%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU734^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 09:36:45 AM UTC 2026
% Result : Theorem 2.72s 1.95s
% Output : Refutation 2.72s
% Verified :
% SZS Type : Refutation
% Derivation depth : 16
% Number of leaves : 14
% Syntax : Number of formulae : 73 ( 15 unt; 0 typ; 3 def)
% Number of atoms : 4588 (1209 equ; 0 cnn)
% Maximal formula atoms : 253 ( 62 avg)
% Number of connectives : 2744 ( 117 ~; 91 |; 802 &; 695 @)
% ( 9 <=>;1030 =>; 0 <=; 0 <~>)
% Maximal formula depth : 261 ( 29 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 455 ( 452 usr; 266 con; 0-4 aty)
% Number of variables : 116 ( 0 ^; 77 !; 39 ?; 116 :)
% 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_302,type,
sP0: $i > $i > $i > $o ).
thf(func_def_303,type,
sP1: $i > $i > $o ).
thf(func_def_304,type,
sP2: $i > $o ).
thf(func_def_305,type,
sP3: $i > $i > $o ).
thf(func_def_306,type,
sP4: $i > $i > $o ).
thf(func_def_308,type,
sP6: $i > $i > $i > $o > $o ).
thf(func_def_312,type,
sP10: $i > $i > $i > $i > $o ).
thf(func_def_314,type,
sK12: $i > $i > ( $i > $i ) > $i ).
thf(func_def_315,type,
sK13: $i > $i ).
thf(func_def_323,type,
sK21: $i > $o ).
thf(func_def_325,type,
sK23: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_326,type,
sK24: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_329,type,
sK27: $i > $o ).
thf(func_def_340,type,
sK38: $i > $i ).
thf(func_def_342,type,
sK40: $i > $i ).
thf(func_def_356,type,
sK54: $i > ( $i > $i ) > $i > $i ).
thf(func_def_358,type,
sK56: $i > $i ).
thf(func_def_363,type,
sK61: $i > $o ).
thf(func_def_391,type,
sK89: $i > $i > $o ).
thf(func_def_396,type,
sK94: $i > $i > $o ).
thf(func_def_398,type,
sK96: $i > $i > ( $i > $i ) > $i ).
thf(func_def_399,type,
sK97: $i > $i ).
thf(func_def_407,type,
sK105: $i > ( $i > $o ) > $i ).
thf(func_def_408,type,
sK106: $i > $o ).
thf(func_def_422,type,
sK120: $i > ( $i > $o ) > $i ).
thf(func_def_423,type,
sK121: $i > $o ).
thf(func_def_436,type,
sK134: $i > $i > $i ).
thf(func_def_443,type,
sK141: $i > $o ).
thf(func_def_451,type,
sK149: $i > $i > $o ).
thf(func_def_467,type,
sK165: ( $i > $o ) > $i > $i > $i ).
thf(func_def_470,type,
sK168: $i > $o ).
thf(func_def_471,type,
sK169: $i > $i > $o ).
thf(func_def_477,type,
sK175: $i > $i > $i ).
thf(func_def_506,type,
sK204: ( $i > $o ) > $i ).
thf(func_def_507,type,
sK205: ( $i > $o ) > $i ).
thf(func_def_508,type,
sK206: $i > $o ).
thf(func_def_513,type,
sK211: $i > $i > $i > $i ).
thf(func_def_525,type,
sK223: $i > $o ).
thf(func_def_547,type,
sK245: $i > $o ).
thf(func_def_548,type,
sK246: $i > $o ).
thf(func_def_549,type,
sK247: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_550,type,
sK248: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_553,type,
sK251: $i > $i > $o ).
thf(func_def_559,type,
sK257: $i > $o ).
thf(func_def_567,type,
sK265: ( $i > $o ) > $i ).
thf(func_def_568,type,
sK266: $i > $o ).
thf(func_def_569,type,
sK267: $i > $i ).
thf(func_def_570,type,
sK268: $i > $i > $i > $i ).
thf(func_def_577,type,
sK275: $i > $o ).
thf(func_def_580,type,
sK278: $i > ( $i > $o ) > $i ).
thf(func_def_590,type,
sK288: $i > $i > $i ).
thf(func_def_595,type,
sK293: $i > $i > $i > $i ).
thf(func_def_596,type,
sK294: $i > $i > $i > $i ).
thf(func_def_606,type,
sK304: $i > $o ).
thf(func_def_607,type,
sK305: $i > $o ).
thf(func_def_608,type,
sK306: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_609,type,
sK307: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_619,type,
sK317: $i > $i > $o ).
thf(func_def_635,type,
sK333: $i > $i > $i > $i ).
thf(func_def_636,type,
sK334: $i > $i > $i > $i ).
thf(func_def_654,type,
sK352: $i > $i > $i ).
thf(func_def_655,type,
sK353: $i > $i > $i ).
thf(func_def_656,type,
sK354: $i > $i > $i ).
thf(func_def_657,type,
sK355: $i > $i > $i ).
thf(func_def_658,type,
sK356: $i > $i > $i ).
thf(func_def_659,type,
sK357: $i > $i > $i > $i ).
thf(func_def_660,type,
sK358: $i > $i ).
thf(func_def_661,type,
sK359: $i > $i ).
thf(func_def_662,type,
sK360: $i > $i ).
thf(func_def_663,type,
sK361: $i > $i ).
thf(func_def_664,type,
sK362: $i > $i > $i > $i ).
thf(func_def_665,type,
sK363: $i > $i > $i > $i ).
thf(func_def_666,type,
sK364: $i > $i > $i ).
thf(func_def_667,type,
sK365: $i > $i > $i ).
thf(func_def_668,type,
sK366: $i > $i > $i ).
thf(func_def_669,type,
sK367: $i > $i ).
thf(func_def_671,type,
sK369: $i > $i ).
thf(func_def_672,type,
sK370: $i > $i ).
thf(func_def_675,type,
sK373: $i > $i > $i ).
thf(func_def_676,type,
sK374: $i > $i > $i ).
thf(func_def_679,type,
sK377: $i > $o ).
thf(func_def_680,type,
sK378: ( $i > $o ) > $i ).
thf(func_def_687,type,
sK385: $i > $i ).
thf(func_def_690,type,
sK388: $i > ( $i > $i ) > $i > $i ).
thf(func_def_697,type,
sK395: $i > $i > $o ).
thf(func_def_698,type,
sK396: $i > $i ).
thf(func_def_699,type,
sK397: $i > $i ).
thf(func_def_700,type,
sK398: ( $i > $i > $o ) > $i > $i ).
thf(func_def_701,type,
sK399: ( $i > $i > $o ) > $i > $i ).
thf(func_def_702,type,
sK400: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_703,type,
sK401: $i > $o ).
thf(func_def_707,type,
sK405: $i > $i ).
thf(func_def_710,type,
sK408: $i > ( $i > $i ) > $i > $i ).
thf(func_def_715,type,
sK413: $i > $i > $i > $i > $i ).
thf(func_def_724,type,
sK422: $i > $i ).
thf(func_def_725,type,
sK423: $i > ( $i > $o ) > $i ).
thf(func_def_726,type,
sK424: $i > $o ).
thf(func_def_734,type,
sK432: $i > $i > $i ).
thf(func_def_746,type,
sK444: $i > $o ).
thf(func_def_764,type,
sK462: $i > $i > $i > $i > $i ).
thf(func_def_773,type,
sK471: $i > $i > $i > $i > $i ).
thf(func_def_776,type,
sK474: $i > $o ).
thf(func_def_777,type,
sK475: $i > $o ).
thf(func_def_778,type,
sK476: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_779,type,
sK477: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_797,type,
sK495: $i > $o ).
thf(func_def_800,type,
sK498: $i > ( $i > $o ) > $i ).
thf(func_def_801,type,
sK499: $i > $i ).
thf(func_def_812,type,
sK510: $i > $o ).
thf(func_def_815,type,
sK513: $i > $i ).
thf(func_def_820,type,
sK518: $i > $o ).
thf(func_def_836,type,
sK534: $i > $o ).
thf(func_def_838,type,
sK536: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_839,type,
sK537: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_848,type,
sK546: $i > $i ).
thf(func_def_850,type,
sK548: $i > $i > $i ).
thf(func_def_882,type,
sK580: $i > $o ).
thf(func_def_884,type,
sK582: $i > ( $i > $o ) > $i ).
thf(func_def_887,type,
sK585: $i > $o ).
thf(func_def_890,type,
sK588: $i > ( $i > $o ) > $i ).
thf(func_def_891,type,
sK589: $i > $o ).
thf(func_def_892,type,
sK590: $i > $i ).
thf(func_def_893,type,
sK591: ( $i > $o ) > $i ).
thf(func_def_914,type,
sK612: $i > $o ).
thf(func_def_917,type,
sK615: $i > ( $i > $o ) > $i ).
thf(func_def_918,type,
sK616: $i > ( $i > $o ) > $i ).
thf(func_def_925,type,
sK623: $i > $o ).
thf(func_def_936,type,
sK634: $i > $o ).
thf(func_def_953,type,
sK651: $i > $i > $i > $i > $i ).
thf(func_def_954,type,
sK652: $i > $i > $i > $i > $i ).
thf(func_def_960,type,
sK658: $i > $i ).
thf(func_def_963,type,
sK661: $i > $i > ( $i > $i ) > $i ).
thf(func_def_970,type,
sK668: $i > $o ).
thf(func_def_972,type,
sK670: ( $i > $o ) > $i > $i ).
thf(func_def_977,type,
sK675: $i > $i > $i ).
thf(func_def_1013,type,
sK711: ( $i > $o ) > $i > $i ).
thf(func_def_1015,type,
sK713: $i > $o ).
thf(func_def_1023,type,
sK721: $i > $i > $i ).
thf(func_def_1030,type,
sK728: $i > $o ).
thf(func_def_1032,type,
sK730: $i > $o ).
thf(func_def_1049,type,
sK747: $i > $o ).
thf(func_def_1052,type,
sK750: $i > ( $i > $o ) > $i ).
thf(func_def_1053,type,
sK751: $i > ( $i > $o ) > $i ).
thf(func_def_1062,type,
sK760: $i > $i > $o ).
thf(func_def_1065,type,
sK763: $i > $i > $i ).
thf(func_def_1076,type,
sK774: $i > $i > $i > $i > $i ).
thf(func_def_1077,type,
sK775: $i > $i > $i > $i > $i ).
thf(func_def_1078,type,
sK776: $i > $i > $i > $i > $i ).
thf(func_def_1079,type,
sK777: $i > $i > $i > $i > $i ).
thf(func_def_1083,type,
sK781: $o > $i > $i > $i ).
thf(func_def_1084,type,
sK782: $i > $i ).
thf(func_def_1095,type,
ph793:
!>[X0: $tType] : X0 ).
thf(f7389,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f3881,f4389,f7273,f7388]) ).
thf(f7388,plain,
spl792_5,
inference(avatar_contradiction_clause,[status(thm)],[f7387]) ).
thf(f7387,plain,
( $false
| spl792_5 ),
inference(trivial_inequality_removal,[status(thm)],[f7386]) ).
thf(f7386,plain,
( ( $true != $true )
| spl792_5 ),
inference(superposition,[status(thm)],[f7298,f3292]) ).
thf(f3292,plain,
binintersectLsub = $true,
inference(cnf_transformation,[status(thm)],[f2181]) ).
thf(f2181,plain,
( ( setminusERneg = $true )
& ( quantDeMorgan4 = $true )
& ( doubleComplementI1 = $true )
& ( doubleComplementSub2 = $true )
& ( subset2powerset = $true )
& ( ubforcartprodlem3 = $true )
& ( infuncsetfunc = $true )
& ( noeltsimpempty = $true )
& ( replAx = $true )
& ( kpairsurjEq = $true )
& ( notsubsetI = $true )
& ( binintersectSubset4 = $true )
& ( lamp = $true )
& ( dpsetconstrERa = $true )
& ( symdiffIneg2 = $true )
& ( setukpairinjR2 = $true )
& ( exu__Cong = $true )
& ( exuI3 = $true )
& ( emptyinunitempty = $true )
& ( emptysetAx = $true )
& ( eqimpsubset2 = $true )
& ( upairinpowunion = $true )
& ( binintersectEL = $true )
& ( complementTI1 = $true )
& ( exuI2 = $true )
& ( kpairp = $true )
& ( setukpairinjR12 = $true )
& ( subsetE = $true )
& ( cartprodsndpairEq = $true )
& ( bs114d = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodpairsurjEq = $true )
& ( foundationAx = $true )
& ( powersetI1 = $true )
& ( setOfPairsIsBReln = $true )
& ( subPowSU = $true )
& ( ubforcartprodlem2 = $true )
& ( notequalI1 = $true )
& ( cartprodsndin = $true )
& ( complementT_lem = $true )
& ( dsetconstrEL = $true )
& ( ex1I = $true )
& ( iftrueProp1 = $true )
& ( cartprodfstpairEq = $true )
& ( eqbreln = $true )
& ( setoftrueEq = $true )
& ( binunionT_lem = $true )
& ( setminusT_lem = $true )
& ( vacuousDall = $true )
& ( sepSubset = $true )
& ( powersetE = $true )
& ( ap2p = $true )
& ( omega__Cong = $true )
& ( prop2setI = $true )
& ( upairsubunion = $true )
& ( dpsetconstrI = $true )
& ( binunionE = $true )
& ( binintersectSubset5 = $true )
& ( kpairiskpair = $true )
& ( exuEu = $true )
& ( setukpairIR = $true )
& ( notinemptyset = $true )
& ( ifp = $true )
& ( cartprodpairmemER = $true )
& ( setukpairinjL2 = $true )
& ( contrasubsetT2 = $true )
& ( powersetAx = $true )
& ( emptysetE = $true )
& ( theeq = $true )
& ( dsetconstrI = $true )
& ( setunionAx = $true )
& ( $true
= ( in @ sK697 @ ( powerset @ sK696 ) ) )
& ( $true
!= ( subset @ ( setminus @ sK696 @ sK697 ) @ ( setminus @ sK696 @ ( binintersect @ sK697 @ sK698 ) ) ) )
& ( $true
= ( in @ sK698 @ ( powerset @ sK696 ) ) )
& ( setunionsingleton = $true )
& ( binunionIL = $true )
& ( setminusSubset1 = $true )
& ( setminusIRneg = $true )
& ( ifSingleton = $true )
& ( setminusLsub = $true )
& ( omegaIndAx = $true )
& ( complementImpComplementIntersect = $true )
& ( emptyset__Cong = $true )
& ( notdexE = $true )
& ( funcext = $true )
& ( powerset__Cong = $true )
& ( subsetTrans = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrEL1 = $true )
& ( binintersectTERcontra = $true )
& ( exuE2 = $true )
& ( contrasubsetT3 = $true )
& ( setminusEL = $true )
& ( powersetI = $true )
& ( iffalseProp2 = $true )
& ( setadjoinIR = $true )
& ( binunionEcases = $true )
& ( cartprodmempair1 = $true )
& ( setunionsingleton1 = $true )
& ( subsetRefl = $true )
& ( binintersectI = $true )
& ( ap2apEq1 = $true )
& ( theprop = $true )
& ( sepInPowerset = $true )
& ( quantDeMorgan2 = $true )
& ( singletonsuniq = $true )
& ( setextAx = $true )
& ( setukpairinjR = $true )
& ( lam2p = $true )
& ( upairsetIR = $true )
& ( emptyinPowerset = $true )
& ( symdiffI2 = $true )
& ( powersetTE1 = $true )
& ( ex1E1 = $true )
& ( symdiffIneg1 = $true )
& ( dsetconstr__Cong = $true )
& ( emptysetsubset = $true )
& ( contrasubsetT = $true )
& ( setextT = $true )
& ( in__Cong = $true )
& ( descrp = $true )
& ( symdiffI1 = $true )
& ( upairsetE = $true )
& ( subbreln = $true )
& ( beta2 = $true )
& ( setunion__Cong = $true )
& ( ksndsingleton = $true )
& ( quantDeMorgan3 = $true )
& ( funcImageSingleton = $true )
& ( setukpairinjL1 = $true )
& ( setadjoinOr = $true )
& ( disjointsetsI1 = $true )
& ( omegaSAx = $true )
& ( eqinunit = $true )
& ( nonemptyI = $true )
& ( binintersectSubset2 = $true )
& ( kfstpairEq = $true )
& ( exuE1 = $true )
& ( prop2set2propI = $true )
& ( singletoninpowerset = $true )
& ( setminusER = $true )
& ( uniqinunit = $true )
& ( dpsetconstrER = $true )
& ( powersetsubset = $true )
& ( inCongP = $true )
& ( binintersectER = $true )
& ( emptyE1 = $true )
& ( setunionE2 = $true )
& ( singletonprop = $true )
& ( setminusI = $true )
& ( symdiffE = $true )
& ( binintersectRsub = $true )
& ( inPowerset = $true )
& ( descr__Cong = $true )
& ( setadjoinAx = $true )
& ( emptyI = $true )
& ( doubleComplementE1 = $true )
& ( setukpairinjL = $true )
& ( binunionLsub = $true )
& ( eta2 = $true )
& ( ex1I2 = $true )
& ( funcGraphProp4 = $true )
& ( subsetI1 = $true )
& ( binunionIR = $true )
& ( setukpairinjR1 = $true )
& ( binintersectSubset3 = $true )
& ( funcext2 = $true )
& ( funcinfuncset = $true )
& ( binintersectSubset1 = $true )
& ( nonemptyI1 = $true )
& ( funcGraphProp3 = $true )
& ( notdallE = $true )
& ( cartprodmempaircEq = $true )
& ( powersetE1 = $true )
& ( dpsetconstrEL2 = $true )
& ( subsetTI = $true )
& ( setextsub = $true )
& ( setadjoinIL = $true )
& ( setadjoin__Cong = $true )
& ( setadjoinSub2 = $true )
& ( subsetE2 = $true )
& ( cartprodfstin = $true )
& ( funcextLem = $true )
& ( iftrueorfalse = $true )
& ( brelnall1 = $true )
& ( upairsetIL = $true )
& ( doubleComplementSub1 = $true )
& ( ap2apEq2 = $true )
& ( binintersectT_lem = $true )
& ( complementTnotintersectT = $true )
& ( contrasubsetT1 = $true )
& ( upairequniteq = $true )
& ( cartprodpairin = $true )
& ( iffalse = $true )
& ( binintersectLsub = $true )
& ( setext = $true )
& ( lam2lamEq = $true )
& ( setminusILneg = $true )
& ( wellorderingAx = $true )
& ( exuI1 = $true )
& ( apProp = $true )
& ( eta1 = $true )
& ( omega0Ax = $true )
& ( notequalI2 = $true )
& ( powersetTI1 = $true )
& ( powersetT_lem = $true )
& ( setukpairIL = $true )
& ( exuE3u = $true )
& ( emptysetimpfalse = $true )
& ( singletoninpowunion = $true )
& ( app = $true )
& ( setadjoinE = $true )
& ( ex1E2 = $true )
& ( ksndpairEq = $true )
& ( binintersectTELcontra = $true )
& ( cartprodmempair = $true )
& ( iftrue = $true )
& ( notinsingleton = $true )
& ( subsetemptysetimpeq = $true )
& ( setukpairinjR11 = $true )
& ( upairset2E = $true )
& ( kfstsingleton = $true )
& ( iffalseProp1 = $true )
& ( secondinupair = $true )
& ( iftrueProp2 = $true )
& ( lamProp = $true )
& ( setunionsingleton2 = $true )
& ( singletonsubset = $true )
& ( doubleComplementEq = $true )
& ( quantDeMorgan1 = $true )
& ( binunionRsub = $true )
& ( nonemptyE1 = $true )
& ( emptyInPowerset = $true )
& ( setbeta = $true )
& ( prop2setE = $true )
& ( dsetconstrER = $true )
& ( beta1 = $true )
& ( singletonsswitch = $true )
& ( setunionE = $true )
& ( funcGraphProp1 = $true )
& ( setadjoinSub = $true )
& ( nonemptyImpWitness = $true )
& ( brelnall2 = $true )
& ( upairset2IR = $true )
& ( eqimpsubset1 = $true )
& ( setminusSubset2 = $true )
& ( setunionI = $true )
& ( setminusELneg = $true )
& ( funcGraphProp2 = $true )
& ( complementTE1 = $true )
& ( subsetI2 = $true )
& ( exuE3e = $true )
& ( ubforcartprodlem1 = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK696,sK697,sK698])],[f949,f2180,f2179]) ).
thf(f2179,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
& ? [X2: $i] :
( ( ( subset @ ( setminus @ X0 @ X1 ) @ ( setminus @ X0 @ ( binintersect @ X1 @ X2 ) ) )
!= $true )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) ) )
=> ( ( $true
= ( in @ sK697 @ ( powerset @ sK696 ) ) )
& ? [X2: $i] :
( ( $true
!= ( subset @ ( setminus @ sK696 @ sK697 ) @ ( setminus @ sK696 @ ( binintersect @ sK697 @ X2 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ sK696 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2180,plain,
( ? [X2: $i] :
( ( $true
!= ( subset @ ( setminus @ sK696 @ sK697 ) @ ( setminus @ sK696 @ ( binintersect @ sK697 @ X2 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ sK696 ) ) ) )
=> ( ( $true
!= ( subset @ ( setminus @ sK696 @ sK697 ) @ ( setminus @ sK696 @ ( binintersect @ sK697 @ sK698 ) ) ) )
& ( $true
= ( in @ sK698 @ ( powerset @ sK696 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f949,plain,
( ( setminusERneg = $true )
& ( quantDeMorgan4 = $true )
& ( doubleComplementI1 = $true )
& ( doubleComplementSub2 = $true )
& ( subset2powerset = $true )
& ( ubforcartprodlem3 = $true )
& ( infuncsetfunc = $true )
& ( noeltsimpempty = $true )
& ( replAx = $true )
& ( kpairsurjEq = $true )
& ( notsubsetI = $true )
& ( binintersectSubset4 = $true )
& ( lamp = $true )
& ( dpsetconstrERa = $true )
& ( symdiffIneg2 = $true )
& ( setukpairinjR2 = $true )
& ( exu__Cong = $true )
& ( exuI3 = $true )
& ( emptyinunitempty = $true )
& ( emptysetAx = $true )
& ( eqimpsubset2 = $true )
& ( upairinpowunion = $true )
& ( binintersectEL = $true )
& ( complementTI1 = $true )
& ( exuI2 = $true )
& ( kpairp = $true )
& ( setukpairinjR12 = $true )
& ( subsetE = $true )
& ( cartprodsndpairEq = $true )
& ( bs114d = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodpairsurjEq = $true )
& ( foundationAx = $true )
& ( powersetI1 = $true )
& ( setOfPairsIsBReln = $true )
& ( subPowSU = $true )
& ( ubforcartprodlem2 = $true )
& ( notequalI1 = $true )
& ( cartprodsndin = $true )
& ( complementT_lem = $true )
& ( dsetconstrEL = $true )
& ( ex1I = $true )
& ( iftrueProp1 = $true )
& ( cartprodfstpairEq = $true )
& ( eqbreln = $true )
& ( setoftrueEq = $true )
& ( binunionT_lem = $true )
& ( setminusT_lem = $true )
& ( vacuousDall = $true )
& ( sepSubset = $true )
& ( powersetE = $true )
& ( ap2p = $true )
& ( omega__Cong = $true )
& ( prop2setI = $true )
& ( upairsubunion = $true )
& ( dpsetconstrI = $true )
& ( binunionE = $true )
& ( binintersectSubset5 = $true )
& ( kpairiskpair = $true )
& ( exuEu = $true )
& ( setukpairIR = $true )
& ( notinemptyset = $true )
& ( ifp = $true )
& ( cartprodpairmemER = $true )
& ( setukpairinjL2 = $true )
& ( contrasubsetT2 = $true )
& ( powersetAx = $true )
& ( emptysetE = $true )
& ( theeq = $true )
& ( dsetconstrI = $true )
& ( setunionAx = $true )
& ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
& ? [X2: $i] :
( ( ( subset @ ( setminus @ X0 @ X1 ) @ ( setminus @ X0 @ ( binintersect @ X1 @ X2 ) ) )
!= $true )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) ) )
& ( setunionsingleton = $true )
& ( binunionIL = $true )
& ( setminusSubset1 = $true )
& ( setminusIRneg = $true )
& ( ifSingleton = $true )
& ( setminusLsub = $true )
& ( omegaIndAx = $true )
& ( complementImpComplementIntersect = $true )
& ( emptyset__Cong = $true )
& ( notdexE = $true )
& ( funcext = $true )
& ( powerset__Cong = $true )
& ( subsetTrans = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrEL1 = $true )
& ( binintersectTERcontra = $true )
& ( exuE2 = $true )
& ( contrasubsetT3 = $true )
& ( setminusEL = $true )
& ( powersetI = $true )
& ( iffalseProp2 = $true )
& ( setadjoinIR = $true )
& ( binunionEcases = $true )
& ( cartprodmempair1 = $true )
& ( setunionsingleton1 = $true )
& ( subsetRefl = $true )
& ( binintersectI = $true )
& ( ap2apEq1 = $true )
& ( theprop = $true )
& ( sepInPowerset = $true )
& ( quantDeMorgan2 = $true )
& ( singletonsuniq = $true )
& ( setextAx = $true )
& ( setukpairinjR = $true )
& ( lam2p = $true )
& ( upairsetIR = $true )
& ( emptyinPowerset = $true )
& ( symdiffI2 = $true )
& ( powersetTE1 = $true )
& ( ex1E1 = $true )
& ( symdiffIneg1 = $true )
& ( dsetconstr__Cong = $true )
& ( emptysetsubset = $true )
& ( contrasubsetT = $true )
& ( setextT = $true )
& ( in__Cong = $true )
& ( descrp = $true )
& ( symdiffI1 = $true )
& ( upairsetE = $true )
& ( subbreln = $true )
& ( beta2 = $true )
& ( setunion__Cong = $true )
& ( ksndsingleton = $true )
& ( quantDeMorgan3 = $true )
& ( funcImageSingleton = $true )
& ( setukpairinjL1 = $true )
& ( setadjoinOr = $true )
& ( disjointsetsI1 = $true )
& ( omegaSAx = $true )
& ( eqinunit = $true )
& ( nonemptyI = $true )
& ( binintersectSubset2 = $true )
& ( kfstpairEq = $true )
& ( exuE1 = $true )
& ( prop2set2propI = $true )
& ( singletoninpowerset = $true )
& ( setminusER = $true )
& ( uniqinunit = $true )
& ( dpsetconstrER = $true )
& ( powersetsubset = $true )
& ( inCongP = $true )
& ( binintersectER = $true )
& ( emptyE1 = $true )
& ( setunionE2 = $true )
& ( singletonprop = $true )
& ( setminusI = $true )
& ( symdiffE = $true )
& ( binintersectRsub = $true )
& ( inPowerset = $true )
& ( descr__Cong = $true )
& ( setadjoinAx = $true )
& ( emptyI = $true )
& ( doubleComplementE1 = $true )
& ( setukpairinjL = $true )
& ( binunionLsub = $true )
& ( eta2 = $true )
& ( ex1I2 = $true )
& ( funcGraphProp4 = $true )
& ( subsetI1 = $true )
& ( binunionIR = $true )
& ( setukpairinjR1 = $true )
& ( binintersectSubset3 = $true )
& ( funcext2 = $true )
& ( funcinfuncset = $true )
& ( binintersectSubset1 = $true )
& ( nonemptyI1 = $true )
& ( funcGraphProp3 = $true )
& ( notdallE = $true )
& ( cartprodmempaircEq = $true )
& ( powersetE1 = $true )
& ( dpsetconstrEL2 = $true )
& ( subsetTI = $true )
& ( setextsub = $true )
& ( setadjoinIL = $true )
& ( setadjoin__Cong = $true )
& ( setadjoinSub2 = $true )
& ( subsetE2 = $true )
& ( cartprodfstin = $true )
& ( funcextLem = $true )
& ( iftrueorfalse = $true )
& ( brelnall1 = $true )
& ( upairsetIL = $true )
& ( doubleComplementSub1 = $true )
& ( ap2apEq2 = $true )
& ( binintersectT_lem = $true )
& ( complementTnotintersectT = $true )
& ( contrasubsetT1 = $true )
& ( upairequniteq = $true )
& ( cartprodpairin = $true )
& ( iffalse = $true )
& ( binintersectLsub = $true )
& ( setext = $true )
& ( lam2lamEq = $true )
& ( setminusILneg = $true )
& ( wellorderingAx = $true )
& ( exuI1 = $true )
& ( apProp = $true )
& ( eta1 = $true )
& ( omega0Ax = $true )
& ( notequalI2 = $true )
& ( powersetTI1 = $true )
& ( powersetT_lem = $true )
& ( setukpairIL = $true )
& ( exuE3u = $true )
& ( emptysetimpfalse = $true )
& ( singletoninpowunion = $true )
& ( app = $true )
& ( setadjoinE = $true )
& ( ex1E2 = $true )
& ( ksndpairEq = $true )
& ( binintersectTELcontra = $true )
& ( cartprodmempair = $true )
& ( iftrue = $true )
& ( notinsingleton = $true )
& ( subsetemptysetimpeq = $true )
& ( setukpairinjR11 = $true )
& ( upairset2E = $true )
& ( kfstsingleton = $true )
& ( iffalseProp1 = $true )
& ( secondinupair = $true )
& ( iftrueProp2 = $true )
& ( lamProp = $true )
& ( setunionsingleton2 = $true )
& ( singletonsubset = $true )
& ( doubleComplementEq = $true )
& ( quantDeMorgan1 = $true )
& ( binunionRsub = $true )
& ( nonemptyE1 = $true )
& ( emptyInPowerset = $true )
& ( setbeta = $true )
& ( prop2setE = $true )
& ( dsetconstrER = $true )
& ( beta1 = $true )
& ( singletonsswitch = $true )
& ( setunionE = $true )
& ( funcGraphProp1 = $true )
& ( setadjoinSub = $true )
& ( nonemptyImpWitness = $true )
& ( brelnall2 = $true )
& ( upairset2IR = $true )
& ( eqimpsubset1 = $true )
& ( setminusSubset2 = $true )
& ( setunionI = $true )
& ( setminusELneg = $true )
& ( funcGraphProp2 = $true )
& ( complementTE1 = $true )
& ( subsetI2 = $true )
& ( exuE3e = $true )
& ( ubforcartprodlem1 = $true ) ),
inference(flattening,[status(thm)],[f948]) ).
thf(f948,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
& ? [X2: $i] :
( ( ( subset @ ( setminus @ X0 @ X1 ) @ ( setminus @ X0 @ ( binintersect @ X1 @ X2 ) ) )
!= $true )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) ) )
& ( complementImpComplementIntersect = $true )
& ( complementTnotintersectT = $true )
& ( doubleComplementEq = $true )
& ( doubleComplementSub2 = $true )
& ( doubleComplementSub1 = $true )
& ( doubleComplementE1 = $true )
& ( doubleComplementI1 = $true )
& ( contrasubsetT3 = $true )
& ( contrasubsetT2 = $true )
& ( contrasubsetT1 = $true )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f403]) ).
thf(f403,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
=> ( ( subset @ ( setminus @ X0 @ X1 ) @ ( setminus @ X0 @ ( binintersect @ X1 @ X2 ) ) )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f402]) ).
thf(f402,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X0 ) )
=> ( subset @ ( setminus @ X0 @ X1 ) @ ( setminus @ X0 @ ( binintersect @ X1 @ X2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f256]) ).
thf(f256,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( subset @ ( setminus @ X3 @ X11 ) @ ( setminus @ X3 @ ( binintersect @ X11 @ X16 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f255]) ).
thf(f255,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( subset @ ( setminus @ X3 @ X11 ) @ ( setminus @ X3 @ ( binintersect @ X11 @ X16 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.XrUd2iUsu2/DTF2THF_28816.p',complementSubsetComplementIntersect) ).
thf(f7298,plain,
( ( binintersectLsub != $true )
| spl792_5 ),
inference(trivial_inequality_removal,[status(thm)],[f7293]) ).
thf(f7293,plain,
( ( binintersectLsub != $true )
| ( $true != $true )
| spl792_5 ),
inference(superposition,[status(thm)],[f3880,f3114]) ).
thf(f3114,plain,
! [X0: $i,X1: $i] :
( ( $true
= ( subset @ ( binintersect @ X1 @ X0 ) @ X1 ) )
| ( binintersectLsub != $true ) ),
inference(cnf_transformation,[status(thm)],[f2036]) ).
thf(f2036,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( subset @ ( binintersect @ X1 @ X0 ) @ X1 ) )
| ( binintersectLsub != $true ) )
& ( ( binintersectLsub = $true )
| ( $true
!= ( subset @ ( binintersect @ sK601 @ sK600 ) @ sK601 ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK600,sK601])],[f2034,f2035]) ).
thf(f2035,plain,
( ? [X2: $i,X3: $i] :
( $true
!= ( subset @ ( binintersect @ X3 @ X2 ) @ X3 ) )
=> ( $true
!= ( subset @ ( binintersect @ sK601 @ sK600 ) @ sK601 ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2034,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( subset @ ( binintersect @ X1 @ X0 ) @ X1 ) )
| ( binintersectLsub != $true ) )
& ( ( binintersectLsub = $true )
| ? [X2: $i,X3: $i] :
( $true
!= ( subset @ ( binintersect @ X3 @ X2 ) @ X3 ) ) ) ),
inference(rectify,[status(thm)],[f2033]) ).
thf(f2033,plain,
( ( ! [X0: $i,X1: $i] :
( $true
= ( subset @ ( binintersect @ X1 @ X0 ) @ X1 ) )
| ( binintersectLsub != $true ) )
& ( ( binintersectLsub = $true )
| ? [X0: $i,X1: $i] :
( $true
!= ( subset @ ( binintersect @ X1 @ X0 ) @ X1 ) ) ) ),
inference(nnf_transformation,[status(thm)],[f269]) ).
thf(f269,plain,
( ! [X0: $i,X1: $i] :
( $true
= ( subset @ ( binintersect @ X1 @ X0 ) @ X1 ) )
<=> ( binintersectLsub = $true ) ),
inference(fool_elimination,[status(thm)],[f268]) ).
thf(f268,plain,
! [X0: $i,X1: $i] :
( ( subset @ ( binintersect @ X1 @ X0 ) @ X1 )
= binintersectLsub ),
inference(rectify,[status(thm)],[f111]) ).
thf(f111,axiom,
! [X4: $i,X3: $i] :
( ( subset @ ( binintersect @ X3 @ X4 ) @ X3 )
= binintersectLsub ),
file('/export/starexec/sandbox/tmp/tmp.XrUd2iUsu2/DTF2THF_28816.p',binintersectLsub) ).
thf(f3880,plain,
( ( $true
!= ( subset @ ( binintersect @ sK697 @ sK698 ) @ sK697 ) )
| spl792_5 ),
inference(avatar_component_clause,[status(thm)],[f3878]) ).
thf(f3878,definition,
( spl792_5
<=> ( $true
= ( subset @ ( binintersect @ sK697 @ sK698 ) @ sK697 ) ) ),
introduced(definition,[new_symbols(naming,[spl792_5])],[avatar_definition]) ).
thf(f7273,plain,
spl792_30,
inference(avatar_contradiction_clause,[status(thm)],[f7272]) ).
thf(f7272,plain,
( $false
| spl792_30 ),
inference(trivial_inequality_removal,[status(thm)],[f7261]) ).
thf(f7261,plain,
( ( $true != $true )
| spl792_30 ),
inference(superposition,[status(thm)],[f4383,f3413]) ).
thf(f3413,plain,
( $true
= ( in @ sK698 @ ( powerset @ sK696 ) ) ),
inference(cnf_transformation,[status(thm)],[f2181]) ).
thf(f4383,plain,
( ( $true
!= ( in @ sK698 @ ( powerset @ sK696 ) ) )
| spl792_30 ),
inference(avatar_component_clause,[status(thm)],[f4381]) ).
thf(f4381,definition,
( spl792_30
<=> ( $true
= ( in @ sK698 @ ( powerset @ sK696 ) ) ) ),
introduced(definition,[new_symbols(naming,[spl792_30])],[avatar_definition]) ).
thf(f4389,plain,
( ~ spl792_30
| spl792_4 ),
inference(avatar_split_clause,[status(thm)],[f4388,f3874,f4381]) ).
thf(f3874,definition,
( spl792_4
<=> ( $true
= ( in @ ( binintersect @ sK697 @ sK698 ) @ ( powerset @ sK696 ) ) ) ),
introduced(definition,[new_symbols(naming,[spl792_4])],[avatar_definition]) ).
thf(f4388,plain,
( ( $true
!= ( in @ sK698 @ ( powerset @ sK696 ) ) )
| spl792_4 ),
inference(trivial_inequality_removal,[status(thm)],[f4387]) ).
thf(f4387,plain,
( ( $true
!= ( in @ sK698 @ ( powerset @ sK696 ) ) )
| ( $true != $true )
| spl792_4 ),
inference(forward_demodulation,[status(thm)],[f4386,f3298]) ).
thf(f3298,plain,
binintersectT_lem = $true,
inference(cnf_transformation,[status(thm)],[f2181]) ).
thf(f4386,plain,
( ( $true
!= ( in @ sK698 @ ( powerset @ sK696 ) ) )
| ( binintersectT_lem != $true )
| spl792_4 ),
inference(trivial_inequality_removal,[status(thm)],[f4385]) ).
thf(f4385,plain,
( ( binintersectT_lem != $true )
| ( $true
!= ( in @ sK698 @ ( powerset @ sK696 ) ) )
| ( $true != $true )
| spl792_4 ),
inference(forward_demodulation,[status(thm)],[f4377,f3415]) ).
thf(f3415,plain,
( $true
= ( in @ sK697 @ ( powerset @ sK696 ) ) ),
inference(cnf_transformation,[status(thm)],[f2181]) ).
thf(f4377,plain,
( ( $true
!= ( in @ sK697 @ ( powerset @ sK696 ) ) )
| ( binintersectT_lem != $true )
| ( $true
!= ( in @ sK698 @ ( powerset @ sK696 ) ) )
| spl792_4 ),
inference(trivial_inequality_removal,[status(thm)],[f4364]) ).
thf(f4364,plain,
( ( $true
!= ( in @ sK697 @ ( powerset @ sK696 ) ) )
| ( binintersectT_lem != $true )
| ( $true
!= ( in @ sK698 @ ( powerset @ sK696 ) ) )
| ( $true != $true )
| spl792_4 ),
inference(superposition,[status(thm)],[f3876,f3034]) ).
thf(f3034,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
= ( in @ ( binintersect @ X1 @ X2 ) @ ( powerset @ X0 ) ) )
| ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) )
| ( binintersectT_lem != $true )
| ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f1937]) ).
thf(f1937,plain,
( ( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
= ( in @ ( binintersect @ X1 @ X2 ) @ ( powerset @ X0 ) ) )
| ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) ) )
| ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) )
| ( binintersectT_lem != $true ) )
& ( ( binintersectT_lem = $true )
| ( ( $true
!= ( in @ ( binintersect @ sK544 @ sK545 ) @ ( powerset @ sK543 ) ) )
& ( $true
= ( in @ sK545 @ ( powerset @ sK543 ) ) )
& ( $true
= ( in @ sK544 @ ( powerset @ sK543 ) ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK543,sK544,sK545])],[f1934,f1936,f1935]) ).
thf(f1935,plain,
( ? [X3: $i,X4: $i] :
( ? [X5: $i] :
( ( $true
!= ( in @ ( binintersect @ X4 @ X5 ) @ ( powerset @ X3 ) ) )
& ( $true
= ( in @ X5 @ ( powerset @ X3 ) ) ) )
& ( ( in @ X4 @ ( powerset @ X3 ) )
= $true ) )
=> ( ? [X5: $i] :
( ( $true
!= ( in @ ( binintersect @ sK544 @ X5 ) @ ( powerset @ sK543 ) ) )
& ( $true
= ( in @ X5 @ ( powerset @ sK543 ) ) ) )
& ( $true
= ( in @ sK544 @ ( powerset @ sK543 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1936,plain,
( ? [X5: $i] :
( ( $true
!= ( in @ ( binintersect @ sK544 @ X5 ) @ ( powerset @ sK543 ) ) )
& ( $true
= ( in @ X5 @ ( powerset @ sK543 ) ) ) )
=> ( ( $true
!= ( in @ ( binintersect @ sK544 @ sK545 ) @ ( powerset @ sK543 ) ) )
& ( $true
= ( in @ sK545 @ ( powerset @ sK543 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1934,plain,
( ( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
= ( in @ ( binintersect @ X1 @ X2 ) @ ( powerset @ X0 ) ) )
| ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) ) )
| ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) )
| ( binintersectT_lem != $true ) )
& ( ( binintersectT_lem = $true )
| ? [X3: $i,X4: $i] :
( ? [X5: $i] :
( ( $true
!= ( in @ ( binintersect @ X4 @ X5 ) @ ( powerset @ X3 ) ) )
& ( $true
= ( in @ X5 @ ( powerset @ X3 ) ) ) )
& ( ( in @ X4 @ ( powerset @ X3 ) )
= $true ) ) ) ),
inference(rectify,[status(thm)],[f1933]) ).
thf(f1933,plain,
( ( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
= ( in @ ( binintersect @ X1 @ X2 ) @ ( powerset @ X0 ) ) )
| ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) ) )
| ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) )
| ( binintersectT_lem != $true ) )
& ( ( binintersectT_lem = $true )
| ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
!= ( in @ ( binintersect @ X1 @ X2 ) @ ( powerset @ X0 ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) )
& ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f1039]) ).
thf(f1039,plain,
( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
= ( in @ ( binintersect @ X1 @ X2 ) @ ( powerset @ X0 ) ) )
| ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) ) )
| ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) )
<=> ( binintersectT_lem = $true ) ),
inference(ennf_transformation,[status(thm)],[f714]) ).
thf(f714,plain,
( ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
=> ( $true
= ( in @ ( binintersect @ X1 @ X2 ) @ ( powerset @ X0 ) ) ) ) )
<=> ( binintersectT_lem = $true ) ),
inference(fool_elimination,[status(thm)],[f713]) ).
thf(f713,plain,
( binintersectT_lem
= ( ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X0 ) )
=> ( in @ ( binintersect @ X1 @ X2 ) @ ( powerset @ X0 ) ) ) ) ) ),
inference(rectify,[status(thm)],[f231]) ).
thf(f231,axiom,
( binintersectT_lem
= ( ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( in @ ( binintersect @ X11 @ X16 ) @ ( powerset @ X3 ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.XrUd2iUsu2/DTF2THF_28816.p',binintersectT_lem) ).
thf(f3876,plain,
( ( $true
!= ( in @ ( binintersect @ sK697 @ sK698 ) @ ( powerset @ sK696 ) ) )
| spl792_4 ),
inference(avatar_component_clause,[status(thm)],[f3874]) ).
thf(f3881,plain,
( ~ spl792_4
| ~ spl792_5 ),
inference(avatar_split_clause,[status(thm)],[f3872,f3878,f3874]) ).
thf(f3872,plain,
( ( $true
!= ( in @ ( binintersect @ sK697 @ sK698 ) @ ( powerset @ sK696 ) ) )
| ( $true
!= ( subset @ ( binintersect @ sK697 @ sK698 ) @ sK697 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3871]) ).
thf(f3871,plain,
( ( $true
!= ( in @ ( binintersect @ sK697 @ sK698 ) @ ( powerset @ sK696 ) ) )
| ( $true
!= ( subset @ ( binintersect @ sK697 @ sK698 ) @ sK697 ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3870,f3421]) ).
thf(f3421,plain,
contrasubsetT2 = $true,
inference(cnf_transformation,[status(thm)],[f2181]) ).
thf(f3870,plain,
( ( contrasubsetT2 != $true )
| ( $true
!= ( subset @ ( binintersect @ sK697 @ sK698 ) @ sK697 ) )
| ( $true
!= ( in @ ( binintersect @ sK697 @ sK698 ) @ ( powerset @ sK696 ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3869]) ).
thf(f3869,plain,
( ( $true != $true )
| ( $true
!= ( in @ ( binintersect @ sK697 @ sK698 ) @ ( powerset @ sK696 ) ) )
| ( contrasubsetT2 != $true )
| ( $true
!= ( subset @ ( binintersect @ sK697 @ sK698 ) @ sK697 ) ) ),
inference(forward_demodulation,[status(thm)],[f3868,f3415]) ).
thf(f3868,plain,
( ( $true
!= ( in @ ( binintersect @ sK697 @ sK698 ) @ ( powerset @ sK696 ) ) )
| ( $true
!= ( in @ sK697 @ ( powerset @ sK696 ) ) )
| ( contrasubsetT2 != $true )
| ( $true
!= ( subset @ ( binintersect @ sK697 @ sK698 ) @ sK697 ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f3859]) ).
thf(f3859,plain,
( ( $true
!= ( subset @ ( binintersect @ sK697 @ sK698 ) @ sK697 ) )
| ( contrasubsetT2 != $true )
| ( $true
!= ( in @ sK697 @ ( powerset @ sK696 ) ) )
| ( $true
!= ( in @ ( binintersect @ sK697 @ sK698 ) @ ( powerset @ sK696 ) ) )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f3414,f2435]) ).
thf(f2435,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( $true
= ( subset @ ( setminus @ X3 @ X5 ) @ ( setminus @ X3 @ X4 ) ) )
| ( contrasubsetT2 != $true )
| ( ( subset @ X4 @ X5 )
!= $true )
| ( ( in @ X4 @ ( powerset @ X3 ) )
!= $true )
| ( $true
!= ( in @ X5 @ ( powerset @ X3 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f1266]) ).
thf(f1266,plain,
( ( ( contrasubsetT2 = $true )
| ( ( $true
= ( in @ sK115 @ ( powerset @ sK113 ) ) )
& ( $true
= ( subset @ sK114 @ sK115 ) )
& ( $true
!= ( subset @ ( setminus @ sK113 @ sK115 ) @ ( setminus @ sK113 @ sK114 ) ) )
& ( $true
= ( in @ sK114 @ ( powerset @ sK113 ) ) ) ) )
& ( ! [X3: $i,X4: $i] :
( ! [X5: $i] :
( ( $true
!= ( in @ X5 @ ( powerset @ X3 ) ) )
| ( ( subset @ X4 @ X5 )
!= $true )
| ( $true
= ( subset @ ( setminus @ X3 @ X5 ) @ ( setminus @ X3 @ X4 ) ) ) )
| ( ( in @ X4 @ ( powerset @ X3 ) )
!= $true ) )
| ( contrasubsetT2 != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK113,sK114,sK115])],[f1263,f1265,f1264]) ).
thf(f1264,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
& ( $true
= ( subset @ X1 @ X2 ) )
& ( $true
!= ( subset @ ( setminus @ X0 @ X2 ) @ ( setminus @ X0 @ X1 ) ) ) )
& ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) )
=> ( ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ sK113 ) ) )
& ( $true
= ( subset @ sK114 @ X2 ) )
& ( $true
!= ( subset @ ( setminus @ sK113 @ X2 ) @ ( setminus @ sK113 @ sK114 ) ) ) )
& ( $true
= ( in @ sK114 @ ( powerset @ sK113 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1265,plain,
( ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ sK113 ) ) )
& ( $true
= ( subset @ sK114 @ X2 ) )
& ( $true
!= ( subset @ ( setminus @ sK113 @ X2 ) @ ( setminus @ sK113 @ sK114 ) ) ) )
=> ( ( $true
= ( in @ sK115 @ ( powerset @ sK113 ) ) )
& ( $true
= ( subset @ sK114 @ sK115 ) )
& ( $true
!= ( subset @ ( setminus @ sK113 @ sK115 ) @ ( setminus @ sK113 @ sK114 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1263,plain,
( ( ( contrasubsetT2 = $true )
| ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
& ( $true
= ( subset @ X1 @ X2 ) )
& ( $true
!= ( subset @ ( setminus @ X0 @ X2 ) @ ( setminus @ X0 @ X1 ) ) ) )
& ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) ) )
& ( ! [X3: $i,X4: $i] :
( ! [X5: $i] :
( ( $true
!= ( in @ X5 @ ( powerset @ X3 ) ) )
| ( ( subset @ X4 @ X5 )
!= $true )
| ( $true
= ( subset @ ( setminus @ X3 @ X5 ) @ ( setminus @ X3 @ X4 ) ) ) )
| ( ( in @ X4 @ ( powerset @ X3 ) )
!= $true ) )
| ( contrasubsetT2 != $true ) ) ),
inference(rectify,[status(thm)],[f1262]) ).
thf(f1262,plain,
( ( ( contrasubsetT2 = $true )
| ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
& ( $true
= ( subset @ X1 @ X2 ) )
& ( $true
!= ( subset @ ( setminus @ X0 @ X2 ) @ ( setminus @ X0 @ X1 ) ) ) )
& ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) ) )
& ( ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) )
| ( $true
!= ( subset @ X1 @ X2 ) )
| ( $true
= ( subset @ ( setminus @ X0 @ X2 ) @ ( setminus @ X0 @ X1 ) ) ) )
| ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) )
| ( contrasubsetT2 != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f1014]) ).
thf(f1014,plain,
( ( contrasubsetT2 = $true )
<=> ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) )
| ( $true
!= ( subset @ X1 @ X2 ) )
| ( $true
= ( subset @ ( setminus @ X0 @ X2 ) @ ( setminus @ X0 @ X1 ) ) ) )
| ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) ) ),
inference(flattening,[status(thm)],[f1013]) ).
thf(f1013,plain,
( ( contrasubsetT2 = $true )
<=> ! [X0: $i,X1: $i] :
( ! [X2: $i] :
( ( $true
= ( subset @ ( setminus @ X0 @ X2 ) @ ( setminus @ X0 @ X1 ) ) )
| ( $true
!= ( subset @ X1 @ X2 ) )
| ( $true
!= ( in @ X2 @ ( powerset @ X0 ) ) ) )
| ( $true
!= ( in @ X1 @ ( powerset @ X0 ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f646]) ).
thf(f646,plain,
( ( contrasubsetT2 = $true )
<=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
=> ( ( $true
= ( subset @ X1 @ X2 ) )
=> ( $true
= ( subset @ ( setminus @ X0 @ X2 ) @ ( setminus @ X0 @ X1 ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f645]) ).
thf(f645,plain,
! [X0: $i,X1: $i] :
( ( ( in @ X1 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X0 ) )
=> ( ( subset @ X1 @ X2 )
=> ( subset @ ( setminus @ X0 @ X2 ) @ ( setminus @ X0 @ X1 ) ) ) ) )
= contrasubsetT2 ),
inference(rectify,[status(thm)],[f246]) ).
thf(f246,axiom,
! [X3: $i,X11: $i] :
( ( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ( ( subset @ X11 @ X16 )
=> ( subset @ ( setminus @ X3 @ X16 ) @ ( setminus @ X3 @ X11 ) ) ) ) )
= contrasubsetT2 ),
file('/export/starexec/sandbox/tmp/tmp.XrUd2iUsu2/DTF2THF_28816.p',contrasubsetT2) ).
thf(f3414,plain,
( $true
!= ( subset @ ( setminus @ sK696 @ sK697 ) @ ( setminus @ sK696 @ ( binintersect @ sK697 @ sK698 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f2181]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SEU734^1 : TPTP v9.2.1. Released v3.7.0.
% 0.00/0.10 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.11/0.29 % Computer : n016.cluster.edu
% 0.11/0.29 % Model : x86_64 x86_64
% 0.11/0.29 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.29 % Memory : 8042.1875MB
% 0.11/0.29 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.29 % CPULimit : 300
% 0.11/0.29 % WCLimit : 300
% 0.11/0.29 % DateTime : Tue Feb 24 12:32:51 EST 2026
% 0.11/0.29 % CPUTime :
% 0.11/0.29 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.37 ---- Original DTF file ---
% 0.15/0.37 thf(spec,logic,$$dhol).
% 0.15/0.37 %------------------------------------------------------------------------------
% 0.15/0.37 % File : SEU734^1 : TPTP v9.2.1. Released v3.7.0.
% 0.15/0.37 % Domain : Set Theory
% 0.15/0.37 % Problem : Typed Set Theory - Laws for Typed Sets
% 0.15/0.37 % Version : Especial.
% 0.15/0.37 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.15/0.37 % subset (setminus A X) (setminus A (binintersect X Y))))
% 0.15/0.37
% 0.15/0.37 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.15/0.37 % Source : [Bro08]
% 0.15/0.37 % Names : ZFC236g [Bro08]
% 0.15/0.37
% 0.15/0.37 % Status : Theorem
% 0.15/0.37 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.18 v7.5.0, 0.14 v7.4.0, 0.22 v7.2.0, 0.12 v7.1.0, 0.38 v7.0.0, 0.29 v6.4.0, 0.33 v6.3.0, 0.40 v6.2.0, 0.29 v6.1.0, 0.43 v5.5.0, 0.50 v5.4.0, 0.60 v5.2.0, 1.00 v3.7.0
% 0.15/0.37 % Syntax : Number of formulae : 543 ( 254 unt; 288 typ; 254 def)
% 0.15/0.37 % Number of atoms : 1461 ( 380 equ; 0 cnn)
% 0.15/0.37 % Maximal formula atoms : 253 ( 5 avg)
% 0.15/0.37 % Number of connectives : 2912 ( 65 ~; 12 |; 44 &;2069 @)
% 0.15/0.37 % ( 14 <=>; 708 =>; 0 <=; 0 <~>)
% 0.15/0.37 % Maximal formula depth : 260 ( 2 avg)
% 0.15/0.37 % Number of types : 2 ( 0 usr)
% 0.15/0.37 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.15/0.37 % Number of symbols : 291 ( 288 usr; 254 con; 0-4 aty)
% 0.15/0.37 % Number of variables : 853 ( 74 ^; 744 !; 35 ?; 853 :)
% 0.15/0.37 % SPC : TH0_THM_EQU_NAR
% 0.15/0.37
% 0.15/0.37 % Comments : http://mathgate.info/detsetitem.php?id=296
% 0.15/0.37 %------------------------------------------------------------------------------
% 0.15/0.37 thf(in_type,type,
% 0.15/0.37 in: $i > $i > $o ).
% 0.15/0.37
% 0.15/0.37 thf(exu_type,type,
% 0.15/0.37 exu: ( $i > $o ) > $o ).
% 0.15/0.37
% 0.15/0.37 thf(exu,definition,
% 0.15/0.37 ( exu
% 0.15/0.37 = ( ^ [Xphi: $i > $o] :
% 0.15/0.37 ? [Xx: $i] :
% 0.15/0.37 ( ( Xphi @ Xx )
% 0.15/0.37 & ! [Xy: $i] :
% 0.15/0.37 ( ( Xphi @ Xy )
% 0.15/0.37 => ( Xx = Xy ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setextAx_type,type,
% 0.15/0.37 setextAx: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setextAx,definition,
% 0.15/0.37 ( setextAx
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 <=> ( in @ Xx @ B ) )
% 0.15/0.37 => ( A = B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(emptyset_type,type,
% 0.15/0.37 emptyset: $i ).
% 0.15/0.37
% 0.15/0.37 thf(emptysetAx_type,type,
% 0.15/0.37 emptysetAx: $o ).
% 0.15/0.37
% 0.15/0.37 thf(emptysetAx,definition,
% 0.15/0.37 ( emptysetAx
% 0.15/0.37 = ( ! [Xx: $i] :
% 0.15/0.37 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoin_type,type,
% 0.15/0.37 setadjoin: $i > $i > $i ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoinAx_type,type,
% 0.15/0.37 setadjoinAx: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoinAx,definition,
% 0.15/0.37 ( setadjoinAx
% 0.15/0.37 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.15/0.37 <=> ( ( Xy = Xx )
% 0.15/0.37 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(powerset_type,type,
% 0.15/0.37 powerset: $i > $i ).
% 0.15/0.37
% 0.15/0.37 thf(powersetAx_type,type,
% 0.15/0.37 powersetAx: $o ).
% 0.15/0.37
% 0.15/0.37 thf(powersetAx,definition,
% 0.15/0.37 ( powersetAx
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( in @ B @ ( powerset @ A ) )
% 0.15/0.37 <=> ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ B )
% 0.15/0.37 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setunion_type,type,
% 0.15/0.37 setunion: $i > $i ).
% 0.15/0.37
% 0.15/0.37 thf(setunionAx_type,type,
% 0.15/0.37 setunionAx: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setunionAx,definition,
% 0.15/0.37 ( setunionAx
% 0.15/0.37 = ( ! [A: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ ( setunion @ A ) )
% 0.15/0.37 <=> ? [B: $i] :
% 0.15/0.37 ( ( in @ Xx @ B )
% 0.15/0.37 & ( in @ B @ A ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(omega_type,type,
% 0.15/0.37 omega: $i ).
% 0.15/0.37
% 0.15/0.37 thf(omega0Ax_type,type,
% 0.15/0.37 omega0Ax: $o ).
% 0.15/0.37
% 0.15/0.37 thf(omega0Ax,definition,
% 0.15/0.37 ( omega0Ax
% 0.15/0.37 = ( in @ emptyset @ omega ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(omegaSAx_type,type,
% 0.15/0.37 omegaSAx: $o ).
% 0.15/0.37
% 0.15/0.37 thf(omegaSAx,definition,
% 0.15/0.37 ( omegaSAx
% 0.15/0.37 = ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ omega )
% 0.15/0.37 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(omegaIndAx_type,type,
% 0.15/0.37 omegaIndAx: $o ).
% 0.15/0.37
% 0.15/0.37 thf(omegaIndAx,definition,
% 0.15/0.37 ( omegaIndAx
% 0.15/0.37 = ( ! [A: $i] :
% 0.15/0.37 ( ( ( in @ emptyset @ A )
% 0.15/0.37 & ! [Xx: $i] :
% 0.15/0.37 ( ( ( in @ Xx @ omega )
% 0.15/0.37 & ( in @ Xx @ A ) )
% 0.15/0.37 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ omega )
% 0.15/0.37 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(replAx_type,type,
% 0.15/0.37 replAx: $o ).
% 0.15/0.37
% 0.15/0.37 thf(replAx,definition,
% 0.15/0.37 ( replAx
% 0.15/0.37 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( exu
% 0.15/0.37 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.15/0.37 => ? [B: $i] :
% 0.15/0.37 ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ B )
% 0.15/0.37 <=> ? [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ A )
% 0.15/0.37 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(foundationAx_type,type,
% 0.15/0.37 foundationAx: $o ).
% 0.15/0.37
% 0.15/0.37 thf(foundationAx,definition,
% 0.15/0.37 ( foundationAx
% 0.15/0.37 = ( ! [A: $i] :
% 0.15/0.37 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.15/0.37 => ? [B: $i] :
% 0.15/0.37 ( ( in @ B @ A )
% 0.15/0.37 & ~ ? [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ B )
% 0.15/0.37 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(wellorderingAx_type,type,
% 0.15/0.37 wellorderingAx: $o ).
% 0.15/0.37
% 0.15/0.37 thf(wellorderingAx,definition,
% 0.15/0.37 ( wellorderingAx
% 0.15/0.37 = ( ! [A: $i] :
% 0.15/0.37 ? [B: $i] :
% 0.15/0.37 ( ! [C: $i] :
% 0.15/0.37 ( ( in @ C @ B )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ C )
% 0.15/0.37 => ( in @ Xx @ A ) ) )
% 0.15/0.37 & ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( ( in @ Xx @ A )
% 0.15/0.37 & ( in @ Xy @ A ) )
% 0.15/0.37 => ( ! [C: $i] :
% 0.15/0.37 ( ( in @ C @ B )
% 0.15/0.37 => ( ( in @ Xx @ C )
% 0.15/0.37 <=> ( in @ Xy @ C ) ) )
% 0.15/0.37 => ( Xx = Xy ) ) )
% 0.15/0.37 & ! [C: $i,D: $i] :
% 0.15/0.37 ( ( ( in @ C @ B )
% 0.15/0.37 & ( in @ D @ B ) )
% 0.15/0.37 => ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ C )
% 0.15/0.37 => ( in @ Xx @ D ) )
% 0.15/0.37 | ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ D )
% 0.15/0.37 => ( in @ Xx @ C ) ) ) )
% 0.15/0.37 & ! [C: $i] :
% 0.15/0.37 ( ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ C )
% 0.15/0.37 => ( in @ Xx @ A ) )
% 0.15/0.37 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.15/0.37 => ? [D: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ D @ B )
% 0.15/0.37 & ( in @ Xx @ C )
% 0.15/0.37 & ~ ? [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ D )
% 0.15/0.37 & ( in @ Xy @ C ) )
% 0.15/0.37 & ! [E: $i] :
% 0.15/0.37 ( ( in @ E @ B )
% 0.15/0.37 => ( ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ E )
% 0.15/0.37 => ( in @ Xy @ D ) )
% 0.15/0.37 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(descr_type,type,
% 0.15/0.37 descr: ( $i > $o ) > $i ).
% 0.15/0.37
% 0.15/0.37 thf(descrp_type,type,
% 0.15/0.37 descrp: $o ).
% 0.15/0.37
% 0.15/0.37 thf(descrp,definition,
% 0.15/0.37 ( descrp
% 0.15/0.37 = ( ! [Xphi: $i > $o] :
% 0.15/0.37 ( ( exu
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.37 => ( Xphi
% 0.15/0.37 @ ( descr
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(dsetconstr_type,type,
% 0.15/0.37 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.15/0.37
% 0.15/0.37 thf(dsetconstrI_type,type,
% 0.15/0.37 dsetconstrI: $o ).
% 0.15/0.37
% 0.15/0.37 thf(dsetconstrI,definition,
% 0.15/0.37 ( dsetconstrI
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( Xphi @ Xx )
% 0.15/0.37 => ( in @ Xx
% 0.15/0.37 @ ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(dsetconstrEL_type,type,
% 0.15/0.37 dsetconstrEL: $o ).
% 0.15/0.37
% 0.15/0.37 thf(dsetconstrEL,definition,
% 0.15/0.37 ( dsetconstrEL
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx
% 0.15/0.37 @ ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.15/0.37 => ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(dsetconstrER_type,type,
% 0.15/0.37 dsetconstrER: $o ).
% 0.15/0.37
% 0.15/0.37 thf(dsetconstrER,definition,
% 0.15/0.37 ( dsetconstrER
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx
% 0.15/0.37 @ ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.15/0.37 => ( Xphi @ Xx ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(exuE1_type,type,
% 0.15/0.37 exuE1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(exuE1,definition,
% 0.15/0.37 ( exuE1
% 0.15/0.37 = ( ! [Xphi: $i > $o] :
% 0.15/0.37 ( ( exu
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.37 => ? [Xx: $i] :
% 0.15/0.37 ( ( Xphi @ Xx )
% 0.15/0.37 & ! [Xy: $i] :
% 0.15/0.37 ( ( Xphi @ Xy )
% 0.15/0.37 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(prop2set_type,type,
% 0.15/0.37 prop2set: $o > $i ).
% 0.15/0.37
% 0.15/0.37 thf(prop2setE_type,type,
% 0.15/0.37 prop2setE: $o ).
% 0.15/0.37
% 0.15/0.37 thf(prop2setE,definition,
% 0.15/0.37 ( prop2setE
% 0.15/0.37 = ( ! [Xphi: $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.15/0.37 => Xphi ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(emptysetE_type,type,
% 0.15/0.37 emptysetE: $o ).
% 0.15/0.37
% 0.15/0.37 thf(emptysetE,definition,
% 0.15/0.37 ( emptysetE
% 0.15/0.37 = ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ emptyset )
% 0.15/0.37 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(emptysetimpfalse_type,type,
% 0.15/0.37 emptysetimpfalse: $o ).
% 0.15/0.37
% 0.15/0.37 thf(emptysetimpfalse,definition,
% 0.15/0.37 ( emptysetimpfalse
% 0.15/0.37 = ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ emptyset )
% 0.15/0.37 => $false ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(notinemptyset_type,type,
% 0.15/0.37 notinemptyset: $o ).
% 0.15/0.37
% 0.15/0.37 thf(notinemptyset,definition,
% 0.15/0.37 ( notinemptyset
% 0.15/0.37 = ( ! [Xx: $i] :
% 0.15/0.37 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(exuE3e_type,type,
% 0.15/0.37 exuE3e: $o ).
% 0.15/0.37
% 0.15/0.37 thf(exuE3e,definition,
% 0.15/0.37 ( exuE3e
% 0.15/0.37 = ( ! [Xphi: $i > $o] :
% 0.15/0.37 ( ( exu
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.37 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setext_type,type,
% 0.15/0.37 setext: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setext,definition,
% 0.15/0.37 ( setext
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ Xx @ B ) )
% 0.15/0.37 => ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ B )
% 0.15/0.37 => ( in @ Xx @ A ) )
% 0.15/0.37 => ( A = B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(emptyI_type,type,
% 0.15/0.37 emptyI: $o ).
% 0.15/0.37
% 0.15/0.37 thf(emptyI,definition,
% 0.15/0.37 ( emptyI
% 0.15/0.37 = ( ! [A: $i] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ~ ( in @ Xx @ A )
% 0.15/0.37 => ( A = emptyset ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(noeltsimpempty_type,type,
% 0.15/0.37 noeltsimpempty: $o ).
% 0.15/0.37
% 0.15/0.37 thf(noeltsimpempty,definition,
% 0.15/0.37 ( noeltsimpempty
% 0.15/0.37 = ( ! [A: $i] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ~ ( in @ Xx @ A )
% 0.15/0.37 => ( A = emptyset ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setbeta_type,type,
% 0.15/0.37 setbeta: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setbeta,definition,
% 0.15/0.37 ( setbeta
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( in @ Xx
% 0.15/0.37 @ ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.15/0.37 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(nonempty_type,type,
% 0.15/0.37 nonempty: $i > $o ).
% 0.15/0.37
% 0.15/0.37 thf(nonempty,definition,
% 0.15/0.37 ( nonempty
% 0.15/0.37 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(nonemptyE1_type,type,
% 0.15/0.37 nonemptyE1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(nonemptyE1,definition,
% 0.15/0.37 ( nonemptyE1
% 0.15/0.37 = ( ! [A: $i] :
% 0.15/0.37 ( ( nonempty @ A )
% 0.15/0.37 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(nonemptyI_type,type,
% 0.15/0.37 nonemptyI: $o ).
% 0.15/0.37
% 0.15/0.37 thf(nonemptyI,definition,
% 0.15/0.37 ( nonemptyI
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( Xphi @ Xx )
% 0.15/0.37 => ( nonempty
% 0.15/0.37 @ ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(nonemptyI1_type,type,
% 0.15/0.37 nonemptyI1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(nonemptyI1,definition,
% 0.15/0.37 ( nonemptyI1
% 0.15/0.37 = ( ! [A: $i] :
% 0.15/0.37 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.15/0.37 => ( nonempty @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoinIL_type,type,
% 0.15/0.37 setadjoinIL: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoinIL,definition,
% 0.15/0.37 ( setadjoinIL
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(emptyinunitempty_type,type,
% 0.15/0.37 emptyinunitempty: $o ).
% 0.15/0.37
% 0.15/0.37 thf(emptyinunitempty,definition,
% 0.15/0.37 ( emptyinunitempty
% 0.15/0.37 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoinIR_type,type,
% 0.15/0.37 setadjoinIR: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoinIR,definition,
% 0.15/0.37 ( setadjoinIR
% 0.15/0.37 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ A )
% 0.15/0.37 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoinE_type,type,
% 0.15/0.37 setadjoinE: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoinE,definition,
% 0.15/0.37 ( setadjoinE
% 0.15/0.37 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.15/0.37 => ! [Xphi: $o] :
% 0.15/0.37 ( ( ( Xy = Xx )
% 0.15/0.37 => Xphi )
% 0.15/0.37 => ( ( ( in @ Xy @ A )
% 0.15/0.37 => Xphi )
% 0.15/0.37 => Xphi ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoinOr_type,type,
% 0.15/0.37 setadjoinOr: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoinOr,definition,
% 0.15/0.37 ( setadjoinOr
% 0.15/0.37 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.15/0.37 => ( ( Xy = Xx )
% 0.15/0.37 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setoftrueEq_type,type,
% 0.15/0.37 setoftrueEq: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setoftrueEq,definition,
% 0.15/0.37 ( setoftrueEq
% 0.15/0.37 = ( ! [A: $i] :
% 0.15/0.37 ( ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xx: $i] : $true )
% 0.15/0.37 = A ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(powersetI_type,type,
% 0.15/0.37 powersetI: $o ).
% 0.15/0.37
% 0.15/0.37 thf(powersetI,definition,
% 0.15/0.37 ( powersetI
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ B )
% 0.15/0.37 => ( in @ Xx @ A ) )
% 0.15/0.37 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(emptyinPowerset_type,type,
% 0.15/0.37 emptyinPowerset: $o ).
% 0.15/0.37
% 0.15/0.37 thf(emptyinPowerset,definition,
% 0.15/0.37 ( emptyinPowerset
% 0.15/0.37 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(emptyInPowerset_type,type,
% 0.15/0.37 emptyInPowerset: $o ).
% 0.15/0.37
% 0.15/0.37 thf(emptyInPowerset,definition,
% 0.15/0.37 ( emptyInPowerset
% 0.15/0.37 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(powersetE_type,type,
% 0.15/0.37 powersetE: $o ).
% 0.15/0.37
% 0.15/0.37 thf(powersetE,definition,
% 0.15/0.37 ( powersetE
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ B @ ( powerset @ A ) )
% 0.15/0.37 => ( ( in @ Xx @ B )
% 0.15/0.37 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setunionI_type,type,
% 0.15/0.37 setunionI: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setunionI,definition,
% 0.15/0.37 ( setunionI
% 0.15/0.37 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.15/0.37 ( ( in @ Xx @ B )
% 0.15/0.37 => ( ( in @ B @ A )
% 0.15/0.37 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setunionE_type,type,
% 0.15/0.37 setunionE: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setunionE,definition,
% 0.15/0.37 ( setunionE
% 0.15/0.37 = ( ! [A: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ ( setunion @ A ) )
% 0.15/0.37 => ! [Xphi: $o] :
% 0.15/0.37 ( ! [B: $i] :
% 0.15/0.37 ( ( in @ Xx @ B )
% 0.15/0.37 => ( ( in @ B @ A )
% 0.15/0.37 => Xphi ) )
% 0.15/0.37 => Xphi ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(subPowSU_type,type,
% 0.15/0.37 subPowSU: $o ).
% 0.15/0.37
% 0.15/0.37 thf(subPowSU,definition,
% 0.15/0.37 ( subPowSU
% 0.15/0.37 = ( ! [A: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(exuE2_type,type,
% 0.15/0.37 exuE2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(exuE2,definition,
% 0.15/0.37 ( exuE2
% 0.15/0.37 = ( ! [Xphi: $i > $o] :
% 0.15/0.37 ( ( exu
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.37 => ? [Xx: $i] :
% 0.15/0.37 ! [Xy: $i] :
% 0.15/0.37 ( ( Xphi @ Xy )
% 0.15/0.37 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(nonemptyImpWitness_type,type,
% 0.15/0.37 nonemptyImpWitness: $o ).
% 0.15/0.37
% 0.15/0.37 thf(nonemptyImpWitness,definition,
% 0.15/0.37 ( nonemptyImpWitness
% 0.15/0.37 = ( ! [A: $i] :
% 0.15/0.37 ( ( nonempty @ A )
% 0.15/0.37 => ? [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 & $true ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(uniqinunit_type,type,
% 0.15/0.37 uniqinunit: $o ).
% 0.15/0.37
% 0.15/0.37 thf(uniqinunit,definition,
% 0.15/0.37 ( uniqinunit
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.15/0.37 => ( Xx = Xy ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(notinsingleton_type,type,
% 0.15/0.37 notinsingleton: $o ).
% 0.15/0.37
% 0.15/0.37 thf(notinsingleton,definition,
% 0.15/0.37 ( notinsingleton
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( Xx != Xy )
% 0.15/0.37 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(eqinunit_type,type,
% 0.15/0.37 eqinunit: $o ).
% 0.15/0.37
% 0.15/0.37 thf(eqinunit,definition,
% 0.15/0.37 ( eqinunit
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( Xx = Xy )
% 0.15/0.37 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(singletonsswitch_type,type,
% 0.15/0.37 singletonsswitch: $o ).
% 0.15/0.37
% 0.15/0.37 thf(singletonsswitch,definition,
% 0.15/0.37 ( singletonsswitch
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.15/0.37 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(upairsetE_type,type,
% 0.15/0.37 upairsetE: $o ).
% 0.15/0.37
% 0.15/0.37 thf(upairsetE,definition,
% 0.15/0.37 ( upairsetE
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.15/0.37 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.15/0.37 => ( ( Xz = Xx )
% 0.15/0.37 | ( Xz = Xy ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(upairsetIL_type,type,
% 0.15/0.37 upairsetIL: $o ).
% 0.15/0.37
% 0.15/0.37 thf(upairsetIL,definition,
% 0.15/0.37 ( upairsetIL
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(upairsetIR_type,type,
% 0.15/0.37 upairsetIR: $o ).
% 0.15/0.37
% 0.15/0.37 thf(upairsetIR,definition,
% 0.15/0.37 ( upairsetIR
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(emptyE1_type,type,
% 0.15/0.37 emptyE1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(emptyE1,definition,
% 0.15/0.37 ( emptyE1
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.37 ( ? [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 & ( Xphi @ Xx ) )
% 0.15/0.37 => ( ( ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.37 = emptyset )
% 0.15/0.37 => $false ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(vacuousDall_type,type,
% 0.15/0.37 vacuousDall: $o ).
% 0.15/0.37
% 0.15/0.37 thf(vacuousDall,definition,
% 0.15/0.37 ( vacuousDall
% 0.15/0.37 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ emptyset )
% 0.15/0.37 => ( Xphi @ Xx ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(quantDeMorgan1_type,type,
% 0.15/0.37 quantDeMorgan1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(quantDeMorgan1,definition,
% 0.15/0.37 ( quantDeMorgan1
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.37 ( ~ ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( Xphi @ Xx ) )
% 0.15/0.37 => ? [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(quantDeMorgan2_type,type,
% 0.15/0.37 quantDeMorgan2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(quantDeMorgan2,definition,
% 0.15/0.37 ( quantDeMorgan2
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ~ ( Xphi @ Xx ) )
% 0.15/0.37 => ~ ? [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(quantDeMorgan3_type,type,
% 0.15/0.37 quantDeMorgan3: $o ).
% 0.15/0.37
% 0.15/0.37 thf(quantDeMorgan3,definition,
% 0.15/0.37 ( quantDeMorgan3
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.37 ( ~ ? [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 & ( Xphi @ Xx ) )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(quantDeMorgan4_type,type,
% 0.15/0.37 quantDeMorgan4: $o ).
% 0.15/0.37
% 0.15/0.37 thf(quantDeMorgan4,definition,
% 0.15/0.37 ( quantDeMorgan4
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.37 ( ? [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 & ~ ( Xphi @ Xx ) )
% 0.15/0.37 => ~ ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(prop2setI_type,type,
% 0.15/0.37 prop2setI: $o ).
% 0.15/0.37
% 0.15/0.37 thf(prop2setI,definition,
% 0.15/0.37 ( prop2setI
% 0.15/0.37 = ( ! [Xphi: $o] :
% 0.15/0.37 ( Xphi
% 0.15/0.37 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(set2prop_type,type,
% 0.15/0.37 set2prop: $i > $o ).
% 0.15/0.37
% 0.15/0.37 thf(prop2set2propI_type,type,
% 0.15/0.37 prop2set2propI: $o ).
% 0.15/0.37
% 0.15/0.37 thf(prop2set2propI,definition,
% 0.15/0.37 ( prop2set2propI
% 0.15/0.37 = ( ! [Xphi: $o] :
% 0.15/0.37 ( Xphi
% 0.15/0.37 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(notdexE_type,type,
% 0.15/0.37 notdexE: $o ).
% 0.15/0.37
% 0.15/0.37 thf(notdexE,definition,
% 0.15/0.37 ( notdexE
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.37 ( ~ ? [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 & ( Xphi @ Xx ) )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(notdallE_type,type,
% 0.15/0.37 notdallE: $o ).
% 0.15/0.37
% 0.15/0.37 thf(notdallE,definition,
% 0.15/0.37 ( notdallE
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.37 ( ~ ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( Xphi @ Xx ) )
% 0.15/0.37 => ? [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(exuI1_type,type,
% 0.15/0.37 exuI1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(exuI1,definition,
% 0.15/0.37 ( exuI1
% 0.15/0.37 = ( ! [Xphi: $i > $o] :
% 0.15/0.37 ( ? [Xx: $i] :
% 0.15/0.37 ( ( Xphi @ Xx )
% 0.15/0.37 & ! [Xy: $i] :
% 0.15/0.37 ( ( Xphi @ Xy )
% 0.15/0.37 => ( Xx = Xy ) ) )
% 0.15/0.37 => ( exu
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(exuI3_type,type,
% 0.15/0.37 exuI3: $o ).
% 0.15/0.37
% 0.15/0.37 thf(exuI3,definition,
% 0.15/0.37 ( exuI3
% 0.15/0.37 = ( ! [Xphi: $i > $o] :
% 0.15/0.37 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.15/0.37 => ( ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( Xphi @ Xx )
% 0.15/0.37 => ( ( Xphi @ Xy )
% 0.15/0.37 => ( Xx = Xy ) ) )
% 0.15/0.37 => ( exu
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(exuI2_type,type,
% 0.15/0.37 exuI2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(exuI2,definition,
% 0.15/0.37 ( exuI2
% 0.15/0.37 = ( ! [Xphi: $i > $o] :
% 0.15/0.37 ( ? [Xx: $i] :
% 0.15/0.37 ! [Xy: $i] :
% 0.15/0.37 ( ( Xphi @ Xy )
% 0.15/0.37 <=> ( Xy = Xx ) )
% 0.15/0.37 => ( exu
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(inCongP_type,type,
% 0.15/0.37 inCongP: $o ).
% 0.15/0.37
% 0.15/0.37 thf(inCongP,definition,
% 0.15/0.37 ( inCongP
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( A = B )
% 0.15/0.37 => ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( Xx = Xy )
% 0.15/0.37 => ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(in__Cong_type,type,
% 0.15/0.37 in__Cong: $o ).
% 0.15/0.37
% 0.15/0.37 thf(in__Cong,definition,
% 0.15/0.37 ( in__Cong
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( A = B )
% 0.15/0.37 => ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( Xx = Xy )
% 0.15/0.37 => ( ( in @ Xx @ A )
% 0.15/0.37 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(exuE3u_type,type,
% 0.15/0.37 exuE3u: $o ).
% 0.15/0.37
% 0.15/0.37 thf(exuE3u,definition,
% 0.15/0.37 ( exuE3u
% 0.15/0.37 = ( ! [Xphi: $i > $o] :
% 0.15/0.37 ( ( exu
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.37 => ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( Xphi @ Xx )
% 0.15/0.37 => ( ( Xphi @ Xy )
% 0.15/0.37 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(exu__Cong_type,type,
% 0.15/0.37 exu__Cong: $o ).
% 0.15/0.37
% 0.15/0.37 thf(exu__Cong,definition,
% 0.15/0.37 ( exu__Cong
% 0.15/0.37 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.15/0.37 ( ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( Xx = Xy )
% 0.15/0.37 => ( ( Xphi @ Xx )
% 0.15/0.37 <=> ( Xpsi @ Xy ) ) )
% 0.15/0.37 => ( ( exu
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.37 <=> ( exu
% 0.15/0.37 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(emptyset__Cong_type,type,
% 0.15/0.37 emptyset__Cong: $o ).
% 0.15/0.37
% 0.15/0.37 thf(emptyset__Cong,definition,
% 0.15/0.37 ( emptyset__Cong
% 0.15/0.37 = ( emptyset = emptyset ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoin__Cong_type,type,
% 0.15/0.37 setadjoin__Cong: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoin__Cong,definition,
% 0.15/0.37 ( setadjoin__Cong
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( Xx = Xy )
% 0.15/0.37 => ! [Xz: $i,Xu: $i] :
% 0.15/0.37 ( ( Xz = Xu )
% 0.15/0.37 => ( ( setadjoin @ Xx @ Xz )
% 0.15/0.37 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(powerset__Cong_type,type,
% 0.15/0.37 powerset__Cong: $o ).
% 0.15/0.37
% 0.15/0.37 thf(powerset__Cong,definition,
% 0.15/0.37 ( powerset__Cong
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( A = B )
% 0.15/0.37 => ( ( powerset @ A )
% 0.15/0.37 = ( powerset @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setunion__Cong_type,type,
% 0.15/0.37 setunion__Cong: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setunion__Cong,definition,
% 0.15/0.37 ( setunion__Cong
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( A = B )
% 0.15/0.37 => ( ( setunion @ A )
% 0.15/0.37 = ( setunion @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(omega__Cong_type,type,
% 0.15/0.37 omega__Cong: $o ).
% 0.15/0.37
% 0.15/0.37 thf(omega__Cong,definition,
% 0.15/0.37 ( omega__Cong
% 0.15/0.37 = ( omega = omega ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(exuEu_type,type,
% 0.15/0.37 exuEu: $o ).
% 0.15/0.37
% 0.15/0.37 thf(exuEu,definition,
% 0.15/0.37 ( exuEu
% 0.15/0.37 = ( ! [Xphi: $i > $o] :
% 0.15/0.37 ( ( exu
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.37 => ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( Xphi @ Xx )
% 0.15/0.37 => ( ( Xphi @ Xy )
% 0.15/0.37 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(descr__Cong_type,type,
% 0.15/0.37 descr__Cong: $o ).
% 0.15/0.37
% 0.15/0.37 thf(descr__Cong,definition,
% 0.15/0.37 ( descr__Cong
% 0.15/0.37 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.15/0.37 ( ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( Xx = Xy )
% 0.15/0.37 => ( ( Xphi @ Xx )
% 0.15/0.37 <=> ( Xpsi @ Xy ) ) )
% 0.15/0.37 => ( ( exu
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.37 => ( ( exu
% 0.15/0.37 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.15/0.37 => ( ( descr
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.37 = ( descr
% 0.15/0.37 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(dsetconstr__Cong_type,type,
% 0.15/0.37 dsetconstr__Cong: $o ).
% 0.15/0.37
% 0.15/0.37 thf(dsetconstr__Cong,definition,
% 0.15/0.37 ( dsetconstr__Cong
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( A = B )
% 0.15/0.37 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( ( Xx = Xy )
% 0.15/0.37 => ( ( Xphi @ Xx )
% 0.15/0.37 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.15/0.37 => ( ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.37 = ( dsetconstr @ B
% 0.15/0.37 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(subset_type,type,
% 0.15/0.37 subset: $i > $i > $o ).
% 0.15/0.37
% 0.15/0.37 thf(disjoint_type,type,
% 0.15/0.37 disjoint: $i > $i > $o ).
% 0.15/0.37
% 0.15/0.37 thf(setsmeet_type,type,
% 0.15/0.37 setsmeet: $i > $i > $o ).
% 0.15/0.37
% 0.15/0.37 thf(subsetI1_type,type,
% 0.15/0.37 subsetI1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(subsetI1,definition,
% 0.15/0.37 ( subsetI1
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ Xx @ B ) )
% 0.15/0.37 => ( subset @ A @ B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(eqimpsubset2_type,type,
% 0.15/0.37 eqimpsubset2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(eqimpsubset2,definition,
% 0.15/0.37 ( eqimpsubset2
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( A = B )
% 0.15/0.37 => ( subset @ B @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(eqimpsubset1_type,type,
% 0.15/0.37 eqimpsubset1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(eqimpsubset1,definition,
% 0.15/0.37 ( eqimpsubset1
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( A = B )
% 0.15/0.37 => ( subset @ A @ B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(subsetI2_type,type,
% 0.15/0.37 subsetI2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(subsetI2,definition,
% 0.15/0.37 ( subsetI2
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ Xx @ B ) )
% 0.15/0.37 => ( subset @ A @ B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(emptysetsubset_type,type,
% 0.15/0.37 emptysetsubset: $o ).
% 0.15/0.37
% 0.15/0.37 thf(emptysetsubset,definition,
% 0.15/0.37 ( emptysetsubset
% 0.15/0.37 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(subsetE_type,type,
% 0.15/0.37 subsetE: $o ).
% 0.15/0.37
% 0.15/0.37 thf(subsetE,definition,
% 0.15/0.37 ( subsetE
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( subset @ A @ B )
% 0.15/0.37 => ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(subsetE2_type,type,
% 0.15/0.37 subsetE2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(subsetE2,definition,
% 0.15/0.37 ( subsetE2
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( subset @ A @ B )
% 0.15/0.37 => ( ~ ( in @ Xx @ B )
% 0.15/0.37 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(notsubsetI_type,type,
% 0.15/0.37 notsubsetI: $o ).
% 0.15/0.37
% 0.15/0.37 thf(notsubsetI,definition,
% 0.15/0.37 ( notsubsetI
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ~ ( in @ Xx @ B )
% 0.15/0.37 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(notequalI1_type,type,
% 0.15/0.37 notequalI1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(notequalI1,definition,
% 0.15/0.37 ( notequalI1
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ~ ( subset @ A @ B )
% 0.15/0.37 => ( A != B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(notequalI2_type,type,
% 0.15/0.37 notequalI2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(notequalI2,definition,
% 0.15/0.37 ( notequalI2
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ~ ( in @ Xx @ B )
% 0.15/0.37 => ( A != B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(subsetRefl_type,type,
% 0.15/0.37 subsetRefl: $o ).
% 0.15/0.37
% 0.15/0.37 thf(subsetRefl,definition,
% 0.15/0.37 ( subsetRefl
% 0.15/0.37 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(subsetTrans_type,type,
% 0.15/0.37 subsetTrans: $o ).
% 0.15/0.37
% 0.15/0.37 thf(subsetTrans,definition,
% 0.15/0.37 ( subsetTrans
% 0.15/0.37 = ( ! [A: $i,B: $i,C: $i] :
% 0.15/0.37 ( ( subset @ A @ B )
% 0.15/0.37 => ( ( subset @ B @ C )
% 0.15/0.37 => ( subset @ A @ C ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoinSub_type,type,
% 0.15/0.37 setadjoinSub: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoinSub,definition,
% 0.15/0.37 ( setadjoinSub
% 0.15/0.37 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoinSub2_type,type,
% 0.15/0.37 setadjoinSub2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setadjoinSub2,definition,
% 0.15/0.37 ( setadjoinSub2
% 0.15/0.37 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.15/0.37 ( ( subset @ A @ B )
% 0.15/0.37 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(subset2powerset_type,type,
% 0.15/0.37 subset2powerset: $o ).
% 0.15/0.37
% 0.15/0.37 thf(subset2powerset,definition,
% 0.15/0.37 ( subset2powerset
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( subset @ A @ B )
% 0.15/0.37 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setextsub_type,type,
% 0.15/0.37 setextsub: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setextsub,definition,
% 0.15/0.37 ( setextsub
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( subset @ A @ B )
% 0.15/0.37 => ( ( subset @ B @ A )
% 0.15/0.37 => ( A = B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(subsetemptysetimpeq_type,type,
% 0.15/0.37 subsetemptysetimpeq: $o ).
% 0.15/0.37
% 0.15/0.37 thf(subsetemptysetimpeq,definition,
% 0.15/0.37 ( subsetemptysetimpeq
% 0.15/0.37 = ( ! [A: $i] :
% 0.15/0.37 ( ( subset @ A @ emptyset )
% 0.15/0.37 => ( A = emptyset ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(powersetI1_type,type,
% 0.15/0.37 powersetI1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(powersetI1,definition,
% 0.15/0.37 ( powersetI1
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( subset @ B @ A )
% 0.15/0.37 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(powersetE1_type,type,
% 0.15/0.37 powersetE1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(powersetE1,definition,
% 0.15/0.37 ( powersetE1
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( in @ B @ ( powerset @ A ) )
% 0.15/0.37 => ( subset @ B @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(inPowerset_type,type,
% 0.15/0.37 inPowerset: $o ).
% 0.15/0.37
% 0.15/0.37 thf(inPowerset,definition,
% 0.15/0.37 ( inPowerset
% 0.15/0.37 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(powersetsubset_type,type,
% 0.15/0.37 powersetsubset: $o ).
% 0.15/0.37
% 0.15/0.37 thf(powersetsubset,definition,
% 0.15/0.37 ( powersetsubset
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( subset @ A @ B )
% 0.15/0.37 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(sepInPowerset_type,type,
% 0.15/0.37 sepInPowerset: $o ).
% 0.15/0.37
% 0.15/0.37 thf(sepInPowerset,definition,
% 0.15/0.37 ( sepInPowerset
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.37 ( in
% 0.15/0.37 @ ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.37 @ ( powerset @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(sepSubset_type,type,
% 0.15/0.37 sepSubset: $o ).
% 0.15/0.37
% 0.15/0.37 thf(sepSubset,definition,
% 0.15/0.37 ( sepSubset
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.37 ( subset
% 0.15/0.37 @ ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.37 @ A ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binunion_type,type,
% 0.15/0.37 binunion: $i > $i > $i ).
% 0.15/0.37
% 0.15/0.37 thf(binunionIL_type,type,
% 0.15/0.37 binunionIL: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binunionIL,definition,
% 0.15/0.37 ( binunionIL
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(upairset2IR_type,type,
% 0.15/0.37 upairset2IR: $o ).
% 0.15/0.37
% 0.15/0.37 thf(upairset2IR,definition,
% 0.15/0.37 ( upairset2IR
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binunionIR_type,type,
% 0.15/0.37 binunionIR: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binunionIR,definition,
% 0.15/0.37 ( binunionIR
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ B )
% 0.15/0.37 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binunionEcases_type,type,
% 0.15/0.37 binunionEcases: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binunionEcases,definition,
% 0.15/0.37 ( binunionEcases
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.15/0.37 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.15/0.37 => ( ( ( in @ Xx @ A )
% 0.15/0.37 => Xphi )
% 0.15/0.37 => ( ( ( in @ Xx @ B )
% 0.15/0.37 => Xphi )
% 0.15/0.37 => Xphi ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binunionE_type,type,
% 0.15/0.37 binunionE: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binunionE,definition,
% 0.15/0.37 ( binunionE
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.15/0.37 => ( ( in @ Xx @ A )
% 0.15/0.37 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binunionLsub_type,type,
% 0.15/0.37 binunionLsub: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binunionLsub,definition,
% 0.15/0.37 ( binunionLsub
% 0.15/0.37 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binunionRsub_type,type,
% 0.15/0.37 binunionRsub: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binunionRsub,definition,
% 0.15/0.37 ( binunionRsub
% 0.15/0.37 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binintersect_type,type,
% 0.15/0.37 binintersect: $i > $i > $i ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectI_type,type,
% 0.15/0.37 binintersectI: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectI,definition,
% 0.15/0.37 ( binintersectI
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( in @ Xx @ B )
% 0.15/0.37 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectSubset5_type,type,
% 0.15/0.37 binintersectSubset5: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectSubset5,definition,
% 0.15/0.37 ( binintersectSubset5
% 0.15/0.37 = ( ! [A: $i,B: $i,C: $i] :
% 0.15/0.37 ( ( subset @ C @ A )
% 0.15/0.37 => ( ( subset @ C @ B )
% 0.15/0.37 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectEL_type,type,
% 0.15/0.37 binintersectEL: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectEL,definition,
% 0.15/0.37 ( binintersectEL
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.15/0.37 => ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectLsub_type,type,
% 0.15/0.37 binintersectLsub: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectLsub,definition,
% 0.15/0.37 ( binintersectLsub
% 0.15/0.37 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectSubset2_type,type,
% 0.15/0.37 binintersectSubset2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectSubset2,definition,
% 0.15/0.37 ( binintersectSubset2
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( subset @ A @ B )
% 0.15/0.37 => ( ( binintersect @ A @ B )
% 0.15/0.37 = A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectSubset3_type,type,
% 0.15/0.37 binintersectSubset3: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectSubset3,definition,
% 0.15/0.37 ( binintersectSubset3
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( ( binintersect @ A @ B )
% 0.15/0.37 = B )
% 0.15/0.37 => ( subset @ B @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectER_type,type,
% 0.15/0.37 binintersectER: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectER,definition,
% 0.15/0.37 ( binintersectER
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.15/0.37 => ( in @ Xx @ B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(disjointsetsI1_type,type,
% 0.15/0.37 disjointsetsI1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(disjointsetsI1,definition,
% 0.15/0.37 ( disjointsetsI1
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ~ ? [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 & ( in @ Xx @ B ) )
% 0.15/0.37 => ( ( binintersect @ A @ B )
% 0.15/0.37 = emptyset ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectRsub_type,type,
% 0.15/0.37 binintersectRsub: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectRsub,definition,
% 0.15/0.37 ( binintersectRsub
% 0.15/0.37 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectSubset4_type,type,
% 0.15/0.37 binintersectSubset4: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectSubset4,definition,
% 0.15/0.37 ( binintersectSubset4
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( subset @ B @ A )
% 0.15/0.37 => ( ( binintersect @ A @ B )
% 0.15/0.37 = B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectSubset1_type,type,
% 0.15/0.37 binintersectSubset1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectSubset1,definition,
% 0.15/0.37 ( binintersectSubset1
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( ( binintersect @ A @ B )
% 0.15/0.37 = A )
% 0.15/0.37 => ( subset @ A @ B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(bs114d_type,type,
% 0.15/0.37 bs114d: $o ).
% 0.15/0.37
% 0.15/0.37 thf(bs114d,definition,
% 0.15/0.37 ( bs114d
% 0.15/0.37 = ( ! [A: $i,B: $i,C: $i] :
% 0.15/0.37 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.15/0.37 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(regular_type,type,
% 0.15/0.37 regular: $i > $o ).
% 0.15/0.37
% 0.15/0.37 thf(setminus_type,type,
% 0.15/0.37 setminus: $i > $i > $i ).
% 0.15/0.37
% 0.15/0.37 thf(setminusI_type,type,
% 0.15/0.37 setminusI: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setminusI,definition,
% 0.15/0.37 ( setminusI
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ~ ( in @ Xx @ B )
% 0.15/0.37 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setminusEL_type,type,
% 0.15/0.37 setminusEL: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setminusEL,definition,
% 0.15/0.37 ( setminusEL
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.15/0.37 => ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setminusER_type,type,
% 0.15/0.37 setminusER: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setminusER,definition,
% 0.15/0.37 ( setminusER
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.15/0.37 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setminusSubset2_type,type,
% 0.15/0.37 setminusSubset2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setminusSubset2,definition,
% 0.15/0.37 ( setminusSubset2
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( subset @ A @ B )
% 0.15/0.37 => ( ( setminus @ A @ B )
% 0.15/0.37 = emptyset ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setminusERneg_type,type,
% 0.15/0.37 setminusERneg: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setminusERneg,definition,
% 0.15/0.37 ( setminusERneg
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.15/0.37 => ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setminusELneg_type,type,
% 0.15/0.37 setminusELneg: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setminusELneg,definition,
% 0.15/0.37 ( setminusELneg
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.15/0.37 => ( ~ ( in @ Xx @ B )
% 0.15/0.37 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setminusILneg_type,type,
% 0.15/0.37 setminusILneg: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setminusILneg,definition,
% 0.15/0.37 ( setminusILneg
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ~ ( in @ Xx @ A )
% 0.15/0.37 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setminusIRneg_type,type,
% 0.15/0.37 setminusIRneg: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setminusIRneg,definition,
% 0.15/0.37 ( setminusIRneg
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ B )
% 0.15/0.37 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setminusLsub_type,type,
% 0.15/0.37 setminusLsub: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setminusLsub,definition,
% 0.15/0.37 ( setminusLsub
% 0.15/0.37 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setminusSubset1_type,type,
% 0.15/0.37 setminusSubset1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setminusSubset1,definition,
% 0.15/0.37 ( setminusSubset1
% 0.15/0.37 = ( ! [A: $i,B: $i] :
% 0.15/0.37 ( ( ( setminus @ A @ B )
% 0.15/0.37 = emptyset )
% 0.15/0.37 => ( subset @ A @ B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(symdiff_type,type,
% 0.15/0.37 symdiff: $i > $i > $i ).
% 0.15/0.37
% 0.15/0.37 thf(symdiffE_type,type,
% 0.15/0.37 symdiffE: $o ).
% 0.15/0.37
% 0.15/0.37 thf(symdiffE,definition,
% 0.15/0.37 ( symdiffE
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.15/0.37 => ! [Xphi: $o] :
% 0.15/0.37 ( ( ( in @ Xx @ A )
% 0.15/0.37 => ( ~ ( in @ Xx @ B )
% 0.15/0.37 => Xphi ) )
% 0.15/0.37 => ( ( ~ ( in @ Xx @ A )
% 0.15/0.37 => ( ( in @ Xx @ B )
% 0.15/0.37 => Xphi ) )
% 0.15/0.37 => Xphi ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(symdiffI1_type,type,
% 0.15/0.37 symdiffI1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(symdiffI1,definition,
% 0.15/0.37 ( symdiffI1
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ~ ( in @ Xx @ B )
% 0.15/0.37 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(symdiffI2_type,type,
% 0.15/0.37 symdiffI2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(symdiffI2,definition,
% 0.15/0.37 ( symdiffI2
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ~ ( in @ Xx @ A )
% 0.15/0.37 => ( ( in @ Xx @ B )
% 0.15/0.37 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(symdiffIneg1_type,type,
% 0.15/0.37 symdiffIneg1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(symdiffIneg1,definition,
% 0.15/0.37 ( symdiffIneg1
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( in @ Xx @ B )
% 0.15/0.37 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(symdiffIneg2_type,type,
% 0.15/0.37 symdiffIneg2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(symdiffIneg2,definition,
% 0.15/0.37 ( symdiffIneg2
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ~ ( in @ Xx @ A )
% 0.15/0.37 => ( ~ ( in @ Xx @ B )
% 0.15/0.37 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(iskpair_type,type,
% 0.15/0.37 iskpair: $i > $o ).
% 0.15/0.37
% 0.15/0.37 thf(secondinupair_type,type,
% 0.15/0.37 secondinupair: $o ).
% 0.15/0.37
% 0.15/0.37 thf(secondinupair,definition,
% 0.15/0.37 ( secondinupair
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairIL_type,type,
% 0.15/0.37 setukpairIL: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairIL,definition,
% 0.15/0.37 ( setukpairIL
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairIR_type,type,
% 0.15/0.37 setukpairIR: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairIR,definition,
% 0.15/0.37 ( setukpairIR
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(kpairiskpair_type,type,
% 0.15/0.37 kpairiskpair: $o ).
% 0.15/0.37
% 0.15/0.37 thf(kpairiskpair,definition,
% 0.15/0.37 ( kpairiskpair
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(kpair_type,type,
% 0.15/0.37 kpair: $i > $i > $i ).
% 0.15/0.37
% 0.15/0.37 thf(kpairp_type,type,
% 0.15/0.37 kpairp: $o ).
% 0.15/0.37
% 0.15/0.37 thf(kpairp,definition,
% 0.15/0.37 ( kpairp
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(cartprod_type,type,
% 0.15/0.37 cartprod: $i > $i > $i ).
% 0.15/0.37
% 0.15/0.37 thf(singletonsubset_type,type,
% 0.15/0.37 singletonsubset: $o ).
% 0.15/0.37
% 0.15/0.37 thf(singletonsubset,definition,
% 0.15/0.37 ( singletonsubset
% 0.15/0.37 = ( ! [A: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(singletoninpowerset_type,type,
% 0.15/0.37 singletoninpowerset: $o ).
% 0.15/0.37
% 0.15/0.37 thf(singletoninpowerset,definition,
% 0.15/0.37 ( singletoninpowerset
% 0.15/0.37 = ( ! [A: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(singletoninpowunion_type,type,
% 0.15/0.37 singletoninpowunion: $o ).
% 0.15/0.37
% 0.15/0.37 thf(singletoninpowunion,definition,
% 0.15/0.37 ( singletoninpowunion
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(upairset2E_type,type,
% 0.15/0.37 upairset2E: $o ).
% 0.15/0.37
% 0.15/0.37 thf(upairset2E,definition,
% 0.15/0.37 ( upairset2E
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.15/0.37 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.15/0.37 => ( ( Xz = Xx )
% 0.15/0.37 | ( Xz = Xy ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(upairsubunion_type,type,
% 0.15/0.37 upairsubunion: $o ).
% 0.15/0.37
% 0.15/0.37 thf(upairsubunion,definition,
% 0.15/0.37 ( upairsubunion
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(upairinpowunion_type,type,
% 0.15/0.37 upairinpowunion: $o ).
% 0.15/0.37
% 0.15/0.37 thf(upairinpowunion,definition,
% 0.15/0.37 ( upairinpowunion
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(ubforcartprodlem1_type,type,
% 0.15/0.37 ubforcartprodlem1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(ubforcartprodlem1,definition,
% 0.15/0.37 ( ubforcartprodlem1
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(ubforcartprodlem2_type,type,
% 0.15/0.37 ubforcartprodlem2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(ubforcartprodlem2,definition,
% 0.15/0.37 ( ubforcartprodlem2
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(ubforcartprodlem3_type,type,
% 0.15/0.37 ubforcartprodlem3: $o ).
% 0.15/0.37
% 0.15/0.37 thf(ubforcartprodlem3,definition,
% 0.15/0.37 ( ubforcartprodlem3
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodpairin_type,type,
% 0.15/0.37 cartprodpairin: $o ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodpairin,definition,
% 0.15/0.37 ( cartprodpairin
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodmempair1_type,type,
% 0.15/0.37 cartprodmempair1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodmempair1,definition,
% 0.15/0.37 ( cartprodmempair1
% 0.15/0.37 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.15/0.37 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.15/0.37 => ? [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 & ? [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 & ( Xu
% 0.15/0.37 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodmempair_type,type,
% 0.15/0.37 cartprodmempair: $o ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodmempair,definition,
% 0.15/0.37 ( cartprodmempair
% 0.15/0.37 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.15/0.37 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.15/0.37 => ( iskpair @ Xu ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setunionE2_type,type,
% 0.15/0.37 setunionE2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setunionE2,definition,
% 0.15/0.37 ( setunionE2
% 0.15/0.37 = ( ! [A: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ ( setunion @ A ) )
% 0.15/0.37 => ? [X: $i] :
% 0.15/0.37 ( ( in @ X @ A )
% 0.15/0.37 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setunionsingleton1_type,type,
% 0.15/0.37 setunionsingleton1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setunionsingleton1,definition,
% 0.15/0.37 ( setunionsingleton1
% 0.15/0.37 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setunionsingleton2_type,type,
% 0.15/0.37 setunionsingleton2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setunionsingleton2,definition,
% 0.15/0.37 ( setunionsingleton2
% 0.15/0.37 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setunionsingleton_type,type,
% 0.15/0.37 setunionsingleton: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setunionsingleton,definition,
% 0.15/0.37 ( setunionsingleton
% 0.15/0.37 = ( ! [Xx: $i] :
% 0.15/0.37 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.15/0.37 = Xx ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(singleton_type,type,
% 0.15/0.37 singleton: $i > $o ).
% 0.15/0.37
% 0.15/0.37 thf(singletonprop_type,type,
% 0.15/0.37 singletonprop: $o ).
% 0.15/0.37
% 0.15/0.37 thf(singletonprop,definition,
% 0.15/0.37 ( singletonprop
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ A )
% 0.15/0.37 => ( ( Xphi @ Xx )
% 0.15/0.37 => ( ( Xphi @ Xy )
% 0.15/0.37 => ( Xx = Xy ) ) ) ) )
% 0.15/0.37 => ( ? [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 & ( Xphi @ Xx ) )
% 0.15/0.37 => ( singleton
% 0.15/0.37 @ ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(ex1_type,type,
% 0.15/0.37 ex1: $i > ( $i > $o ) > $o ).
% 0.15/0.37
% 0.15/0.37 thf(ex1,definition,
% 0.15/0.37 ( ex1
% 0.15/0.37 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.15/0.37 ( singleton
% 0.15/0.37 @ ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(ex1E1_type,type,
% 0.15/0.37 ex1E1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(ex1E1,definition,
% 0.15/0.37 ( ex1E1
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.37 ( ( ex1 @ A
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.37 => ? [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(ex1I_type,type,
% 0.15/0.37 ex1I: $o ).
% 0.15/0.37
% 0.15/0.37 thf(ex1I,definition,
% 0.15/0.37 ( ex1I
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( Xphi @ Xx )
% 0.15/0.37 => ( ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ A )
% 0.15/0.37 => ( ( Xphi @ Xy )
% 0.15/0.37 => ( Xy = Xx ) ) )
% 0.15/0.37 => ( ex1 @ A
% 0.15/0.37 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(ex1I2_type,type,
% 0.15/0.37 ex1I2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(ex1I2,definition,
% 0.15/0.37 ( ex1I2
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ A )
% 0.15/0.37 => ( ( Xphi @ Xx )
% 0.15/0.37 => ( ( Xphi @ Xy )
% 0.15/0.37 => ( Xx = Xy ) ) ) ) )
% 0.15/0.37 => ( ? [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 & ( Xphi @ Xx ) )
% 0.15/0.37 => ( ex1 @ A
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(singletonsuniq_type,type,
% 0.15/0.37 singletonsuniq: $o ).
% 0.15/0.37
% 0.15/0.37 thf(singletonsuniq,definition,
% 0.15/0.37 ( singletonsuniq
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( ( setadjoin @ Xx @ emptyset )
% 0.15/0.37 = ( setadjoin @ Xy @ emptyset ) )
% 0.15/0.37 => ( Xx = Xy ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(atmost1p_type,type,
% 0.15/0.37 atmost1p: $i > $o ).
% 0.15/0.37
% 0.15/0.37 thf(atleast2p_type,type,
% 0.15/0.37 atleast2p: $i > $o ).
% 0.15/0.37
% 0.15/0.37 thf(atmost2p_type,type,
% 0.15/0.37 atmost2p: $i > $o ).
% 0.15/0.37
% 0.15/0.37 thf(upairsetp_type,type,
% 0.15/0.37 upairsetp: $i > $o ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairinjL1_type,type,
% 0.15/0.37 setukpairinjL1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairinjL1,definition,
% 0.15/0.37 ( setukpairinjL1
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.15/0.37 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.15/0.37 => ( Xx = Xz ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(kfstsingleton_type,type,
% 0.15/0.37 kfstsingleton: $o ).
% 0.15/0.37
% 0.15/0.37 thf(kfstsingleton,definition,
% 0.15/0.37 ( kfstsingleton
% 0.15/0.37 = ( ! [Xu: $i] :
% 0.15/0.37 ( ( iskpair @ Xu )
% 0.15/0.37 => ( singleton
% 0.15/0.37 @ ( dsetconstr @ ( setunion @ Xu )
% 0.15/0.37 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(theprop_type,type,
% 0.15/0.37 theprop: $o ).
% 0.15/0.37
% 0.15/0.37 thf(theprop,definition,
% 0.15/0.37 ( theprop
% 0.15/0.37 = ( ! [X: $i] :
% 0.15/0.37 ( ( singleton @ X )
% 0.15/0.37 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(kfst_type,type,
% 0.15/0.37 kfst: $i > $i ).
% 0.15/0.37
% 0.15/0.37 thf(kfstpairEq_type,type,
% 0.15/0.37 kfstpairEq: $o ).
% 0.15/0.37
% 0.15/0.37 thf(kfstpairEq,definition,
% 0.15/0.37 ( kfstpairEq
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.15/0.37 = Xx ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodfstin_type,type,
% 0.15/0.37 cartprodfstin: $o ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodfstin,definition,
% 0.15/0.37 ( cartprodfstin
% 0.15/0.37 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.15/0.37 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.15/0.37 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairinjL2_type,type,
% 0.15/0.37 setukpairinjL2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairinjL2,definition,
% 0.15/0.37 ( setukpairinjL2
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.15/0.37 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.15/0.37 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.15/0.37 => ( Xx = Xz ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairinjL_type,type,
% 0.15/0.37 setukpairinjL: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairinjL,definition,
% 0.15/0.37 ( setukpairinjL
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.15/0.37 ( ( ( kpair @ Xx @ Xy )
% 0.15/0.37 = ( kpair @ Xz @ Xu ) )
% 0.15/0.37 => ( Xx = Xz ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairinjR11_type,type,
% 0.15/0.37 setukpairinjR11: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairinjR11,definition,
% 0.15/0.37 ( setukpairinjR11
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( Xx = Xy )
% 0.15/0.37 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.15/0.37 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairinjR12_type,type,
% 0.15/0.37 setukpairinjR12: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairinjR12,definition,
% 0.15/0.37 ( setukpairinjR12
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( Xx = Xy )
% 0.15/0.37 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.15/0.37 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairinjR1_type,type,
% 0.15/0.37 setukpairinjR1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairinjR1,definition,
% 0.15/0.37 ( setukpairinjR1
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.15/0.37 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.15/0.37 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.15/0.37 => ( ( Xz = Xu )
% 0.15/0.37 => ( Xy = Xu ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(upairequniteq_type,type,
% 0.15/0.37 upairequniteq: $o ).
% 0.15/0.37
% 0.15/0.37 thf(upairequniteq,definition,
% 0.15/0.37 ( upairequniteq
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.15/0.37 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.15/0.37 = ( setadjoin @ Xz @ emptyset ) )
% 0.15/0.37 => ( Xx = Xy ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairinjR2_type,type,
% 0.15/0.37 setukpairinjR2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairinjR2,definition,
% 0.15/0.37 ( setukpairinjR2
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.15/0.37 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.15/0.37 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.15/0.37 => ( Xy = Xu ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairinjR_type,type,
% 0.15/0.37 setukpairinjR: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setukpairinjR,definition,
% 0.15/0.37 ( setukpairinjR
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.15/0.37 ( ( ( kpair @ Xx @ Xy )
% 0.15/0.37 = ( kpair @ Xz @ Xu ) )
% 0.15/0.37 => ( Xy = Xu ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(ksndsingleton_type,type,
% 0.15/0.37 ksndsingleton: $o ).
% 0.15/0.37
% 0.15/0.37 thf(ksndsingleton,definition,
% 0.15/0.37 ( ksndsingleton
% 0.15/0.37 = ( ! [Xu: $i] :
% 0.15/0.37 ( ( iskpair @ Xu )
% 0.15/0.37 => ( singleton
% 0.15/0.37 @ ( dsetconstr @ ( setunion @ Xu )
% 0.15/0.37 @ ^ [Xx: $i] :
% 0.15/0.37 ( Xu
% 0.15/0.37 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(ksnd_type,type,
% 0.15/0.37 ksnd: $i > $i ).
% 0.15/0.37
% 0.15/0.37 thf(ksndpairEq_type,type,
% 0.15/0.37 ksndpairEq: $o ).
% 0.15/0.37
% 0.15/0.37 thf(ksndpairEq,definition,
% 0.15/0.37 ( ksndpairEq
% 0.15/0.37 = ( ! [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.15/0.37 = Xy ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(kpairsurjEq_type,type,
% 0.15/0.37 kpairsurjEq: $o ).
% 0.15/0.37
% 0.15/0.37 thf(kpairsurjEq,definition,
% 0.15/0.37 ( kpairsurjEq
% 0.15/0.37 = ( ! [Xu: $i] :
% 0.15/0.37 ( ( iskpair @ Xu )
% 0.15/0.37 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.15/0.37 = Xu ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodsndin_type,type,
% 0.15/0.37 cartprodsndin: $o ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodsndin,definition,
% 0.15/0.37 ( cartprodsndin
% 0.15/0.37 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.15/0.37 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.15/0.37 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodpairmemEL_type,type,
% 0.15/0.37 cartprodpairmemEL: $o ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodpairmemEL,definition,
% 0.15/0.37 ( cartprodpairmemEL
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.15/0.37 => ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodpairmemER_type,type,
% 0.15/0.37 cartprodpairmemER: $o ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodpairmemER,definition,
% 0.15/0.37 ( cartprodpairmemER
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.15/0.37 => ( in @ Xy @ B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodmempaircEq_type,type,
% 0.15/0.37 cartprodmempaircEq: $o ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodmempaircEq,definition,
% 0.15/0.37 ( cartprodmempaircEq
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( ( kpair @ Xx @ Xy )
% 0.15/0.37 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodfstpairEq_type,type,
% 0.15/0.37 cartprodfstpairEq: $o ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodfstpairEq,definition,
% 0.15/0.37 ( cartprodfstpairEq
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.15/0.37 = Xx ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodsndpairEq_type,type,
% 0.15/0.37 cartprodsndpairEq: $o ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodsndpairEq,definition,
% 0.15/0.37 ( cartprodsndpairEq
% 0.15/0.37 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.15/0.37 = Xy ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodpairsurjEq_type,type,
% 0.15/0.37 cartprodpairsurjEq: $o ).
% 0.15/0.37
% 0.15/0.37 thf(cartprodpairsurjEq,definition,
% 0.15/0.37 ( cartprodpairsurjEq
% 0.15/0.37 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.15/0.37 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.15/0.37 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.15/0.37 = Xu ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(breln_type,type,
% 0.15/0.37 breln: $i > $i > $i > $o ).
% 0.15/0.37
% 0.15/0.37 thf(breln,definition,
% 0.15/0.37 ( breln
% 0.15/0.37 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(dpsetconstr_type,type,
% 0.15/0.37 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.15/0.37
% 0.15/0.37 thf(dpsetconstrI_type,type,
% 0.15/0.37 dpsetconstrI: $o ).
% 0.15/0.37
% 0.15/0.37 thf(dpsetconstrI,definition,
% 0.15/0.37 ( dpsetconstrI
% 0.15/0.37 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( ( Xphi @ Xx @ Xy )
% 0.15/0.37 => ( in @ ( kpair @ Xx @ Xy )
% 0.15/0.37 @ ( dpsetconstr @ A @ B
% 0.15/0.37 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(dpsetconstrSub_type,type,
% 0.15/0.37 dpsetconstrSub: $o ).
% 0.15/0.37
% 0.15/0.37 thf(dpsetconstrSub,definition,
% 0.15/0.37 ( dpsetconstrSub
% 0.15/0.37 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.15/0.37 ( subset
% 0.15/0.37 @ ( dpsetconstr @ A @ B
% 0.15/0.37 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.15/0.37 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setOfPairsIsBReln_type,type,
% 0.15/0.37 setOfPairsIsBReln: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setOfPairsIsBReln,definition,
% 0.15/0.37 ( setOfPairsIsBReln
% 0.15/0.37 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.15/0.37 ( breln @ A @ B
% 0.15/0.37 @ ( dpsetconstr @ A @ B
% 0.15/0.37 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(dpsetconstrERa_type,type,
% 0.15/0.37 dpsetconstrERa: $o ).
% 0.15/0.37
% 0.15/0.37 thf(dpsetconstrERa,definition,
% 0.15/0.37 ( dpsetconstrERa
% 0.15/0.37 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.15/0.37 @ ( dpsetconstr @ A @ B
% 0.15/0.37 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.15/0.37 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(dpsetconstrEL1_type,type,
% 0.15/0.37 dpsetconstrEL1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(dpsetconstrEL1,definition,
% 0.15/0.37 ( dpsetconstrEL1
% 0.15/0.37 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( in @ ( kpair @ Xx @ Xy )
% 0.15/0.37 @ ( dpsetconstr @ A @ B
% 0.15/0.37 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.15/0.37 => ( in @ Xx @ A ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(dpsetconstrEL2_type,type,
% 0.15/0.37 dpsetconstrEL2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(dpsetconstrEL2,definition,
% 0.15/0.37 ( dpsetconstrEL2
% 0.15/0.37 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( in @ ( kpair @ Xx @ Xy )
% 0.15/0.37 @ ( dpsetconstr @ A @ B
% 0.15/0.37 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.15/0.37 => ( in @ Xy @ B ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(dpsetconstrER_type,type,
% 0.15/0.37 dpsetconstrER: $o ).
% 0.15/0.37
% 0.15/0.37 thf(dpsetconstrER,definition,
% 0.15/0.37 ( dpsetconstrER
% 0.15/0.37 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( in @ ( kpair @ Xx @ Xy )
% 0.15/0.37 @ ( dpsetconstr @ A @ B
% 0.15/0.37 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.15/0.37 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(func_type,type,
% 0.15/0.37 func: $i > $i > $i > $o ).
% 0.15/0.37
% 0.15/0.37 thf(funcSet_type,type,
% 0.15/0.37 funcSet: $i > $i > $i ).
% 0.15/0.37
% 0.15/0.37 thf(funcImageSingleton_type,type,
% 0.15/0.37 funcImageSingleton: $o ).
% 0.15/0.37
% 0.15/0.37 thf(funcImageSingleton,definition,
% 0.15/0.37 ( funcImageSingleton
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( func @ A @ B @ Xf )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( singleton
% 0.15/0.37 @ ( dsetconstr @ B
% 0.15/0.37 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(apProp_type,type,
% 0.15/0.37 apProp: $o ).
% 0.15/0.37
% 0.15/0.37 thf(apProp,definition,
% 0.15/0.37 ( apProp
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( func @ A @ B @ Xf )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in
% 0.15/0.37 @ ( setunion
% 0.15/0.37 @ ( dsetconstr @ B
% 0.15/0.37 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.15/0.37 @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(ap_type,type,
% 0.15/0.37 ap: $i > $i > $i > $i > $i ).
% 0.15/0.37
% 0.15/0.37 thf(app_type,type,
% 0.15/0.37 app: $o ).
% 0.15/0.37
% 0.15/0.37 thf(app,definition,
% 0.15/0.37 ( app
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( func @ A @ B @ Xf )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(infuncsetfunc_type,type,
% 0.15/0.37 infuncsetfunc: $o ).
% 0.15/0.37
% 0.15/0.37 thf(infuncsetfunc,definition,
% 0.15/0.37 ( infuncsetfunc
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.37 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(ap2p_type,type,
% 0.15/0.37 ap2p: $o ).
% 0.15/0.37
% 0.15/0.37 thf(ap2p,definition,
% 0.15/0.37 ( ap2p
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(funcinfuncset_type,type,
% 0.15/0.37 funcinfuncset: $o ).
% 0.15/0.37
% 0.15/0.37 thf(funcinfuncset,definition,
% 0.15/0.37 ( funcinfuncset
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( func @ A @ B @ Xf )
% 0.15/0.37 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(lamProp_type,type,
% 0.15/0.37 lamProp: $o ).
% 0.15/0.37
% 0.15/0.37 thf(lamProp,definition,
% 0.15/0.37 ( lamProp
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.37 => ( func @ A @ B
% 0.15/0.37 @ ( dpsetconstr @ A @ B
% 0.15/0.37 @ ^ [Xx: $i,Xy: $i] :
% 0.15/0.37 ( ( Xf @ Xx )
% 0.15/0.37 = Xy ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(lam_type,type,
% 0.15/0.37 lam: $i > $i > ( $i > $i ) > $i ).
% 0.15/0.37
% 0.15/0.37 thf(lamp_type,type,
% 0.15/0.37 lamp: $o ).
% 0.15/0.37
% 0.15/0.37 thf(lamp,definition,
% 0.15/0.37 ( lamp
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.37 => ( func @ A @ B
% 0.15/0.37 @ ( lam @ A @ B
% 0.15/0.37 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(lam2p_type,type,
% 0.15/0.37 lam2p: $o ).
% 0.15/0.37
% 0.15/0.37 thf(lam2p,definition,
% 0.15/0.37 ( lam2p
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.37 => ( in
% 0.15/0.37 @ ( lam @ A @ B
% 0.15/0.37 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.15/0.37 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(brelnall1_type,type,
% 0.15/0.37 brelnall1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(brelnall1,definition,
% 0.15/0.37 ( brelnall1
% 0.15/0.37 = ( ! [A: $i,B: $i,R: $i] :
% 0.15/0.37 ( ( breln @ A @ B @ R )
% 0.15/0.37 => ! [Xphi: $i > $o] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.37 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ R )
% 0.15/0.37 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(brelnall2_type,type,
% 0.15/0.37 brelnall2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(brelnall2,definition,
% 0.15/0.37 ( brelnall2
% 0.15/0.37 = ( ! [A: $i,B: $i,R: $i] :
% 0.15/0.37 ( ( breln @ A @ B @ R )
% 0.15/0.37 => ! [Xphi: $i > $o] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.37 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ R )
% 0.15/0.37 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(ex1E2_type,type,
% 0.15/0.37 ex1E2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(ex1E2,definition,
% 0.15/0.37 ( ex1E2
% 0.15/0.37 = ( ! [A: $i,Xphi: $i > $o] :
% 0.15/0.37 ( ( ex1 @ A
% 0.15/0.37 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ A )
% 0.15/0.37 => ( ( Xphi @ Xx )
% 0.15/0.37 => ( ( Xphi @ Xy )
% 0.15/0.37 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(funcGraphProp1_type,type,
% 0.15/0.37 funcGraphProp1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(funcGraphProp1,definition,
% 0.15/0.37 ( funcGraphProp1
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( func @ A @ B @ Xf )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(funcGraphProp3_type,type,
% 0.15/0.37 funcGraphProp3: $o ).
% 0.15/0.37
% 0.15/0.37 thf(funcGraphProp3,definition,
% 0.15/0.37 ( funcGraphProp3
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(funcGraphProp2_type,type,
% 0.15/0.37 funcGraphProp2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(funcGraphProp2,definition,
% 0.15/0.37 ( funcGraphProp2
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( func @ A @ B @ Xf )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.15/0.37 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.37 = Xy ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(funcextLem_type,type,
% 0.15/0.37 funcextLem: $o ).
% 0.15/0.37
% 0.15/0.37 thf(funcextLem,definition,
% 0.15/0.37 ( funcextLem
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( func @ A @ B @ Xf )
% 0.15/0.37 => ! [Xg: $i] :
% 0.15/0.37 ( ( func @ A @ B @ Xg )
% 0.15/0.37 => ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.37 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.15/0.37 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(funcGraphProp4_type,type,
% 0.15/0.37 funcGraphProp4: $o ).
% 0.15/0.37
% 0.15/0.37 thf(funcGraphProp4,definition,
% 0.15/0.37 ( funcGraphProp4
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.15/0.37 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.37 = Xy ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(subbreln_type,type,
% 0.15/0.37 subbreln: $o ).
% 0.15/0.37
% 0.15/0.37 thf(subbreln,definition,
% 0.15/0.37 ( subbreln
% 0.15/0.37 = ( ! [A: $i,B: $i,R: $i] :
% 0.15/0.37 ( ( breln @ A @ B @ R )
% 0.15/0.37 => ! [S: $i] :
% 0.15/0.37 ( ( breln @ A @ B @ S )
% 0.15/0.37 => ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.37 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.15/0.37 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(eqbreln_type,type,
% 0.15/0.37 eqbreln: $o ).
% 0.15/0.37
% 0.15/0.37 thf(eqbreln,definition,
% 0.15/0.37 ( eqbreln
% 0.15/0.37 = ( ! [A: $i,B: $i,R: $i] :
% 0.15/0.37 ( ( breln @ A @ B @ R )
% 0.15/0.37 => ! [S: $i] :
% 0.15/0.37 ( ( breln @ A @ B @ S )
% 0.15/0.37 => ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.15/0.37 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.15/0.37 => ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ B )
% 0.15/0.37 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.15/0.37 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.15/0.37 => ( R = S ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(funcext_type,type,
% 0.15/0.37 funcext: $o ).
% 0.15/0.37
% 0.15/0.37 thf(funcext,definition,
% 0.15/0.37 ( funcext
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( func @ A @ B @ Xf )
% 0.15/0.37 => ! [Xg: $i] :
% 0.15/0.37 ( ( func @ A @ B @ Xg )
% 0.15/0.37 => ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.37 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.15/0.37 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(funcext2_type,type,
% 0.15/0.37 funcext2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(funcext2,definition,
% 0.15/0.37 ( funcext2
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.37 => ! [Xg: $i] :
% 0.15/0.37 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.15/0.37 => ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.37 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.15/0.37 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(ap2apEq1_type,type,
% 0.15/0.37 ap2apEq1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(ap2apEq1,definition,
% 0.15/0.37 ( ap2apEq1
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.37 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(ap2apEq2_type,type,
% 0.15/0.37 ap2apEq2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(ap2apEq2,definition,
% 0.15/0.37 ( ap2apEq2
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( func @ A @ B @ Xf )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.15/0.37 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(beta1_type,type,
% 0.15/0.37 beta1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(beta1,definition,
% 0.15/0.37 ( beta1
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( ap @ A @ B
% 0.15/0.37 @ ( lam @ A @ B
% 0.15/0.37 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.15/0.37 @ Xx )
% 0.15/0.37 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(eta1_type,type,
% 0.15/0.37 eta1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(eta1,definition,
% 0.15/0.37 ( eta1
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( func @ A @ B @ Xf )
% 0.15/0.37 => ( ( lam @ A @ B
% 0.15/0.37 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.15/0.37 = Xf ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(lam2lamEq_type,type,
% 0.15/0.37 lam2lamEq: $o ).
% 0.15/0.37
% 0.15/0.37 thf(lam2lamEq,definition,
% 0.15/0.37 ( lam2lamEq
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.37 => ( ( lam @ A @ B
% 0.15/0.37 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.15/0.37 = ( lam @ A @ B
% 0.15/0.37 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(beta2_type,type,
% 0.15/0.37 beta2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(beta2,definition,
% 0.15/0.37 ( beta2
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.15/0.37 ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( ap @ A @ B
% 0.15/0.37 @ ( lam @ A @ B
% 0.15/0.37 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.15/0.37 @ Xx )
% 0.15/0.37 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(eta2_type,type,
% 0.15/0.37 eta2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(eta2,definition,
% 0.15/0.37 ( eta2
% 0.15/0.37 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.15/0.37 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.15/0.37 => ( ( lam @ A @ B
% 0.15/0.37 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.15/0.37 = Xf ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(iffalseProp1_type,type,
% 0.15/0.37 iffalseProp1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(iffalseProp1,definition,
% 0.15/0.37 ( iffalseProp1
% 0.15/0.37 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ A )
% 0.15/0.37 => ( ~ Xphi
% 0.15/0.37 => ( in @ Xy
% 0.15/0.37 @ ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xz: $i] :
% 0.15/0.37 ( ( Xphi
% 0.15/0.37 & ( Xz = Xx ) )
% 0.15/0.37 | ( ~ Xphi
% 0.15/0.37 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(iffalseProp2_type,type,
% 0.15/0.37 iffalseProp2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(iffalseProp2,definition,
% 0.15/0.37 ( iffalseProp2
% 0.15/0.37 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ A )
% 0.15/0.37 => ( ~ Xphi
% 0.15/0.37 => ( ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xz: $i] :
% 0.15/0.37 ( ( Xphi
% 0.15/0.37 & ( Xz = Xx ) )
% 0.15/0.37 | ( ~ Xphi
% 0.15/0.37 & ( Xz = Xy ) ) ) )
% 0.15/0.37 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(iftrueProp1_type,type,
% 0.15/0.37 iftrueProp1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(iftrueProp1,definition,
% 0.15/0.37 ( iftrueProp1
% 0.15/0.37 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ A )
% 0.15/0.37 => ( Xphi
% 0.15/0.37 => ( in @ Xx
% 0.15/0.37 @ ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xz: $i] :
% 0.15/0.37 ( ( Xphi
% 0.15/0.37 & ( Xz = Xx ) )
% 0.15/0.37 | ( ~ Xphi
% 0.15/0.37 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(iftrueProp2_type,type,
% 0.15/0.37 iftrueProp2: $o ).
% 0.15/0.37
% 0.15/0.37 thf(iftrueProp2,definition,
% 0.15/0.37 ( iftrueProp2
% 0.15/0.37 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ A )
% 0.15/0.37 => ( Xphi
% 0.15/0.37 => ( ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xz: $i] :
% 0.15/0.37 ( ( Xphi
% 0.15/0.37 & ( Xz = Xx ) )
% 0.15/0.37 | ( ~ Xphi
% 0.15/0.37 & ( Xz = Xy ) ) ) )
% 0.15/0.37 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(ifSingleton_type,type,
% 0.15/0.37 ifSingleton: $o ).
% 0.15/0.37
% 0.15/0.37 thf(ifSingleton,definition,
% 0.15/0.37 ( ifSingleton
% 0.15/0.37 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ A )
% 0.15/0.37 => ( singleton
% 0.15/0.37 @ ( dsetconstr @ A
% 0.15/0.37 @ ^ [Xz: $i] :
% 0.15/0.37 ( ( Xphi
% 0.15/0.37 & ( Xz = Xx ) )
% 0.15/0.37 | ( ~ Xphi
% 0.15/0.37 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(if_type,type,
% 0.15/0.37 if: $i > $o > $i > $i > $i ).
% 0.15/0.37
% 0.15/0.37 thf(ifp_type,type,
% 0.15/0.37 ifp: $o ).
% 0.15/0.37
% 0.15/0.37 thf(ifp,definition,
% 0.15/0.37 ( ifp
% 0.15/0.37 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ A )
% 0.15/0.37 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(theeq_type,type,
% 0.15/0.37 theeq: $o ).
% 0.15/0.37
% 0.15/0.37 thf(theeq,definition,
% 0.15/0.37 ( theeq
% 0.15/0.37 = ( ! [X: $i] :
% 0.15/0.37 ( ( singleton @ X )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ X )
% 0.15/0.37 => ( ( setunion @ X )
% 0.15/0.37 = Xx ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(iftrue_type,type,
% 0.15/0.37 iftrue: $o ).
% 0.15/0.37
% 0.15/0.37 thf(iftrue,definition,
% 0.15/0.37 ( iftrue
% 0.15/0.37 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ A )
% 0.15/0.37 => ( Xphi
% 0.15/0.37 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.15/0.37 = Xx ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(iffalse_type,type,
% 0.15/0.37 iffalse: $o ).
% 0.15/0.37
% 0.15/0.37 thf(iffalse,definition,
% 0.15/0.37 ( iffalse
% 0.15/0.37 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ A )
% 0.15/0.37 => ( ~ Xphi
% 0.15/0.37 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.15/0.37 = Xy ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(iftrueorfalse_type,type,
% 0.15/0.37 iftrueorfalse: $o ).
% 0.15/0.37
% 0.15/0.37 thf(iftrueorfalse,definition,
% 0.15/0.37 ( iftrueorfalse
% 0.15/0.37 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ! [Xy: $i] :
% 0.15/0.37 ( ( in @ Xy @ A )
% 0.15/0.37 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectT_lem_type,type,
% 0.15/0.37 binintersectT_lem: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectT_lem,definition,
% 0.15/0.37 ( binintersectT_lem
% 0.15/0.37 = ( ! [A: $i,X: $i] :
% 0.15/0.37 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.37 => ! [Y: $i] :
% 0.15/0.37 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.37 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binunionT_lem_type,type,
% 0.15/0.37 binunionT_lem: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binunionT_lem,definition,
% 0.15/0.37 ( binunionT_lem
% 0.15/0.37 = ( ! [A: $i,X: $i] :
% 0.15/0.37 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.37 => ! [Y: $i] :
% 0.15/0.37 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.37 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(powersetT_lem_type,type,
% 0.15/0.37 powersetT_lem: $o ).
% 0.15/0.37
% 0.15/0.37 thf(powersetT_lem,definition,
% 0.15/0.37 ( powersetT_lem
% 0.15/0.37 = ( ! [A: $i,X: $i] :
% 0.15/0.37 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.37 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setminusT_lem_type,type,
% 0.15/0.37 setminusT_lem: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setminusT_lem,definition,
% 0.15/0.37 ( setminusT_lem
% 0.15/0.37 = ( ! [A: $i,X: $i] :
% 0.15/0.37 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.37 => ! [Y: $i] :
% 0.15/0.37 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.37 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(complementT_lem_type,type,
% 0.15/0.37 complementT_lem: $o ).
% 0.15/0.37
% 0.15/0.37 thf(complementT_lem,definition,
% 0.15/0.37 ( complementT_lem
% 0.15/0.37 = ( ! [A: $i,X: $i] :
% 0.15/0.37 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.37 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(setextT_type,type,
% 0.15/0.37 setextT: $o ).
% 0.15/0.37
% 0.15/0.37 thf(setextT,definition,
% 0.15/0.37 ( setextT
% 0.15/0.37 = ( ! [A: $i,X: $i] :
% 0.15/0.37 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.37 => ! [Y: $i] :
% 0.15/0.37 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.37 => ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( in @ Xx @ X )
% 0.15/0.37 => ( in @ Xx @ Y ) ) )
% 0.15/0.37 => ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( in @ Xx @ Y )
% 0.15/0.37 => ( in @ Xx @ X ) ) )
% 0.15/0.37 => ( X = Y ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(subsetTI_type,type,
% 0.15/0.37 subsetTI: $o ).
% 0.15/0.37
% 0.15/0.37 thf(subsetTI,definition,
% 0.15/0.37 ( subsetTI
% 0.15/0.37 = ( ! [A: $i,X: $i] :
% 0.15/0.37 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.37 => ! [Y: $i] :
% 0.15/0.37 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.37 => ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( in @ Xx @ X )
% 0.15/0.37 => ( in @ Xx @ Y ) ) )
% 0.15/0.37 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(powersetTI1_type,type,
% 0.15/0.37 powersetTI1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(powersetTI1,definition,
% 0.15/0.37 ( powersetTI1
% 0.15/0.37 = ( ! [A: $i,X: $i] :
% 0.15/0.37 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.37 => ! [Y: $i] :
% 0.15/0.37 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.37 => ( ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( in @ Xx @ X )
% 0.15/0.37 => ( in @ Xx @ Y ) ) )
% 0.15/0.37 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(powersetTE1_type,type,
% 0.15/0.37 powersetTE1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(powersetTE1,definition,
% 0.15/0.37 ( powersetTE1
% 0.15/0.37 = ( ! [A: $i,X: $i] :
% 0.15/0.37 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.37 => ! [Y: $i] :
% 0.15/0.37 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( in @ X @ ( powerset @ Y ) )
% 0.15/0.37 => ( ( in @ Xx @ X )
% 0.15/0.37 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(complementTI1_type,type,
% 0.15/0.37 complementTI1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(complementTI1,definition,
% 0.15/0.37 ( complementTI1
% 0.15/0.37 = ( ! [A: $i,X: $i] :
% 0.15/0.37 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ( in @ Xx @ X )
% 0.15/0.37 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(complementTE1_type,type,
% 0.15/0.37 complementTE1: $o ).
% 0.15/0.37
% 0.15/0.37 thf(complementTE1,definition,
% 0.15/0.37 ( complementTE1
% 0.15/0.37 = ( ! [A: $i,X: $i] :
% 0.15/0.37 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.37 => ! [Xx: $i] :
% 0.15/0.37 ( ( in @ Xx @ A )
% 0.15/0.37 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.15/0.37 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectTELcontra_type,type,
% 0.15/0.37 binintersectTELcontra: $o ).
% 0.15/0.37
% 0.15/0.37 thf(binintersectTELcontra,definition,
% 0.15/0.37 ( binintersectTELcontra
% 0.15/0.37 = ( ! [A: $i,X: $i] :
% 0.15/0.37 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.37 => ! [Y: $i] :
% 0.15/0.37 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.38 => ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ~ ( in @ Xx @ X )
% 0.15/0.38 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectTERcontra_type,type,
% 0.15/0.38 binintersectTERcontra: $o ).
% 0.15/0.38
% 0.15/0.38 thf(binintersectTERcontra,definition,
% 0.15/0.38 ( binintersectTERcontra
% 0.15/0.38 = ( ! [A: $i,X: $i] :
% 0.15/0.38 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.38 => ! [Y: $i] :
% 0.15/0.38 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.38 => ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ~ ( in @ Xx @ Y )
% 0.15/0.38 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(contrasubsetT_type,type,
% 0.15/0.38 contrasubsetT: $o ).
% 0.15/0.38
% 0.15/0.38 thf(contrasubsetT,definition,
% 0.15/0.38 ( contrasubsetT
% 0.15/0.38 = ( ! [A: $i,X: $i] :
% 0.15/0.38 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.38 => ! [Y: $i] :
% 0.15/0.38 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.38 => ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.15/0.38 => ( ( in @ Xx @ Y )
% 0.15/0.38 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(contrasubsetT1_type,type,
% 0.15/0.38 contrasubsetT1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(contrasubsetT1,definition,
% 0.15/0.38 ( contrasubsetT1
% 0.15/0.38 = ( ! [A: $i,X: $i] :
% 0.15/0.38 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.38 => ! [Y: $i] :
% 0.15/0.38 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.38 => ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ( subset @ X @ Y )
% 0.15/0.38 => ( ~ ( in @ Xx @ Y )
% 0.15/0.38 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(contrasubsetT2_type,type,
% 0.15/0.38 contrasubsetT2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(contrasubsetT2,definition,
% 0.15/0.38 ( contrasubsetT2
% 0.15/0.38 = ( ! [A: $i,X: $i] :
% 0.15/0.38 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.38 => ! [Y: $i] :
% 0.15/0.38 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.38 => ( ( subset @ X @ Y )
% 0.15/0.38 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(contrasubsetT3_type,type,
% 0.15/0.38 contrasubsetT3: $o ).
% 0.15/0.38
% 0.15/0.38 thf(contrasubsetT3,definition,
% 0.15/0.38 ( contrasubsetT3
% 0.15/0.38 = ( ! [A: $i,X: $i] :
% 0.15/0.38 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.38 => ! [Y: $i] :
% 0.15/0.38 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.38 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.15/0.38 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(doubleComplementI1_type,type,
% 0.15/0.38 doubleComplementI1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(doubleComplementI1,definition,
% 0.15/0.38 ( doubleComplementI1
% 0.15/0.38 = ( ! [A: $i,X: $i] :
% 0.15/0.38 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.38 => ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ( in @ Xx @ X )
% 0.15/0.38 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(doubleComplementE1_type,type,
% 0.15/0.38 doubleComplementE1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(doubleComplementE1,definition,
% 0.15/0.38 ( doubleComplementE1
% 0.15/0.38 = ( ! [A: $i,X: $i] :
% 0.15/0.38 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.38 => ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.15/0.38 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(doubleComplementSub1_type,type,
% 0.15/0.38 doubleComplementSub1: $o ).
% 0.15/0.38
% 0.15/0.38 thf(doubleComplementSub1,definition,
% 0.15/0.38 ( doubleComplementSub1
% 0.15/0.38 = ( ! [A: $i,X: $i] :
% 0.15/0.38 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.38 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(doubleComplementSub2_type,type,
% 0.15/0.38 doubleComplementSub2: $o ).
% 0.15/0.38
% 0.15/0.38 thf(doubleComplementSub2,definition,
% 0.15/0.38 ( doubleComplementSub2
% 0.15/0.38 = ( ! [A: $i,X: $i] :
% 0.15/0.38 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.38 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(doubleComplementEq_type,type,
% 0.15/0.38 doubleComplementEq: $o ).
% 0.15/0.38
% 0.15/0.38 thf(doubleComplementEq,definition,
% 0.15/0.38 ( doubleComplementEq
% 0.15/0.38 = ( ! [A: $i,X: $i] :
% 0.15/0.38 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.38 => ( X
% 0.15/0.38 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(complementTnotintersectT_type,type,
% 0.15/0.38 complementTnotintersectT: $o ).
% 0.15/0.38
% 0.15/0.38 thf(complementTnotintersectT,definition,
% 0.15/0.38 ( complementTnotintersectT
% 0.15/0.38 = ( ! [A: $i,X: $i] :
% 0.15/0.38 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.38 => ! [Y: $i] :
% 0.15/0.38 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.38 => ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.15/0.38 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(complementImpComplementIntersect_type,type,
% 0.15/0.38 complementImpComplementIntersect: $o ).
% 0.15/0.38
% 0.15/0.38 thf(complementImpComplementIntersect,definition,
% 0.15/0.38 ( complementImpComplementIntersect
% 0.15/0.38 = ( ! [A: $i,X: $i] :
% 0.15/0.38 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.38 => ! [Y: $i] :
% 0.15/0.38 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.38 => ! [Xx: $i] :
% 0.15/0.38 ( ( in @ Xx @ A )
% 0.15/0.38 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.15/0.38 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 thf(complementSubsetComplementIntersect,conjecture,
% 0.15/0.38 ( setextAx
% 0.15/0.38 => ( emptysetAx
% 0.15/0.38 => ( setadjoinAx
% 0.15/0.38 => ( powersetAx
% 0.15/0.38 => ( setunionAx
% 0.15/0.38 => ( omega0Ax
% 0.15/0.38 => ( omegaSAx
% 0.15/0.38 => ( omegaIndAx
% 0.15/0.38 => ( replAx
% 0.15/0.38 => ( foundationAx
% 0.15/0.38 => ( wellorderingAx
% 0.15/0.38 => ( descrp
% 0.15/0.38 => ( dsetconstrI
% 0.15/0.38 => ( dsetconstrEL
% 0.15/0.38 => ( dsetconstrER
% 0.15/0.38 => ( exuE1
% 0.15/0.38 => ( prop2setE
% 0.15/0.38 => ( emptysetE
% 0.15/0.38 => ( emptysetimpfalse
% 0.15/0.38 => ( notinemptyset
% 0.15/0.38 => ( exuE3e
% 0.15/0.38 => ( setext
% 0.15/0.38 => ( emptyI
% 0.15/0.38 => ( noeltsimpempty
% 0.15/0.38 => ( setbeta
% 0.15/0.38 => ( nonemptyE1
% 0.15/0.38 => ( nonemptyI
% 0.15/0.38 => ( nonemptyI1
% 0.15/0.38 => ( setadjoinIL
% 0.15/0.38 => ( emptyinunitempty
% 0.15/0.38 => ( setadjoinIR
% 0.15/0.38 => ( setadjoinE
% 0.15/0.38 => ( setadjoinOr
% 0.15/0.38 => ( setoftrueEq
% 0.15/0.38 => ( powersetI
% 0.15/0.38 => ( emptyinPowerset
% 0.15/0.38 => ( emptyInPowerset
% 0.15/0.38 => ( powersetE
% 0.15/0.38 => ( setunionI
% 0.15/0.38 => ( setunionE
% 0.15/0.38 => ( subPowSU
% 0.15/0.38 => ( exuE2
% 0.15/0.38 => ( nonemptyImpWitness
% 0.15/0.38 => ( uniqinunit
% 0.15/0.38 => ( notinsingleton
% 0.15/0.38 => ( eqinunit
% 0.15/0.38 => ( singletonsswitch
% 0.15/0.38 => ( upairsetE
% 0.15/0.38 => ( upairsetIL
% 0.15/0.38 => ( upairsetIR
% 0.15/0.38 => ( emptyE1
% 0.15/0.38 => ( vacuousDall
% 0.15/0.38 => ( quantDeMorgan1
% 0.15/0.38 => ( quantDeMorgan2
% 0.15/0.38 => ( quantDeMorgan3
% 0.15/0.38 => ( quantDeMorgan4
% 0.15/0.38 => ( prop2setI
% 0.15/0.38 => ( prop2set2propI
% 0.15/0.38 => ( notdexE
% 0.15/0.38 => ( notdallE
% 0.15/0.38 => ( exuI1
% 0.15/0.38 => ( exuI3
% 0.15/0.38 => ( exuI2
% 0.15/0.38 => ( inCongP
% 0.15/0.38 => ( in__Cong
% 0.15/0.38 => ( exuE3u
% 0.15/0.38 => ( exu__Cong
% 0.15/0.38 => ( emptyset__Cong
% 0.15/0.38 => ( setadjoin__Cong
% 0.15/0.38 => ( powerset__Cong
% 0.15/0.38 => ( setunion__Cong
% 0.15/0.38 => ( omega__Cong
% 0.15/0.38 => ( exuEu
% 0.15/0.38 => ( descr__Cong
% 0.15/0.38 => ( dsetconstr__Cong
% 0.15/0.38 => ( subsetI1
% 0.15/0.38 => ( eqimpsubset2
% 0.15/0.38 => ( eqimpsubset1
% 0.15/0.38 => ( subsetI2
% 0.15/0.38 => ( emptysetsubset
% 0.15/0.38 => ( subsetE
% 0.15/0.38 => ( subsetE2
% 0.15/0.38 => ( notsubsetI
% 0.15/0.38 => ( notequalI1
% 0.15/0.38 => ( notequalI2
% 0.15/0.38 => ( subsetRefl
% 0.15/0.38 => ( subsetTrans
% 0.15/0.38 => ( setadjoinSub
% 0.15/0.38 => ( setadjoinSub2
% 0.15/0.38 => ( subset2powerset
% 0.15/0.38 => ( setextsub
% 0.15/0.38 => ( subsetemptysetimpeq
% 0.15/0.38 => ( powersetI1
% 0.15/0.38 => ( powersetE1
% 0.15/0.38 => ( inPowerset
% 0.15/0.38 => ( powersetsubset
% 0.15/0.38 => ( sepInPowerset
% 0.15/0.38 => ( sepSubset
% 0.15/0.38 => ( binunionIL
% 0.15/0.38 => ( upairset2IR
% 0.15/0.38 => ( binunionIR
% 0.15/0.38 => ( binunionEcases
% 0.15/0.38 => ( binunionE
% 0.15/0.38 => ( binunionLsub
% 0.15/0.38 => ( binunionRsub
% 0.15/0.38 => ( binintersectI
% 0.15/0.38 => ( binintersectSubset5
% 0.15/0.38 => ( binintersectEL
% 0.15/0.38 => ( binintersectLsub
% 0.15/0.38 => ( binintersectSubset2
% 0.15/0.38 => ( binintersectSubset3
% 0.15/0.38 => ( binintersectER
% 0.15/0.38 => ( disjointsetsI1
% 0.15/0.38 => ( binintersectRsub
% 0.15/0.38 => ( binintersectSubset4
% 0.15/0.38 => ( binintersectSubset1
% 0.15/0.38 => ( bs114d
% 0.15/0.38 => ( setminusI
% 0.15/0.38 => ( setminusEL
% 0.15/0.38 => ( setminusER
% 0.15/0.38 => ( setminusSubset2
% 0.15/0.38 => ( setminusERneg
% 0.15/0.38 => ( setminusELneg
% 0.15/0.38 => ( setminusILneg
% 0.15/0.38 => ( setminusIRneg
% 0.15/0.38 => ( setminusLsub
% 0.15/0.38 => ( setminusSubset1
% 0.15/0.38 => ( symdiffE
% 0.15/0.38 => ( symdiffI1
% 0.15/0.38 => ( symdiffI2
% 0.15/0.38 => ( symdiffIneg1
% 0.15/0.38 => ( symdiffIneg2
% 0.15/0.38 => ( secondinupair
% 0.15/0.38 => ( setukpairIL
% 0.15/0.38 => ( setukpairIR
% 0.15/0.38 => ( kpairiskpair
% 0.15/0.38 => ( kpairp
% 0.15/0.38 => ( singletonsubset
% 0.15/0.38 => ( singletoninpowerset
% 0.15/0.38 => ( singletoninpowunion
% 0.15/0.38 => ( upairset2E
% 0.15/0.38 => ( upairsubunion
% 0.15/0.38 => ( upairinpowunion
% 0.15/0.38 => ( ubforcartprodlem1
% 0.15/0.38 => ( ubforcartprodlem2
% 0.15/0.38 => ( ubforcartprodlem3
% 0.15/0.38 => ( cartprodpairin
% 0.15/0.38 => ( cartprodmempair1
% 0.15/0.38 => ( cartprodmempair
% 0.15/0.38 => ( setunionE2
% 0.15/0.38 => ( setunionsingleton1
% 0.15/0.38 => ( setunionsingleton2
% 0.15/0.38 => ( setunionsingleton
% 0.15/0.38 => ( singletonprop
% 0.15/0.38 => ( ex1E1
% 0.15/0.38 => ( ex1I
% 0.15/0.38 => ( ex1I2
% 0.15/0.38 => ( singletonsuniq
% 0.15/0.38 => ( setukpairinjL1
% 0.15/0.38 => ( kfstsingleton
% 0.15/0.38 => ( theprop
% 0.15/0.38 => ( kfstpairEq
% 0.15/0.38 => ( cartprodfstin
% 0.15/0.38 => ( setukpairinjL2
% 0.15/0.38 => ( setukpairinjL
% 0.15/0.38 => ( setukpairinjR11
% 0.15/0.38 => ( setukpairinjR12
% 0.15/0.38 => ( setukpairinjR1
% 0.15/0.38 => ( upairequniteq
% 0.15/0.38 => ( setukpairinjR2
% 0.15/0.38 => ( setukpairinjR
% 0.15/0.38 => ( ksndsingleton
% 0.15/0.38 => ( ksndpairEq
% 0.15/0.38 => ( kpairsurjEq
% 0.15/0.38 => ( cartprodsndin
% 0.15/0.38 => ( cartprodpairmemEL
% 0.15/0.38 => ( cartprodpairmemER
% 0.15/0.38 => ( cartprodmempaircEq
% 0.15/0.38 => ( cartprodfstpairEq
% 0.15/0.38 => ( cartprodsndpairEq
% 0.15/0.38 => ( cartprodpairsurjEq
% 0.15/0.38 => ( dpsetconstrI
% 0.15/0.38 => ( dpsetconstrSub
% 0.15/0.38 => ( setOfPairsIsBReln
% 0.15/0.38 => ( dpsetconstrERa
% 0.15/0.38 => ( dpsetconstrEL1
% 0.15/0.38 => ( dpsetconstrEL2
% 0.15/0.38 => ( dpsetconstrER
% 0.15/0.38 => ( funcImageSingleton
% 0.15/0.38 => ( apProp
% 0.15/0.38 => ( app
% 0.15/0.38 => ( infuncsetfunc
% 0.15/0.38 => ( ap2p
% 0.15/0.38 => ( funcinfuncset
% 0.15/0.38 => ( lamProp
% 0.15/0.38 => ( lamp
% 0.15/0.38 => ( lam2p
% 0.15/0.38 => ( brelnall1
% 0.15/0.38 => ( brelnall2
% 0.15/0.38 => ( ex1E2
% 0.15/0.38 => ( funcGraphProp1
% 0.15/0.38 => ( funcGraphProp3
% 0.15/0.38 => ( funcGraphProp2
% 0.15/0.38 => ( funcextLem
% 0.15/0.38 => ( funcGraphProp4
% 0.15/0.38 => ( subbreln
% 0.15/0.38 => ( eqbreln
% 0.15/0.38 => ( funcext
% 0.15/0.38 => ( funcext2
% 0.15/0.38 => ( ap2apEq1
% 0.15/0.38 => ( ap2apEq2
% 0.15/0.38 => ( beta1
% 0.15/0.38 => ( eta1
% 0.15/0.38 => ( lam2lamEq
% 0.15/0.38 => ( beta2
% 0.15/0.38 => ( eta2
% 0.15/0.38 => ( iffalseProp1
% 0.15/0.38 => ( iffalseProp2
% 0.15/0.38 => ( iftrueProp1
% 0.15/0.38 => ( iftrueProp2
% 0.15/0.38 => ( ifSingleton
% 0.15/0.38 => ( ifp
% 0.15/0.38 => ( theeq
% 0.15/0.38 => ( iftrue
% 0.15/0.38 => ( iffalse
% 0.15/0.38 => ( iftrueorfalse
% 0.15/0.38 => ( binintersectT_lem
% 0.15/0.38 => ( binunionT_lem
% 0.15/0.38 => ( powersetT_lem
% 0.15/0.38 => ( setminusT_lem
% 0.15/0.38 => ( complementT_lem
% 0.15/0.38 => ( setextT
% 0.15/0.38 => ( subsetTI
% 0.15/0.38 => ( powersetTI1
% 0.15/0.38 => ( powersetTE1
% 0.15/0.38 => ( complementTI1
% 0.15/0.38 => ( complementTE1
% 0.15/0.38 => ( binintersectTELcontra
% 0.15/0.38 => ( binintersectTERcontra
% 0.15/0.38 => ( contrasubsetT
% 0.15/0.38 => ( contrasubsetT1
% 0.15/0.38 => ( contrasubsetT2
% 0.15/0.38 => ( contrasubsetT3
% 0.15/0.38 => ( doubleComplementI1
% 0.15/0.38 => ( doubleComplementE1
% 0.15/0.38 => ( doubleComplementSub1
% 0.15/0.38 => ( doubleComplementSub2
% 0.15/0.38 => ( doubleComplementEq
% 0.15/0.38 => ( complementTnotintersectT
% 0.15/0.38 => ( complementImpComplementIntersect
% 0.15/0.38 => ! [A: $i,X: $i] :
% 0.15/0.38 ( ( in @ X @ ( powerset @ A ) )
% 0.15/0.38 => ! [Y: $i] :
% 0.15/0.38 ( ( in @ Y @ ( powerset @ A ) )
% 0.15/0.38 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.15/0.38
% 0.15/0.38 %------------------------------------------------------------------------------
% 0.15/0.38 ------------------------
% 2.61/1.45 ---- Embedded in THF ---
% 2.61/1.45 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.61/1.45 %%% Generated on Tue Feb 24 12:32:51 EST 2026
% 2.61/1.45 %%% using '$$dhol' embedding, version 1.3.0.
% 2.61/1.45 %%% Logic specification used:
% 2.61/1.45 %%% thf(spec, logic, $$dhol).
% 2.61/1.45
% 2.61/1.45 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.XrUd2iUsu2/DTF2DTF28816.p
% See solution above
% 2.61/1.45 ------------------------
% 2.61/1.48 ---- Cleaned THF ---
% 2.61/1.48 thf(in_type,type,
% 2.61/1.48 in: $i > $i > $o ).
% 2.61/1.48
% 2.61/1.48 thf(exu_type,type,
% 2.61/1.48 exu: ( $i > $o ) > $o ).
% 2.61/1.48
% 2.61/1.48 thf(exu,definition,
% 2.61/1.48 ( exu
% 2.61/1.48 = ( ^ [Xphi: $i > $o] :
% 2.61/1.48 ? [Xx: $i] :
% 2.61/1.48 ( ( Xphi @ Xx )
% 2.61/1.48 & ! [Xy: $i] :
% 2.61/1.48 ( ( Xphi @ Xy )
% 2.61/1.48 => ( Xx = Xy ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setextAx_type,type,
% 2.61/1.48 setextAx: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setextAx,definition,
% 2.61/1.48 ( setextAx
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 <=> ( in @ Xx @ B ) )
% 2.61/1.48 => ( A = B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(emptyset_type,type,
% 2.61/1.48 emptyset: $i ).
% 2.61/1.48
% 2.61/1.48 thf(emptysetAx_type,type,
% 2.61/1.48 emptysetAx: $o ).
% 2.61/1.48
% 2.61/1.48 thf(emptysetAx,definition,
% 2.61/1.48 ( emptysetAx
% 2.61/1.48 = ( ! [Xx: $i] :
% 2.61/1.48 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoin_type,type,
% 2.61/1.48 setadjoin: $i > $i > $i ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoinAx_type,type,
% 2.61/1.48 setadjoinAx: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoinAx,definition,
% 2.61/1.48 ( setadjoinAx
% 2.61/1.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.61/1.48 <=> ( ( Xy = Xx )
% 2.61/1.48 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(powerset_type,type,
% 2.61/1.48 powerset: $i > $i ).
% 2.61/1.48
% 2.61/1.48 thf(powersetAx_type,type,
% 2.61/1.48 powersetAx: $o ).
% 2.61/1.48
% 2.61/1.48 thf(powersetAx,definition,
% 2.61/1.48 ( powersetAx
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( in @ B @ ( powerset @ A ) )
% 2.61/1.48 <=> ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ B )
% 2.61/1.48 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setunion_type,type,
% 2.61/1.48 setunion: $i > $i ).
% 2.61/1.48
% 2.61/1.48 thf(setunionAx_type,type,
% 2.61/1.48 setunionAx: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setunionAx,definition,
% 2.61/1.48 ( setunionAx
% 2.61/1.48 = ( ! [A: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ ( setunion @ A ) )
% 2.61/1.48 <=> ? [B: $i] :
% 2.61/1.48 ( ( in @ Xx @ B )
% 2.61/1.48 & ( in @ B @ A ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(omega_type,type,
% 2.61/1.48 omega: $i ).
% 2.61/1.48
% 2.61/1.48 thf(omega0Ax_type,type,
% 2.61/1.48 omega0Ax: $o ).
% 2.61/1.48
% 2.61/1.48 thf(omega0Ax,definition,
% 2.61/1.48 ( omega0Ax
% 2.61/1.48 = ( in @ emptyset @ omega ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(omegaSAx_type,type,
% 2.61/1.48 omegaSAx: $o ).
% 2.61/1.48
% 2.61/1.48 thf(omegaSAx,definition,
% 2.61/1.48 ( omegaSAx
% 2.61/1.48 = ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ omega )
% 2.61/1.48 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(omegaIndAx_type,type,
% 2.61/1.48 omegaIndAx: $o ).
% 2.61/1.48
% 2.61/1.48 thf(omegaIndAx,definition,
% 2.61/1.48 ( omegaIndAx
% 2.61/1.48 = ( ! [A: $i] :
% 2.61/1.48 ( ( ( in @ emptyset @ A )
% 2.61/1.48 & ! [Xx: $i] :
% 2.61/1.48 ( ( ( in @ Xx @ omega )
% 2.61/1.48 & ( in @ Xx @ A ) )
% 2.61/1.48 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ omega )
% 2.61/1.48 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(replAx_type,type,
% 2.61/1.48 replAx: $o ).
% 2.61/1.48
% 2.61/1.48 thf(replAx,definition,
% 2.61/1.48 ( replAx
% 2.61/1.48 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( exu
% 2.61/1.48 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.61/1.48 => ? [B: $i] :
% 2.61/1.48 ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ B )
% 2.61/1.48 <=> ? [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ A )
% 2.61/1.48 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(foundationAx_type,type,
% 2.61/1.48 foundationAx: $o ).
% 2.61/1.48
% 2.61/1.48 thf(foundationAx,definition,
% 2.61/1.48 ( foundationAx
% 2.61/1.48 = ( ! [A: $i] :
% 2.61/1.48 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.61/1.48 => ? [B: $i] :
% 2.61/1.48 ( ( in @ B @ A )
% 2.61/1.48 & ~ ? [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ B )
% 2.61/1.48 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(wellorderingAx_type,type,
% 2.61/1.48 wellorderingAx: $o ).
% 2.61/1.48
% 2.61/1.48 thf(wellorderingAx,definition,
% 2.61/1.48 ( wellorderingAx
% 2.61/1.48 = ( ! [A: $i] :
% 2.61/1.48 ? [B: $i] :
% 2.61/1.48 ( ! [C: $i] :
% 2.61/1.48 ( ( in @ C @ B )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ C )
% 2.61/1.48 => ( in @ Xx @ A ) ) )
% 2.61/1.48 & ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( ( in @ Xx @ A )
% 2.61/1.48 & ( in @ Xy @ A ) )
% 2.61/1.48 => ( ! [C: $i] :
% 2.61/1.48 ( ( in @ C @ B )
% 2.61/1.48 => ( ( in @ Xx @ C )
% 2.61/1.48 <=> ( in @ Xy @ C ) ) )
% 2.61/1.48 => ( Xx = Xy ) ) )
% 2.61/1.48 & ! [C: $i,D: $i] :
% 2.61/1.48 ( ( ( in @ C @ B )
% 2.61/1.48 & ( in @ D @ B ) )
% 2.61/1.48 => ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ C )
% 2.61/1.48 => ( in @ Xx @ D ) )
% 2.61/1.48 | ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ D )
% 2.61/1.48 => ( in @ Xx @ C ) ) ) )
% 2.61/1.48 & ! [C: $i] :
% 2.61/1.48 ( ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ C )
% 2.61/1.48 => ( in @ Xx @ A ) )
% 2.61/1.48 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.61/1.48 => ? [D: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ D @ B )
% 2.61/1.48 & ( in @ Xx @ C )
% 2.61/1.48 & ~ ? [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ D )
% 2.61/1.48 & ( in @ Xy @ C ) )
% 2.61/1.48 & ! [E: $i] :
% 2.61/1.48 ( ( in @ E @ B )
% 2.61/1.48 => ( ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ E )
% 2.61/1.48 => ( in @ Xy @ D ) )
% 2.61/1.48 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(descr_type,type,
% 2.61/1.48 descr: ( $i > $o ) > $i ).
% 2.61/1.48
% 2.61/1.48 thf(descrp_type,type,
% 2.61/1.48 descrp: $o ).
% 2.61/1.48
% 2.61/1.48 thf(descrp,definition,
% 2.61/1.48 ( descrp
% 2.61/1.48 = ( ! [Xphi: $i > $o] :
% 2.61/1.48 ( ( exu
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.48 => ( Xphi
% 2.61/1.48 @ ( descr
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(dsetconstr_type,type,
% 2.61/1.48 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.61/1.48
% 2.61/1.48 thf(dsetconstrI_type,type,
% 2.61/1.48 dsetconstrI: $o ).
% 2.61/1.48
% 2.61/1.48 thf(dsetconstrI,definition,
% 2.61/1.48 ( dsetconstrI
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( Xphi @ Xx )
% 2.61/1.48 => ( in @ Xx
% 2.61/1.48 @ ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(dsetconstrEL_type,type,
% 2.61/1.48 dsetconstrEL: $o ).
% 2.61/1.48
% 2.61/1.48 thf(dsetconstrEL,definition,
% 2.61/1.48 ( dsetconstrEL
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx
% 2.61/1.48 @ ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.61/1.48 => ( in @ Xx @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(dsetconstrER_type,type,
% 2.61/1.48 dsetconstrER: $o ).
% 2.61/1.48
% 2.61/1.48 thf(dsetconstrER,definition,
% 2.61/1.48 ( dsetconstrER
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx
% 2.61/1.48 @ ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.61/1.48 => ( Xphi @ Xx ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(exuE1_type,type,
% 2.61/1.48 exuE1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(exuE1,definition,
% 2.61/1.48 ( exuE1
% 2.61/1.48 = ( ! [Xphi: $i > $o] :
% 2.61/1.48 ( ( exu
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.48 => ? [Xx: $i] :
% 2.61/1.48 ( ( Xphi @ Xx )
% 2.61/1.48 & ! [Xy: $i] :
% 2.61/1.48 ( ( Xphi @ Xy )
% 2.61/1.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(prop2set_type,type,
% 2.61/1.48 prop2set: $o > $i ).
% 2.61/1.48
% 2.61/1.48 thf(prop2setE_type,type,
% 2.61/1.48 prop2setE: $o ).
% 2.61/1.48
% 2.61/1.48 thf(prop2setE,definition,
% 2.61/1.48 ( prop2setE
% 2.61/1.48 = ( ! [Xphi: $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.61/1.48 => Xphi ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(emptysetE_type,type,
% 2.61/1.48 emptysetE: $o ).
% 2.61/1.48
% 2.61/1.48 thf(emptysetE,definition,
% 2.61/1.48 ( emptysetE
% 2.61/1.48 = ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ emptyset )
% 2.61/1.48 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(emptysetimpfalse_type,type,
% 2.61/1.48 emptysetimpfalse: $o ).
% 2.61/1.48
% 2.61/1.48 thf(emptysetimpfalse,definition,
% 2.61/1.48 ( emptysetimpfalse
% 2.61/1.48 = ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ emptyset )
% 2.61/1.48 => $false ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(notinemptyset_type,type,
% 2.61/1.48 notinemptyset: $o ).
% 2.61/1.48
% 2.61/1.48 thf(notinemptyset,definition,
% 2.61/1.48 ( notinemptyset
% 2.61/1.48 = ( ! [Xx: $i] :
% 2.61/1.48 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(exuE3e_type,type,
% 2.61/1.48 exuE3e: $o ).
% 2.61/1.48
% 2.61/1.48 thf(exuE3e,definition,
% 2.61/1.48 ( exuE3e
% 2.61/1.48 = ( ! [Xphi: $i > $o] :
% 2.61/1.48 ( ( exu
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.48 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setext_type,type,
% 2.61/1.48 setext: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setext,definition,
% 2.61/1.48 ( setext
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ Xx @ B ) )
% 2.61/1.48 => ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ B )
% 2.61/1.48 => ( in @ Xx @ A ) )
% 2.61/1.48 => ( A = B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(emptyI_type,type,
% 2.61/1.48 emptyI: $o ).
% 2.61/1.48
% 2.61/1.48 thf(emptyI,definition,
% 2.61/1.48 ( emptyI
% 2.61/1.48 = ( ! [A: $i] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ~ ( in @ Xx @ A )
% 2.61/1.48 => ( A = emptyset ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(noeltsimpempty_type,type,
% 2.61/1.48 noeltsimpempty: $o ).
% 2.61/1.48
% 2.61/1.48 thf(noeltsimpempty,definition,
% 2.61/1.48 ( noeltsimpempty
% 2.61/1.48 = ( ! [A: $i] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ~ ( in @ Xx @ A )
% 2.61/1.48 => ( A = emptyset ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setbeta_type,type,
% 2.61/1.48 setbeta: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setbeta,definition,
% 2.61/1.48 ( setbeta
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( in @ Xx
% 2.61/1.48 @ ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.61/1.48 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(nonempty_type,type,
% 2.61/1.48 nonempty: $i > $o ).
% 2.61/1.48
% 2.61/1.48 thf(nonempty,definition,
% 2.61/1.48 ( nonempty
% 2.61/1.48 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(nonemptyE1_type,type,
% 2.61/1.48 nonemptyE1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(nonemptyE1,definition,
% 2.61/1.48 ( nonemptyE1
% 2.61/1.48 = ( ! [A: $i] :
% 2.61/1.48 ( ( nonempty @ A )
% 2.61/1.48 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(nonemptyI_type,type,
% 2.61/1.48 nonemptyI: $o ).
% 2.61/1.48
% 2.61/1.48 thf(nonemptyI,definition,
% 2.61/1.48 ( nonemptyI
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( Xphi @ Xx )
% 2.61/1.48 => ( nonempty
% 2.61/1.48 @ ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(nonemptyI1_type,type,
% 2.61/1.48 nonemptyI1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(nonemptyI1,definition,
% 2.61/1.48 ( nonemptyI1
% 2.61/1.48 = ( ! [A: $i] :
% 2.61/1.48 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.61/1.48 => ( nonempty @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoinIL_type,type,
% 2.61/1.48 setadjoinIL: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoinIL,definition,
% 2.61/1.48 ( setadjoinIL
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(emptyinunitempty_type,type,
% 2.61/1.48 emptyinunitempty: $o ).
% 2.61/1.48
% 2.61/1.48 thf(emptyinunitempty,definition,
% 2.61/1.48 ( emptyinunitempty
% 2.61/1.48 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoinIR_type,type,
% 2.61/1.48 setadjoinIR: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoinIR,definition,
% 2.61/1.48 ( setadjoinIR
% 2.61/1.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ A )
% 2.61/1.48 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoinE_type,type,
% 2.61/1.48 setadjoinE: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoinE,definition,
% 2.61/1.48 ( setadjoinE
% 2.61/1.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.61/1.48 => ! [Xphi: $o] :
% 2.61/1.48 ( ( ( Xy = Xx )
% 2.61/1.48 => Xphi )
% 2.61/1.48 => ( ( ( in @ Xy @ A )
% 2.61/1.48 => Xphi )
% 2.61/1.48 => Xphi ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoinOr_type,type,
% 2.61/1.48 setadjoinOr: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoinOr,definition,
% 2.61/1.48 ( setadjoinOr
% 2.61/1.48 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.61/1.48 => ( ( Xy = Xx )
% 2.61/1.48 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setoftrueEq_type,type,
% 2.61/1.48 setoftrueEq: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setoftrueEq,definition,
% 2.61/1.48 ( setoftrueEq
% 2.61/1.48 = ( ! [A: $i] :
% 2.61/1.48 ( ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xx: $i] : $true )
% 2.61/1.48 = A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(powersetI_type,type,
% 2.61/1.48 powersetI: $o ).
% 2.61/1.48
% 2.61/1.48 thf(powersetI,definition,
% 2.61/1.48 ( powersetI
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ B )
% 2.61/1.48 => ( in @ Xx @ A ) )
% 2.61/1.48 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(emptyinPowerset_type,type,
% 2.61/1.48 emptyinPowerset: $o ).
% 2.61/1.48
% 2.61/1.48 thf(emptyinPowerset,definition,
% 2.61/1.48 ( emptyinPowerset
% 2.61/1.48 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(emptyInPowerset_type,type,
% 2.61/1.48 emptyInPowerset: $o ).
% 2.61/1.48
% 2.61/1.48 thf(emptyInPowerset,definition,
% 2.61/1.48 ( emptyInPowerset
% 2.61/1.48 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(powersetE_type,type,
% 2.61/1.48 powersetE: $o ).
% 2.61/1.48
% 2.61/1.48 thf(powersetE,definition,
% 2.61/1.48 ( powersetE
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ B @ ( powerset @ A ) )
% 2.61/1.48 => ( ( in @ Xx @ B )
% 2.61/1.48 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setunionI_type,type,
% 2.61/1.48 setunionI: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setunionI,definition,
% 2.61/1.48 ( setunionI
% 2.61/1.48 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.61/1.48 ( ( in @ Xx @ B )
% 2.61/1.48 => ( ( in @ B @ A )
% 2.61/1.48 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setunionE_type,type,
% 2.61/1.48 setunionE: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setunionE,definition,
% 2.61/1.48 ( setunionE
% 2.61/1.48 = ( ! [A: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ ( setunion @ A ) )
% 2.61/1.48 => ! [Xphi: $o] :
% 2.61/1.48 ( ! [B: $i] :
% 2.61/1.48 ( ( in @ Xx @ B )
% 2.61/1.48 => ( ( in @ B @ A )
% 2.61/1.48 => Xphi ) )
% 2.61/1.48 => Xphi ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(subPowSU_type,type,
% 2.61/1.48 subPowSU: $o ).
% 2.61/1.48
% 2.61/1.48 thf(subPowSU,definition,
% 2.61/1.48 ( subPowSU
% 2.61/1.48 = ( ! [A: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(exuE2_type,type,
% 2.61/1.48 exuE2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(exuE2,definition,
% 2.61/1.48 ( exuE2
% 2.61/1.48 = ( ! [Xphi: $i > $o] :
% 2.61/1.48 ( ( exu
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.48 => ? [Xx: $i] :
% 2.61/1.48 ! [Xy: $i] :
% 2.61/1.48 ( ( Xphi @ Xy )
% 2.61/1.48 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(nonemptyImpWitness_type,type,
% 2.61/1.48 nonemptyImpWitness: $o ).
% 2.61/1.48
% 2.61/1.48 thf(nonemptyImpWitness,definition,
% 2.61/1.48 ( nonemptyImpWitness
% 2.61/1.48 = ( ! [A: $i] :
% 2.61/1.48 ( ( nonempty @ A )
% 2.61/1.48 => ? [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 & $true ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(uniqinunit_type,type,
% 2.61/1.48 uniqinunit: $o ).
% 2.61/1.48
% 2.61/1.48 thf(uniqinunit,definition,
% 2.61/1.48 ( uniqinunit
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.61/1.48 => ( Xx = Xy ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(notinsingleton_type,type,
% 2.61/1.48 notinsingleton: $o ).
% 2.61/1.48
% 2.61/1.48 thf(notinsingleton,definition,
% 2.61/1.48 ( notinsingleton
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( Xx != Xy )
% 2.61/1.48 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(eqinunit_type,type,
% 2.61/1.48 eqinunit: $o ).
% 2.61/1.48
% 2.61/1.48 thf(eqinunit,definition,
% 2.61/1.48 ( eqinunit
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( Xx = Xy )
% 2.61/1.48 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(singletonsswitch_type,type,
% 2.61/1.48 singletonsswitch: $o ).
% 2.61/1.48
% 2.61/1.48 thf(singletonsswitch,definition,
% 2.61/1.48 ( singletonsswitch
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.61/1.48 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(upairsetE_type,type,
% 2.61/1.48 upairsetE: $o ).
% 2.61/1.48
% 2.61/1.48 thf(upairsetE,definition,
% 2.61/1.48 ( upairsetE
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.61/1.48 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.61/1.48 => ( ( Xz = Xx )
% 2.61/1.48 | ( Xz = Xy ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(upairsetIL_type,type,
% 2.61/1.48 upairsetIL: $o ).
% 2.61/1.48
% 2.61/1.48 thf(upairsetIL,definition,
% 2.61/1.48 ( upairsetIL
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(upairsetIR_type,type,
% 2.61/1.48 upairsetIR: $o ).
% 2.61/1.48
% 2.61/1.48 thf(upairsetIR,definition,
% 2.61/1.48 ( upairsetIR
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(emptyE1_type,type,
% 2.61/1.48 emptyE1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(emptyE1,definition,
% 2.61/1.48 ( emptyE1
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.48 ( ? [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 & ( Xphi @ Xx ) )
% 2.61/1.48 => ( ( ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.48 = emptyset )
% 2.61/1.48 => $false ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(vacuousDall_type,type,
% 2.61/1.48 vacuousDall: $o ).
% 2.61/1.48
% 2.61/1.48 thf(vacuousDall,definition,
% 2.61/1.48 ( vacuousDall
% 2.61/1.48 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ emptyset )
% 2.61/1.48 => ( Xphi @ Xx ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(quantDeMorgan1_type,type,
% 2.61/1.48 quantDeMorgan1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(quantDeMorgan1,definition,
% 2.61/1.48 ( quantDeMorgan1
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.48 ( ~ ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( Xphi @ Xx ) )
% 2.61/1.48 => ? [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(quantDeMorgan2_type,type,
% 2.61/1.48 quantDeMorgan2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(quantDeMorgan2,definition,
% 2.61/1.48 ( quantDeMorgan2
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ~ ( Xphi @ Xx ) )
% 2.61/1.48 => ~ ? [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(quantDeMorgan3_type,type,
% 2.61/1.48 quantDeMorgan3: $o ).
% 2.61/1.48
% 2.61/1.48 thf(quantDeMorgan3,definition,
% 2.61/1.48 ( quantDeMorgan3
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.48 ( ~ ? [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 & ( Xphi @ Xx ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(quantDeMorgan4_type,type,
% 2.61/1.48 quantDeMorgan4: $o ).
% 2.61/1.48
% 2.61/1.48 thf(quantDeMorgan4,definition,
% 2.61/1.48 ( quantDeMorgan4
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.48 ( ? [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 & ~ ( Xphi @ Xx ) )
% 2.61/1.48 => ~ ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(prop2setI_type,type,
% 2.61/1.48 prop2setI: $o ).
% 2.61/1.48
% 2.61/1.48 thf(prop2setI,definition,
% 2.61/1.48 ( prop2setI
% 2.61/1.48 = ( ! [Xphi: $o] :
% 2.61/1.48 ( Xphi
% 2.61/1.48 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(set2prop_type,type,
% 2.61/1.48 set2prop: $i > $o ).
% 2.61/1.48
% 2.61/1.48 thf(prop2set2propI_type,type,
% 2.61/1.48 prop2set2propI: $o ).
% 2.61/1.48
% 2.61/1.48 thf(prop2set2propI,definition,
% 2.61/1.48 ( prop2set2propI
% 2.61/1.48 = ( ! [Xphi: $o] :
% 2.61/1.48 ( Xphi
% 2.61/1.48 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(notdexE_type,type,
% 2.61/1.48 notdexE: $o ).
% 2.61/1.48
% 2.61/1.48 thf(notdexE,definition,
% 2.61/1.48 ( notdexE
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.48 ( ~ ? [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 & ( Xphi @ Xx ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(notdallE_type,type,
% 2.61/1.48 notdallE: $o ).
% 2.61/1.48
% 2.61/1.48 thf(notdallE,definition,
% 2.61/1.48 ( notdallE
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.48 ( ~ ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( Xphi @ Xx ) )
% 2.61/1.48 => ? [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(exuI1_type,type,
% 2.61/1.48 exuI1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(exuI1,definition,
% 2.61/1.48 ( exuI1
% 2.61/1.48 = ( ! [Xphi: $i > $o] :
% 2.61/1.48 ( ? [Xx: $i] :
% 2.61/1.48 ( ( Xphi @ Xx )
% 2.61/1.48 & ! [Xy: $i] :
% 2.61/1.48 ( ( Xphi @ Xy )
% 2.61/1.48 => ( Xx = Xy ) ) )
% 2.61/1.48 => ( exu
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(exuI3_type,type,
% 2.61/1.48 exuI3: $o ).
% 2.61/1.48
% 2.61/1.48 thf(exuI3,definition,
% 2.61/1.48 ( exuI3
% 2.61/1.48 = ( ! [Xphi: $i > $o] :
% 2.61/1.48 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.61/1.48 => ( ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( Xphi @ Xx )
% 2.61/1.48 => ( ( Xphi @ Xy )
% 2.61/1.48 => ( Xx = Xy ) ) )
% 2.61/1.48 => ( exu
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(exuI2_type,type,
% 2.61/1.48 exuI2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(exuI2,definition,
% 2.61/1.48 ( exuI2
% 2.61/1.48 = ( ! [Xphi: $i > $o] :
% 2.61/1.48 ( ? [Xx: $i] :
% 2.61/1.48 ! [Xy: $i] :
% 2.61/1.48 ( ( Xphi @ Xy )
% 2.61/1.48 <=> ( Xy = Xx ) )
% 2.61/1.48 => ( exu
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(inCongP_type,type,
% 2.61/1.48 inCongP: $o ).
% 2.61/1.48
% 2.61/1.48 thf(inCongP,definition,
% 2.61/1.48 ( inCongP
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( A = B )
% 2.61/1.48 => ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( Xx = Xy )
% 2.61/1.48 => ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(in__Cong_type,type,
% 2.61/1.48 in__Cong: $o ).
% 2.61/1.48
% 2.61/1.48 thf(in__Cong,definition,
% 2.61/1.48 ( in__Cong
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( A = B )
% 2.61/1.48 => ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( Xx = Xy )
% 2.61/1.48 => ( ( in @ Xx @ A )
% 2.61/1.48 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(exuE3u_type,type,
% 2.61/1.48 exuE3u: $o ).
% 2.61/1.48
% 2.61/1.48 thf(exuE3u,definition,
% 2.61/1.48 ( exuE3u
% 2.61/1.48 = ( ! [Xphi: $i > $o] :
% 2.61/1.48 ( ( exu
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.48 => ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( Xphi @ Xx )
% 2.61/1.48 => ( ( Xphi @ Xy )
% 2.61/1.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(exu__Cong_type,type,
% 2.61/1.48 exu__Cong: $o ).
% 2.61/1.48
% 2.61/1.48 thf(exu__Cong,definition,
% 2.61/1.48 ( exu__Cong
% 2.61/1.48 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.61/1.48 ( ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( Xx = Xy )
% 2.61/1.48 => ( ( Xphi @ Xx )
% 2.61/1.48 <=> ( Xpsi @ Xy ) ) )
% 2.61/1.48 => ( ( exu
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.48 <=> ( exu
% 2.61/1.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(emptyset__Cong_type,type,
% 2.61/1.48 emptyset__Cong: $o ).
% 2.61/1.48
% 2.61/1.48 thf(emptyset__Cong,definition,
% 2.61/1.48 ( emptyset__Cong
% 2.61/1.48 = ( emptyset = emptyset ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoin__Cong_type,type,
% 2.61/1.48 setadjoin__Cong: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoin__Cong,definition,
% 2.61/1.48 ( setadjoin__Cong
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( Xx = Xy )
% 2.61/1.48 => ! [Xz: $i,Xu: $i] :
% 2.61/1.48 ( ( Xz = Xu )
% 2.61/1.48 => ( ( setadjoin @ Xx @ Xz )
% 2.61/1.48 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(powerset__Cong_type,type,
% 2.61/1.48 powerset__Cong: $o ).
% 2.61/1.48
% 2.61/1.48 thf(powerset__Cong,definition,
% 2.61/1.48 ( powerset__Cong
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( A = B )
% 2.61/1.48 => ( ( powerset @ A )
% 2.61/1.48 = ( powerset @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setunion__Cong_type,type,
% 2.61/1.48 setunion__Cong: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setunion__Cong,definition,
% 2.61/1.48 ( setunion__Cong
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( A = B )
% 2.61/1.48 => ( ( setunion @ A )
% 2.61/1.48 = ( setunion @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(omega__Cong_type,type,
% 2.61/1.48 omega__Cong: $o ).
% 2.61/1.48
% 2.61/1.48 thf(omega__Cong,definition,
% 2.61/1.48 ( omega__Cong
% 2.61/1.48 = ( omega = omega ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(exuEu_type,type,
% 2.61/1.48 exuEu: $o ).
% 2.61/1.48
% 2.61/1.48 thf(exuEu,definition,
% 2.61/1.48 ( exuEu
% 2.61/1.48 = ( ! [Xphi: $i > $o] :
% 2.61/1.48 ( ( exu
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.48 => ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( Xphi @ Xx )
% 2.61/1.48 => ( ( Xphi @ Xy )
% 2.61/1.48 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(descr__Cong_type,type,
% 2.61/1.48 descr__Cong: $o ).
% 2.61/1.48
% 2.61/1.48 thf(descr__Cong,definition,
% 2.61/1.48 ( descr__Cong
% 2.61/1.48 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.61/1.48 ( ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( Xx = Xy )
% 2.61/1.48 => ( ( Xphi @ Xx )
% 2.61/1.48 <=> ( Xpsi @ Xy ) ) )
% 2.61/1.48 => ( ( exu
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.48 => ( ( exu
% 2.61/1.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.61/1.48 => ( ( descr
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.48 = ( descr
% 2.61/1.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(dsetconstr__Cong_type,type,
% 2.61/1.48 dsetconstr__Cong: $o ).
% 2.61/1.48
% 2.61/1.48 thf(dsetconstr__Cong,definition,
% 2.61/1.48 ( dsetconstr__Cong
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( A = B )
% 2.61/1.48 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( ( Xx = Xy )
% 2.61/1.48 => ( ( Xphi @ Xx )
% 2.61/1.48 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.61/1.48 => ( ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.48 = ( dsetconstr @ B
% 2.61/1.48 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(subset_type,type,
% 2.61/1.48 subset: $i > $i > $o ).
% 2.61/1.48
% 2.61/1.48 thf(disjoint_type,type,
% 2.61/1.48 disjoint: $i > $i > $o ).
% 2.61/1.48
% 2.61/1.48 thf(setsmeet_type,type,
% 2.61/1.48 setsmeet: $i > $i > $o ).
% 2.61/1.48
% 2.61/1.48 thf(subsetI1_type,type,
% 2.61/1.48 subsetI1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(subsetI1,definition,
% 2.61/1.48 ( subsetI1
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ Xx @ B ) )
% 2.61/1.48 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(eqimpsubset2_type,type,
% 2.61/1.48 eqimpsubset2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(eqimpsubset2,definition,
% 2.61/1.48 ( eqimpsubset2
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( A = B )
% 2.61/1.48 => ( subset @ B @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(eqimpsubset1_type,type,
% 2.61/1.48 eqimpsubset1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(eqimpsubset1,definition,
% 2.61/1.48 ( eqimpsubset1
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( A = B )
% 2.61/1.48 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(subsetI2_type,type,
% 2.61/1.48 subsetI2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(subsetI2,definition,
% 2.61/1.48 ( subsetI2
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ Xx @ B ) )
% 2.61/1.48 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(emptysetsubset_type,type,
% 2.61/1.48 emptysetsubset: $o ).
% 2.61/1.48
% 2.61/1.48 thf(emptysetsubset,definition,
% 2.61/1.48 ( emptysetsubset
% 2.61/1.48 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(subsetE_type,type,
% 2.61/1.48 subsetE: $o ).
% 2.61/1.48
% 2.61/1.48 thf(subsetE,definition,
% 2.61/1.48 ( subsetE
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( subset @ A @ B )
% 2.61/1.48 => ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(subsetE2_type,type,
% 2.61/1.48 subsetE2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(subsetE2,definition,
% 2.61/1.48 ( subsetE2
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( subset @ A @ B )
% 2.61/1.48 => ( ~ ( in @ Xx @ B )
% 2.61/1.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(notsubsetI_type,type,
% 2.61/1.48 notsubsetI: $o ).
% 2.61/1.48
% 2.61/1.48 thf(notsubsetI,definition,
% 2.61/1.48 ( notsubsetI
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ~ ( in @ Xx @ B )
% 2.61/1.48 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(notequalI1_type,type,
% 2.61/1.48 notequalI1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(notequalI1,definition,
% 2.61/1.48 ( notequalI1
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ~ ( subset @ A @ B )
% 2.61/1.48 => ( A != B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(notequalI2_type,type,
% 2.61/1.48 notequalI2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(notequalI2,definition,
% 2.61/1.48 ( notequalI2
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ~ ( in @ Xx @ B )
% 2.61/1.48 => ( A != B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(subsetRefl_type,type,
% 2.61/1.48 subsetRefl: $o ).
% 2.61/1.48
% 2.61/1.48 thf(subsetRefl,definition,
% 2.61/1.48 ( subsetRefl
% 2.61/1.48 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(subsetTrans_type,type,
% 2.61/1.48 subsetTrans: $o ).
% 2.61/1.48
% 2.61/1.48 thf(subsetTrans,definition,
% 2.61/1.48 ( subsetTrans
% 2.61/1.48 = ( ! [A: $i,B: $i,C: $i] :
% 2.61/1.48 ( ( subset @ A @ B )
% 2.61/1.48 => ( ( subset @ B @ C )
% 2.61/1.48 => ( subset @ A @ C ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoinSub_type,type,
% 2.61/1.48 setadjoinSub: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoinSub,definition,
% 2.61/1.48 ( setadjoinSub
% 2.61/1.48 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoinSub2_type,type,
% 2.61/1.48 setadjoinSub2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setadjoinSub2,definition,
% 2.61/1.48 ( setadjoinSub2
% 2.61/1.48 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.61/1.48 ( ( subset @ A @ B )
% 2.61/1.48 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(subset2powerset_type,type,
% 2.61/1.48 subset2powerset: $o ).
% 2.61/1.48
% 2.61/1.48 thf(subset2powerset,definition,
% 2.61/1.48 ( subset2powerset
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( subset @ A @ B )
% 2.61/1.48 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setextsub_type,type,
% 2.61/1.48 setextsub: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setextsub,definition,
% 2.61/1.48 ( setextsub
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( subset @ A @ B )
% 2.61/1.48 => ( ( subset @ B @ A )
% 2.61/1.48 => ( A = B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(subsetemptysetimpeq_type,type,
% 2.61/1.48 subsetemptysetimpeq: $o ).
% 2.61/1.48
% 2.61/1.48 thf(subsetemptysetimpeq,definition,
% 2.61/1.48 ( subsetemptysetimpeq
% 2.61/1.48 = ( ! [A: $i] :
% 2.61/1.48 ( ( subset @ A @ emptyset )
% 2.61/1.48 => ( A = emptyset ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(powersetI1_type,type,
% 2.61/1.48 powersetI1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(powersetI1,definition,
% 2.61/1.48 ( powersetI1
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( subset @ B @ A )
% 2.61/1.48 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(powersetE1_type,type,
% 2.61/1.48 powersetE1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(powersetE1,definition,
% 2.61/1.48 ( powersetE1
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( in @ B @ ( powerset @ A ) )
% 2.61/1.48 => ( subset @ B @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(inPowerset_type,type,
% 2.61/1.48 inPowerset: $o ).
% 2.61/1.48
% 2.61/1.48 thf(inPowerset,definition,
% 2.61/1.48 ( inPowerset
% 2.61/1.48 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(powersetsubset_type,type,
% 2.61/1.48 powersetsubset: $o ).
% 2.61/1.48
% 2.61/1.48 thf(powersetsubset,definition,
% 2.61/1.48 ( powersetsubset
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( subset @ A @ B )
% 2.61/1.48 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(sepInPowerset_type,type,
% 2.61/1.48 sepInPowerset: $o ).
% 2.61/1.48
% 2.61/1.48 thf(sepInPowerset,definition,
% 2.61/1.48 ( sepInPowerset
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.48 ( in
% 2.61/1.48 @ ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.48 @ ( powerset @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(sepSubset_type,type,
% 2.61/1.48 sepSubset: $o ).
% 2.61/1.48
% 2.61/1.48 thf(sepSubset,definition,
% 2.61/1.48 ( sepSubset
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.48 ( subset
% 2.61/1.48 @ ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.48 @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binunion_type,type,
% 2.61/1.48 binunion: $i > $i > $i ).
% 2.61/1.48
% 2.61/1.48 thf(binunionIL_type,type,
% 2.61/1.48 binunionIL: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binunionIL,definition,
% 2.61/1.48 ( binunionIL
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(upairset2IR_type,type,
% 2.61/1.48 upairset2IR: $o ).
% 2.61/1.48
% 2.61/1.48 thf(upairset2IR,definition,
% 2.61/1.48 ( upairset2IR
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binunionIR_type,type,
% 2.61/1.48 binunionIR: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binunionIR,definition,
% 2.61/1.48 ( binunionIR
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ B )
% 2.61/1.48 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binunionEcases_type,type,
% 2.61/1.48 binunionEcases: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binunionEcases,definition,
% 2.61/1.48 ( binunionEcases
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.61/1.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.61/1.48 => ( ( ( in @ Xx @ A )
% 2.61/1.48 => Xphi )
% 2.61/1.48 => ( ( ( in @ Xx @ B )
% 2.61/1.48 => Xphi )
% 2.61/1.48 => Xphi ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binunionE_type,type,
% 2.61/1.48 binunionE: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binunionE,definition,
% 2.61/1.48 ( binunionE
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.61/1.48 => ( ( in @ Xx @ A )
% 2.61/1.48 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binunionLsub_type,type,
% 2.61/1.48 binunionLsub: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binunionLsub,definition,
% 2.61/1.48 ( binunionLsub
% 2.61/1.48 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binunionRsub_type,type,
% 2.61/1.48 binunionRsub: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binunionRsub,definition,
% 2.61/1.48 ( binunionRsub
% 2.61/1.48 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binintersect_type,type,
% 2.61/1.48 binintersect: $i > $i > $i ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectI_type,type,
% 2.61/1.48 binintersectI: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectI,definition,
% 2.61/1.48 ( binintersectI
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( in @ Xx @ B )
% 2.61/1.48 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectSubset5_type,type,
% 2.61/1.48 binintersectSubset5: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectSubset5,definition,
% 2.61/1.48 ( binintersectSubset5
% 2.61/1.48 = ( ! [A: $i,B: $i,C: $i] :
% 2.61/1.48 ( ( subset @ C @ A )
% 2.61/1.48 => ( ( subset @ C @ B )
% 2.61/1.48 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectEL_type,type,
% 2.61/1.48 binintersectEL: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectEL,definition,
% 2.61/1.48 ( binintersectEL
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.61/1.48 => ( in @ Xx @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectLsub_type,type,
% 2.61/1.48 binintersectLsub: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectLsub,definition,
% 2.61/1.48 ( binintersectLsub
% 2.61/1.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectSubset2_type,type,
% 2.61/1.48 binintersectSubset2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectSubset2,definition,
% 2.61/1.48 ( binintersectSubset2
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( subset @ A @ B )
% 2.61/1.48 => ( ( binintersect @ A @ B )
% 2.61/1.48 = A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectSubset3_type,type,
% 2.61/1.48 binintersectSubset3: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectSubset3,definition,
% 2.61/1.48 ( binintersectSubset3
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( ( binintersect @ A @ B )
% 2.61/1.48 = B )
% 2.61/1.48 => ( subset @ B @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectER_type,type,
% 2.61/1.48 binintersectER: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectER,definition,
% 2.61/1.48 ( binintersectER
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.61/1.48 => ( in @ Xx @ B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(disjointsetsI1_type,type,
% 2.61/1.48 disjointsetsI1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(disjointsetsI1,definition,
% 2.61/1.48 ( disjointsetsI1
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ~ ? [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 & ( in @ Xx @ B ) )
% 2.61/1.48 => ( ( binintersect @ A @ B )
% 2.61/1.48 = emptyset ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectRsub_type,type,
% 2.61/1.48 binintersectRsub: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectRsub,definition,
% 2.61/1.48 ( binintersectRsub
% 2.61/1.48 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectSubset4_type,type,
% 2.61/1.48 binintersectSubset4: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectSubset4,definition,
% 2.61/1.48 ( binintersectSubset4
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( subset @ B @ A )
% 2.61/1.48 => ( ( binintersect @ A @ B )
% 2.61/1.48 = B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectSubset1_type,type,
% 2.61/1.48 binintersectSubset1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectSubset1,definition,
% 2.61/1.48 ( binintersectSubset1
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( ( binintersect @ A @ B )
% 2.61/1.48 = A )
% 2.61/1.48 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(bs114d_type,type,
% 2.61/1.48 bs114d: $o ).
% 2.61/1.48
% 2.61/1.48 thf(bs114d,definition,
% 2.61/1.48 ( bs114d
% 2.61/1.48 = ( ! [A: $i,B: $i,C: $i] :
% 2.61/1.48 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.61/1.48 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(regular_type,type,
% 2.61/1.48 regular: $i > $o ).
% 2.61/1.48
% 2.61/1.48 thf(setminus_type,type,
% 2.61/1.48 setminus: $i > $i > $i ).
% 2.61/1.48
% 2.61/1.48 thf(setminusI_type,type,
% 2.61/1.48 setminusI: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setminusI,definition,
% 2.61/1.48 ( setminusI
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ~ ( in @ Xx @ B )
% 2.61/1.48 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setminusEL_type,type,
% 2.61/1.48 setminusEL: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setminusEL,definition,
% 2.61/1.48 ( setminusEL
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.61/1.48 => ( in @ Xx @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setminusER_type,type,
% 2.61/1.48 setminusER: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setminusER,definition,
% 2.61/1.48 ( setminusER
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.61/1.48 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setminusSubset2_type,type,
% 2.61/1.48 setminusSubset2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setminusSubset2,definition,
% 2.61/1.48 ( setminusSubset2
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( subset @ A @ B )
% 2.61/1.48 => ( ( setminus @ A @ B )
% 2.61/1.48 = emptyset ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setminusERneg_type,type,
% 2.61/1.48 setminusERneg: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setminusERneg,definition,
% 2.61/1.48 ( setminusERneg
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.61/1.48 => ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setminusELneg_type,type,
% 2.61/1.48 setminusELneg: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setminusELneg,definition,
% 2.61/1.48 ( setminusELneg
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.61/1.48 => ( ~ ( in @ Xx @ B )
% 2.61/1.48 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setminusILneg_type,type,
% 2.61/1.48 setminusILneg: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setminusILneg,definition,
% 2.61/1.48 ( setminusILneg
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ~ ( in @ Xx @ A )
% 2.61/1.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setminusIRneg_type,type,
% 2.61/1.48 setminusIRneg: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setminusIRneg,definition,
% 2.61/1.48 ( setminusIRneg
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ B )
% 2.61/1.48 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setminusLsub_type,type,
% 2.61/1.48 setminusLsub: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setminusLsub,definition,
% 2.61/1.48 ( setminusLsub
% 2.61/1.48 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setminusSubset1_type,type,
% 2.61/1.48 setminusSubset1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setminusSubset1,definition,
% 2.61/1.48 ( setminusSubset1
% 2.61/1.48 = ( ! [A: $i,B: $i] :
% 2.61/1.48 ( ( ( setminus @ A @ B )
% 2.61/1.48 = emptyset )
% 2.61/1.48 => ( subset @ A @ B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(symdiff_type,type,
% 2.61/1.48 symdiff: $i > $i > $i ).
% 2.61/1.48
% 2.61/1.48 thf(symdiffE_type,type,
% 2.61/1.48 symdiffE: $o ).
% 2.61/1.48
% 2.61/1.48 thf(symdiffE,definition,
% 2.61/1.48 ( symdiffE
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.61/1.48 => ! [Xphi: $o] :
% 2.61/1.48 ( ( ( in @ Xx @ A )
% 2.61/1.48 => ( ~ ( in @ Xx @ B )
% 2.61/1.48 => Xphi ) )
% 2.61/1.48 => ( ( ~ ( in @ Xx @ A )
% 2.61/1.48 => ( ( in @ Xx @ B )
% 2.61/1.48 => Xphi ) )
% 2.61/1.48 => Xphi ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(symdiffI1_type,type,
% 2.61/1.48 symdiffI1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(symdiffI1,definition,
% 2.61/1.48 ( symdiffI1
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ~ ( in @ Xx @ B )
% 2.61/1.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(symdiffI2_type,type,
% 2.61/1.48 symdiffI2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(symdiffI2,definition,
% 2.61/1.48 ( symdiffI2
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ~ ( in @ Xx @ A )
% 2.61/1.48 => ( ( in @ Xx @ B )
% 2.61/1.48 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(symdiffIneg1_type,type,
% 2.61/1.48 symdiffIneg1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(symdiffIneg1,definition,
% 2.61/1.48 ( symdiffIneg1
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( in @ Xx @ B )
% 2.61/1.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(symdiffIneg2_type,type,
% 2.61/1.48 symdiffIneg2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(symdiffIneg2,definition,
% 2.61/1.48 ( symdiffIneg2
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ~ ( in @ Xx @ A )
% 2.61/1.48 => ( ~ ( in @ Xx @ B )
% 2.61/1.48 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(iskpair_type,type,
% 2.61/1.48 iskpair: $i > $o ).
% 2.61/1.48
% 2.61/1.48 thf(secondinupair_type,type,
% 2.61/1.48 secondinupair: $o ).
% 2.61/1.48
% 2.61/1.48 thf(secondinupair,definition,
% 2.61/1.48 ( secondinupair
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairIL_type,type,
% 2.61/1.48 setukpairIL: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairIL,definition,
% 2.61/1.48 ( setukpairIL
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairIR_type,type,
% 2.61/1.48 setukpairIR: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairIR,definition,
% 2.61/1.48 ( setukpairIR
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(kpairiskpair_type,type,
% 2.61/1.48 kpairiskpair: $o ).
% 2.61/1.48
% 2.61/1.48 thf(kpairiskpair,definition,
% 2.61/1.48 ( kpairiskpair
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(kpair_type,type,
% 2.61/1.48 kpair: $i > $i > $i ).
% 2.61/1.48
% 2.61/1.48 thf(kpairp_type,type,
% 2.61/1.48 kpairp: $o ).
% 2.61/1.48
% 2.61/1.48 thf(kpairp,definition,
% 2.61/1.48 ( kpairp
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(cartprod_type,type,
% 2.61/1.48 cartprod: $i > $i > $i ).
% 2.61/1.48
% 2.61/1.48 thf(singletonsubset_type,type,
% 2.61/1.48 singletonsubset: $o ).
% 2.61/1.48
% 2.61/1.48 thf(singletonsubset,definition,
% 2.61/1.48 ( singletonsubset
% 2.61/1.48 = ( ! [A: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(singletoninpowerset_type,type,
% 2.61/1.48 singletoninpowerset: $o ).
% 2.61/1.48
% 2.61/1.48 thf(singletoninpowerset,definition,
% 2.61/1.48 ( singletoninpowerset
% 2.61/1.48 = ( ! [A: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(singletoninpowunion_type,type,
% 2.61/1.48 singletoninpowunion: $o ).
% 2.61/1.48
% 2.61/1.48 thf(singletoninpowunion,definition,
% 2.61/1.48 ( singletoninpowunion
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(upairset2E_type,type,
% 2.61/1.48 upairset2E: $o ).
% 2.61/1.48
% 2.61/1.48 thf(upairset2E,definition,
% 2.61/1.48 ( upairset2E
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.61/1.48 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.61/1.48 => ( ( Xz = Xx )
% 2.61/1.48 | ( Xz = Xy ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(upairsubunion_type,type,
% 2.61/1.48 upairsubunion: $o ).
% 2.61/1.48
% 2.61/1.48 thf(upairsubunion,definition,
% 2.61/1.48 ( upairsubunion
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(upairinpowunion_type,type,
% 2.61/1.48 upairinpowunion: $o ).
% 2.61/1.48
% 2.61/1.48 thf(upairinpowunion,definition,
% 2.61/1.48 ( upairinpowunion
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(ubforcartprodlem1_type,type,
% 2.61/1.48 ubforcartprodlem1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(ubforcartprodlem1,definition,
% 2.61/1.48 ( ubforcartprodlem1
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(ubforcartprodlem2_type,type,
% 2.61/1.48 ubforcartprodlem2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(ubforcartprodlem2,definition,
% 2.61/1.48 ( ubforcartprodlem2
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(ubforcartprodlem3_type,type,
% 2.61/1.48 ubforcartprodlem3: $o ).
% 2.61/1.48
% 2.61/1.48 thf(ubforcartprodlem3,definition,
% 2.61/1.48 ( ubforcartprodlem3
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodpairin_type,type,
% 2.61/1.48 cartprodpairin: $o ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodpairin,definition,
% 2.61/1.48 ( cartprodpairin
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodmempair1_type,type,
% 2.61/1.48 cartprodmempair1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodmempair1,definition,
% 2.61/1.48 ( cartprodmempair1
% 2.61/1.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.61/1.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.61/1.48 => ? [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 & ? [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 & ( Xu
% 2.61/1.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodmempair_type,type,
% 2.61/1.48 cartprodmempair: $o ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodmempair,definition,
% 2.61/1.48 ( cartprodmempair
% 2.61/1.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.61/1.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.61/1.48 => ( iskpair @ Xu ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setunionE2_type,type,
% 2.61/1.48 setunionE2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setunionE2,definition,
% 2.61/1.48 ( setunionE2
% 2.61/1.48 = ( ! [A: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ ( setunion @ A ) )
% 2.61/1.48 => ? [X: $i] :
% 2.61/1.48 ( ( in @ X @ A )
% 2.61/1.48 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setunionsingleton1_type,type,
% 2.61/1.48 setunionsingleton1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setunionsingleton1,definition,
% 2.61/1.48 ( setunionsingleton1
% 2.61/1.48 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setunionsingleton2_type,type,
% 2.61/1.48 setunionsingleton2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setunionsingleton2,definition,
% 2.61/1.48 ( setunionsingleton2
% 2.61/1.48 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setunionsingleton_type,type,
% 2.61/1.48 setunionsingleton: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setunionsingleton,definition,
% 2.61/1.48 ( setunionsingleton
% 2.61/1.48 = ( ! [Xx: $i] :
% 2.61/1.48 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.61/1.48 = Xx ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(singleton_type,type,
% 2.61/1.48 singleton: $i > $o ).
% 2.61/1.48
% 2.61/1.48 thf(singletonprop_type,type,
% 2.61/1.48 singletonprop: $o ).
% 2.61/1.48
% 2.61/1.48 thf(singletonprop,definition,
% 2.61/1.48 ( singletonprop
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ A )
% 2.61/1.48 => ( ( Xphi @ Xx )
% 2.61/1.48 => ( ( Xphi @ Xy )
% 2.61/1.48 => ( Xx = Xy ) ) ) ) )
% 2.61/1.48 => ( ? [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 & ( Xphi @ Xx ) )
% 2.61/1.48 => ( singleton
% 2.61/1.48 @ ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(ex1_type,type,
% 2.61/1.48 ex1: $i > ( $i > $o ) > $o ).
% 2.61/1.48
% 2.61/1.48 thf(ex1,definition,
% 2.61/1.48 ( ex1
% 2.61/1.48 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.61/1.48 ( singleton
% 2.61/1.48 @ ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(ex1E1_type,type,
% 2.61/1.48 ex1E1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(ex1E1,definition,
% 2.61/1.48 ( ex1E1
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.48 ( ( ex1 @ A
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.48 => ? [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(ex1I_type,type,
% 2.61/1.48 ex1I: $o ).
% 2.61/1.48
% 2.61/1.48 thf(ex1I,definition,
% 2.61/1.48 ( ex1I
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( Xphi @ Xx )
% 2.61/1.48 => ( ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ A )
% 2.61/1.48 => ( ( Xphi @ Xy )
% 2.61/1.48 => ( Xy = Xx ) ) )
% 2.61/1.48 => ( ex1 @ A
% 2.61/1.48 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(ex1I2_type,type,
% 2.61/1.48 ex1I2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(ex1I2,definition,
% 2.61/1.48 ( ex1I2
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ A )
% 2.61/1.48 => ( ( Xphi @ Xx )
% 2.61/1.48 => ( ( Xphi @ Xy )
% 2.61/1.48 => ( Xx = Xy ) ) ) ) )
% 2.61/1.48 => ( ? [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 & ( Xphi @ Xx ) )
% 2.61/1.48 => ( ex1 @ A
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(singletonsuniq_type,type,
% 2.61/1.48 singletonsuniq: $o ).
% 2.61/1.48
% 2.61/1.48 thf(singletonsuniq,definition,
% 2.61/1.48 ( singletonsuniq
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( ( setadjoin @ Xx @ emptyset )
% 2.61/1.48 = ( setadjoin @ Xy @ emptyset ) )
% 2.61/1.48 => ( Xx = Xy ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(atmost1p_type,type,
% 2.61/1.48 atmost1p: $i > $o ).
% 2.61/1.48
% 2.61/1.48 thf(atleast2p_type,type,
% 2.61/1.48 atleast2p: $i > $o ).
% 2.61/1.48
% 2.61/1.48 thf(atmost2p_type,type,
% 2.61/1.48 atmost2p: $i > $o ).
% 2.61/1.48
% 2.61/1.48 thf(upairsetp_type,type,
% 2.61/1.48 upairsetp: $i > $o ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairinjL1_type,type,
% 2.61/1.48 setukpairinjL1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairinjL1,definition,
% 2.61/1.48 ( setukpairinjL1
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.61/1.48 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.61/1.48 => ( Xx = Xz ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(kfstsingleton_type,type,
% 2.61/1.48 kfstsingleton: $o ).
% 2.61/1.48
% 2.61/1.48 thf(kfstsingleton,definition,
% 2.61/1.48 ( kfstsingleton
% 2.61/1.48 = ( ! [Xu: $i] :
% 2.61/1.48 ( ( iskpair @ Xu )
% 2.61/1.48 => ( singleton
% 2.61/1.48 @ ( dsetconstr @ ( setunion @ Xu )
% 2.61/1.48 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(theprop_type,type,
% 2.61/1.48 theprop: $o ).
% 2.61/1.48
% 2.61/1.48 thf(theprop,definition,
% 2.61/1.48 ( theprop
% 2.61/1.48 = ( ! [X: $i] :
% 2.61/1.48 ( ( singleton @ X )
% 2.61/1.48 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(kfst_type,type,
% 2.61/1.48 kfst: $i > $i ).
% 2.61/1.48
% 2.61/1.48 thf(kfstpairEq_type,type,
% 2.61/1.48 kfstpairEq: $o ).
% 2.61/1.48
% 2.61/1.48 thf(kfstpairEq,definition,
% 2.61/1.48 ( kfstpairEq
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.61/1.48 = Xx ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodfstin_type,type,
% 2.61/1.48 cartprodfstin: $o ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodfstin,definition,
% 2.61/1.48 ( cartprodfstin
% 2.61/1.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.61/1.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.61/1.48 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairinjL2_type,type,
% 2.61/1.48 setukpairinjL2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairinjL2,definition,
% 2.61/1.48 ( setukpairinjL2
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.61/1.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.61/1.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.61/1.48 => ( Xx = Xz ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairinjL_type,type,
% 2.61/1.48 setukpairinjL: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairinjL,definition,
% 2.61/1.48 ( setukpairinjL
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.61/1.48 ( ( ( kpair @ Xx @ Xy )
% 2.61/1.48 = ( kpair @ Xz @ Xu ) )
% 2.61/1.48 => ( Xx = Xz ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairinjR11_type,type,
% 2.61/1.48 setukpairinjR11: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairinjR11,definition,
% 2.61/1.48 ( setukpairinjR11
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( Xx = Xy )
% 2.61/1.48 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.61/1.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairinjR12_type,type,
% 2.61/1.48 setukpairinjR12: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairinjR12,definition,
% 2.61/1.48 ( setukpairinjR12
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( Xx = Xy )
% 2.61/1.48 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.61/1.48 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairinjR1_type,type,
% 2.61/1.48 setukpairinjR1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairinjR1,definition,
% 2.61/1.48 ( setukpairinjR1
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.61/1.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.61/1.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.61/1.48 => ( ( Xz = Xu )
% 2.61/1.48 => ( Xy = Xu ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(upairequniteq_type,type,
% 2.61/1.48 upairequniteq: $o ).
% 2.61/1.48
% 2.61/1.48 thf(upairequniteq,definition,
% 2.61/1.48 ( upairequniteq
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.61/1.48 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.61/1.48 = ( setadjoin @ Xz @ emptyset ) )
% 2.61/1.48 => ( Xx = Xy ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairinjR2_type,type,
% 2.61/1.48 setukpairinjR2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairinjR2,definition,
% 2.61/1.48 ( setukpairinjR2
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.61/1.48 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.61/1.48 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.61/1.48 => ( Xy = Xu ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairinjR_type,type,
% 2.61/1.48 setukpairinjR: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setukpairinjR,definition,
% 2.61/1.48 ( setukpairinjR
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.61/1.48 ( ( ( kpair @ Xx @ Xy )
% 2.61/1.48 = ( kpair @ Xz @ Xu ) )
% 2.61/1.48 => ( Xy = Xu ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(ksndsingleton_type,type,
% 2.61/1.48 ksndsingleton: $o ).
% 2.61/1.48
% 2.61/1.48 thf(ksndsingleton,definition,
% 2.61/1.48 ( ksndsingleton
% 2.61/1.48 = ( ! [Xu: $i] :
% 2.61/1.48 ( ( iskpair @ Xu )
% 2.61/1.48 => ( singleton
% 2.61/1.48 @ ( dsetconstr @ ( setunion @ Xu )
% 2.61/1.48 @ ^ [Xx: $i] :
% 2.61/1.48 ( Xu
% 2.61/1.48 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(ksnd_type,type,
% 2.61/1.48 ksnd: $i > $i ).
% 2.61/1.48
% 2.61/1.48 thf(ksndpairEq_type,type,
% 2.61/1.48 ksndpairEq: $o ).
% 2.61/1.48
% 2.61/1.48 thf(ksndpairEq,definition,
% 2.61/1.48 ( ksndpairEq
% 2.61/1.48 = ( ! [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.61/1.48 = Xy ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(kpairsurjEq_type,type,
% 2.61/1.48 kpairsurjEq: $o ).
% 2.61/1.48
% 2.61/1.48 thf(kpairsurjEq,definition,
% 2.61/1.48 ( kpairsurjEq
% 2.61/1.48 = ( ! [Xu: $i] :
% 2.61/1.48 ( ( iskpair @ Xu )
% 2.61/1.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.61/1.48 = Xu ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodsndin_type,type,
% 2.61/1.48 cartprodsndin: $o ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodsndin,definition,
% 2.61/1.48 ( cartprodsndin
% 2.61/1.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.61/1.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.61/1.48 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodpairmemEL_type,type,
% 2.61/1.48 cartprodpairmemEL: $o ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodpairmemEL,definition,
% 2.61/1.48 ( cartprodpairmemEL
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.61/1.48 => ( in @ Xx @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodpairmemER_type,type,
% 2.61/1.48 cartprodpairmemER: $o ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodpairmemER,definition,
% 2.61/1.48 ( cartprodpairmemER
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.61/1.48 => ( in @ Xy @ B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodmempaircEq_type,type,
% 2.61/1.48 cartprodmempaircEq: $o ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodmempaircEq,definition,
% 2.61/1.48 ( cartprodmempaircEq
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( ( kpair @ Xx @ Xy )
% 2.61/1.48 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodfstpairEq_type,type,
% 2.61/1.48 cartprodfstpairEq: $o ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodfstpairEq,definition,
% 2.61/1.48 ( cartprodfstpairEq
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.61/1.48 = Xx ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodsndpairEq_type,type,
% 2.61/1.48 cartprodsndpairEq: $o ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodsndpairEq,definition,
% 2.61/1.48 ( cartprodsndpairEq
% 2.61/1.48 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.61/1.48 = Xy ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodpairsurjEq_type,type,
% 2.61/1.48 cartprodpairsurjEq: $o ).
% 2.61/1.48
% 2.61/1.48 thf(cartprodpairsurjEq,definition,
% 2.61/1.48 ( cartprodpairsurjEq
% 2.61/1.48 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.61/1.48 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.61/1.48 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.61/1.48 = Xu ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(breln_type,type,
% 2.61/1.48 breln: $i > $i > $i > $o ).
% 2.61/1.48
% 2.61/1.48 thf(breln,definition,
% 2.61/1.48 ( breln
% 2.61/1.48 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(dpsetconstr_type,type,
% 2.61/1.48 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.61/1.48
% 2.61/1.48 thf(dpsetconstrI_type,type,
% 2.61/1.48 dpsetconstrI: $o ).
% 2.61/1.48
% 2.61/1.48 thf(dpsetconstrI,definition,
% 2.61/1.48 ( dpsetconstrI
% 2.61/1.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( ( Xphi @ Xx @ Xy )
% 2.61/1.48 => ( in @ ( kpair @ Xx @ Xy )
% 2.61/1.48 @ ( dpsetconstr @ A @ B
% 2.61/1.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(dpsetconstrSub_type,type,
% 2.61/1.48 dpsetconstrSub: $o ).
% 2.61/1.48
% 2.61/1.48 thf(dpsetconstrSub,definition,
% 2.61/1.48 ( dpsetconstrSub
% 2.61/1.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.61/1.48 ( subset
% 2.61/1.48 @ ( dpsetconstr @ A @ B
% 2.61/1.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.61/1.48 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setOfPairsIsBReln_type,type,
% 2.61/1.48 setOfPairsIsBReln: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setOfPairsIsBReln,definition,
% 2.61/1.48 ( setOfPairsIsBReln
% 2.61/1.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.61/1.48 ( breln @ A @ B
% 2.61/1.48 @ ( dpsetconstr @ A @ B
% 2.61/1.48 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(dpsetconstrERa_type,type,
% 2.61/1.48 dpsetconstrERa: $o ).
% 2.61/1.48
% 2.61/1.48 thf(dpsetconstrERa,definition,
% 2.61/1.48 ( dpsetconstrERa
% 2.61/1.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.61/1.48 @ ( dpsetconstr @ A @ B
% 2.61/1.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.61/1.48 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(dpsetconstrEL1_type,type,
% 2.61/1.48 dpsetconstrEL1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(dpsetconstrEL1,definition,
% 2.61/1.48 ( dpsetconstrEL1
% 2.61/1.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( in @ ( kpair @ Xx @ Xy )
% 2.61/1.48 @ ( dpsetconstr @ A @ B
% 2.61/1.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.61/1.48 => ( in @ Xx @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(dpsetconstrEL2_type,type,
% 2.61/1.48 dpsetconstrEL2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(dpsetconstrEL2,definition,
% 2.61/1.48 ( dpsetconstrEL2
% 2.61/1.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( in @ ( kpair @ Xx @ Xy )
% 2.61/1.48 @ ( dpsetconstr @ A @ B
% 2.61/1.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.61/1.48 => ( in @ Xy @ B ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(dpsetconstrER_type,type,
% 2.61/1.48 dpsetconstrER: $o ).
% 2.61/1.48
% 2.61/1.48 thf(dpsetconstrER,definition,
% 2.61/1.48 ( dpsetconstrER
% 2.61/1.48 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( in @ ( kpair @ Xx @ Xy )
% 2.61/1.48 @ ( dpsetconstr @ A @ B
% 2.61/1.48 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.61/1.48 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(func_type,type,
% 2.61/1.48 func: $i > $i > $i > $o ).
% 2.61/1.48
% 2.61/1.48 thf(funcSet_type,type,
% 2.61/1.48 funcSet: $i > $i > $i ).
% 2.61/1.48
% 2.61/1.48 thf(funcImageSingleton_type,type,
% 2.61/1.48 funcImageSingleton: $o ).
% 2.61/1.48
% 2.61/1.48 thf(funcImageSingleton,definition,
% 2.61/1.48 ( funcImageSingleton
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( func @ A @ B @ Xf )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( singleton
% 2.61/1.48 @ ( dsetconstr @ B
% 2.61/1.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(apProp_type,type,
% 2.61/1.48 apProp: $o ).
% 2.61/1.48
% 2.61/1.48 thf(apProp,definition,
% 2.61/1.48 ( apProp
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( func @ A @ B @ Xf )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in
% 2.61/1.48 @ ( setunion
% 2.61/1.48 @ ( dsetconstr @ B
% 2.61/1.48 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.61/1.48 @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(ap_type,type,
% 2.61/1.48 ap: $i > $i > $i > $i > $i ).
% 2.61/1.48
% 2.61/1.48 thf(app_type,type,
% 2.61/1.48 app: $o ).
% 2.61/1.48
% 2.61/1.48 thf(app,definition,
% 2.61/1.48 ( app
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( func @ A @ B @ Xf )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(infuncsetfunc_type,type,
% 2.61/1.48 infuncsetfunc: $o ).
% 2.61/1.48
% 2.61/1.48 thf(infuncsetfunc,definition,
% 2.61/1.48 ( infuncsetfunc
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.61/1.48 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(ap2p_type,type,
% 2.61/1.48 ap2p: $o ).
% 2.61/1.48
% 2.61/1.48 thf(ap2p,definition,
% 2.61/1.48 ( ap2p
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(funcinfuncset_type,type,
% 2.61/1.48 funcinfuncset: $o ).
% 2.61/1.48
% 2.61/1.48 thf(funcinfuncset,definition,
% 2.61/1.48 ( funcinfuncset
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( func @ A @ B @ Xf )
% 2.61/1.48 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(lamProp_type,type,
% 2.61/1.48 lamProp: $o ).
% 2.61/1.48
% 2.61/1.48 thf(lamProp,definition,
% 2.61/1.48 ( lamProp
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.61/1.48 => ( func @ A @ B
% 2.61/1.48 @ ( dpsetconstr @ A @ B
% 2.61/1.48 @ ^ [Xx: $i,Xy: $i] :
% 2.61/1.48 ( ( Xf @ Xx )
% 2.61/1.48 = Xy ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(lam_type,type,
% 2.61/1.48 lam: $i > $i > ( $i > $i ) > $i ).
% 2.61/1.48
% 2.61/1.48 thf(lamp_type,type,
% 2.61/1.48 lamp: $o ).
% 2.61/1.48
% 2.61/1.48 thf(lamp,definition,
% 2.61/1.48 ( lamp
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.61/1.48 => ( func @ A @ B
% 2.61/1.48 @ ( lam @ A @ B
% 2.61/1.48 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(lam2p_type,type,
% 2.61/1.48 lam2p: $o ).
% 2.61/1.48
% 2.61/1.48 thf(lam2p,definition,
% 2.61/1.48 ( lam2p
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.61/1.48 => ( in
% 2.61/1.48 @ ( lam @ A @ B
% 2.61/1.48 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.61/1.48 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(brelnall1_type,type,
% 2.61/1.48 brelnall1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(brelnall1,definition,
% 2.61/1.48 ( brelnall1
% 2.61/1.48 = ( ! [A: $i,B: $i,R: $i] :
% 2.61/1.48 ( ( breln @ A @ B @ R )
% 2.61/1.48 => ! [Xphi: $i > $o] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.61/1.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ R )
% 2.61/1.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(brelnall2_type,type,
% 2.61/1.48 brelnall2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(brelnall2,definition,
% 2.61/1.48 ( brelnall2
% 2.61/1.48 = ( ! [A: $i,B: $i,R: $i] :
% 2.61/1.48 ( ( breln @ A @ B @ R )
% 2.61/1.48 => ! [Xphi: $i > $o] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.61/1.48 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ R )
% 2.61/1.48 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(ex1E2_type,type,
% 2.61/1.48 ex1E2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(ex1E2,definition,
% 2.61/1.48 ( ex1E2
% 2.61/1.48 = ( ! [A: $i,Xphi: $i > $o] :
% 2.61/1.48 ( ( ex1 @ A
% 2.61/1.48 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ A )
% 2.61/1.48 => ( ( Xphi @ Xx )
% 2.61/1.48 => ( ( Xphi @ Xy )
% 2.61/1.48 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(funcGraphProp1_type,type,
% 2.61/1.48 funcGraphProp1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(funcGraphProp1,definition,
% 2.61/1.48 ( funcGraphProp1
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( func @ A @ B @ Xf )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(funcGraphProp3_type,type,
% 2.61/1.48 funcGraphProp3: $o ).
% 2.61/1.48
% 2.61/1.48 thf(funcGraphProp3,definition,
% 2.61/1.48 ( funcGraphProp3
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(funcGraphProp2_type,type,
% 2.61/1.48 funcGraphProp2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(funcGraphProp2,definition,
% 2.61/1.48 ( funcGraphProp2
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( func @ A @ B @ Xf )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.61/1.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.61/1.48 = Xy ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(funcextLem_type,type,
% 2.61/1.48 funcextLem: $o ).
% 2.61/1.48
% 2.61/1.48 thf(funcextLem,definition,
% 2.61/1.48 ( funcextLem
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( func @ A @ B @ Xf )
% 2.61/1.48 => ! [Xg: $i] :
% 2.61/1.48 ( ( func @ A @ B @ Xg )
% 2.61/1.48 => ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.61/1.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.61/1.48 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(funcGraphProp4_type,type,
% 2.61/1.48 funcGraphProp4: $o ).
% 2.61/1.48
% 2.61/1.48 thf(funcGraphProp4,definition,
% 2.61/1.48 ( funcGraphProp4
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.61/1.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.61/1.48 = Xy ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(subbreln_type,type,
% 2.61/1.48 subbreln: $o ).
% 2.61/1.48
% 2.61/1.48 thf(subbreln,definition,
% 2.61/1.48 ( subbreln
% 2.61/1.48 = ( ! [A: $i,B: $i,R: $i] :
% 2.61/1.48 ( ( breln @ A @ B @ R )
% 2.61/1.48 => ! [S: $i] :
% 2.61/1.48 ( ( breln @ A @ B @ S )
% 2.61/1.48 => ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.61/1.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.61/1.48 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(eqbreln_type,type,
% 2.61/1.48 eqbreln: $o ).
% 2.61/1.48
% 2.61/1.48 thf(eqbreln,definition,
% 2.61/1.48 ( eqbreln
% 2.61/1.48 = ( ! [A: $i,B: $i,R: $i] :
% 2.61/1.48 ( ( breln @ A @ B @ R )
% 2.61/1.48 => ! [S: $i] :
% 2.61/1.48 ( ( breln @ A @ B @ S )
% 2.61/1.48 => ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.61/1.48 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.61/1.48 => ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ B )
% 2.61/1.48 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.61/1.48 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.61/1.48 => ( R = S ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(funcext_type,type,
% 2.61/1.48 funcext: $o ).
% 2.61/1.48
% 2.61/1.48 thf(funcext,definition,
% 2.61/1.48 ( funcext
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( func @ A @ B @ Xf )
% 2.61/1.48 => ! [Xg: $i] :
% 2.61/1.48 ( ( func @ A @ B @ Xg )
% 2.61/1.48 => ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.61/1.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.61/1.48 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(funcext2_type,type,
% 2.61/1.48 funcext2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(funcext2,definition,
% 2.61/1.48 ( funcext2
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.61/1.48 => ! [Xg: $i] :
% 2.61/1.48 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.61/1.48 => ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.61/1.48 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.61/1.48 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(ap2apEq1_type,type,
% 2.61/1.48 ap2apEq1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(ap2apEq1,definition,
% 2.61/1.48 ( ap2apEq1
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.61/1.48 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(ap2apEq2_type,type,
% 2.61/1.48 ap2apEq2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(ap2apEq2,definition,
% 2.61/1.48 ( ap2apEq2
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( func @ A @ B @ Xf )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.61/1.48 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(beta1_type,type,
% 2.61/1.48 beta1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(beta1,definition,
% 2.61/1.48 ( beta1
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( ap @ A @ B
% 2.61/1.48 @ ( lam @ A @ B
% 2.61/1.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.61/1.48 @ Xx )
% 2.61/1.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(eta1_type,type,
% 2.61/1.48 eta1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(eta1,definition,
% 2.61/1.48 ( eta1
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( func @ A @ B @ Xf )
% 2.61/1.48 => ( ( lam @ A @ B
% 2.61/1.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.61/1.48 = Xf ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(lam2lamEq_type,type,
% 2.61/1.48 lam2lamEq: $o ).
% 2.61/1.48
% 2.61/1.48 thf(lam2lamEq,definition,
% 2.61/1.48 ( lam2lamEq
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.61/1.48 => ( ( lam @ A @ B
% 2.61/1.48 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.61/1.48 = ( lam @ A @ B
% 2.61/1.48 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(beta2_type,type,
% 2.61/1.48 beta2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(beta2,definition,
% 2.61/1.48 ( beta2
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.61/1.48 ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( ap @ A @ B
% 2.61/1.48 @ ( lam @ A @ B
% 2.61/1.48 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.61/1.48 @ Xx )
% 2.61/1.48 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(eta2_type,type,
% 2.61/1.48 eta2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(eta2,definition,
% 2.61/1.48 ( eta2
% 2.61/1.48 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.61/1.48 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.61/1.48 => ( ( lam @ A @ B
% 2.61/1.48 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.61/1.48 = Xf ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(iffalseProp1_type,type,
% 2.61/1.48 iffalseProp1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(iffalseProp1,definition,
% 2.61/1.48 ( iffalseProp1
% 2.61/1.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ A )
% 2.61/1.48 => ( ~ Xphi
% 2.61/1.48 => ( in @ Xy
% 2.61/1.48 @ ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xz: $i] :
% 2.61/1.48 ( ( Xphi
% 2.61/1.48 & ( Xz = Xx ) )
% 2.61/1.48 | ( ~ Xphi
% 2.61/1.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(iffalseProp2_type,type,
% 2.61/1.48 iffalseProp2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(iffalseProp2,definition,
% 2.61/1.48 ( iffalseProp2
% 2.61/1.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ A )
% 2.61/1.48 => ( ~ Xphi
% 2.61/1.48 => ( ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xz: $i] :
% 2.61/1.48 ( ( Xphi
% 2.61/1.48 & ( Xz = Xx ) )
% 2.61/1.48 | ( ~ Xphi
% 2.61/1.48 & ( Xz = Xy ) ) ) )
% 2.61/1.48 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(iftrueProp1_type,type,
% 2.61/1.48 iftrueProp1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(iftrueProp1,definition,
% 2.61/1.48 ( iftrueProp1
% 2.61/1.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ A )
% 2.61/1.48 => ( Xphi
% 2.61/1.48 => ( in @ Xx
% 2.61/1.48 @ ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xz: $i] :
% 2.61/1.48 ( ( Xphi
% 2.61/1.48 & ( Xz = Xx ) )
% 2.61/1.48 | ( ~ Xphi
% 2.61/1.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(iftrueProp2_type,type,
% 2.61/1.48 iftrueProp2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(iftrueProp2,definition,
% 2.61/1.48 ( iftrueProp2
% 2.61/1.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ A )
% 2.61/1.48 => ( Xphi
% 2.61/1.48 => ( ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xz: $i] :
% 2.61/1.48 ( ( Xphi
% 2.61/1.48 & ( Xz = Xx ) )
% 2.61/1.48 | ( ~ Xphi
% 2.61/1.48 & ( Xz = Xy ) ) ) )
% 2.61/1.48 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(ifSingleton_type,type,
% 2.61/1.48 ifSingleton: $o ).
% 2.61/1.48
% 2.61/1.48 thf(ifSingleton,definition,
% 2.61/1.48 ( ifSingleton
% 2.61/1.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ A )
% 2.61/1.48 => ( singleton
% 2.61/1.48 @ ( dsetconstr @ A
% 2.61/1.48 @ ^ [Xz: $i] :
% 2.61/1.48 ( ( Xphi
% 2.61/1.48 & ( Xz = Xx ) )
% 2.61/1.48 | ( ~ Xphi
% 2.61/1.48 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(if_type,type,
% 2.61/1.48 if: $i > $o > $i > $i > $i ).
% 2.61/1.48
% 2.61/1.48 thf(ifp_type,type,
% 2.61/1.48 ifp: $o ).
% 2.61/1.48
% 2.61/1.48 thf(ifp,definition,
% 2.61/1.48 ( ifp
% 2.61/1.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ A )
% 2.61/1.48 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(theeq_type,type,
% 2.61/1.48 theeq: $o ).
% 2.61/1.48
% 2.61/1.48 thf(theeq,definition,
% 2.61/1.48 ( theeq
% 2.61/1.48 = ( ! [X: $i] :
% 2.61/1.48 ( ( singleton @ X )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ X )
% 2.61/1.48 => ( ( setunion @ X )
% 2.61/1.48 = Xx ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(iftrue_type,type,
% 2.61/1.48 iftrue: $o ).
% 2.61/1.48
% 2.61/1.48 thf(iftrue,definition,
% 2.61/1.48 ( iftrue
% 2.61/1.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ A )
% 2.61/1.48 => ( Xphi
% 2.61/1.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.61/1.48 = Xx ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(iffalse_type,type,
% 2.61/1.48 iffalse: $o ).
% 2.61/1.48
% 2.61/1.48 thf(iffalse,definition,
% 2.61/1.48 ( iffalse
% 2.61/1.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ A )
% 2.61/1.48 => ( ~ Xphi
% 2.61/1.48 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.61/1.48 = Xy ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(iftrueorfalse_type,type,
% 2.61/1.48 iftrueorfalse: $o ).
% 2.61/1.48
% 2.61/1.48 thf(iftrueorfalse,definition,
% 2.61/1.48 ( iftrueorfalse
% 2.61/1.48 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ! [Xy: $i] :
% 2.61/1.48 ( ( in @ Xy @ A )
% 2.61/1.48 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectT_lem_type,type,
% 2.61/1.48 binintersectT_lem: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectT_lem,definition,
% 2.61/1.48 ( binintersectT_lem
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Y: $i] :
% 2.61/1.48 ( ( in @ Y @ ( powerset @ A ) )
% 2.61/1.48 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binunionT_lem_type,type,
% 2.61/1.48 binunionT_lem: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binunionT_lem,definition,
% 2.61/1.48 ( binunionT_lem
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Y: $i] :
% 2.61/1.48 ( ( in @ Y @ ( powerset @ A ) )
% 2.61/1.48 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(powersetT_lem_type,type,
% 2.61/1.48 powersetT_lem: $o ).
% 2.61/1.48
% 2.61/1.48 thf(powersetT_lem,definition,
% 2.61/1.48 ( powersetT_lem
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setminusT_lem_type,type,
% 2.61/1.48 setminusT_lem: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setminusT_lem,definition,
% 2.61/1.48 ( setminusT_lem
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Y: $i] :
% 2.61/1.48 ( ( in @ Y @ ( powerset @ A ) )
% 2.61/1.48 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(complementT_lem_type,type,
% 2.61/1.48 complementT_lem: $o ).
% 2.61/1.48
% 2.61/1.48 thf(complementT_lem,definition,
% 2.61/1.48 ( complementT_lem
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(setextT_type,type,
% 2.61/1.48 setextT: $o ).
% 2.61/1.48
% 2.61/1.48 thf(setextT,definition,
% 2.61/1.48 ( setextT
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Y: $i] :
% 2.61/1.48 ( ( in @ Y @ ( powerset @ A ) )
% 2.61/1.48 => ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( in @ Xx @ X )
% 2.61/1.48 => ( in @ Xx @ Y ) ) )
% 2.61/1.48 => ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( in @ Xx @ Y )
% 2.61/1.48 => ( in @ Xx @ X ) ) )
% 2.61/1.48 => ( X = Y ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(subsetTI_type,type,
% 2.61/1.48 subsetTI: $o ).
% 2.61/1.48
% 2.61/1.48 thf(subsetTI,definition,
% 2.61/1.48 ( subsetTI
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Y: $i] :
% 2.61/1.48 ( ( in @ Y @ ( powerset @ A ) )
% 2.61/1.48 => ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( in @ Xx @ X )
% 2.61/1.48 => ( in @ Xx @ Y ) ) )
% 2.61/1.48 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(powersetTI1_type,type,
% 2.61/1.48 powersetTI1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(powersetTI1,definition,
% 2.61/1.48 ( powersetTI1
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Y: $i] :
% 2.61/1.48 ( ( in @ Y @ ( powerset @ A ) )
% 2.61/1.48 => ( ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( in @ Xx @ X )
% 2.61/1.48 => ( in @ Xx @ Y ) ) )
% 2.61/1.48 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(powersetTE1_type,type,
% 2.61/1.48 powersetTE1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(powersetTE1,definition,
% 2.61/1.48 ( powersetTE1
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Y: $i] :
% 2.61/1.48 ( ( in @ Y @ ( powerset @ A ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( in @ X @ ( powerset @ Y ) )
% 2.61/1.48 => ( ( in @ Xx @ X )
% 2.61/1.48 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(complementTI1_type,type,
% 2.61/1.48 complementTI1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(complementTI1,definition,
% 2.61/1.48 ( complementTI1
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( in @ Xx @ X )
% 2.61/1.48 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(complementTE1_type,type,
% 2.61/1.48 complementTE1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(complementTE1,definition,
% 2.61/1.48 ( complementTE1
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.61/1.48 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectTELcontra_type,type,
% 2.61/1.48 binintersectTELcontra: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectTELcontra,definition,
% 2.61/1.48 ( binintersectTELcontra
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Y: $i] :
% 2.61/1.48 ( ( in @ Y @ ( powerset @ A ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ~ ( in @ Xx @ X )
% 2.61/1.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectTERcontra_type,type,
% 2.61/1.48 binintersectTERcontra: $o ).
% 2.61/1.48
% 2.61/1.48 thf(binintersectTERcontra,definition,
% 2.61/1.48 ( binintersectTERcontra
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Y: $i] :
% 2.61/1.48 ( ( in @ Y @ ( powerset @ A ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ~ ( in @ Xx @ Y )
% 2.61/1.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(contrasubsetT_type,type,
% 2.61/1.48 contrasubsetT: $o ).
% 2.61/1.48
% 2.61/1.48 thf(contrasubsetT,definition,
% 2.61/1.48 ( contrasubsetT
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Y: $i] :
% 2.61/1.48 ( ( in @ Y @ ( powerset @ A ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 2.61/1.48 => ( ( in @ Xx @ Y )
% 2.61/1.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(contrasubsetT1_type,type,
% 2.61/1.48 contrasubsetT1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(contrasubsetT1,definition,
% 2.61/1.48 ( contrasubsetT1
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Y: $i] :
% 2.61/1.48 ( ( in @ Y @ ( powerset @ A ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( subset @ X @ Y )
% 2.61/1.48 => ( ~ ( in @ Xx @ Y )
% 2.61/1.48 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(contrasubsetT2_type,type,
% 2.61/1.48 contrasubsetT2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(contrasubsetT2,definition,
% 2.61/1.48 ( contrasubsetT2
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Y: $i] :
% 2.61/1.48 ( ( in @ Y @ ( powerset @ A ) )
% 2.61/1.48 => ( ( subset @ X @ Y )
% 2.61/1.48 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(contrasubsetT3_type,type,
% 2.61/1.48 contrasubsetT3: $o ).
% 2.61/1.48
% 2.61/1.48 thf(contrasubsetT3,definition,
% 2.61/1.48 ( contrasubsetT3
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Y: $i] :
% 2.61/1.48 ( ( in @ Y @ ( powerset @ A ) )
% 2.61/1.48 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 2.61/1.48 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(doubleComplementI1_type,type,
% 2.61/1.48 doubleComplementI1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(doubleComplementI1,definition,
% 2.61/1.48 ( doubleComplementI1
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( in @ Xx @ X )
% 2.61/1.48 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(doubleComplementE1_type,type,
% 2.61/1.48 doubleComplementE1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(doubleComplementE1,definition,
% 2.61/1.48 ( doubleComplementE1
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 2.61/1.48 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(doubleComplementSub1_type,type,
% 2.61/1.48 doubleComplementSub1: $o ).
% 2.61/1.48
% 2.61/1.48 thf(doubleComplementSub1,definition,
% 2.61/1.48 ( doubleComplementSub1
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(doubleComplementSub2_type,type,
% 2.61/1.48 doubleComplementSub2: $o ).
% 2.61/1.48
% 2.61/1.48 thf(doubleComplementSub2,definition,
% 2.61/1.48 ( doubleComplementSub2
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(doubleComplementEq_type,type,
% 2.61/1.48 doubleComplementEq: $o ).
% 2.61/1.48
% 2.61/1.48 thf(doubleComplementEq,definition,
% 2.61/1.48 ( doubleComplementEq
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ( X
% 2.61/1.48 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(complementTnotintersectT_type,type,
% 2.61/1.48 complementTnotintersectT: $o ).
% 2.61/1.48
% 2.61/1.48 thf(complementTnotintersectT,definition,
% 2.61/1.48 ( complementTnotintersectT
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Y: $i] :
% 2.61/1.48 ( ( in @ Y @ ( powerset @ A ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.61/1.48 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(complementImpComplementIntersect_type,type,
% 2.61/1.48 complementImpComplementIntersect: $o ).
% 2.61/1.48
% 2.61/1.48 thf(complementImpComplementIntersect,definition,
% 2.61/1.48 ( complementImpComplementIntersect
% 2.61/1.48 = ( ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Y: $i] :
% 2.61/1.48 ( ( in @ Y @ ( powerset @ A ) )
% 2.61/1.48 => ! [Xx: $i] :
% 2.61/1.48 ( ( in @ Xx @ A )
% 2.61/1.48 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 2.61/1.48 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(complementSubsetComplementIntersect,conjecture,
% 2.61/1.48 ( setextAx
% 2.61/1.48 => ( emptysetAx
% 2.61/1.48 => ( setadjoinAx
% 2.61/1.48 => ( powersetAx
% 2.61/1.48 => ( setunionAx
% 2.61/1.48 => ( omega0Ax
% 2.61/1.48 => ( omegaSAx
% 2.61/1.48 => ( omegaIndAx
% 2.61/1.48 => ( replAx
% 2.61/1.48 => ( foundationAx
% 2.61/1.48 => ( wellorderingAx
% 2.61/1.48 => ( descrp
% 2.61/1.48 => ( dsetconstrI
% 2.61/1.48 => ( dsetconstrEL
% 2.61/1.48 => ( dsetconstrER
% 2.61/1.48 => ( exuE1
% 2.61/1.48 => ( prop2setE
% 2.61/1.48 => ( emptysetE
% 2.61/1.48 => ( emptysetimpfalse
% 2.61/1.48 => ( notinemptyset
% 2.61/1.48 => ( exuE3e
% 2.61/1.48 => ( setext
% 2.61/1.48 => ( emptyI
% 2.61/1.48 => ( noeltsimpempty
% 2.61/1.48 => ( setbeta
% 2.61/1.48 => ( nonemptyE1
% 2.61/1.48 => ( nonemptyI
% 2.61/1.48 => ( nonemptyI1
% 2.61/1.48 => ( setadjoinIL
% 2.61/1.48 => ( emptyinunitempty
% 2.61/1.48 => ( setadjoinIR
% 2.61/1.48 => ( setadjoinE
% 2.61/1.48 => ( setadjoinOr
% 2.61/1.48 => ( setoftrueEq
% 2.61/1.48 => ( powersetI
% 2.61/1.48 => ( emptyinPowerset
% 2.61/1.48 => ( emptyInPowerset
% 2.61/1.48 => ( powersetE
% 2.61/1.48 => ( setunionI
% 2.61/1.48 => ( setunionE
% 2.61/1.48 => ( subPowSU
% 2.61/1.48 => ( exuE2
% 2.61/1.48 => ( nonemptyImpWitness
% 2.61/1.48 => ( uniqinunit
% 2.61/1.48 => ( notinsingleton
% 2.61/1.48 => ( eqinunit
% 2.61/1.48 => ( singletonsswitch
% 2.61/1.48 => ( upairsetE
% 2.61/1.48 => ( upairsetIL
% 2.61/1.48 => ( upairsetIR
% 2.61/1.48 => ( emptyE1
% 2.61/1.48 => ( vacuousDall
% 2.61/1.48 => ( quantDeMorgan1
% 2.61/1.48 => ( quantDeMorgan2
% 2.61/1.48 => ( quantDeMorgan3
% 2.61/1.48 => ( quantDeMorgan4
% 2.61/1.48 => ( prop2setI
% 2.61/1.48 => ( prop2set2propI
% 2.61/1.48 => ( notdexE
% 2.61/1.48 => ( notdallE
% 2.61/1.48 => ( exuI1
% 2.61/1.48 => ( exuI3
% 2.61/1.48 => ( exuI2
% 2.61/1.48 => ( inCongP
% 2.61/1.48 => ( in__Cong
% 2.61/1.48 => ( exuE3u
% 2.61/1.48 => ( exu__Cong
% 2.61/1.48 => ( emptyset__Cong
% 2.61/1.48 => ( setadjoin__Cong
% 2.61/1.48 => ( powerset__Cong
% 2.61/1.48 => ( setunion__Cong
% 2.61/1.48 => ( omega__Cong
% 2.61/1.48 => ( exuEu
% 2.61/1.48 => ( descr__Cong
% 2.61/1.48 => ( dsetconstr__Cong
% 2.61/1.48 => ( subsetI1
% 2.61/1.48 => ( eqimpsubset2
% 2.61/1.48 => ( eqimpsubset1
% 2.61/1.48 => ( subsetI2
% 2.61/1.48 => ( emptysetsubset
% 2.61/1.48 => ( subsetE
% 2.61/1.48 => ( subsetE2
% 2.61/1.48 => ( notsubsetI
% 2.61/1.48 => ( notequalI1
% 2.61/1.48 => ( notequalI2
% 2.61/1.48 => ( subsetRefl
% 2.61/1.48 => ( subsetTrans
% 2.61/1.48 => ( setadjoinSub
% 2.61/1.48 => ( setadjoinSub2
% 2.61/1.48 => ( subset2powerset
% 2.61/1.48 => ( setextsub
% 2.61/1.48 => ( subsetemptysetimpeq
% 2.61/1.48 => ( powersetI1
% 2.61/1.48 => ( powersetE1
% 2.61/1.48 => ( inPowerset
% 2.61/1.48 => ( powersetsubset
% 2.61/1.48 => ( sepInPowerset
% 2.61/1.48 => ( sepSubset
% 2.61/1.48 => ( binunionIL
% 2.61/1.48 => ( upairset2IR
% 2.61/1.48 => ( binunionIR
% 2.61/1.48 => ( binunionEcases
% 2.61/1.48 => ( binunionE
% 2.61/1.48 => ( binunionLsub
% 2.61/1.48 => ( binunionRsub
% 2.61/1.48 => ( binintersectI
% 2.61/1.48 => ( binintersectSubset5
% 2.61/1.48 => ( binintersectEL
% 2.61/1.48 => ( binintersectLsub
% 2.61/1.48 => ( binintersectSubset2
% 2.61/1.48 => ( binintersectSubset3
% 2.61/1.48 => ( binintersectER
% 2.61/1.48 => ( disjointsetsI1
% 2.61/1.48 => ( binintersectRsub
% 2.61/1.48 => ( binintersectSubset4
% 2.61/1.48 => ( binintersectSubset1
% 2.61/1.48 => ( bs114d
% 2.61/1.48 => ( setminusI
% 2.61/1.48 => ( setminusEL
% 2.61/1.48 => ( setminusER
% 2.61/1.48 => ( setminusSubset2
% 2.61/1.48 => ( setminusERneg
% 2.61/1.48 => ( setminusELneg
% 2.61/1.48 => ( setminusILneg
% 2.61/1.48 => ( setminusIRneg
% 2.61/1.48 => ( setminusLsub
% 2.61/1.48 => ( setminusSubset1
% 2.61/1.48 => ( symdiffE
% 2.61/1.48 => ( symdiffI1
% 2.61/1.48 => ( symdiffI2
% 2.61/1.48 => ( symdiffIneg1
% 2.61/1.48 => ( symdiffIneg2
% 2.61/1.48 => ( secondinupair
% 2.61/1.48 => ( setukpairIL
% 2.61/1.48 => ( setukpairIR
% 2.61/1.48 => ( kpairiskpair
% 2.61/1.48 => ( kpairp
% 2.61/1.48 => ( singletonsubset
% 2.61/1.48 => ( singletoninpowerset
% 2.61/1.48 => ( singletoninpowunion
% 2.61/1.48 => ( upairset2E
% 2.61/1.48 => ( upairsubunion
% 2.61/1.48 => ( upairinpowunion
% 2.61/1.48 => ( ubforcartprodlem1
% 2.61/1.48 => ( ubforcartprodlem2
% 2.61/1.48 => ( ubforcartprodlem3
% 2.61/1.48 => ( cartprodpairin
% 2.61/1.48 => ( cartprodmempair1
% 2.61/1.48 => ( cartprodmempair
% 2.61/1.48 => ( setunionE2
% 2.61/1.48 => ( setunionsingleton1
% 2.61/1.48 => ( setunionsingleton2
% 2.61/1.48 => ( setunionsingleton
% 2.61/1.48 => ( singletonprop
% 2.61/1.48 => ( ex1E1
% 2.61/1.48 => ( ex1I
% 2.61/1.48 => ( ex1I2
% 2.61/1.48 => ( singletonsuniq
% 2.61/1.48 => ( setukpairinjL1
% 2.61/1.48 => ( kfstsingleton
% 2.61/1.48 => ( theprop
% 2.61/1.48 => ( kfstpairEq
% 2.61/1.48 => ( cartprodfstin
% 2.61/1.48 => ( setukpairinjL2
% 2.61/1.48 => ( setukpairinjL
% 2.61/1.48 => ( setukpairinjR11
% 2.61/1.48 => ( setukpairinjR12
% 2.61/1.48 => ( setukpairinjR1
% 2.61/1.48 => ( upairequniteq
% 2.61/1.48 => ( setukpairinjR2
% 2.61/1.48 => ( setukpairinjR
% 2.61/1.48 => ( ksndsingleton
% 2.61/1.48 => ( ksndpairEq
% 2.61/1.48 => ( kpairsurjEq
% 2.61/1.48 => ( cartprodsndin
% 2.61/1.48 => ( cartprodpairmemEL
% 2.61/1.48 => ( cartprodpairmemER
% 2.61/1.48 => ( cartprodmempaircEq
% 2.61/1.48 => ( cartprodfstpairEq
% 2.61/1.48 => ( cartprodsndpairEq
% 2.61/1.48 => ( cartprodpairsurjEq
% 2.61/1.48 => ( dpsetconstrI
% 2.61/1.48 => ( dpsetconstrSub
% 2.61/1.48 => ( setOfPairsIsBReln
% 2.61/1.48 => ( dpsetconstrERa
% 2.61/1.48 => ( dpsetconstrEL1
% 2.61/1.48 => ( dpsetconstrEL2
% 2.61/1.48 => ( dpsetconstrER
% 2.61/1.48 => ( funcImageSingleton
% 2.61/1.48 => ( apProp
% 2.61/1.48 => ( app
% 2.61/1.48 => ( infuncsetfunc
% 2.61/1.48 => ( ap2p
% 2.61/1.48 => ( funcinfuncset
% 2.61/1.48 => ( lamProp
% 2.61/1.48 => ( lamp
% 2.61/1.48 => ( lam2p
% 2.61/1.48 => ( brelnall1
% 2.61/1.48 => ( brelnall2
% 2.61/1.48 => ( ex1E2
% 2.61/1.48 => ( funcGraphProp1
% 2.61/1.48 => ( funcGraphProp3
% 2.61/1.48 => ( funcGraphProp2
% 2.61/1.48 => ( funcextLem
% 2.61/1.48 => ( funcGraphProp4
% 2.61/1.48 => ( subbreln
% 2.61/1.48 => ( eqbreln
% 2.61/1.48 => ( funcext
% 2.61/1.48 => ( funcext2
% 2.61/1.48 => ( ap2apEq1
% 2.61/1.48 => ( ap2apEq2
% 2.61/1.48 => ( beta1
% 2.61/1.48 => ( eta1
% 2.61/1.48 => ( lam2lamEq
% 2.61/1.48 => ( beta2
% 2.61/1.48 => ( eta2
% 2.61/1.48 => ( iffalseProp1
% 2.61/1.48 => ( iffalseProp2
% 2.61/1.48 => ( iftrueProp1
% 2.61/1.48 => ( iftrueProp2
% 2.61/1.48 => ( ifSingleton
% 2.61/1.48 => ( ifp
% 2.61/1.48 => ( theeq
% 2.61/1.48 => ( iftrue
% 2.61/1.48 => ( iffalse
% 2.61/1.48 => ( iftrueorfalse
% 2.61/1.48 => ( binintersectT_lem
% 2.61/1.48 => ( binunionT_lem
% 2.61/1.48 => ( powersetT_lem
% 2.61/1.48 => ( setminusT_lem
% 2.61/1.48 => ( complementT_lem
% 2.61/1.48 => ( setextT
% 2.61/1.48 => ( subsetTI
% 2.61/1.48 => ( powersetTI1
% 2.61/1.48 => ( powersetTE1
% 2.61/1.48 => ( complementTI1
% 2.61/1.48 => ( complementTE1
% 2.61/1.48 => ( binintersectTELcontra
% 2.61/1.48 => ( binintersectTERcontra
% 2.61/1.48 => ( contrasubsetT
% 2.61/1.48 => ( contrasubsetT1
% 2.61/1.48 => ( contrasubsetT2
% 2.61/1.48 => ( contrasubsetT3
% 2.61/1.48 => ( doubleComplementI1
% 2.61/1.48 => ( doubleComplementE1
% 2.61/1.48 => ( doubleComplementSub1
% 2.61/1.48 => ( doubleComplementSub2
% 2.61/1.48 => ( doubleComplementEq
% 2.61/1.48 => ( complementTnotintersectT
% 2.61/1.48 => ( complementImpComplementIntersect
% 2.61/1.48 => ! [A: $i,X: $i] :
% 2.61/1.48 ( ( in @ X @ ( powerset @ A ) )
% 2.61/1.48 => ! [Y: $i] :
% 2.61/1.48 ( ( in @ Y @ ( powerset @ A ) )
% 2.61/1.48 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.61/1.48 ------------------------
% 2.72/1.51 % (28918)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_28816 for (2999ds/183Mi)
% 2.72/1.51 % (28921)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_28816 for (2999ds/2Mi)
% 2.72/1.51 % (28922)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_28816 for (2999ds/2Mi)
% 2.72/1.51 % (28919)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_28816 for (2999ds/4Mi)
% 2.72/1.51 % (28920)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_28816 for (2999ds/27Mi)
% 2.72/1.51 % (28923)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_28816 for (2999ds/275Mi)
% 2.72/1.51 % (28924)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_28816 for (2999ds/18Mi)
% 2.72/1.52 % (28921)Instruction limit reached!
% 2.72/1.52 % (28921)------------------------------
% 2.72/1.52 % (28921)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.52 % (28921)Termination reason: Unknown
% 2.72/1.52 % (28922)Instruction limit reached!
% 2.72/1.52 % (28922)------------------------------
% 2.72/1.52 % (28922)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.52 % (28922)Termination reason: Unknown
% 2.72/1.52 % (28922)Termination phase: shuffling
% 2.72/1.52
% 2.72/1.52 % (28922)Memory used [KB]: 1407
% 2.72/1.52 % (28922)Time elapsed: 0.003 s
% 2.72/1.52 % (28922)Instructions burned: 3 (million)
% 2.72/1.52 % (28922)------------------------------
% 2.72/1.52 % (28922)------------------------------
% 2.72/1.52 % (28921)Termination phase: shuffling
% 2.72/1.52
% 2.72/1.52 % (28921)Memory used [KB]: 1407
% 2.72/1.52 % (28921)Time elapsed: 0.003 s
% 2.72/1.52 % (28921)Instructions burned: 3 (million)
% 2.72/1.52 % (28921)------------------------------
% 2.72/1.52 % (28921)------------------------------
% 2.72/1.52 % (28919)Instruction limit reached!
% 2.72/1.52 % (28919)------------------------------
% 2.72/1.52 % (28919)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.52 % (28919)Termination reason: Unknown
% 2.72/1.52 % (28919)Termination phase: shuffling
% 2.72/1.52
% 2.72/1.52 % (28919)Memory used [KB]: 1535
% 2.72/1.52 % (28919)Time elapsed: 0.004 s
% 2.72/1.52 % (28919)Instructions burned: 5 (million)
% 2.72/1.52 % (28919)------------------------------
% 2.72/1.52 % (28919)------------------------------
% 2.72/1.52 % (28924)Instruction limit reached!
% 2.72/1.52 % (28924)------------------------------
% 2.72/1.52 % (28924)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.52 % (28924)Termination reason: Unknown
% 2.72/1.52 % (28924)Termination phase: shuffling
% 2.72/1.52
% 2.72/1.52 % (28924)Memory used [KB]: 1791
% 2.72/1.52 % (28924)Time elapsed: 0.011 s
% 2.72/1.52 % (28924)Instructions burned: 19 (million)
% 2.72/1.52 % (28924)------------------------------
% 2.72/1.52 % (28924)------------------------------
% 2.72/1.53 % (28920)Instruction limit reached!
% 2.72/1.53 % (28920)------------------------------
% 2.72/1.53 % (28920)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.53 % (28920)Termination reason: Unknown
% 2.72/1.53 % (28920)Termination phase: shuffling
% 2.72/1.53
% 2.72/1.53 % (28920)Memory used [KB]: 1918
% 2.72/1.53 % (28920)Time elapsed: 0.014 s
% 2.72/1.53 % (28920)Instructions burned: 27 (million)
% 2.72/1.53 % (28920)------------------------------
% 2.72/1.53 % (28920)------------------------------
% 2.72/1.53 % (28926)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_28816 for (2999ds/37Mi)
% 2.72/1.53 % (28925)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_28816 for (2999ds/3Mi)
% 2.72/1.53 % (28925)Instruction limit reached!
% 2.72/1.53 % (28925)------------------------------
% 2.72/1.53 % (28925)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.53 % (28925)Termination reason: Unknown
% 2.72/1.53 % (28925)Termination phase: shuffling
% 2.72/1.53
% 2.72/1.53 % (28925)Memory used [KB]: 1535
% 2.72/1.53 % (28925)Time elapsed: 0.003 s
% 2.72/1.53 % (28925)Instructions burned: 3 (million)
% 2.72/1.53 % (28925)------------------------------
% 2.72/1.53 % (28925)------------------------------
% 2.72/1.53 % (28927)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_28816 for (2999ds/15Mi)
% 2.72/1.54 % (28927)Instruction limit reached!
% 2.72/1.54 % (28927)------------------------------
% 2.72/1.54 % (28927)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.54 % (28927)Termination reason: Unknown
% 2.72/1.54 % (28927)Termination phase: shuffling
% 2.72/1.54 % (28928)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_28816 for (2999ds/3Mi)
% 2.72/1.54
% 2.72/1.54 % (28927)Memory used [KB]: 1663
% 2.72/1.54 % (28927)Time elapsed: 0.009 s
% 2.72/1.54 % (28927)Instructions burned: 15 (million)
% 2.72/1.54 % (28927)------------------------------
% 2.72/1.54 % (28927)------------------------------
% 2.72/1.54 % (28928)Instruction limit reached!
% 2.72/1.54 % (28928)------------------------------
% 2.72/1.54 % (28928)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.54 % (28928)Termination reason: Unknown
% 2.72/1.54 % (28928)Termination phase: shuffling
% 2.72/1.54
% 2.72/1.54 % (28928)Memory used [KB]: 1535
% 2.72/1.54 % (28928)Time elapsed: 0.003 s
% 2.72/1.54 % (28928)Instructions burned: 3 (million)
% 2.72/1.54 % (28928)------------------------------
% 2.72/1.54 % (28928)------------------------------
% 2.72/1.54 % (28929)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_28816 for (2999ds/1041Mi)
% 2.72/1.55 % (28926)Instruction limit reached!
% 2.72/1.55 % (28926)------------------------------
% 2.72/1.55 % (28926)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.55 % (28926)Termination reason: Unknown
% 2.72/1.55 % (28926)Termination phase: shuffling
% 2.72/1.55
% 2.72/1.55 % (28926)Memory used [KB]: 2302
% 2.72/1.55 % (28926)Time elapsed: 0.018 s
% 2.72/1.55 % (28926)Instructions burned: 37 (million)
% 2.72/1.55 % (28926)------------------------------
% 2.72/1.55 % (28926)------------------------------
% 2.72/1.55 % (28930)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_28816 for (2999ds/7Mi)
% 2.72/1.55 % (28930)Instruction limit reached!
% 2.72/1.55 % (28930)------------------------------
% 2.72/1.55 % (28930)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.55 % (28930)Termination reason: Unknown
% 2.72/1.55 % (28930)Termination phase: shuffling
% 2.72/1.55
% 2.72/1.55 % (28930)Memory used [KB]: 1663
% 2.72/1.55 % (28930)Time elapsed: 0.005 s
% 2.72/1.55 % (28930)Instructions burned: 8 (million)
% 2.72/1.55 % (28930)------------------------------
% 2.72/1.55 % (28930)------------------------------
% 2.72/1.56 % (28918)Instruction limit reached!
% 2.72/1.56 % (28918)------------------------------
% 2.72/1.56 % (28918)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.56 % (28918)Termination reason: Unknown
% 2.72/1.56 % (28918)Termination phase: Saturation
% 2.72/1.56
% 2.72/1.56 % (28918)Memory used [KB]: 7675
% 2.72/1.56 % (28918)Time elapsed: 0.050 s
% 2.72/1.56 % (28918)Instructions burned: 184 (million)
% 2.72/1.56 % (28918)------------------------------
% 2.72/1.56 % (28918)------------------------------
% 2.72/1.56 % (28933)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_28816 for (2999ds/3Mi)
% 2.72/1.56 % (28933)Instruction limit reached!
% 2.72/1.56 % (28933)------------------------------
% 2.72/1.56 % (28933)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.56 % (28933)Termination reason: Unknown
% 2.72/1.56 % (28933)Termination phase: shuffling
% 2.72/1.56
% 2.72/1.56 % (28933)Memory used [KB]: 1535
% 2.72/1.56 % (28933)Time elapsed: 0.003 s
% 2.72/1.56 % (28933)Instructions burned: 3 (million)
% 2.72/1.56 % (28933)------------------------------
% 2.72/1.56 % (28933)------------------------------
% 2.72/1.56 % (28932)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_28816 for (2999ds/16Mi)
% 2.72/1.56 % (28934)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_28816 for (2999ds/3Mi)
% 2.72/1.57 % (28934)Instruction limit reached!
% 2.72/1.57 % (28934)------------------------------
% 2.72/1.57 % (28934)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.57 % (28934)Termination reason: Unknown
% 2.72/1.57 % (28934)Termination phase: shuffling
% 2.72/1.57
% 2.72/1.57 % (28934)Memory used [KB]: 1407
% 2.72/1.57 % (28934)Time elapsed: 0.003 s
% 2.72/1.57 % (28934)Instructions burned: 3 (million)
% 2.72/1.57 % (28934)------------------------------
% 2.72/1.57 % (28934)------------------------------
% 2.72/1.57 % (28936)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_28816 for (2999ds/3Mi)
% 2.72/1.57 % (28936)Instruction limit reached!
% 2.72/1.57 % (28936)------------------------------
% 2.72/1.57 % (28936)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.57 % (28936)Termination reason: Unknown
% 2.72/1.57 % (28936)Termination phase: shuffling
% 2.72/1.57
% 2.72/1.57 % (28936)Memory used [KB]: 1535
% 2.72/1.57 % (28936)Time elapsed: 0.002 s
% 2.72/1.57 % (28936)Instructions burned: 3 (million)
% 2.72/1.57 % (28936)------------------------------
% 2.72/1.57 % (28936)------------------------------
% 2.72/1.57 % (28935)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_28816 for (2999ds/7Mi)
% 2.72/1.57 % (28932)Instruction limit reached!
% 2.72/1.57 % (28932)------------------------------
% 2.72/1.57 % (28932)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.57 % (28932)Termination reason: Unknown
% 2.72/1.57 % (28932)Termination phase: shuffling
% 2.72/1.57
% 2.72/1.57 % (28932)Memory used [KB]: 1791
% 2.72/1.57 % (28932)Time elapsed: 0.009 s
% 2.72/1.57 % (28932)Instructions burned: 17 (million)
% 2.72/1.57 % (28932)------------------------------
% 2.72/1.57 % (28932)------------------------------
% 2.72/1.57 % (28935)Instruction limit reached!
% 2.72/1.57 % (28935)------------------------------
% 2.72/1.57 % (28935)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.57 % (28935)Termination reason: Unknown
% 2.72/1.57 % (28935)Termination phase: shuffling
% 2.72/1.57
% 2.72/1.57 % (28935)Memory used [KB]: 1535
% 2.72/1.57 % (28935)Time elapsed: 0.005 s
% 2.72/1.57 % (28935)Instructions burned: 8 (million)
% 2.72/1.57 % (28935)------------------------------
% 2.72/1.57 % (28935)------------------------------
% 2.72/1.58 % (28937)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_28816 for (2999ds/4Mi)
% 2.72/1.58 % (28939)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_28816 for (2999ds/710Mi)
% 2.72/1.58 % (28937)Instruction limit reached!
% 2.72/1.58 % (28937)------------------------------
% 2.72/1.58 % (28937)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.58 % (28937)Termination reason: Unknown
% 2.72/1.58 % (28937)Termination phase: shuffling
% 2.72/1.58
% 2.72/1.58 % (28937)Memory used [KB]: 1535
% 2.72/1.58 % (28937)Time elapsed: 0.004 s
% 2.72/1.58 % (28937)Instructions burned: 6 (million)
% 2.72/1.58 % (28937)------------------------------
% 2.72/1.58 % (28937)------------------------------
% 2.72/1.58 % (28938)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_28816 for (2999ds/18Mi)
% 2.72/1.59 % (28940)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_28816 for (2999ds/6Mi)
% 2.72/1.59 % (28940)Instruction limit reached!
% 2.72/1.59 % (28940)------------------------------
% 2.72/1.59 % (28940)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.59 % (28940)Termination reason: Unknown
% 2.72/1.59 % (28940)Termination phase: shuffling
% 2.72/1.59
% 2.72/1.59 % (28940)Memory used [KB]: 1535
% 2.72/1.59 % (28940)Time elapsed: 0.004 s
% 2.72/1.59 % (28940)Instructions burned: 6 (million)
% 2.72/1.59 % (28940)------------------------------
% 2.72/1.59 % (28940)------------------------------
% 2.72/1.59 % (28938)Instruction limit reached!
% 2.72/1.59 % (28938)------------------------------
% 2.72/1.59 % (28938)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.59 % (28938)Termination reason: Unknown
% 2.72/1.59 % (28938)Termination phase: shuffling
% 2.72/1.59
% 2.72/1.59 % (28938)Memory used [KB]: 1791
% 2.72/1.59 % (28938)Time elapsed: 0.010 s
% 2.72/1.59 % (28938)Instructions burned: 19 (million)
% 2.72/1.59 % (28938)------------------------------
% 2.72/1.59 % (28938)------------------------------
% 2.72/1.59 % (28942)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_28816 for (2999ds/21Mi)
% 2.72/1.60 % (28942)Instruction limit reached!
% 2.72/1.60 % (28942)------------------------------
% 2.72/1.60 % (28942)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.60 % (28942)Termination reason: Unknown
% 2.72/1.60 % (28942)Termination phase: shuffling
% 2.72/1.60
% 2.72/1.60 % (28942)Memory used [KB]: 1791
% 2.72/1.60 % (28942)Time elapsed: 0.011 s
% 2.72/1.60 % (28942)Instructions burned: 21 (million)
% 2.72/1.60 % (28942)------------------------------
% 2.72/1.60 % (28942)------------------------------
% 2.72/1.61 % (28941)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_28816 for (2999ds/902Mi)
% 2.72/1.61 % (28943)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_28816 for (2999ds/5Mi)
% 2.72/1.61 % (28944)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_28816 for (2999ds/6Mi)
% 2.72/1.61 % (28943)Instruction limit reached!
% 2.72/1.61 % (28943)------------------------------
% 2.72/1.61 % (28943)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.61 % (28943)Termination reason: Unknown
% 2.72/1.61 % (28943)Termination phase: shuffling
% 2.72/1.61
% 2.72/1.61 % (28943)Memory used [KB]: 1535
% 2.72/1.61 % (28943)Time elapsed: 0.004 s
% 2.72/1.61 % (28943)Instructions burned: 6 (million)
% 2.72/1.61 % (28943)------------------------------
% 2.72/1.61 % (28943)------------------------------
% 2.72/1.61 % (28944)Instruction limit reached!
% 2.72/1.61 % (28944)------------------------------
% 2.72/1.61 % (28944)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.61 % (28944)Termination reason: Unknown
% 2.72/1.61 % (28944)Termination phase: shuffling
% 2.72/1.61
% 2.72/1.61 % (28944)Memory used [KB]: 1535
% 2.72/1.61 % (28944)Time elapsed: 0.004 s
% 2.72/1.61 % (28944)Instructions burned: 6 (million)
% 2.72/1.61 % (28944)------------------------------
% 2.72/1.61 % (28944)------------------------------
% 2.72/1.62 % (28945)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_28816 for (2998ds/377Mi)
% 2.72/1.62 % (28947)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_28816 for (2998ds/19Mi)
% 2.72/1.63 % (28946)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_28816 for (2998ds/779Mi)
% 2.72/1.63 % (28947)Instruction limit reached!
% 2.72/1.63 % (28947)------------------------------
% 2.72/1.63 % (28947)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.63 % (28947)Termination reason: Unknown
% 2.72/1.63 % (28947)Termination phase: shuffling
% 2.72/1.63
% 2.72/1.63 % (28947)Memory used [KB]: 1791
% 2.72/1.63 % (28947)Time elapsed: 0.010 s
% 2.72/1.63 % (28947)Instructions burned: 19 (million)
% 2.72/1.63 % (28947)------------------------------
% 2.72/1.63 % (28947)------------------------------
% 2.72/1.64 % (28923)Instruction limit reached!
% 2.72/1.64 % (28923)------------------------------
% 2.72/1.64 % (28923)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.64 % (28923)Termination reason: Unknown
% 2.72/1.64 % (28923)Termination phase: Saturation
% 2.72/1.64
% 2.72/1.64 % (28923)Memory used [KB]: 9594
% 2.72/1.64 % (28923)Time elapsed: 0.131 s
% 2.72/1.64 % (28923)Instructions burned: 275 (million)
% 2.72/1.64 % (28923)------------------------------
% 2.72/1.64 % (28923)------------------------------
% 2.72/1.65 % (28948)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_28816 for (2998ds/879Mi)
% 2.72/1.66 % (28949)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_28816 for (2998ds/17Mi)
% 2.72/1.67 % (28949)Instruction limit reached!
% 2.72/1.67 % (28949)------------------------------
% 2.72/1.67 % (28949)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.67 % (28949)Termination reason: Unknown
% 2.72/1.67 % (28949)Termination phase: shuffling
% 2.72/1.67
% 2.72/1.67 % (28949)Memory used [KB]: 1791
% 2.72/1.67 % (28949)Time elapsed: 0.010 s
% 2.72/1.67 % (28949)Instructions burned: 18 (million)
% 2.72/1.67 % (28949)------------------------------
% 2.72/1.67 % (28949)------------------------------
% 2.72/1.69 % (28950)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_28816 for (2998ds/3Mi)
% 2.72/1.69 % (28950)Instruction limit reached!
% 2.72/1.69 % (28950)------------------------------
% 2.72/1.69 % (28950)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.69 % (28950)Termination reason: Unknown
% 2.72/1.69 % (28950)Termination phase: shuffling
% 2.72/1.69
% 2.72/1.69 % (28950)Memory used [KB]: 1407
% 2.72/1.69 % (28950)Time elapsed: 0.003 s
% 2.72/1.69 % (28950)Instructions burned: 3 (million)
% 2.72/1.69 % (28950)------------------------------
% 2.72/1.69 % (28950)------------------------------
% 2.72/1.71 % (28951)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_28816 for (2998ds/30Mi)
% 2.72/1.72 % (28951)Instruction limit reached!
% 2.72/1.72 % (28951)------------------------------
% 2.72/1.72 % (28951)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.72 % (28951)Termination reason: Unknown
% 2.72/1.72 % (28951)Termination phase: shuffling
% 2.72/1.72
% 2.72/1.72 % (28951)Memory used [KB]: 2046
% 2.72/1.72 % (28951)Time elapsed: 0.016 s
% 2.72/1.72 % (28951)Instructions burned: 31 (million)
% 2.72/1.72 % (28951)------------------------------
% 2.72/1.72 % (28951)------------------------------
% 2.72/1.74 % (28952)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_28816 for (2997ds/127Mi)
% 2.72/1.79 % (28945)Instruction limit reached!
% 2.72/1.79 % (28945)------------------------------
% 2.72/1.79 % (28945)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.79 % (28945)Termination reason: Unknown
% 2.72/1.79 % (28945)Termination phase: Saturation
% 2.72/1.79
% 2.72/1.79 % (28945)Memory used [KB]: 9338
% 2.72/1.79 % (28945)Time elapsed: 0.175 s
% 2.72/1.79 % (28945)Instructions burned: 378 (million)
% 2.72/1.79 % (28945)------------------------------
% 2.72/1.79 % (28945)------------------------------
% 2.72/1.79 % (28952)Instruction limit reached!
% 2.72/1.79 % (28952)------------------------------
% 2.72/1.79 % (28952)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.79 % (28952)Termination reason: Unknown
% 2.72/1.79 % (28952)Termination phase: Property scanning
% 2.72/1.79
% 2.72/1.79 % (28952)Memory used [KB]: 2686
% 2.72/1.79 % (28952)Time elapsed: 0.058 s
% 2.72/1.79 % (28952)Instructions burned: 128 (million)
% 2.72/1.79 % (28952)------------------------------
% 2.72/1.79 % (28952)------------------------------
% 2.72/1.81 % (28954)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_28816 for (2997ds/3Mi)
% 2.72/1.81 % (28954)Instruction limit reached!
% 2.72/1.81 % (28954)------------------------------
% 2.72/1.81 % (28954)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.81 % (28954)Termination reason: Unknown
% 2.72/1.81 % (28954)Termination phase: shuffling
% 2.72/1.81
% 2.72/1.81 % (28954)Memory used [KB]: 1535
% 2.72/1.81 % (28954)Time elapsed: 0.004 s
% 2.72/1.81 % (28954)Instructions burned: 4 (million)
% 2.72/1.81 % (28954)------------------------------
% 2.72/1.81 % (28954)------------------------------
% 2.72/1.81 % (28939)Instruction limit reached!
% 2.72/1.81 % (28939)------------------------------
% 2.72/1.81 % (28939)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.81 % (28939)Termination reason: Unknown
% 2.72/1.81 % (28939)Termination phase: Saturation
% 2.72/1.81
% 2.72/1.81 % (28939)Memory used [KB]: 10490
% 2.72/1.81 % (28939)Time elapsed: 0.238 s
% 2.72/1.81 % (28939)Instructions burned: 711 (million)
% 2.72/1.81 % (28939)------------------------------
% 2.72/1.81 % (28939)------------------------------
% 2.72/1.82 % (28953)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_28816 for (2997ds/100Mi)
% 2.72/1.83 % (28956)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_28816 for (2996ds/86Mi)
% 2.72/1.83 % (28955)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_28816 for (2996ds/20Mi)
% 2.72/1.84 % (28953)Instruction limit reached!
% 2.72/1.84 % (28953)------------------------------
% 2.72/1.84 % (28953)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.84 % (28953)Termination reason: Unknown
% 2.72/1.84 % (28955)Instruction limit reached!
% 2.72/1.84 % (28955)------------------------------
% 2.72/1.84 % (28955)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.84 % (28955)Termination reason: Unknown
% 2.72/1.84 % (28955)Termination phase: shuffling
% 2.72/1.84
% 2.72/1.84 % (28955)Memory used [KB]: 1791
% 2.72/1.84 % (28955)Time elapsed: 0.012 s
% 2.72/1.84 % (28955)Instructions burned: 20 (million)
% 2.72/1.84 % (28955)------------------------------
% 2.72/1.84 % (28955)------------------------------
% 2.72/1.84 % (28953)Termination phase: Function definition elimination
% 2.72/1.84
% 2.72/1.84 % (28953)Memory used [KB]: 2558
% 2.72/1.84 % (28953)Time elapsed: 0.027 s
% 2.72/1.84 % (28953)Instructions burned: 102 (million)
% 2.72/1.84 % (28953)------------------------------
% 2.72/1.84 % (28953)------------------------------
% 2.72/1.85 % (28957)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_28816 for (2996ds/107Mi)
% 2.72/1.86 % (28958)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_28816 for (2996ds/39Mi)
% 2.72/1.87 % (28956)Instruction limit reached!
% 2.72/1.87 % (28956)------------------------------
% 2.72/1.87 % (28956)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.87 % (28956)Termination reason: Unknown
% 2.72/1.87 % (28956)Termination phase: Property scanning
% 2.72/1.87
% 2.72/1.87 % (28956)Memory used [KB]: 2558
% 2.72/1.87 % (28956)Time elapsed: 0.040 s
% 2.72/1.87 % (28956)Instructions burned: 88 (million)
% 2.72/1.87 % (28956)------------------------------
% 2.72/1.87 % (28956)------------------------------
% 2.72/1.87 % (28958)Instruction limit reached!
% 2.72/1.87 % (28958)------------------------------
% 2.72/1.87 % (28958)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.87 % (28958)Termination reason: Unknown
% 2.72/1.87 % (28958)Termination phase: shuffling
% 2.72/1.87
% 2.72/1.87 % (28958)Memory used [KB]: 2302
% 2.72/1.87 % (28958)Time elapsed: 0.012 s
% 2.72/1.87 % (28958)Instructions burned: 42 (million)
% 2.72/1.87 % (28958)------------------------------
% 2.72/1.87 % (28958)------------------------------
% 2.72/1.88 % (28959)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_28816 for (2996ds/448Mi)
% 2.72/1.89 % (28960)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_28816 for (2996ds/46Mi)
% 2.72/1.90 % (28957)Instruction limit reached!
% 2.72/1.90 % (28957)------------------------------
% 2.72/1.90 % (28957)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.90 % (28957)Termination reason: Unknown
% 2.72/1.90 % (28957)Termination phase: Property scanning
% 2.72/1.90
% 2.72/1.90 % (28957)Memory used [KB]: 2558
% 2.72/1.90 % (28957)Time elapsed: 0.029 s
% 2.72/1.90 % (28957)Instructions burned: 107 (million)
% 2.72/1.90 % (28957)------------------------------
% 2.72/1.90 % (28957)------------------------------
% 2.72/1.91 % (28961)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_28816 for (2996ds/98Mi)
% 2.72/1.91 % (28960)Instruction limit reached!
% 2.72/1.91 % (28960)------------------------------
% 2.72/1.91 % (28960)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.91 % (28960)Termination reason: Unknown
% 2.72/1.91 % (28960)Termination phase: Property scanning
% 2.72/1.91
% 2.72/1.91 % (28960)Memory used [KB]: 2430
% 2.72/1.91 % (28960)Time elapsed: 0.023 s
% 2.72/1.91 % (28960)Instructions burned: 48 (million)
% 2.72/1.91 % (28960)------------------------------
% 2.72/1.91 % (28960)------------------------------
% 2.72/1.93 % (28962)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_28816 for (2995ds/507Mi)
% 2.72/1.93 % (28946)Instruction limit reached!
% 2.72/1.93 % (28946)------------------------------
% 2.72/1.93 % (28946)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.93 % (28946)Termination reason: Unknown
% 2.72/1.93 % (28946)Termination phase: Saturation
% 2.72/1.93
% 2.72/1.93 % (28946)Memory used [KB]: 7547
% 2.72/1.93 % (28946)Time elapsed: 0.301 s
% 2.72/1.93 % (28946)Instructions burned: 781 (million)
% 2.72/1.93 % (28946)------------------------------
% 2.72/1.93 % (28946)------------------------------
% 2.72/1.93 % (28961)Instruction limit reached!
% 2.72/1.93 % (28961)------------------------------
% 2.72/1.93 % (28961)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.93 % (28961)Termination reason: Unknown
% 2.72/1.93 % (28961)Termination phase: Property scanning
% 2.72/1.93
% 2.72/1.93 % (28961)Memory used [KB]: 2430
% 2.72/1.93 % (28961)Time elapsed: 0.026 s
% 2.72/1.93 % (28961)Instructions burned: 100 (million)
% 2.72/1.93 % (28961)------------------------------
% 2.72/1.93 % (28961)------------------------------
% 2.72/1.94 % (28941)First to succeed.
% 2.72/1.94 % (28964)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_28816 for (2995ds/39Mi)
% 2.72/1.94 % (28963)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_28816 for (2995ds/149Mi)
% 2.72/1.94 % (28929)Instruction limit reached!
% 2.72/1.94 % (28929)------------------------------
% 2.72/1.94 % (28929)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.94 % (28929)Termination reason: Unknown
% 2.72/1.94 % (28929)Termination phase: Saturation
% 2.72/1.94
% 2.72/1.94 % (28929)Memory used [KB]: 14839
% 2.72/1.94 % (28929)Time elapsed: 0.404 s
% 2.72/1.94 % (28929)Instructions burned: 1043 (million)
% 2.72/1.94 % (28929)------------------------------
% 2.72/1.94 % (28929)------------------------------
% 2.72/1.95 % (28941)Refutation found. Thanks to Tanya!
% 2.72/1.95 % SZS status Theorem for DTF2THF_28816
% 2.72/1.95 % SZS output start Proof for DTF2THF_28816
% 2.72/1.95 thf(func_def_0, type, in: $i > $i > $o).
% 2.72/1.95 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.72/1.95 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.72/1.95 thf(func_def_8, type, powerset: $i > $i).
% 2.72/1.95 thf(func_def_10, type, setunion: $i > $i).
% 2.72/1.95 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.72/1.95 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.72/1.95 thf(func_def_26, type, prop2set: $o > $i).
% 2.72/1.95 thf(func_def_36, type, nonempty: $i > $o).
% 2.72/1.95 thf(func_def_69, type, set2prop: $i > $o).
% 2.72/1.95 thf(func_def_88, type, subset: $i > $i > $o).
% 2.72/1.95 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.72/1.95 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.72/1.95 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.72/1.95 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.72/1.95 thf(func_def_135, type, regular: $i > $o).
% 2.72/1.95 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.72/1.95 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.72/1.95 thf(func_def_153, type, iskpair: $i > $o).
% 2.72/1.95 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.72/1.95 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.72/1.95 thf(func_def_177, type, singleton: $i > $o).
% 2.72/1.95 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.72/1.95 thf(func_def_184, type, atmost1p: $i > $o).
% 2.72/1.95 thf(func_def_185, type, atleast2p: $i > $o).
% 2.72/1.95 thf(func_def_186, type, atmost2p: $i > $o).
% 2.72/1.95 thf(func_def_187, type, upairsetp: $i > $o).
% 2.72/1.95 thf(func_def_191, type, kfst: $i > $i).
% 2.72/1.95 thf(func_def_203, type, ksnd: $i > $i).
% 2.72/1.95 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.72/1.95 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.72/1.95 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.72/1.95 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.72/1.95 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.72/1.95 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.72/1.95 thf(func_def_302, type, sP0: $i > $i > $i > $o).
% 2.72/1.95 thf(func_def_303, type, sP1: $i > $i > $o).
% 2.72/1.95 thf(func_def_304, type, sP2: $i > $o).
% 2.72/1.95 thf(func_def_305, type, sP3: $i > $i > $o).
% 2.72/1.95 thf(func_def_306, type, sP4: $i > $i > $o).
% 2.72/1.95 thf(func_def_308, type, sP6: $i > $i > $i > $o > $o).
% 2.72/1.95 thf(func_def_312, type, sP10: $i > $i > $i > $i > $o).
% 2.72/1.95 thf(func_def_314, type, sK12: $i > $i > ($i > $i) > $i).
% 2.72/1.95 thf(func_def_315, type, sK13: $i > $i).
% 2.72/1.95 thf(func_def_323, type, sK21: $i > $o).
% 2.72/1.95 thf(func_def_325, type, sK23: ($i > $o) > $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_326, type, sK24: ($i > $o) > $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_329, type, sK27: $i > $o).
% 2.72/1.95 thf(func_def_340, type, sK38: $i > $i).
% 2.72/1.95 thf(func_def_342, type, sK40: $i > $i).
% 2.72/1.95 thf(func_def_356, type, sK54: $i > ($i > $i) > $i > $i).
% 2.72/1.95 thf(func_def_358, type, sK56: $i > $i).
% 2.72/1.95 thf(func_def_363, type, sK61: $i > $o).
% 2.72/1.95 thf(func_def_391, type, sK89: $i > $i > $o).
% 2.72/1.95 thf(func_def_396, type, sK94: $i > $i > $o).
% 2.72/1.95 thf(func_def_398, type, sK96: $i > $i > ($i > $i) > $i).
% 2.72/1.95 thf(func_def_399, type, sK97: $i > $i).
% 2.72/1.95 thf(func_def_407, type, sK105: $i > ($i > $o) > $i).
% 2.72/1.95 thf(func_def_408, type, sK106: $i > $o).
% 2.72/1.95 thf(func_def_422, type, sK120: $i > ($i > $o) > $i).
% 2.72/1.95 thf(func_def_423, type, sK121: $i > $o).
% 2.72/1.95 thf(func_def_436, type, sK134: $i > $i > $i).
% 2.72/1.95 thf(func_def_443, type, sK141: $i > $o).
% 2.72/1.95 thf(func_def_451, type, sK149: $i > $i > $o).
% 2.72/1.95 thf(func_def_467, type, sK165: ($i > $o) > $i > $i > $i).
% 2.72/1.95 thf(func_def_470, type, sK168: $i > $o).
% 2.72/1.95 thf(func_def_471, type, sK169: $i > $i > $o).
% 2.72/1.95 thf(func_def_477, type, sK175: $i > $i > $i).
% 2.72/1.95 thf(func_def_506, type, sK204: ($i > $o) > $i).
% 2.72/1.95 thf(func_def_507, type, sK205: ($i > $o) > $i).
% 2.72/1.95 thf(func_def_508, type, sK206: $i > $o).
% 2.72/1.95 thf(func_def_513, type, sK211: $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_525, type, sK223: $i > $o).
% 2.72/1.95 thf(func_def_547, type, sK245: $i > $o).
% 2.72/1.95 thf(func_def_548, type, sK246: $i > $o).
% 2.72/1.95 thf(func_def_549, type, sK247: ($i > $o) > ($i > $o) > $i).
% 2.72/1.95 thf(func_def_550, type, sK248: ($i > $o) > ($i > $o) > $i).
% 2.72/1.95 thf(func_def_553, type, sK251: $i > $i > $o).
% 2.72/1.95 thf(func_def_559, type, sK257: $i > $o).
% 2.72/1.95 thf(func_def_567, type, sK265: ($i > $o) > $i).
% 2.72/1.95 thf(func_def_568, type, sK266: $i > $o).
% 2.72/1.95 thf(func_def_569, type, sK267: $i > $i).
% 2.72/1.95 thf(func_def_570, type, sK268: $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_577, type, sK275: $i > $o).
% 2.72/1.95 thf(func_def_580, type, sK278: $i > ($i > $o) > $i).
% 2.72/1.95 thf(func_def_590, type, sK288: $i > $i > $i).
% 2.72/1.95 thf(func_def_595, type, sK293: $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_596, type, sK294: $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_606, type, sK304: $i > $o).
% 2.72/1.95 thf(func_def_607, type, sK305: $i > $o).
% 2.72/1.95 thf(func_def_608, type, sK306: ($i > $o) > ($i > $o) > $i).
% 2.72/1.95 thf(func_def_609, type, sK307: ($i > $o) > ($i > $o) > $i).
% 2.72/1.95 thf(func_def_619, type, sK317: $i > $i > $o).
% 2.72/1.95 thf(func_def_635, type, sK333: $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_636, type, sK334: $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_654, type, sK352: $i > $i > $i).
% 2.72/1.95 thf(func_def_655, type, sK353: $i > $i > $i).
% 2.72/1.95 thf(func_def_656, type, sK354: $i > $i > $i).
% 2.72/1.95 thf(func_def_657, type, sK355: $i > $i > $i).
% 2.72/1.95 thf(func_def_658, type, sK356: $i > $i > $i).
% 2.72/1.95 thf(func_def_659, type, sK357: $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_660, type, sK358: $i > $i).
% 2.72/1.95 thf(func_def_661, type, sK359: $i > $i).
% 2.72/1.95 thf(func_def_662, type, sK360: $i > $i).
% 2.72/1.95 thf(func_def_663, type, sK361: $i > $i).
% 2.72/1.95 thf(func_def_664, type, sK362: $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_665, type, sK363: $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_666, type, sK364: $i > $i > $i).
% 2.72/1.95 thf(func_def_667, type, sK365: $i > $i > $i).
% 2.72/1.95 thf(func_def_668, type, sK366: $i > $i > $i).
% 2.72/1.95 thf(func_def_669, type, sK367: $i > $i).
% 2.72/1.95 thf(func_def_671, type, sK369: $i > $i).
% 2.72/1.95 thf(func_def_672, type, sK370: $i > $i).
% 2.72/1.95 thf(func_def_675, type, sK373: $i > $i > $i).
% 2.72/1.95 thf(func_def_676, type, sK374: $i > $i > $i).
% 2.72/1.95 thf(func_def_679, type, sK377: $i > $o).
% 2.72/1.95 thf(func_def_680, type, sK378: ($i > $o) > $i).
% 2.72/1.95 thf(func_def_687, type, sK385: $i > $i).
% 2.72/1.95 thf(func_def_690, type, sK388: $i > ($i > $i) > $i > $i).
% 2.72/1.95 thf(func_def_697, type, sK395: $i > $i > $o).
% 2.72/1.95 thf(func_def_698, type, sK396: $i > $i).
% 2.72/1.95 thf(func_def_699, type, sK397: $i > $i).
% 2.72/1.95 thf(func_def_700, type, sK398: ($i > $i > $o) > $i > $i).
% 2.72/1.95 thf(func_def_701, type, sK399: ($i > $i > $o) > $i > $i).
% 2.72/1.95 thf(func_def_702, type, sK400: $i > ($i > $i > $o) > $i > $i).
% 2.72/1.95 thf(func_def_703, type, sK401: $i > $o).
% 2.72/1.95 thf(func_def_707, type, sK405: $i > $i).
% 2.72/1.95 thf(func_def_710, type, sK408: $i > ($i > $i) > $i > $i).
% 2.72/1.95 thf(func_def_715, type, sK413: $i > $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_724, type, sK422: $i > $i).
% 2.72/1.95 thf(func_def_725, type, sK423: $i > ($i > $o) > $i).
% 2.72/1.95 thf(func_def_726, type, sK424: $i > $o).
% 2.72/1.95 thf(func_def_734, type, sK432: $i > $i > $i).
% 2.72/1.95 thf(func_def_746, type, sK444: $i > $o).
% 2.72/1.95 thf(func_def_764, type, sK462: $i > $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_773, type, sK471: $i > $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_776, type, sK474: $i > $o).
% 2.72/1.95 thf(func_def_777, type, sK475: $i > $o).
% 2.72/1.95 thf(func_def_778, type, sK476: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.72/1.95 thf(func_def_779, type, sK477: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.72/1.95 thf(func_def_797, type, sK495: $i > $o).
% 2.72/1.95 thf(func_def_800, type, sK498: $i > ($i > $o) > $i).
% 2.72/1.95 thf(func_def_801, type, sK499: $i > $i).
% 2.72/1.95 thf(func_def_812, type, sK510: $i > $o).
% 2.72/1.95 thf(func_def_815, type, sK513: $i > $i).
% 2.72/1.95 thf(func_def_820, type, sK518: $i > $o).
% 2.72/1.95 thf(func_def_836, type, sK534: $i > $o).
% 2.72/1.95 thf(func_def_838, type, sK536: ($i > $o) > $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_839, type, sK537: ($i > $o) > $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_848, type, sK546: $i > $i).
% 2.72/1.95 thf(func_def_850, type, sK548: $i > $i > $i).
% 2.72/1.95 thf(func_def_882, type, sK580: $i > $o).
% 2.72/1.95 thf(func_def_884, type, sK582: $i > ($i > $o) > $i).
% 2.72/1.95 thf(func_def_887, type, sK585: $i > $o).
% 2.72/1.95 thf(func_def_890, type, sK588: $i > ($i > $o) > $i).
% 2.72/1.95 thf(func_def_891, type, sK589: $i > $o).
% 2.72/1.95 thf(func_def_892, type, sK590: $i > $i).
% 2.72/1.95 thf(func_def_893, type, sK591: ($i > $o) > $i).
% 2.72/1.95 thf(func_def_914, type, sK612: $i > $o).
% 2.72/1.95 thf(func_def_917, type, sK615: $i > ($i > $o) > $i).
% 2.72/1.95 thf(func_def_918, type, sK616: $i > ($i > $o) > $i).
% 2.72/1.95 thf(func_def_925, type, sK623: $i > $o).
% 2.72/1.95 thf(func_def_936, type, sK634: $i > $o).
% 2.72/1.95 thf(func_def_953, type, sK651: $i > $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_954, type, sK652: $i > $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_960, type, sK658: $i > $i).
% 2.72/1.95 thf(func_def_963, type, sK661: $i > $i > ($i > $i) > $i).
% 2.72/1.95 thf(func_def_970, type, sK668: $i > $o).
% 2.72/1.95 thf(func_def_972, type, sK670: ($i > $o) > $i > $i).
% 2.72/1.95 thf(func_def_977, type, sK675: $i > $i > $i).
% 2.72/1.95 thf(func_def_1013, type, sK711: ($i > $o) > $i > $i).
% 2.72/1.95 thf(func_def_1015, type, sK713: $i > $o).
% 2.72/1.95 thf(func_def_1023, type, sK721: $i > $i > $i).
% 2.72/1.95 thf(func_def_1030, type, sK728: $i > $o).
% 2.72/1.95 thf(func_def_1032, type, sK730: $i > $o).
% 2.72/1.95 thf(func_def_1049, type, sK747: $i > $o).
% 2.72/1.95 thf(func_def_1052, type, sK750: $i > ($i > $o) > $i).
% 2.72/1.95 thf(func_def_1053, type, sK751: $i > ($i > $o) > $i).
% 2.72/1.95 thf(func_def_1062, type, sK760: $i > $i > $o).
% 2.72/1.95 thf(func_def_1065, type, sK763: $i > $i > $i).
% 2.72/1.95 thf(func_def_1076, type, sK774: $i > $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_1077, type, sK775: $i > $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_1078, type, sK776: $i > $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_1079, type, sK777: $i > $i > $i > $i > $i).
% 2.72/1.95 thf(func_def_1083, type, sK781: $o > $i > $i > $i).
% 2.72/1.95 thf(func_def_1084, type, sK782: $i > $i).
% 2.72/1.95 thf(func_def_1095, type, ph793: !>[X0: $tType]:(X0)).
% 2.72/1.95 thf(f7389,plain,(
% 2.72/1.95 $false),
% 2.72/1.95 inference(avatar_sat_refutation,[status(thm)],[f3881,f4389,f7273,f7388])).
% 2.72/1.95 thf(f7388,plain,(
% 2.72/1.95 spl792_5),
% 2.72/1.95 inference(avatar_contradiction_clause,[status(thm)],[f7387])).
% 2.72/1.95 thf(f7387,plain,(
% 2.72/1.95 $false | spl792_5),
% 2.72/1.95 inference(trivial_inequality_removal,[status(thm)],[f7386])).
% 2.72/1.95 thf(f7386,plain,(
% 2.72/1.95 ($true != $true) | spl792_5),
% 2.72/1.95 inference(superposition,[status(thm)],[f7298,f3292])).
% 2.72/1.95 thf(f3292,plain,(
% 2.72/1.95 (binintersectLsub = $true)),
% 2.72/1.95 inference(cnf_transformation,[status(thm)],[f2181])).
% 2.72/1.95 thf(f2181,plain,(
% 2.72/1.95 (setminusERneg = $true) & (quantDeMorgan4 = $true) & (doubleComplementI1 = $true) & (doubleComplementSub2 = $true) & (subset2powerset = $true) & (ubforcartprodlem3 = $true) & (infuncsetfunc = $true) & (noeltsimpempty = $true) & (replAx = $true) & (kpairsurjEq = $true) & (notsubsetI = $true) & (binintersectSubset4 = $true) & (lamp = $true) & (dpsetconstrERa = $true) & (symdiffIneg2 = $true) & (setukpairinjR2 = $true) & (exu__Cong = $true) & (exuI3 = $true) & (emptyinunitempty = $true) & (emptysetAx = $true) & (eqimpsubset2 = $true) & (upairinpowunion = $true) & (binintersectEL = $true) & (complementTI1 = $true) & (exuI2 = $true) & (kpairp = $true) & (setukpairinjR12 = $true) & (subsetE = $true) & (cartprodsndpairEq = $true) & (bs114d = $true) & (cartprodpairmemEL = $true) & (cartprodpairsurjEq = $true) & (foundationAx = $true) & (powersetI1 = $true) & (setOfPairsIsBReln = $true) & (subPowSU = $true) & (ubforcartprodlem2 = $true) & (notequalI1 = $true) & (cartprodsndin = $true) & (complementT_lem = $true) & (dsetconstrEL = $true) & (ex1I = $true) & (iftrueProp1 = $true) & (cartprodfstpairEq = $true) & (eqbreln = $true) & (setoftrueEq = $true) & (binunionT_lem = $true) & (setminusT_lem = $true) & (vacuousDall = $true) & (sepSubset = $true) & (powersetE = $true) & (ap2p = $true) & (omega__Cong = $true) & (prop2setI = $true) & (upairsubunion = $true) & (dpsetconstrI = $true) & (binunionE = $true) & (binintersectSubset5 = $true) & (kpairiskpair = $true) & (exuEu = $true) & (setukpairIR = $true) & (notinemptyset = $true) & (ifp = $true) & (cartprodpairmemER = $true) & (setukpairinjL2 = $true) & (contrasubsetT2 = $true) & (powersetAx = $true) & (emptysetE = $true) & (theeq = $true) & (dsetconstrI = $true) & (setunionAx = $true) & (($true = (in @ sK697 @ (powerset @ sK696))) & (($true != (subset @ (setminus @ sK696 @ sK697) @ (setminus @ sK696 @ (binintersect @ sK697 @ sK698)))) & ($true = (in @ sK698 @ (powerset @ sK696))))) & (setunionsingleton = $true) & (binunionIL = $true) & (setminusSubset1 = $true) & (setminusIRneg = $true) & (ifSingleton = $true) & (setminusLsub = $true) & (omegaIndAx = $true) & (complementImpComplementIntersect = $true) & (emptyset__Cong = $true) & (notdexE = $true) & (funcext = $true) & (powerset__Cong = $true) & (subsetTrans = $true) & (dpsetconstrSub = $true) & (dpsetconstrEL1 = $true) & (binintersectTERcontra = $true) & (exuE2 = $true) & (contrasubsetT3 = $true) & (setminusEL = $true) & (powersetI = $true) & (iffalseProp2 = $true) & (setadjoinIR = $true) & (binunionEcases = $true) & (cartprodmempair1 = $true) & (setunionsingleton1 = $true) & (subsetRefl = $true) & (binintersectI = $true) & (ap2apEq1 = $true) & (theprop = $true) & (sepInPowerset = $true) & (quantDeMorgan2 = $true) & (singletonsuniq = $true) & (setextAx = $true) & (setukpairinjR = $true) & (lam2p = $true) & (upairsetIR = $true) & (emptyinPowerset = $true) & (symdiffI2 = $true) & (powersetTE1 = $true) & (ex1E1 = $true) & (symdiffIneg1 = $true) & (dsetconstr__Cong = $true) & (emptysetsubset = $true) & (contrasubsetT = $true) & (setextT = $true) & (in__Cong = $true) & (descrp = $true) & (symdiffI1 = $true) & (upairsetE = $true) & (subbreln = $true) & (beta2 = $true) & (setunion__Cong = $true) & (ksndsingleton = $true) & (quantDeMorgan3 = $true) & (funcImageSingleton = $true) & (setukpairinjL1 = $true) & (setadjoinOr = $true) & (disjointsetsI1 = $true) & (omegaSAx = $true) & (eqinunit = $true) & (nonemptyI = $true) & (binintersectSubset2 = $true) & (kfstpairEq = $true) & (exuE1 = $true) & (prop2set2propI = $true) & (singletoninpowerset = $true) & (setminusER = $true) & (uniqinunit = $true) & (dpsetconstrER = $true) & (powersetsubset = $true) & (inCongP = $true) & (binintersectER = $true) & (emptyE1 = $true) & (setunionE2 = $true) & (singletonprop = $true) & (setminusI = $true) & (symdiffE = $true) & (binintersectRsub = $true) & (inPowerset = $true) & (descr__Cong = $true) & (setadjoinAx = $true) & (emptyI = $true) & (doubleComplementE1 = $true) & (setukpairinjL = $true) & (binunionLsub = $true) & (eta2 = $true) & (ex1I2 = $true) & (funcGraphProp4 = $true) & (subsetI1 = $true) & (binunionIR = $true) & (setukpairinjR1 = $true) & (binintersectSubset3 = $true) & (funcext2 = $true) & (funcinfuncset = $true) & (binintersectSubset1 = $true) & (nonemptyI1 = $true) & (funcGraphProp3 = $true) & (notdallE = $true) & (cartprodmempaircEq = $true) & (powersetE1 = $true) & (dpsetconstrEL2 = $true) & (subsetTI = $true) & (setextsub = $true) & (setadjoinIL = $true) & (setadjoin__Cong = $true) & (setadjoinSub2 = $true) & (subsetE2 = $true) & (cartprodfstin = $true) & (funcextLem = $true) & (iftrueorfalse = $true) & (brelnall1 = $true) & (upairsetIL = $true) & (doubleComplementSub1 = $true) & (ap2apEq2 = $true) & (binintersectT_lem = $true) & (complementTnotintersectT = $true) & (contrasubsetT1 = $true) & (upairequniteq = $true) & (cartprodpairin = $true) & (iffalse = $true) & (binintersectLsub = $true) & (setext = $true) & (lam2lamEq = $true) & (setminusILneg = $true) & (wellorderingAx = $true) & (exuI1 = $true) & (apProp = $true) & (eta1 = $true) & (omega0Ax = $true) & (notequalI2 = $true) & (powersetTI1 = $true) & (powersetT_lem = $true) & (setukpairIL = $true) & (exuE3u = $true) & (emptysetimpfalse = $true) & (singletoninpowunion = $true) & (app = $true) & (setadjoinE = $true) & (ex1E2 = $true) & (ksndpairEq = $true) & (binintersectTELcontra = $true) & (cartprodmempair = $true) & (iftrue = $true) & (notinsingleton = $true) & (subsetemptysetimpeq = $true) & (setukpairinjR11 = $true) & (upairset2E = $true) & (kfstsingleton = $true) & (iffalseProp1 = $true) & (secondinupair = $true) & (iftrueProp2 = $true) & (lamProp = $true) & (setunionsingleton2 = $true) & (singletonsubset = $true) & (doubleComplementEq = $true) & (quantDeMorgan1 = $true) & (binunionRsub = $true) & (nonemptyE1 = $true) & (emptyInPowerset = $true) & (setbeta = $true) & (prop2setE = $true) & (dsetconstrER = $true) & (beta1 = $true) & (singletonsswitch = $true) & (setunionE = $true) & (funcGraphProp1 = $true) & (setadjoinSub = $true) & (nonemptyImpWitness = $true) & (brelnall2 = $true) & (upairset2IR = $true) & (eqimpsubset1 = $true) & (setminusSubset2 = $true) & (setunionI = $true) & (setminusELneg = $true) & (funcGraphProp2 = $true) & (complementTE1 = $true) & (subsetI2 = $true) & (exuE3e = $true) & (ubforcartprodlem1 = $true)),
% 2.72/1.95 inference(skolemisation,[status(esa),new_symbols(skolem,[sK696,sK697,sK698])],[f949,f2180,f2179])).
% 2.72/1.95 thf(f2179,plain,(
% 2.72/1.95 ? [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) & ? [X2 : $i] : (((subset @ (setminus @ X0 @ X1) @ (setminus @ X0 @ (binintersect @ X1 @ X2))) != $true) & ($true = (in @ X2 @ (powerset @ X0))))) => (($true = (in @ sK697 @ (powerset @ sK696))) & ? [X2 : $i] : (($true != (subset @ (setminus @ sK696 @ sK697) @ (setminus @ sK696 @ (binintersect @ sK697 @ X2)))) & ($true = (in @ X2 @ (powerset @ sK696)))))),
% 2.72/1.95 introduced(definition,[],[choice_axiom])).
% 2.72/1.95 thf(f2180,plain,(
% 2.72/1.95 ? [X2 : $i] : (($true != (subset @ (setminus @ sK696 @ sK697) @ (setminus @ sK696 @ (binintersect @ sK697 @ X2)))) & ($true = (in @ X2 @ (powerset @ sK696)))) => (($true != (subset @ (setminus @ sK696 @ sK697) @ (setminus @ sK696 @ (binintersect @ sK697 @ sK698)))) & ($true = (in @ sK698 @ (powerset @ sK696))))),
% 2.72/1.95 introduced(definition,[],[choice_axiom])).
% 2.72/1.95 thf(f949,plain,(
% 2.72/1.95 (setminusERneg = $true) & (quantDeMorgan4 = $true) & (doubleComplementI1 = $true) & (doubleComplementSub2 = $true) & (subset2powerset = $true) & (ubforcartprodlem3 = $true) & (infuncsetfunc = $true) & (noeltsimpempty = $true) & (replAx = $true) & (kpairsurjEq = $true) & (notsubsetI = $true) & (binintersectSubset4 = $true) & (lamp = $true) & (dpsetconstrERa = $true) & (symdiffIneg2 = $true) & (setukpairinjR2 = $true) & (exu__Cong = $true) & (exuI3 = $true) & (emptyinunitempty = $true) & (emptysetAx = $true) & (eqimpsubset2 = $true) & (upairinpowunion = $true) & (binintersectEL = $true) & (complementTI1 = $true) & (exuI2 = $true) & (kpairp = $true) & (setukpairinjR12 = $true) & (subsetE = $true) & (cartprodsndpairEq = $true) & (bs114d = $true) & (cartprodpairmemEL = $true) & (cartprodpairsurjEq = $true) & (foundationAx = $true) & (powersetI1 = $true) & (setOfPairsIsBReln = $true) & (subPowSU = $true) & (ubforcartprodlem2 = $true) & (notequalI1 = $true) & (cartprodsndin = $true) & (complementT_lem = $true) & (dsetconstrEL = $true) & (ex1I = $true) & (iftrueProp1 = $true) & (cartprodfstpairEq = $true) & (eqbreln = $true) & (setoftrueEq = $true) & (binunionT_lem = $true) & (setminusT_lem = $true) & (vacuousDall = $true) & (sepSubset = $true) & (powersetE = $true) & (ap2p = $true) & (omega__Cong = $true) & (prop2setI = $true) & (upairsubunion = $true) & (dpsetconstrI = $true) & (binunionE = $true) & (binintersectSubset5 = $true) & (kpairiskpair = $true) & (exuEu = $true) & (setukpairIR = $true) & (notinemptyset = $true) & (ifp = $true) & (cartprodpairmemER = $true) & (setukpairinjL2 = $true) & (contrasubsetT2 = $true) & (powersetAx = $true) & (emptysetE = $true) & (theeq = $true) & (dsetconstrI = $true) & (setunionAx = $true) & ? [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) & ? [X2 : $i] : (((subset @ (setminus @ X0 @ X1) @ (setminus @ X0 @ (binintersect @ X1 @ X2))) != $true) & ($true = (in @ X2 @ (powerset @ X0))))) & (setunionsingleton = $true) & (binunionIL = $true) & (setminusSubset1 = $true) & (setminusIRneg = $true) & (ifSingleton = $true) & (setminusLsub = $true) & (omegaIndAx = $true) & (complementImpComplementIntersect = $true) & (emptyset__Cong = $true) & (notdexE = $true) & (funcext = $true) & (powerset__Cong = $true) & (subsetTrans = $true) & (dpsetconstrSub = $true) & (dpsetconstrEL1 = $true) & (binintersectTERcontra = $true) & (exuE2 = $true) & (contrasubsetT3 = $true) & (setminusEL = $true) & (powersetI = $true) & (iffalseProp2 = $true) & (setadjoinIR = $true) & (binunionEcases = $true) & (cartprodmempair1 = $true) & (setunionsingleton1 = $true) & (subsetRefl = $true) & (binintersectI = $true) & (ap2apEq1 = $true) & (theprop = $true) & (sepInPowerset = $true) & (quantDeMorgan2 = $true) & (singletonsuniq = $true) & (setextAx = $true) & (setukpairinjR = $true) & (lam2p = $true) & (upairsetIR = $true) & (emptyinPowerset = $true) & (symdiffI2 = $true) & (powersetTE1 = $true) & (ex1E1 = $true) & (symdiffIneg1 = $true) & (dsetconstr__Cong = $true) & (emptysetsubset = $true) & (contrasubsetT = $true) & (setextT = $true) & (in__Cong = $true) & (descrp = $true) & (symdiffI1 = $true) & (upairsetE = $true) & (subbreln = $true) & (beta2 = $true) & (setunion__Cong = $true) & (ksndsingleton = $true) & (quantDeMorgan3 = $true) & (funcImageSingleton = $true) & (setukpairinjL1 = $true) & (setadjoinOr = $true) & (disjointsetsI1 = $true) & (omegaSAx = $true) & (eqinunit = $true) & (nonemptyI = $true) & (binintersectSubset2 = $true) & (kfstpairEq = $true) & (exuE1 = $true) & (prop2set2propI = $true) & (singletoninpowerset = $true) & (setminusER = $true) & (uniqinunit = $true) & (dpsetconstrER = $true) & (powersetsubset = $true) & (inCongP = $true) & (binintersectER = $true) & (emptyE1 = $true) & (setunionE2 = $true) & (singletonprop = $true) & (setminusI = $true) & (symdiffE = $true) & (binintersectRsub = $true) & (inPowerset = $true) & (descr__Cong = $true) & (setadjoinAx = $true) & (emptyI = $true) & (doubleComplementE1 = $true) & (setukpairinjL = $true) & (binunionLsub = $true) & (eta2 = $true) & (ex1I2 = $true) & (funcGraphProp4 = $true) & (subsetI1 = $true) & (binunionIR = $true) & (setukpairinjR1 = $true) & (binintersectSubset3 = $true) & (funcext2 = $true) & (funcinfuncset = $true) & (binintersectSubset1 = $true) & (nonemptyI1 = $true) & (funcGraphProp3 = $true) & (notdallE = $true) & (cartprodmempaircEq = $true) & (powersetE1 = $true) & (dpsetconstrEL2 = $true) & (subsetTI = $true) & (setextsub = $true) & (setadjoinIL = $true) & (setadjoin__Cong = $true) & (setadjoinSub2 = $true) & (subsetE2 = $true) & (cartprodfstin = $true) & (funcextLem = $true) & (iftrueorfalse = $true) & (brelnall1 = $true) & (upairsetIL = $true) & (doubleComplementSub1 = $true) & (ap2apEq2 = $true) & (binintersectT_lem = $true) & (complementTnotintersectT = $true) & (contrasubsetT1 = $true) & (upairequniteq = $true) & (cartprodpairin = $true) & (iffalse = $true) & (binintersectLsub = $true) & (setext = $true) & (lam2lamEq = $true) & (setminusILneg = $true) & (wellorderingAx = $true) & (exuI1 = $true) & (apProp = $true) & (eta1 = $true) & (omega0Ax = $true) & (notequalI2 = $true) & (powersetTI1 = $true) & (powersetT_lem = $true) & (setukpairIL = $true) & (exuE3u = $true) & (emptysetimpfalse = $true) & (singletoninpowunion = $true) & (app = $true) & (setadjoinE = $true) & (ex1E2 = $true) & (ksndpairEq = $true) & (binintersectTELcontra = $true) & (cartprodmempair = $true) & (iftrue = $true) & (notinsingleton = $true) & (subsetemptysetimpeq = $true) & (setukpairinjR11 = $true) & (upairset2E = $true) & (kfstsingleton = $true) & (iffalseProp1 = $true) & (secondinupair = $true) & (iftrueProp2 = $true) & (lamProp = $true) & (setunionsingleton2 = $true) & (singletonsubset = $true) & (doubleComplementEq = $true) & (quantDeMorgan1 = $true) & (binunionRsub = $true) & (nonemptyE1 = $true) & (emptyInPowerset = $true) & (setbeta = $true) & (prop2setE = $true) & (dsetconstrER = $true) & (beta1 = $true) & (singletonsswitch = $true) & (setunionE = $true) & (funcGraphProp1 = $true) & (setadjoinSub = $true) & (nonemptyImpWitness = $true) & (brelnall2 = $true) & (upairset2IR = $true) & (eqimpsubset1 = $true) & (setminusSubset2 = $true) & (setunionI = $true) & (setminusELneg = $true) & (funcGraphProp2 = $true) & (complementTE1 = $true) & (subsetI2 = $true) & (exuE3e = $true) & (ubforcartprodlem1 = $true)),
% 2.72/1.95 inference(flattening,[status(thm)],[f948])).
% 2.72/1.95 thf(f948,plain,(
% 2.72/1.95 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) & ? [X2 : $i] : (((subset @ (setminus @ X0 @ X1) @ (setminus @ X0 @ (binintersect @ X1 @ X2))) != $true) & ($true = (in @ X2 @ (powerset @ X0))))) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 2.72/1.95 inference(ennf_transformation,[status(thm)],[f403])).
% 2.72/1.95 thf(f403,plain,(
% 2.72/1.95 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) => ((subset @ (setminus @ X0 @ X1) @ (setminus @ X0 @ (binintersect @ X1 @ X2))) = $true)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.72/1.95 inference(fool_elimination,[status(thm)],[f402])).
% 2.72/1.95 thf(f402,plain,(
% 2.72/1.95 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => ! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X0)) => (subset @ (setminus @ X0 @ X1) @ (setminus @ X0 @ (binintersect @ X1 @ X2)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.72/1.95 inference(rectify,[status(thm)],[f256])).
% 2.72/1.95 thf(f256,negated_conjecture,(
% 2.72/1.95 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => (subset @ (setminus @ X3 @ X11) @ (setminus @ X3 @ (binintersect @ X11 @ X16)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.72/1.95 inference(negated_conjecture,[status(cth)],[f255])).
% 2.72/1.95 thf(f255,conjecture,(
% 2.72/1.95 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => (subset @ (setminus @ X3 @ X11) @ (setminus @ X3 @ (binintersect @ X11 @ X16))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.72/1.95 file('/export/starexec/sandbox/tmp/tmp.XrUd2iUsu2/DTF2THF_28816.p',complementSubsetComplementIntersect)).
% 2.72/1.95 thf(f7298,plain,(
% 2.72/1.95 (binintersectLsub != $true) | spl792_5),
% 2.72/1.95 inference(trivial_inequality_removal,[status(thm)],[f7293])).
% 2.72/1.95 thf(f7293,plain,(
% 2.72/1.95 (binintersectLsub != $true) | ($true != $true) | spl792_5),
% 2.72/1.95 inference(superposition,[status(thm)],[f3880,f3114])).
% 2.72/1.95 thf(f3114,plain,(
% 2.72/1.95 ( ! [X0 : $i,X1 : $i] : (($true = (subset @ (binintersect @ X1 @ X0) @ X1)) | (binintersectLsub != $true)) )),
% 2.72/1.95 inference(cnf_transformation,[status(thm)],[f2036])).
% 2.72/1.95 thf(f2036,plain,(
% 2.72/1.95 (! [X0 : $i,X1 : $i] : ($true = (subset @ (binintersect @ X1 @ X0) @ X1)) | (binintersectLsub != $true)) & ((binintersectLsub = $true) | ($true != (subset @ (binintersect @ sK601 @ sK600) @ sK601)))),
% 2.72/1.95 inference(skolemisation,[status(esa),new_symbols(skolem,[sK600,sK601])],[f2034,f2035])).
% 2.72/1.95 thf(f2035,plain,(
% 2.72/1.95 ? [X2 : $i,X3 : $i] : ($true != (subset @ (binintersect @ X3 @ X2) @ X3)) => ($true != (subset @ (binintersect @ sK601 @ sK600) @ sK601))),
% 2.72/1.95 introduced(definition,[],[choice_axiom])).
% 2.72/1.95 thf(f2034,plain,(
% 2.72/1.95 (! [X0 : $i,X1 : $i] : ($true = (subset @ (binintersect @ X1 @ X0) @ X1)) | (binintersectLsub != $true)) & ((binintersectLsub = $true) | ? [X2 : $i,X3 : $i] : ($true != (subset @ (binintersect @ X3 @ X2) @ X3)))),
% 2.72/1.95 inference(rectify,[status(thm)],[f2033])).
% 2.72/1.95 thf(f2033,plain,(
% 2.72/1.95 (! [X0 : $i,X1 : $i] : ($true = (subset @ (binintersect @ X1 @ X0) @ X1)) | (binintersectLsub != $true)) & ((binintersectLsub = $true) | ? [X0 : $i,X1 : $i] : ($true != (subset @ (binintersect @ X1 @ X0) @ X1)))),
% 2.72/1.95 inference(nnf_transformation,[status(thm)],[f269])).
% 2.72/1.95 thf(f269,plain,(
% 2.72/1.95 ! [X0 : $i,X1 : $i] : ($true = (subset @ (binintersect @ X1 @ X0) @ X1)) <=> (binintersectLsub = $true)),
% 2.72/1.95 inference(fool_elimination,[status(thm)],[f268])).
% 2.72/1.95 thf(f268,plain,(
% 2.72/1.95 (! [X0 : $i,X1 : $i] : (subset @ (binintersect @ X1 @ X0) @ X1) = binintersectLsub)),
% 2.72/1.95 inference(rectify,[status(thm)],[f111])).
% 2.72/1.95 thf(f111,axiom,(
% 2.72/1.95 (! [X4 : $i,X3 : $i] : (subset @ (binintersect @ X3 @ X4) @ X3) = binintersectLsub)),
% 2.72/1.95 file('/export/starexec/sandbox/tmp/tmp.XrUd2iUsu2/DTF2THF_28816.p',binintersectLsub)).
% 2.72/1.95 thf(f3880,plain,(
% 2.72/1.95 ($true != (subset @ (binintersect @ sK697 @ sK698) @ sK697)) | spl792_5),
% 2.72/1.95 inference(avatar_component_clause,[status(thm)],[f3878])).
% 2.72/1.95 thf(f3878,definition,(
% 2.72/1.95 spl792_5 <=> ($true = (subset @ (binintersect @ sK697 @ sK698) @ sK697))),
% 2.72/1.95 introduced(definition,[new_symbols(naming,[spl792_5])],[avatar_definition])).
% 2.72/1.95 thf(f7273,plain,(
% 2.72/1.95 spl792_30),
% 2.72/1.95 inference(avatar_contradiction_clause,[status(thm)],[f7272])).
% 2.72/1.95 thf(f7272,plain,(
% 2.72/1.95 $false | spl792_30),
% 2.72/1.95 inference(trivial_inequality_removal,[status(thm)],[f7261])).
% 2.72/1.95 thf(f7261,plain,(
% 2.72/1.95 ($true != $true) | spl792_30),
% 2.72/1.95 inference(superposition,[status(thm)],[f4383,f3413])).
% 2.72/1.95 thf(f3413,plain,(
% 2.72/1.95 ($true = (in @ sK698 @ (powerset @ sK696)))),
% 2.72/1.95 inference(cnf_transformation,[status(thm)],[f2181])).
% 2.72/1.95 thf(f4383,plain,(
% 2.72/1.95 ($true != (in @ sK698 @ (powerset @ sK696))) | spl792_30),
% 2.72/1.95 inference(avatar_component_clause,[status(thm)],[f4381])).
% 2.72/1.95 thf(f4381,definition,(
% 2.72/1.95 spl792_30 <=> ($true = (in @ sK698 @ (powerset @ sK696)))),
% 2.72/1.95 introduced(definition,[new_symbols(naming,[spl792_30])],[avatar_definition])).
% 2.72/1.95 thf(f4389,plain,(
% 2.72/1.95 ~spl792_30 | spl792_4),
% 2.72/1.95 inference(avatar_split_clause,[status(thm)],[f4388,f3874,f4381])).
% 2.72/1.95 thf(f3874,definition,(
% 2.72/1.95 spl792_4 <=> ($true = (in @ (binintersect @ sK697 @ sK698) @ (powerset @ sK696)))),
% 2.72/1.95 introduced(definition,[new_symbols(naming,[spl792_4])],[avatar_definition])).
% 2.72/1.95 thf(f4388,plain,(
% 2.72/1.95 ($true != (in @ sK698 @ (powerset @ sK696))) | spl792_4),
% 2.72/1.95 inference(trivial_inequality_removal,[status(thm)],[f4387])).
% 2.72/1.95 thf(f4387,plain,(
% 2.72/1.95 ($true != (in @ sK698 @ (powerset @ sK696))) | ($true != $true) | spl792_4),
% 2.72/1.95 inference(forward_demodulation,[status(thm)],[f4386,f3298])).
% 2.72/1.95 thf(f3298,plain,(
% 2.72/1.95 (binintersectT_lem = $true)),
% 2.72/1.95 inference(cnf_transformation,[status(thm)],[f2181])).
% 2.72/1.95 thf(f4386,plain,(
% 2.72/1.95 ($true != (in @ sK698 @ (powerset @ sK696))) | (binintersectT_lem != $true) | spl792_4),
% 2.72/1.95 inference(trivial_inequality_removal,[status(thm)],[f4385])).
% 2.72/1.95 thf(f4385,plain,(
% 2.72/1.95 (binintersectT_lem != $true) | ($true != (in @ sK698 @ (powerset @ sK696))) | ($true != $true) | spl792_4),
% 2.72/1.95 inference(forward_demodulation,[status(thm)],[f4377,f3415])).
% 2.72/1.95 thf(f3415,plain,(
% 2.72/1.95 ($true = (in @ sK697 @ (powerset @ sK696)))),
% 2.72/1.95 inference(cnf_transformation,[status(thm)],[f2181])).
% 2.72/1.95 thf(f4377,plain,(
% 2.72/1.95 ($true != (in @ sK697 @ (powerset @ sK696))) | (binintersectT_lem != $true) | ($true != (in @ sK698 @ (powerset @ sK696))) | spl792_4),
% 2.72/1.95 inference(trivial_inequality_removal,[status(thm)],[f4364])).
% 2.72/1.95 thf(f4364,plain,(
% 2.72/1.95 ($true != (in @ sK697 @ (powerset @ sK696))) | (binintersectT_lem != $true) | ($true != (in @ sK698 @ (powerset @ sK696))) | ($true != $true) | spl792_4),
% 2.72/1.95 inference(superposition,[status(thm)],[f3876,f3034])).
% 2.72/1.95 thf(f3034,plain,(
% 2.72/1.95 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true = (in @ (binintersect @ X1 @ X2) @ (powerset @ X0))) | ($true != (in @ X2 @ (powerset @ X0))) | (binintersectT_lem != $true) | ($true != (in @ X1 @ (powerset @ X0)))) )),
% 2.72/1.95 inference(cnf_transformation,[status(thm)],[f1937])).
% 2.72/1.95 thf(f1937,plain,(
% 2.72/1.95 (! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true = (in @ (binintersect @ X1 @ X2) @ (powerset @ X0))) | ($true != (in @ X2 @ (powerset @ X0)))) | ($true != (in @ X1 @ (powerset @ X0)))) | (binintersectT_lem != $true)) & ((binintersectT_lem = $true) | ((($true != (in @ (binintersect @ sK544 @ sK545) @ (powerset @ sK543))) & ($true = (in @ sK545 @ (powerset @ sK543)))) & ($true = (in @ sK544 @ (powerset @ sK543)))))),
% 2.72/1.95 inference(skolemisation,[status(esa),new_symbols(skolem,[sK543,sK544,sK545])],[f1934,f1936,f1935])).
% 2.72/1.95 thf(f1935,plain,(
% 2.72/1.95 ? [X3 : $i,X4 : $i] : (? [X5 : $i] : (($true != (in @ (binintersect @ X4 @ X5) @ (powerset @ X3))) & ($true = (in @ X5 @ (powerset @ X3)))) & ((in @ X4 @ (powerset @ X3)) = $true)) => (? [X5 : $i] : (($true != (in @ (binintersect @ sK544 @ X5) @ (powerset @ sK543))) & ($true = (in @ X5 @ (powerset @ sK543)))) & ($true = (in @ sK544 @ (powerset @ sK543))))),
% 2.72/1.95 introduced(definition,[],[choice_axiom])).
% 2.72/1.95 thf(f1936,plain,(
% 2.72/1.95 ? [X5 : $i] : (($true != (in @ (binintersect @ sK544 @ X5) @ (powerset @ sK543))) & ($true = (in @ X5 @ (powerset @ sK543)))) => (($true != (in @ (binintersect @ sK544 @ sK545) @ (powerset @ sK543))) & ($true = (in @ sK545 @ (powerset @ sK543))))),
% 2.72/1.95 introduced(definition,[],[choice_axiom])).
% 2.72/1.95 thf(f1934,plain,(
% 2.72/1.95 (! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true = (in @ (binintersect @ X1 @ X2) @ (powerset @ X0))) | ($true != (in @ X2 @ (powerset @ X0)))) | ($true != (in @ X1 @ (powerset @ X0)))) | (binintersectT_lem != $true)) & ((binintersectT_lem = $true) | ? [X3 : $i,X4 : $i] : (? [X5 : $i] : (($true != (in @ (binintersect @ X4 @ X5) @ (powerset @ X3))) & ($true = (in @ X5 @ (powerset @ X3)))) & ((in @ X4 @ (powerset @ X3)) = $true)))),
% 2.72/1.95 inference(rectify,[status(thm)],[f1933])).
% 2.72/1.95 thf(f1933,plain,(
% 2.72/1.95 (! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true = (in @ (binintersect @ X1 @ X2) @ (powerset @ X0))) | ($true != (in @ X2 @ (powerset @ X0)))) | ($true != (in @ X1 @ (powerset @ X0)))) | (binintersectT_lem != $true)) & ((binintersectT_lem = $true) | ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true != (in @ (binintersect @ X1 @ X2) @ (powerset @ X0))) & ($true = (in @ X2 @ (powerset @ X0)))) & ($true = (in @ X1 @ (powerset @ X0)))))),
% 2.72/1.95 inference(nnf_transformation,[status(thm)],[f1039])).
% 2.72/1.95 thf(f1039,plain,(
% 2.72/1.95 ! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true = (in @ (binintersect @ X1 @ X2) @ (powerset @ X0))) | ($true != (in @ X2 @ (powerset @ X0)))) | ($true != (in @ X1 @ (powerset @ X0)))) <=> (binintersectT_lem = $true)),
% 2.72/1.95 inference(ennf_transformation,[status(thm)],[f714])).
% 2.72/1.95 thf(f714,plain,(
% 2.72/1.95 ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) => ($true = (in @ (binintersect @ X1 @ X2) @ (powerset @ X0))))) <=> (binintersectT_lem = $true)),
% 2.72/1.95 inference(fool_elimination,[status(thm)],[f713])).
% 2.72/1.95 thf(f713,plain,(
% 2.72/1.95 (binintersectT_lem = ! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X0)) => (in @ (binintersect @ X1 @ X2) @ (powerset @ X0)))))),
% 2.72/1.95 inference(rectify,[status(thm)],[f231])).
% 2.72/1.95 thf(f231,axiom,(
% 2.72/1.95 (binintersectT_lem = ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => (in @ (binintersect @ X11 @ X16) @ (powerset @ X3)))))),
% 2.72/1.95 file('/export/starexec/sandbox/tmp/tmp.XrUd2iUsu2/DTF2THF_28816.p',binintersectT_lem)).
% 2.72/1.95 thf(f3876,plain,(
% 2.72/1.95 ($true != (in @ (binintersect @ sK697 @ sK698) @ (powerset @ sK696))) | spl792_4),
% 2.72/1.95 inference(avatar_component_clause,[status(thm)],[f3874])).
% 2.72/1.95 thf(f3881,plain,(
% 2.72/1.95 ~spl792_4 | ~spl792_5),
% 2.72/1.95 inference(avatar_split_clause,[status(thm)],[f3872,f3878,f3874])).
% 2.72/1.95 thf(f3872,plain,(
% 2.72/1.95 ($true != (in @ (binintersect @ sK697 @ sK698) @ (powerset @ sK696))) | ($true != (subset @ (binintersect @ sK697 @ sK698) @ sK697))),
% 2.72/1.95 inference(trivial_inequality_removal,[status(thm)],[f3871])).
% 2.72/1.95 thf(f3871,plain,(
% 2.72/1.95 ($true != (in @ (binintersect @ sK697 @ sK698) @ (powerset @ sK696))) | ($true != (subset @ (binintersect @ sK697 @ sK698) @ sK697)) | ($true != $true)),
% 2.72/1.95 inference(forward_demodulation,[status(thm)],[f3870,f3421])).
% 2.72/1.95 thf(f3421,plain,(
% 2.72/1.95 (contrasubsetT2 = $true)),
% 2.72/1.95 inference(cnf_transformation,[status(thm)],[f2181])).
% 2.72/1.95 thf(f3870,plain,(
% 2.72/1.95 (contrasubsetT2 != $true) | ($true != (subset @ (binintersect @ sK697 @ sK698) @ sK697)) | ($true != (in @ (binintersect @ sK697 @ sK698) @ (powerset @ sK696)))),
% 2.72/1.95 inference(trivial_inequality_removal,[status(thm)],[f3869])).
% 2.72/1.95 thf(f3869,plain,(
% 2.72/1.95 ($true != $true) | ($true != (in @ (binintersect @ sK697 @ sK698) @ (powerset @ sK696))) | (contrasubsetT2 != $true) | ($true != (subset @ (binintersect @ sK697 @ sK698) @ sK697))),
% 2.72/1.95 inference(forward_demodulation,[status(thm)],[f3868,f3415])).
% 2.72/1.95 thf(f3868,plain,(
% 2.72/1.95 ($true != (in @ (binintersect @ sK697 @ sK698) @ (powerset @ sK696))) | ($true != (in @ sK697 @ (powerset @ sK696))) | (contrasubsetT2 != $true) | ($true != (subset @ (binintersect @ sK697 @ sK698) @ sK697))),
% 2.72/1.95 inference(trivial_inequality_removal,[status(thm)],[f3859])).
% 2.72/1.95 thf(f3859,plain,(
% 2.72/1.95 ($true != (subset @ (binintersect @ sK697 @ sK698) @ sK697)) | (contrasubsetT2 != $true) | ($true != (in @ sK697 @ (powerset @ sK696))) | ($true != (in @ (binintersect @ sK697 @ sK698) @ (powerset @ sK696))) | ($true != $true)),
% 2.72/1.95 inference(superposition,[status(thm)],[f3414,f2435])).
% 2.72/1.95 thf(f2435,plain,(
% 2.72/1.95 ( ! [X3 : $i,X4 : $i,X5 : $i] : (($true = (subset @ (setminus @ X3 @ X5) @ (setminus @ X3 @ X4))) | (contrasubsetT2 != $true) | ((subset @ X4 @ X5) != $true) | ((in @ X4 @ (powerset @ X3)) != $true) | ($true != (in @ X5 @ (powerset @ X3)))) )),
% 2.72/1.95 inference(cnf_transformation,[status(thm)],[f1266])).
% 2.72/1.95 thf(f1266,plain,(
% 2.72/1.95 ((contrasubsetT2 = $true) | ((($true = (in @ sK115 @ (powerset @ sK113))) & ($true = (subset @ sK114 @ sK115)) & ($true != (subset @ (setminus @ sK113 @ sK115) @ (setminus @ sK113 @ sK114)))) & ($true = (in @ sK114 @ (powerset @ sK113))))) & (! [X3 : $i,X4 : $i] : (! [X5 : $i] : (($true != (in @ X5 @ (powerset @ X3))) | ((subset @ X4 @ X5) != $true) | ($true = (subset @ (setminus @ X3 @ X5) @ (setminus @ X3 @ X4)))) | ((in @ X4 @ (powerset @ X3)) != $true)) | (contrasubsetT2 != $true))),
% 2.72/1.95 inference(skolemisation,[status(esa),new_symbols(skolem,[sK113,sK114,sK115])],[f1263,f1265,f1264])).
% 2.72/1.95 thf(f1264,plain,(
% 2.72/1.95 ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) & ($true = (subset @ X1 @ X2)) & ($true != (subset @ (setminus @ X0 @ X2) @ (setminus @ X0 @ X1)))) & ($true = (in @ X1 @ (powerset @ X0)))) => (? [X2 : $i] : (($true = (in @ X2 @ (powerset @ sK113))) & ($true = (subset @ sK114 @ X2)) & ($true != (subset @ (setminus @ sK113 @ X2) @ (setminus @ sK113 @ sK114)))) & ($true = (in @ sK114 @ (powerset @ sK113))))),
% 2.72/1.95 introduced(definition,[],[choice_axiom])).
% 2.72/1.95 thf(f1265,plain,(
% 2.72/1.95 ? [X2 : $i] : (($true = (in @ X2 @ (powerset @ sK113))) & ($true = (subset @ sK114 @ X2)) & ($true != (subset @ (setminus @ sK113 @ X2) @ (setminus @ sK113 @ sK114)))) => (($true = (in @ sK115 @ (powerset @ sK113))) & ($true = (subset @ sK114 @ sK115)) & ($true != (subset @ (setminus @ sK113 @ sK115) @ (setminus @ sK113 @ sK114))))),
% 2.72/1.95 introduced(definition,[],[choice_axiom])).
% 2.72/1.95 thf(f1263,plain,(
% 2.72/1.95 ((contrasubsetT2 = $true) | ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) & ($true = (subset @ X1 @ X2)) & ($true != (subset @ (setminus @ X0 @ X2) @ (setminus @ X0 @ X1)))) & ($true = (in @ X1 @ (powerset @ X0))))) & (! [X3 : $i,X4 : $i] : (! [X5 : $i] : (($true != (in @ X5 @ (powerset @ X3))) | ((subset @ X4 @ X5) != $true) | ($true = (subset @ (setminus @ X3 @ X5) @ (setminus @ X3 @ X4)))) | ((in @ X4 @ (powerset @ X3)) != $true)) | (contrasubsetT2 != $true))),
% 2.72/1.95 inference(rectify,[status(thm)],[f1262])).
% 2.72/1.95 thf(f1262,plain,(
% 2.72/1.95 ((contrasubsetT2 = $true) | ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) & ($true = (subset @ X1 @ X2)) & ($true != (subset @ (setminus @ X0 @ X2) @ (setminus @ X0 @ X1)))) & ($true = (in @ X1 @ (powerset @ X0))))) & (! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true != (in @ X2 @ (powerset @ X0))) | ($true != (subset @ X1 @ X2)) | ($true = (subset @ (setminus @ X0 @ X2) @ (setminus @ X0 @ X1)))) | ($true != (in @ X1 @ (powerset @ X0)))) | (contrasubsetT2 != $true))),
% 2.72/1.95 inference(nnf_transformation,[status(thm)],[f1014])).
% 2.72/1.95 thf(f1014,plain,(
% 2.72/1.95 (contrasubsetT2 = $true) <=> ! [X0 : $i,X1 : $i] : (! [X2 : $i] : (($true != (in @ X2 @ (powerset @ X0))) | ($true != (subset @ X1 @ X2)) | ($true = (subset @ (setminus @ X0 @ X2) @ (setminus @ X0 @ X1)))) | ($true != (in @ X1 @ (powerset @ X0))))),
% 2.72/1.95 inference(flattening,[status(thm)],[f1013])).
% 2.72/1.95 thf(f1013,plain,(
% 2.72/1.95 (contrasubsetT2 = $true) <=> ! [X0 : $i,X1 : $i] : (! [X2 : $i] : ((($true = (subset @ (setminus @ X0 @ X2) @ (setminus @ X0 @ X1))) | ($true != (subset @ X1 @ X2))) | ($true != (in @ X2 @ (powerset @ X0)))) | ($true != (in @ X1 @ (powerset @ X0))))),
% 2.72/1.95 inference(ennf_transformation,[status(thm)],[f646])).
% 2.72/1.95 thf(f646,plain,(
% 2.72/1.95 (contrasubsetT2 = $true) <=> ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) => (($true = (subset @ X1 @ X2)) => ($true = (subset @ (setminus @ X0 @ X2) @ (setminus @ X0 @ X1))))))),
% 2.72/1.95 inference(fool_elimination,[status(thm)],[f645])).
% 2.72/1.95 thf(f645,plain,(
% 2.72/1.95 (! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X0)) => ((subset @ X1 @ X2) => (subset @ (setminus @ X0 @ X2) @ (setminus @ X0 @ X1))))) = contrasubsetT2)),
% 2.72/1.95 inference(rectify,[status(thm)],[f246])).
% 2.72/1.95 thf(f246,axiom,(
% 2.72/1.95 (! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ((subset @ X11 @ X16) => (subset @ (setminus @ X3 @ X16) @ (setminus @ X3 @ X11))))) = contrasubsetT2)),
% 2.72/1.95 file('/export/starexec/sandbox/tmp/tmp.XrUd2iUsu2/DTF2THF_28816.p',contrasubsetT2)).
% 2.72/1.95 thf(f3414,plain,(
% 2.72/1.95 ($true != (subset @ (setminus @ sK696 @ sK697) @ (setminus @ sK696 @ (binintersect @ sK697 @ sK698))))),
% 2.72/1.95 inference(cnf_transformation,[status(thm)],[f2181])).
% 2.72/1.95 % SZS output end Proof for DTF2THF_28816
% 2.72/1.95 % (28941)------------------------------
% 2.72/1.95 % (28941)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.95 % (28941)Termination reason: Refutation
% 2.72/1.95
% 2.72/1.95 % (28941)Memory used [KB]: 11641
% 2.72/1.95 % (28941)Time elapsed: 0.366 s
% 2.72/1.95 % (28941)Instructions burned: 695 (million)
% 2.72/1.95 % (28941)------------------------------
% 2.72/1.95 % (28941)------------------------------
% 2.72/1.95 % (28917)Success in time 0.464 s
%------------------------------------------------------------------------------