSubset2
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ( subset @ A @ B )
% 2.94/1.63 => ( ( binintersect @ A @ B )
% 2.94/1.63 = A ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectSubset3_type,type,
% 2.94/1.63 binintersectSubset3: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectSubset3,definition,
% 2.94/1.63 ( binintersectSubset3
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ( ( binintersect @ A @ B )
% 2.94/1.63 = B )
% 2.94/1.63 => ( subset @ B @ A ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectER_type,type,
% 2.94/1.63 binintersectER: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectER,definition,
% 2.94/1.63 ( binintersectER
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.94/1.63 => ( in @ Xx @ B ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(disjointsetsI1_type,type,
% 2.94/1.63 disjointsetsI1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(disjointsetsI1,definition,
% 2.94/1.63 ( disjointsetsI1
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ~ ? [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 & ( in @ Xx @ B ) )
% 2.94/1.63 => ( ( binintersect @ A @ B )
% 2.94/1.63 = emptyset ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectRsub_type,type,
% 2.94/1.63 binintersectRsub: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectRsub,definition,
% 2.94/1.63 ( binintersectRsub
% 2.94/1.63 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectSubset4_type,type,
% 2.94/1.63 binintersectSubset4: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectSubset4,definition,
% 2.94/1.63 ( binintersectSubset4
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ( subset @ B @ A )
% 2.94/1.63 => ( ( binintersect @ A @ B )
% 2.94/1.63 = B ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectSubset1_type,type,
% 2.94/1.63 binintersectSubset1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectSubset1,definition,
% 2.94/1.63 ( binintersectSubset1
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ( ( binintersect @ A @ B )
% 2.94/1.63 = A )
% 2.94/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(bs114d_type,type,
% 2.94/1.63 bs114d: $o ).
% 2.94/1.63
% 2.94/1.63 thf(bs114d,definition,
% 2.94/1.63 ( bs114d
% 2.94/1.63 = ( ! [A: $i,B: $i,C: $i] :
% 2.94/1.63 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.94/1.63 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(regular_type,type,
% 2.94/1.63 regular: $i > $o ).
% 2.94/1.63
% 2.94/1.63 thf(setminus_type,type,
% 2.94/1.63 setminus: $i > $i > $i ).
% 2.94/1.63
% 2.94/1.63 thf(setminusI_type,type,
% 2.94/1.63 setminusI: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setminusI,definition,
% 2.94/1.63 ( setminusI
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( ~ ( in @ Xx @ B )
% 2.94/1.63 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setminusEL_type,type,
% 2.94/1.63 setminusEL: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setminusEL,definition,
% 2.94/1.63 ( setminusEL
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.94/1.63 => ( in @ Xx @ A ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setminusER_type,type,
% 2.94/1.63 setminusER: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setminusER,definition,
% 2.94/1.63 ( setminusER
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.94/1.63 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setminusSubset2_type,type,
% 2.94/1.63 setminusSubset2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setminusSubset2,definition,
% 2.94/1.63 ( setminusSubset2
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ( subset @ A @ B )
% 2.94/1.63 => ( ( setminus @ A @ B )
% 2.94/1.63 = emptyset ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setminusERneg_type,type,
% 2.94/1.63 setminusERneg: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setminusERneg,definition,
% 2.94/1.63 ( setminusERneg
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.94/1.63 => ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setminusELneg_type,type,
% 2.94/1.63 setminusELneg: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setminusELneg,definition,
% 2.94/1.63 ( setminusELneg
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.94/1.63 => ( ~ ( in @ Xx @ B )
% 2.94/1.63 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setminusILneg_type,type,
% 2.94/1.63 setminusILneg: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setminusILneg,definition,
% 2.94/1.63 ( setminusILneg
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ~ ( in @ Xx @ A )
% 2.94/1.63 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setminusIRneg_type,type,
% 2.94/1.63 setminusIRneg: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setminusIRneg,definition,
% 2.94/1.63 ( setminusIRneg
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ B )
% 2.94/1.63 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setminusLsub_type,type,
% 2.94/1.63 setminusLsub: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setminusLsub,definition,
% 2.94/1.63 ( setminusLsub
% 2.94/1.63 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setminusSubset1_type,type,
% 2.94/1.63 setminusSubset1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setminusSubset1,definition,
% 2.94/1.63 ( setminusSubset1
% 2.94/1.63 = ( ! [A: $i,B: $i] :
% 2.94/1.63 ( ( ( setminus @ A @ B )
% 2.94/1.63 = emptyset )
% 2.94/1.63 => ( subset @ A @ B ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(symdiff_type,type,
% 2.94/1.63 symdiff: $i > $i > $i ).
% 2.94/1.63
% 2.94/1.63 thf(symdiffE_type,type,
% 2.94/1.63 symdiffE: $o ).
% 2.94/1.63
% 2.94/1.63 thf(symdiffE,definition,
% 2.94/1.63 ( symdiffE
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.94/1.63 => ! [Xphi: $o] :
% 2.94/1.63 ( ( ( in @ Xx @ A )
% 2.94/1.63 => ( ~ ( in @ Xx @ B )
% 2.94/1.63 => Xphi ) )
% 2.94/1.63 => ( ( ~ ( in @ Xx @ A )
% 2.94/1.63 => ( ( in @ Xx @ B )
% 2.94/1.63 => Xphi ) )
% 2.94/1.63 => Xphi ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(symdiffI1_type,type,
% 2.94/1.63 symdiffI1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(symdiffI1,definition,
% 2.94/1.63 ( symdiffI1
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( ~ ( in @ Xx @ B )
% 2.94/1.63 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(symdiffI2_type,type,
% 2.94/1.63 symdiffI2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(symdiffI2,definition,
% 2.94/1.63 ( symdiffI2
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ~ ( in @ Xx @ A )
% 2.94/1.63 => ( ( in @ Xx @ B )
% 2.94/1.63 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(symdiffIneg1_type,type,
% 2.94/1.63 symdiffIneg1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(symdiffIneg1,definition,
% 2.94/1.63 ( symdiffIneg1
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( ( in @ Xx @ B )
% 2.94/1.63 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(symdiffIneg2_type,type,
% 2.94/1.63 symdiffIneg2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(symdiffIneg2,definition,
% 2.94/1.63 ( symdiffIneg2
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ~ ( in @ Xx @ A )
% 2.94/1.63 => ( ~ ( in @ Xx @ B )
% 2.94/1.63 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(iskpair_type,type,
% 2.94/1.63 iskpair: $i > $o ).
% 2.94/1.63
% 2.94/1.63 thf(secondinupair_type,type,
% 2.94/1.63 secondinupair: $o ).
% 2.94/1.63
% 2.94/1.63 thf(secondinupair,definition,
% 2.94/1.63 ( secondinupair
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairIL_type,type,
% 2.94/1.63 setukpairIL: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairIL,definition,
% 2.94/1.63 ( setukpairIL
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairIR_type,type,
% 2.94/1.63 setukpairIR: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairIR,definition,
% 2.94/1.63 ( setukpairIR
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(kpairiskpair_type,type,
% 2.94/1.63 kpairiskpair: $o ).
% 2.94/1.63
% 2.94/1.63 thf(kpairiskpair,definition,
% 2.94/1.63 ( kpairiskpair
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(kpair_type,type,
% 2.94/1.63 kpair: $i > $i > $i ).
% 2.94/1.63
% 2.94/1.63 thf(kpairp_type,type,
% 2.94/1.63 kpairp: $o ).
% 2.94/1.63
% 2.94/1.63 thf(kpairp,definition,
% 2.94/1.63 ( kpairp
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(cartprod_type,type,
% 2.94/1.63 cartprod: $i > $i > $i ).
% 2.94/1.63
% 2.94/1.63 thf(singletonsubset_type,type,
% 2.94/1.63 singletonsubset: $o ).
% 2.94/1.63
% 2.94/1.63 thf(singletonsubset,definition,
% 2.94/1.63 ( singletonsubset
% 2.94/1.63 = ( ! [A: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(singletoninpowerset_type,type,
% 2.94/1.63 singletoninpowerset: $o ).
% 2.94/1.63
% 2.94/1.63 thf(singletoninpowerset,definition,
% 2.94/1.63 ( singletoninpowerset
% 2.94/1.63 = ( ! [A: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(singletoninpowunion_type,type,
% 2.94/1.63 singletoninpowunion: $o ).
% 2.94/1.63
% 2.94/1.63 thf(singletoninpowunion,definition,
% 2.94/1.63 ( singletoninpowunion
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(upairset2E_type,type,
% 2.94/1.63 upairset2E: $o ).
% 2.94/1.63
% 2.94/1.63 thf(upairset2E,definition,
% 2.94/1.63 ( upairset2E
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.94/1.63 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.94/1.63 => ( ( Xz = Xx )
% 2.94/1.63 | ( Xz = Xy ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(upairsubunion_type,type,
% 2.94/1.63 upairsubunion: $o ).
% 2.94/1.63
% 2.94/1.63 thf(upairsubunion,definition,
% 2.94/1.63 ( upairsubunion
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(upairinpowunion_type,type,
% 2.94/1.63 upairinpowunion: $o ).
% 2.94/1.63
% 2.94/1.63 thf(upairinpowunion,definition,
% 2.94/1.63 ( upairinpowunion
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(ubforcartprodlem1_type,type,
% 2.94/1.63 ubforcartprodlem1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(ubforcartprodlem1,definition,
% 2.94/1.63 ( ubforcartprodlem1
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(ubforcartprodlem2_type,type,
% 2.94/1.63 ubforcartprodlem2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(ubforcartprodlem2,definition,
% 2.94/1.63 ( ubforcartprodlem2
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(ubforcartprodlem3_type,type,
% 2.94/1.63 ubforcartprodlem3: $o ).
% 2.94/1.63
% 2.94/1.63 thf(ubforcartprodlem3,definition,
% 2.94/1.63 ( ubforcartprodlem3
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodpairin_type,type,
% 2.94/1.63 cartprodpairin: $o ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodpairin,definition,
% 2.94/1.63 ( cartprodpairin
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodmempair1_type,type,
% 2.94/1.63 cartprodmempair1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodmempair1,definition,
% 2.94/1.63 ( cartprodmempair1
% 2.94/1.63 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.94/1.63 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.94/1.63 => ? [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 & ? [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 & ( Xu
% 2.94/1.63 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodmempair_type,type,
% 2.94/1.63 cartprodmempair: $o ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodmempair,definition,
% 2.94/1.63 ( cartprodmempair
% 2.94/1.63 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.94/1.63 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.94/1.63 => ( iskpair @ Xu ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setunionE2_type,type,
% 2.94/1.63 setunionE2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setunionE2,definition,
% 2.94/1.63 ( setunionE2
% 2.94/1.63 = ( ! [A: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ ( setunion @ A ) )
% 2.94/1.63 => ? [X: $i] :
% 2.94/1.63 ( ( in @ X @ A )
% 2.94/1.63 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setunionsingleton1_type,type,
% 2.94/1.63 setunionsingleton1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setunionsingleton1,definition,
% 2.94/1.63 ( setunionsingleton1
% 2.94/1.63 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setunionsingleton2_type,type,
% 2.94/1.63 setunionsingleton2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setunionsingleton2,definition,
% 2.94/1.63 ( setunionsingleton2
% 2.94/1.63 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setunionsingleton_type,type,
% 2.94/1.63 setunionsingleton: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setunionsingleton,definition,
% 2.94/1.63 ( setunionsingleton
% 2.94/1.63 = ( ! [Xx: $i] :
% 2.94/1.63 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.94/1.63 = Xx ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(singleton_type,type,
% 2.94/1.63 singleton: $i > $o ).
% 2.94/1.63
% 2.94/1.63 thf(singletonprop_type,type,
% 2.94/1.63 singletonprop: $o ).
% 2.94/1.63
% 2.94/1.63 thf(singletonprop,definition,
% 2.94/1.63 ( singletonprop
% 2.94/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.94/1.63 ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ A )
% 2.94/1.63 => ( ( Xphi @ Xx )
% 2.94/1.63 => ( ( Xphi @ Xy )
% 2.94/1.63 => ( Xx = Xy ) ) ) ) )
% 2.94/1.63 => ( ? [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 & ( Xphi @ Xx ) )
% 2.94/1.63 => ( singleton
% 2.94/1.63 @ ( dsetconstr @ A
% 2.94/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(ex1_type,type,
% 2.94/1.63 ex1: $i > ( $i > $o ) > $o ).
% 2.94/1.63
% 2.94/1.63 thf(ex1,definition,
% 2.94/1.63 ( ex1
% 2.94/1.63 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.94/1.63 ( singleton
% 2.94/1.63 @ ( dsetconstr @ A
% 2.94/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(ex1E1_type,type,
% 2.94/1.63 ex1E1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(ex1E1,definition,
% 2.94/1.63 ( ex1E1
% 2.94/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.94/1.63 ( ( ex1 @ A
% 2.94/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.94/1.63 => ? [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(ex1I_type,type,
% 2.94/1.63 ex1I: $o ).
% 2.94/1.63
% 2.94/1.63 thf(ex1I,definition,
% 2.94/1.63 ( ex1I
% 2.94/1.63 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( ( Xphi @ Xx )
% 2.94/1.63 => ( ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ A )
% 2.94/1.63 => ( ( Xphi @ Xy )
% 2.94/1.63 => ( Xy = Xx ) ) )
% 2.94/1.63 => ( ex1 @ A
% 2.94/1.63 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(ex1I2_type,type,
% 2.94/1.63 ex1I2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(ex1I2,definition,
% 2.94/1.63 ( ex1I2
% 2.94/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.94/1.63 ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ A )
% 2.94/1.63 => ( ( Xphi @ Xx )
% 2.94/1.63 => ( ( Xphi @ Xy )
% 2.94/1.63 => ( Xx = Xy ) ) ) ) )
% 2.94/1.63 => ( ? [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 & ( Xphi @ Xx ) )
% 2.94/1.63 => ( ex1 @ A
% 2.94/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(singletonsuniq_type,type,
% 2.94/1.63 singletonsuniq: $o ).
% 2.94/1.63
% 2.94/1.63 thf(singletonsuniq,definition,
% 2.94/1.63 ( singletonsuniq
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.94/1.63 ( ( ( setadjoin @ Xx @ emptyset )
% 2.94/1.63 = ( setadjoin @ Xy @ emptyset ) )
% 2.94/1.63 => ( Xx = Xy ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(atmost1p_type,type,
% 2.94/1.63 atmost1p: $i > $o ).
% 2.94/1.63
% 2.94/1.63 thf(atleast2p_type,type,
% 2.94/1.63 atleast2p: $i > $o ).
% 2.94/1.63
% 2.94/1.63 thf(atmost2p_type,type,
% 2.94/1.63 atmost2p: $i > $o ).
% 2.94/1.63
% 2.94/1.63 thf(upairsetp_type,type,
% 2.94/1.63 upairsetp: $i > $o ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairinjL1_type,type,
% 2.94/1.63 setukpairinjL1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairinjL1,definition,
% 2.94/1.63 ( setukpairinjL1
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.94/1.63 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.94/1.63 => ( Xx = Xz ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(kfstsingleton_type,type,
% 2.94/1.63 kfstsingleton: $o ).
% 2.94/1.63
% 2.94/1.63 thf(kfstsingleton,definition,
% 2.94/1.63 ( kfstsingleton
% 2.94/1.63 = ( ! [Xu: $i] :
% 2.94/1.63 ( ( iskpair @ Xu )
% 2.94/1.63 => ( singleton
% 2.94/1.63 @ ( dsetconstr @ ( setunion @ Xu )
% 2.94/1.63 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(theprop_type,type,
% 2.94/1.63 theprop: $o ).
% 2.94/1.63
% 2.94/1.63 thf(theprop,definition,
% 2.94/1.63 ( theprop
% 2.94/1.63 = ( ! [X: $i] :
% 2.94/1.63 ( ( singleton @ X )
% 2.94/1.63 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(kfst_type,type,
% 2.94/1.63 kfst: $i > $i ).
% 2.94/1.63
% 2.94/1.63 thf(kfstpairEq_type,type,
% 2.94/1.63 kfstpairEq: $o ).
% 2.94/1.63
% 2.94/1.63 thf(kfstpairEq,definition,
% 2.94/1.63 ( kfstpairEq
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.94/1.63 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.94/1.63 = Xx ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodfstin_type,type,
% 2.94/1.63 cartprodfstin: $o ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodfstin,definition,
% 2.94/1.63 ( cartprodfstin
% 2.94/1.63 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.94/1.63 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.94/1.63 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairinjL2_type,type,
% 2.94/1.63 setukpairinjL2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairinjL2,definition,
% 2.94/1.63 ( setukpairinjL2
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.94/1.63 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.94/1.63 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.94/1.63 => ( Xx = Xz ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairinjL_type,type,
% 2.94/1.63 setukpairinjL: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairinjL,definition,
% 2.94/1.63 ( setukpairinjL
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.94/1.63 ( ( ( kpair @ Xx @ Xy )
% 2.94/1.63 = ( kpair @ Xz @ Xu ) )
% 2.94/1.63 => ( Xx = Xz ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairinjR11_type,type,
% 2.94/1.63 setukpairinjR11: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairinjR11,definition,
% 2.94/1.63 ( setukpairinjR11
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.94/1.63 ( ( Xx = Xy )
% 2.94/1.63 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.94/1.63 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairinjR12_type,type,
% 2.94/1.63 setukpairinjR12: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairinjR12,definition,
% 2.94/1.63 ( setukpairinjR12
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.94/1.63 ( ( Xx = Xy )
% 2.94/1.63 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.94/1.63 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairinjR1_type,type,
% 2.94/1.63 setukpairinjR1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairinjR1,definition,
% 2.94/1.63 ( setukpairinjR1
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.94/1.63 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.94/1.63 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.94/1.63 => ( ( Xz = Xu )
% 2.94/1.63 => ( Xy = Xu ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(upairequniteq_type,type,
% 2.94/1.63 upairequniteq: $o ).
% 2.94/1.63
% 2.94/1.63 thf(upairequniteq,definition,
% 2.94/1.63 ( upairequniteq
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.94/1.63 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.94/1.63 = ( setadjoin @ Xz @ emptyset ) )
% 2.94/1.63 => ( Xx = Xy ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairinjR2_type,type,
% 2.94/1.63 setukpairinjR2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairinjR2,definition,
% 2.94/1.63 ( setukpairinjR2
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.94/1.63 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.94/1.63 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.94/1.63 => ( Xy = Xu ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairinjR_type,type,
% 2.94/1.63 setukpairinjR: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setukpairinjR,definition,
% 2.94/1.63 ( setukpairinjR
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.94/1.63 ( ( ( kpair @ Xx @ Xy )
% 2.94/1.63 = ( kpair @ Xz @ Xu ) )
% 2.94/1.63 => ( Xy = Xu ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(ksndsingleton_type,type,
% 2.94/1.63 ksndsingleton: $o ).
% 2.94/1.63
% 2.94/1.63 thf(ksndsingleton,definition,
% 2.94/1.63 ( ksndsingleton
% 2.94/1.63 = ( ! [Xu: $i] :
% 2.94/1.63 ( ( iskpair @ Xu )
% 2.94/1.63 => ( singleton
% 2.94/1.63 @ ( dsetconstr @ ( setunion @ Xu )
% 2.94/1.63 @ ^ [Xx: $i] :
% 2.94/1.63 ( Xu
% 2.94/1.63 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(ksnd_type,type,
% 2.94/1.63 ksnd: $i > $i ).
% 2.94/1.63
% 2.94/1.63 thf(ksndpairEq_type,type,
% 2.94/1.63 ksndpairEq: $o ).
% 2.94/1.63
% 2.94/1.63 thf(ksndpairEq,definition,
% 2.94/1.63 ( ksndpairEq
% 2.94/1.63 = ( ! [Xx: $i,Xy: $i] :
% 2.94/1.63 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.94/1.63 = Xy ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(kpairsurjEq_type,type,
% 2.94/1.63 kpairsurjEq: $o ).
% 2.94/1.63
% 2.94/1.63 thf(kpairsurjEq,definition,
% 2.94/1.63 ( kpairsurjEq
% 2.94/1.63 = ( ! [Xu: $i] :
% 2.94/1.63 ( ( iskpair @ Xu )
% 2.94/1.63 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.94/1.63 = Xu ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodsndin_type,type,
% 2.94/1.63 cartprodsndin: $o ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodsndin,definition,
% 2.94/1.63 ( cartprodsndin
% 2.94/1.63 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.94/1.63 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.94/1.63 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodpairmemEL_type,type,
% 2.94/1.63 cartprodpairmemEL: $o ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodpairmemEL,definition,
% 2.94/1.63 ( cartprodpairmemEL
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.94/1.63 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.94/1.63 => ( in @ Xx @ A ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodpairmemER_type,type,
% 2.94/1.63 cartprodpairmemER: $o ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodpairmemER,definition,
% 2.94/1.63 ( cartprodpairmemER
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.94/1.63 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.94/1.63 => ( in @ Xy @ B ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodmempaircEq_type,type,
% 2.94/1.63 cartprodmempaircEq: $o ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodmempaircEq,definition,
% 2.94/1.63 ( cartprodmempaircEq
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( ( kpair @ Xx @ Xy )
% 2.94/1.63 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodfstpairEq_type,type,
% 2.94/1.63 cartprodfstpairEq: $o ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodfstpairEq,definition,
% 2.94/1.63 ( cartprodfstpairEq
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.94/1.63 = Xx ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodsndpairEq_type,type,
% 2.94/1.63 cartprodsndpairEq: $o ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodsndpairEq,definition,
% 2.94/1.63 ( cartprodsndpairEq
% 2.94/1.63 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.94/1.63 = Xy ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodpairsurjEq_type,type,
% 2.94/1.63 cartprodpairsurjEq: $o ).
% 2.94/1.63
% 2.94/1.63 thf(cartprodpairsurjEq,definition,
% 2.94/1.63 ( cartprodpairsurjEq
% 2.94/1.63 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.94/1.63 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.94/1.63 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.94/1.63 = Xu ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(breln_type,type,
% 2.94/1.63 breln: $i > $i > $i > $o ).
% 2.94/1.63
% 2.94/1.63 thf(breln,definition,
% 2.94/1.63 ( breln
% 2.94/1.63 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(dpsetconstr_type,type,
% 2.94/1.63 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.94/1.63
% 2.94/1.63 thf(dpsetconstrI_type,type,
% 2.94/1.63 dpsetconstrI: $o ).
% 2.94/1.63
% 2.94/1.63 thf(dpsetconstrI,definition,
% 2.94/1.63 ( dpsetconstrI
% 2.94/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( ( Xphi @ Xx @ Xy )
% 2.94/1.63 => ( in @ ( kpair @ Xx @ Xy )
% 2.94/1.63 @ ( dpsetconstr @ A @ B
% 2.94/1.63 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(dpsetconstrSub_type,type,
% 2.94/1.63 dpsetconstrSub: $o ).
% 2.94/1.63
% 2.94/1.63 thf(dpsetconstrSub,definition,
% 2.94/1.63 ( dpsetconstrSub
% 2.94/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.94/1.63 ( subset
% 2.94/1.63 @ ( dpsetconstr @ A @ B
% 2.94/1.63 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.94/1.63 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setOfPairsIsBReln_type,type,
% 2.94/1.63 setOfPairsIsBReln: $o ).
% 2.94/1.63
% 2.94/1.63 thf(setOfPairsIsBReln,definition,
% 2.94/1.63 ( setOfPairsIsBReln
% 2.94/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.94/1.63 ( breln @ A @ B
% 2.94/1.63 @ ( dpsetconstr @ A @ B
% 2.94/1.63 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(dpsetconstrERa_type,type,
% 2.94/1.63 dpsetconstrERa: $o ).
% 2.94/1.63
% 2.94/1.63 thf(dpsetconstrERa,definition,
% 2.94/1.63 ( dpsetconstrERa
% 2.94/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.94/1.63 @ ( dpsetconstr @ A @ B
% 2.94/1.63 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.94/1.63 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(dpsetconstrEL1_type,type,
% 2.94/1.63 dpsetconstrEL1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(dpsetconstrEL1,definition,
% 2.94/1.63 ( dpsetconstrEL1
% 2.94/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.94/1.63 ( ( in @ ( kpair @ Xx @ Xy )
% 2.94/1.63 @ ( dpsetconstr @ A @ B
% 2.94/1.63 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.94/1.63 => ( in @ Xx @ A ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(dpsetconstrEL2_type,type,
% 2.94/1.63 dpsetconstrEL2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(dpsetconstrEL2,definition,
% 2.94/1.63 ( dpsetconstrEL2
% 2.94/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.94/1.63 ( ( in @ ( kpair @ Xx @ Xy )
% 2.94/1.63 @ ( dpsetconstr @ A @ B
% 2.94/1.63 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.94/1.63 => ( in @ Xy @ B ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(dpsetconstrER_type,type,
% 2.94/1.63 dpsetconstrER: $o ).
% 2.94/1.63
% 2.94/1.63 thf(dpsetconstrER,definition,
% 2.94/1.63 ( dpsetconstrER
% 2.94/1.63 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.94/1.63 ( ( in @ ( kpair @ Xx @ Xy )
% 2.94/1.63 @ ( dpsetconstr @ A @ B
% 2.94/1.63 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.94/1.63 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(func_type,type,
% 2.94/1.63 func: $i > $i > $i > $o ).
% 2.94/1.63
% 2.94/1.63 thf(funcSet_type,type,
% 2.94/1.63 funcSet: $i > $i > $i ).
% 2.94/1.63
% 2.94/1.63 thf(funcImageSingleton_type,type,
% 2.94/1.63 funcImageSingleton: $o ).
% 2.94/1.63
% 2.94/1.63 thf(funcImageSingleton,definition,
% 2.94/1.63 ( funcImageSingleton
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( func @ A @ B @ Xf )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( singleton
% 2.94/1.63 @ ( dsetconstr @ B
% 2.94/1.63 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(apProp_type,type,
% 2.94/1.63 apProp: $o ).
% 2.94/1.63
% 2.94/1.63 thf(apProp,definition,
% 2.94/1.63 ( apProp
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( func @ A @ B @ Xf )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( in
% 2.94/1.63 @ ( setunion
% 2.94/1.63 @ ( dsetconstr @ B
% 2.94/1.63 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.94/1.63 @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(ap_type,type,
% 2.94/1.63 ap: $i > $i > $i > $i > $i ).
% 2.94/1.63
% 2.94/1.63 thf(app_type,type,
% 2.94/1.63 app: $o ).
% 2.94/1.63
% 2.94/1.63 thf(app,definition,
% 2.94/1.63 ( app
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( func @ A @ B @ Xf )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(infuncsetfunc_type,type,
% 2.94/1.63 infuncsetfunc: $o ).
% 2.94/1.63
% 2.94/1.63 thf(infuncsetfunc,definition,
% 2.94/1.63 ( infuncsetfunc
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.94/1.63 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(ap2p_type,type,
% 2.94/1.63 ap2p: $o ).
% 2.94/1.63
% 2.94/1.63 thf(ap2p,definition,
% 2.94/1.63 ( ap2p
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(funcinfuncset_type,type,
% 2.94/1.63 funcinfuncset: $o ).
% 2.94/1.63
% 2.94/1.63 thf(funcinfuncset,definition,
% 2.94/1.63 ( funcinfuncset
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( func @ A @ B @ Xf )
% 2.94/1.63 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(lamProp_type,type,
% 2.94/1.63 lamProp: $o ).
% 2.94/1.63
% 2.94/1.63 thf(lamProp,definition,
% 2.94/1.63 ( lamProp
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.94/1.63 ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.94/1.63 => ( func @ A @ B
% 2.94/1.63 @ ( dpsetconstr @ A @ B
% 2.94/1.63 @ ^ [Xx: $i,Xy: $i] :
% 2.94/1.63 ( ( Xf @ Xx )
% 2.94/1.63 = Xy ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(lam_type,type,
% 2.94/1.63 lam: $i > $i > ( $i > $i ) > $i ).
% 2.94/1.63
% 2.94/1.63 thf(lamp_type,type,
% 2.94/1.63 lamp: $o ).
% 2.94/1.63
% 2.94/1.63 thf(lamp,definition,
% 2.94/1.63 ( lamp
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.94/1.63 ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.94/1.63 => ( func @ A @ B
% 2.94/1.63 @ ( lam @ A @ B
% 2.94/1.63 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(lam2p_type,type,
% 2.94/1.63 lam2p: $o ).
% 2.94/1.63
% 2.94/1.63 thf(lam2p,definition,
% 2.94/1.63 ( lam2p
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.94/1.63 ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.94/1.63 => ( in
% 2.94/1.63 @ ( lam @ A @ B
% 2.94/1.63 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.94/1.63 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(brelnall1_type,type,
% 2.94/1.63 brelnall1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(brelnall1,definition,
% 2.94/1.63 ( brelnall1
% 2.94/1.63 = ( ! [A: $i,B: $i,R: $i] :
% 2.94/1.63 ( ( breln @ A @ B @ R )
% 2.94/1.63 => ! [Xphi: $i > $o] :
% 2.94/1.63 ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.94/1.63 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ R )
% 2.94/1.63 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(brelnall2_type,type,
% 2.94/1.63 brelnall2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(brelnall2,definition,
% 2.94/1.63 ( brelnall2
% 2.94/1.63 = ( ! [A: $i,B: $i,R: $i] :
% 2.94/1.63 ( ( breln @ A @ B @ R )
% 2.94/1.63 => ! [Xphi: $i > $o] :
% 2.94/1.63 ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.94/1.63 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ R )
% 2.94/1.63 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(ex1E2_type,type,
% 2.94/1.63 ex1E2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(ex1E2,definition,
% 2.94/1.63 ( ex1E2
% 2.94/1.63 = ( ! [A: $i,Xphi: $i > $o] :
% 2.94/1.63 ( ( ex1 @ A
% 2.94/1.63 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ A )
% 2.94/1.63 => ( ( Xphi @ Xx )
% 2.94/1.63 => ( ( Xphi @ Xy )
% 2.94/1.63 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(funcGraphProp1_type,type,
% 2.94/1.63 funcGraphProp1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(funcGraphProp1,definition,
% 2.94/1.63 ( funcGraphProp1
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( func @ A @ B @ Xf )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(funcGraphProp3_type,type,
% 2.94/1.63 funcGraphProp3: $o ).
% 2.94/1.63
% 2.94/1.63 thf(funcGraphProp3,definition,
% 2.94/1.63 ( funcGraphProp3
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(funcGraphProp2_type,type,
% 2.94/1.63 funcGraphProp2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(funcGraphProp2,definition,
% 2.94/1.63 ( funcGraphProp2
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( func @ A @ B @ Xf )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.94/1.63 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.94/1.63 = Xy ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(funcextLem_type,type,
% 2.94/1.63 funcextLem: $o ).
% 2.94/1.63
% 2.94/1.63 thf(funcextLem,definition,
% 2.94/1.63 ( funcextLem
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( func @ A @ B @ Xf )
% 2.94/1.63 => ! [Xg: $i] :
% 2.94/1.63 ( ( func @ A @ B @ Xg )
% 2.94/1.63 => ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.94/1.63 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.94/1.63 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(funcGraphProp4_type,type,
% 2.94/1.63 funcGraphProp4: $o ).
% 2.94/1.63
% 2.94/1.63 thf(funcGraphProp4,definition,
% 2.94/1.63 ( funcGraphProp4
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.94/1.63 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.94/1.63 = Xy ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(subbreln_type,type,
% 2.94/1.63 subbreln: $o ).
% 2.94/1.63
% 2.94/1.63 thf(subbreln,definition,
% 2.94/1.63 ( subbreln
% 2.94/1.63 = ( ! [A: $i,B: $i,R: $i] :
% 2.94/1.63 ( ( breln @ A @ B @ R )
% 2.94/1.63 => ! [S: $i] :
% 2.94/1.63 ( ( breln @ A @ B @ S )
% 2.94/1.63 => ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.94/1.63 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.94/1.63 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(eqbreln_type,type,
% 2.94/1.63 eqbreln: $o ).
% 2.94/1.63
% 2.94/1.63 thf(eqbreln,definition,
% 2.94/1.63 ( eqbreln
% 2.94/1.63 = ( ! [A: $i,B: $i,R: $i] :
% 2.94/1.63 ( ( breln @ A @ B @ R )
% 2.94/1.63 => ! [S: $i] :
% 2.94/1.63 ( ( breln @ A @ B @ S )
% 2.94/1.63 => ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.94/1.63 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.94/1.63 => ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ B )
% 2.94/1.63 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.94/1.63 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.94/1.63 => ( R = S ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(funcext_type,type,
% 2.94/1.63 funcext: $o ).
% 2.94/1.63
% 2.94/1.63 thf(funcext,definition,
% 2.94/1.63 ( funcext
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( func @ A @ B @ Xf )
% 2.94/1.63 => ! [Xg: $i] :
% 2.94/1.63 ( ( func @ A @ B @ Xg )
% 2.94/1.63 => ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.94/1.63 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.94/1.63 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(funcext2_type,type,
% 2.94/1.63 funcext2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(funcext2,definition,
% 2.94/1.63 ( funcext2
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.94/1.63 => ! [Xg: $i] :
% 2.94/1.63 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.94/1.63 => ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.94/1.63 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.94/1.63 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(ap2apEq1_type,type,
% 2.94/1.63 ap2apEq1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(ap2apEq1,definition,
% 2.94/1.63 ( ap2apEq1
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.94/1.63 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(ap2apEq2_type,type,
% 2.94/1.63 ap2apEq2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(ap2apEq2,definition,
% 2.94/1.63 ( ap2apEq2
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( func @ A @ B @ Xf )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.94/1.63 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(beta1_type,type,
% 2.94/1.63 beta1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(beta1,definition,
% 2.94/1.63 ( beta1
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.94/1.63 ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( ( ap @ A @ B
% 2.94/1.63 @ ( lam @ A @ B
% 2.94/1.63 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.94/1.63 @ Xx )
% 2.94/1.63 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(eta1_type,type,
% 2.94/1.63 eta1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(eta1,definition,
% 2.94/1.63 ( eta1
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( func @ A @ B @ Xf )
% 2.94/1.63 => ( ( lam @ A @ B
% 2.94/1.63 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.94/1.63 = Xf ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(lam2lamEq_type,type,
% 2.94/1.63 lam2lamEq: $o ).
% 2.94/1.63
% 2.94/1.63 thf(lam2lamEq,definition,
% 2.94/1.63 ( lam2lamEq
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.94/1.63 ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.94/1.63 => ( ( lam @ A @ B
% 2.94/1.63 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.94/1.63 = ( lam @ A @ B
% 2.94/1.63 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(beta2_type,type,
% 2.94/1.63 beta2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(beta2,definition,
% 2.94/1.63 ( beta2
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.94/1.63 ( ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ( ( ap @ A @ B
% 2.94/1.63 @ ( lam @ A @ B
% 2.94/1.63 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.94/1.63 @ Xx )
% 2.94/1.63 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(eta2_type,type,
% 2.94/1.63 eta2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(eta2,definition,
% 2.94/1.63 ( eta2
% 2.94/1.63 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.94/1.63 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.94/1.63 => ( ( lam @ A @ B
% 2.94/1.63 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.94/1.63 = Xf ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(iffalseProp1_type,type,
% 2.94/1.63 iffalseProp1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(iffalseProp1,definition,
% 2.94/1.63 ( iffalseProp1
% 2.94/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ A )
% 2.94/1.63 => ( ~ Xphi
% 2.94/1.63 => ( in @ Xy
% 2.94/1.63 @ ( dsetconstr @ A
% 2.94/1.63 @ ^ [Xz: $i] :
% 2.94/1.63 ( ( Xphi
% 2.94/1.63 & ( Xz = Xx ) )
% 2.94/1.63 | ( ~ Xphi
% 2.94/1.63 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(iffalseProp2_type,type,
% 2.94/1.63 iffalseProp2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(iffalseProp2,definition,
% 2.94/1.63 ( iffalseProp2
% 2.94/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ A )
% 2.94/1.63 => ( ~ Xphi
% 2.94/1.63 => ( ( dsetconstr @ A
% 2.94/1.63 @ ^ [Xz: $i] :
% 2.94/1.63 ( ( Xphi
% 2.94/1.63 & ( Xz = Xx ) )
% 2.94/1.63 | ( ~ Xphi
% 2.94/1.63 & ( Xz = Xy ) ) ) )
% 2.94/1.63 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(iftrueProp1_type,type,
% 2.94/1.63 iftrueProp1: $o ).
% 2.94/1.63
% 2.94/1.63 thf(iftrueProp1,definition,
% 2.94/1.63 ( iftrueProp1
% 2.94/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ A )
% 2.94/1.63 => ( Xphi
% 2.94/1.63 => ( in @ Xx
% 2.94/1.63 @ ( dsetconstr @ A
% 2.94/1.63 @ ^ [Xz: $i] :
% 2.94/1.63 ( ( Xphi
% 2.94/1.63 & ( Xz = Xx ) )
% 2.94/1.63 | ( ~ Xphi
% 2.94/1.63 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(iftrueProp2_type,type,
% 2.94/1.63 iftrueProp2: $o ).
% 2.94/1.63
% 2.94/1.63 thf(iftrueProp2,definition,
% 2.94/1.63 ( iftrueProp2
% 2.94/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ A )
% 2.94/1.63 => ( Xphi
% 2.94/1.63 => ( ( dsetconstr @ A
% 2.94/1.63 @ ^ [Xz: $i] :
% 2.94/1.63 ( ( Xphi
% 2.94/1.63 & ( Xz = Xx ) )
% 2.94/1.63 | ( ~ Xphi
% 2.94/1.63 & ( Xz = Xy ) ) ) )
% 2.94/1.63 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(ifSingleton_type,type,
% 2.94/1.63 ifSingleton: $o ).
% 2.94/1.63
% 2.94/1.63 thf(ifSingleton,definition,
% 2.94/1.63 ( ifSingleton
% 2.94/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ A )
% 2.94/1.63 => ( singleton
% 2.94/1.63 @ ( dsetconstr @ A
% 2.94/1.63 @ ^ [Xz: $i] :
% 2.94/1.63 ( ( Xphi
% 2.94/1.63 & ( Xz = Xx ) )
% 2.94/1.63 | ( ~ Xphi
% 2.94/1.63 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(if_type,type,
% 2.94/1.63 if: $i > $o > $i > $i > $i ).
% 2.94/1.63
% 2.94/1.63 thf(ifp_type,type,
% 2.94/1.63 ifp: $o ).
% 2.94/1.63
% 2.94/1.63 thf(ifp,definition,
% 2.94/1.63 ( ifp
% 2.94/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ A )
% 2.94/1.63 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(theeq_type,type,
% 2.94/1.63 theeq: $o ).
% 2.94/1.63
% 2.94/1.63 thf(theeq,definition,
% 2.94/1.63 ( theeq
% 2.94/1.63 = ( ! [X: $i] :
% 2.94/1.63 ( ( singleton @ X )
% 2.94/1.63 => ! [Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ X )
% 2.94/1.63 => ( ( setunion @ X )
% 2.94/1.63 = Xx ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(iftrue_type,type,
% 2.94/1.63 iftrue: $o ).
% 2.94/1.63
% 2.94/1.63 thf(iftrue,definition,
% 2.94/1.63 ( iftrue
% 2.94/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ A )
% 2.94/1.63 => ( Xphi
% 2.94/1.63 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.94/1.63 = Xx ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(iffalse_type,type,
% 2.94/1.63 iffalse: $o ).
% 2.94/1.63
% 2.94/1.63 thf(iffalse,definition,
% 2.94/1.63 ( iffalse
% 2.94/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ A )
% 2.94/1.63 => ( ~ Xphi
% 2.94/1.63 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.94/1.63 = Xy ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(iftrueorfalse_type,type,
% 2.94/1.63 iftrueorfalse: $o ).
% 2.94/1.63
% 2.94/1.63 thf(iftrueorfalse,definition,
% 2.94/1.63 ( iftrueorfalse
% 2.94/1.63 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.94/1.63 ( ( in @ Xx @ A )
% 2.94/1.63 => ! [Xy: $i] :
% 2.94/1.63 ( ( in @ Xy @ A )
% 2.94/1.63 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectT_lem_type,type,
% 2.94/1.63 binintersectT_lem: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binintersectT_lem,definition,
% 2.94/1.63 ( binintersectT_lem
% 2.94/1.63 = ( ! [A: $i,X: $i] :
% 2.94/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.94/1.63 => ! [Y: $i] :
% 2.94/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.94/1.63 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(binunionT_lem_type,type,
% 2.94/1.63 binunionT_lem: $o ).
% 2.94/1.63
% 2.94/1.63 thf(binunionT_lem,definition,
% 2.94/1.63 ( binunionT_lem
% 2.94/1.63 = ( ! [A: $i,X: $i] :
% 2.94/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.94/1.63 => ! [Y: $i] :
% 2.94/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.94/1.63 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(powersetT_lem_type,type,
% 2.94/1.63 powersetT_lem: $o ).
% 2.94/1.63
% 2.94/1.63 thf(powersetT_lem,definition,
% 2.94/1.63 ( powersetT_lem
% 2.94/1.63 = ( ! [A: $i,X: $i] :
% 2.94/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.94/1.63 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.94/1.63
% 2.94/1.63 thf(setminusT_lem,conjecture,
% 2.94/1.63 ( setextAx
% 2.94/1.63 => ( emptysetAx
% 2.94/1.63 => ( setadjoinAx
% 2.94/1.63 => ( powersetAx
% 2.94/1.63 => ( setunionAx
% 2.94/1.63 => ( omega0Ax
% 2.94/1.63 => ( omegaSAx
% 2.94/1.63 => ( omegaIndAx
% 2.94/1.63 => ( replAx
% 2.94/1.63 => ( foundationAx
% 2.94/1.63 => ( wellorderingAx
% 2.94/1.63 => ( descrp
% 2.94/1.63 => ( dsetconstrI
% 2.94/1.63 => ( dsetconstrEL
% 2.94/1.63 => ( dsetconstrER
% 2.94/1.63 => ( exuE1
% 2.94/1.63 => ( prop2setE
% 2.94/1.63 => ( emptysetE
% 2.94/1.63 => ( emptysetimpfalse
% 2.94/1.63 => ( notinemptyset
% 2.94/1.63 => ( exuE3e
% 2.94/1.63 => ( setext
% 2.94/1.63 => ( emptyI
% 2.94/1.63 => ( noeltsimpempty
% 2.94/1.63 => ( setbeta
% 2.94/1.63 => ( nonemptyE1
% 2.94/1.63 => ( nonemptyI
% 2.94/1.63 => ( nonemptyI1
% 2.94/1.63 => ( setadjoinIL
% 2.94/1.63 => ( emptyinunitempty
% 2.94/1.63 => ( setadjoinIR
% 2.94/1.63 => ( setadjoinE
% 2.94/1.63 => ( setadjoinOr
% 2.94/1.63 => ( setoftrueEq
% 2.94/1.63 => ( powersetI
% 2.94/1.63 => ( emptyinPowerset
% 2.94/1.63 => ( emptyInPowerset
% 2.94/1.63 => ( powersetE
% 2.94/1.63 => ( setunionI
% 2.94/1.63 => ( setunionE
% 2.94/1.63 => ( subPowSU
% 2.94/1.63 => ( exuE2
% 2.94/1.63 => ( nonemptyImpWitness
% 2.94/1.63 => ( uniqinunit
% 2.94/1.63 => ( notinsingleton
% 2.94/1.63 => ( eqinunit
% 2.94/1.63 => ( singletonsswitch
% 2.94/1.63 => ( upairsetE
% 2.94/1.63 => ( upairsetIL
% 2.94/1.63 => ( upairsetIR
% 2.94/1.63 => ( emptyE1
% 2.94/1.63 => ( vacuousDall
% 2.94/1.63 => ( quantDeMorgan1
% 2.94/1.63 => ( quantDeMorgan2
% 2.94/1.63 => ( quantDeMorgan3
% 2.94/1.63 => ( quantDeMorgan4
% 2.94/1.63 => ( prop2setI
% 2.94/1.63 => ( prop2set2propI
% 2.94/1.63 => ( notdexE
% 2.94/1.63 => ( notdallE
% 2.94/1.63 => ( exuI1
% 2.94/1.63 => ( exuI3
% 2.94/1.63 => ( exuI2
% 2.94/1.63 => ( inCongP
% 2.94/1.63 => ( in__Cong
% 2.94/1.63 => ( exuE3u
% 2.94/1.63 => ( exu__Cong
% 2.94/1.63 => ( emptyset__Cong
% 2.94/1.63 => ( setadjoin__Cong
% 2.94/1.63 => ( powerset__Cong
% 2.94/1.63 => ( setunion__Cong
% 2.94/1.63 => ( omega__Cong
% 2.94/1.63 => ( exuEu
% 2.94/1.63 => ( descr__Cong
% 2.94/1.63 => ( dsetconstr__Cong
% 2.94/1.63 => ( subsetI1
% 2.94/1.63 => ( eqimpsubset2
% 2.94/1.63 => ( eqimpsubset1
% 2.94/1.63 => ( subsetI2
% 2.94/1.63 => ( emptysetsubset
% 2.94/1.63 => ( subsetE
% 2.94/1.63 => ( subsetE2
% 2.94/1.63 => ( notsubsetI
% 2.94/1.63 => ( notequalI1
% 2.94/1.63 => ( notequalI2
% 2.94/1.63 => ( subsetRefl
% 2.94/1.63 => ( subsetTrans
% 2.94/1.63 => ( setadjoinSub
% 2.94/1.63 => ( setadjoinSub2
% 2.94/1.63 => ( subset2powerset
% 2.94/1.63 => ( setextsub
% 2.94/1.63 => ( subsetemptysetimpeq
% 2.94/1.63 => ( powersetI1
% 2.94/1.63 => ( powersetE1
% 2.94/1.63 => ( inPowerset
% 2.94/1.63 => ( powersetsubset
% 2.94/1.63 => ( sepInPowerset
% 2.94/1.63 => ( sepSubset
% 2.94/1.63 => ( binunionIL
% 2.94/1.63 => ( upairset2IR
% 2.94/1.63 => ( binunionIR
% 2.94/1.63 => ( binunionEcases
% 2.94/1.63 => ( binunionE
% 2.94/1.63 => ( binunionLsub
% 2.94/1.63 => ( binunionRsub
% 2.94/1.63 => ( binintersectI
% 2.94/1.63 => ( binintersectSubset5
% 2.94/1.63 => ( binintersectEL
% 2.94/1.63 => ( binintersectLsub
% 2.94/1.63 => ( binintersectSubset2
% 2.94/1.63 => ( binintersectSubset3
% 2.94/1.63 => ( binintersectER
% 2.94/1.63 => ( disjointsetsI1
% 2.94/1.63 => ( binintersectRsub
% 2.94/1.63 => ( binintersectSubset4
% 2.94/1.63 => ( binintersectSubset1
% 2.94/1.63 => ( bs114d
% 2.94/1.63 => ( setminusI
% 2.94/1.63 => ( setminusEL
% 2.94/1.63 => ( setminusER
% 2.94/1.63 => ( setminusSubset2
% 2.94/1.63 => ( setminusERneg
% 2.94/1.63 => ( setminusELneg
% 2.94/1.63 => ( setminusILneg
% 2.94/1.63 => ( setminusIRneg
% 2.94/1.63 => ( setminusLsub
% 2.94/1.63 => ( setminusSubset1
% 2.94/1.63 => ( symdiffE
% 2.94/1.63 => ( symdiffI1
% 2.94/1.63 => ( symdiffI2
% 2.94/1.63 => ( symdiffIneg1
% 2.94/1.63 => ( symdiffIneg2
% 2.94/1.63 => ( secondinupair
% 2.94/1.63 => ( setukpairIL
% 2.94/1.63 => ( setukpairIR
% 2.94/1.63 => ( kpairiskpair
% 2.94/1.63 => ( kpairp
% 2.94/1.63 => ( singletonsubset
% 2.94/1.63 => ( singletoninpowerset
% 2.94/1.63 => ( singletoninpowunion
% 2.94/1.63 => ( upairset2E
% 2.94/1.63 => ( upairsubunion
% 2.94/1.63 => ( upairinpowunion
% 2.94/1.63 => ( ubforcartprodlem1
% 2.94/1.63 => ( ubforcartprodlem2
% 2.94/1.63 => ( ubforcartprodlem3
% 2.94/1.63 => ( cartprodpairin
% 2.94/1.63 => ( cartprodmempair1
% 2.94/1.63 => ( cartprodmempair
% 2.94/1.63 => ( setunionE2
% 2.94/1.63 => ( setunionsingleton1
% 2.94/1.63 => ( setunionsingleton2
% 2.94/1.63 => ( setunionsingleton
% 2.94/1.63 => ( singletonprop
% 2.94/1.63 => ( ex1E1
% 2.94/1.63 => ( ex1I
% 2.94/1.63 => ( ex1I2
% 2.94/1.63 => ( singletonsuniq
% 2.94/1.63 => ( setukpairinjL1
% 2.94/1.63 => ( kfstsingleton
% 2.94/1.63 => ( theprop
% 2.94/1.63 => ( kfstpairEq
% 2.94/1.63 => ( cartprodfstin
% 2.94/1.63 => ( setukpairinjL2
% 2.94/1.63 => ( setukpairinjL
% 2.94/1.63 => ( setukpairinjR11
% 2.94/1.63 => ( setukpairinjR12
% 2.94/1.63 => ( setukpairinjR1
% 2.94/1.63 => ( upairequniteq
% 2.94/1.63 => ( setukpairinjR2
% 2.94/1.63 => ( setukpairinjR
% 2.94/1.63 => ( ksndsingleton
% 2.94/1.63 => ( ksndpairEq
% 2.94/1.63 => ( kpairsurjEq
% 2.94/1.63 => ( cartprodsndin
% 2.94/1.63 => ( cartprodpairmemEL
% 2.94/1.63 => ( cartprodpairmemER
% 2.94/1.63 => ( cartprodmempaircEq
% 2.94/1.63 => ( cartprodfstpairEq
% 2.94/1.63 => ( cartprodsndpairEq
% 2.94/1.63 => ( cartprodpairsurjEq
% 2.94/1.63 => ( dpsetconstrI
% 2.94/1.63 => ( dpsetconstrSub
% 2.94/1.63 => ( setOfPairsIsBReln
% 2.94/1.63 => ( dpsetconstrERa
% 2.94/1.63 => ( dpsetconstrEL1
% 2.94/1.63 => ( dpsetconstrEL2
% 2.94/1.63 => ( dpsetconstrER
% 2.94/1.63 => ( funcImageSingleton
% 2.94/1.63 => ( apProp
% 2.94/1.63 => ( app
% 2.94/1.63 => ( infuncsetfunc
% 2.94/1.63 => ( ap2p
% 2.94/1.63 => ( funcinfuncset
% 2.94/1.63 => ( lamProp
% 2.94/1.63 => ( lamp
% 2.94/1.63 => ( lam2p
% 2.94/1.63 => ( brelnall1
% 2.94/1.63 => ( brelnall2
% 2.94/1.63 => ( ex1E2
% 2.94/1.63 => ( funcGraphProp1
% 2.94/1.63 => ( funcGraphProp3
% 2.94/1.63 => ( funcGraphProp2
% 2.94/1.63 => ( funcextLem
% 2.94/1.63 => ( funcGraphProp4
% 2.94/1.63 => ( subbreln
% 2.94/1.63 => ( eqbreln
% 2.94/1.63 => ( funcext
% 2.94/1.63 => ( funcext2
% 2.94/1.63 => ( ap2apEq1
% 2.94/1.63 => ( ap2apEq2
% 2.94/1.63 => ( beta1
% 2.94/1.63 => ( eta1
% 2.94/1.63 => ( lam2lamEq
% 2.94/1.63 => ( beta2
% 2.94/1.63 => ( eta2
% 2.94/1.63 => ( iffalseProp1
% 2.94/1.63 => ( iffalseProp2
% 2.94/1.63 => ( iftrueProp1
% 2.94/1.63 => ( iftrueProp2
% 2.94/1.63 => ( ifSingleton
% 2.94/1.63 => ( ifp
% 2.94/1.63 => ( theeq
% 2.94/1.63 => ( iftrue
% 2.94/1.63 => ( iffalse
% 2.94/1.63 => ( iftrueorfalse
% 2.94/1.63 => ( binintersectT_lem
% 2.94/1.63 => ( binunionT_lem
% 2.94/1.63 => ( powersetT_lem
% 2.94/1.63 => ! [A: $i,X: $i] :
% 2.94/1.63 ( ( in @ X @ ( powerset @ A ) )
% 2.94/1.63 => ! [Y: $i] :
% 2.94/1.63 ( ( in @ Y @ ( powerset @ A ) )
% 2.94/1.63 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.94/1.63 ------------------------
% 2.94/1.66 % (22311)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_22036 for (2999ds/4Mi)
% 2.94/1.66 % (22314)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_22036 for (2999ds/2Mi)
% 2.94/1.66 % (22310)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22036 for (2999ds/183Mi)
% 2.94/1.66 % (22314)Instruction limit reached!
% 2.94/1.66 % (22314)------------------------------
% 2.94/1.66 % (22314)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.66 % (22314)Termination reason: Unknown
% 2.94/1.66 % (22314)Termination phase: shuffling
% 2.94/1.66
% 2.94/1.66 % (22314)Memory used [KB]: 1407
% 2.94/1.66 % (22314)Time elapsed: 0.002 s
% 2.94/1.66 % (22314)Instructions burned: 3 (million)
% 2.94/1.66 % (22314)------------------------------
% 2.94/1.66 % (22314)------------------------------
% 2.94/1.66 % (22311)Instruction limit reached!
% 2.94/1.66 % (22311)------------------------------
% 2.94/1.66 % (22311)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.66 % (22311)Termination reason: Unknown
% 2.94/1.66 % (22311)Termination phase: shuffling
% 2.94/1.66
% 2.94/1.66 % (22311)Memory used [KB]: 1407
% 2.94/1.66 % (22311)Time elapsed: 0.003 s
% 2.94/1.66 % (22311)Instructions burned: 5 (million)
% 2.94/1.66 % (22311)------------------------------
% 2.94/1.66 % (22311)------------------------------
% 2.94/1.67 % (22312)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_22036 for (2999ds/27Mi)
% 2.94/1.67 % (22313)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22036 for (2999ds/2Mi)
% 2.94/1.67 % (22315)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_22036 for (2999ds/275Mi)
% 2.94/1.67 % (22316)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22036 for (2999ds/18Mi)
% 2.94/1.67 % (22313)Instruction limit reached!
% 2.94/1.67 % (22313)------------------------------
% 2.94/1.67 % (22313)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.67 % (22313)Termination reason: Unknown
% 2.94/1.67 % (22313)Termination phase: shuffling
% 2.94/1.67
% 2.94/1.67 % (22313)Memory used [KB]: 1407
% 2.94/1.67 % (22313)Time elapsed: 0.004 s
% 2.94/1.67 % (22313)Instructions burned: 3 (million)
% 2.94/1.67 % (22313)------------------------------
% 2.94/1.67 % (22313)------------------------------
% 2.94/1.67 % (22317)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22036 for (2999ds/3Mi)
% 2.94/1.67 % (22318)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_22036 for (2999ds/37Mi)
% 2.94/1.67 % (22317)Instruction limit reached!
% 2.94/1.67 % (22317)------------------------------
% 2.94/1.67 % (22317)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.67 % (22317)Termination reason: Unknown
% 2.94/1.67 % (22317)Termination phase: shuffling
% 2.94/1.67
% 2.94/1.67 % (22317)Memory used [KB]: 1407
% 2.94/1.67 % (22317)Time elapsed: 0.003 s
% 2.94/1.67 % (22317)Instructions burned: 4 (million)
% 2.94/1.67 % (22317)------------------------------
% 2.94/1.67 % (22317)------------------------------
% 2.94/1.68 % (22316)Instruction limit reached!
% 2.94/1.68 % (22316)------------------------------
% 2.94/1.68 % (22316)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.68 % (22316)Termination reason: Unknown
% 2.94/1.68 % (22316)Termination phase: shuffling
% 2.94/1.68
% 2.94/1.68 % (22316)Memory used [KB]: 1791
% 2.94/1.68 % (22316)Time elapsed: 0.013 s
% 2.94/1.68 % (22316)Instructions burned: 18 (million)
% 2.94/1.68 % (22316)------------------------------
% 2.94/1.68 % (22316)------------------------------
% 2.94/1.68 % (22312)Instruction limit reached!
% 2.94/1.68 % (22312)------------------------------
% 2.94/1.68 % (22312)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.68 % (22312)Termination reason: Unknown
% 2.94/1.68 % (22312)Termination phase: shuffling
% 2.94/1.68
% 2.94/1.68 % (22312)Memory used [KB]: 1918
% 2.94/1.68 % (22312)Time elapsed: 0.016 s
% 2.94/1.68 % (22312)Instructions burned: 27 (million)
% 2.94/1.68 % (22312)------------------------------
% 2.94/1.68 % (22312)------------------------------
% 2.94/1.68 % (22320)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_22036 for (2999ds/3Mi)
% 2.94/1.68 % (22318)Instruction limit reached!
% 2.94/1.68 % (22318)------------------------------
% 2.94/1.68 % (22318)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.68 % (22318)Termination reason: Unknown
% 2.94/1.68 % (22318)Termination phase: shuffling
% 2.94/1.68
% 2.94/1.68 % (22318)Memory used [KB]: 2174
% 2.94/1.68 % (22318)Time elapsed: 0.013 s
% 2.94/1.68 % (22318)Instructions burned: 39 (million)
% 2.94/1.68 % (22318)------------------------------
% 2.94/1.68 % (22318)------------------------------
% 2.94/1.68 % (22320)Instruction limit reached!
% 2.94/1.68 % (22320)------------------------------
% 2.94/1.68 % (22320)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.68 % (22320)Termination reason: Unknown
% 2.94/1.68 % (22320)Termination phase: shuffling
% 2.94/1.68
% 2.94/1.68 % (22320)Memory used [KB]: 1407
% 2.94/1.68 % (22320)Time elapsed: 0.003 s
% 2.94/1.68 % (22320)Instructions burned: 4 (million)
% 2.94/1.68 % (22320)------------------------------
% 2.94/1.68 % (22320)------------------------------
% 2.94/1.69 % (22319)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_22036 for (2999ds/15Mi)
% 2.94/1.69 % (22323)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_22036 for (2999ds/16Mi)
% 2.94/1.70 % (22324)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_22036 for (2999ds/3Mi)
% 2.94/1.70 % (22319)Instruction limit reached!
% 2.94/1.70 % (22319)------------------------------
% 2.94/1.70 % (22319)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.70 % (22319)Termination reason: Unknown
% 2.94/1.70 % (22319)Termination phase: shuffling
% 2.94/1.70
% 2.94/1.70 % (22319)Memory used [KB]: 1663
% 2.94/1.70 % (22319)Time elapsed: 0.010 s
% 2.94/1.70 % (22319)Instructions burned: 15 (million)
% 2.94/1.70 % (22319)------------------------------
% 2.94/1.70 % (22319)------------------------------
% 2.97/1.70 % (22324)Instruction limit reached!
% 2.97/1.70 % (22324)------------------------------
% 2.97/1.70 % (22324)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.70 % (22324)Termination reason: Unknown
% 2.97/1.70 % (22324)Termination phase: shuffling
% 2.97/1.70
% 2.97/1.70 % (22324)Memory used [KB]: 1407
% 2.97/1.70 % (22324)Time elapsed: 0.003 s
% 2.97/1.70 % (22324)Instructions burned: 4 (million)
% 2.97/1.70 % (22324)------------------------------
% 2.97/1.70 % (22324)------------------------------
% 2.97/1.70 % (22321)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_22036 for (2999ds/1041Mi)
% 2.97/1.70 % (22322)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_22036 for (2999ds/7Mi)
% 2.97/1.70 % (22323)Instruction limit reached!
% 2.97/1.70 % (22323)------------------------------
% 2.97/1.70 % (22323)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.70 % (22323)Termination reason: Unknown
% 2.97/1.70 % (22323)Termination phase: shuffling
% 2.97/1.70
% 2.97/1.70 % (22323)Memory used [KB]: 1791
% 2.97/1.70 % (22323)Time elapsed: 0.007 s
% 2.97/1.70 % (22323)Instructions burned: 18 (million)
% 2.97/1.70 % (22323)------------------------------
% 2.97/1.70 % (22323)------------------------------
% 2.97/1.70 % (22322)Instruction limit reached!
% 2.97/1.70 % (22322)------------------------------
% 2.97/1.70 % (22322)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.70 % (22322)Termination reason: Unknown
% 2.97/1.70 % (22322)Termination phase: shuffling
% 2.97/1.70
% 2.97/1.70 % (22322)Memory used [KB]: 1535
% 2.97/1.70 % (22322)Time elapsed: 0.005 s
% 2.97/1.70 % (22322)Instructions burned: 7 (million)
% 2.97/1.70 % (22322)------------------------------
% 2.97/1.70 % (22322)------------------------------
% 2.97/1.71 % (22326)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_22036 for (2999ds/7Mi)
% 2.97/1.71 % (22325)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_22036 for (2999ds/3Mi)
% 2.97/1.71 % (22326)Instruction limit reached!
% 2.97/1.71 % (22326)------------------------------
% 2.97/1.71 % (22326)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.71 % (22326)Termination reason: Unknown
% 2.97/1.71 % (22326)Termination phase: shuffling
% 2.97/1.71
% 2.97/1.71 % (22326)Memory used [KB]: 1535
% 2.97/1.71 % (22326)Time elapsed: 0.004 s
% 2.97/1.71 % (22326)Instructions burned: 8 (million)
% 2.97/1.71 % (22326)------------------------------
% 2.97/1.71 % (22326)------------------------------
% 2.97/1.71 % (22325)Instruction limit reached!
% 2.97/1.71 % (22325)------------------------------
% 2.97/1.71 % (22325)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.71 % (22325)Termination reason: Unknown
% 2.97/1.71 % (22325)Termination phase: shuffling
% 2.97/1.71
% 2.97/1.71 % (22325)Memory used [KB]: 1407
% 2.97/1.71 % (22325)Time elapsed: 0.002 s
% 2.97/1.71 % (22325)Instructions burned: 3 (million)
% 2.97/1.71 % (22325)------------------------------
% 2.97/1.71 % (22325)------------------------------
% 2.97/1.71 % (22327)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_22036 for (2999ds/3Mi)
% 2.97/1.71 % (22327)Instruction limit reached!
% 2.97/1.71 % (22327)------------------------------
% 2.97/1.71 % (22327)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.71 % (22327)Termination reason: Unknown
% 2.97/1.71 % (22327)Termination phase: shuffling
% 2.97/1.71
% 2.97/1.71 % (22327)Memory used [KB]: 1535
% 2.97/1.71 % (22327)Time elapsed: 0.003 s
% 2.97/1.71 % (22327)Instructions burned: 6 (million)
% 2.97/1.71 % (22327)------------------------------
% 2.97/1.71 % (22327)------------------------------
% 2.97/1.71 % (22310)Instruction limit reached!
% 2.97/1.71 % (22310)------------------------------
% 2.97/1.71 % (22310)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.71 % (22310)Termination reason: Unknown
% 2.97/1.71 % (22310)Termination phase: Saturation
% 2.97/1.71
% 2.97/1.71 % (22310)Memory used [KB]: 7547
% 2.97/1.71 % (22310)Time elapsed: 0.053 s
% 2.97/1.71 % (22310)Instructions burned: 184 (million)
% 2.97/1.71 % (22310)------------------------------
% 2.97/1.71 % (22310)------------------------------
% 2.97/1.72 % (22328)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_22036 for (2999ds/4Mi)
% 2.97/1.72 % (22328)Instruction limit reached!
% 2.97/1.72 % (22328)------------------------------
% 2.97/1.72 % (22328)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.72 % (22328)Termination reason: Unknown
% 2.97/1.72 % (22328)Termination phase: shuffling
% 2.97/1.72
% 2.97/1.72 % (22328)Memory used [KB]: 1535
% 2.97/1.72 % (22328)Time elapsed: 0.003 s
% 2.97/1.72 % (22328)Instructions burned: 6 (million)
% 2.97/1.72 % (22328)------------------------------
% 2.97/1.72 % (22328)------------------------------
% 2.97/1.72 % (22329)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_22036 for (2999ds/18Mi)
% 2.97/1.72 % (22330)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_22036 for (2999ds/710Mi)
% 2.97/1.72 % (22331)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_22036 for (2999ds/6Mi)
% 2.97/1.72 % (22331)Instruction limit reached!
% 2.97/1.72 % (22331)------------------------------
% 2.97/1.72 % (22331)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.72 % (22331)Termination reason: Unknown
% 2.97/1.72 % (22331)Termination phase: shuffling
% 2.97/1.72
% 2.97/1.72 % (22331)Memory used [KB]: 1535
% 2.97/1.72 % (22331)Time elapsed: 0.003 s
% 2.97/1.72 % (22331)Instructions burned: 7 (million)
% 2.97/1.72 % (22331)------------------------------
% 2.97/1.72 % (22331)------------------------------
% 2.97/1.72 % (22332)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_22036 for (2999ds/902Mi)
% 2.97/1.72 % (22329)Instruction limit reached!
% 2.97/1.72 % (22329)------------------------------
% 2.97/1.72 % (22329)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.72 % (22329)Termination reason: Unknown
% 2.97/1.72 % (22329)Termination phase: shuffling
% 2.97/1.72
% 2.97/1.72 % (22329)Memory used [KB]: 1791
% 2.97/1.72 % (22329)Time elapsed: 0.007 s
% 2.97/1.72 % (22329)Instructions burned: 18 (million)
% 2.97/1.72 % (22329)------------------------------
% 2.97/1.72 % (22329)------------------------------
% 2.97/1.73 % (22333)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_22036 for (2999ds/21Mi)
% 2.97/1.73 % (22334)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_22036 for (2999ds/5Mi)
% 2.97/1.73 % (22334)Instruction limit reached!
% 2.97/1.73 % (22334)------------------------------
% 2.97/1.73 % (22334)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.73 % (22334)Termination reason: Unknown
% 2.97/1.73 % (22334)Termination phase: shuffling
% 2.97/1.73
% 2.97/1.73 % (22334)Memory used [KB]: 1535
% 2.97/1.73 % (22334)Time elapsed: 0.003 s
% 2.97/1.73 % (22334)Instructions burned: 7 (million)
% 2.97/1.73 % (22334)------------------------------
% 2.97/1.73 % (22334)------------------------------
% 2.97/1.73 % (22335)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_22036 for (2998ds/6Mi)
% 2.97/1.73 % (22333)Instruction limit reached!
% 2.97/1.73 % (22333)------------------------------
% 2.97/1.73 % (22333)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.73 % (22333)Termination reason: Unknown
% 2.97/1.73 % (22333)Termination phase: shuffling
% 2.97/1.73
% 2.97/1.73 % (22333)Memory used [KB]: 1791
% 2.97/1.73 % (22333)Time elapsed: 0.008 s
% 2.97/1.73 % (22333)Instructions burned: 23 (million)
% 2.97/1.73 % (22333)------------------------------
% 2.97/1.73 % (22333)------------------------------
% 2.97/1.74 % (22335)Instruction limit reached!
% 2.97/1.74 % (22335)------------------------------
% 2.97/1.74 % (22335)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.74 % (22335)Termination reason: Unknown
% 2.97/1.74 % (22335)Termination phase: shuffling
% 2.97/1.74
% 2.97/1.74 % (22335)Memory used [KB]: 1535
% 2.97/1.74 % (22335)Time elapsed: 0.004 s
% 2.97/1.74 % (22335)Instructions burned: 8 (million)
% 2.97/1.74 % (22335)------------------------------
% 2.97/1.74 % (22335)------------------------------
% 2.97/1.74 % (22336)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_22036 for (2998ds/377Mi)
% 2.97/1.74 % (22337)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_22036 for (2998ds/779Mi)
% 2.97/1.75 % (22338)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_22036 for (2998ds/19Mi)
% 2.97/1.75 % (22338)Instruction limit reached!
% 2.97/1.75 % (22338)------------------------------
% 2.97/1.75 % (22338)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.75 % (22338)Termination reason: Unknown
% 2.97/1.75 % (22338)Termination phase: shuffling
% 2.97/1.75
% 2.97/1.75 % (22338)Memory used [KB]: 1791
% 2.97/1.75 % (22338)Time elapsed: 0.007 s
% 2.97/1.75 % (22338)Instructions burned: 21 (million)
% 2.97/1.75 % (22338)------------------------------
% 2.97/1.75 % (22338)------------------------------
% 2.97/1.76 % (22339)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_22036 for (2998ds/879Mi)
% 2.97/1.79 % (22315)Instruction limit reached!
% 2.97/1.79 % (22315)------------------------------
% 2.97/1.79 % (22315)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.79 % (22315)Termination reason: Unknown
% 2.97/1.79 % (22315)Termination phase: Saturation
% 2.97/1.79
% 2.97/1.79 % (22315)Memory used [KB]: 9083
% 2.97/1.79 % (22315)Time elapsed: 0.126 s
% 2.97/1.79 % (22315)Instructions burned: 277 (million)
% 2.97/1.79 % (22315)------------------------------
% 2.97/1.79 % (22315)------------------------------
% 2.97/1.80 % (22340)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_22036 for (2998ds/17Mi)
% 2.97/1.81 % (22340)Instruction limit reached!
% 2.97/1.81 % (22340)------------------------------
% 2.97/1.81 % (22340)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.81 % (22340)Termination reason: Unknown
% 2.97/1.81 % (22340)Termination phase: shuffling
% 2.97/1.81
% 2.97/1.81 % (22340)Memory used [KB]: 1791
% 2.97/1.81 % (22340)Time elapsed: 0.007 s
% 2.97/1.81 % (22340)Instructions burned: 20 (million)
% 2.97/1.81 % (22340)------------------------------
% 2.97/1.81 % (22340)------------------------------
% 2.97/1.82 % (22341)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_22036 for (2998ds/3Mi)
% 2.97/1.82 % (22341)Instruction limit reached!
% 2.97/1.82 % (22341)------------------------------
% 2.97/1.82 % (22341)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.82 % (22341)Termination reason: Unknown
% 2.97/1.82 % (22341)Termination phase: shuffling
% 2.97/1.82
% 2.97/1.82 % (22341)Memory used [KB]: 1407
% 2.97/1.82 % (22341)Time elapsed: 0.003 s
% 2.97/1.82 % (22341)Instructions burned: 4 (million)
% 2.97/1.82 % (22341)------------------------------
% 2.97/1.82 % (22341)------------------------------
% 2.97/1.83 % (22342)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_22036 for (2998ds/30Mi)
% 2.97/1.84 % (22342)Instruction limit reached!
% 2.97/1.84 % (22342)------------------------------
% 2.97/1.84 % (22342)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.84 % (22342)Termination reason: Unknown
% 2.97/1.84 % (22342)Termination phase: shuffling
% 2.97/1.84
% 2.97/1.84 % (22342)Memory used [KB]: 2046
% 2.97/1.84 % (22342)Time elapsed: 0.011 s
% 2.97/1.84 % (22342)Instructions burned: 31 (million)
% 2.97/1.84 % (22342)------------------------------
% 2.97/1.84 % (22342)------------------------------
% 2.97/1.85 % (22336)Instruction limit reached!
% 2.97/1.85 % (22336)------------------------------
% 2.97/1.85 % (22336)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.85 % (22336)Termination reason: Unknown
% 2.97/1.85 % (22336)Termination phase: Saturation
% 2.97/1.85
% 2.97/1.85 % (22336)Memory used [KB]: 8699
% 2.97/1.85 % (22336)Time elapsed: 0.107 s
% 2.97/1.85 % (22336)Instructions burned: 379 (million)
% 2.97/1.85 % (22336)------------------------------
% 2.97/1.85 % (22336)------------------------------
% 2.97/1.85 % (22343)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_22036 for (2997ds/127Mi)
% 2.97/1.86 % (22344)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_22036 for (2997ds/100Mi)
% 2.97/1.89 % (22343)Instruction limit reached!
% 2.97/1.89 % (22343)------------------------------
% 2.97/1.89 % (22343)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.89 % (22343)Termination reason: Unknown
% 2.97/1.89 % (22343)Termination phase: Property scanning
% 2.97/1.89
% 2.97/1.89 % (22343)Memory used [KB]: 2558
% 2.97/1.89 % (22343)Time elapsed: 0.037 s
% 2.97/1.89 % (22343)Instructions burned: 128 (million)
% 2.97/1.89 % (22343)------------------------------
% 2.97/1.89 % (22343)------------------------------
% 2.97/1.89 % (22344)Instruction limit reached!
% 2.97/1.89 % (22344)------------------------------
% 2.97/1.89 % (22344)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.89 % (22344)Termination reason: Unknown
% 2.97/1.89 % (22344)Termination phase: Property scanning
% 2.97/1.89
% 2.97/1.89 % (22344)Memory used [KB]: 2558
% 2.97/1.89 % (22344)Time elapsed: 0.028 s
% 2.97/1.89 % (22344)Instructions burned: 101 (million)
% 2.97/1.89 % (22344)------------------------------
% 2.97/1.89 % (22344)------------------------------
% 2.97/1.90 % (22346)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_22036 for (2997ds/20Mi)
% 2.97/1.90 % (22345)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_22036 for (2997ds/3Mi)
% 2.97/1.90 % (22345)Instruction limit reached!
% 2.97/1.90 % (22345)------------------------------
% 2.97/1.90 % (22345)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.90 % (22345)Termination reason: Unknown
% 2.97/1.90 % (22345)Termination phase: shuffling
% 2.97/1.90
% 2.97/1.90 % (22345)Memory used [KB]: 1407
% 2.97/1.90 % (22345)Time elapsed: 0.002 s
% 2.97/1.90 % (22345)Instructions burned: 3 (million)
% 2.97/1.90 % (22345)------------------------------
% 2.97/1.90 % (22345)------------------------------
% 2.97/1.90 % (22346)Instruction limit reached!
% 2.97/1.90 % (22346)------------------------------
% 2.97/1.90 % (22346)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.90 % (22346)Termination reason: Unknown
% 2.97/1.90 % (22346)Termination phase: shuffling
% 2.97/1.90
% 2.97/1.90 % (22346)Memory used [KB]: 1791
% 2.97/1.90 % (22346)Time elapsed: 0.008 s
% 2.97/1.90 % (22346)Instructions burned: 22 (million)
% 2.97/1.90 % (22346)------------------------------
% 2.97/1.90 % (22346)------------------------------
% 2.97/1.91 % (22347)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_22036 for (2997ds/86Mi)
% 2.97/1.91 % (22348)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_22036 for (2997ds/107Mi)
% 2.97/1.93 % (22347)Instruction limit reached!
% 2.97/1.93 % (22347)------------------------------
% 2.97/1.93 % (22347)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.93 % (22347)Termination reason: Unknown
% 2.97/1.93 % (22347)Termination phase: Property scanning
% 2.97/1.93
% 2.97/1.93 % (22347)Memory used [KB]: 2430
% 2.97/1.93 % (22347)Time elapsed: 0.026 s
% 2.97/1.93 % (22347)Instructions burned: 87 (million)
% 2.97/1.93 % (22347)------------------------------
% 2.97/1.93 % (22347)------------------------------
% 2.97/1.94 % (22348)Instruction limit reached!
% 2.97/1.94 % (22348)------------------------------
% 2.97/1.94 % (22348)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.94 % (22348)Termination reason: Unknown
% 2.97/1.94 % (22348)Termination phase: Function definition elimination
% 2.97/1.94
% 2.97/1.94 % (22348)Memory used [KB]: 2430
% 2.97/1.94 % (22348)Time elapsed: 0.029 s
% 2.97/1.94 % (22348)Instructions burned: 110 (million)
% 2.97/1.94 % (22348)------------------------------
% 2.97/1.94 % (22348)------------------------------
% 2.97/1.94 % (22349)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_22036 for (2997ds/39Mi)
% 2.97/1.95 % (22350)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_22036 for (2996ds/448Mi)
% 2.97/1.95 % (22349)Instruction limit reached!
% 2.97/1.95 % (22349)------------------------------
% 2.97/1.95 % (22349)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.95 % (22349)Termination reason: Unknown
% 2.97/1.95 % (22349)Termination phase: shuffling
% 2.97/1.95
% 2.97/1.95 % (22349)Memory used [KB]: 2174
% 2.97/1.95 % (22349)Time elapsed: 0.013 s
% 2.97/1.95 % (22349)Instructions burned: 39 (million)
% 2.97/1.95 % (22349)------------------------------
% 2.97/1.95 % (22349)------------------------------
% 2.97/1.95 % (22337)Instruction limit reached!
% 2.97/1.95 % (22337)------------------------------
% 2.97/1.95 % (22337)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.95 % (22337)Termination reason: Unknown
% 2.97/1.95 % (22337)Termination phase: Saturation
% 2.97/1.95
% 2.97/1.95 % (22337)Memory used [KB]: 7419
% 2.97/1.95 % (22337)Time elapsed: 0.210 s
% 2.97/1.95 % (22337)Instructions burned: 780 (million)
% 2.97/1.95 % (22337)------------------------------
% 2.97/1.95 % (22337)------------------------------
% 2.97/1.96 % (22330)Instruction limit reached!
% 2.97/1.96 % (22330)------------------------------
% 2.97/1.96 % (22330)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.96 % (22330)Termination reason: Unknown
% 2.97/1.96 % (22330)Termination phase: Saturation
% 2.97/1.96
% 2.97/1.96 % (22330)Memory used [KB]: 10106
% 2.97/1.96 % (22330)Time elapsed: 0.238 s
% 2.97/1.96 % (22330)Instructions burned: 711 (million)
% 2.97/1.96 % (22330)------------------------------
% 2.97/1.96 % (22330)------------------------------
% 2.97/1.96 % (22351)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_22036 for (2996ds/46Mi)
% 2.97/1.96 % (22352)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_22036 for (2996ds/98Mi)
% 2.97/1.97 % (22353)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_22036 for (2996ds/507Mi)
% 2.97/1.98 % (22351)Instruction limit reached!
% 2.97/1.98 % (22351)------------------------------
% 2.97/1.98 % (22351)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.98 % (22351)Termination reason: Unknown
% 2.97/1.98 % (22351)Termination phase: Property scanning
% 2.97/1.98
% 2.97/1.98 % (22351)Memory used [KB]: 2302
% 2.97/1.98 % (22351)Time elapsed: 0.016 s
% 2.97/1.98 % (22351)Instructions burned: 49 (million)
% 2.97/1.98 % (22351)------------------------------
% 2.97/1.98 % (22351)------------------------------
% 2.97/1.98 % (22332)First to succeed.
% 2.97/1.98 % (22332)Refutation found. Thanks to Tanya!
% 2.97/1.98 % SZS status Theorem for DTF2THF_22036
% 2.97/1.98 % SZS output start Proof for DTF2THF_22036
% 2.97/1.98 thf(func_def_0, type, in: $i > $i > $o).
% 2.97/1.98 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.97/1.98 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.97/1.98 thf(func_def_8, type, powerset: $i > $i).
% 2.97/1.98 thf(func_def_10, type, setunion: $i > $i).
% 2.97/1.98 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.97/1.98 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.97/1.98 thf(func_def_26, type, prop2set: $o > $i).
% 2.97/1.98 thf(func_def_36, type, nonempty: $i > $o).
% 2.97/1.98 thf(func_def_69, type, set2prop: $i > $o).
% 2.97/1.98 thf(func_def_88, type, subset: $i > $i > $o).
% 2.97/1.98 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.97/1.98 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.97/1.98 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.97/1.98 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.97/1.98 thf(func_def_135, type, regular: $i > $o).
% 2.97/1.98 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.97/1.98 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.97/1.98 thf(func_def_153, type, iskpair: $i > $o).
% 2.97/1.98 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.97/1.98 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.97/1.98 thf(func_def_177, type, singleton: $i > $o).
% 2.97/1.98 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.97/1.98 thf(func_def_184, type, atmost1p: $i > $o).
% 2.97/1.98 thf(func_def_185, type, atleast2p: $i > $o).
% 2.97/1.98 thf(func_def_186, type, atmost2p: $i > $o).
% 2.97/1.98 thf(func_def_187, type, upairsetp: $i > $o).
% 2.97/1.98 thf(func_def_191, type, kfst: $i > $i).
% 2.97/1.98 thf(func_def_203, type, ksnd: $i > $i).
% 2.97/1.98 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.97/1.98 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.97/1.98 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.97/1.98 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.97/1.98 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.97/1.98 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.97/1.98 thf(func_def_284, type, sP3: $i > $i > $i > $o > $o).
% 2.97/1.98 thf(func_def_285, type, sP4: $i > $i > $i > $i > $o).
% 2.97/1.98 thf(func_def_287, type, sP6: $i > $i > $o).
% 2.97/1.98 thf(func_def_288, type, sP7: $i > $o).
% 2.97/1.98 thf(func_def_289, type, sP8: $i > $i > $o).
% 2.97/1.98 thf(func_def_290, type, sP9: $i > $i > $o).
% 2.97/1.98 thf(func_def_292, type, sK11: $i > $i > ($i > $o) > $i).
% 2.97/1.98 thf(func_def_293, type, sK12: $i > $o).
% 2.97/1.98 thf(func_def_303, type, sK22: ($i > $o) > ($i > $o) > $i).
% 2.97/1.98 thf(func_def_304, type, sK23: ($i > $o) > ($i > $o) > $i).
% 2.97/1.98 thf(func_def_305, type, sK24: $i > $o).
% 2.97/1.98 thf(func_def_306, type, sK25: $i > $o).
% 2.97/1.98 thf(func_def_309, type, sK28: $i > $i).
% 2.97/1.98 thf(func_def_310, type, sK29: ($i > $i) > $i > $i > $i).
% 2.97/1.98 thf(func_def_318, type, sK37: $i > $i > $o).
% 2.97/1.98 thf(func_def_325, type, sK44: $i > $i > $o).
% 2.97/1.98 thf(func_def_327, type, sK46: $i > $i).
% 2.97/1.98 thf(func_def_328, type, sK47: $i > $i).
% 2.97/1.98 thf(func_def_329, type, sK48: $i > ($i > $i > $o) > $i).
% 2.97/1.98 thf(func_def_330, type, sK49: $i > ($i > $i > $o) > $i).
% 2.97/1.98 thf(func_def_331, type, sK50: $i > $i > ($i > $i > $o) > $i).
% 2.97/1.98 thf(func_def_336, type, sK55: $i > ($i > $o) > $i).
% 2.97/1.98 thf(func_def_337, type, sK56: $i > $o).
% 2.97/1.98 thf(func_def_346, type, sK65: $i > ($i > $o) > $i).
% 2.97/1.98 thf(func_def_347, type, sK66: $i > $o).
% 2.97/1.98 thf(func_def_357, type, sK76: $i > $i).
% 2.97/1.98 thf(func_def_359, type, sK78: $i > $o).
% 2.97/1.98 thf(func_def_368, type, sK87: $i > $i > $i).
% 2.97/1.98 thf(func_def_376, type, sK95: ($i > $o) > $i).
% 2.97/1.98 thf(func_def_377, type, sK96: ($i > $o) > $i).
% 2.97/1.98 thf(func_def_378, type, sK97: $i > $o).
% 2.97/1.98 thf(func_def_403, type, sK122: $i > $o).
% 2.97/1.98 thf(func_def_408, type, sK127: $i > $o).
% 2.97/1.98 thf(func_def_410, type, sK129: ($i > $o) > $i > $i).
% 2.97/1.98 thf(func_def_415, type, sK134: $i > $i > $i).
% 2.97/1.98 thf(func_def_416, type, sK135: $i > $i).
% 2.97/1.98 thf(func_def_419, type, sK138: $i > $i > ($i > $i) > $i).
% 2.97/1.98 thf(func_def_425, type, sK144: $i > $i > $o).
% 2.97/1.98 thf(func_def_434, type, sK153: $i > $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_447, type, sK166: $i > $o).
% 2.97/1.98 thf(func_def_448, type, sK167: $i > $o).
% 2.97/1.98 thf(func_def_449, type, sK168: ($i > $o) > ($i > $o) > $i).
% 2.97/1.98 thf(func_def_450, type, sK169: ($i > $o) > ($i > $o) > $i).
% 2.97/1.98 thf(func_def_454, type, sK173: $i > $i > $o).
% 2.97/1.98 thf(func_def_465, type, sK184: $i > $o).
% 2.97/1.98 thf(func_def_471, type, sK190: ($i > $o) > $i).
% 2.97/1.98 thf(func_def_472, type, sK191: $i > $o).
% 2.97/1.98 thf(func_def_473, type, sK192: $i > $i).
% 2.97/1.98 thf(func_def_480, type, sK199: $i > $i > $i).
% 2.97/1.98 thf(func_def_499, type, sK218: $i > $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_507, type, sK226: $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_508, type, sK227: $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_512, type, sK231: $i > $o).
% 2.97/1.98 thf(func_def_533, type, sK252: $i > $o).
% 2.97/1.98 thf(func_def_535, type, sK254: ($i > $o) > $i > $i).
% 2.97/1.98 thf(func_def_536, type, sK255: ($i > $o) > $i > $i).
% 2.97/1.98 thf(func_def_540, type, sK259: $i > $i > $i).
% 2.97/1.98 thf(func_def_550, type, sK269: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.97/1.98 thf(func_def_551, type, sK270: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.97/1.98 thf(func_def_554, type, sK273: $i > $o).
% 2.97/1.98 thf(func_def_555, type, sK274: $i > $o).
% 2.97/1.98 thf(func_def_561, type, sK280: $i > $i).
% 2.97/1.98 thf(func_def_563, type, sK282: $o > $i > $i > $i).
% 2.97/1.98 thf(func_def_577, type, sK296: $i > $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_578, type, sK297: $i > $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_584, type, sK303: $i > $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_598, type, sK317: $i > $i > $o).
% 2.97/1.98 thf(func_def_609, type, sK328: ($i > $o) > $i).
% 2.97/1.98 thf(func_def_610, type, sK329: $i > $o).
% 2.97/1.98 thf(func_def_611, type, sK330: $i > $i).
% 2.97/1.98 thf(func_def_613, type, sK332: $i > $o).
% 2.97/1.98 thf(func_def_621, type, sK340: $i > $o).
% 2.97/1.98 thf(func_def_640, type, sK359: $i > $i > $i).
% 2.97/1.98 thf(func_def_643, type, sK362: $i > $o).
% 2.97/1.98 thf(func_def_646, type, sK365: $i > $i > $o).
% 2.97/1.98 thf(func_def_654, type, sK373: $i > $o).
% 2.97/1.98 thf(func_def_657, type, sK376: $i > ($i > $o) > $i).
% 2.97/1.98 thf(func_def_658, type, sK377: $i > $o).
% 2.97/1.98 thf(func_def_687, type, sK406: $i > $i).
% 2.97/1.98 thf(func_def_688, type, sK407: ($i > $i) > $i > $i > $i).
% 2.97/1.98 thf(func_def_693, type, sK412: $i > $i).
% 2.97/1.98 thf(func_def_694, type, sK413: $i > $i > $i).
% 2.97/1.98 thf(func_def_704, type, sK423: $i > $o).
% 2.97/1.98 thf(func_def_707, type, sK426: ($i > $o) > $i > $i).
% 2.97/1.98 thf(func_def_709, type, sK428: $i > $o).
% 2.97/1.98 thf(func_def_714, type, sK433: $i > ($i > $o) > $i).
% 2.97/1.98 thf(func_def_715, type, sK434: $i > $o).
% 2.97/1.98 thf(func_def_718, type, sK437: $i > $i > $i).
% 2.97/1.98 thf(func_def_726, type, sK445: ($i > $o) > $i).
% 2.97/1.98 thf(func_def_727, type, sK446: $i > $o).
% 2.97/1.98 thf(func_def_735, type, sK454: $i > $i > ($i > $i) > $i).
% 2.97/1.98 thf(func_def_736, type, sK455: $i > $i).
% 2.97/1.98 thf(func_def_761, type, sK480: $i > $i).
% 2.97/1.98 thf(func_def_769, type, sK488: $i > $i).
% 2.97/1.98 thf(func_def_778, type, sK497: $i > $i > $o).
% 2.97/1.98 thf(func_def_787, type, sK506: $i > $i > $i).
% 2.97/1.98 thf(func_def_788, type, sK507: $i > $i > $i).
% 2.97/1.98 thf(func_def_794, type, sK513: $i > $o).
% 2.97/1.98 thf(func_def_796, type, sK515: ($i > $o) > $i > $i).
% 2.97/1.98 thf(func_def_797, type, sK516: $i > $o).
% 2.97/1.98 thf(func_def_801, type, sK520: $i > $o).
% 2.97/1.98 thf(func_def_810, type, sK529: $i > $i > $i).
% 2.97/1.98 thf(func_def_816, type, sK535: $i > $o).
% 2.97/1.98 thf(func_def_818, type, sK537: ($i > $o) > $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_819, type, sK538: ($i > $o) > $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_836, type, sK555: $i > $o).
% 2.97/1.98 thf(func_def_837, type, sK556: $i > $o).
% 2.97/1.98 thf(func_def_850, type, sK569: $i > ($i > $o) > $i).
% 2.97/1.98 thf(func_def_851, type, sK570: $i > $o).
% 2.97/1.98 thf(func_def_853, type, sK572: $i > $i).
% 2.97/1.98 thf(func_def_856, type, sK575: $i > $i).
% 2.97/1.98 thf(func_def_858, type, sK577: $i > $i).
% 2.97/1.98 thf(func_def_861, type, sK580: $i > ($i > $i) > $i > $i).
% 2.97/1.98 thf(func_def_873, type, sK592: $i > $i > $o).
% 2.97/1.98 thf(func_def_884, type, sK603: ($i > $o) > $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_885, type, sK604: ($i > $o) > $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_889, type, sK608: $i > $o).
% 2.97/1.98 thf(func_def_895, type, sK614: $i > $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_896, type, sK615: $i > $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_897, type, sK616: $i > $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_898, type, sK617: $i > $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_899, type, sK618: $i > $i > $i).
% 2.97/1.98 thf(func_def_900, type, sK619: $i > $i > $i).
% 2.97/1.98 thf(func_def_901, type, sK620: $i > $i > $i).
% 2.97/1.98 thf(func_def_902, type, sK621: $i > $i > $i).
% 2.97/1.98 thf(func_def_903, type, sK622: $i > $i > $i).
% 2.97/1.98 thf(func_def_904, type, sK623: $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_905, type, sK624: $i > $i).
% 2.97/1.98 thf(func_def_906, type, sK625: $i > $i).
% 2.97/1.98 thf(func_def_907, type, sK626: $i > $i).
% 2.97/1.98 thf(func_def_908, type, sK627: $i > $i).
% 2.97/1.98 thf(func_def_909, type, sK628: $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_910, type, sK629: $i > $i > $i > $i).
% 2.97/1.98 thf(func_def_911, type, sK630: $i > $i > $i).
% 2.97/1.98 thf(func_def_912, type, sK631: $i > $i > $i).
% 2.97/1.98 thf(func_def_913, type, sK632: $i > $i > $i).
% 2.97/1.98 thf(func_def_915, type, sK634: $i > $i).
% 2.97/1.98 thf(func_def_916, type, sK635: $i > $i).
% 2.97/1.98 thf(func_def_917, type, sK636: $i > $i).
% 2.97/1.98 thf(func_def_934, type, sK653: $i > $i > ($i > $i) > $i).
% 2.97/1.98 thf(func_def_935, type, sK654: $i > $i).
% 2.97/1.98 thf(func_def_942, type, sK661: ($i > $o) > $i > $i).
% 2.97/1.98 thf(func_def_944, type, sK663: $i > $o).
% 2.97/1.98 thf(func_def_962, type, sK681: $i > $o).
% 2.97/1.98 thf(func_def_964, type, sK683: ($i > $o) > $i > $i).
% 2.97/1.98 thf(func_def_965, type, sK684: ($i > $o) > $i > $i).
% 2.97/1.98 thf(func_def_1003, type, ph722: !>[X0: $tType]:(X0)).
% 2.97/1.98 thf(f9119,plain,(
% 2.97/1.98 $false),
% 2.97/1.98 inference(avatar_sat_refutation,[status(thm)],[f7069,f9118])).
% 2.97/1.98 thf(f9118,plain,(
% 2.97/1.98 ~spl721_127),
% 2.97/1.98 inference(avatar_split_clause,[status(thm)],[f6759,f6940])).
% 2.97/1.98 thf(f6940,definition,(
% 2.97/1.98 spl721_127 <=> ($true = (subset @ sK74 @ sK73))),
% 2.97/1.98 introduced(definition,[new_symbols(naming,[spl721_127])],[avatar_definition])).
% 2.97/1.98 thf(f6759,plain,(
% 2.97/1.98 ($true != (subset @ sK74 @ sK73))),
% 2.97/1.98 inference(trivial_inequality_removal,[status(thm)],[f6758])).
% 2.97/1.98 thf(f6758,plain,(
% 2.97/1.98 ($true != (subset @ sK74 @ sK73)) | ($true != $true)),
% 2.97/1.98 inference(forward_demodulation,[status(thm)],[f6622,f2311])).
% 2.97/1.98 thf(f2311,plain,(
% 2.97/1.98 (setminusLsub = $true)),
% 2.97/1.98 inference(cnf_transformation,[status(thm)],[f1106])).
% 2.97/1.98 thf(f1106,plain,(
% 2.97/1.98 (eta2 = $true) & (brelnall1 = $true) & (setbeta = $true) & (upairequniteq = $true) & (notequalI1 = $true) & (exuE3e = $true) & (ubforcartprodlem1 = $true) & (cartprodfstin = $true) & (binintersectSubset2 = $true) & (powersetI = $true) & (setadjoinSub = $true) & (setadjoinE = $true) & (omegaSAx = $true) & (upairinpowunion = $true) & (symdiffI2 = $true) & (emptysetsubset = $true) & (cartprodpairmemER = $true) & (setukpairinjR12 = $true) & (binintersectSubset1 = $true) & (emptyinPowerset = $true) & (subsetTrans = $true) & (notdallE = $true) & (descrp = $true) & (kpairiskpair = $true) & (funcext = $true) & (lam2lamEq = $true) & (beta1 = $true) & (setunion__Cong = $true) & (sepInPowerset = $true) & (prop2set2propI = $true) & (dsetconstrEL = $true) & (prop2setI = $true) & (cartprodmempair1 = $true) & (brelnall2 = $true) & (setminusIRneg = $true) & (binintersectRsub = $true) & (setunionsingleton2 = $true) & (omega__Cong = $true) & (setukpairinjR11 = $true) & (emptyInPowerset = $true) & (iftrueorfalse = $true) & (setminusSubset2 = $true) & (powersetE = $true) & (nonemptyI1 = $true) & (setextAx = $true) & (inCongP = $true) & (setunionE = $true) & (emptyE1 = $true) & (setminusELneg = $true) & (binunionIL = $true) & (upairsetIL = $true) & (cartprodpairmemEL = $true) & (vacuousDall = $true) & (ifp = $true) & (ap2apEq1 = $true) & (funcGraphProp4 = $true) & (dsetconstrI = $true) & (eqinunit = $true) & (binunionEcases = $true) & (replAx = $true) & (setoftrueEq = $true) & (bs114d = $true) & (symdiffI1 = $true) & (ex1I2 = $true) & (secondinupair = $true) & (ubforcartprodlem2 = $true) & (subsetE2 = $true) & (lam2p = $true) & (setukpairinjR = $true) & (ap2p = $true) & (subset2powerset = $true) & (ex1E1 = $true) & (setOfPairsIsBReln = $true) & (binintersectT_lem = $true) & (setadjoinSub2 = $true) & (setminusER = $true) & (eqbreln = $true) & (singletoninpowunion = $true) & (kfstpairEq = $true) & (quantDeMorgan4 = $true) & (cartprodmempaircEq = $true) & (cartprodmempair = $true) & (subsetE = $true) & (singletonsswitch = $true) & (setunionI = $true) & (descr__Cong = $true) & (lamProp = $true) & (symdiffE = $true) & (cartprodfstpairEq = $true) & (setminusSubset1 = $true) & (dsetconstr__Cong = $true) & (subsetRefl = $true) & (ksndpairEq = $true) & (funcGraphProp2 = $true) & (setminusEL = $true) & (subsetI1 = $true) & (ksndsingleton = $true) & (in__Cong = $true) & (setadjoin__Cong = $true) & (subsetemptysetimpeq = $true) & (setminusLsub = $true) & (inPowerset = $true) & (funcGraphProp1 = $true) & (singletonsuniq = $true) & (dpsetconstrEL1 = $true) & (eqimpsubset1 = $true) & (setadjoinAx = $true) & (iftrue = $true) & (upairset2IR = $true) & (apProp = $true) & (funcext2 = $true) & (cartprodpairsurjEq = $true) & (binintersectER = $true) & (cartprodsndpairEq = $true) & (setunionsingleton1 = $true) & (exuE2 = $true) & (symdiffIneg2 = $true) & (quantDeMorgan2 = $true) & (emptyI = $true) & (binintersectSubset4 = $true) & (setminusI = $true) & (setunionAx = $true) & (powersetI1 = $true) & (($true = (in @ sK74 @ (powerset @ sK73))) & (($true = (in @ sK75 @ (powerset @ sK73))) & ((in @ (setminus @ sK74 @ sK75) @ (powerset @ sK73)) != $true))) & (dpsetconstrEL2 = $true) & (quantDeMorgan3 = $true) & (nonemptyImpWitness = $true) & (dpsetconstrER = $true) & (binintersectI = $true) & (binintersectLsub = $true) & (powerset__Cong = $true) & (omega0Ax = $true) & (iffalse = $true) & (dpsetconstrERa = $true) & (infuncsetfunc = $true) & (binintersectSubset3 = $true) & (omegaIndAx = $true) & (notequalI2 = $true) & (emptysetE = $true) & (funcextLem = $true) & (dpsetconstrI = $true) & (subsetI2 = $true) & (funcinfuncset = $true) & (setext = $true) & (quantDeMorgan1 = $true) & (notsubsetI = $true) & (setadjoinIL = $true) & (binunionLsub = $true) & (exuE3u = $true) & (subbreln = $true) & (nonemptyE1 = $true) & (emptyset__Cong = $true) & (ubforcartprodlem3 = $true) & (cartprodpairin = $true) & (iftrueProp1 = $true) & (disjointsetsI1 = $true) & (eta1 = $true) & (funcImageSingleton = $true) & (kpairsurjEq = $true) & (setukpairinjR1 = $true) & (ex1E2 = $true) & (cartprodsndin = $true) & (lamp = $true) & (setminusILneg = $true) & (iffalseProp2 = $true) & (binunionRsub = $true) & (setukpairIR = $true) & (singletonprop = $true) & (notinsingleton = $true) & (ex1I = $true) & (wellorderingAx = $true) & (theeq = $true) & (theprop = $true) & (setukpairinjL = $true) & (iftrueProp2 = $true) & (setukpairinjR2 = $true) & (exuE1 = $true) & (setunionsingleton = $true) & (binunionIR = $true) & (prop2setE = $true) & (setadjoinOr = $true) & (setukpairinjL2 = $true) & (ifSingleton = $true) & (eqimpsubset2 = $true) & (notinemptyset = $true) & (nonemptyI = $true) & (exu__Cong = $true) & (emptysetAx = $true) & (noeltsimpempty = $true) & (dsetconstrER = $true) & (powersetsubset = $true) & (sepSubset = $true) & (setextsub = $true) & (iffalseProp1 = $true) & (exuI2 = $true) & (powersetE1 = $true) & (binintersectSubset5 = $true) & (setadjoinIR = $true) & (powersetAx = $true) & (upairsetIR = $true) & (notdexE = $true) & (setukpairIL = $true) & (subPowSU = $true) & (setminusERneg = $true) & (binunionT_lem = $true) & (exuI3 = $true) & (symdiffIneg1 = $true) & (dpsetconstrSub = $true) & (singletoninpowerset = $true) & (setunionE2 = $true) & (kpairp = $true) & (binunionE = $true) & (singletonsubset = $true) & (emptysetimpfalse = $true) & (uniqinunit = $true) & (kfstsingleton = $true) & (foundationAx = $true) & (ap2apEq2 = $true) & (app = $true) & (emptyinunitempty = $true) & (upairsetE = $true) & (exuI1 = $true) & (setukpairinjL1 = $true) & (binintersectEL = $true) & (funcGraphProp3 = $true) & (upairset2E = $true) & (upairsubunion = $true) & (exuEu = $true) & (powersetT_lem = $true) & (beta2 = $true)),
% 2.97/1.98 inference(skolemisation,[status(esa),new_symbols(skolem,[sK73,sK74,sK75])],[f831,f1105,f1104])).
% 2.97/1.98 thf(f1104,plain,(
% 2.97/1.98 ? [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) & ? [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) & ($true != (in @ (setminus @ X1 @ X2) @ (powerset @ X0))))) => (($true = (in @ sK74 @ (powerset @ sK73))) & ? [X2 : $i] : (($true = (in @ X2 @ (powerset @ sK73))) & ($true != (in @ (setminus @ sK74 @ X2) @ (powerset @ sK73)))))),
% 2.97/1.98 introduced(definition,[],[choice_axiom])).
% 2.97/1.98 thf(f1105,plain,(
% 2.97/1.98 ? [X2 : $i] : (($true = (in @ X2 @ (powerset @ sK73))) & ($true != (in @ (setminus @ sK74 @ X2) @ (powerset @ sK73)))) => (($true = (in @ sK75 @ (powerset @ sK73))) & ((in @ (setminus @ sK74 @ sK75) @ (powerset @ sK73)) != $true))),
% 2.97/1.98 introduced(definition,[],[choice_axiom])).
% 2.97/1.98 thf(f831,plain,(
% 2.97/1.98 (eta2 = $true) & (brelnall1 = $true) & (setbeta = $true) & (upairequniteq = $true) & (notequalI1 = $true) & (exuE3e = $true) & (ubforcartprodlem1 = $true) & (cartprodfstin = $true) & (binintersectSubset2 = $true) & (powersetI = $true) & (setadjoinSub = $true) & (setadjoinE = $true) & (omegaSAx = $true) & (upairinpowunion = $true) & (symdiffI2 = $true) & (emptysetsubset = $true) & (cartprodpairmemER = $true) & (setukpairinjR12 = $true) & (binintersectSubset1 = $true) & (emptyinPowerset = $true) & (subsetTrans = $true) & (notdallE = $true) & (descrp = $true) & (kpairiskpair = $true) & (funcext = $true) & (lam2lamEq = $true) & (beta1 = $true) & (setunion__Cong = $true) & (sepInPowerset = $true) & (prop2set2propI = $true) & (dsetconstrEL = $true) & (prop2setI = $true) & (cartprodmempair1 = $true) & (brelnall2 = $true) & (setminusIRneg = $true) & (binintersectRsub = $true) & (setunionsingleton2 = $true) & (omega__Cong = $true) & (setukpairinjR11 = $true) & (emptyInPowerset = $true) & (iftrueorfalse = $true) & (setminusSubset2 = $true) & (powersetE = $true) & (nonemptyI1 = $true) & (setextAx = $true) & (inCongP = $true) & (setunionE = $true) & (emptyE1 = $true) & (setminusELneg = $true) & (binunionIL = $true) & (upairsetIL = $true) & (cartprodpairmemEL = $true) & (vacuousDall = $true) & (ifp = $true) & (ap2apEq1 = $true) & (funcGraphProp4 = $true) & (dsetconstrI = $true) & (eqinunit = $true) & (binunionEcases = $true) & (replAx = $true) & (setoftrueEq = $true) & (bs114d = $true) & (symdiffI1 = $true) & (ex1I2 = $true) & (secondinupair = $true) & (ubforcartprodlem2 = $true) & (subsetE2 = $true) & (lam2p = $true) & (setukpairinjR = $true) & (ap2p = $true) & (subset2powerset = $true) & (ex1E1 = $true) & (setOfPairsIsBReln = $true) & (binintersectT_lem = $true) & (setadjoinSub2 = $true) & (setminusER = $true) & (eqbreln = $true) & (singletoninpowunion = $true) & (kfstpairEq = $true) & (quantDeMorgan4 = $true) & (cartprodmempaircEq = $true) & (cartprodmempair = $true) & (subsetE = $true) & (singletonsswitch = $true) & (setunionI = $true) & (descr__Cong = $true) & (lamProp = $true) & (symdiffE = $true) & (cartprodfstpairEq = $true) & (setminusSubset1 = $true) & (dsetconstr__Cong = $true) & (subsetRefl = $true) & (ksndpairEq = $true) & (funcGraphProp2 = $true) & (setminusEL = $true) & (subsetI1 = $true) & (ksndsingleton = $true) & (in__Cong = $true) & (setadjoin__Cong = $true) & (subsetemptysetimpeq = $true) & (setminusLsub = $true) & (inPowerset = $true) & (funcGraphProp1 = $true) & (singletonsuniq = $true) & (dpsetconstrEL1 = $true) & (eqimpsubset1 = $true) & (setadjoinAx = $true) & (iftrue = $true) & (upairset2IR = $true) & (apProp = $true) & (funcext2 = $true) & (cartprodpairsurjEq = $true) & (binintersectER = $true) & (cartprodsndpairEq = $true) & (setunionsingleton1 = $true) & (exuE2 = $true) & (symdiffIneg2 = $true) & (quantDeMorgan2 = $true) & (emptyI = $true) & (binintersectSubset4 = $true) & (setminusI = $true) & (setunionAx = $true) & (powersetI1 = $true) & ? [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) & ? [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) & ($true != (in @ (setminus @ X1 @ X2) @ (powerset @ X0))))) & (dpsetconstrEL2 = $true) & (quantDeMorgan3 = $true) & (nonemptyImpWitness = $true) & (dpsetconstrER = $true) & (binintersectI = $true) & (binintersectLsub = $true) & (powerset__Cong = $true) & (omega0Ax = $true) & (iffalse = $true) & (dpsetconstrERa = $true) & (infuncsetfunc = $true) & (binintersectSubset3 = $true) & (omegaIndAx = $true) & (notequalI2 = $true) & (emptysetE = $true) & (funcextLem = $true) & (dpsetconstrI = $true) & (subsetI2 = $true) & (funcinfuncset = $true) & (setext = $true) & (quantDeMorgan1 = $true) & (notsubsetI = $true) & (setadjoinIL = $true) & (binunionLsub = $true) & (exuE3u = $true) & (subbreln = $true) & (nonemptyE1 = $true) & (emptyset__Cong = $true) & (ubforcartprodlem3 = $true) & (cartprodpairin = $true) & (iftrueProp1 = $true) & (disjointsetsI1 = $true) & (eta1 = $true) & (funcImageSingleton = $true) & (kpairsurjEq = $true) & (setukpairinjR1 = $true) & (ex1E2 = $true) & (cartprodsndin = $true) & (lamp = $true) & (setminusILneg = $true) & (iffalseProp2 = $true) & (binunionRsub = $true) & (setukpairIR = $true) & (singletonprop = $true) & (notinsingleton = $true) & (ex1I = $true) & (wellorderingAx = $true) & (theeq = $true) & (theprop = $true) & (setukpairinjL = $true) & (iftrueProp2 = $true) & (setukpairinjR2 = $true) & (exuE1 = $true) & (setunionsingleton = $true) & (binunionIR = $true) & (prop2setE = $true) & (setadjoinOr = $true) & (setukpairinjL2 = $true) & (ifSingleton = $true) & (eqimpsubset2 = $true) & (notinemptyset = $true) & (nonemptyI = $true) & (exu__Cong = $true) & (emptysetAx = $true) & (noeltsimpempty = $true) & (dsetconstrER = $true) & (powersetsubset = $true) & (sepSubset = $true) & (setextsub = $true) & (iffalseProp1 = $true) & (exuI2 = $true) & (powersetE1 = $true) & (binintersectSubset5 = $true) & (setadjoinIR = $true) & (powersetAx = $true) & (upairsetIR = $true) & (notdexE = $true) & (setukpairIL = $true) & (subPowSU = $true) & (setminusERneg = $true) & (binunionT_lem = $true) & (exuI3 = $true) & (symdiffIneg1 = $true) & (dpsetconstrSub = $true) & (singletoninpowerset = $true) & (setunionE2 = $true) & (kpairp = $true) & (binunionE = $true) & (singletonsubset = $true) & (emptysetimpfalse = $true) & (uniqinunit = $true) & (kfstsingleton = $true) & (foundationAx = $true) & (ap2apEq2 = $true) & (app = $true) & (emptyinunitempty = $true) & (upairsetE = $true) & (exuI1 = $true) & (setukpairinjL1 = $true) & (binintersectEL = $true) & (funcGraphProp3 = $true) & (upairset2E = $true) & (upairsubunion = $true) & (exuEu = $true) & (powersetT_lem = $true) & (beta2 = $true)),
% 2.97/1.98 inference(flattening,[status(thm)],[f830])).
% 2.97/1.98 thf(f830,plain,(
% 2.97/1.98 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) & ? [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) & ($true != (in @ (setminus @ X1 @ X2) @ (powerset @ X0))))) & (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)),
% 2.97/1.98 inference(ennf_transformation,[status(thm)],[f316])).
% 2.97/1.98 thf(f316,plain,(
% 2.97/1.98 ~((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) => ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) => ($true = (in @ (setminus @ X1 @ X2) @ (powerset @ X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.97/1.98 inference(fool_elimination,[status(thm)],[f315])).
% 2.97/1.98 thf(f315,plain,(
% 2.97/1.98 ~(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 => ! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X0)) => (in @ (setminus @ X1 @ X2) @ (powerset @ X0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.97/1.98 inference(rectify,[status(thm)],[f235])).
% 2.97/1.98 thf(f235,negated_conjecture,(
% 2.97/1.98 ~(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 => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => (in @ (setminus @ X11 @ X16) @ (powerset @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.97/1.98 inference(negated_conjecture,[status(cth)],[f234])).
% 2.97/1.98 thf(f234,conjecture,(
% 2.97/1.98 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 => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => (in @ (setminus @ X11 @ X16) @ (powerset @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.97/1.98 file('/export/starexec/sandbox2/tmp/tmp.9EhP1wCwEA/DTF2THF_22036.p',setminusT_lem)).
% 2.97/1.98 thf(f6622,plain,(
% 2.97/1.98 (setminusLsub != $true) | ($true != (subset @ sK74 @ sK73))),
% 2.97/1.98 inference(trivial_inequality_removal,[status(thm)],[f6591])).
% 2.97/1.98 thf(f6591,plain,(
% 2.97/1.98 (setminusLsub != $true) | ($true != $true) | ($true != (subset @ sK74 @ sK73))),
% 2.97/1.98 inference(superposition,[status(thm)],[f3574,f2124])).
% 2.97/1.98 thf(f2124,plain,(
% 2.97/1.98 ( ! [X0 : $i,X1 : $i] : (($true = (subset @ (setminus @ X1 @ X0) @ X1)) | (setminusLsub != $true)) )),
% 2.97/1.98 inference(cnf_transformation,[status(thm)],[f1039])).
% 2.97/1.98 thf(f1039,plain,(
% 2.97/1.98 (! [X0 : $i,X1 : $i] : ($true = (subset @ (setminus @ X1 @ X0) @ X1)) | (setminusLsub != $true)) & ((setminusLsub = $true) | ($true != (subset @ (setminus @ sK32 @ sK31) @ sK32)))),
% 2.97/1.98 inference(skolemisation,[status(esa),new_symbols(skolem,[sK31,sK32])],[f1037,f1038])).
% 2.97/1.98 thf(f1038,plain,(
% 2.97/1.98 ? [X2 : $i,X3 : $i] : ($true != (subset @ (setminus @ X3 @ X2) @ X3)) => ($true != (subset @ (setminus @ sK32 @ sK31) @ sK32))),
% 2.97/1.98 introduced(definition,[],[choice_axiom])).
% 2.97/1.98 thf(f1037,plain,(
% 2.97/1.98 (! [X0 : $i,X1 : $i] : ($true = (subset @ (setminus @ X1 @ X0) @ X1)) | (setminusLsub != $true)) & ((setminusLsub = $true) | ? [X2 : $i,X3 : $i] : ($true != (subset @ (setminus @ X3 @ X2) @ X3)))),
% 2.97/1.98 inference(rectify,[status(thm)],[f1036])).
% 2.97/1.98 thf(f1036,plain,(
% 2.97/1.98 (! [X0 : $i,X1 : $i] : ($true = (subset @ (setminus @ X1 @ X0) @ X1)) | (setminusLsub != $true)) & ((setminusLsub = $true) | ? [X0 : $i,X1 : $i] : ($true != (subset @ (setminus @ X1 @ X0) @ X1)))),
% 2.97/1.98 inference(nnf_transformation,[status(thm)],[f475])).
% 2.97/1.98 thf(f475,plain,(
% 2.97/1.98 ! [X0 : $i,X1 : $i] : ($true = (subset @ (setminus @ X1 @ X0) @ X1)) <=> (setminusLsub = $true)),
% 2.97/1.98 inference(fool_elimination,[status(thm)],[f474])).
% 2.97/1.98 thf(f474,plain,(
% 2.97/1.98 (setminusLsub = ! [X0 : $i,X1 : $i] : (subset @ (setminus @ X1 @ X0) @ X1))),
% 2.97/1.98 inference(rectify,[status(thm)],[f128])).
% 2.97/1.98 thf(f128,axiom,(
% 2.97/1.98 (setminusLsub = ! [X4 : $i,X3 : $i] : (subset @ (setminus @ X3 @ X4) @ X3))),
% 2.97/1.98 file('/export/starexec/sandbox2/tmp/tmp.9EhP1wCwEA/DTF2THF_22036.p',setminusLsub)).
% 2.97/1.98 thf(f3574,plain,(
% 2.97/1.98 ( ! [X0 : $i] : (($true != (subset @ (setminus @ sK74 @ sK75) @ X0)) | ($true != (subset @ X0 @ sK73))) )),
% 2.97/1.98 inference(trivial_inequality_removal,[status(thm)],[f3573])).
% 2.97/1.98 thf(f3573,plain,(
% 2.97/1.98 ( ! [X0 : $i] : (($true != (subset @ (setminus @ sK74 @ sK75) @ X0)) | ($true != $true) | ($true != (subset @ X0 @ sK73))) )),
% 2.97/1.98 inference(forward_demodulation,[status(thm)],[f3559,f2391])).
% 2.97/1.98 thf(f2391,plain,(
% 2.97/1.98 (subsetTrans = $true)),
% 2.97/1.98 inference(cnf_transformation,[status(thm)],[f1106])).
% 2.97/1.98 thf(f3559,plain,(
% 2.97/1.98 ( ! [X0 : $i] : ((subsetTrans != $true) | ($true != (subset @ (setminus @ sK74 @ sK75) @ X0)) | ($true != (subset @ X0 @ sK73))) )),
% 2.97/1.98 inference(trivial_inequality_removal,[status(thm)],[f3554])).
% 2.97/1.98 thf(f3554,plain,(
% 2.97/1.98 ( ! [X0 : $i] : ((subsetTrans != $true) | ($true != (subset @ (setminus @ sK74 @ sK75) @ X0)) | ($true != (subset @ X0 @ sK73)) | ($true != $true)) )),
% 2.97/1.98 inference(superposition,[status(thm)],[f3506,f2951])).
% 2.97/1.98 thf(f2951,plain,(
% 2.97/1.98 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true = (subset @ X0 @ X1)) | (subsetTrans != $true) | ($true != (subset @ X2 @ X1)) | ($true != (subset @ X0 @ X2))) )),
% 2.97/1.98 inference(cnf_transformation,[status(thm)],[f1764])).
% 2.97/1.98 thf(f1764,plain,(
% 2.97/1.98 (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (subset @ X2 @ X1)) | ($true != (subset @ X0 @ X2)) | ($true = (subset @ X0 @ X1))) | (subsetTrans != $true)) & ((subsetTrans = $true) | (($true = (subset @ sK502 @ sK501)) & ($true = (subset @ sK500 @ sK502)) & ((subset @ sK500 @ sK501) != $true)))),
% 2.97/1.98 inference(skolemisation,[status(esa),new_symbols(skolem,[sK500,sK501,sK502])],[f1762,f1763])).
% 2.97/1.98 thf(f1763,plain,(
% 2.97/1.98 ? [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X5 @ X4) = $true) & ((subset @ X3 @ X5) = $true) & ((subset @ X3 @ X4) != $true)) => (($true = (subset @ sK502 @ sK501)) & ($true = (subset @ sK500 @ sK502)) & ((subset @ sK500 @ sK501) != $true))),
% 2.97/1.98 introduced(definition,[],[choice_axiom])).
% 2.97/1.98 thf(f1762,plain,(
% 2.97/1.98 (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (subset @ X2 @ X1)) | ($true != (subset @ X0 @ X2)) | ($true = (subset @ X0 @ X1))) | (subsetTrans != $true)) & ((subsetTrans = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (((subset @ X5 @ X4) = $true) & ((subset @ X3 @ X5) = $true) & ((subset @ X3 @ X4) != $true)))),
% 2.97/1.98 inference(rectify,[status(thm)],[f1761])).
% 2.97/1.98 thf(f1761,plain,(
% 2.97/1.98 (! [X0 : $i,X1 : $i,X2 : $i] : (($true != (subset @ X2 @ X1)) | ($true != (subset @ X0 @ X2)) | ($true = (subset @ X0 @ X1))) | (subsetTrans != $true)) & ((subsetTrans = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true = (subset @ X2 @ X1)) & ($true = (subset @ X0 @ X2)) & ($true != (subset @ X0 @ X1))))),
% 2.97/1.98 inference(nnf_transformation,[status(thm)],[f891])).
% 2.97/1.98 thf(f891,plain,(
% 2.97/1.98 ! [X0 : $i,X1 : $i,X2 : $i] : (($true != (subset @ X2 @ X1)) | ($true != (subset @ X0 @ X2)) | ($true = (subset @ X0 @ X1))) <=> (subsetTrans = $true)),
% 2.97/1.98 inference(flattening,[status(thm)],[f890])).
% 2.97/1.98 thf(f890,plain,(
% 2.97/1.98 (subsetTrans = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : ((($true = (subset @ X0 @ X1)) | ($true != (subset @ X2 @ X1))) | ($true != (subset @ X0 @ X2)))),
% 2.97/1.98 inference(ennf_transformation,[status(thm)],[f574])).
% 2.97/1.98 thf(f574,plain,(
% 2.97/1.98 (subsetTrans = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (subset @ X0 @ X2)) => (($true = (subset @ X2 @ X1)) => ($true = (subset @ X0 @ X1))))),
% 2.97/1.98 inference(fool_elimination,[status(thm)],[f573])).
% 2.97/1.98 thf(f573,plain,(
% 2.97/1.98 (! [X0 : $i,X1 : $i,X2 : $i] : ((subset @ X0 @ X2) => ((subset @ X2 @ X1) => (subset @ X0 @ X1))) = subsetTrans)),
% 2.97/1.98 inference(rectify,[status(thm)],[f89])).
% 2.97/1.98 thf(f89,axiom,(
% 2.97/1.98 (! [X3 : $i,X5 : $i,X4 : $i] : ((subset @ X3 @ X4) => ((subset @ X4 @ X5) => (subset @ X3 @ X5))) = subsetTrans)),
% 2.97/1.98 file('/export/starexec/sandbox2/tmp/tmp.9EhP1wCwEA/DTF2THF_22036.p',subsetTrans)).
% 2.97/1.98 thf(f3506,plain,(
% 2.97/1.98 ($true != (subset @ (setminus @ sK74 @ sK75) @ sK73))),
% 2.97/1.98 inference(trivial_inequality_removal,[status(thm)],[f3505])).
% 2.97/1.98 thf(f3505,plain,(
% 2.97/1.98 ($true != $true) | ($true != (subset @ (setminus @ sK74 @ sK75) @ sK73))),
% 2.97/1.98 inference(forward_demodulation,[status(thm)],[f3487,f2341])).
% 2.97/1.98 thf(f2341,plain,(
% 2.97/1.98 (subset2powerset = $true)),
% 2.97/1.98 inference(cnf_transformation,[status(thm)],[f1106])).
% 2.97/1.98 thf(f3487,plain,(
% 2.97/1.98 (subset2powerset != $true) | ($true != (subset @ (setminus @ sK74 @ sK75) @ sK73))),
% 2.97/1.98 inference(trivial_inequality_removal,[status(thm)],[f3479])).
% 2.97/1.98 thf(f3479,plain,(
% 2.97/1.98 ($true != $true) | ($true != (subset @ (setminus @ sK74 @ sK75) @ sK73)) | (subset2powerset != $true)),
% 2.97/1.98 inference(superposition,[status(thm)],[f2286,f2581])).
% 2.97/1.98 thf(f2581,plain,(
% 2.97/1.98 ( ! [X0 : $i,X1 : $i] : (($true = (in @ X0 @ (powerset @ X1))) | (subset2powerset != $true) | ($true != (subset @ X0 @ X1))) )),
% 2.97/1.98 inference(cnf_transformation,[status(thm)],[f1322])).
% 2.97/1.98 thf(f1322,plain,(
% 2.97/1.98 (! [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) | ($true = (in @ X0 @ (powerset @ X1)))) | (subset2powerset != $true)) & ((subset2powerset = $true) | (($true = (subset @ sK208 @ sK209)) & ($true != (in @ sK208 @ (powerset @ sK209)))))),
% 2.97/1.98 inference(skolemisation,[status(esa),new_symbols(skolem,[sK208,sK209])],[f1320,f1321])).
% 2.97/1.98 thf(f1321,plain,(
% 2.97/1.98 ? [X2 : $i,X3 : $i] : (($true = (subset @ X2 @ X3)) & ((in @ X2 @ (powerset @ X3)) != $true)) => (($true = (subset @ sK208 @ sK209)) & ($true != (in @ sK208 @ (powerset @ sK209))))),
% 2.97/1.98 introduced(definition,[],[choice_axiom])).
% 2.97/1.98 thf(f1320,plain,(
% 2.97/1.98 (! [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) | ($true = (in @ X0 @ (powerset @ X1)))) | (subset2powerset != $true)) & ((subset2powerset = $true) | ? [X2 : $i,X3 : $i] : (($true = (subset @ X2 @ X3)) & ((in @ X2 @ (powerset @ X3)) != $true)))),
% 2.97/1.98 inference(rectify,[status(thm)],[f1319])).
% 2.97/1.98 thf(f1319,plain,(
% 2.97/1.98 (! [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) | ($true = (in @ X0 @ (powerset @ X1)))) | (subset2powerset != $true)) & ((subset2powerset = $true) | ? [X0 : $i,X1 : $i] : (($true = (subset @ X0 @ X1)) & ($true != (in @ X0 @ (powerset @ X1)))))),
% 2.97/1.98 inference(nnf_transformation,[status(thm)],[f917])).
% 2.97/1.98 thf(f917,plain,(
% 2.97/1.98 ! [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) | ($true = (in @ X0 @ (powerset @ X1)))) <=> (subset2powerset = $true)),
% 2.97/1.98 inference(ennf_transformation,[status(thm)],[f377])).
% 2.97/1.98 thf(f377,plain,(
% 2.97/1.98 ! [X0 : $i,X1 : $i] : (($true = (subset @ X0 @ X1)) => ($true = (in @ X0 @ (powerset @ X1)))) <=> (subset2powerset = $true)),
% 2.97/1.98 inference(fool_elimination,[status(thm)],[f376])).
% 2.97/1.98 thf(f376,plain,(
% 2.97/1.98 (! [X0 : $i,X1 : $i] : ((subset @ X0 @ X1) => (in @ X0 @ (powerset @ X1))) = subset2powerset)),
% 2.97/1.98 inference(rectify,[status(thm)],[f92])).
% 2.97/1.98 thf(f92,axiom,(
% 2.97/1.98 (! [X3 : $i,X4 : $i] : ((subset @ X3 @ X4) => (in @ X3 @ (powerset @ X4))) = subset2powerset)),
% 2.97/1.98 file('/export/starexec/sandbox2/tmp/tmp.9EhP1wCwEA/DTF2THF_22036.p',subset2powerset)).
% 2.97/1.98 thf(f2286,plain,(
% 2.97/1.98 ((in @ (setminus @ sK74 @ sK75) @ (powerset @ sK73)) != $true)),
% 2.97/1.98 inference(cnf_transformation,[status(thm)],[f1106])).
% 2.97/1.98 thf(f7069,plain,(
% 2.97/1.98 spl721_127),
% 2.97/1.98 inference(avatar_split_clause,[status(thm)],[f7068,f6940])).
% 2.97/1.98 thf(f7068,plain,(
% 2.97/1.98 ($true = (subset @ sK74 @ sK73))),
% 2.97/1.98 inference(trivial_inequality_removal,[status(thm)],[f7067])).
% 2.97/1.98 thf(f7067,plain,(
% 2.97/1.98 ($true != $true) | ($true = (subset @ sK74 @ sK73))),
% 2.97/1.98 inference(forward_demodulation,[status(thm)],[f7063,f2268])).
% 2.97/1.98 thf(f2268,plain,(
% 2.97/1.98 (subsetI2 = $true)),
% 2.97/1.98 inference(cnf_transformation,[status(thm)],[f1106])).
% 2.97/1.98 thf(f7063,plain,(
% 2.97/1.98 (subsetI2 != $true) | ($true = (subset @ sK74 @ sK73))),
% 2.97/1.98 inference(trivial_inequality_removal,[status(thm)],[f7062])).
% 2.97/1.98 thf(f7062,plain,(
% 2.97/1.98 ($true = (subset @ sK74 @ sK73)) | ($true != $true) | (subsetI2 != $true)),
% 2.97/1.98 inference(duplicate_literal_removal,[status(thm)],[f7054])).
% 2.97/1.98 thf(f7054,plain,(
% 2.97/1.98 ($true = (subset @ sK74 @ sK73)) | ($true = (subset @ sK74 @ sK73)) | ($true != $true) | (subsetI2 != $true)),
% 2.97/1.98 inference(superposition,[status(thm)],[f3877,f2771])).
% 2.97/1.98 thf(f2771,plain,(
% 2.97/1.98 ( ! [X0 : $i,X1 : $i] : (($true = (in @ (sK359 @ X1 @ X0) @ X0)) | (subsetI2 != $true) | ($true = (subset @ X0 @ X1))) )),
% 2.97/1.98 inference(cnf_transformation,[status(thm)],[f1546])).
% 2.97/1.98 thf(f1546,plain,(
% 2.97/1.98 (! [X0 : $i,X1 : $i] : (($true = (subset @ X0 @ X1)) | (($true = (in @ (sK359 @ X1 @ X0) @ X0)) & ($true != (in @ (sK359 @ X1 @ X0) @ X1)))) | (subsetI2 != $true)) & ((subsetI2 = $true) | (($true != (subset @ sK360 @ sK361)) & ! [X5 : $i] : (($true != (in @ X5 @ sK360)) | ($true = (in @ X5 @ sK361)))))),
% 2.97/1.98 inference(skolemisation,[status(esa),new_symbols(skolem,[sK359,sK360,sK361])],[f1543,f1545,f1544])).
% 2.97/1.98 thf(f1544,plain,(
% 2.97/1.98 ! [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true = (in @ X2 @ X0)) & ($true != (in @ X2 @ X1))) => (($true = (in @ (sK359 @ X1 @ X0) @ X0)) & ($true != (in @ (sK359 @ X1 @ X0) @ X1))))),
% 2.97/1.98 introduced(definition,[],[choice_axiom])).
% 2.97/1.98 thf(f1545,plain,(
% 2.97/1.98 ? [X3 : $i,X4 : $i] : (((subset @ X3 @ X4) != $true) & ! [X5 : $i] : (($true != (in @ X5 @ X3)) | ((in @ X5 @ X4) = $true))) => (($true != (subset @ sK360 @ sK361)) & ! [X5 : $i] : (($true != (in @ X5 @ sK360)) | ($true = (in @ X5 @ sK361))))),
% 2.97/1.98 introduced(definition,[],[choice_axiom])).
% 2.97/1.98 thf(f1543,plain,(
% 2.97/1.98 (! [X0 : $i,X1 : $i] : (($true = (subset @ X0 @ X1)) | ? [X2 : $i] : (($true = (in @ X2 @ X0)) & ($true != (in @ X2 @ X1)))) | (subsetI2 != $true)) & ((subsetI2 = $true) | ? [X3 : $i,X4 : $i] : (((subset @ X3 @ X4) != $true) & ! [X5 : $i] : (($true != (in @ X5 @ X3)) | ((in @ X5 @ X4) = $true))))),
% 2.97/1.98 inference(rectify,[status(thm)],[f1542])).
% 2.97/1.98 thf(f1542,plain,(
% 2.97/1.98 (! [X0 : $i,X1 : $i] : (($true = (subset @ X0 @ X1)) | ? [X2 : $i] : (($true = (in @ X2 @ X0)) & ($true != (in @ X2 @ X1)))) | (subsetI2 != $true)) & ((subsetI2 = $true) | ? [X0 : $i,X1 : $i] : (($true != (subset @ X0 @ X1)) & ! [X2 : $i] : (($true != (in @ X2 @ X0)) | ($true = (in @ X2 @ X1)))))),
% 2.97/1.98 inference(nnf_transformation,[status(thm)],[f801])).
% 2.97/1.98 thf(f801,plain,(
% 2.97/1.98 ! [X0 : $i,X1 : $i] : (($true = (subset @ X0 @ X1)) | ? [X2 : $i] : (($true = (in @ X2 @ X0)) & ($true != (in @ X2 @ X1)))) <=> (subsetI2 = $true)),
% 2.97/1.98 inference(ennf_transformation,[status(thm)],[f264])).
% 2.97/1.98 thf(f264,plain,(
% 2.97/1.98 (subsetI2 = $true) <=> ! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true = (in @ X2 @ X0)) => ($true = (in @ X2 @ X1))) => ($true = (subset @ X0 @ X1)))),
% 2.97/1.98 inference(fool_elimination,[status(thm)],[f263])).
% 2.97/1.98 thf(f263,plain,(
% 2.97/1.98 (! [X0 : $i,X1 : $i] : (! [X2 : $i] : ((in @ X2 @ X0) => (in @ X2 @ X1)) => (subset @ X0 @ X1)) = subsetI2)),
% 2.97/1.98 inference(rectify,[status(thm)],[f81])).
% 2.97/1.98 thf(f81,axiom,(
% 2.97/1.98 (! [X3 : $i,X4 : $i] : (! [X1 : $i] : ((in @ X1 @ X3) => (in @ X1 @ X4)) => (subset @ X3 @ X4)) = subsetI2)),
% 2.97/1.98 file('/export/starexec/sandbox2/tmp/tmp.9EhP1wCwEA/DTF2THF_22036.p',subsetI2)).
% 2.97/1.98 thf(f3877,plain,(
% 2.97/1.98 ( ! [X0 : $i] : (($true != (in @ (sK359 @ sK73 @ X0) @ sK74)) | ($true = (subset @ X0 @ sK73))) )),
% 2.97/1.98 inference(trivial_inequality_removal,[status(thm)],[f3876])).
% 2.97/1.98 thf(f3876,plain,(
% 2.97/1.98 ( ! [X0 : $i] : (($true != (in @ (sK359 @ sK73 @ X0) @ sK74)) | ($true != $true) | ($true = (subset @ X0 @ sK73))) )),
% 2.97/1.98 inference(forward_demodulation,[status(thm)],[f3851,f2268])).
% 2.97/1.98 thf(f3851,plain,(
% 2.97/1.98 ( ! [X0 : $i] : (($true != (in @ (sK359 @ sK73 @ X0) @ sK74)) | (subsetI2 != $true) | ($true = (subset @ X0 @ sK73))) )),
% 2.97/1.98 inference(trivial_inequality_removal,[status(thm)],[f3828])).
% 2.97/1.98 thf(f3828,plain,(
% 2.97/1.98 ( ! [X0 : $i] : (($true != $true) | ($true != (in @ (sK359 @ sK73 @ X0) @ sK74)) | ($true = (subset @ X0 @ sK73)) | (subsetI2 != $true)) )),
% 2.97/1.98 inference(superposition,[status(thm)],[f2770,f3476])).
% 2.97/1.98 thf(f3476,plain,(
% 2.97/1.98 ( ! [X0 : $i] : (($true = (in @ X0 @ sK73)) | ($true != (in @ X0 @ sK74))) )),
% 2.97/1.98 inference(trivial_inequality_removal,[status(thm)],[f3475])).
% 2.97/1.98 thf(f3475,plain,(
% 2.97/1.98 ( ! [X0 : $i] : (($true = (in @ X0 @ sK73)) | ($true != $true) | ($true != (in @ X0 @ sK74))) )),
% 2.97/1.98 inference(forward_demodulation,[status(thm)],[f3457,f2211])).
% 2.97/1.98 thf(f2211,plain,(
% 2.97/1.98 (powersetAx = $true)),
% 2.97/1.98 inference(cnf_transformation,[status(thm)],[f1106])).
% 2.97/1.98 thf(f3457,plain,(
% 2.97/1.98 ( ! [X0 : $i] : ((powersetAx != $true) | ($true != (in @ X0 @ sK74)) | ($true = (in @ X0 @ sK73))) )),
% 2.97/1.98 inference(trivial_inequality_removal,[status(thm)],[f3453])).
% 2.97/1.98 thf(f3453,plain,(
% 2.97/1.98 ( ! [X0 : $i] : (($true != $true) | ($true = (in @ X0 @ sK73)) | ($true != (in @ X0 @ sK74)) | (powersetAx != $true)) )),
% 2.97/1.98 inference(superposition,[status(thm)],[f2485,f2288])).
% 2.97/1.98 thf(f2288,plain,(
% 2.97/1.98 ($true = (in @ sK74 @ (powerset @ sK73)))),
% 2.97/1.98 inference(cnf_transformation,[status(thm)],[f1106])).
% 2.97/1.98 thf(f2485,plain,(
% 2.97/1.98 ( ! [X7 : $i,X4 : $i,X5 : $i] : (((in @ X5 @ (powerset @ X4)) != $true) | (powersetAx != $true) | ((in @ X7 @ X4) = $true) | ($true != (in @ X7 @ X5))) )),
% 2.97/1.98 inference(cnf_transformation,[status(thm)],[f1209])).
% 2.97/1.98 thf(f1209,plain,(
% 2.97/1.98 ((powersetAx = $true) | (((($true != (in @ sK133 @ sK131)) & ($true = (in @ sK133 @ sK132))) | ($true != (in @ sK132 @ (powerset @ sK131)))) & (! [X3 : $i] : (($true = (in @ X3 @ sK131)) | ($true != (in @ X3 @ sK132))) | ($true = (in @ sK132 @ (powerset @ sK131)))))) & (! [X4 : $i,X5 : $i] : ((((in @ X5 @ (powerset @ X4)) = $true) | (($true != (in @ (sK134 @ X5 @ X4) @ X4)) & ($true = (in @ (sK134 @ X5 @ X4) @ X5)))) & (! [X7 : $i] : (((in @ X7 @ X4) = $true) | ($true != (in @ X7 @ X5))) | ((in @ X5 @ (powerset @ X4)) != $true))) | (powersetAx != $true))),
% 2.97/1.98 inference(skolemisation,[status(esa),new_symbols(skolem,[sK131,sK132,sK133,sK134])],[f1205,f1208,f1207,f1206])).
% 2.97/1.98 thf(f1206,plain,(
% 2.97/1.98 ? [X0 : $i,X1 : $i] : ((? [X2 : $i] : (($true != (in @ X2 @ X0)) & ($true = (in @ X2 @ X1))) | ($true != (in @ X1 @ (powerset @ X0)))) & (! [X3 : $i] : (($true = (in @ X3 @ X0)) | ($true != (in @ X3 @ X1))) | ($true = (in @ X1 @ (powerset @ X0))))) => ((? [X2 : $i] : (($true != (in @ X2 @ sK131)) & ($true = (in @ X2 @ sK132))) | ($true != (in @ sK132 @ (powerset @ sK131)))) & (! [X3 : $i] : (($true = (in @ X3 @ sK131)) | ($true != (in @ X3 @ sK132))) | ($true = (in @ sK132 @ (powerset @ sK131)))))),
% 2.97/1.98 introduced(definition,[],[choice_axiom])).
% 2.97/1.98 thf(f1207,plain,(
% 2.97/1.98 ? [X2 : $i] : (($true != (in @ X2 @ sK131)) & ($true = (in @ X2 @ sK132))) => (($true != (in @ sK133 @ sK131)) & ($true = (in @ sK133 @ sK132)))),
% 2.97/1.98 introduced(definition,[],[choice_axiom])).
% 2.97/1.98 thf(f1208,plain,(
% 2.97/1.98 ! [X4 : $i,X5 : $i] : (? [X6 : $i] : (((in @ X6 @ X4) != $true) & ($true = (in @ X6 @ X5))) => (($true != (in @ (sK134 @ X5 @ X4) @ X4)) & ($true = (in @ (sK134 @ X5 @ X4) @ X5))))),
% 2.97/1.98 introduced(definition,[],[choice_axiom])).
% 2.97/1.98 thf(f1205,plain,(
% 2.97/1.98 ((powersetAx = $true) | ? [X0 : $i,X1 : $i] : ((? [X2 : $i] : (($true != (in @ X2 @ X0)) & ($true = (in @ X2 @ X1))) | ($true != (in @ X1 @ (powerset @ X0)))) & (! [X3 : $i] : (($true = (in @ X3 @ X0)) | ($true != (in @ X3 @ X1))) | ($true = (in @ X1 @ (powerset @ X0)))))) & (! [X4 : $i,X5 : $i] : ((((in @ X5 @ (powerset @ X4)) = $true) | ? [X6 : $i] : (((in @ X6 @ X4) != $true) & ($true = (in @ X6 @ X5)))) & (! [X7 : $i] : (((in @ X7 @ X4) = $true) | ($true != (in @ X7 @ X5))) | ((in @ X5 @ (powerset @ X4)) != $true))) | (powersetAx != $true))),
% 2.97/1.98 inference(rectify,[status(thm)],[f1204])).
% 2.97/1.98 thf(f1204,plain,(
% 2.97/1.98 ((powersetAx = $true) | ? [X0 : $i,X1 : $i] : ((? [X2 : $i] : (($true != (in @ X2 @ X0)) & ($true = (in @ X2 @ X1))) | ($true != (in @ X1 @ (powerset @ X0)))) & (! [X2 : $i] : (($true = (in @ X2 @ X0)) | ($true != (in @ X2 @ X1))) | ($true = (in @ X1 @ (powerset @ X0)))))) & (! [X0 : $i,X1 : $i] : ((($true = (in @ X1 @ (powerset @ X0))) | ? [X2 : $i] : (($true != (in @ X2 @ X0)) & ($true = (in @ X2 @ X1)))) & (! [X2 : $i] : (($true = (in @ X2 @ X0)) | ($true != (in @ X2 @ X1))) | ($true != (in @ X1 @ (powerset @ X0))))) | (powersetAx != $true))),
% 2.97/1.98 inference(nnf_transformation,[status(thm)],[f918])).
% 2.97/1.98 thf(f918,plain,(
% 2.97/1.98 (powersetAx = $true) <=> ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) <=> ! [X2 : $i] : (($true = (in @ X2 @ X0)) | ($true != (in @ X2 @ X1))))),
% 2.97/1.98 inference(ennf_transformation,[status(thm)],[f545])).
% 2.97/1.98 thf(f545,plain,(
% 2.97/1.98 ! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true = (in @ X2 @ X1)) => ($true = (in @ X2 @ X0))) <=> ($true = (in @ X1 @ (powerset @ X0)))) <=> (powersetAx = $true)),
% 2.97/1.98 inference(fool_elimination,[status(thm)],[f544])).
% 2.97/1.98 thf(f544,plain,(
% 2.97/1.98 (powersetAx = ! [X0 : $i,X1 : $i] : (! [X2 : $i] : ((in @ X2 @ X1) => (in @ X2 @ X0)) <=> (in @ X1 @ (powerset @ X0))))),
% 2.97/1.98 inference(rectify,[status(thm)],[f5])).
% 2.97/1.98 thf(f5,axiom,(
% 2.97/1.98 (powersetAx = ! [X3 : $i,X4 : $i] : (! [X1 : $i] : ((in @ X1 @ X4) => (in @ X1 @ X3)) <=> (in @ X4 @ (powerset @ X3))))),
% 2.97/1.98 file('/export/starexec/sandbox2/tmp/tmp.9EhP1wCwEA/DTF2THF_22036.p',powersetAx)).
% 2.97/1.98 thf(f2770,plain,(
% 2.97/1.98 ( ! [X0 : $i,X1 : $i] : (($true != (in @ (sK359 @ X1 @ X0) @ X1)) | ($true = (subset @ X0 @ X1)) | (subsetI2 != $true)) )),
% 2.97/1.98 inference(cnf_transformation,[status(thm)],[f1546])).
% 2.97/1.98 % SZS output end Proof for DTF2THF_22036
% 2.97/1.98 % (22332)------------------------------
% 2.97/1.98 % (22332)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.97/1.98 % (22332)Termination reason: Refutation
% 2.97/1.98
% 2.97/1.98 % (22332)Memory used [KB]: 11257
% 2.97/1.98 % (22332)Time elapsed: 0.281 s
% 2.97/1.98 % (22332)Instructions burned: 779 (million)
% 2.97/1.98 % (22332)------------------------------
% 2.97/1.98 % (22332)------------------------------
% 2.97/1.98 % (22309)Success in time 0.346 s
%------------------------------------------------------------------------------