%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU720^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n011.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 09:36:39 AM UTC 2026
% Result : Theorem 2.20s 1.66s
% Output : Refutation 2.20s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 7
% Syntax : Number of formulae : 41 ( 9 unt; 0 typ; 2 def)
% Number of atoms : 4674 (1288 equ; 0 cnn)
% Maximal formula atoms : 240 ( 114 avg)
% Number of connectives : 2329 ( 65 ~; 34 |; 740 &; 278 @)
% ( 6 <=>;1206 =>; 0 <=; 0 <~>)
% Maximal formula depth : 248 ( 51 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 435 ( 432 usr; 246 con; 0-4 aty)
% Number of variables : 65 ( 0 ^; 45 !; 20 ?; 65 :)
% 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_290,type,
sP2: $o > $i > $i > $i > $o ).
thf(func_def_291,type,
sP3: $i > $i > $o ).
thf(func_def_292,type,
sP4: $i > $o ).
thf(func_def_293,type,
sP5: $i > $i > $o ).
thf(func_def_294,type,
sP6: $i > $i > $o ).
thf(func_def_296,type,
sP8: $i > $i > $i > $i > $o ).
thf(func_def_299,type,
sP11: $i > $i > $i > $o ).
thf(func_def_304,type,
sK16: $i > $o ).
thf(func_def_305,type,
sK17: $i > $o ).
thf(func_def_315,type,
sK27: $i > $i ).
thf(func_def_318,type,
sK30: $i > ( $i > $i ) > $i > $i ).
thf(func_def_340,type,
sK52: $i > $i > $o ).
thf(func_def_370,type,
sK82: $i > $i > $o ).
thf(func_def_378,type,
sK90: $i > $i ).
thf(func_def_382,type,
sK94: $i > $i > $o ).
thf(func_def_389,type,
sK101: $i > $o ).
thf(func_def_413,type,
sK125: $i > $o ).
thf(func_def_415,type,
sK127: ( $i > $o ) > $i > $i ).
thf(func_def_419,type,
sK131: $i > $o ).
thf(func_def_422,type,
sK134: $i > ( $i > $o ) > $i ).
thf(func_def_423,type,
sK135: $i > ( $i > $o ) > $i ).
thf(func_def_432,type,
sK144: $i > $i > ( $i > $i ) > $i ).
thf(func_def_433,type,
sK145: $i > $i ).
thf(func_def_439,type,
sK151: $i > $o ).
thf(func_def_451,type,
sK163: $i > $i > $i > $i > $i ).
thf(func_def_452,type,
sK164: $i > $i > $i ).
thf(func_def_455,type,
sK167: $i > $i ).
thf(func_def_457,type,
sK169: $i > $i ).
thf(func_def_466,type,
sK178: $i > $o ).
thf(func_def_468,type,
sK180: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_469,type,
sK181: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_470,type,
sK182: $i > $i > $i ).
thf(func_def_471,type,
sK183: $i > $i > $i ).
thf(func_def_479,type,
sK191: $i > $i ).
thf(func_def_482,type,
sK194: $i > $i > $i ).
thf(func_def_485,type,
sK197: ( $i > $o ) > $i ).
thf(func_def_486,type,
sK198: $i > $o ).
thf(func_def_487,type,
sK199: $i > $i ).
thf(func_def_496,type,
sK208: $i > $o ).
thf(func_def_500,type,
sK212: $i > $o ).
thf(func_def_503,type,
sK215: $i > ( $i > $o ) > $i > $i ).
thf(func_def_505,type,
sK217: $i > $o ).
thf(func_def_520,type,
sK232: ( $i > $i ) > $i > $i > $i ).
thf(func_def_523,type,
sK235: $i > $i ).
thf(func_def_527,type,
sK239: $i > $i > $i > $i ).
thf(func_def_552,type,
sK264: $i > $o ).
thf(func_def_572,type,
sK284: $i > $o ).
thf(func_def_574,type,
sK286: ( $i > $o ) > $i > $i ).
thf(func_def_584,type,
sK296: $i > $i > $i > $i ).
thf(func_def_591,type,
sK303: $i > ( $i > $o ) > $i ).
thf(func_def_592,type,
sK304: $i > $o ).
thf(func_def_603,type,
sK315: $i > $i > $i ).
thf(func_def_610,type,
sK322: $i > $i > $i > $i > $i ).
thf(func_def_615,type,
sK327: $i > $i > $i > $i ).
thf(func_def_616,type,
sK328: $i > $i > $i > $i ).
thf(func_def_639,type,
sK351: $i > $o ).
thf(func_def_660,type,
sK372: ( $i > $o ) > $i ).
thf(func_def_661,type,
sK373: $i > $o ).
thf(func_def_662,type,
sK374: $i > $i ).
thf(func_def_669,type,
sK381: $i > $i ).
thf(func_def_670,type,
sK382: ( $i > $i ) > $i > $i > $i ).
thf(func_def_684,type,
sK396: $i > $o ).
thf(func_def_687,type,
sK399: $i > $i ).
thf(func_def_703,type,
sK415: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_704,type,
sK416: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_705,type,
sK417: $i > $o ).
thf(func_def_706,type,
sK418: $i > $o ).
thf(func_def_707,type,
sK419: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_708,type,
sK420: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_712,type,
sK424: $i > $o ).
thf(func_def_717,type,
sK429: $i > $i > $i ).
thf(func_def_718,type,
sK430: $i > $i > $i ).
thf(func_def_719,type,
sK431: $i > $i > $i ).
thf(func_def_720,type,
sK432: $i > $i > $i ).
thf(func_def_721,type,
sK433: $i > $i > $i > $i ).
thf(func_def_722,type,
sK434: $i > $i ).
thf(func_def_723,type,
sK435: $i > $i ).
thf(func_def_724,type,
sK436: $i > $i ).
thf(func_def_725,type,
sK437: $i > $i ).
thf(func_def_726,type,
sK438: $i > $i > $i > $i ).
thf(func_def_727,type,
sK439: $i > $i > $i > $i ).
thf(func_def_728,type,
sK440: $i > $i > $i ).
thf(func_def_729,type,
sK441: $i > $i > $i ).
thf(func_def_730,type,
sK442: $i > $i > $i ).
thf(func_def_732,type,
sK444: $i > $i ).
thf(func_def_733,type,
sK445: $i > $i ).
thf(func_def_734,type,
sK446: $i > $i ).
thf(func_def_735,type,
sK447: $i > $i > $i ).
thf(func_def_736,type,
sK448: $i > $i > $i ).
thf(func_def_741,type,
sK453: $i > $o ).
thf(func_def_745,type,
sK457: $i > $i > $i > $i > $i ).
thf(func_def_746,type,
sK458: $i > $i > $i > $i > $i ).
thf(func_def_757,type,
sK469: $i > $o ).
thf(func_def_760,type,
sK472: $i > ( $i > $o ) > $i ).
thf(func_def_761,type,
sK473: $i > $o ).
thf(func_def_763,type,
sK475: $i > ( $i > $o ) > $i ).
thf(func_def_764,type,
sK476: $i > $o ).
thf(func_def_765,type,
sK477: ( $i > $o ) > $i ).
thf(func_def_778,type,
sK490: $i > $i > $i ).
thf(func_def_796,type,
sK508: $i > $i > ( $i > $i ) > $i ).
thf(func_def_797,type,
sK509: $i > $i ).
thf(func_def_813,type,
sK525: $i > $i > $o ).
thf(func_def_819,type,
sK531: $i > ( $i > $o ) > $i ).
thf(func_def_820,type,
sK532: $i > $o ).
thf(func_def_829,type,
sK541: $i > $i > $i > $i > $i ).
thf(func_def_856,type,
sK568: $i > $i ).
thf(func_def_860,type,
sK572: $o > $i > $i > $i ).
thf(func_def_871,type,
sK583: $i > $o ).
thf(func_def_886,type,
sK598: $i > $i ).
thf(func_def_887,type,
sK599: ( $i > $i ) > $i > $i > $i ).
thf(func_def_890,type,
sK602: $i > $i > $i ).
thf(func_def_897,type,
sK609: $i > $o ).
thf(func_def_900,type,
sK612: $i > ( $i > $o ) > $i ).
thf(func_def_901,type,
sK613: $i > ( $i > $o ) > $i ).
thf(func_def_916,type,
sK628: $i > $o ).
thf(func_def_921,type,
sK633: $i > $i > $i > $i > $i ).
thf(func_def_922,type,
sK634: $i > $i > $i > $i > $i ).
thf(func_def_923,type,
sK635: $i > $i > $i > $i > $i ).
thf(func_def_924,type,
sK636: $i > $i > $i > $i > $i ).
thf(func_def_931,type,
sK643: $i > $i > $i ).
thf(func_def_934,type,
sK646: $i > $i > $o ).
thf(func_def_937,type,
sK649: $i > $o ).
thf(func_def_939,type,
sK651: $i > ( $i > $o ) > $i ).
thf(func_def_944,type,
sK656: $i > $o ).
thf(func_def_948,type,
sK660: $i > $i > $o ).
thf(func_def_950,type,
sK662: $i > $i ).
thf(func_def_951,type,
sK663: $i > $i ).
thf(func_def_952,type,
sK664: $i > ( $i > $i > $o ) > $i ).
thf(func_def_953,type,
sK665: $i > ( $i > $i > $o ) > $i ).
thf(func_def_954,type,
sK666: $i > $i > ( $i > $i > $o ) > $i ).
thf(func_def_971,type,
sK683: $i > $i > $i ).
thf(func_def_972,type,
sK684: $i > $i > $o ).
thf(func_def_986,type,
sK698: $i > $i > $o ).
thf(func_def_990,type,
sK702: $i > $i > $i > $i ).
thf(func_def_994,type,
sK706: $i > $i > $i > $i ).
thf(func_def_995,type,
sK707: ( $i > $o ) > $i ).
thf(func_def_996,type,
sK708: ( $i > $o ) > $i ).
thf(func_def_997,type,
sK709: $i > $o ).
thf(func_def_1001,type,
sK713: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_1002,type,
sK714: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_1005,type,
sK717: $i > $o ).
thf(func_def_1006,type,
sK718: $i > $o ).
thf(func_def_1007,type,
sK719: $i > ( $i > $o ) > $i ).
thf(func_def_1008,type,
sK720: $i > $o ).
thf(func_def_1011,type,
sK723: $i > $i ).
thf(func_def_1014,type,
sK726: $i > ( $i > $o ) > $i ).
thf(func_def_1015,type,
sK727: $i > $o ).
thf(func_def_1031,type,
sK743: $i > $o ).
thf(func_def_1032,type,
sK744: $i > $o ).
thf(func_def_1033,type,
sK745: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1034,type,
sK746: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_1036,type,
ph748:
!>[X0: $tType] : X0 ).
thf(f3729,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f3667,f3692,f3715]) ).
thf(f3715,plain,
~ spl747_6,
inference(avatar_contradiction_clause,[status(thm)],[f3714]) ).
thf(f3714,plain,
( $false
| ~ spl747_6 ),
inference(trivial_inequality_removal,[status(thm)],[f3709]) ).
thf(f3709,plain,
( ( $true != $true )
| ~ spl747_6 ),
inference(superposition,[status(thm)],[f2382,f3647]) ).
thf(f3647,plain,
( ( ( in @ sK130 @ sK128 )
= $true )
| ~ spl747_6 ),
inference(avatar_component_clause,[status(thm)],[f3645]) ).
thf(f3645,definition,
( spl747_6
<=> ( ( in @ sK130 @ sK128 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl747_6])],[avatar_definition]) ).
thf(f2382,plain,
( ( in @ sK130 @ sK128 )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1229]) ).
thf(f1229,plain,
( ( exuI2 = $true )
& ( app = $true )
& ( prop2setE = $true )
& ( setoftrueEq = $true )
& ( powersetTE1 = $true )
& ( exuI1 = $true )
& ( notinemptyset = $true )
& ( funcext = $true )
& ( cartprodpairmemEL = $true )
& ( binintersectSubset2 = $true )
& ( setadjoinIR = $true )
& ( iftrueProp1 = $true )
& ( iftrueProp2 = $true )
& ( omega0Ax = $true )
& ( emptysetimpfalse = $true )
& ( binintersectI = $true )
& ( setunionsingleton1 = $true )
& ( cartprodmempair1 = $true )
& ( cartprodsndpairEq = $true )
& ( exuE3e = $true )
& ( prop2setI = $true )
& ( kfstsingleton = $true )
& ( singletoninpowunion = $true )
& ( binintersectLsub = $true )
& ( nonemptyI1 = $true )
& ( omegaIndAx = $true )
& ( funcImageSingleton = $true )
& ( powersetE1 = $true )
& ( nonemptyE1 = $true )
& ( cartprodmempair = $true )
& ( setadjoin__Cong = $true )
& ( setextAx = $true )
& ( ex1E1 = $true )
& ( funcGraphProp4 = $true )
& ( notdallE = $true )
& ( noeltsimpempty = $true )
& ( setminusIRneg = $true )
& ( upairsetIR = $true )
& ( funcextLem = $true )
& ( subbreln = $true )
& ( funcGraphProp1 = $true )
& ( setunionE2 = $true )
& ( quantDeMorgan1 = $true )
& ( setukpairinjL1 = $true )
& ( dpsetconstrEL2 = $true )
& ( emptyset__Cong = $true )
& ( subsetTrans = $true )
& ( ex1I2 = $true )
& ( powersetTI1 = $true )
& ( singletonsubset = $true )
& ( notequalI1 = $true )
& ( theeq = $true )
& ( setukpairinjR12 = $true )
& ( ifp = $true )
& ( setunionAx = $true )
& ( cartprodfstin = $true )
& ( setbeta = $true )
& ( apProp = $true )
& ( subset2powerset = $true )
& ( setminusI = $true )
& ( lam2lamEq = $true )
& ( setminusELneg = $true )
& ( powersetI = $true )
& ( nonemptyI = $true )
& ( iftrue = $true )
& ( singletoninpowerset = $true )
& ( singletonprop = $true )
& ( notsubsetI = $true )
& ( eqbreln = $true )
& ( funcGraphProp2 = $true )
& ( binintersectSubset3 = $true )
& ( exuI3 = $true )
& ( emptyI = $true )
& ( powersetAx = $true )
& ( symdiffE = $true )
& ( wellorderingAx = $true )
& ( ap2apEq1 = $true )
& ( setminusERneg = $true )
& ( descrp = $true )
& ( exuE1 = $true )
& ( setadjoinE = $true )
& ( notdexE = $true )
& ( setminusILneg = $true )
& ( powersetsubset = $true )
& ( nonemptyImpWitness = $true )
& ( subsetTI = $true )
& ( setadjoinAx = $true )
& ( setext = $true )
& ( funcinfuncset = $true )
& ( setminusSubset1 = $true )
& ( singletonsuniq = $true )
& ( emptyInPowerset = $true )
& ( lamp = $true )
& ( disjointsetsI1 = $true )
& ( exuE2 = $true )
& ( binintersectEL = $true )
& ( subsetE = $true )
& ( iffalseProp1 = $true )
& ( sepInPowerset = $true )
& ( binunionE = $true )
& ( setminusLsub = $true )
& ( setunionsingleton = $true )
& ( binunionIL = $true )
& ( inCongP = $true )
& ( dsetconstrER = $true )
& ( binintersectER = $true )
& ( subPowSU = $true )
& ( sepSubset = $true )
& ( setextT = $true )
& ( prop2set2propI = $true )
& ( theprop = $true )
& ( binunionLsub = $true )
& ( eta2 = $true )
& ( ap2apEq2 = $true )
& ( dpsetconstrEL1 = $true )
& ( setadjoinIL = $true )
& ( setukpairinjL2 = $true )
& ( emptyinPowerset = $true )
& ( cartprodpairin = $true )
& ( ap2p = $true )
& ( omega__Cong = $true )
& ( setminusT_lem = $true )
& ( iffalse = $true )
& ( uniqinunit = $true )
& ( exu__Cong = $true )
& ( infuncsetfunc = $true )
& ( cartprodpairsurjEq = $true )
& ( upairinpowunion = $true )
& ( quantDeMorgan2 = $true )
& ( dpsetconstrERa = $true )
& ( setukpairinjR11 = $true )
& ( ksndsingleton = $true )
& ( omegaSAx = $true )
& ( symdiffIneg2 = $true )
& ( inPowerset = $true )
& ( setadjoinSub = $true )
& ( eta1 = $true )
& ( symdiffI1 = $true )
& ( beta1 = $true )
& ( setadjoinSub2 = $true )
& ( eqimpsubset2 = $true )
& ( funcGraphProp3 = $true )
& ( foundationAx = $true )
& ( setukpairinjR = $true )
& ( cartprodsndin = $true )
& ( upairsetE = $true )
& ( upairequniteq = $true )
& ( dpsetconstrER = $true )
& ( ubforcartprodlem3 = $true )
& ( setunionI = $true )
& ( eqimpsubset1 = $true )
& ( funcext2 = $true )
& ( cartprodmempaircEq = $true )
& ( iffalseProp2 = $true )
& ( notinsingleton = $true )
& ( setukpairIL = $true )
& ( kpairsurjEq = $true )
& ( kpairp = $true )
& ( subsetRefl = $true )
& ( binintersectSubset5 = $true )
& ( setukpairIR = $true )
& ( emptyinunitempty = $true )
& ( in__Cong = $true )
& ( secondinupair = $true )
& ( setminusSubset2 = $true )
& ( quantDeMorgan3 = $true )
& ( binintersectT_lem = $true )
& ( binintersectSubset1 = $true )
& ( upairsetIL = $true )
& ( dsetconstr__Cong = $true )
& ( setminusER = $true )
& ( eqinunit = $true )
& ( beta2 = $true )
& ( setunion__Cong = $true )
& ( setukpairinjR2 = $true )
& ( setunionE = $true )
& ( ubforcartprodlem2 = $true )
& ( powersetI1 = $true )
& ( ifSingleton = $true )
& ( ( in @ sK130 @ ( setminus @ sK129 @ sK128 ) )
!= $true )
& ( ( in @ sK130 @ sK128 )
!= $true )
& ( ( in @ sK130 @ sK129 )
= $true )
& ( ( in @ sK128 @ ( powerset @ sK129 ) )
= $true )
& ( upairset2IR = $true )
& ( upairsubunion = $true )
& ( cartprodfstpairEq = $true )
& ( subsetE2 = $true )
& ( brelnall2 = $true )
& ( setminusEL = $true )
& ( symdiffI2 = $true )
& ( subsetI2 = $true )
& ( subsetI1 = $true )
& ( brelnall1 = $true )
& ( complementTI1 = $true )
& ( emptysetAx = $true )
& ( setunionsingleton2 = $true )
& ( lam2p = $true )
& ( ex1E2 = $true )
& ( bs114d = $true )
& ( binunionIR = $true )
& ( kpairiskpair = $true )
& ( ksndpairEq = $true )
& ( binunionEcases = $true )
& ( dpsetconstrI = $true )
& ( binintersectSubset4 = $true )
& ( notequalI2 = $true )
& ( vacuousDall = $true )
& ( powersetE = $true )
& ( iftrueorfalse = $true )
& ( setextsub = $true )
& ( ubforcartprodlem1 = $true )
& ( emptysetE = $true )
& ( cartprodpairmemER = $true )
& ( binunionRsub = $true )
& ( powerset__Cong = $true )
& ( quantDeMorgan4 = $true )
& ( setadjoinOr = $true )
& ( dpsetconstrSub = $true )
& ( powersetT_lem = $true )
& ( kfstpairEq = $true )
& ( complementT_lem = $true )
& ( binintersectRsub = $true )
& ( binunionT_lem = $true )
& ( symdiffIneg1 = $true )
& ( setOfPairsIsBReln = $true )
& ( singletonsswitch = $true )
& ( exuE3u = $true )
& ( lamProp = $true )
& ( dsetconstrI = $true )
& ( subsetemptysetimpeq = $true )
& ( upairset2E = $true )
& ( exuEu = $true )
& ( emptysetsubset = $true )
& ( descr__Cong = $true )
& ( ex1I = $true )
& ( replAx = $true )
& ( emptyE1 = $true )
& ( dsetconstrEL = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjL = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK128,sK129,sK130])],[f862,f1228,f1227]) ).
thf(f1227,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
!= ( in @ X2 @ ( setminus @ X1 @ X0 ) ) )
& ( ( in @ X2 @ X0 )
!= $true )
& ( ( in @ X2 @ X1 )
= $true ) )
& ( ( in @ X0 @ ( powerset @ X1 ) )
= $true ) )
=> ( ? [X2: $i] :
( ( ( in @ X2 @ ( setminus @ sK129 @ sK128 ) )
!= $true )
& ( ( in @ X2 @ sK128 )
!= $true )
& ( ( in @ X2 @ sK129 )
= $true ) )
& ( ( in @ sK128 @ ( powerset @ sK129 ) )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1228,plain,
( ? [X2: $i] :
( ( ( in @ X2 @ ( setminus @ sK129 @ sK128 ) )
!= $true )
& ( ( in @ X2 @ sK128 )
!= $true )
& ( ( in @ X2 @ sK129 )
= $true ) )
=> ( ( ( in @ sK130 @ ( setminus @ sK129 @ sK128 ) )
!= $true )
& ( ( in @ sK130 @ sK128 )
!= $true )
& ( ( in @ sK130 @ sK129 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f862,plain,
( ( exuI2 = $true )
& ( app = $true )
& ( prop2setE = $true )
& ( setoftrueEq = $true )
& ( powersetTE1 = $true )
& ( exuI1 = $true )
& ( notinemptyset = $true )
& ( funcext = $true )
& ( cartprodpairmemEL = $true )
& ( binintersectSubset2 = $true )
& ( setadjoinIR = $true )
& ( iftrueProp1 = $true )
& ( iftrueProp2 = $true )
& ( omega0Ax = $true )
& ( emptysetimpfalse = $true )
& ( binintersectI = $true )
& ( setunionsingleton1 = $true )
& ( cartprodmempair1 = $true )
& ( cartprodsndpairEq = $true )
& ( exuE3e = $true )
& ( prop2setI = $true )
& ( kfstsingleton = $true )
& ( singletoninpowunion = $true )
& ( binintersectLsub = $true )
& ( nonemptyI1 = $true )
& ( omegaIndAx = $true )
& ( funcImageSingleton = $true )
& ( powersetE1 = $true )
& ( nonemptyE1 = $true )
& ( cartprodmempair = $true )
& ( setadjoin__Cong = $true )
& ( setextAx = $true )
& ( ex1E1 = $true )
& ( funcGraphProp4 = $true )
& ( notdallE = $true )
& ( noeltsimpempty = $true )
& ( setminusIRneg = $true )
& ( upairsetIR = $true )
& ( funcextLem = $true )
& ( subbreln = $true )
& ( funcGraphProp1 = $true )
& ( setunionE2 = $true )
& ( quantDeMorgan1 = $true )
& ( setukpairinjL1 = $true )
& ( dpsetconstrEL2 = $true )
& ( emptyset__Cong = $true )
& ( subsetTrans = $true )
& ( ex1I2 = $true )
& ( powersetTI1 = $true )
& ( singletonsubset = $true )
& ( notequalI1 = $true )
& ( theeq = $true )
& ( setukpairinjR12 = $true )
& ( ifp = $true )
& ( setunionAx = $true )
& ( cartprodfstin = $true )
& ( setbeta = $true )
& ( apProp = $true )
& ( subset2powerset = $true )
& ( setminusI = $true )
& ( lam2lamEq = $true )
& ( setminusELneg = $true )
& ( powersetI = $true )
& ( nonemptyI = $true )
& ( iftrue = $true )
& ( singletoninpowerset = $true )
& ( singletonprop = $true )
& ( notsubsetI = $true )
& ( eqbreln = $true )
& ( funcGraphProp2 = $true )
& ( binintersectSubset3 = $true )
& ( exuI3 = $true )
& ( emptyI = $true )
& ( powersetAx = $true )
& ( symdiffE = $true )
& ( wellorderingAx = $true )
& ( ap2apEq1 = $true )
& ( setminusERneg = $true )
& ( descrp = $true )
& ( exuE1 = $true )
& ( setadjoinE = $true )
& ( notdexE = $true )
& ( setminusILneg = $true )
& ( powersetsubset = $true )
& ( nonemptyImpWitness = $true )
& ( subsetTI = $true )
& ( setadjoinAx = $true )
& ( setext = $true )
& ( funcinfuncset = $true )
& ( setminusSubset1 = $true )
& ( singletonsuniq = $true )
& ( emptyInPowerset = $true )
& ( lamp = $true )
& ( disjointsetsI1 = $true )
& ( exuE2 = $true )
& ( binintersectEL = $true )
& ( subsetE = $true )
& ( iffalseProp1 = $true )
& ( sepInPowerset = $true )
& ( binunionE = $true )
& ( setminusLsub = $true )
& ( setunionsingleton = $true )
& ( binunionIL = $true )
& ( inCongP = $true )
& ( dsetconstrER = $true )
& ( binintersectER = $true )
& ( subPowSU = $true )
& ( sepSubset = $true )
& ( setextT = $true )
& ( prop2set2propI = $true )
& ( theprop = $true )
& ( binunionLsub = $true )
& ( eta2 = $true )
& ( ap2apEq2 = $true )
& ( dpsetconstrEL1 = $true )
& ( setadjoinIL = $true )
& ( setukpairinjL2 = $true )
& ( emptyinPowerset = $true )
& ( cartprodpairin = $true )
& ( ap2p = $true )
& ( omega__Cong = $true )
& ( setminusT_lem = $true )
& ( iffalse = $true )
& ( uniqinunit = $true )
& ( exu__Cong = $true )
& ( infuncsetfunc = $true )
& ( cartprodpairsurjEq = $true )
& ( upairinpowunion = $true )
& ( quantDeMorgan2 = $true )
& ( dpsetconstrERa = $true )
& ( setukpairinjR11 = $true )
& ( ksndsingleton = $true )
& ( omegaSAx = $true )
& ( symdiffIneg2 = $true )
& ( inPowerset = $true )
& ( setadjoinSub = $true )
& ( eta1 = $true )
& ( symdiffI1 = $true )
& ( beta1 = $true )
& ( setadjoinSub2 = $true )
& ( eqimpsubset2 = $true )
& ( funcGraphProp3 = $true )
& ( foundationAx = $true )
& ( setukpairinjR = $true )
& ( cartprodsndin = $true )
& ( upairsetE = $true )
& ( upairequniteq = $true )
& ( dpsetconstrER = $true )
& ( ubforcartprodlem3 = $true )
& ( setunionI = $true )
& ( eqimpsubset1 = $true )
& ( funcext2 = $true )
& ( cartprodmempaircEq = $true )
& ( iffalseProp2 = $true )
& ( notinsingleton = $true )
& ( setukpairIL = $true )
& ( kpairsurjEq = $true )
& ( kpairp = $true )
& ( subsetRefl = $true )
& ( binintersectSubset5 = $true )
& ( setukpairIR = $true )
& ( emptyinunitempty = $true )
& ( in__Cong = $true )
& ( secondinupair = $true )
& ( setminusSubset2 = $true )
& ( quantDeMorgan3 = $true )
& ( binintersectT_lem = $true )
& ( binintersectSubset1 = $true )
& ( upairsetIL = $true )
& ( dsetconstr__Cong = $true )
& ( setminusER = $true )
& ( eqinunit = $true )
& ( beta2 = $true )
& ( setunion__Cong = $true )
& ( setukpairinjR2 = $true )
& ( setunionE = $true )
& ( ubforcartprodlem2 = $true )
& ( powersetI1 = $true )
& ( ifSingleton = $true )
& ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( $true
!= ( in @ X2 @ ( setminus @ X1 @ X0 ) ) )
& ( ( in @ X2 @ X0 )
!= $true )
& ( ( in @ X2 @ X1 )
= $true ) )
& ( ( in @ X0 @ ( powerset @ X1 ) )
= $true ) )
& ( upairset2IR = $true )
& ( upairsubunion = $true )
& ( cartprodfstpairEq = $true )
& ( subsetE2 = $true )
& ( brelnall2 = $true )
& ( setminusEL = $true )
& ( symdiffI2 = $true )
& ( subsetI2 = $true )
& ( subsetI1 = $true )
& ( brelnall1 = $true )
& ( complementTI1 = $true )
& ( emptysetAx = $true )
& ( setunionsingleton2 = $true )
& ( lam2p = $true )
& ( ex1E2 = $true )
& ( bs114d = $true )
& ( binunionIR = $true )
& ( kpairiskpair = $true )
& ( ksndpairEq = $true )
& ( binunionEcases = $true )
& ( dpsetconstrI = $true )
& ( binintersectSubset4 = $true )
& ( notequalI2 = $true )
& ( vacuousDall = $true )
& ( powersetE = $true )
& ( iftrueorfalse = $true )
& ( setextsub = $true )
& ( ubforcartprodlem1 = $true )
& ( emptysetE = $true )
& ( cartprodpairmemER = $true )
& ( binunionRsub = $true )
& ( powerset__Cong = $true )
& ( quantDeMorgan4 = $true )
& ( setadjoinOr = $true )
& ( dpsetconstrSub = $true )
& ( powersetT_lem = $true )
& ( kfstpairEq = $true )
& ( complementT_lem = $true )
& ( binintersectRsub = $true )
& ( binunionT_lem = $true )
& ( symdiffIneg1 = $true )
& ( setOfPairsIsBReln = $true )
& ( singletonsswitch = $true )
& ( exuE3u = $true )
& ( lamProp = $true )
& ( dsetconstrI = $true )
& ( subsetemptysetimpeq = $true )
& ( upairset2E = $true )
& ( exuEu = $true )
& ( emptysetsubset = $true )
& ( descr__Cong = $true )
& ( ex1I = $true )
& ( replAx = $true )
& ( emptyE1 = $true )
& ( dsetconstrEL = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjL = $true ) ),
inference(flattening,[status(thm)],[f861]) ).
thf(f861,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ( ( in @ X2 @ X0 )
!= $true )
& ( $true
!= ( in @ X2 @ ( setminus @ X1 @ X0 ) ) )
& ( ( in @ X2 @ X1 )
= $true ) )
& ( ( in @ X0 @ ( powerset @ X1 ) )
= $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)],[f742]) ).
thf(f742,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 )
=> ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
=> ! [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
=> ( ( $true
!= ( in @ X2 @ ( setminus @ X1 @ X0 ) ) )
=> ( ( in @ X2 @ X0 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(flattening,[status(thm)],[f527]) ).
thf(f527,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 )
=> ! [X0: $i,X1: $i] :
( ( ( in @ X0 @ ( powerset @ X1 ) )
= $true )
=> ! [X2: $i] :
( ( ( in @ X2 @ X1 )
= $true )
=> ( ( $true
!= ( in @ X2 @ ( setminus @ X1 @ X0 ) ) )
=> ( ( in @ X2 @ X0 )
= $true ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f526]) ).
thf(f526,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
=> ! [X0: $i,X1: $i] :
( ( in @ X0 @ ( powerset @ X1 ) )
=> ! [X2: $i] :
( ( in @ X2 @ X1 )
=> ( ~ ( in @ X2 @ ( setminus @ X1 @ X0 ) )
=> ( in @ X2 @ X0 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f242]) ).
thf(f242,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
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ~ ( in @ X1 @ ( setminus @ X3 @ X11 ) )
=> ( in @ X1 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f241]) ).
thf(f241,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
=> ! [X11: $i,X3: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ~ ( in @ X1 @ ( setminus @ X3 @ X11 ) )
=> ( in @ X1 @ X11 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.SCLFyHJTHZ/DTF2THF_24371.p',complementTE1) ).
thf(f3692,plain,
spl747_5,
inference(avatar_contradiction_clause,[status(thm)],[f3691]) ).
thf(f3691,plain,
( $false
| spl747_5 ),
inference(trivial_inequality_removal,[status(thm)],[f3681]) ).
thf(f3681,plain,
( ( $true != $true )
| spl747_5 ),
inference(superposition,[status(thm)],[f3643,f2381]) ).
thf(f2381,plain,
( ( in @ sK130 @ sK129 )
= $true ),
inference(cnf_transformation,[status(thm)],[f1229]) ).
thf(f3643,plain,
( ( ( in @ sK130 @ sK129 )
!= $true )
| spl747_5 ),
inference(avatar_component_clause,[status(thm)],[f3641]) ).
thf(f3641,definition,
( spl747_5
<=> ( ( in @ sK130 @ sK129 )
= $true ) ),
introduced(definition,[new_symbols(naming,[spl747_5])],[avatar_definition]) ).
thf(f3667,plain,
( spl747_6
| ~ spl747_5 ),
inference(avatar_split_clause,[status(thm)],[f3666,f3641,f3645]) ).
thf(f3666,plain,
( ( ( in @ sK130 @ sK128 )
= $true )
| ( ( in @ sK130 @ sK129 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3665]) ).
thf(f3665,plain,
( ( ( in @ sK130 @ sK128 )
= $true )
| ( ( in @ sK130 @ sK129 )
!= $true )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f3633,f2503]) ).
thf(f2503,plain,
setminusI = $true,
inference(cnf_transformation,[status(thm)],[f1229]) ).
thf(f3633,plain,
( ( setminusI != $true )
| ( ( in @ sK130 @ sK128 )
= $true )
| ( ( in @ sK130 @ sK129 )
!= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f3624]) ).
thf(f3624,plain,
( ( setminusI != $true )
| ( $true != $true )
| ( ( in @ sK130 @ sK128 )
= $true )
| ( ( in @ sK130 @ sK129 )
!= $true ) ),
inference(superposition,[status(thm)],[f2383,f3132]) ).
thf(f3132,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ ( setminus @ X5 @ X3 ) )
= $true )
| ( setminusI != $true )
| ( ( in @ X4 @ X3 )
= $true )
| ( ( in @ X4 @ X5 )
!= $true ) ),
inference(cnf_transformation,[status(thm)],[f1883]) ).
thf(f1883,plain,
( ( ( setminusI = $true )
| ( ( ( in @ sK561 @ sK560 )
!= $true )
& ( $true
!= ( in @ sK561 @ ( setminus @ sK562 @ sK560 ) ) )
& ( ( in @ sK561 @ sK562 )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ X3 )
= $true )
| ( ( in @ X4 @ ( setminus @ X5 @ X3 ) )
= $true )
| ( ( in @ X4 @ X5 )
!= $true ) )
| ( setminusI != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK560,sK561,sK562])],[f1881,f1882]) ).
thf(f1882,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ X0 ) )
& ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
!= $true )
& ( ( in @ X1 @ X2 )
= $true ) )
=> ( ( ( in @ sK561 @ sK560 )
!= $true )
& ( $true
!= ( in @ sK561 @ ( setminus @ sK562 @ sK560 ) ) )
& ( ( in @ sK561 @ sK562 )
= $true ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1881,plain,
( ( ( setminusI = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ X0 ) )
& ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
!= $true )
& ( ( in @ X1 @ X2 )
= $true ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( ( in @ X4 @ X3 )
= $true )
| ( ( in @ X4 @ ( setminus @ X5 @ X3 ) )
= $true )
| ( ( in @ X4 @ X5 )
!= $true ) )
| ( setminusI != $true ) ) ),
inference(rectify,[status(thm)],[f1880]) ).
thf(f1880,plain,
( ( ( setminusI = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X1 @ X0 ) )
& ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
!= $true )
& ( ( in @ X1 @ X2 )
= $true ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
| ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true )
| ( ( in @ X1 @ X2 )
!= $true ) )
| ( setminusI != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f956]) ).
thf(f956,plain,
( ( setminusI = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X1 @ X0 ) )
| ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true )
| ( ( in @ X1 @ X2 )
!= $true ) ) ),
inference(flattening,[status(thm)],[f955]) ).
thf(f955,plain,
( ( setminusI = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true )
| ( $true
= ( in @ X1 @ X0 ) )
| ( ( in @ X1 @ X2 )
!= $true ) ) ),
inference(ennf_transformation,[status(thm)],[f744]) ).
thf(f744,plain,
( ( setminusI = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
=> ( ( $true
!= ( in @ X1 @ X0 ) )
=> ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true ) ) ) ),
inference(flattening,[status(thm)],[f535]) ).
thf(f535,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
= $true )
=> ( ( $true
!= ( in @ X1 @ X0 ) )
=> ( ( in @ X1 @ ( setminus @ X2 @ X0 ) )
= $true ) ) )
<=> ( setminusI = $true ) ),
inference(fool_elimination,[status(thm)],[f534]) ).
thf(f534,plain,
! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ X2 )
=> ( ~ ( in @ X1 @ X0 )
=> ( in @ X1 @ ( setminus @ X2 @ X0 ) ) ) )
= setminusI ),
inference(rectify,[status(thm)],[f120]) ).
thf(f120,axiom,
! [X4: $i,X1: $i,X3: $i] :
( ( ( in @ X1 @ X3 )
=> ( ~ ( in @ X1 @ X4 )
=> ( in @ X1 @ ( setminus @ X3 @ X4 ) ) ) )
= setminusI ),
file('/export/starexec/sandbox2/tmp/tmp.SCLFyHJTHZ/DTF2THF_24371.p',setminusI) ).
thf(f2383,plain,
( ( in @ sK130 @ ( setminus @ sK129 @ sK128 ) )
!= $true ),
inference(cnf_transformation,[status(thm)],[f1229]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : SEU720^1 : TPTP v9.2.1. Released v3.7.0.
% 0.11/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.14/0.32 % Computer : n011.cluster.edu
% 0.14/0.32 % Model : x86_64 x86_64
% 0.14/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.32 % Memory : 8042.1875MB
% 0.14/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.32 % CPULimit : 300
% 0.14/0.32 % WCLimit : 300
% 0.14/0.32 % DateTime : Tue Feb 24 12:31:32 EST 2026
% 0.14/0.32 % CPUTime :
% 0.14/0.32 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.42 ---- Original DTF file ---
% 0.16/0.42 thf(spec,logic,$$dhol).
% 0.16/0.42 %------------------------------------------------------------------------------
% 0.16/0.42 % File : SEU720^1 : TPTP v9.2.1. Released v3.7.0.
% 0.16/0.42 % Domain : Set Theory
% 0.16/0.42 % Problem : Typed Set Theory - Laws for Typed Sets
% 0.16/0.42 % Version : Especial.
% 0.16/0.42 % English : (! A:i.! X:i.in X (powerset A) -> (! x:i.in x A ->
% 0.16/0.42 % ~(in x (setminus A X)) -> in x X))
% 0.16/0.42
% 0.16/0.42 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.16/0.42 % Source : [Bro08]
% 0.16/0.42 % Names : ZFC222g [Bro08]
% 0.16/0.42
% 0.16/0.42 % Status : Theorem
% 0.16/0.42 % Rating : 0.11 v9.1.0, 0.12 v9.0.0, 0.20 v8.2.0, 0.31 v8.1.0, 0.09 v7.5.0, 0.00 v7.4.0, 0.11 v7.2.0, 0.00 v7.1.0, 0.25 v7.0.0, 0.14 v6.4.0, 0.17 v6.3.0, 0.20 v6.2.0, 0.14 v6.1.0, 0.29 v5.5.0, 0.33 v5.4.0, 0.40 v5.2.0, 0.60 v5.1.0, 0.80 v5.0.0, 0.60 v4.1.0, 0.67 v3.7.0
% 0.16/0.42 % Syntax : Number of formulae : 515 ( 240 unt; 274 typ; 240 def)
% 0.16/0.42 % Number of atoms : 1362 ( 365 equ; 0 cnn)
% 0.16/0.42 % Maximal formula atoms : 240 ( 5 avg)
% 0.16/0.42 % Number of connectives : 2662 ( 57 ~; 12 |; 44 &;1884 @)
% 0.16/0.42 % ( 14 <=>; 651 =>; 0 <=; 0 <~>)
% 0.16/0.42 % Maximal formula depth : 247 ( 2 avg)
% 0.16/0.42 % Number of types : 2 ( 0 usr)
% 0.16/0.42 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.16/0.42 % Number of symbols : 277 ( 274 usr; 240 con; 0-4 aty)
% 0.16/0.42 % Number of variables : 808 ( 74 ^; 699 !; 35 ?; 808 :)
% 0.16/0.42 % SPC : TH0_THM_EQU_NAR
% 0.16/0.42
% 0.16/0.42 % Comments : http://mathgate.info/detsetitem.php?id=280
% 0.16/0.42 %------------------------------------------------------------------------------
% 0.16/0.42 thf(in_type,type,
% 0.16/0.42 in: $i > $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(exu_type,type,
% 0.16/0.42 exu: ( $i > $o ) > $o ).
% 0.16/0.42
% 0.16/0.42 thf(exu,definition,
% 0.16/0.42 ( exu
% 0.16/0.42 = ( ^ [Xphi: $i > $o] :
% 0.16/0.42 ? [Xx: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 & ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setextAx_type,type,
% 0.16/0.42 setextAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setextAx,definition,
% 0.16/0.42 ( setextAx
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 <=> ( in @ Xx @ B ) )
% 0.16/0.42 => ( A = B ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyset_type,type,
% 0.16/0.42 emptyset: $i ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetAx_type,type,
% 0.16/0.42 emptysetAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetAx,definition,
% 0.16/0.42 ( emptysetAx
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoin_type,type,
% 0.16/0.42 setadjoin: $i > $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinAx_type,type,
% 0.16/0.42 setadjoinAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinAx,definition,
% 0.16/0.42 ( setadjoinAx
% 0.16/0.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.42 <=> ( ( Xy = Xx )
% 0.16/0.42 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powerset_type,type,
% 0.16/0.42 powerset: $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(powersetAx_type,type,
% 0.16/0.42 powersetAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetAx,definition,
% 0.16/0.42 ( powersetAx
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.42 <=> ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunion_type,type,
% 0.16/0.42 setunion: $i > $i ).
% 0.16/0.42
% 0.16/0.42 thf(setunionAx_type,type,
% 0.16/0.42 setunionAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionAx,definition,
% 0.16/0.42 ( setunionAx
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.42 <=> ? [B: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 & ( in @ B @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(omega_type,type,
% 0.16/0.42 omega: $i ).
% 0.16/0.42
% 0.16/0.42 thf(omega0Ax_type,type,
% 0.16/0.42 omega0Ax: $o ).
% 0.16/0.42
% 0.16/0.42 thf(omega0Ax,definition,
% 0.16/0.42 ( omega0Ax
% 0.16/0.42 = ( in @ emptyset @ omega ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(omegaSAx_type,type,
% 0.16/0.42 omegaSAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(omegaSAx,definition,
% 0.16/0.42 ( omegaSAx
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ omega )
% 0.16/0.42 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(omegaIndAx_type,type,
% 0.16/0.42 omegaIndAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(omegaIndAx,definition,
% 0.16/0.42 ( omegaIndAx
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( ( in @ emptyset @ A )
% 0.16/0.42 & ! [Xx: $i] :
% 0.16/0.42 ( ( ( in @ Xx @ omega )
% 0.16/0.42 & ( in @ Xx @ A ) )
% 0.16/0.42 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ omega )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(replAx_type,type,
% 0.16/0.42 replAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(replAx,definition,
% 0.16/0.42 ( replAx
% 0.16/0.42 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( exu
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.16/0.42 => ? [B: $i] :
% 0.16/0.42 ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 <=> ? [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(foundationAx_type,type,
% 0.16/0.42 foundationAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(foundationAx,definition,
% 0.16/0.42 ( foundationAx
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.42 => ? [B: $i] :
% 0.16/0.42 ( ( in @ B @ A )
% 0.16/0.42 & ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(wellorderingAx_type,type,
% 0.16/0.42 wellorderingAx: $o ).
% 0.16/0.42
% 0.16/0.42 thf(wellorderingAx,definition,
% 0.16/0.42 ( wellorderingAx
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ? [B: $i] :
% 0.16/0.42 ( ! [C: $i] :
% 0.16/0.42 ( ( in @ C @ B )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ C )
% 0.16/0.42 => ( in @ Xx @ A ) ) )
% 0.16/0.42 & ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( ( in @ Xx @ A )
% 0.16/0.42 & ( in @ Xy @ A ) )
% 0.16/0.42 => ( ! [C: $i] :
% 0.16/0.42 ( ( in @ C @ B )
% 0.16/0.42 => ( ( in @ Xx @ C )
% 0.16/0.42 <=> ( in @ Xy @ C ) ) )
% 0.16/0.42 => ( Xx = Xy ) ) )
% 0.16/0.42 & ! [C: $i,D: $i] :
% 0.16/0.42 ( ( ( in @ C @ B )
% 0.16/0.42 & ( in @ D @ B ) )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ C )
% 0.16/0.42 => ( in @ Xx @ D ) )
% 0.16/0.42 | ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ D )
% 0.16/0.42 => ( in @ Xx @ C ) ) ) )
% 0.16/0.42 & ! [C: $i] :
% 0.16/0.42 ( ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ C )
% 0.16/0.42 => ( in @ Xx @ A ) )
% 0.16/0.42 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.16/0.42 => ? [D: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ D @ B )
% 0.16/0.42 & ( in @ Xx @ C )
% 0.16/0.42 & ~ ? [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ D )
% 0.16/0.42 & ( in @ Xy @ C ) )
% 0.16/0.42 & ! [E: $i] :
% 0.16/0.42 ( ( in @ E @ B )
% 0.16/0.42 => ( ! [Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ E )
% 0.16/0.42 => ( in @ Xy @ D ) )
% 0.16/0.42 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(descr_type,type,
% 0.16/0.42 descr: ( $i > $o ) > $i ).
% 0.16/0.42
% 0.16/0.42 thf(descrp_type,type,
% 0.16/0.42 descrp: $o ).
% 0.16/0.42
% 0.16/0.42 thf(descrp,definition,
% 0.16/0.42 ( descrp
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ( Xphi
% 0.16/0.42 @ ( descr
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstr_type,type,
% 0.16/0.42 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrI_type,type,
% 0.16/0.42 dsetconstrI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrI,definition,
% 0.16/0.42 ( dsetconstrI
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 => ( in @ Xx
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrEL_type,type,
% 0.16/0.42 dsetconstrEL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrEL,definition,
% 0.16/0.42 ( dsetconstrEL
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrER_type,type,
% 0.16/0.42 dsetconstrER: $o ).
% 0.16/0.42
% 0.16/0.42 thf(dsetconstrER,definition,
% 0.16/0.42 ( dsetconstrER
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuE1_type,type,
% 0.16/0.42 exuE1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuE1,definition,
% 0.16/0.42 ( exuE1
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 & ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(prop2set_type,type,
% 0.16/0.42 prop2set: $o > $i ).
% 0.16/0.42
% 0.16/0.42 thf(prop2setE_type,type,
% 0.16/0.42 prop2setE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(prop2setE,definition,
% 0.16/0.42 ( prop2setE
% 0.16/0.42 = ( ! [Xphi: $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.16/0.42 => Xphi ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetE_type,type,
% 0.16/0.42 emptysetE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetE,definition,
% 0.16/0.42 ( emptysetE
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ emptyset )
% 0.16/0.42 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetimpfalse_type,type,
% 0.16/0.42 emptysetimpfalse: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptysetimpfalse,definition,
% 0.16/0.42 ( emptysetimpfalse
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ emptyset )
% 0.16/0.42 => $false ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notinemptyset_type,type,
% 0.16/0.42 notinemptyset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notinemptyset,definition,
% 0.16/0.42 ( notinemptyset
% 0.16/0.42 = ( ! [Xx: $i] :
% 0.16/0.42 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuE3e_type,type,
% 0.16/0.42 exuE3e: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuE3e,definition,
% 0.16/0.42 ( exuE3e
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setext_type,type,
% 0.16/0.42 setext: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setext,definition,
% 0.16/0.42 ( setext
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ B ) )
% 0.16/0.42 => ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ A ) )
% 0.16/0.42 => ( A = B ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyI_type,type,
% 0.16/0.42 emptyI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyI,definition,
% 0.16/0.42 ( emptyI
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ~ ( in @ Xx @ A )
% 0.16/0.42 => ( A = emptyset ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(noeltsimpempty_type,type,
% 0.16/0.42 noeltsimpempty: $o ).
% 0.16/0.42
% 0.16/0.42 thf(noeltsimpempty,definition,
% 0.16/0.42 ( noeltsimpempty
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ~ ( in @ Xx @ A )
% 0.16/0.42 => ( A = emptyset ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setbeta_type,type,
% 0.16/0.42 setbeta: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setbeta,definition,
% 0.16/0.42 ( setbeta
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( in @ Xx
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.16/0.42 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(nonempty_type,type,
% 0.16/0.42 nonempty: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(nonempty,definition,
% 0.16/0.42 ( nonempty
% 0.16/0.42 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyE1_type,type,
% 0.16/0.42 nonemptyE1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyE1,definition,
% 0.16/0.42 ( nonemptyE1
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( nonempty @ A )
% 0.16/0.42 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyI_type,type,
% 0.16/0.42 nonemptyI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyI,definition,
% 0.16/0.42 ( nonemptyI
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 => ( nonempty
% 0.16/0.42 @ ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyI1_type,type,
% 0.16/0.42 nonemptyI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyI1,definition,
% 0.16/0.42 ( nonemptyI1
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.16/0.42 => ( nonempty @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinIL_type,type,
% 0.16/0.42 setadjoinIL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinIL,definition,
% 0.16/0.42 ( setadjoinIL
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyinunitempty_type,type,
% 0.16/0.42 emptyinunitempty: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyinunitempty,definition,
% 0.16/0.42 ( emptyinunitempty
% 0.16/0.42 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinIR_type,type,
% 0.16/0.42 setadjoinIR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinIR,definition,
% 0.16/0.42 ( setadjoinIR
% 0.16/0.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ A )
% 0.16/0.42 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinE_type,type,
% 0.16/0.42 setadjoinE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinE,definition,
% 0.16/0.42 ( setadjoinE
% 0.16/0.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.42 => ! [Xphi: $o] :
% 0.16/0.42 ( ( ( Xy = Xx )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => ( ( ( in @ Xy @ A )
% 0.16/0.42 => Xphi )
% 0.16/0.42 => Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinOr_type,type,
% 0.16/0.42 setadjoinOr: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoinOr,definition,
% 0.16/0.42 ( setadjoinOr
% 0.16/0.42 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.16/0.42 => ( ( Xy = Xx )
% 0.16/0.42 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setoftrueEq_type,type,
% 0.16/0.42 setoftrueEq: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setoftrueEq,definition,
% 0.16/0.42 ( setoftrueEq
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : $true )
% 0.16/0.42 = A ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetI_type,type,
% 0.16/0.42 powersetI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetI,definition,
% 0.16/0.42 ( powersetI
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ A ) )
% 0.16/0.42 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyinPowerset_type,type,
% 0.16/0.42 emptyinPowerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyinPowerset,definition,
% 0.16/0.42 ( emptyinPowerset
% 0.16/0.42 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyInPowerset_type,type,
% 0.16/0.42 emptyInPowerset: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyInPowerset,definition,
% 0.16/0.42 ( emptyInPowerset
% 0.16/0.42 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(powersetE_type,type,
% 0.16/0.42 powersetE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(powersetE,definition,
% 0.16/0.42 ( powersetE
% 0.16/0.42 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.42 => ( ( in @ Xx @ B )
% 0.16/0.42 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunionI_type,type,
% 0.16/0.42 setunionI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionI,definition,
% 0.16/0.42 ( setunionI
% 0.16/0.42 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( ( in @ B @ A )
% 0.16/0.42 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setunionE_type,type,
% 0.16/0.42 setunionE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setunionE,definition,
% 0.16/0.42 ( setunionE
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.42 => ! [Xphi: $o] :
% 0.16/0.42 ( ! [B: $i] :
% 0.16/0.42 ( ( in @ Xx @ B )
% 0.16/0.42 => ( ( in @ B @ A )
% 0.16/0.42 => Xphi ) )
% 0.16/0.42 => Xphi ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(subPowSU_type,type,
% 0.16/0.42 subPowSU: $o ).
% 0.16/0.42
% 0.16/0.42 thf(subPowSU,definition,
% 0.16/0.42 ( subPowSU
% 0.16/0.42 = ( ! [A: $i,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuE2_type,type,
% 0.16/0.42 exuE2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuE2,definition,
% 0.16/0.42 ( exuE2
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyImpWitness_type,type,
% 0.16/0.42 nonemptyImpWitness: $o ).
% 0.16/0.42
% 0.16/0.42 thf(nonemptyImpWitness,definition,
% 0.16/0.42 ( nonemptyImpWitness
% 0.16/0.42 = ( ! [A: $i] :
% 0.16/0.42 ( ( nonempty @ A )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & $true ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(uniqinunit_type,type,
% 0.16/0.42 uniqinunit: $o ).
% 0.16/0.42
% 0.16/0.42 thf(uniqinunit,definition,
% 0.16/0.42 ( uniqinunit
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notinsingleton_type,type,
% 0.16/0.42 notinsingleton: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notinsingleton,definition,
% 0.16/0.42 ( notinsingleton
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx != Xy )
% 0.16/0.42 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(eqinunit_type,type,
% 0.16/0.42 eqinunit: $o ).
% 0.16/0.42
% 0.16/0.42 thf(eqinunit,definition,
% 0.16/0.42 ( eqinunit
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsswitch_type,type,
% 0.16/0.42 singletonsswitch: $o ).
% 0.16/0.42
% 0.16/0.42 thf(singletonsswitch,definition,
% 0.16/0.42 ( singletonsswitch
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.42 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetE_type,type,
% 0.16/0.42 upairsetE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetE,definition,
% 0.16/0.42 ( upairsetE
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.42 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.42 => ( ( Xz = Xx )
% 0.16/0.42 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetIL_type,type,
% 0.16/0.42 upairsetIL: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetIL,definition,
% 0.16/0.42 ( upairsetIL
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetIR_type,type,
% 0.16/0.42 upairsetIR: $o ).
% 0.16/0.42
% 0.16/0.42 thf(upairsetIR,definition,
% 0.16/0.42 ( upairsetIR
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyE1_type,type,
% 0.16/0.42 emptyE1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyE1,definition,
% 0.16/0.42 ( emptyE1
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ( ( ( dsetconstr @ A
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 = emptyset )
% 0.16/0.42 => $false ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(vacuousDall_type,type,
% 0.16/0.42 vacuousDall: $o ).
% 0.16/0.42
% 0.16/0.42 thf(vacuousDall,definition,
% 0.16/0.42 ( vacuousDall
% 0.16/0.42 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ emptyset )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan1_type,type,
% 0.16/0.42 quantDeMorgan1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan1,definition,
% 0.16/0.42 ( quantDeMorgan1
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ~ ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan2_type,type,
% 0.16/0.42 quantDeMorgan2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan2,definition,
% 0.16/0.42 ( quantDeMorgan2
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ~ ( Xphi @ Xx ) )
% 0.16/0.42 => ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan3_type,type,
% 0.16/0.42 quantDeMorgan3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan3,definition,
% 0.16/0.42 ( quantDeMorgan3
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan4_type,type,
% 0.16/0.42 quantDeMorgan4: $o ).
% 0.16/0.42
% 0.16/0.42 thf(quantDeMorgan4,definition,
% 0.16/0.42 ( quantDeMorgan4
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ~ ( Xphi @ Xx ) )
% 0.16/0.42 => ~ ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(prop2setI_type,type,
% 0.16/0.42 prop2setI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(prop2setI,definition,
% 0.16/0.42 ( prop2setI
% 0.16/0.42 = ( ! [Xphi: $o] :
% 0.16/0.42 ( Xphi
% 0.16/0.42 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(set2prop_type,type,
% 0.16/0.42 set2prop: $i > $o ).
% 0.16/0.42
% 0.16/0.42 thf(prop2set2propI_type,type,
% 0.16/0.42 prop2set2propI: $o ).
% 0.16/0.42
% 0.16/0.42 thf(prop2set2propI,definition,
% 0.16/0.42 ( prop2set2propI
% 0.16/0.42 = ( ! [Xphi: $o] :
% 0.16/0.42 ( Xphi
% 0.16/0.42 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notdexE_type,type,
% 0.16/0.42 notdexE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notdexE,definition,
% 0.16/0.42 ( notdexE
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ~ ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(notdallE_type,type,
% 0.16/0.42 notdallE: $o ).
% 0.16/0.42
% 0.16/0.42 thf(notdallE,definition,
% 0.16/0.42 ( notdallE
% 0.16/0.42 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.42 ( ~ ! [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 => ( Xphi @ Xx ) )
% 0.16/0.42 => ? [Xx: $i] :
% 0.16/0.42 ( ( in @ Xx @ A )
% 0.16/0.42 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuI1_type,type,
% 0.16/0.42 exuI1: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuI1,definition,
% 0.16/0.42 ( exuI1
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 & ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) )
% 0.16/0.42 => ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuI3_type,type,
% 0.16/0.42 exuI3: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuI3,definition,
% 0.16/0.42 ( exuI3
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.16/0.42 => ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) )
% 0.16/0.42 => ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuI2_type,type,
% 0.16/0.42 exuI2: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuI2,definition,
% 0.16/0.42 ( exuI2
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ? [Xx: $i] :
% 0.16/0.42 ! [Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xy )
% 0.16/0.42 <=> ( Xy = Xx ) )
% 0.16/0.42 => ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(inCongP_type,type,
% 0.16/0.42 inCongP: $o ).
% 0.16/0.42
% 0.16/0.42 thf(inCongP,definition,
% 0.16/0.42 ( inCongP
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(in__Cong_type,type,
% 0.16/0.42 in__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(in__Cong,definition,
% 0.16/0.42 ( in__Cong
% 0.16/0.42 = ( ! [A: $i,B: $i] :
% 0.16/0.42 ( ( A = B )
% 0.16/0.42 => ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( in @ Xx @ A )
% 0.16/0.42 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exuE3u_type,type,
% 0.16/0.42 exuE3u: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exuE3u,definition,
% 0.16/0.42 ( exuE3u
% 0.16/0.42 = ( ! [Xphi: $i > $o] :
% 0.16/0.42 ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 => ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xphi @ Xx )
% 0.16/0.42 => ( ( Xphi @ Xy )
% 0.16/0.42 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(exu__Cong_type,type,
% 0.16/0.42 exu__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(exu__Cong,definition,
% 0.16/0.42 ( exu__Cong
% 0.16/0.42 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.42 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.42 ( ( Xx = Xy )
% 0.16/0.42 => ( ( Xphi @ Xx )
% 0.16/0.42 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.42 => ( ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.42 <=> ( exu
% 0.16/0.42 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(emptyset__Cong_type,type,
% 0.16/0.42 emptyset__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(emptyset__Cong,definition,
% 0.16/0.42 ( emptyset__Cong
% 0.16/0.42 = ( emptyset = emptyset ) ) ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoin__Cong_type,type,
% 0.16/0.42 setadjoin__Cong: $o ).
% 0.16/0.42
% 0.16/0.42 thf(setadjoin__Cong,definition,
% 0.16/0.42 ( setadjoin__Cong
% 0.16/0.42 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xx = Xy )
% 0.16/0.43 => ! [Xz: $i,Xu: $i] :
% 0.16/0.43 ( ( Xz = Xu )
% 0.16/0.43 => ( ( setadjoin @ Xx @ Xz )
% 0.16/0.43 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powerset__Cong_type,type,
% 0.16/0.43 powerset__Cong: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powerset__Cong,definition,
% 0.16/0.43 ( powerset__Cong
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( A = B )
% 0.16/0.43 => ( ( powerset @ A )
% 0.16/0.43 = ( powerset @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunion__Cong_type,type,
% 0.16/0.43 setunion__Cong: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunion__Cong,definition,
% 0.16/0.43 ( setunion__Cong
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( A = B )
% 0.16/0.43 => ( ( setunion @ A )
% 0.16/0.43 = ( setunion @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(omega__Cong_type,type,
% 0.16/0.43 omega__Cong: $o ).
% 0.16/0.43
% 0.16/0.43 thf(omega__Cong,definition,
% 0.16/0.43 ( omega__Cong
% 0.16/0.43 = ( omega = omega ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(exuEu_type,type,
% 0.16/0.43 exuEu: $o ).
% 0.16/0.43
% 0.16/0.43 thf(exuEu,definition,
% 0.16/0.43 ( exuEu
% 0.16/0.43 = ( ! [Xphi: $i > $o] :
% 0.16/0.43 ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xphi @ Xx )
% 0.16/0.43 => ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(descr__Cong_type,type,
% 0.16/0.43 descr__Cong: $o ).
% 0.16/0.43
% 0.16/0.43 thf(descr__Cong,definition,
% 0.16/0.43 ( descr__Cong
% 0.16/0.43 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.43 ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xx = Xy )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 <=> ( Xpsi @ Xy ) ) )
% 0.16/0.43 => ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ( ( exu
% 0.16/0.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.16/0.43 => ( ( descr
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 = ( descr
% 0.16/0.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstr__Cong_type,type,
% 0.16/0.43 dsetconstr__Cong: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dsetconstr__Cong,definition,
% 0.16/0.43 ( dsetconstr__Cong
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( A = B )
% 0.16/0.43 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( Xx = Xy )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.16/0.43 => ( ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 = ( dsetconstr @ B
% 0.16/0.43 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subset_type,type,
% 0.16/0.43 subset: $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(disjoint_type,type,
% 0.16/0.43 disjoint: $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(setsmeet_type,type,
% 0.16/0.43 setsmeet: $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetI1_type,type,
% 0.16/0.43 subsetI1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetI1,definition,
% 0.16/0.43 ( subsetI1
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ Xx @ B ) )
% 0.16/0.43 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(eqimpsubset2_type,type,
% 0.16/0.43 eqimpsubset2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(eqimpsubset2,definition,
% 0.16/0.43 ( eqimpsubset2
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( A = B )
% 0.16/0.43 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(eqimpsubset1_type,type,
% 0.16/0.43 eqimpsubset1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(eqimpsubset1,definition,
% 0.16/0.43 ( eqimpsubset1
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( A = B )
% 0.16/0.43 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subsetI2_type,type,
% 0.16/0.43 subsetI2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetI2,definition,
% 0.16/0.43 ( subsetI2
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ Xx @ B ) )
% 0.16/0.43 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetsubset_type,type,
% 0.16/0.43 emptysetsubset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(emptysetsubset,definition,
% 0.16/0.43 ( emptysetsubset
% 0.16/0.43 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subsetE_type,type,
% 0.16/0.43 subsetE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetE,definition,
% 0.16/0.43 ( subsetE
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subsetE2_type,type,
% 0.16/0.43 subsetE2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetE2,definition,
% 0.16/0.43 ( subsetE2
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( ~ ( in @ Xx @ B )
% 0.16/0.43 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(notsubsetI_type,type,
% 0.16/0.43 notsubsetI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(notsubsetI,definition,
% 0.16/0.43 ( notsubsetI
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ~ ( in @ Xx @ B )
% 0.16/0.43 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(notequalI1_type,type,
% 0.16/0.43 notequalI1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(notequalI1,definition,
% 0.16/0.43 ( notequalI1
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ~ ( subset @ A @ B )
% 0.16/0.43 => ( A != B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(notequalI2_type,type,
% 0.16/0.43 notequalI2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(notequalI2,definition,
% 0.16/0.43 ( notequalI2
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ~ ( in @ Xx @ B )
% 0.16/0.43 => ( A != B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subsetRefl_type,type,
% 0.16/0.43 subsetRefl: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetRefl,definition,
% 0.16/0.43 ( subsetRefl
% 0.16/0.43 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subsetTrans_type,type,
% 0.16/0.43 subsetTrans: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetTrans,definition,
% 0.16/0.43 ( subsetTrans
% 0.16/0.43 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( ( subset @ B @ C )
% 0.16/0.43 => ( subset @ A @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinSub_type,type,
% 0.16/0.43 setadjoinSub: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinSub,definition,
% 0.16/0.43 ( setadjoinSub
% 0.16/0.43 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinSub2_type,type,
% 0.16/0.43 setadjoinSub2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setadjoinSub2,definition,
% 0.16/0.43 ( setadjoinSub2
% 0.16/0.43 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subset2powerset_type,type,
% 0.16/0.43 subset2powerset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subset2powerset,definition,
% 0.16/0.43 ( subset2powerset
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setextsub_type,type,
% 0.16/0.43 setextsub: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setextsub,definition,
% 0.16/0.43 ( setextsub
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( ( subset @ B @ A )
% 0.16/0.43 => ( A = B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subsetemptysetimpeq_type,type,
% 0.16/0.43 subsetemptysetimpeq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetemptysetimpeq,definition,
% 0.16/0.43 ( subsetemptysetimpeq
% 0.16/0.43 = ( ! [A: $i] :
% 0.16/0.43 ( ( subset @ A @ emptyset )
% 0.16/0.43 => ( A = emptyset ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powersetI1_type,type,
% 0.16/0.43 powersetI1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetI1,definition,
% 0.16/0.43 ( powersetI1
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( subset @ B @ A )
% 0.16/0.43 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powersetE1_type,type,
% 0.16/0.43 powersetE1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetE1,definition,
% 0.16/0.43 ( powersetE1
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( in @ B @ ( powerset @ A ) )
% 0.16/0.43 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(inPowerset_type,type,
% 0.16/0.43 inPowerset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(inPowerset,definition,
% 0.16/0.43 ( inPowerset
% 0.16/0.43 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powersetsubset_type,type,
% 0.16/0.43 powersetsubset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetsubset,definition,
% 0.16/0.43 ( powersetsubset
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sepInPowerset_type,type,
% 0.16/0.43 sepInPowerset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(sepInPowerset,definition,
% 0.16/0.43 ( sepInPowerset
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( in
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 @ ( powerset @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(sepSubset_type,type,
% 0.16/0.43 sepSubset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(sepSubset,definition,
% 0.16/0.43 ( sepSubset
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( subset
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binunion_type,type,
% 0.16/0.43 binunion: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(binunionIL_type,type,
% 0.16/0.43 binunionIL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binunionIL,definition,
% 0.16/0.43 ( binunionIL
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(upairset2IR_type,type,
% 0.16/0.43 upairset2IR: $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairset2IR,definition,
% 0.16/0.43 ( upairset2IR
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binunionIR_type,type,
% 0.16/0.43 binunionIR: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binunionIR,definition,
% 0.16/0.43 ( binunionIR
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binunionEcases_type,type,
% 0.16/0.43 binunionEcases: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binunionEcases,definition,
% 0.16/0.43 ( binunionEcases
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.16/0.43 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.43 => ( ( ( in @ Xx @ A )
% 0.16/0.43 => Xphi )
% 0.16/0.43 => ( ( ( in @ Xx @ B )
% 0.16/0.43 => Xphi )
% 0.16/0.43 => Xphi ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binunionE_type,type,
% 0.16/0.43 binunionE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binunionE,definition,
% 0.16/0.43 ( binunionE
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.16/0.43 => ( ( in @ Xx @ A )
% 0.16/0.43 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binunionLsub_type,type,
% 0.16/0.43 binunionLsub: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binunionLsub,definition,
% 0.16/0.43 ( binunionLsub
% 0.16/0.43 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binunionRsub_type,type,
% 0.16/0.43 binunionRsub: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binunionRsub,definition,
% 0.16/0.43 ( binunionRsub
% 0.16/0.43 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersect_type,type,
% 0.16/0.43 binintersect: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectI_type,type,
% 0.16/0.43 binintersectI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectI,definition,
% 0.16/0.43 ( binintersectI
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset5_type,type,
% 0.16/0.43 binintersectSubset5: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset5,definition,
% 0.16/0.43 ( binintersectSubset5
% 0.16/0.43 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.43 ( ( subset @ C @ A )
% 0.16/0.43 => ( ( subset @ C @ B )
% 0.16/0.43 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectEL_type,type,
% 0.16/0.43 binintersectEL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectEL,definition,
% 0.16/0.43 ( binintersectEL
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectLsub_type,type,
% 0.16/0.43 binintersectLsub: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectLsub,definition,
% 0.16/0.43 ( binintersectLsub
% 0.16/0.43 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset2_type,type,
% 0.16/0.43 binintersectSubset2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset2,definition,
% 0.16/0.43 ( binintersectSubset2
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( ( binintersect @ A @ B )
% 0.16/0.43 = A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset3_type,type,
% 0.16/0.43 binintersectSubset3: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset3,definition,
% 0.16/0.43 ( binintersectSubset3
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( ( binintersect @ A @ B )
% 0.16/0.43 = B )
% 0.16/0.43 => ( subset @ B @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectER_type,type,
% 0.16/0.43 binintersectER: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectER,definition,
% 0.16/0.43 ( binintersectER
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.16/0.43 => ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(disjointsetsI1_type,type,
% 0.16/0.43 disjointsetsI1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(disjointsetsI1,definition,
% 0.16/0.43 ( disjointsetsI1
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ~ ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ( in @ Xx @ B ) )
% 0.16/0.43 => ( ( binintersect @ A @ B )
% 0.16/0.43 = emptyset ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectRsub_type,type,
% 0.16/0.43 binintersectRsub: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectRsub,definition,
% 0.16/0.43 ( binintersectRsub
% 0.16/0.43 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset4_type,type,
% 0.16/0.43 binintersectSubset4: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset4,definition,
% 0.16/0.43 ( binintersectSubset4
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( subset @ B @ A )
% 0.16/0.43 => ( ( binintersect @ A @ B )
% 0.16/0.43 = B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset1_type,type,
% 0.16/0.43 binintersectSubset1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectSubset1,definition,
% 0.16/0.43 ( binintersectSubset1
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( ( binintersect @ A @ B )
% 0.16/0.43 = A )
% 0.16/0.43 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(bs114d_type,type,
% 0.16/0.43 bs114d: $o ).
% 0.16/0.43
% 0.16/0.43 thf(bs114d,definition,
% 0.16/0.43 ( bs114d
% 0.16/0.43 = ( ! [A: $i,B: $i,C: $i] :
% 0.16/0.43 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.16/0.43 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(regular_type,type,
% 0.16/0.43 regular: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminus_type,type,
% 0.16/0.43 setminus: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(setminusI_type,type,
% 0.16/0.43 setminusI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusI,definition,
% 0.16/0.43 ( setminusI
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ~ ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusEL_type,type,
% 0.16/0.43 setminusEL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusEL,definition,
% 0.16/0.43 ( setminusEL
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusER_type,type,
% 0.16/0.43 setminusER: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusER,definition,
% 0.16/0.43 ( setminusER
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.43 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusSubset2_type,type,
% 0.16/0.43 setminusSubset2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusSubset2,definition,
% 0.16/0.43 ( setminusSubset2
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( subset @ A @ B )
% 0.16/0.43 => ( ( setminus @ A @ B )
% 0.16/0.43 = emptyset ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusERneg_type,type,
% 0.16/0.43 setminusERneg: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusERneg,definition,
% 0.16/0.43 ( setminusERneg
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.43 => ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusELneg_type,type,
% 0.16/0.43 setminusELneg: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusELneg,definition,
% 0.16/0.43 ( setminusELneg
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.16/0.43 => ( ~ ( in @ Xx @ B )
% 0.16/0.43 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusILneg_type,type,
% 0.16/0.43 setminusILneg: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusILneg,definition,
% 0.16/0.43 ( setminusILneg
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ~ ( in @ Xx @ A )
% 0.16/0.43 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusIRneg_type,type,
% 0.16/0.43 setminusIRneg: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusIRneg,definition,
% 0.16/0.43 ( setminusIRneg
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ B )
% 0.16/0.43 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusLsub_type,type,
% 0.16/0.43 setminusLsub: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusLsub,definition,
% 0.16/0.43 ( setminusLsub
% 0.16/0.43 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusSubset1_type,type,
% 0.16/0.43 setminusSubset1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusSubset1,definition,
% 0.16/0.43 ( setminusSubset1
% 0.16/0.43 = ( ! [A: $i,B: $i] :
% 0.16/0.43 ( ( ( setminus @ A @ B )
% 0.16/0.43 = emptyset )
% 0.16/0.43 => ( subset @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(symdiff_type,type,
% 0.16/0.43 symdiff: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffE_type,type,
% 0.16/0.43 symdiffE: $o ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffE,definition,
% 0.16/0.43 ( symdiffE
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.16/0.43 => ! [Xphi: $o] :
% 0.16/0.43 ( ( ( in @ Xx @ A )
% 0.16/0.43 => ( ~ ( in @ Xx @ B )
% 0.16/0.43 => Xphi ) )
% 0.16/0.43 => ( ( ~ ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ Xx @ B )
% 0.16/0.43 => Xphi ) )
% 0.16/0.43 => Xphi ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffI1_type,type,
% 0.16/0.43 symdiffI1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffI1,definition,
% 0.16/0.43 ( symdiffI1
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ~ ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffI2_type,type,
% 0.16/0.43 symdiffI2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffI2,definition,
% 0.16/0.43 ( symdiffI2
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ~ ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ Xx @ B )
% 0.16/0.43 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffIneg1_type,type,
% 0.16/0.43 symdiffIneg1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffIneg1,definition,
% 0.16/0.43 ( symdiffIneg1
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ Xx @ B )
% 0.16/0.43 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffIneg2_type,type,
% 0.16/0.43 symdiffIneg2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(symdiffIneg2,definition,
% 0.16/0.43 ( symdiffIneg2
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ~ ( in @ Xx @ A )
% 0.16/0.43 => ( ~ ( in @ Xx @ B )
% 0.16/0.43 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(iskpair_type,type,
% 0.16/0.43 iskpair: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(secondinupair_type,type,
% 0.16/0.43 secondinupair: $o ).
% 0.16/0.43
% 0.16/0.43 thf(secondinupair,definition,
% 0.16/0.43 ( secondinupair
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairIL_type,type,
% 0.16/0.43 setukpairIL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairIL,definition,
% 0.16/0.43 ( setukpairIL
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairIR_type,type,
% 0.16/0.43 setukpairIR: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairIR,definition,
% 0.16/0.43 ( setukpairIR
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(kpairiskpair_type,type,
% 0.16/0.43 kpairiskpair: $o ).
% 0.16/0.43
% 0.16/0.43 thf(kpairiskpair,definition,
% 0.16/0.43 ( kpairiskpair
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(kpair_type,type,
% 0.16/0.43 kpair: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(kpairp_type,type,
% 0.16/0.43 kpairp: $o ).
% 0.16/0.43
% 0.16/0.43 thf(kpairp,definition,
% 0.16/0.43 ( kpairp
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprod_type,type,
% 0.16/0.43 cartprod: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(singletonsubset_type,type,
% 0.16/0.43 singletonsubset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(singletonsubset,definition,
% 0.16/0.43 ( singletonsubset
% 0.16/0.43 = ( ! [A: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(singletoninpowerset_type,type,
% 0.16/0.43 singletoninpowerset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(singletoninpowerset,definition,
% 0.16/0.43 ( singletoninpowerset
% 0.16/0.43 = ( ! [A: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(singletoninpowunion_type,type,
% 0.16/0.43 singletoninpowunion: $o ).
% 0.16/0.43
% 0.16/0.43 thf(singletoninpowunion,definition,
% 0.16/0.43 ( singletoninpowunion
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(upairset2E_type,type,
% 0.16/0.43 upairset2E: $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairset2E,definition,
% 0.16/0.43 ( upairset2E
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.43 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.16/0.43 => ( ( Xz = Xx )
% 0.16/0.43 | ( Xz = Xy ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(upairsubunion_type,type,
% 0.16/0.43 upairsubunion: $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairsubunion,definition,
% 0.16/0.43 ( upairsubunion
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(upairinpowunion_type,type,
% 0.16/0.43 upairinpowunion: $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairinpowunion,definition,
% 0.16/0.43 ( upairinpowunion
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ubforcartprodlem1_type,type,
% 0.16/0.43 ubforcartprodlem1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ubforcartprodlem1,definition,
% 0.16/0.43 ( ubforcartprodlem1
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ubforcartprodlem2_type,type,
% 0.16/0.43 ubforcartprodlem2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ubforcartprodlem2,definition,
% 0.16/0.43 ( ubforcartprodlem2
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ubforcartprodlem3_type,type,
% 0.16/0.43 ubforcartprodlem3: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ubforcartprodlem3,definition,
% 0.16/0.43 ( ubforcartprodlem3
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodpairin_type,type,
% 0.16/0.43 cartprodpairin: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodpairin,definition,
% 0.16/0.43 ( cartprodpairin
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodmempair1_type,type,
% 0.16/0.43 cartprodmempair1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodmempair1,definition,
% 0.16/0.43 ( cartprodmempair1
% 0.16/0.43 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.43 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.43 => ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ? [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 & ( Xu
% 0.16/0.43 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodmempair_type,type,
% 0.16/0.43 cartprodmempair: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodmempair,definition,
% 0.16/0.43 ( cartprodmempair
% 0.16/0.43 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.43 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.43 => ( iskpair @ Xu ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunionE2_type,type,
% 0.16/0.43 setunionE2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunionE2,definition,
% 0.16/0.43 ( setunionE2
% 0.16/0.43 = ( ! [A: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ ( setunion @ A ) )
% 0.16/0.43 => ? [X: $i] :
% 0.16/0.43 ( ( in @ X @ A )
% 0.16/0.43 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunionsingleton1_type,type,
% 0.16/0.43 setunionsingleton1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunionsingleton1,definition,
% 0.16/0.43 ( setunionsingleton1
% 0.16/0.43 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunionsingleton2_type,type,
% 0.16/0.43 setunionsingleton2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunionsingleton2,definition,
% 0.16/0.43 ( setunionsingleton2
% 0.16/0.43 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setunionsingleton_type,type,
% 0.16/0.43 setunionsingleton: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setunionsingleton,definition,
% 0.16/0.43 ( setunionsingleton
% 0.16/0.43 = ( ! [Xx: $i] :
% 0.16/0.43 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.16/0.43 = Xx ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(singleton_type,type,
% 0.16/0.43 singleton: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(singletonprop_type,type,
% 0.16/0.43 singletonprop: $o ).
% 0.16/0.43
% 0.16/0.43 thf(singletonprop,definition,
% 0.16/0.43 ( singletonprop
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 => ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) )
% 0.16/0.43 => ( ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ( Xphi @ Xx ) )
% 0.16/0.43 => ( singleton
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ex1_type,type,
% 0.16/0.43 ex1: $i > ( $i > $o ) > $o ).
% 0.16/0.43
% 0.16/0.43 thf(ex1,definition,
% 0.16/0.43 ( ex1
% 0.16/0.43 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( singleton
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ex1E1_type,type,
% 0.16/0.43 ex1E1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ex1E1,definition,
% 0.16/0.43 ( ex1E1
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( ( ex1 @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ex1I_type,type,
% 0.16/0.43 ex1I: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ex1I,definition,
% 0.16/0.43 ( ex1I
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 => ( ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xy = Xx ) ) )
% 0.16/0.43 => ( ex1 @ A
% 0.16/0.43 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ex1I2_type,type,
% 0.16/0.43 ex1I2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ex1I2,definition,
% 0.16/0.43 ( ex1I2
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 => ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) )
% 0.16/0.43 => ( ? [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 & ( Xphi @ Xx ) )
% 0.16/0.43 => ( ex1 @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(singletonsuniq_type,type,
% 0.16/0.43 singletonsuniq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(singletonsuniq,definition,
% 0.16/0.43 ( singletonsuniq
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( ( setadjoin @ Xx @ emptyset )
% 0.16/0.43 = ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(atmost1p_type,type,
% 0.16/0.43 atmost1p: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(atleast2p_type,type,
% 0.16/0.43 atleast2p: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(atmost2p_type,type,
% 0.16/0.43 atmost2p: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairsetp_type,type,
% 0.16/0.43 upairsetp: $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjL1_type,type,
% 0.16/0.43 setukpairinjL1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjL1,definition,
% 0.16/0.43 ( setukpairinjL1
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.43 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.43 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(kfstsingleton_type,type,
% 0.16/0.43 kfstsingleton: $o ).
% 0.16/0.43
% 0.16/0.43 thf(kfstsingleton,definition,
% 0.16/0.43 ( kfstsingleton
% 0.16/0.43 = ( ! [Xu: $i] :
% 0.16/0.43 ( ( iskpair @ Xu )
% 0.16/0.43 => ( singleton
% 0.16/0.43 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.43 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(theprop_type,type,
% 0.16/0.43 theprop: $o ).
% 0.16/0.43
% 0.16/0.43 thf(theprop,definition,
% 0.16/0.43 ( theprop
% 0.16/0.43 = ( ! [X: $i] :
% 0.16/0.43 ( ( singleton @ X )
% 0.16/0.43 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(kfst_type,type,
% 0.16/0.43 kfst: $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(kfstpairEq_type,type,
% 0.16/0.43 kfstpairEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(kfstpairEq,definition,
% 0.16/0.43 ( kfstpairEq
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.43 = Xx ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodfstin_type,type,
% 0.16/0.43 cartprodfstin: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodfstin,definition,
% 0.16/0.43 ( cartprodfstin
% 0.16/0.43 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.43 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.43 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjL2_type,type,
% 0.16/0.43 setukpairinjL2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjL2,definition,
% 0.16/0.43 ( setukpairinjL2
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.43 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.43 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.43 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjL_type,type,
% 0.16/0.43 setukpairinjL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjL,definition,
% 0.16/0.43 ( setukpairinjL
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.43 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.43 = ( kpair @ Xz @ Xu ) )
% 0.16/0.43 => ( Xx = Xz ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR11_type,type,
% 0.16/0.43 setukpairinjR11: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR11,definition,
% 0.16/0.43 ( setukpairinjR11
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xx = Xy )
% 0.16/0.43 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.43 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR12_type,type,
% 0.16/0.43 setukpairinjR12: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR12,definition,
% 0.16/0.43 ( setukpairinjR12
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xx = Xy )
% 0.16/0.43 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.43 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR1_type,type,
% 0.16/0.43 setukpairinjR1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR1,definition,
% 0.16/0.43 ( setukpairinjR1
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.43 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.43 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.43 => ( ( Xz = Xu )
% 0.16/0.43 => ( Xy = Xu ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(upairequniteq_type,type,
% 0.16/0.43 upairequniteq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(upairequniteq,definition,
% 0.16/0.43 ( upairequniteq
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.16/0.43 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.16/0.43 = ( setadjoin @ Xz @ emptyset ) )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR2_type,type,
% 0.16/0.43 setukpairinjR2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR2,definition,
% 0.16/0.43 ( setukpairinjR2
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.43 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.16/0.43 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.16/0.43 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR_type,type,
% 0.16/0.43 setukpairinjR: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setukpairinjR,definition,
% 0.16/0.43 ( setukpairinjR
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.16/0.43 ( ( ( kpair @ Xx @ Xy )
% 0.16/0.43 = ( kpair @ Xz @ Xu ) )
% 0.16/0.43 => ( Xy = Xu ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ksndsingleton_type,type,
% 0.16/0.43 ksndsingleton: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ksndsingleton,definition,
% 0.16/0.43 ( ksndsingleton
% 0.16/0.43 = ( ! [Xu: $i] :
% 0.16/0.43 ( ( iskpair @ Xu )
% 0.16/0.43 => ( singleton
% 0.16/0.43 @ ( dsetconstr @ ( setunion @ Xu )
% 0.16/0.43 @ ^ [Xx: $i] :
% 0.16/0.43 ( Xu
% 0.16/0.43 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ksnd_type,type,
% 0.16/0.43 ksnd: $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(ksndpairEq_type,type,
% 0.16/0.43 ksndpairEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ksndpairEq,definition,
% 0.16/0.43 ( ksndpairEq
% 0.16/0.43 = ( ! [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.43 = Xy ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(kpairsurjEq_type,type,
% 0.16/0.43 kpairsurjEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(kpairsurjEq,definition,
% 0.16/0.43 ( kpairsurjEq
% 0.16/0.43 = ( ! [Xu: $i] :
% 0.16/0.43 ( ( iskpair @ Xu )
% 0.16/0.43 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.43 = Xu ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodsndin_type,type,
% 0.16/0.43 cartprodsndin: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodsndin,definition,
% 0.16/0.43 ( cartprodsndin
% 0.16/0.43 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.43 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.43 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodpairmemEL_type,type,
% 0.16/0.43 cartprodpairmemEL: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodpairmemEL,definition,
% 0.16/0.43 ( cartprodpairmemEL
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodpairmemER_type,type,
% 0.16/0.43 cartprodpairmemER: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodpairmemER,definition,
% 0.16/0.43 ( cartprodpairmemER
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.16/0.43 => ( in @ Xy @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodmempaircEq_type,type,
% 0.16/0.43 cartprodmempaircEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodmempaircEq,definition,
% 0.16/0.43 ( cartprodmempaircEq
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( kpair @ Xx @ Xy )
% 0.16/0.43 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodfstpairEq_type,type,
% 0.16/0.43 cartprodfstpairEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodfstpairEq,definition,
% 0.16/0.43 ( cartprodfstpairEq
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.16/0.43 = Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodsndpairEq_type,type,
% 0.16/0.43 cartprodsndpairEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodsndpairEq,definition,
% 0.16/0.43 ( cartprodsndpairEq
% 0.16/0.43 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.16/0.43 = Xy ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodpairsurjEq_type,type,
% 0.16/0.43 cartprodpairsurjEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(cartprodpairsurjEq,definition,
% 0.16/0.43 ( cartprodpairsurjEq
% 0.16/0.43 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.16/0.43 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.16/0.43 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.16/0.43 = Xu ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(breln_type,type,
% 0.16/0.43 breln: $i > $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(breln,definition,
% 0.16/0.43 ( breln
% 0.16/0.43 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstr_type,type,
% 0.16/0.43 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstrI_type,type,
% 0.16/0.43 dpsetconstrI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstrI,definition,
% 0.16/0.43 ( dpsetconstrI
% 0.16/0.43 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( Xphi @ Xx @ Xy )
% 0.16/0.43 => ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.43 @ ( dpsetconstr @ A @ B
% 0.16/0.43 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstrSub_type,type,
% 0.16/0.43 dpsetconstrSub: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstrSub,definition,
% 0.16/0.43 ( dpsetconstrSub
% 0.16/0.43 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.43 ( subset
% 0.16/0.43 @ ( dpsetconstr @ A @ B
% 0.16/0.43 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.16/0.43 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setOfPairsIsBReln_type,type,
% 0.16/0.43 setOfPairsIsBReln: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setOfPairsIsBReln,definition,
% 0.16/0.43 ( setOfPairsIsBReln
% 0.16/0.43 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.16/0.43 ( breln @ A @ B
% 0.16/0.43 @ ( dpsetconstr @ A @ B
% 0.16/0.43 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstrERa_type,type,
% 0.16/0.43 dpsetconstrERa: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstrERa,definition,
% 0.16/0.43 ( dpsetconstrERa
% 0.16/0.43 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.43 @ ( dpsetconstr @ A @ B
% 0.16/0.43 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.43 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstrEL1_type,type,
% 0.16/0.43 dpsetconstrEL1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstrEL1,definition,
% 0.16/0.43 ( dpsetconstrEL1
% 0.16/0.43 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.43 @ ( dpsetconstr @ A @ B
% 0.16/0.43 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.43 => ( in @ Xx @ A ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstrEL2_type,type,
% 0.16/0.43 dpsetconstrEL2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstrEL2,definition,
% 0.16/0.43 ( dpsetconstrEL2
% 0.16/0.43 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.43 @ ( dpsetconstr @ A @ B
% 0.16/0.43 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.43 => ( in @ Xy @ B ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstrER_type,type,
% 0.16/0.43 dpsetconstrER: $o ).
% 0.16/0.43
% 0.16/0.43 thf(dpsetconstrER,definition,
% 0.16/0.43 ( dpsetconstrER
% 0.16/0.43 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( in @ ( kpair @ Xx @ Xy )
% 0.16/0.43 @ ( dpsetconstr @ A @ B
% 0.16/0.43 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.16/0.43 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(func_type,type,
% 0.16/0.43 func: $i > $i > $i > $o ).
% 0.16/0.43
% 0.16/0.43 thf(funcSet_type,type,
% 0.16/0.43 funcSet: $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(funcImageSingleton_type,type,
% 0.16/0.43 funcImageSingleton: $o ).
% 0.16/0.43
% 0.16/0.43 thf(funcImageSingleton,definition,
% 0.16/0.43 ( funcImageSingleton
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( func @ A @ B @ Xf )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( singleton
% 0.16/0.43 @ ( dsetconstr @ B
% 0.16/0.43 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(apProp_type,type,
% 0.16/0.43 apProp: $o ).
% 0.16/0.43
% 0.16/0.43 thf(apProp,definition,
% 0.16/0.43 ( apProp
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( func @ A @ B @ Xf )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in
% 0.16/0.43 @ ( setunion
% 0.16/0.43 @ ( dsetconstr @ B
% 0.16/0.43 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.16/0.43 @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ap_type,type,
% 0.16/0.43 ap: $i > $i > $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(app_type,type,
% 0.16/0.43 app: $o ).
% 0.16/0.43
% 0.16/0.43 thf(app,definition,
% 0.16/0.43 ( app
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( func @ A @ B @ Xf )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(infuncsetfunc_type,type,
% 0.16/0.43 infuncsetfunc: $o ).
% 0.16/0.43
% 0.16/0.43 thf(infuncsetfunc,definition,
% 0.16/0.43 ( infuncsetfunc
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.43 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ap2p_type,type,
% 0.16/0.43 ap2p: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ap2p,definition,
% 0.16/0.43 ( ap2p
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(funcinfuncset_type,type,
% 0.16/0.43 funcinfuncset: $o ).
% 0.16/0.43
% 0.16/0.43 thf(funcinfuncset,definition,
% 0.16/0.43 ( funcinfuncset
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( func @ A @ B @ Xf )
% 0.16/0.43 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(lamProp_type,type,
% 0.16/0.43 lamProp: $o ).
% 0.16/0.43
% 0.16/0.43 thf(lamProp,definition,
% 0.16/0.43 ( lamProp
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.43 => ( func @ A @ B
% 0.16/0.43 @ ( dpsetconstr @ A @ B
% 0.16/0.43 @ ^ [Xx: $i,Xy: $i] :
% 0.16/0.43 ( ( Xf @ Xx )
% 0.16/0.43 = Xy ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(lam_type,type,
% 0.16/0.43 lam: $i > $i > ( $i > $i ) > $i ).
% 0.16/0.43
% 0.16/0.43 thf(lamp_type,type,
% 0.16/0.43 lamp: $o ).
% 0.16/0.43
% 0.16/0.43 thf(lamp,definition,
% 0.16/0.43 ( lamp
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.43 => ( func @ A @ B
% 0.16/0.43 @ ( lam @ A @ B
% 0.16/0.43 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(lam2p_type,type,
% 0.16/0.43 lam2p: $o ).
% 0.16/0.43
% 0.16/0.43 thf(lam2p,definition,
% 0.16/0.43 ( lam2p
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.43 => ( in
% 0.16/0.43 @ ( lam @ A @ B
% 0.16/0.43 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.16/0.43 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(brelnall1_type,type,
% 0.16/0.43 brelnall1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(brelnall1,definition,
% 0.16/0.43 ( brelnall1
% 0.16/0.43 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.43 ( ( breln @ A @ B @ R )
% 0.16/0.43 => ! [Xphi: $i > $o] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.43 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ R )
% 0.16/0.43 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(brelnall2_type,type,
% 0.16/0.43 brelnall2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(brelnall2,definition,
% 0.16/0.43 ( brelnall2
% 0.16/0.43 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.43 ( ( breln @ A @ B @ R )
% 0.16/0.43 => ! [Xphi: $i > $o] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.43 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ R )
% 0.16/0.43 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ex1E2_type,type,
% 0.16/0.43 ex1E2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ex1E2,definition,
% 0.16/0.43 ( ex1E2
% 0.16/0.43 = ( ! [A: $i,Xphi: $i > $o] :
% 0.16/0.43 ( ( ex1 @ A
% 0.16/0.43 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( ( Xphi @ Xx )
% 0.16/0.43 => ( ( Xphi @ Xy )
% 0.16/0.43 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(funcGraphProp1_type,type,
% 0.16/0.43 funcGraphProp1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(funcGraphProp1,definition,
% 0.16/0.43 ( funcGraphProp1
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( func @ A @ B @ Xf )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(funcGraphProp3_type,type,
% 0.16/0.43 funcGraphProp3: $o ).
% 0.16/0.43
% 0.16/0.43 thf(funcGraphProp3,definition,
% 0.16/0.43 ( funcGraphProp3
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(funcGraphProp2_type,type,
% 0.16/0.43 funcGraphProp2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(funcGraphProp2,definition,
% 0.16/0.43 ( funcGraphProp2
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( func @ A @ B @ Xf )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.16/0.43 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.43 = Xy ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(funcextLem_type,type,
% 0.16/0.43 funcextLem: $o ).
% 0.16/0.43
% 0.16/0.43 thf(funcextLem,definition,
% 0.16/0.43 ( funcextLem
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( func @ A @ B @ Xf )
% 0.16/0.43 => ! [Xg: $i] :
% 0.16/0.43 ( ( func @ A @ B @ Xg )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.43 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.16/0.43 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(funcGraphProp4_type,type,
% 0.16/0.43 funcGraphProp4: $o ).
% 0.16/0.43
% 0.16/0.43 thf(funcGraphProp4,definition,
% 0.16/0.43 ( funcGraphProp4
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.16/0.43 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.43 = Xy ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subbreln_type,type,
% 0.16/0.43 subbreln: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subbreln,definition,
% 0.16/0.43 ( subbreln
% 0.16/0.43 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.43 ( ( breln @ A @ B @ R )
% 0.16/0.43 => ! [S: $i] :
% 0.16/0.43 ( ( breln @ A @ B @ S )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.43 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.16/0.43 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(eqbreln_type,type,
% 0.16/0.43 eqbreln: $o ).
% 0.16/0.43
% 0.16/0.43 thf(eqbreln,definition,
% 0.16/0.43 ( eqbreln
% 0.16/0.43 = ( ! [A: $i,B: $i,R: $i] :
% 0.16/0.43 ( ( breln @ A @ B @ R )
% 0.16/0.43 => ! [S: $i] :
% 0.16/0.43 ( ( breln @ A @ B @ S )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.16/0.43 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ B )
% 0.16/0.43 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.16/0.43 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.16/0.43 => ( R = S ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(funcext_type,type,
% 0.16/0.43 funcext: $o ).
% 0.16/0.43
% 0.16/0.43 thf(funcext,definition,
% 0.16/0.43 ( funcext
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( func @ A @ B @ Xf )
% 0.16/0.43 => ! [Xg: $i] :
% 0.16/0.43 ( ( func @ A @ B @ Xg )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.43 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.16/0.43 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(funcext2_type,type,
% 0.16/0.43 funcext2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(funcext2,definition,
% 0.16/0.43 ( funcext2
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.43 => ! [Xg: $i] :
% 0.16/0.43 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.43 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.16/0.43 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ap2apEq1_type,type,
% 0.16/0.43 ap2apEq1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ap2apEq1,definition,
% 0.16/0.43 ( ap2apEq1
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.43 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ap2apEq2_type,type,
% 0.16/0.43 ap2apEq2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ap2apEq2,definition,
% 0.16/0.43 ( ap2apEq2
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( func @ A @ B @ Xf )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.16/0.43 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(beta1_type,type,
% 0.16/0.43 beta1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(beta1,definition,
% 0.16/0.43 ( beta1
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( ap @ A @ B
% 0.16/0.43 @ ( lam @ A @ B
% 0.16/0.43 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.16/0.43 @ Xx )
% 0.16/0.43 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(eta1_type,type,
% 0.16/0.43 eta1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(eta1,definition,
% 0.16/0.43 ( eta1
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( func @ A @ B @ Xf )
% 0.16/0.43 => ( ( lam @ A @ B
% 0.16/0.43 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.16/0.43 = Xf ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(lam2lamEq_type,type,
% 0.16/0.43 lam2lamEq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(lam2lamEq,definition,
% 0.16/0.43 ( lam2lamEq
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.43 => ( ( lam @ A @ B
% 0.16/0.43 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.16/0.43 = ( lam @ A @ B
% 0.16/0.43 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(beta2_type,type,
% 0.16/0.43 beta2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(beta2,definition,
% 0.16/0.43 ( beta2
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.16/0.43 ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( ap @ A @ B
% 0.16/0.43 @ ( lam @ A @ B
% 0.16/0.43 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.16/0.43 @ Xx )
% 0.16/0.43 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(eta2_type,type,
% 0.16/0.43 eta2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(eta2,definition,
% 0.16/0.43 ( eta2
% 0.16/0.43 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.16/0.43 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.16/0.43 => ( ( lam @ A @ B
% 0.16/0.43 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.16/0.43 = Xf ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(iffalseProp1_type,type,
% 0.16/0.43 iffalseProp1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(iffalseProp1,definition,
% 0.16/0.43 ( iffalseProp1
% 0.16/0.43 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( ~ Xphi
% 0.16/0.43 => ( in @ Xy
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xz: $i] :
% 0.16/0.43 ( ( Xphi
% 0.16/0.43 & ( Xz = Xx ) )
% 0.16/0.43 | ( ~ Xphi
% 0.16/0.43 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(iffalseProp2_type,type,
% 0.16/0.43 iffalseProp2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(iffalseProp2,definition,
% 0.16/0.43 ( iffalseProp2
% 0.16/0.43 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( ~ Xphi
% 0.16/0.43 => ( ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xz: $i] :
% 0.16/0.43 ( ( Xphi
% 0.16/0.43 & ( Xz = Xx ) )
% 0.16/0.43 | ( ~ Xphi
% 0.16/0.43 & ( Xz = Xy ) ) ) )
% 0.16/0.43 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(iftrueProp1_type,type,
% 0.16/0.43 iftrueProp1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(iftrueProp1,definition,
% 0.16/0.43 ( iftrueProp1
% 0.16/0.43 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( Xphi
% 0.16/0.43 => ( in @ Xx
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xz: $i] :
% 0.16/0.43 ( ( Xphi
% 0.16/0.43 & ( Xz = Xx ) )
% 0.16/0.43 | ( ~ Xphi
% 0.16/0.43 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(iftrueProp2_type,type,
% 0.16/0.43 iftrueProp2: $o ).
% 0.16/0.43
% 0.16/0.43 thf(iftrueProp2,definition,
% 0.16/0.43 ( iftrueProp2
% 0.16/0.43 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( Xphi
% 0.16/0.43 => ( ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xz: $i] :
% 0.16/0.43 ( ( Xphi
% 0.16/0.43 & ( Xz = Xx ) )
% 0.16/0.43 | ( ~ Xphi
% 0.16/0.43 & ( Xz = Xy ) ) ) )
% 0.16/0.43 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(ifSingleton_type,type,
% 0.16/0.43 ifSingleton: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ifSingleton,definition,
% 0.16/0.43 ( ifSingleton
% 0.16/0.43 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( singleton
% 0.16/0.43 @ ( dsetconstr @ A
% 0.16/0.43 @ ^ [Xz: $i] :
% 0.16/0.43 ( ( Xphi
% 0.16/0.43 & ( Xz = Xx ) )
% 0.16/0.43 | ( ~ Xphi
% 0.16/0.43 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(if_type,type,
% 0.16/0.43 if: $i > $o > $i > $i > $i ).
% 0.16/0.43
% 0.16/0.43 thf(ifp_type,type,
% 0.16/0.43 ifp: $o ).
% 0.16/0.43
% 0.16/0.43 thf(ifp,definition,
% 0.16/0.43 ( ifp
% 0.16/0.43 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(theeq_type,type,
% 0.16/0.43 theeq: $o ).
% 0.16/0.43
% 0.16/0.43 thf(theeq,definition,
% 0.16/0.43 ( theeq
% 0.16/0.43 = ( ! [X: $i] :
% 0.16/0.43 ( ( singleton @ X )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ X )
% 0.16/0.43 => ( ( setunion @ X )
% 0.16/0.43 = Xx ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(iftrue_type,type,
% 0.16/0.43 iftrue: $o ).
% 0.16/0.43
% 0.16/0.43 thf(iftrue,definition,
% 0.16/0.43 ( iftrue
% 0.16/0.43 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( Xphi
% 0.16/0.43 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.16/0.43 = Xx ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(iffalse_type,type,
% 0.16/0.43 iffalse: $o ).
% 0.16/0.43
% 0.16/0.43 thf(iffalse,definition,
% 0.16/0.43 ( iffalse
% 0.16/0.43 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( ~ Xphi
% 0.16/0.43 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.16/0.43 = Xy ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(iftrueorfalse_type,type,
% 0.16/0.43 iftrueorfalse: $o ).
% 0.16/0.43
% 0.16/0.43 thf(iftrueorfalse,definition,
% 0.16/0.43 ( iftrueorfalse
% 0.16/0.43 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ! [Xy: $i] :
% 0.16/0.43 ( ( in @ Xy @ A )
% 0.16/0.43 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectT_lem_type,type,
% 0.16/0.43 binintersectT_lem: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binintersectT_lem,definition,
% 0.16/0.43 ( binintersectT_lem
% 0.16/0.43 = ( ! [A: $i,X: $i] :
% 0.16/0.43 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.43 => ! [Y: $i] :
% 0.16/0.43 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.43 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(binunionT_lem_type,type,
% 0.16/0.43 binunionT_lem: $o ).
% 0.16/0.43
% 0.16/0.43 thf(binunionT_lem,definition,
% 0.16/0.43 ( binunionT_lem
% 0.16/0.43 = ( ! [A: $i,X: $i] :
% 0.16/0.43 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.43 => ! [Y: $i] :
% 0.16/0.43 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.43 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powersetT_lem_type,type,
% 0.16/0.43 powersetT_lem: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetT_lem,definition,
% 0.16/0.43 ( powersetT_lem
% 0.16/0.43 = ( ! [A: $i,X: $i] :
% 0.16/0.43 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.43 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setminusT_lem_type,type,
% 0.16/0.43 setminusT_lem: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setminusT_lem,definition,
% 0.16/0.43 ( setminusT_lem
% 0.16/0.43 = ( ! [A: $i,X: $i] :
% 0.16/0.43 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.43 => ! [Y: $i] :
% 0.16/0.43 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.43 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(complementT_lem_type,type,
% 0.16/0.43 complementT_lem: $o ).
% 0.16/0.43
% 0.16/0.43 thf(complementT_lem,definition,
% 0.16/0.43 ( complementT_lem
% 0.16/0.43 = ( ! [A: $i,X: $i] :
% 0.16/0.43 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.43 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(setextT_type,type,
% 0.16/0.43 setextT: $o ).
% 0.16/0.43
% 0.16/0.43 thf(setextT,definition,
% 0.16/0.43 ( setextT
% 0.16/0.43 = ( ! [A: $i,X: $i] :
% 0.16/0.43 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.43 => ! [Y: $i] :
% 0.16/0.43 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ Xx @ X )
% 0.16/0.43 => ( in @ Xx @ Y ) ) )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ Xx @ Y )
% 0.16/0.43 => ( in @ Xx @ X ) ) )
% 0.16/0.43 => ( X = Y ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(subsetTI_type,type,
% 0.16/0.43 subsetTI: $o ).
% 0.16/0.43
% 0.16/0.43 thf(subsetTI,definition,
% 0.16/0.43 ( subsetTI
% 0.16/0.43 = ( ! [A: $i,X: $i] :
% 0.16/0.43 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.43 => ! [Y: $i] :
% 0.16/0.43 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ Xx @ X )
% 0.16/0.43 => ( in @ Xx @ Y ) ) )
% 0.16/0.43 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powersetTI1_type,type,
% 0.16/0.43 powersetTI1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetTI1,definition,
% 0.16/0.43 ( powersetTI1
% 0.16/0.43 = ( ! [A: $i,X: $i] :
% 0.16/0.43 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.43 => ! [Y: $i] :
% 0.16/0.43 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.43 => ( ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ Xx @ X )
% 0.16/0.43 => ( in @ Xx @ Y ) ) )
% 0.16/0.43 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(powersetTE1_type,type,
% 0.16/0.43 powersetTE1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(powersetTE1,definition,
% 0.16/0.43 ( powersetTE1
% 0.16/0.43 = ( ! [A: $i,X: $i] :
% 0.16/0.43 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.43 => ! [Y: $i] :
% 0.16/0.43 ( ( in @ Y @ ( powerset @ A ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ X @ ( powerset @ Y ) )
% 0.16/0.43 => ( ( in @ Xx @ X )
% 0.16/0.43 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(complementTI1_type,type,
% 0.16/0.43 complementTI1: $o ).
% 0.16/0.43
% 0.16/0.43 thf(complementTI1,definition,
% 0.16/0.43 ( complementTI1
% 0.16/0.43 = ( ! [A: $i,X: $i] :
% 0.16/0.43 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ( in @ Xx @ X )
% 0.16/0.43 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 thf(complementTE1,conjecture,
% 0.16/0.43 ( setextAx
% 0.16/0.43 => ( emptysetAx
% 0.16/0.43 => ( setadjoinAx
% 0.16/0.43 => ( powersetAx
% 0.16/0.43 => ( setunionAx
% 0.16/0.43 => ( omega0Ax
% 0.16/0.43 => ( omegaSAx
% 0.16/0.43 => ( omegaIndAx
% 0.16/0.43 => ( replAx
% 0.16/0.43 => ( foundationAx
% 0.16/0.43 => ( wellorderingAx
% 0.16/0.43 => ( descrp
% 0.16/0.43 => ( dsetconstrI
% 0.16/0.43 => ( dsetconstrEL
% 0.16/0.43 => ( dsetconstrER
% 0.16/0.43 => ( exuE1
% 0.16/0.43 => ( prop2setE
% 0.16/0.43 => ( emptysetE
% 0.16/0.43 => ( emptysetimpfalse
% 0.16/0.43 => ( notinemptyset
% 0.16/0.43 => ( exuE3e
% 0.16/0.43 => ( setext
% 0.16/0.43 => ( emptyI
% 0.16/0.43 => ( noeltsimpempty
% 0.16/0.43 => ( setbeta
% 0.16/0.43 => ( nonemptyE1
% 0.16/0.43 => ( nonemptyI
% 0.16/0.43 => ( nonemptyI1
% 0.16/0.43 => ( setadjoinIL
% 0.16/0.43 => ( emptyinunitempty
% 0.16/0.43 => ( setadjoinIR
% 0.16/0.43 => ( setadjoinE
% 0.16/0.43 => ( setadjoinOr
% 0.16/0.43 => ( setoftrueEq
% 0.16/0.43 => ( powersetI
% 0.16/0.43 => ( emptyinPowerset
% 0.16/0.43 => ( emptyInPowerset
% 0.16/0.43 => ( powersetE
% 0.16/0.43 => ( setunionI
% 0.16/0.43 => ( setunionE
% 0.16/0.43 => ( subPowSU
% 0.16/0.43 => ( exuE2
% 0.16/0.43 => ( nonemptyImpWitness
% 0.16/0.43 => ( uniqinunit
% 0.16/0.43 => ( notinsingleton
% 0.16/0.43 => ( eqinunit
% 0.16/0.43 => ( singletonsswitch
% 0.16/0.43 => ( upairsetE
% 0.16/0.43 => ( upairsetIL
% 0.16/0.43 => ( upairsetIR
% 0.16/0.43 => ( emptyE1
% 0.16/0.43 => ( vacuousDall
% 0.16/0.43 => ( quantDeMorgan1
% 0.16/0.43 => ( quantDeMorgan2
% 0.16/0.43 => ( quantDeMorgan3
% 0.16/0.43 => ( quantDeMorgan4
% 0.16/0.43 => ( prop2setI
% 0.16/0.43 => ( prop2set2propI
% 0.16/0.43 => ( notdexE
% 0.16/0.43 => ( notdallE
% 0.16/0.43 => ( exuI1
% 0.16/0.43 => ( exuI3
% 0.16/0.43 => ( exuI2
% 0.16/0.43 => ( inCongP
% 0.16/0.43 => ( in__Cong
% 0.16/0.43 => ( exuE3u
% 0.16/0.43 => ( exu__Cong
% 0.16/0.43 => ( emptyset__Cong
% 0.16/0.43 => ( setadjoin__Cong
% 0.16/0.43 => ( powerset__Cong
% 0.16/0.43 => ( setunion__Cong
% 0.16/0.43 => ( omega__Cong
% 0.16/0.43 => ( exuEu
% 0.16/0.43 => ( descr__Cong
% 0.16/0.43 => ( dsetconstr__Cong
% 0.16/0.43 => ( subsetI1
% 0.16/0.43 => ( eqimpsubset2
% 0.16/0.43 => ( eqimpsubset1
% 0.16/0.43 => ( subsetI2
% 0.16/0.43 => ( emptysetsubset
% 0.16/0.43 => ( subsetE
% 0.16/0.43 => ( subsetE2
% 0.16/0.43 => ( notsubsetI
% 0.16/0.43 => ( notequalI1
% 0.16/0.43 => ( notequalI2
% 0.16/0.43 => ( subsetRefl
% 0.16/0.43 => ( subsetTrans
% 0.16/0.43 => ( setadjoinSub
% 0.16/0.43 => ( setadjoinSub2
% 0.16/0.43 => ( subset2powerset
% 0.16/0.43 => ( setextsub
% 0.16/0.43 => ( subsetemptysetimpeq
% 0.16/0.43 => ( powersetI1
% 0.16/0.43 => ( powersetE1
% 0.16/0.43 => ( inPowerset
% 0.16/0.43 => ( powersetsubset
% 0.16/0.43 => ( sepInPowerset
% 0.16/0.43 => ( sepSubset
% 0.16/0.43 => ( binunionIL
% 0.16/0.43 => ( upairset2IR
% 0.16/0.43 => ( binunionIR
% 0.16/0.43 => ( binunionEcases
% 0.16/0.43 => ( binunionE
% 0.16/0.43 => ( binunionLsub
% 0.16/0.43 => ( binunionRsub
% 0.16/0.43 => ( binintersectI
% 0.16/0.43 => ( binintersectSubset5
% 0.16/0.43 => ( binintersectEL
% 0.16/0.43 => ( binintersectLsub
% 0.16/0.43 => ( binintersectSubset2
% 0.16/0.43 => ( binintersectSubset3
% 0.16/0.43 => ( binintersectER
% 0.16/0.43 => ( disjointsetsI1
% 0.16/0.43 => ( binintersectRsub
% 0.16/0.43 => ( binintersectSubset4
% 0.16/0.43 => ( binintersectSubset1
% 0.16/0.43 => ( bs114d
% 0.16/0.43 => ( setminusI
% 0.16/0.43 => ( setminusEL
% 0.16/0.43 => ( setminusER
% 0.16/0.43 => ( setminusSubset2
% 0.16/0.43 => ( setminusERneg
% 0.16/0.43 => ( setminusELneg
% 0.16/0.43 => ( setminusILneg
% 0.16/0.43 => ( setminusIRneg
% 0.16/0.43 => ( setminusLsub
% 0.16/0.43 => ( setminusSubset1
% 0.16/0.43 => ( symdiffE
% 0.16/0.43 => ( symdiffI1
% 0.16/0.43 => ( symdiffI2
% 0.16/0.43 => ( symdiffIneg1
% 0.16/0.43 => ( symdiffIneg2
% 0.16/0.43 => ( secondinupair
% 0.16/0.43 => ( setukpairIL
% 0.16/0.43 => ( setukpairIR
% 0.16/0.43 => ( kpairiskpair
% 0.16/0.43 => ( kpairp
% 0.16/0.43 => ( singletonsubset
% 0.16/0.43 => ( singletoninpowerset
% 0.16/0.43 => ( singletoninpowunion
% 0.16/0.43 => ( upairset2E
% 0.16/0.43 => ( upairsubunion
% 0.16/0.43 => ( upairinpowunion
% 0.16/0.43 => ( ubforcartprodlem1
% 0.16/0.43 => ( ubforcartprodlem2
% 0.16/0.43 => ( ubforcartprodlem3
% 0.16/0.43 => ( cartprodpairin
% 0.16/0.43 => ( cartprodmempair1
% 0.16/0.43 => ( cartprodmempair
% 0.16/0.43 => ( setunionE2
% 0.16/0.43 => ( setunionsingleton1
% 0.16/0.43 => ( setunionsingleton2
% 0.16/0.43 => ( setunionsingleton
% 0.16/0.43 => ( singletonprop
% 0.16/0.43 => ( ex1E1
% 0.16/0.43 => ( ex1I
% 0.16/0.43 => ( ex1I2
% 0.16/0.43 => ( singletonsuniq
% 0.16/0.43 => ( setukpairinjL1
% 0.16/0.43 => ( kfstsingleton
% 0.16/0.43 => ( theprop
% 0.16/0.43 => ( kfstpairEq
% 0.16/0.43 => ( cartprodfstin
% 0.16/0.43 => ( setukpairinjL2
% 0.16/0.43 => ( setukpairinjL
% 0.16/0.43 => ( setukpairinjR11
% 0.16/0.43 => ( setukpairinjR12
% 0.16/0.43 => ( setukpairinjR1
% 0.16/0.43 => ( upairequniteq
% 0.16/0.43 => ( setukpairinjR2
% 0.16/0.43 => ( setukpairinjR
% 0.16/0.43 => ( ksndsingleton
% 0.16/0.43 => ( ksndpairEq
% 0.16/0.43 => ( kpairsurjEq
% 0.16/0.43 => ( cartprodsndin
% 0.16/0.43 => ( cartprodpairmemEL
% 0.16/0.43 => ( cartprodpairmemER
% 0.16/0.43 => ( cartprodmempaircEq
% 0.16/0.43 => ( cartprodfstpairEq
% 0.16/0.43 => ( cartprodsndpairEq
% 0.16/0.43 => ( cartprodpairsurjEq
% 0.16/0.43 => ( dpsetconstrI
% 0.16/0.43 => ( dpsetconstrSub
% 0.16/0.43 => ( setOfPairsIsBReln
% 0.16/0.43 => ( dpsetconstrERa
% 0.16/0.43 => ( dpsetconstrEL1
% 0.16/0.43 => ( dpsetconstrEL2
% 0.16/0.43 => ( dpsetconstrER
% 0.16/0.43 => ( funcImageSingleton
% 0.16/0.43 => ( apProp
% 0.16/0.43 => ( app
% 0.16/0.43 => ( infuncsetfunc
% 0.16/0.43 => ( ap2p
% 0.16/0.43 => ( funcinfuncset
% 0.16/0.43 => ( lamProp
% 0.16/0.43 => ( lamp
% 0.16/0.43 => ( lam2p
% 0.16/0.43 => ( brelnall1
% 0.16/0.43 => ( brelnall2
% 0.16/0.43 => ( ex1E2
% 0.16/0.43 => ( funcGraphProp1
% 0.16/0.43 => ( funcGraphProp3
% 0.16/0.43 => ( funcGraphProp2
% 0.16/0.43 => ( funcextLem
% 0.16/0.43 => ( funcGraphProp4
% 0.16/0.43 => ( subbreln
% 0.16/0.43 => ( eqbreln
% 0.16/0.43 => ( funcext
% 0.16/0.43 => ( funcext2
% 0.16/0.43 => ( ap2apEq1
% 0.16/0.43 => ( ap2apEq2
% 0.16/0.43 => ( beta1
% 0.16/0.43 => ( eta1
% 0.16/0.43 => ( lam2lamEq
% 0.16/0.43 => ( beta2
% 0.16/0.43 => ( eta2
% 0.16/0.43 => ( iffalseProp1
% 0.16/0.43 => ( iffalseProp2
% 0.16/0.43 => ( iftrueProp1
% 0.16/0.43 => ( iftrueProp2
% 0.16/0.43 => ( ifSingleton
% 0.16/0.43 => ( ifp
% 0.16/0.43 => ( theeq
% 0.16/0.43 => ( iftrue
% 0.16/0.43 => ( iffalse
% 0.16/0.43 => ( iftrueorfalse
% 0.16/0.43 => ( binintersectT_lem
% 0.16/0.43 => ( binunionT_lem
% 0.16/0.43 => ( powersetT_lem
% 0.16/0.43 => ( setminusT_lem
% 0.16/0.43 => ( complementT_lem
% 0.16/0.43 => ( setextT
% 0.16/0.43 => ( subsetTI
% 0.16/0.43 => ( powersetTI1
% 0.16/0.43 => ( powersetTE1
% 0.16/0.43 => ( complementTI1
% 0.16/0.43 => ! [A: $i,X: $i] :
% 0.16/0.43 ( ( in @ X @ ( powerset @ A ) )
% 0.16/0.43 => ! [Xx: $i] :
% 0.16/0.43 ( ( in @ Xx @ A )
% 0.16/0.43 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.16/0.43 => ( in @ Xx @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.16/0.43
% 0.16/0.43 %------------------------------------------------------------------------------
% 0.16/0.43 ------------------------
% 2.07/1.43 ---- Embedded in THF ---
% 2.07/1.43 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.07/1.43 %%% Generated on Tue Feb 24 12:31:33 EST 2026
% 2.07/1.43 %%% using '$$dhol' embedding, version 1.3.0.
% 2.07/1.43 %%% Logic specification used:
% 2.07/1.43 %%% thf(spec, logic, $$dhol).
% 2.07/1.43
% 2.07/1.43 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.SCLFyHJTHZ/DTF2DTF24371.p
% See solution above
% 2.07/1.43 ------------------------
% 2.07/1.46 ---- Cleaned THF ---
% 2.07/1.46 thf(in_type,type,
% 2.07/1.46 in: $i > $i > $o ).
% 2.07/1.46
% 2.07/1.46 thf(exu_type,type,
% 2.07/1.46 exu: ( $i > $o ) > $o ).
% 2.07/1.46
% 2.07/1.46 thf(exu,definition,
% 2.07/1.46 ( exu
% 2.07/1.46 = ( ^ [Xphi: $i > $o] :
% 2.07/1.46 ? [Xx: $i] :
% 2.07/1.46 ( ( Xphi @ Xx )
% 2.07/1.46 & ! [Xy: $i] :
% 2.07/1.46 ( ( Xphi @ Xy )
% 2.07/1.46 => ( Xx = Xy ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setextAx_type,type,
% 2.07/1.46 setextAx: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setextAx,definition,
% 2.07/1.46 ( setextAx
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 <=> ( in @ Xx @ B ) )
% 2.07/1.46 => ( A = B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(emptyset_type,type,
% 2.07/1.46 emptyset: $i ).
% 2.07/1.46
% 2.07/1.46 thf(emptysetAx_type,type,
% 2.07/1.46 emptysetAx: $o ).
% 2.07/1.46
% 2.07/1.46 thf(emptysetAx,definition,
% 2.07/1.46 ( emptysetAx
% 2.07/1.46 = ( ! [Xx: $i] :
% 2.07/1.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoin_type,type,
% 2.07/1.46 setadjoin: $i > $i > $i ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoinAx_type,type,
% 2.07/1.46 setadjoinAx: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoinAx,definition,
% 2.07/1.46 ( setadjoinAx
% 2.07/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.07/1.46 <=> ( ( Xy = Xx )
% 2.07/1.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(powerset_type,type,
% 2.07/1.46 powerset: $i > $i ).
% 2.07/1.46
% 2.07/1.46 thf(powersetAx_type,type,
% 2.07/1.46 powersetAx: $o ).
% 2.07/1.46
% 2.07/1.46 thf(powersetAx,definition,
% 2.07/1.46 ( powersetAx
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( in @ B @ ( powerset @ A ) )
% 2.07/1.46 <=> ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ B )
% 2.07/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setunion_type,type,
% 2.07/1.46 setunion: $i > $i ).
% 2.07/1.46
% 2.07/1.46 thf(setunionAx_type,type,
% 2.07/1.46 setunionAx: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setunionAx,definition,
% 2.07/1.46 ( setunionAx
% 2.07/1.46 = ( ! [A: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ ( setunion @ A ) )
% 2.07/1.46 <=> ? [B: $i] :
% 2.07/1.46 ( ( in @ Xx @ B )
% 2.07/1.46 & ( in @ B @ A ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(omega_type,type,
% 2.07/1.46 omega: $i ).
% 2.07/1.46
% 2.07/1.46 thf(omega0Ax_type,type,
% 2.07/1.46 omega0Ax: $o ).
% 2.07/1.46
% 2.07/1.46 thf(omega0Ax,definition,
% 2.07/1.46 ( omega0Ax
% 2.07/1.46 = ( in @ emptyset @ omega ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(omegaSAx_type,type,
% 2.07/1.46 omegaSAx: $o ).
% 2.07/1.46
% 2.07/1.46 thf(omegaSAx,definition,
% 2.07/1.46 ( omegaSAx
% 2.07/1.46 = ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ omega )
% 2.07/1.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(omegaIndAx_type,type,
% 2.07/1.46 omegaIndAx: $o ).
% 2.07/1.46
% 2.07/1.46 thf(omegaIndAx,definition,
% 2.07/1.46 ( omegaIndAx
% 2.07/1.46 = ( ! [A: $i] :
% 2.07/1.46 ( ( ( in @ emptyset @ A )
% 2.07/1.46 & ! [Xx: $i] :
% 2.07/1.46 ( ( ( in @ Xx @ omega )
% 2.07/1.46 & ( in @ Xx @ A ) )
% 2.07/1.46 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ omega )
% 2.07/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(replAx_type,type,
% 2.07/1.46 replAx: $o ).
% 2.07/1.46
% 2.07/1.46 thf(replAx,definition,
% 2.07/1.46 ( replAx
% 2.07/1.46 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( exu
% 2.07/1.46 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 2.07/1.46 => ? [B: $i] :
% 2.07/1.46 ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ B )
% 2.07/1.46 <=> ? [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ A )
% 2.07/1.46 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(foundationAx_type,type,
% 2.07/1.46 foundationAx: $o ).
% 2.07/1.46
% 2.07/1.46 thf(foundationAx,definition,
% 2.07/1.46 ( foundationAx
% 2.07/1.46 = ( ! [A: $i] :
% 2.07/1.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.07/1.46 => ? [B: $i] :
% 2.07/1.46 ( ( in @ B @ A )
% 2.07/1.46 & ~ ? [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ B )
% 2.07/1.46 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(wellorderingAx_type,type,
% 2.07/1.46 wellorderingAx: $o ).
% 2.07/1.46
% 2.07/1.46 thf(wellorderingAx,definition,
% 2.07/1.46 ( wellorderingAx
% 2.07/1.46 = ( ! [A: $i] :
% 2.07/1.46 ? [B: $i] :
% 2.07/1.46 ( ! [C: $i] :
% 2.07/1.46 ( ( in @ C @ B )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ C )
% 2.07/1.46 => ( in @ Xx @ A ) ) )
% 2.07/1.46 & ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( ( in @ Xx @ A )
% 2.07/1.46 & ( in @ Xy @ A ) )
% 2.07/1.46 => ( ! [C: $i] :
% 2.07/1.46 ( ( in @ C @ B )
% 2.07/1.46 => ( ( in @ Xx @ C )
% 2.07/1.46 <=> ( in @ Xy @ C ) ) )
% 2.07/1.46 => ( Xx = Xy ) ) )
% 2.07/1.46 & ! [C: $i,D: $i] :
% 2.07/1.46 ( ( ( in @ C @ B )
% 2.07/1.46 & ( in @ D @ B ) )
% 2.07/1.46 => ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ C )
% 2.07/1.46 => ( in @ Xx @ D ) )
% 2.07/1.46 | ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ D )
% 2.07/1.46 => ( in @ Xx @ C ) ) ) )
% 2.07/1.46 & ! [C: $i] :
% 2.07/1.46 ( ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ C )
% 2.07/1.46 => ( in @ Xx @ A ) )
% 2.07/1.46 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 2.07/1.46 => ? [D: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ D @ B )
% 2.07/1.46 & ( in @ Xx @ C )
% 2.07/1.46 & ~ ? [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ D )
% 2.07/1.46 & ( in @ Xy @ C ) )
% 2.07/1.46 & ! [E: $i] :
% 2.07/1.46 ( ( in @ E @ B )
% 2.07/1.46 => ( ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ E )
% 2.07/1.46 => ( in @ Xy @ D ) )
% 2.07/1.46 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(descr_type,type,
% 2.07/1.46 descr: ( $i > $o ) > $i ).
% 2.07/1.46
% 2.07/1.46 thf(descrp_type,type,
% 2.07/1.46 descrp: $o ).
% 2.07/1.46
% 2.07/1.46 thf(descrp,definition,
% 2.07/1.46 ( descrp
% 2.07/1.46 = ( ! [Xphi: $i > $o] :
% 2.07/1.46 ( ( exu
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.07/1.46 => ( Xphi
% 2.07/1.46 @ ( descr
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(dsetconstr_type,type,
% 2.07/1.46 dsetconstr: $i > ( $i > $o ) > $i ).
% 2.07/1.46
% 2.07/1.46 thf(dsetconstrI_type,type,
% 2.07/1.46 dsetconstrI: $o ).
% 2.07/1.46
% 2.07/1.46 thf(dsetconstrI,definition,
% 2.07/1.46 ( dsetconstrI
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ( Xphi @ Xx )
% 2.07/1.46 => ( in @ Xx
% 2.07/1.46 @ ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(dsetconstrEL_type,type,
% 2.07/1.46 dsetconstrEL: $o ).
% 2.07/1.46
% 2.07/1.46 thf(dsetconstrEL,definition,
% 2.07/1.46 ( dsetconstrEL
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx
% 2.07/1.46 @ ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.07/1.46 => ( in @ Xx @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(dsetconstrER_type,type,
% 2.07/1.46 dsetconstrER: $o ).
% 2.07/1.46
% 2.07/1.46 thf(dsetconstrER,definition,
% 2.07/1.46 ( dsetconstrER
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx
% 2.07/1.46 @ ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.07/1.46 => ( Xphi @ Xx ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(exuE1_type,type,
% 2.07/1.46 exuE1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(exuE1,definition,
% 2.07/1.46 ( exuE1
% 2.07/1.46 = ( ! [Xphi: $i > $o] :
% 2.07/1.46 ( ( exu
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.07/1.46 => ? [Xx: $i] :
% 2.07/1.46 ( ( Xphi @ Xx )
% 2.07/1.46 & ! [Xy: $i] :
% 2.07/1.46 ( ( Xphi @ Xy )
% 2.07/1.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(prop2set_type,type,
% 2.07/1.46 prop2set: $o > $i ).
% 2.07/1.46
% 2.07/1.46 thf(prop2setE_type,type,
% 2.07/1.46 prop2setE: $o ).
% 2.07/1.46
% 2.07/1.46 thf(prop2setE,definition,
% 2.07/1.46 ( prop2setE
% 2.07/1.46 = ( ! [Xphi: $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 2.07/1.46 => Xphi ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(emptysetE_type,type,
% 2.07/1.46 emptysetE: $o ).
% 2.07/1.46
% 2.07/1.46 thf(emptysetE,definition,
% 2.07/1.46 ( emptysetE
% 2.07/1.46 = ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ emptyset )
% 2.07/1.46 => ! [Xphi: $o] : Xphi ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(emptysetimpfalse_type,type,
% 2.07/1.46 emptysetimpfalse: $o ).
% 2.07/1.46
% 2.07/1.46 thf(emptysetimpfalse,definition,
% 2.07/1.46 ( emptysetimpfalse
% 2.07/1.46 = ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ emptyset )
% 2.07/1.46 => $false ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(notinemptyset_type,type,
% 2.07/1.46 notinemptyset: $o ).
% 2.07/1.46
% 2.07/1.46 thf(notinemptyset,definition,
% 2.07/1.46 ( notinemptyset
% 2.07/1.46 = ( ! [Xx: $i] :
% 2.07/1.46 ~ ( in @ Xx @ emptyset ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(exuE3e_type,type,
% 2.07/1.46 exuE3e: $o ).
% 2.07/1.46
% 2.07/1.46 thf(exuE3e,definition,
% 2.07/1.46 ( exuE3e
% 2.07/1.46 = ( ! [Xphi: $i > $o] :
% 2.07/1.46 ( ( exu
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.07/1.46 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setext_type,type,
% 2.07/1.46 setext: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setext,definition,
% 2.07/1.46 ( setext
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ Xx @ B ) )
% 2.07/1.46 => ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ B )
% 2.07/1.46 => ( in @ Xx @ A ) )
% 2.07/1.46 => ( A = B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(emptyI_type,type,
% 2.07/1.46 emptyI: $o ).
% 2.07/1.46
% 2.07/1.46 thf(emptyI,definition,
% 2.07/1.46 ( emptyI
% 2.07/1.46 = ( ! [A: $i] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ~ ( in @ Xx @ A )
% 2.07/1.46 => ( A = emptyset ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(noeltsimpempty_type,type,
% 2.07/1.46 noeltsimpempty: $o ).
% 2.07/1.46
% 2.07/1.46 thf(noeltsimpempty,definition,
% 2.07/1.46 ( noeltsimpempty
% 2.07/1.46 = ( ! [A: $i] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ~ ( in @ Xx @ A )
% 2.07/1.46 => ( A = emptyset ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setbeta_type,type,
% 2.07/1.46 setbeta: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setbeta,definition,
% 2.07/1.46 ( setbeta
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ( in @ Xx
% 2.07/1.46 @ ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 2.07/1.46 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(nonempty_type,type,
% 2.07/1.46 nonempty: $i > $o ).
% 2.07/1.46
% 2.07/1.46 thf(nonempty,definition,
% 2.07/1.46 ( nonempty
% 2.07/1.46 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(nonemptyE1_type,type,
% 2.07/1.46 nonemptyE1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(nonemptyE1,definition,
% 2.07/1.46 ( nonemptyE1
% 2.07/1.46 = ( ! [A: $i] :
% 2.07/1.46 ( ( nonempty @ A )
% 2.07/1.46 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(nonemptyI_type,type,
% 2.07/1.46 nonemptyI: $o ).
% 2.07/1.46
% 2.07/1.46 thf(nonemptyI,definition,
% 2.07/1.46 ( nonemptyI
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ( Xphi @ Xx )
% 2.07/1.46 => ( nonempty
% 2.07/1.46 @ ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(nonemptyI1_type,type,
% 2.07/1.46 nonemptyI1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(nonemptyI1,definition,
% 2.07/1.46 ( nonemptyI1
% 2.07/1.46 = ( ! [A: $i] :
% 2.07/1.46 ( ? [Xx: $i] : ( in @ Xx @ A )
% 2.07/1.46 => ( nonempty @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoinIL_type,type,
% 2.07/1.46 setadjoinIL: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoinIL,definition,
% 2.07/1.46 ( setadjoinIL
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(emptyinunitempty_type,type,
% 2.07/1.46 emptyinunitempty: $o ).
% 2.07/1.46
% 2.07/1.46 thf(emptyinunitempty,definition,
% 2.07/1.46 ( emptyinunitempty
% 2.07/1.46 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoinIR_type,type,
% 2.07/1.46 setadjoinIR: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoinIR,definition,
% 2.07/1.46 ( setadjoinIR
% 2.07/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ A )
% 2.07/1.46 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoinE_type,type,
% 2.07/1.46 setadjoinE: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoinE,definition,
% 2.07/1.46 ( setadjoinE
% 2.07/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.07/1.46 => ! [Xphi: $o] :
% 2.07/1.46 ( ( ( Xy = Xx )
% 2.07/1.46 => Xphi )
% 2.07/1.46 => ( ( ( in @ Xy @ A )
% 2.07/1.46 => Xphi )
% 2.07/1.46 => Xphi ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoinOr_type,type,
% 2.07/1.46 setadjoinOr: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoinOr,definition,
% 2.07/1.46 ( setadjoinOr
% 2.07/1.46 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 2.07/1.46 => ( ( Xy = Xx )
% 2.07/1.46 | ( in @ Xy @ A ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setoftrueEq_type,type,
% 2.07/1.46 setoftrueEq: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setoftrueEq,definition,
% 2.07/1.46 ( setoftrueEq
% 2.07/1.46 = ( ! [A: $i] :
% 2.07/1.46 ( ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xx: $i] : $true )
% 2.07/1.46 = A ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(powersetI_type,type,
% 2.07/1.46 powersetI: $o ).
% 2.07/1.46
% 2.07/1.46 thf(powersetI,definition,
% 2.07/1.46 ( powersetI
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ B )
% 2.07/1.46 => ( in @ Xx @ A ) )
% 2.07/1.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(emptyinPowerset_type,type,
% 2.07/1.46 emptyinPowerset: $o ).
% 2.07/1.46
% 2.07/1.46 thf(emptyinPowerset,definition,
% 2.07/1.46 ( emptyinPowerset
% 2.07/1.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(emptyInPowerset_type,type,
% 2.07/1.46 emptyInPowerset: $o ).
% 2.07/1.46
% 2.07/1.46 thf(emptyInPowerset,definition,
% 2.07/1.46 ( emptyInPowerset
% 2.07/1.46 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(powersetE_type,type,
% 2.07/1.46 powersetE: $o ).
% 2.07/1.46
% 2.07/1.46 thf(powersetE,definition,
% 2.07/1.46 ( powersetE
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ B @ ( powerset @ A ) )
% 2.07/1.46 => ( ( in @ Xx @ B )
% 2.07/1.46 => ( in @ Xx @ A ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setunionI_type,type,
% 2.07/1.46 setunionI: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setunionI,definition,
% 2.07/1.46 ( setunionI
% 2.07/1.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.07/1.46 ( ( in @ Xx @ B )
% 2.07/1.46 => ( ( in @ B @ A )
% 2.07/1.46 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setunionE_type,type,
% 2.07/1.46 setunionE: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setunionE,definition,
% 2.07/1.46 ( setunionE
% 2.07/1.46 = ( ! [A: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ ( setunion @ A ) )
% 2.07/1.46 => ! [Xphi: $o] :
% 2.07/1.46 ( ! [B: $i] :
% 2.07/1.46 ( ( in @ Xx @ B )
% 2.07/1.46 => ( ( in @ B @ A )
% 2.07/1.46 => Xphi ) )
% 2.07/1.46 => Xphi ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(subPowSU_type,type,
% 2.07/1.46 subPowSU: $o ).
% 2.07/1.46
% 2.07/1.46 thf(subPowSU,definition,
% 2.07/1.46 ( subPowSU
% 2.07/1.46 = ( ! [A: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(exuE2_type,type,
% 2.07/1.46 exuE2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(exuE2,definition,
% 2.07/1.46 ( exuE2
% 2.07/1.46 = ( ! [Xphi: $i > $o] :
% 2.07/1.46 ( ( exu
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.07/1.46 => ? [Xx: $i] :
% 2.07/1.46 ! [Xy: $i] :
% 2.07/1.46 ( ( Xphi @ Xy )
% 2.07/1.46 <=> ( Xy = Xx ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(nonemptyImpWitness_type,type,
% 2.07/1.46 nonemptyImpWitness: $o ).
% 2.07/1.46
% 2.07/1.46 thf(nonemptyImpWitness,definition,
% 2.07/1.46 ( nonemptyImpWitness
% 2.07/1.46 = ( ! [A: $i] :
% 2.07/1.46 ( ( nonempty @ A )
% 2.07/1.46 => ? [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 & $true ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(uniqinunit_type,type,
% 2.07/1.46 uniqinunit: $o ).
% 2.07/1.46
% 2.07/1.46 thf(uniqinunit,definition,
% 2.07/1.46 ( uniqinunit
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.07/1.46 => ( Xx = Xy ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(notinsingleton_type,type,
% 2.07/1.46 notinsingleton: $o ).
% 2.07/1.46
% 2.07/1.46 thf(notinsingleton,definition,
% 2.07/1.46 ( notinsingleton
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( Xx != Xy )
% 2.07/1.46 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(eqinunit_type,type,
% 2.07/1.46 eqinunit: $o ).
% 2.07/1.46
% 2.07/1.46 thf(eqinunit,definition,
% 2.07/1.46 ( eqinunit
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( Xx = Xy )
% 2.07/1.46 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(singletonsswitch_type,type,
% 2.07/1.46 singletonsswitch: $o ).
% 2.07/1.46
% 2.07/1.46 thf(singletonsswitch,definition,
% 2.07/1.46 ( singletonsswitch
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.07/1.46 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(upairsetE_type,type,
% 2.07/1.46 upairsetE: $o ).
% 2.07/1.46
% 2.07/1.46 thf(upairsetE,definition,
% 2.07/1.46 ( upairsetE
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.07/1.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.07/1.46 => ( ( Xz = Xx )
% 2.07/1.46 | ( Xz = Xy ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(upairsetIL_type,type,
% 2.07/1.46 upairsetIL: $o ).
% 2.07/1.46
% 2.07/1.46 thf(upairsetIL,definition,
% 2.07/1.46 ( upairsetIL
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(upairsetIR_type,type,
% 2.07/1.46 upairsetIR: $o ).
% 2.07/1.46
% 2.07/1.46 thf(upairsetIR,definition,
% 2.07/1.46 ( upairsetIR
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(emptyE1_type,type,
% 2.07/1.46 emptyE1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(emptyE1,definition,
% 2.07/1.46 ( emptyE1
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.07/1.46 ( ? [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 & ( Xphi @ Xx ) )
% 2.07/1.46 => ( ( ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.07/1.46 = emptyset )
% 2.07/1.46 => $false ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(vacuousDall_type,type,
% 2.07/1.46 vacuousDall: $o ).
% 2.07/1.46
% 2.07/1.46 thf(vacuousDall,definition,
% 2.07/1.46 ( vacuousDall
% 2.07/1.46 = ( ! [Xphi: $i > $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ emptyset )
% 2.07/1.46 => ( Xphi @ Xx ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(quantDeMorgan1_type,type,
% 2.07/1.46 quantDeMorgan1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(quantDeMorgan1,definition,
% 2.07/1.46 ( quantDeMorgan1
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.07/1.46 ( ~ ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( Xphi @ Xx ) )
% 2.07/1.46 => ? [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(quantDeMorgan2_type,type,
% 2.07/1.46 quantDeMorgan2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(quantDeMorgan2,definition,
% 2.07/1.46 ( quantDeMorgan2
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ~ ( Xphi @ Xx ) )
% 2.07/1.46 => ~ ? [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(quantDeMorgan3_type,type,
% 2.07/1.46 quantDeMorgan3: $o ).
% 2.07/1.46
% 2.07/1.46 thf(quantDeMorgan3,definition,
% 2.07/1.46 ( quantDeMorgan3
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.07/1.46 ( ~ ? [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 & ( Xphi @ Xx ) )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(quantDeMorgan4_type,type,
% 2.07/1.46 quantDeMorgan4: $o ).
% 2.07/1.46
% 2.07/1.46 thf(quantDeMorgan4,definition,
% 2.07/1.46 ( quantDeMorgan4
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.07/1.46 ( ? [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 & ~ ( Xphi @ Xx ) )
% 2.07/1.46 => ~ ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( Xphi @ Xx ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(prop2setI_type,type,
% 2.07/1.46 prop2setI: $o ).
% 2.07/1.46
% 2.07/1.46 thf(prop2setI,definition,
% 2.07/1.46 ( prop2setI
% 2.07/1.46 = ( ! [Xphi: $o] :
% 2.07/1.46 ( Xphi
% 2.07/1.46 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(set2prop_type,type,
% 2.07/1.46 set2prop: $i > $o ).
% 2.07/1.46
% 2.07/1.46 thf(prop2set2propI_type,type,
% 2.07/1.46 prop2set2propI: $o ).
% 2.07/1.46
% 2.07/1.46 thf(prop2set2propI,definition,
% 2.07/1.46 ( prop2set2propI
% 2.07/1.46 = ( ! [Xphi: $o] :
% 2.07/1.46 ( Xphi
% 2.07/1.46 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(notdexE_type,type,
% 2.07/1.46 notdexE: $o ).
% 2.07/1.46
% 2.07/1.46 thf(notdexE,definition,
% 2.07/1.46 ( notdexE
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.07/1.46 ( ~ ? [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 & ( Xphi @ Xx ) )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(notdallE_type,type,
% 2.07/1.46 notdallE: $o ).
% 2.07/1.46
% 2.07/1.46 thf(notdallE,definition,
% 2.07/1.46 ( notdallE
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.07/1.46 ( ~ ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( Xphi @ Xx ) )
% 2.07/1.46 => ? [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(exuI1_type,type,
% 2.07/1.46 exuI1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(exuI1,definition,
% 2.07/1.46 ( exuI1
% 2.07/1.46 = ( ! [Xphi: $i > $o] :
% 2.07/1.46 ( ? [Xx: $i] :
% 2.07/1.46 ( ( Xphi @ Xx )
% 2.07/1.46 & ! [Xy: $i] :
% 2.07/1.46 ( ( Xphi @ Xy )
% 2.07/1.46 => ( Xx = Xy ) ) )
% 2.07/1.46 => ( exu
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(exuI3_type,type,
% 2.07/1.46 exuI3: $o ).
% 2.07/1.46
% 2.07/1.46 thf(exuI3,definition,
% 2.07/1.46 ( exuI3
% 2.07/1.46 = ( ! [Xphi: $i > $o] :
% 2.07/1.46 ( ? [Xx: $i] : ( Xphi @ Xx )
% 2.07/1.46 => ( ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( Xphi @ Xx )
% 2.07/1.46 => ( ( Xphi @ Xy )
% 2.07/1.46 => ( Xx = Xy ) ) )
% 2.07/1.46 => ( exu
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(exuI2_type,type,
% 2.07/1.46 exuI2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(exuI2,definition,
% 2.07/1.46 ( exuI2
% 2.07/1.46 = ( ! [Xphi: $i > $o] :
% 2.07/1.46 ( ? [Xx: $i] :
% 2.07/1.46 ! [Xy: $i] :
% 2.07/1.46 ( ( Xphi @ Xy )
% 2.07/1.46 <=> ( Xy = Xx ) )
% 2.07/1.46 => ( exu
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(inCongP_type,type,
% 2.07/1.46 inCongP: $o ).
% 2.07/1.46
% 2.07/1.46 thf(inCongP,definition,
% 2.07/1.46 ( inCongP
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( A = B )
% 2.07/1.46 => ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( Xx = Xy )
% 2.07/1.46 => ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(in__Cong_type,type,
% 2.07/1.46 in__Cong: $o ).
% 2.07/1.46
% 2.07/1.46 thf(in__Cong,definition,
% 2.07/1.46 ( in__Cong
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( A = B )
% 2.07/1.46 => ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( Xx = Xy )
% 2.07/1.46 => ( ( in @ Xx @ A )
% 2.07/1.46 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(exuE3u_type,type,
% 2.07/1.46 exuE3u: $o ).
% 2.07/1.46
% 2.07/1.46 thf(exuE3u,definition,
% 2.07/1.46 ( exuE3u
% 2.07/1.46 = ( ! [Xphi: $i > $o] :
% 2.07/1.46 ( ( exu
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.07/1.46 => ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( Xphi @ Xx )
% 2.07/1.46 => ( ( Xphi @ Xy )
% 2.07/1.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(exu__Cong_type,type,
% 2.07/1.46 exu__Cong: $o ).
% 2.07/1.46
% 2.07/1.46 thf(exu__Cong,definition,
% 2.07/1.46 ( exu__Cong
% 2.07/1.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.07/1.46 ( ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( Xx = Xy )
% 2.07/1.46 => ( ( Xphi @ Xx )
% 2.07/1.46 <=> ( Xpsi @ Xy ) ) )
% 2.07/1.46 => ( ( exu
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.07/1.46 <=> ( exu
% 2.07/1.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(emptyset__Cong_type,type,
% 2.07/1.46 emptyset__Cong: $o ).
% 2.07/1.46
% 2.07/1.46 thf(emptyset__Cong,definition,
% 2.07/1.46 ( emptyset__Cong
% 2.07/1.46 = ( emptyset = emptyset ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoin__Cong_type,type,
% 2.07/1.46 setadjoin__Cong: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoin__Cong,definition,
% 2.07/1.46 ( setadjoin__Cong
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( Xx = Xy )
% 2.07/1.46 => ! [Xz: $i,Xu: $i] :
% 2.07/1.46 ( ( Xz = Xu )
% 2.07/1.46 => ( ( setadjoin @ Xx @ Xz )
% 2.07/1.46 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(powerset__Cong_type,type,
% 2.07/1.46 powerset__Cong: $o ).
% 2.07/1.46
% 2.07/1.46 thf(powerset__Cong,definition,
% 2.07/1.46 ( powerset__Cong
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( A = B )
% 2.07/1.46 => ( ( powerset @ A )
% 2.07/1.46 = ( powerset @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setunion__Cong_type,type,
% 2.07/1.46 setunion__Cong: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setunion__Cong,definition,
% 2.07/1.46 ( setunion__Cong
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( A = B )
% 2.07/1.46 => ( ( setunion @ A )
% 2.07/1.46 = ( setunion @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(omega__Cong_type,type,
% 2.07/1.46 omega__Cong: $o ).
% 2.07/1.46
% 2.07/1.46 thf(omega__Cong,definition,
% 2.07/1.46 ( omega__Cong
% 2.07/1.46 = ( omega = omega ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(exuEu_type,type,
% 2.07/1.46 exuEu: $o ).
% 2.07/1.46
% 2.07/1.46 thf(exuEu,definition,
% 2.07/1.46 ( exuEu
% 2.07/1.46 = ( ! [Xphi: $i > $o] :
% 2.07/1.46 ( ( exu
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.07/1.46 => ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( Xphi @ Xx )
% 2.07/1.46 => ( ( Xphi @ Xy )
% 2.07/1.46 => ( Xx = Xy ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(descr__Cong_type,type,
% 2.07/1.46 descr__Cong: $o ).
% 2.07/1.46
% 2.07/1.46 thf(descr__Cong,definition,
% 2.07/1.46 ( descr__Cong
% 2.07/1.46 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.07/1.46 ( ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( Xx = Xy )
% 2.07/1.46 => ( ( Xphi @ Xx )
% 2.07/1.46 <=> ( Xpsi @ Xy ) ) )
% 2.07/1.46 => ( ( exu
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.07/1.46 => ( ( exu
% 2.07/1.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 2.07/1.46 => ( ( descr
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.07/1.46 = ( descr
% 2.07/1.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(dsetconstr__Cong_type,type,
% 2.07/1.46 dsetconstr__Cong: $o ).
% 2.07/1.46
% 2.07/1.46 thf(dsetconstr__Cong,definition,
% 2.07/1.46 ( dsetconstr__Cong
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( A = B )
% 2.07/1.46 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( ( Xx = Xy )
% 2.07/1.46 => ( ( Xphi @ Xx )
% 2.07/1.46 <=> ( Xpsi @ Xy ) ) ) ) )
% 2.07/1.46 => ( ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.07/1.46 = ( dsetconstr @ B
% 2.07/1.46 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(subset_type,type,
% 2.07/1.46 subset: $i > $i > $o ).
% 2.07/1.46
% 2.07/1.46 thf(disjoint_type,type,
% 2.07/1.46 disjoint: $i > $i > $o ).
% 2.07/1.46
% 2.07/1.46 thf(setsmeet_type,type,
% 2.07/1.46 setsmeet: $i > $i > $o ).
% 2.07/1.46
% 2.07/1.46 thf(subsetI1_type,type,
% 2.07/1.46 subsetI1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(subsetI1,definition,
% 2.07/1.46 ( subsetI1
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ Xx @ B ) )
% 2.07/1.46 => ( subset @ A @ B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(eqimpsubset2_type,type,
% 2.07/1.46 eqimpsubset2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(eqimpsubset2,definition,
% 2.07/1.46 ( eqimpsubset2
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( A = B )
% 2.07/1.46 => ( subset @ B @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(eqimpsubset1_type,type,
% 2.07/1.46 eqimpsubset1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(eqimpsubset1,definition,
% 2.07/1.46 ( eqimpsubset1
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( A = B )
% 2.07/1.46 => ( subset @ A @ B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(subsetI2_type,type,
% 2.07/1.46 subsetI2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(subsetI2,definition,
% 2.07/1.46 ( subsetI2
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ Xx @ B ) )
% 2.07/1.46 => ( subset @ A @ B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(emptysetsubset_type,type,
% 2.07/1.46 emptysetsubset: $o ).
% 2.07/1.46
% 2.07/1.46 thf(emptysetsubset,definition,
% 2.07/1.46 ( emptysetsubset
% 2.07/1.46 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(subsetE_type,type,
% 2.07/1.46 subsetE: $o ).
% 2.07/1.46
% 2.07/1.46 thf(subsetE,definition,
% 2.07/1.46 ( subsetE
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( subset @ A @ B )
% 2.07/1.46 => ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(subsetE2_type,type,
% 2.07/1.46 subsetE2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(subsetE2,definition,
% 2.07/1.46 ( subsetE2
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( subset @ A @ B )
% 2.07/1.46 => ( ~ ( in @ Xx @ B )
% 2.07/1.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(notsubsetI_type,type,
% 2.07/1.46 notsubsetI: $o ).
% 2.07/1.46
% 2.07/1.46 thf(notsubsetI,definition,
% 2.07/1.46 ( notsubsetI
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ~ ( in @ Xx @ B )
% 2.07/1.46 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(notequalI1_type,type,
% 2.07/1.46 notequalI1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(notequalI1,definition,
% 2.07/1.46 ( notequalI1
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ~ ( subset @ A @ B )
% 2.07/1.46 => ( A != B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(notequalI2_type,type,
% 2.07/1.46 notequalI2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(notequalI2,definition,
% 2.07/1.46 ( notequalI2
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ~ ( in @ Xx @ B )
% 2.07/1.46 => ( A != B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(subsetRefl_type,type,
% 2.07/1.46 subsetRefl: $o ).
% 2.07/1.46
% 2.07/1.46 thf(subsetRefl,definition,
% 2.07/1.46 ( subsetRefl
% 2.07/1.46 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(subsetTrans_type,type,
% 2.07/1.46 subsetTrans: $o ).
% 2.07/1.46
% 2.07/1.46 thf(subsetTrans,definition,
% 2.07/1.46 ( subsetTrans
% 2.07/1.46 = ( ! [A: $i,B: $i,C: $i] :
% 2.07/1.46 ( ( subset @ A @ B )
% 2.07/1.46 => ( ( subset @ B @ C )
% 2.07/1.46 => ( subset @ A @ C ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoinSub_type,type,
% 2.07/1.46 setadjoinSub: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoinSub,definition,
% 2.07/1.46 ( setadjoinSub
% 2.07/1.46 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoinSub2_type,type,
% 2.07/1.46 setadjoinSub2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setadjoinSub2,definition,
% 2.07/1.46 ( setadjoinSub2
% 2.07/1.46 = ( ! [A: $i,Xx: $i,B: $i] :
% 2.07/1.46 ( ( subset @ A @ B )
% 2.07/1.46 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(subset2powerset_type,type,
% 2.07/1.46 subset2powerset: $o ).
% 2.07/1.46
% 2.07/1.46 thf(subset2powerset,definition,
% 2.07/1.46 ( subset2powerset
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( subset @ A @ B )
% 2.07/1.46 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setextsub_type,type,
% 2.07/1.46 setextsub: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setextsub,definition,
% 2.07/1.46 ( setextsub
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( subset @ A @ B )
% 2.07/1.46 => ( ( subset @ B @ A )
% 2.07/1.46 => ( A = B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(subsetemptysetimpeq_type,type,
% 2.07/1.46 subsetemptysetimpeq: $o ).
% 2.07/1.46
% 2.07/1.46 thf(subsetemptysetimpeq,definition,
% 2.07/1.46 ( subsetemptysetimpeq
% 2.07/1.46 = ( ! [A: $i] :
% 2.07/1.46 ( ( subset @ A @ emptyset )
% 2.07/1.46 => ( A = emptyset ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(powersetI1_type,type,
% 2.07/1.46 powersetI1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(powersetI1,definition,
% 2.07/1.46 ( powersetI1
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( subset @ B @ A )
% 2.07/1.46 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(powersetE1_type,type,
% 2.07/1.46 powersetE1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(powersetE1,definition,
% 2.07/1.46 ( powersetE1
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( in @ B @ ( powerset @ A ) )
% 2.07/1.46 => ( subset @ B @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(inPowerset_type,type,
% 2.07/1.46 inPowerset: $o ).
% 2.07/1.46
% 2.07/1.46 thf(inPowerset,definition,
% 2.07/1.46 ( inPowerset
% 2.07/1.46 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(powersetsubset_type,type,
% 2.07/1.46 powersetsubset: $o ).
% 2.07/1.46
% 2.07/1.46 thf(powersetsubset,definition,
% 2.07/1.46 ( powersetsubset
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( subset @ A @ B )
% 2.07/1.46 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(sepInPowerset_type,type,
% 2.07/1.46 sepInPowerset: $o ).
% 2.07/1.46
% 2.07/1.46 thf(sepInPowerset,definition,
% 2.07/1.46 ( sepInPowerset
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.07/1.46 ( in
% 2.07/1.46 @ ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.07/1.46 @ ( powerset @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(sepSubset_type,type,
% 2.07/1.46 sepSubset: $o ).
% 2.07/1.46
% 2.07/1.46 thf(sepSubset,definition,
% 2.07/1.46 ( sepSubset
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.07/1.46 ( subset
% 2.07/1.46 @ ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.07/1.46 @ A ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binunion_type,type,
% 2.07/1.46 binunion: $i > $i > $i ).
% 2.07/1.46
% 2.07/1.46 thf(binunionIL_type,type,
% 2.07/1.46 binunionIL: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binunionIL,definition,
% 2.07/1.46 ( binunionIL
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(upairset2IR_type,type,
% 2.07/1.46 upairset2IR: $o ).
% 2.07/1.46
% 2.07/1.46 thf(upairset2IR,definition,
% 2.07/1.46 ( upairset2IR
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binunionIR_type,type,
% 2.07/1.46 binunionIR: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binunionIR,definition,
% 2.07/1.46 ( binunionIR
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ B )
% 2.07/1.46 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binunionEcases_type,type,
% 2.07/1.46 binunionEcases: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binunionEcases,definition,
% 2.07/1.46 ( binunionEcases
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 2.07/1.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.07/1.46 => ( ( ( in @ Xx @ A )
% 2.07/1.46 => Xphi )
% 2.07/1.46 => ( ( ( in @ Xx @ B )
% 2.07/1.46 => Xphi )
% 2.07/1.46 => Xphi ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binunionE_type,type,
% 2.07/1.46 binunionE: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binunionE,definition,
% 2.07/1.46 ( binunionE
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 2.07/1.46 => ( ( in @ Xx @ A )
% 2.07/1.46 | ( in @ Xx @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binunionLsub_type,type,
% 2.07/1.46 binunionLsub: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binunionLsub,definition,
% 2.07/1.46 ( binunionLsub
% 2.07/1.46 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binunionRsub_type,type,
% 2.07/1.46 binunionRsub: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binunionRsub,definition,
% 2.07/1.46 ( binunionRsub
% 2.07/1.46 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binintersect_type,type,
% 2.07/1.46 binintersect: $i > $i > $i ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectI_type,type,
% 2.07/1.46 binintersectI: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectI,definition,
% 2.07/1.46 ( binintersectI
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ( in @ Xx @ B )
% 2.07/1.46 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectSubset5_type,type,
% 2.07/1.46 binintersectSubset5: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectSubset5,definition,
% 2.07/1.46 ( binintersectSubset5
% 2.07/1.46 = ( ! [A: $i,B: $i,C: $i] :
% 2.07/1.46 ( ( subset @ C @ A )
% 2.07/1.46 => ( ( subset @ C @ B )
% 2.07/1.46 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectEL_type,type,
% 2.07/1.46 binintersectEL: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectEL,definition,
% 2.07/1.46 ( binintersectEL
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.07/1.46 => ( in @ Xx @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectLsub_type,type,
% 2.07/1.46 binintersectLsub: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectLsub,definition,
% 2.07/1.46 ( binintersectLsub
% 2.07/1.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectSubset2_type,type,
% 2.07/1.46 binintersectSubset2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectSubset2,definition,
% 2.07/1.46 ( binintersectSubset2
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( subset @ A @ B )
% 2.07/1.46 => ( ( binintersect @ A @ B )
% 2.07/1.46 = A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectSubset3_type,type,
% 2.07/1.46 binintersectSubset3: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectSubset3,definition,
% 2.07/1.46 ( binintersectSubset3
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( ( binintersect @ A @ B )
% 2.07/1.46 = B )
% 2.07/1.46 => ( subset @ B @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectER_type,type,
% 2.07/1.46 binintersectER: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectER,definition,
% 2.07/1.46 ( binintersectER
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 2.07/1.46 => ( in @ Xx @ B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(disjointsetsI1_type,type,
% 2.07/1.46 disjointsetsI1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(disjointsetsI1,definition,
% 2.07/1.46 ( disjointsetsI1
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ~ ? [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 & ( in @ Xx @ B ) )
% 2.07/1.46 => ( ( binintersect @ A @ B )
% 2.07/1.46 = emptyset ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectRsub_type,type,
% 2.07/1.46 binintersectRsub: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectRsub,definition,
% 2.07/1.46 ( binintersectRsub
% 2.07/1.46 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectSubset4_type,type,
% 2.07/1.46 binintersectSubset4: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectSubset4,definition,
% 2.07/1.46 ( binintersectSubset4
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( subset @ B @ A )
% 2.07/1.46 => ( ( binintersect @ A @ B )
% 2.07/1.46 = B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectSubset1_type,type,
% 2.07/1.46 binintersectSubset1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectSubset1,definition,
% 2.07/1.46 ( binintersectSubset1
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( ( binintersect @ A @ B )
% 2.07/1.46 = A )
% 2.07/1.46 => ( subset @ A @ B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(bs114d_type,type,
% 2.07/1.46 bs114d: $o ).
% 2.07/1.46
% 2.07/1.46 thf(bs114d,definition,
% 2.07/1.46 ( bs114d
% 2.07/1.46 = ( ! [A: $i,B: $i,C: $i] :
% 2.07/1.46 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 2.07/1.46 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(regular_type,type,
% 2.07/1.46 regular: $i > $o ).
% 2.07/1.46
% 2.07/1.46 thf(setminus_type,type,
% 2.07/1.46 setminus: $i > $i > $i ).
% 2.07/1.46
% 2.07/1.46 thf(setminusI_type,type,
% 2.07/1.46 setminusI: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setminusI,definition,
% 2.07/1.46 ( setminusI
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ~ ( in @ Xx @ B )
% 2.07/1.46 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setminusEL_type,type,
% 2.07/1.46 setminusEL: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setminusEL,definition,
% 2.07/1.46 ( setminusEL
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.07/1.46 => ( in @ Xx @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setminusER_type,type,
% 2.07/1.46 setminusER: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setminusER,definition,
% 2.07/1.46 ( setminusER
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 2.07/1.46 => ~ ( in @ Xx @ B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setminusSubset2_type,type,
% 2.07/1.46 setminusSubset2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setminusSubset2,definition,
% 2.07/1.46 ( setminusSubset2
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( subset @ A @ B )
% 2.07/1.46 => ( ( setminus @ A @ B )
% 2.07/1.46 = emptyset ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setminusERneg_type,type,
% 2.07/1.46 setminusERneg: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setminusERneg,definition,
% 2.07/1.46 ( setminusERneg
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.07/1.46 => ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ Xx @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setminusELneg_type,type,
% 2.07/1.46 setminusELneg: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setminusELneg,definition,
% 2.07/1.46 ( setminusELneg
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 2.07/1.46 => ( ~ ( in @ Xx @ B )
% 2.07/1.46 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setminusILneg_type,type,
% 2.07/1.46 setminusILneg: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setminusILneg,definition,
% 2.07/1.46 ( setminusILneg
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ~ ( in @ Xx @ A )
% 2.07/1.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setminusIRneg_type,type,
% 2.07/1.46 setminusIRneg: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setminusIRneg,definition,
% 2.07/1.46 ( setminusIRneg
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ B )
% 2.07/1.46 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setminusLsub_type,type,
% 2.07/1.46 setminusLsub: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setminusLsub,definition,
% 2.07/1.46 ( setminusLsub
% 2.07/1.46 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setminusSubset1_type,type,
% 2.07/1.46 setminusSubset1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setminusSubset1,definition,
% 2.07/1.46 ( setminusSubset1
% 2.07/1.46 = ( ! [A: $i,B: $i] :
% 2.07/1.46 ( ( ( setminus @ A @ B )
% 2.07/1.46 = emptyset )
% 2.07/1.46 => ( subset @ A @ B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(symdiff_type,type,
% 2.07/1.46 symdiff: $i > $i > $i ).
% 2.07/1.46
% 2.07/1.46 thf(symdiffE_type,type,
% 2.07/1.46 symdiffE: $o ).
% 2.07/1.46
% 2.07/1.46 thf(symdiffE,definition,
% 2.07/1.46 ( symdiffE
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 2.07/1.46 => ! [Xphi: $o] :
% 2.07/1.46 ( ( ( in @ Xx @ A )
% 2.07/1.46 => ( ~ ( in @ Xx @ B )
% 2.07/1.46 => Xphi ) )
% 2.07/1.46 => ( ( ~ ( in @ Xx @ A )
% 2.07/1.46 => ( ( in @ Xx @ B )
% 2.07/1.46 => Xphi ) )
% 2.07/1.46 => Xphi ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(symdiffI1_type,type,
% 2.07/1.46 symdiffI1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(symdiffI1,definition,
% 2.07/1.46 ( symdiffI1
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ~ ( in @ Xx @ B )
% 2.07/1.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(symdiffI2_type,type,
% 2.07/1.46 symdiffI2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(symdiffI2,definition,
% 2.07/1.46 ( symdiffI2
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ~ ( in @ Xx @ A )
% 2.07/1.46 => ( ( in @ Xx @ B )
% 2.07/1.46 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(symdiffIneg1_type,type,
% 2.07/1.46 symdiffIneg1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(symdiffIneg1,definition,
% 2.07/1.46 ( symdiffIneg1
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ( in @ Xx @ B )
% 2.07/1.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(symdiffIneg2_type,type,
% 2.07/1.46 symdiffIneg2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(symdiffIneg2,definition,
% 2.07/1.46 ( symdiffIneg2
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ~ ( in @ Xx @ A )
% 2.07/1.46 => ( ~ ( in @ Xx @ B )
% 2.07/1.46 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(iskpair_type,type,
% 2.07/1.46 iskpair: $i > $o ).
% 2.07/1.46
% 2.07/1.46 thf(secondinupair_type,type,
% 2.07/1.46 secondinupair: $o ).
% 2.07/1.46
% 2.07/1.46 thf(secondinupair,definition,
% 2.07/1.46 ( secondinupair
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairIL_type,type,
% 2.07/1.46 setukpairIL: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairIL,definition,
% 2.07/1.46 ( setukpairIL
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairIR_type,type,
% 2.07/1.46 setukpairIR: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairIR,definition,
% 2.07/1.46 ( setukpairIR
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(kpairiskpair_type,type,
% 2.07/1.46 kpairiskpair: $o ).
% 2.07/1.46
% 2.07/1.46 thf(kpairiskpair,definition,
% 2.07/1.46 ( kpairiskpair
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(kpair_type,type,
% 2.07/1.46 kpair: $i > $i > $i ).
% 2.07/1.46
% 2.07/1.46 thf(kpairp_type,type,
% 2.07/1.46 kpairp: $o ).
% 2.07/1.46
% 2.07/1.46 thf(kpairp,definition,
% 2.07/1.46 ( kpairp
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(cartprod_type,type,
% 2.07/1.46 cartprod: $i > $i > $i ).
% 2.07/1.46
% 2.07/1.46 thf(singletonsubset_type,type,
% 2.07/1.46 singletonsubset: $o ).
% 2.07/1.46
% 2.07/1.46 thf(singletonsubset,definition,
% 2.07/1.46 ( singletonsubset
% 2.07/1.46 = ( ! [A: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(singletoninpowerset_type,type,
% 2.07/1.46 singletoninpowerset: $o ).
% 2.07/1.46
% 2.07/1.46 thf(singletoninpowerset,definition,
% 2.07/1.46 ( singletoninpowerset
% 2.07/1.46 = ( ! [A: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(singletoninpowunion_type,type,
% 2.07/1.46 singletoninpowunion: $o ).
% 2.07/1.46
% 2.07/1.46 thf(singletoninpowunion,definition,
% 2.07/1.46 ( singletoninpowunion
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(upairset2E_type,type,
% 2.07/1.46 upairset2E: $o ).
% 2.07/1.46
% 2.07/1.46 thf(upairset2E,definition,
% 2.07/1.46 ( upairset2E
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.07/1.46 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 2.07/1.46 => ( ( Xz = Xx )
% 2.07/1.46 | ( Xz = Xy ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(upairsubunion_type,type,
% 2.07/1.46 upairsubunion: $o ).
% 2.07/1.46
% 2.07/1.46 thf(upairsubunion,definition,
% 2.07/1.46 ( upairsubunion
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(upairinpowunion_type,type,
% 2.07/1.46 upairinpowunion: $o ).
% 2.07/1.46
% 2.07/1.46 thf(upairinpowunion,definition,
% 2.07/1.46 ( upairinpowunion
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(ubforcartprodlem1_type,type,
% 2.07/1.46 ubforcartprodlem1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(ubforcartprodlem1,definition,
% 2.07/1.46 ( ubforcartprodlem1
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(ubforcartprodlem2_type,type,
% 2.07/1.46 ubforcartprodlem2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(ubforcartprodlem2,definition,
% 2.07/1.46 ( ubforcartprodlem2
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(ubforcartprodlem3_type,type,
% 2.07/1.46 ubforcartprodlem3: $o ).
% 2.07/1.46
% 2.07/1.46 thf(ubforcartprodlem3,definition,
% 2.07/1.46 ( ubforcartprodlem3
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodpairin_type,type,
% 2.07/1.46 cartprodpairin: $o ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodpairin,definition,
% 2.07/1.46 ( cartprodpairin
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodmempair1_type,type,
% 2.07/1.46 cartprodmempair1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodmempair1,definition,
% 2.07/1.46 ( cartprodmempair1
% 2.07/1.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.07/1.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.07/1.46 => ? [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 & ? [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 & ( Xu
% 2.07/1.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodmempair_type,type,
% 2.07/1.46 cartprodmempair: $o ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodmempair,definition,
% 2.07/1.46 ( cartprodmempair
% 2.07/1.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.07/1.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.07/1.46 => ( iskpair @ Xu ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setunionE2_type,type,
% 2.07/1.46 setunionE2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setunionE2,definition,
% 2.07/1.46 ( setunionE2
% 2.07/1.46 = ( ! [A: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ ( setunion @ A ) )
% 2.07/1.46 => ? [X: $i] :
% 2.07/1.46 ( ( in @ X @ A )
% 2.07/1.46 & ( in @ Xx @ X ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setunionsingleton1_type,type,
% 2.07/1.46 setunionsingleton1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setunionsingleton1,definition,
% 2.07/1.46 ( setunionsingleton1
% 2.07/1.46 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setunionsingleton2_type,type,
% 2.07/1.46 setunionsingleton2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setunionsingleton2,definition,
% 2.07/1.46 ( setunionsingleton2
% 2.07/1.46 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setunionsingleton_type,type,
% 2.07/1.46 setunionsingleton: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setunionsingleton,definition,
% 2.07/1.46 ( setunionsingleton
% 2.07/1.46 = ( ! [Xx: $i] :
% 2.07/1.46 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 2.07/1.46 = Xx ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(singleton_type,type,
% 2.07/1.46 singleton: $i > $o ).
% 2.07/1.46
% 2.07/1.46 thf(singletonprop_type,type,
% 2.07/1.46 singletonprop: $o ).
% 2.07/1.46
% 2.07/1.46 thf(singletonprop,definition,
% 2.07/1.46 ( singletonprop
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ A )
% 2.07/1.46 => ( ( Xphi @ Xx )
% 2.07/1.46 => ( ( Xphi @ Xy )
% 2.07/1.46 => ( Xx = Xy ) ) ) ) )
% 2.07/1.46 => ( ? [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 & ( Xphi @ Xx ) )
% 2.07/1.46 => ( singleton
% 2.07/1.46 @ ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(ex1_type,type,
% 2.07/1.46 ex1: $i > ( $i > $o ) > $o ).
% 2.07/1.46
% 2.07/1.46 thf(ex1,definition,
% 2.07/1.46 ( ex1
% 2.07/1.46 = ( ^ [A: $i,Xphi: $i > $o] :
% 2.07/1.46 ( singleton
% 2.07/1.46 @ ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(ex1E1_type,type,
% 2.07/1.46 ex1E1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(ex1E1,definition,
% 2.07/1.46 ( ex1E1
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.07/1.46 ( ( ex1 @ A
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.07/1.46 => ? [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 & ( Xphi @ Xx ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(ex1I_type,type,
% 2.07/1.46 ex1I: $o ).
% 2.07/1.46
% 2.07/1.46 thf(ex1I,definition,
% 2.07/1.46 ( ex1I
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ( Xphi @ Xx )
% 2.07/1.46 => ( ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ A )
% 2.07/1.46 => ( ( Xphi @ Xy )
% 2.07/1.46 => ( Xy = Xx ) ) )
% 2.07/1.46 => ( ex1 @ A
% 2.07/1.46 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(ex1I2_type,type,
% 2.07/1.46 ex1I2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(ex1I2,definition,
% 2.07/1.46 ( ex1I2
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ A )
% 2.07/1.46 => ( ( Xphi @ Xx )
% 2.07/1.46 => ( ( Xphi @ Xy )
% 2.07/1.46 => ( Xx = Xy ) ) ) ) )
% 2.07/1.46 => ( ? [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 & ( Xphi @ Xx ) )
% 2.07/1.46 => ( ex1 @ A
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(singletonsuniq_type,type,
% 2.07/1.46 singletonsuniq: $o ).
% 2.07/1.46
% 2.07/1.46 thf(singletonsuniq,definition,
% 2.07/1.46 ( singletonsuniq
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( ( setadjoin @ Xx @ emptyset )
% 2.07/1.46 = ( setadjoin @ Xy @ emptyset ) )
% 2.07/1.46 => ( Xx = Xy ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(atmost1p_type,type,
% 2.07/1.46 atmost1p: $i > $o ).
% 2.07/1.46
% 2.07/1.46 thf(atleast2p_type,type,
% 2.07/1.46 atleast2p: $i > $o ).
% 2.07/1.46
% 2.07/1.46 thf(atmost2p_type,type,
% 2.07/1.46 atmost2p: $i > $o ).
% 2.07/1.46
% 2.07/1.46 thf(upairsetp_type,type,
% 2.07/1.46 upairsetp: $i > $o ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairinjL1_type,type,
% 2.07/1.46 setukpairinjL1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairinjL1,definition,
% 2.07/1.46 ( setukpairinjL1
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.07/1.46 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 2.07/1.46 => ( Xx = Xz ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(kfstsingleton_type,type,
% 2.07/1.46 kfstsingleton: $o ).
% 2.07/1.46
% 2.07/1.46 thf(kfstsingleton,definition,
% 2.07/1.46 ( kfstsingleton
% 2.07/1.46 = ( ! [Xu: $i] :
% 2.07/1.46 ( ( iskpair @ Xu )
% 2.07/1.46 => ( singleton
% 2.07/1.46 @ ( dsetconstr @ ( setunion @ Xu )
% 2.07/1.46 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(theprop_type,type,
% 2.07/1.46 theprop: $o ).
% 2.07/1.46
% 2.07/1.46 thf(theprop,definition,
% 2.07/1.46 ( theprop
% 2.07/1.46 = ( ! [X: $i] :
% 2.07/1.46 ( ( singleton @ X )
% 2.07/1.46 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(kfst_type,type,
% 2.07/1.46 kfst: $i > $i ).
% 2.07/1.46
% 2.07/1.46 thf(kfstpairEq_type,type,
% 2.07/1.46 kfstpairEq: $o ).
% 2.07/1.46
% 2.07/1.46 thf(kfstpairEq,definition,
% 2.07/1.46 ( kfstpairEq
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.07/1.46 = Xx ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodfstin_type,type,
% 2.07/1.46 cartprodfstin: $o ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodfstin,definition,
% 2.07/1.46 ( cartprodfstin
% 2.07/1.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.07/1.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.07/1.46 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairinjL2_type,type,
% 2.07/1.46 setukpairinjL2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairinjL2,definition,
% 2.07/1.46 ( setukpairinjL2
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.07/1.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.07/1.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.07/1.46 => ( Xx = Xz ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairinjL_type,type,
% 2.07/1.46 setukpairinjL: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairinjL,definition,
% 2.07/1.46 ( setukpairinjL
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.07/1.46 ( ( ( kpair @ Xx @ Xy )
% 2.07/1.46 = ( kpair @ Xz @ Xu ) )
% 2.07/1.46 => ( Xx = Xz ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairinjR11_type,type,
% 2.07/1.46 setukpairinjR11: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairinjR11,definition,
% 2.07/1.46 ( setukpairinjR11
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( Xx = Xy )
% 2.07/1.46 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.07/1.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairinjR12_type,type,
% 2.07/1.46 setukpairinjR12: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairinjR12,definition,
% 2.07/1.46 ( setukpairinjR12
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( Xx = Xy )
% 2.07/1.46 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.07/1.46 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairinjR1_type,type,
% 2.07/1.46 setukpairinjR1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairinjR1,definition,
% 2.07/1.46 ( setukpairinjR1
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.07/1.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.07/1.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.07/1.46 => ( ( Xz = Xu )
% 2.07/1.46 => ( Xy = Xu ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(upairequniteq_type,type,
% 2.07/1.46 upairequniteq: $o ).
% 2.07/1.46
% 2.07/1.46 thf(upairequniteq,definition,
% 2.07/1.46 ( upairequniteq
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 2.07/1.46 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 2.07/1.46 = ( setadjoin @ Xz @ emptyset ) )
% 2.07/1.46 => ( Xx = Xy ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairinjR2_type,type,
% 2.07/1.46 setukpairinjR2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairinjR2,definition,
% 2.07/1.46 ( setukpairinjR2
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.07/1.46 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 2.07/1.46 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 2.07/1.46 => ( Xy = Xu ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairinjR_type,type,
% 2.07/1.46 setukpairinjR: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setukpairinjR,definition,
% 2.07/1.46 ( setukpairinjR
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 2.07/1.46 ( ( ( kpair @ Xx @ Xy )
% 2.07/1.46 = ( kpair @ Xz @ Xu ) )
% 2.07/1.46 => ( Xy = Xu ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(ksndsingleton_type,type,
% 2.07/1.46 ksndsingleton: $o ).
% 2.07/1.46
% 2.07/1.46 thf(ksndsingleton,definition,
% 2.07/1.46 ( ksndsingleton
% 2.07/1.46 = ( ! [Xu: $i] :
% 2.07/1.46 ( ( iskpair @ Xu )
% 2.07/1.46 => ( singleton
% 2.07/1.46 @ ( dsetconstr @ ( setunion @ Xu )
% 2.07/1.46 @ ^ [Xx: $i] :
% 2.07/1.46 ( Xu
% 2.07/1.46 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(ksnd_type,type,
% 2.07/1.46 ksnd: $i > $i ).
% 2.07/1.46
% 2.07/1.46 thf(ksndpairEq_type,type,
% 2.07/1.46 ksndpairEq: $o ).
% 2.07/1.46
% 2.07/1.46 thf(ksndpairEq,definition,
% 2.07/1.46 ( ksndpairEq
% 2.07/1.46 = ( ! [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.07/1.46 = Xy ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(kpairsurjEq_type,type,
% 2.07/1.46 kpairsurjEq: $o ).
% 2.07/1.46
% 2.07/1.46 thf(kpairsurjEq,definition,
% 2.07/1.46 ( kpairsurjEq
% 2.07/1.46 = ( ! [Xu: $i] :
% 2.07/1.46 ( ( iskpair @ Xu )
% 2.07/1.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.07/1.46 = Xu ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodsndin_type,type,
% 2.07/1.46 cartprodsndin: $o ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodsndin,definition,
% 2.07/1.46 ( cartprodsndin
% 2.07/1.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.07/1.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.07/1.46 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodpairmemEL_type,type,
% 2.07/1.46 cartprodpairmemEL: $o ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodpairmemEL,definition,
% 2.07/1.46 ( cartprodpairmemEL
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.07/1.46 => ( in @ Xx @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodpairmemER_type,type,
% 2.07/1.46 cartprodpairmemER: $o ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodpairmemER,definition,
% 2.07/1.46 ( cartprodpairmemER
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 2.07/1.46 => ( in @ Xy @ B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodmempaircEq_type,type,
% 2.07/1.46 cartprodmempaircEq: $o ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodmempaircEq,definition,
% 2.07/1.46 ( cartprodmempaircEq
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( ( kpair @ Xx @ Xy )
% 2.07/1.46 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodfstpairEq_type,type,
% 2.07/1.46 cartprodfstpairEq: $o ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodfstpairEq,definition,
% 2.07/1.46 ( cartprodfstpairEq
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 2.07/1.46 = Xx ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodsndpairEq_type,type,
% 2.07/1.46 cartprodsndpairEq: $o ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodsndpairEq,definition,
% 2.07/1.46 ( cartprodsndpairEq
% 2.07/1.46 = ( ! [A: $i,B: $i,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 2.07/1.46 = Xy ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodpairsurjEq_type,type,
% 2.07/1.46 cartprodpairsurjEq: $o ).
% 2.07/1.46
% 2.07/1.46 thf(cartprodpairsurjEq,definition,
% 2.07/1.46 ( cartprodpairsurjEq
% 2.07/1.46 = ( ! [A: $i,B: $i,Xu: $i] :
% 2.07/1.46 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 2.07/1.46 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 2.07/1.46 = Xu ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(breln_type,type,
% 2.07/1.46 breln: $i > $i > $i > $o ).
% 2.07/1.46
% 2.07/1.46 thf(breln,definition,
% 2.07/1.46 ( breln
% 2.07/1.46 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(dpsetconstr_type,type,
% 2.07/1.46 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 2.07/1.46
% 2.07/1.46 thf(dpsetconstrI_type,type,
% 2.07/1.46 dpsetconstrI: $o ).
% 2.07/1.46
% 2.07/1.46 thf(dpsetconstrI,definition,
% 2.07/1.46 ( dpsetconstrI
% 2.07/1.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( ( Xphi @ Xx @ Xy )
% 2.07/1.46 => ( in @ ( kpair @ Xx @ Xy )
% 2.07/1.46 @ ( dpsetconstr @ A @ B
% 2.07/1.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(dpsetconstrSub_type,type,
% 2.07/1.46 dpsetconstrSub: $o ).
% 2.07/1.46
% 2.07/1.46 thf(dpsetconstrSub,definition,
% 2.07/1.46 ( dpsetconstrSub
% 2.07/1.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.07/1.46 ( subset
% 2.07/1.46 @ ( dpsetconstr @ A @ B
% 2.07/1.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 2.07/1.46 @ ( cartprod @ A @ B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(setOfPairsIsBReln_type,type,
% 2.07/1.46 setOfPairsIsBReln: $o ).
% 2.07/1.46
% 2.07/1.46 thf(setOfPairsIsBReln,definition,
% 2.07/1.46 ( setOfPairsIsBReln
% 2.07/1.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 2.07/1.46 ( breln @ A @ B
% 2.07/1.46 @ ( dpsetconstr @ A @ B
% 2.07/1.46 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(dpsetconstrERa_type,type,
% 2.07/1.46 dpsetconstrERa: $o ).
% 2.07/1.46
% 2.07/1.46 thf(dpsetconstrERa,definition,
% 2.07/1.46 ( dpsetconstrERa
% 2.07/1.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( ( in @ ( kpair @ Xx @ Xy )
% 2.07/1.46 @ ( dpsetconstr @ A @ B
% 2.07/1.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.07/1.46 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(dpsetconstrEL1_type,type,
% 2.07/1.46 dpsetconstrEL1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(dpsetconstrEL1,definition,
% 2.07/1.46 ( dpsetconstrEL1
% 2.07/1.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( in @ ( kpair @ Xx @ Xy )
% 2.07/1.46 @ ( dpsetconstr @ A @ B
% 2.07/1.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.07/1.46 => ( in @ Xx @ A ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(dpsetconstrEL2_type,type,
% 2.07/1.46 dpsetconstrEL2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(dpsetconstrEL2,definition,
% 2.07/1.46 ( dpsetconstrEL2
% 2.07/1.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( in @ ( kpair @ Xx @ Xy )
% 2.07/1.46 @ ( dpsetconstr @ A @ B
% 2.07/1.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.07/1.46 => ( in @ Xy @ B ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(dpsetconstrER_type,type,
% 2.07/1.46 dpsetconstrER: $o ).
% 2.07/1.46
% 2.07/1.46 thf(dpsetconstrER,definition,
% 2.07/1.46 ( dpsetconstrER
% 2.07/1.46 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( in @ ( kpair @ Xx @ Xy )
% 2.07/1.46 @ ( dpsetconstr @ A @ B
% 2.07/1.46 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 2.07/1.46 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(func_type,type,
% 2.07/1.46 func: $i > $i > $i > $o ).
% 2.07/1.46
% 2.07/1.46 thf(funcSet_type,type,
% 2.07/1.46 funcSet: $i > $i > $i ).
% 2.07/1.46
% 2.07/1.46 thf(funcImageSingleton_type,type,
% 2.07/1.46 funcImageSingleton: $o ).
% 2.07/1.46
% 2.07/1.46 thf(funcImageSingleton,definition,
% 2.07/1.46 ( funcImageSingleton
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( func @ A @ B @ Xf )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( singleton
% 2.07/1.46 @ ( dsetconstr @ B
% 2.07/1.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(apProp_type,type,
% 2.07/1.46 apProp: $o ).
% 2.07/1.46
% 2.07/1.46 thf(apProp,definition,
% 2.07/1.46 ( apProp
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( func @ A @ B @ Xf )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in
% 2.07/1.46 @ ( setunion
% 2.07/1.46 @ ( dsetconstr @ B
% 2.07/1.46 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 2.07/1.46 @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(ap_type,type,
% 2.07/1.46 ap: $i > $i > $i > $i > $i ).
% 2.07/1.46
% 2.07/1.46 thf(app_type,type,
% 2.07/1.46 app: $o ).
% 2.07/1.46
% 2.07/1.46 thf(app,definition,
% 2.07/1.46 ( app
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( func @ A @ B @ Xf )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(infuncsetfunc_type,type,
% 2.07/1.46 infuncsetfunc: $o ).
% 2.07/1.46
% 2.07/1.46 thf(infuncsetfunc,definition,
% 2.07/1.46 ( infuncsetfunc
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.07/1.46 => ( func @ A @ B @ Xf ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(ap2p_type,type,
% 2.07/1.46 ap2p: $o ).
% 2.07/1.46
% 2.07/1.46 thf(ap2p,definition,
% 2.07/1.46 ( ap2p
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(funcinfuncset_type,type,
% 2.07/1.46 funcinfuncset: $o ).
% 2.07/1.46
% 2.07/1.46 thf(funcinfuncset,definition,
% 2.07/1.46 ( funcinfuncset
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( func @ A @ B @ Xf )
% 2.07/1.46 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(lamProp_type,type,
% 2.07/1.46 lamProp: $o ).
% 2.07/1.46
% 2.07/1.46 thf(lamProp,definition,
% 2.07/1.46 ( lamProp
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.07/1.46 => ( func @ A @ B
% 2.07/1.46 @ ( dpsetconstr @ A @ B
% 2.07/1.46 @ ^ [Xx: $i,Xy: $i] :
% 2.07/1.46 ( ( Xf @ Xx )
% 2.07/1.46 = Xy ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(lam_type,type,
% 2.07/1.46 lam: $i > $i > ( $i > $i ) > $i ).
% 2.07/1.46
% 2.07/1.46 thf(lamp_type,type,
% 2.07/1.46 lamp: $o ).
% 2.07/1.46
% 2.07/1.46 thf(lamp,definition,
% 2.07/1.46 ( lamp
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.07/1.46 => ( func @ A @ B
% 2.07/1.46 @ ( lam @ A @ B
% 2.07/1.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(lam2p_type,type,
% 2.07/1.46 lam2p: $o ).
% 2.07/1.46
% 2.07/1.46 thf(lam2p,definition,
% 2.07/1.46 ( lam2p
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.07/1.46 => ( in
% 2.07/1.46 @ ( lam @ A @ B
% 2.07/1.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.07/1.46 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(brelnall1_type,type,
% 2.07/1.46 brelnall1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(brelnall1,definition,
% 2.07/1.46 ( brelnall1
% 2.07/1.46 = ( ! [A: $i,B: $i,R: $i] :
% 2.07/1.46 ( ( breln @ A @ B @ R )
% 2.07/1.46 => ! [Xphi: $i > $o] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.07/1.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ R )
% 2.07/1.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(brelnall2_type,type,
% 2.07/1.46 brelnall2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(brelnall2,definition,
% 2.07/1.46 ( brelnall2
% 2.07/1.46 = ( ! [A: $i,B: $i,R: $i] :
% 2.07/1.46 ( ( breln @ A @ B @ R )
% 2.07/1.46 => ! [Xphi: $i > $o] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.07/1.46 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ R )
% 2.07/1.46 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(ex1E2_type,type,
% 2.07/1.46 ex1E2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(ex1E2,definition,
% 2.07/1.46 ( ex1E2
% 2.07/1.46 = ( ! [A: $i,Xphi: $i > $o] :
% 2.07/1.46 ( ( ex1 @ A
% 2.07/1.46 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ A )
% 2.07/1.46 => ( ( Xphi @ Xx )
% 2.07/1.46 => ( ( Xphi @ Xy )
% 2.07/1.46 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(funcGraphProp1_type,type,
% 2.07/1.46 funcGraphProp1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(funcGraphProp1,definition,
% 2.07/1.46 ( funcGraphProp1
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( func @ A @ B @ Xf )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(funcGraphProp3_type,type,
% 2.07/1.46 funcGraphProp3: $o ).
% 2.07/1.46
% 2.07/1.46 thf(funcGraphProp3,definition,
% 2.07/1.46 ( funcGraphProp3
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(funcGraphProp2_type,type,
% 2.07/1.46 funcGraphProp2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(funcGraphProp2,definition,
% 2.07/1.46 ( funcGraphProp2
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( func @ A @ B @ Xf )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.07/1.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.07/1.46 = Xy ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(funcextLem_type,type,
% 2.07/1.46 funcextLem: $o ).
% 2.07/1.46
% 2.07/1.46 thf(funcextLem,definition,
% 2.07/1.46 ( funcextLem
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( func @ A @ B @ Xf )
% 2.07/1.46 => ! [Xg: $i] :
% 2.07/1.46 ( ( func @ A @ B @ Xg )
% 2.07/1.46 => ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.07/1.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 2.07/1.46 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(funcGraphProp4_type,type,
% 2.07/1.46 funcGraphProp4: $o ).
% 2.07/1.46
% 2.07/1.46 thf(funcGraphProp4,definition,
% 2.07/1.46 ( funcGraphProp4
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 2.07/1.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.07/1.46 = Xy ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(subbreln_type,type,
% 2.07/1.46 subbreln: $o ).
% 2.07/1.46
% 2.07/1.46 thf(subbreln,definition,
% 2.07/1.46 ( subbreln
% 2.07/1.46 = ( ! [A: $i,B: $i,R: $i] :
% 2.07/1.46 ( ( breln @ A @ B @ R )
% 2.07/1.46 => ! [S: $i] :
% 2.07/1.46 ( ( breln @ A @ B @ S )
% 2.07/1.46 => ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.07/1.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.07/1.46 => ( subset @ R @ S ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(eqbreln_type,type,
% 2.07/1.46 eqbreln: $o ).
% 2.07/1.46
% 2.07/1.46 thf(eqbreln,definition,
% 2.07/1.46 ( eqbreln
% 2.07/1.46 = ( ! [A: $i,B: $i,R: $i] :
% 2.07/1.46 ( ( breln @ A @ B @ R )
% 2.07/1.46 => ! [S: $i] :
% 2.07/1.46 ( ( breln @ A @ B @ S )
% 2.07/1.46 => ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 2.07/1.46 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 2.07/1.46 => ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ B )
% 2.07/1.46 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 2.07/1.46 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 2.07/1.46 => ( R = S ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(funcext_type,type,
% 2.07/1.46 funcext: $o ).
% 2.07/1.46
% 2.07/1.46 thf(funcext,definition,
% 2.07/1.46 ( funcext
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( func @ A @ B @ Xf )
% 2.07/1.46 => ! [Xg: $i] :
% 2.07/1.46 ( ( func @ A @ B @ Xg )
% 2.07/1.46 => ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.07/1.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.07/1.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(funcext2_type,type,
% 2.07/1.46 funcext2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(funcext2,definition,
% 2.07/1.46 ( funcext2
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.07/1.46 => ! [Xg: $i] :
% 2.07/1.46 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 2.07/1.46 => ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.07/1.46 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 2.07/1.46 => ( Xf = Xg ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(ap2apEq1_type,type,
% 2.07/1.46 ap2apEq1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(ap2apEq1,definition,
% 2.07/1.46 ( ap2apEq1
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.07/1.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(ap2apEq2_type,type,
% 2.07/1.46 ap2apEq2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(ap2apEq2,definition,
% 2.07/1.46 ( ap2apEq2
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( func @ A @ B @ Xf )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ( ap @ A @ B @ Xf @ Xx )
% 2.07/1.46 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(beta1_type,type,
% 2.07/1.46 beta1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(beta1,definition,
% 2.07/1.46 ( beta1
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ( ap @ A @ B
% 2.07/1.46 @ ( lam @ A @ B
% 2.07/1.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.07/1.46 @ Xx )
% 2.07/1.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(eta1_type,type,
% 2.07/1.46 eta1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(eta1,definition,
% 2.07/1.46 ( eta1
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( func @ A @ B @ Xf )
% 2.07/1.46 => ( ( lam @ A @ B
% 2.07/1.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.07/1.46 = Xf ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(lam2lamEq_type,type,
% 2.07/1.46 lam2lamEq: $o ).
% 2.07/1.46
% 2.07/1.46 thf(lam2lamEq,definition,
% 2.07/1.46 ( lam2lamEq
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.07/1.46 => ( ( lam @ A @ B
% 2.07/1.46 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 2.07/1.46 = ( lam @ A @ B
% 2.07/1.46 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(beta2_type,type,
% 2.07/1.46 beta2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(beta2,definition,
% 2.07/1.46 ( beta2
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 2.07/1.46 ( ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( in @ ( Xf @ Xx ) @ B ) )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ( ( ap @ A @ B
% 2.07/1.46 @ ( lam @ A @ B
% 2.07/1.46 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 2.07/1.46 @ Xx )
% 2.07/1.46 = ( Xf @ Xx ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(eta2_type,type,
% 2.07/1.46 eta2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(eta2,definition,
% 2.07/1.46 ( eta2
% 2.07/1.46 = ( ! [A: $i,B: $i,Xf: $i] :
% 2.07/1.46 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 2.07/1.46 => ( ( lam @ A @ B
% 2.07/1.46 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 2.07/1.46 = Xf ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(iffalseProp1_type,type,
% 2.07/1.46 iffalseProp1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(iffalseProp1,definition,
% 2.07/1.46 ( iffalseProp1
% 2.07/1.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ A )
% 2.07/1.46 => ( ~ Xphi
% 2.07/1.46 => ( in @ Xy
% 2.07/1.46 @ ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xz: $i] :
% 2.07/1.46 ( ( Xphi
% 2.07/1.46 & ( Xz = Xx ) )
% 2.07/1.46 | ( ~ Xphi
% 2.07/1.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(iffalseProp2_type,type,
% 2.07/1.46 iffalseProp2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(iffalseProp2,definition,
% 2.07/1.46 ( iffalseProp2
% 2.07/1.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ A )
% 2.07/1.46 => ( ~ Xphi
% 2.07/1.46 => ( ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xz: $i] :
% 2.07/1.46 ( ( Xphi
% 2.07/1.46 & ( Xz = Xx ) )
% 2.07/1.46 | ( ~ Xphi
% 2.07/1.46 & ( Xz = Xy ) ) ) )
% 2.07/1.46 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(iftrueProp1_type,type,
% 2.07/1.46 iftrueProp1: $o ).
% 2.07/1.46
% 2.07/1.46 thf(iftrueProp1,definition,
% 2.07/1.46 ( iftrueProp1
% 2.07/1.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ A )
% 2.07/1.46 => ( Xphi
% 2.07/1.46 => ( in @ Xx
% 2.07/1.46 @ ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xz: $i] :
% 2.07/1.46 ( ( Xphi
% 2.07/1.46 & ( Xz = Xx ) )
% 2.07/1.46 | ( ~ Xphi
% 2.07/1.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(iftrueProp2_type,type,
% 2.07/1.46 iftrueProp2: $o ).
% 2.07/1.46
% 2.07/1.46 thf(iftrueProp2,definition,
% 2.07/1.46 ( iftrueProp2
% 2.07/1.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ A )
% 2.07/1.46 => ( Xphi
% 2.07/1.46 => ( ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xz: $i] :
% 2.07/1.46 ( ( Xphi
% 2.07/1.46 & ( Xz = Xx ) )
% 2.07/1.46 | ( ~ Xphi
% 2.07/1.46 & ( Xz = Xy ) ) ) )
% 2.07/1.46 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(ifSingleton_type,type,
% 2.07/1.46 ifSingleton: $o ).
% 2.07/1.46
% 2.07/1.46 thf(ifSingleton,definition,
% 2.07/1.46 ( ifSingleton
% 2.07/1.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ A )
% 2.07/1.46 => ( singleton
% 2.07/1.46 @ ( dsetconstr @ A
% 2.07/1.46 @ ^ [Xz: $i] :
% 2.07/1.46 ( ( Xphi
% 2.07/1.46 & ( Xz = Xx ) )
% 2.07/1.46 | ( ~ Xphi
% 2.07/1.46 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(if_type,type,
% 2.07/1.46 if: $i > $o > $i > $i > $i ).
% 2.07/1.46
% 2.07/1.46 thf(ifp_type,type,
% 2.07/1.46 ifp: $o ).
% 2.07/1.46
% 2.07/1.46 thf(ifp,definition,
% 2.07/1.46 ( ifp
% 2.07/1.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ A )
% 2.07/1.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(theeq_type,type,
% 2.07/1.46 theeq: $o ).
% 2.07/1.46
% 2.07/1.46 thf(theeq,definition,
% 2.07/1.46 ( theeq
% 2.07/1.46 = ( ! [X: $i] :
% 2.07/1.46 ( ( singleton @ X )
% 2.07/1.46 => ! [Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ X )
% 2.07/1.46 => ( ( setunion @ X )
% 2.07/1.46 = Xx ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(iftrue_type,type,
% 2.07/1.46 iftrue: $o ).
% 2.07/1.46
% 2.07/1.46 thf(iftrue,definition,
% 2.07/1.46 ( iftrue
% 2.07/1.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ A )
% 2.07/1.46 => ( Xphi
% 2.07/1.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.07/1.46 = Xx ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(iffalse_type,type,
% 2.07/1.46 iffalse: $o ).
% 2.07/1.46
% 2.07/1.46 thf(iffalse,definition,
% 2.07/1.46 ( iffalse
% 2.07/1.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ A )
% 2.07/1.46 => ( ~ Xphi
% 2.07/1.46 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 2.07/1.46 = Xy ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(iftrueorfalse_type,type,
% 2.07/1.46 iftrueorfalse: $o ).
% 2.07/1.46
% 2.07/1.46 thf(iftrueorfalse,definition,
% 2.07/1.46 ( iftrueorfalse
% 2.07/1.46 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 2.07/1.46 ( ( in @ Xx @ A )
% 2.07/1.46 => ! [Xy: $i] :
% 2.07/1.46 ( ( in @ Xy @ A )
% 2.07/1.46 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectT_lem_type,type,
% 2.07/1.46 binintersectT_lem: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binintersectT_lem,definition,
% 2.07/1.46 ( binintersectT_lem
% 2.07/1.46 = ( ! [A: $i,X: $i] :
% 2.07/1.46 ( ( in @ X @ ( powerset @ A ) )
% 2.07/1.46 => ! [Y: $i] :
% 2.07/1.46 ( ( in @ Y @ ( powerset @ A ) )
% 2.07/1.46 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(binunionT_lem_type,type,
% 2.07/1.46 binunionT_lem: $o ).
% 2.07/1.46
% 2.07/1.46 thf(binunionT_lem,definition,
% 2.07/1.46 ( binunionT_lem
% 2.07/1.46 = ( ! [A: $i,X: $i] :
% 2.07/1.46 ( ( in @ X @ ( powerset @ A ) )
% 2.07/1.46 => ! [Y: $i] :
% 2.07/1.46 ( ( in @ Y @ ( powerset @ A ) )
% 2.07/1.46 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.07/1.46
% 2.07/1.46 thf(powersetT_lem_type,type,
% 2.07/1.46 powersetT_lem: $o ).
% 2.07/1.46
% 2.07/1.46 thf(powersetT_lem,definition,
% 2.07/1.46 ( powersetT_lem
% 2.07/1.46 = ( ! [A: $i,X: $i] :
% 2.07/1.46 ( ( in @ X @ ( powerset @ A ) )
% 2.07/1.47 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 2.07/1.47
% 2.07/1.47 thf(setminusT_lem_type,type,
% 2.07/1.47 setminusT_lem: $o ).
% 2.07/1.47
% 2.07/1.47 thf(setminusT_lem,definition,
% 2.07/1.47 ( setminusT_lem
% 2.07/1.47 = ( ! [A: $i,X: $i] :
% 2.07/1.47 ( ( in @ X @ ( powerset @ A ) )
% 2.07/1.47 => ! [Y: $i] :
% 2.07/1.47 ( ( in @ Y @ ( powerset @ A ) )
% 2.07/1.47 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 2.07/1.47
% 2.07/1.47 thf(complementT_lem_type,type,
% 2.07/1.47 complementT_lem: $o ).
% 2.07/1.47
% 2.07/1.47 thf(complementT_lem,definition,
% 2.07/1.47 ( complementT_lem
% 2.07/1.47 = ( ! [A: $i,X: $i] :
% 2.07/1.47 ( ( in @ X @ ( powerset @ A ) )
% 2.07/1.47 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 2.07/1.47
% 2.07/1.47 thf(setextT_type,type,
% 2.07/1.47 setextT: $o ).
% 2.07/1.47
% 2.07/1.47 thf(setextT,definition,
% 2.07/1.47 ( setextT
% 2.07/1.47 = ( ! [A: $i,X: $i] :
% 2.07/1.47 ( ( in @ X @ ( powerset @ A ) )
% 2.07/1.47 => ! [Y: $i] :
% 2.07/1.47 ( ( in @ Y @ ( powerset @ A ) )
% 2.07/1.47 => ( ! [Xx: $i] :
% 2.07/1.47 ( ( in @ Xx @ A )
% 2.07/1.47 => ( ( in @ Xx @ X )
% 2.07/1.47 => ( in @ Xx @ Y ) ) )
% 2.07/1.47 => ( ! [Xx: $i] :
% 2.07/1.47 ( ( in @ Xx @ A )
% 2.07/1.47 => ( ( in @ Xx @ Y )
% 2.07/1.47 => ( in @ Xx @ X ) ) )
% 2.07/1.47 => ( X = Y ) ) ) ) ) ) ) ).
% 2.07/1.47
% 2.07/1.47 thf(subsetTI_type,type,
% 2.07/1.47 subsetTI: $o ).
% 2.07/1.47
% 2.07/1.47 thf(subsetTI,definition,
% 2.07/1.47 ( subsetTI
% 2.07/1.47 = ( ! [A: $i,X: $i] :
% 2.07/1.47 ( ( in @ X @ ( powerset @ A ) )
% 2.07/1.47 => ! [Y: $i] :
% 2.07/1.47 ( ( in @ Y @ ( powerset @ A ) )
% 2.07/1.47 => ( ! [Xx: $i] :
% 2.07/1.47 ( ( in @ Xx @ A )
% 2.07/1.47 => ( ( in @ Xx @ X )
% 2.07/1.47 => ( in @ Xx @ Y ) ) )
% 2.07/1.47 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 2.07/1.47
% 2.07/1.47 thf(powersetTI1_type,type,
% 2.07/1.47 powersetTI1: $o ).
% 2.07/1.47
% 2.07/1.47 thf(powersetTI1,definition,
% 2.07/1.47 ( powersetTI1
% 2.07/1.47 = ( ! [A: $i,X: $i] :
% 2.07/1.47 ( ( in @ X @ ( powerset @ A ) )
% 2.07/1.47 => ! [Y: $i] :
% 2.07/1.47 ( ( in @ Y @ ( powerset @ A ) )
% 2.07/1.47 => ( ! [Xx: $i] :
% 2.07/1.47 ( ( in @ Xx @ A )
% 2.07/1.47 => ( ( in @ Xx @ X )
% 2.07/1.47 => ( in @ Xx @ Y ) ) )
% 2.07/1.47 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 2.07/1.47
% 2.07/1.47 thf(powersetTE1_type,type,
% 2.07/1.47 powersetTE1: $o ).
% 2.07/1.47
% 2.07/1.47 thf(powersetTE1,definition,
% 2.07/1.47 ( powersetTE1
% 2.07/1.47 = ( ! [A: $i,X: $i] :
% 2.07/1.47 ( ( in @ X @ ( powerset @ A ) )
% 2.07/1.47 => ! [Y: $i] :
% 2.07/1.47 ( ( in @ Y @ ( powerset @ A ) )
% 2.07/1.47 => ! [Xx: $i] :
% 2.07/1.47 ( ( in @ Xx @ A )
% 2.07/1.47 => ( ( in @ X @ ( powerset @ Y ) )
% 2.07/1.47 => ( ( in @ Xx @ X )
% 2.07/1.47 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 2.07/1.47
% 2.07/1.47 thf(complementTI1_type,type,
% 2.07/1.47 complementTI1: $o ).
% 2.07/1.47
% 2.07/1.47 thf(complementTI1,definition,
% 2.07/1.47 ( complementTI1
% 2.07/1.47 = ( ! [A: $i,X: $i] :
% 2.07/1.47 ( ( in @ X @ ( powerset @ A ) )
% 2.07/1.47 => ! [Xx: $i] :
% 2.07/1.47 ( ( in @ Xx @ A )
% 2.07/1.47 => ( ( in @ Xx @ X )
% 2.07/1.47 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 2.07/1.47
% 2.07/1.47 thf(complementTE1,conjecture,
% 2.07/1.47 ( setextAx
% 2.07/1.47 => ( emptysetAx
% 2.07/1.47 => ( setadjoinAx
% 2.07/1.47 => ( powersetAx
% 2.07/1.47 => ( setunionAx
% 2.07/1.47 => ( omega0Ax
% 2.07/1.47 => ( omegaSAx
% 2.07/1.47 => ( omegaIndAx
% 2.07/1.47 => ( replAx
% 2.07/1.47 => ( foundationAx
% 2.07/1.47 => ( wellorderingAx
% 2.07/1.47 => ( descrp
% 2.07/1.47 => ( dsetconstrI
% 2.07/1.47 => ( dsetconstrEL
% 2.07/1.47 => ( dsetconstrER
% 2.07/1.47 => ( exuE1
% 2.07/1.47 => ( prop2setE
% 2.07/1.47 => ( emptysetE
% 2.07/1.47 => ( emptysetimpfalse
% 2.07/1.47 => ( notinemptyset
% 2.07/1.47 => ( exuE3e
% 2.07/1.47 => ( setext
% 2.07/1.47 => ( emptyI
% 2.07/1.47 => ( noeltsimpempty
% 2.07/1.47 => ( setbeta
% 2.07/1.47 => ( nonemptyE1
% 2.07/1.47 => ( nonemptyI
% 2.07/1.47 => ( nonemptyI1
% 2.07/1.47 => ( setadjoinIL
% 2.07/1.47 => ( emptyinunitempty
% 2.07/1.47 => ( setadjoinIR
% 2.07/1.47 => ( setadjoinE
% 2.07/1.47 => ( setadjoinOr
% 2.07/1.47 => ( setoftrueEq
% 2.07/1.47 => ( powersetI
% 2.07/1.47 => ( emptyinPowerset
% 2.07/1.47 => ( emptyInPowerset
% 2.07/1.47 => ( powersetE
% 2.07/1.47 => ( setunionI
% 2.07/1.47 => ( setunionE
% 2.07/1.47 => ( subPowSU
% 2.07/1.47 => ( exuE2
% 2.07/1.47 => ( nonemptyImpWitness
% 2.07/1.47 => ( uniqinunit
% 2.07/1.47 => ( notinsingleton
% 2.07/1.47 => ( eqinunit
% 2.07/1.47 => ( singletonsswitch
% 2.07/1.47 => ( upairsetE
% 2.07/1.47 => ( upairsetIL
% 2.07/1.47 => ( upairsetIR
% 2.07/1.47 => ( emptyE1
% 2.07/1.47 => ( vacuousDall
% 2.07/1.47 => ( quantDeMorgan1
% 2.07/1.47 => ( quantDeMorgan2
% 2.07/1.47 => ( quantDeMorgan3
% 2.07/1.47 => ( quantDeMorgan4
% 2.07/1.47 => ( prop2setI
% 2.07/1.47 => ( prop2set2propI
% 2.07/1.47 => ( notdexE
% 2.07/1.47 => ( notdallE
% 2.07/1.47 => ( exuI1
% 2.07/1.47 => ( exuI3
% 2.07/1.47 => ( exuI2
% 2.07/1.47 => ( inCongP
% 2.07/1.47 => ( in__Cong
% 2.07/1.47 => ( exuE3u
% 2.07/1.47 => ( exu__Cong
% 2.07/1.47 => ( emptyset__Cong
% 2.07/1.47 => ( setadjoin__Cong
% 2.07/1.47 => ( powerset__Cong
% 2.07/1.47 => ( setunion__Cong
% 2.07/1.47 => ( omega__Cong
% 2.07/1.47 => ( exuEu
% 2.07/1.47 => ( descr__Cong
% 2.07/1.47 => ( dsetconstr__Cong
% 2.07/1.47 => ( subsetI1
% 2.07/1.47 => ( eqimpsubset2
% 2.07/1.47 => ( eqimpsubset1
% 2.07/1.47 => ( subsetI2
% 2.07/1.47 => ( emptysetsubset
% 2.07/1.47 => ( subsetE
% 2.07/1.47 => ( subsetE2
% 2.07/1.47 => ( notsubsetI
% 2.07/1.47 => ( notequalI1
% 2.07/1.47 => ( notequalI2
% 2.07/1.47 => ( subsetRefl
% 2.07/1.47 => ( subsetTrans
% 2.07/1.47 => ( setadjoinSub
% 2.07/1.47 => ( setadjoinSub2
% 2.07/1.47 => ( subset2powerset
% 2.07/1.47 => ( setextsub
% 2.07/1.47 => ( subsetemptysetimpeq
% 2.07/1.47 => ( powersetI1
% 2.07/1.47 => ( powersetE1
% 2.07/1.47 => ( inPowerset
% 2.07/1.47 => ( powersetsubset
% 2.07/1.47 => ( sepInPowerset
% 2.07/1.47 => ( sepSubset
% 2.07/1.47 => ( binunionIL
% 2.07/1.47 => ( upairset2IR
% 2.07/1.47 => ( binunionIR
% 2.07/1.47 => ( binunionEcases
% 2.07/1.47 => ( binunionE
% 2.07/1.47 => ( binunionLsub
% 2.07/1.47 => ( binunionRsub
% 2.07/1.47 => ( binintersectI
% 2.07/1.47 => ( binintersectSubset5
% 2.07/1.47 => ( binintersectEL
% 2.07/1.47 => ( binintersectLsub
% 2.07/1.47 => ( binintersectSubset2
% 2.07/1.47 => ( binintersectSubset3
% 2.07/1.47 => ( binintersectER
% 2.07/1.47 => ( disjointsetsI1
% 2.07/1.47 => ( binintersectRsub
% 2.07/1.47 => ( binintersectSubset4
% 2.07/1.47 => ( binintersectSubset1
% 2.07/1.47 => ( bs114d
% 2.07/1.47 => ( setminusI
% 2.07/1.47 => ( setminusEL
% 2.07/1.47 => ( setminusER
% 2.07/1.47 => ( setminusSubset2
% 2.07/1.47 => ( setminusERneg
% 2.07/1.47 => ( setminusELneg
% 2.07/1.47 => ( setminusILneg
% 2.07/1.47 => ( setminusIRneg
% 2.07/1.47 => ( setminusLsub
% 2.07/1.47 => ( setminusSubset1
% 2.07/1.47 => ( symdiffE
% 2.07/1.47 => ( symdiffI1
% 2.07/1.47 => ( symdiffI2
% 2.07/1.47 => ( symdiffIneg1
% 2.07/1.47 => ( symdiffIneg2
% 2.07/1.47 => ( secondinupair
% 2.07/1.47 => ( setukpairIL
% 2.07/1.47 => ( setukpairIR
% 2.07/1.47 => ( kpairiskpair
% 2.07/1.47 => ( kpairp
% 2.07/1.47 => ( singletonsubset
% 2.07/1.47 => ( singletoninpowerset
% 2.07/1.47 => ( singletoninpowunion
% 2.07/1.47 => ( upairset2E
% 2.07/1.47 => ( upairsubunion
% 2.07/1.47 => ( upairinpowunion
% 2.07/1.47 => ( ubforcartprodlem1
% 2.07/1.47 => ( ubforcartprodlem2
% 2.07/1.47 => ( ubforcartprodlem3
% 2.07/1.47 => ( cartprodpairin
% 2.07/1.47 => ( cartprodmempair1
% 2.07/1.47 => ( cartprodmempair
% 2.07/1.47 => ( setunionE2
% 2.07/1.47 => ( setunionsingleton1
% 2.07/1.47 => ( setunionsingleton2
% 2.07/1.47 => ( setunionsingleton
% 2.07/1.47 => ( singletonprop
% 2.07/1.47 => ( ex1E1
% 2.07/1.47 => ( ex1I
% 2.07/1.47 => ( ex1I2
% 2.07/1.47 => ( singletonsuniq
% 2.07/1.47 => ( setukpairinjL1
% 2.07/1.47 => ( kfstsingleton
% 2.07/1.47 => ( theprop
% 2.07/1.47 => ( kfstpairEq
% 2.07/1.47 => ( cartprodfstin
% 2.07/1.47 => ( setukpairinjL2
% 2.07/1.47 => ( setukpairinjL
% 2.07/1.47 => ( setukpairinjR11
% 2.07/1.47 => ( setukpairinjR12
% 2.07/1.47 => ( setukpairinjR1
% 2.07/1.47 => ( upairequniteq
% 2.07/1.47 => ( setukpairinjR2
% 2.07/1.47 => ( setukpairinjR
% 2.07/1.47 => ( ksndsingleton
% 2.07/1.47 => ( ksndpairEq
% 2.07/1.47 => ( kpairsurjEq
% 2.07/1.47 => ( cartprodsndin
% 2.07/1.47 => ( cartprodpairmemEL
% 2.07/1.47 => ( cartprodpairmemER
% 2.07/1.47 => ( cartprodmempaircEq
% 2.07/1.47 => ( cartprodfstpairEq
% 2.07/1.47 => ( cartprodsndpairEq
% 2.07/1.47 => ( cartprodpairsurjEq
% 2.07/1.47 => ( dpsetconstrI
% 2.07/1.47 => ( dpsetconstrSub
% 2.07/1.47 => ( setOfPairsIsBReln
% 2.07/1.47 => ( dpsetconstrERa
% 2.07/1.47 => ( dpsetconstrEL1
% 2.07/1.47 => ( dpsetconstrEL2
% 2.07/1.47 => ( dpsetconstrER
% 2.07/1.47 => ( funcImageSingleton
% 2.07/1.47 => ( apProp
% 2.07/1.47 => ( app
% 2.07/1.47 => ( infuncsetfunc
% 2.07/1.47 => ( ap2p
% 2.07/1.47 => ( funcinfuncset
% 2.07/1.47 => ( lamProp
% 2.07/1.47 => ( lamp
% 2.07/1.47 => ( lam2p
% 2.07/1.47 => ( brelnall1
% 2.07/1.47 => ( brelnall2
% 2.07/1.47 => ( ex1E2
% 2.07/1.47 => ( funcGraphProp1
% 2.07/1.47 => ( funcGraphProp3
% 2.07/1.47 => ( funcGraphProp2
% 2.07/1.47 => ( funcextLem
% 2.07/1.47 => ( funcGraphProp4
% 2.07/1.47 => ( subbreln
% 2.07/1.47 => ( eqbreln
% 2.07/1.47 => ( funcext
% 2.07/1.47 => ( funcext2
% 2.07/1.47 => ( ap2apEq1
% 2.07/1.47 => ( ap2apEq2
% 2.07/1.47 => ( beta1
% 2.07/1.47 => ( eta1
% 2.07/1.47 => ( lam2lamEq
% 2.07/1.47 => ( beta2
% 2.07/1.47 => ( eta2
% 2.07/1.47 => ( iffalseProp1
% 2.07/1.47 => ( iffalseProp2
% 2.07/1.47 => ( iftrueProp1
% 2.07/1.47 => ( iftrueProp2
% 2.07/1.47 => ( ifSingleton
% 2.07/1.47 => ( ifp
% 2.07/1.47 => ( theeq
% 2.07/1.47 => ( iftrue
% 2.07/1.47 => ( iffalse
% 2.07/1.47 => ( iftrueorfalse
% 2.07/1.47 => ( binintersectT_lem
% 2.07/1.47 => ( binunionT_lem
% 2.07/1.47 => ( powersetT_lem
% 2.07/1.47 => ( setminusT_lem
% 2.07/1.47 => ( complementT_lem
% 2.07/1.47 => ( setextT
% 2.07/1.47 => ( subsetTI
% 2.07/1.47 => ( powersetTI1
% 2.07/1.47 => ( powersetTE1
% 2.07/1.47 => ( complementTI1
% 2.07/1.47 => ! [A: $i,X: $i] :
% 2.07/1.47 ( ( in @ X @ ( powerset @ A ) )
% 2.07/1.47 => ! [Xx: $i] :
% 2.07/1.47 ( ( in @ Xx @ A )
% 2.07/1.47 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 2.07/1.47 => ( in @ Xx @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 2.07/1.47 ------------------------
% 2.07/1.49 % (24633)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_24371 for (2999ds/4Mi)
% 2.07/1.49 % (24637)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_24371 for (2999ds/275Mi)
% 2.07/1.49 % (24639)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_24371 for (2999ds/18Mi)
% 2.07/1.49 % (24633)Instruction limit reached!
% 2.07/1.49 % (24633)------------------------------
% 2.07/1.49 % (24633)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.07/1.49 % (24633)Termination reason: Unknown
% 2.07/1.49 % (24633)Termination phase: shuffling
% 2.07/1.49
% 2.07/1.49 % (24633)Memory used [KB]: 1535
% 2.07/1.49 % (24633)Time elapsed: 0.003 s
% 2.07/1.49 % (24633)Instructions burned: 5 (million)
% 2.07/1.49 % (24633)------------------------------
% 2.07/1.49 % (24633)------------------------------
% 2.07/1.49 % (24639)Instruction limit reached!
% 2.07/1.49 % (24639)------------------------------
% 2.07/1.49 % (24639)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.07/1.49 % (24639)Termination reason: Unknown
% 2.07/1.49 % (24639)Termination phase: shuffling
% 2.07/1.49
% 2.07/1.49 % (24639)Memory used [KB]: 1663
% 2.07/1.49 % (24639)Time elapsed: 0.007 s
% 2.07/1.49 % (24639)Instructions burned: 18 (million)
% 2.07/1.49 % (24639)------------------------------
% 2.07/1.49 % (24639)------------------------------
% 2.07/1.49 % (24634)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_24371 for (2999ds/27Mi)
% 2.07/1.49 % (24636)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_24371 for (2999ds/2Mi)
% 2.07/1.49 % (24635)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_24371 for (2999ds/2Mi)
% 2.07/1.49 % (24632)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_24371 for (2999ds/183Mi)
% 2.07/1.50 % (24635)Instruction limit reached!
% 2.07/1.50 % (24635)------------------------------
% 2.07/1.50 % (24635)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.07/1.50 % (24635)Termination reason: Unknown
% 2.07/1.50 % (24635)Termination phase: shuffling
% 2.07/1.50
% 2.07/1.50 % (24635)Memory used [KB]: 1407
% 2.07/1.50 % (24635)Time elapsed: 0.003 s
% 2.07/1.50 % (24635)Instructions burned: 3 (million)
% 2.07/1.50 % (24635)------------------------------
% 2.07/1.50 % (24635)------------------------------
% 2.07/1.50 % (24636)Instruction limit reached!
% 2.07/1.50 % (24636)------------------------------
% 2.07/1.50 % (24636)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.07/1.50 % (24636)Termination reason: Unknown
% 2.07/1.50 % (24636)Termination phase: shuffling
% 2.07/1.50
% 2.07/1.50 % (24636)Memory used [KB]: 1407
% 2.07/1.50 % (24636)Time elapsed: 0.004 s
% 2.07/1.50 % (24636)Instructions burned: 3 (million)
% 2.07/1.50 % (24636)------------------------------
% 2.07/1.50 % (24636)------------------------------
% 2.20/1.50 % (24645)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24371 for (2999ds/3Mi)
% 2.20/1.50 % (24645)Instruction limit reached!
% 2.20/1.50 % (24645)------------------------------
% 2.20/1.50 % (24645)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.50 % (24645)Termination reason: Unknown
% 2.20/1.50 % (24645)Termination phase: shuffling
% 2.20/1.50
% 2.20/1.50 % (24645)Memory used [KB]: 1407
% 2.20/1.50 % (24645)Time elapsed: 0.002 s
% 2.20/1.50 % (24645)Instructions burned: 3 (million)
% 2.20/1.50 % (24645)------------------------------
% 2.20/1.50 % (24645)------------------------------
% 2.20/1.50 % (24648)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_24371 for (2999ds/37Mi)
% 2.20/1.51 % (24653)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_24371 for (2999ds/1041Mi)
% 2.20/1.51 % (24634)Instruction limit reached!
% 2.20/1.51 % (24634)------------------------------
% 2.20/1.51 % (24634)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.51 % (24634)Termination reason: Unknown
% 2.20/1.51 % (24634)Termination phase: shuffling
% 2.20/1.51
% 2.20/1.51 % (24634)Memory used [KB]: 1918
% 2.20/1.51 % (24634)Time elapsed: 0.016 s
% 2.20/1.51 % (24634)Instructions burned: 27 (million)
% 2.20/1.51 % (24634)------------------------------
% 2.20/1.51 % (24634)------------------------------
% 2.20/1.51 % (24648)Instruction limit reached!
% 2.20/1.51 % (24648)------------------------------
% 2.20/1.51 % (24648)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.51 % (24648)Termination reason: Unknown
% 2.20/1.51 % (24648)Termination phase: shuffling
% 2.20/1.51
% 2.20/1.51 % (24648)Memory used [KB]: 2174
% 2.20/1.51 % (24648)Time elapsed: 0.011 s
% 2.20/1.51 % (24648)Instructions burned: 37 (million)
% 2.20/1.51 % (24648)------------------------------
% 2.20/1.51 % (24648)------------------------------
% 2.20/1.51 % (24651)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_24371 for (2999ds/3Mi)
% 2.20/1.51 % (24650)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_24371 for (2999ds/15Mi)
% 2.20/1.52 % (24651)Instruction limit reached!
% 2.20/1.52 % (24651)------------------------------
% 2.20/1.52 % (24651)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.52 % (24651)Termination reason: Unknown
% 2.20/1.52 % (24651)Termination phase: shuffling
% 2.20/1.52
% 2.20/1.52 % (24651)Memory used [KB]: 1407
% 2.20/1.52 % (24651)Time elapsed: 0.004 s
% 2.20/1.52 % (24651)Instructions burned: 3 (million)
% 2.20/1.52 % (24651)------------------------------
% 2.20/1.52 % (24651)------------------------------
% 2.20/1.52 % (24658)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_24371 for (2999ds/16Mi)
% 2.20/1.52 % (24650)Instruction limit reached!
% 2.20/1.52 % (24650)------------------------------
% 2.20/1.52 % (24650)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.52 % (24650)Termination reason: Unknown
% 2.20/1.52 % (24650)Termination phase: shuffling
% 2.20/1.52
% 2.20/1.52 % (24650)Memory used [KB]: 1663
% 2.20/1.52 % (24650)Time elapsed: 0.012 s
% 2.20/1.52 % (24650)Instructions burned: 16 (million)
% 2.20/1.52 % (24650)------------------------------
% 2.20/1.52 % (24650)------------------------------
% 2.20/1.53 % (24658)Instruction limit reached!
% 2.20/1.53 % (24658)------------------------------
% 2.20/1.53 % (24658)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.53 % (24658)Termination reason: Unknown
% 2.20/1.53 % (24658)Termination phase: shuffling
% 2.20/1.53
% 2.20/1.53 % (24658)Memory used [KB]: 1791
% 2.20/1.53 % (24658)Time elapsed: 0.006 s
% 2.20/1.53 % (24658)Instructions burned: 17 (million)
% 2.20/1.53 % (24658)------------------------------
% 2.20/1.53 % (24658)------------------------------
% 2.20/1.53 % (24657)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_24371 for (2999ds/7Mi)
% 2.20/1.53 % (24657)Instruction limit reached!
% 2.20/1.53 % (24657)------------------------------
% 2.20/1.53 % (24657)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.53 % (24657)Termination reason: Unknown
% 2.20/1.53 % (24657)Termination phase: shuffling
% 2.20/1.53
% 2.20/1.53 % (24657)Memory used [KB]: 1535
% 2.20/1.53 % (24657)Time elapsed: 0.006 s
% 2.20/1.53 % (24657)Instructions burned: 7 (million)
% 2.20/1.53 % (24657)------------------------------
% 2.20/1.53 % (24657)------------------------------
% 2.20/1.53 % (24660)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_24371 for (2999ds/3Mi)
% 2.20/1.53 % (24660)Instruction limit reached!
% 2.20/1.53 % (24660)------------------------------
% 2.20/1.53 % (24660)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.53 % (24660)Termination reason: Unknown
% 2.20/1.53 % (24660)Termination phase: shuffling
% 2.20/1.53
% 2.20/1.53 % (24660)Memory used [KB]: 1407
% 2.20/1.53 % (24660)Time elapsed: 0.004 s
% 2.20/1.53 % (24660)Instructions burned: 3 (million)
% 2.20/1.53 % (24660)------------------------------
% 2.20/1.53 % (24660)------------------------------
% 2.20/1.54 % (24665)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_24371 for (2999ds/7Mi)
% 2.20/1.54 % (24665)Instruction limit reached!
% 2.20/1.54 % (24665)------------------------------
% 2.20/1.54 % (24665)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.54 % (24665)Termination reason: Unknown
% 2.20/1.54 % (24665)Termination phase: shuffling
% 2.20/1.54
% 2.20/1.54 % (24665)Memory used [KB]: 1535
% 2.20/1.54 % (24665)Time elapsed: 0.003 s
% 2.20/1.54 % (24665)Instructions burned: 7 (million)
% 2.20/1.54 % (24665)------------------------------
% 2.20/1.54 % (24665)------------------------------
% 2.20/1.54 % (24664)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_24371 for (2999ds/3Mi)
% 2.20/1.54 % (24664)Instruction limit reached!
% 2.20/1.54 % (24664)------------------------------
% 2.20/1.54 % (24664)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.54 % (24664)Termination reason: Unknown
% 2.20/1.54 % (24664)Termination phase: shuffling
% 2.20/1.54
% 2.20/1.54 % (24664)Memory used [KB]: 1407
% 2.20/1.54 % (24664)Time elapsed: 0.004 s
% 2.20/1.54 % (24664)Instructions burned: 3 (million)
% 2.20/1.54 % (24664)------------------------------
% 2.20/1.54 % (24664)------------------------------
% 2.20/1.55 % (24671)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_24371 for (2999ds/18Mi)
% 2.20/1.55 % (24668)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_24371 for (2999ds/3Mi)
% 2.20/1.55 % (24668)Instruction limit reached!
% 2.20/1.55 % (24668)------------------------------
% 2.20/1.55 % (24668)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.55 % (24668)Termination reason: Unknown
% 2.20/1.55 % (24668)Termination phase: shuffling
% 2.20/1.55
% 2.20/1.55 % (24668)Memory used [KB]: 1407
% 2.20/1.55 % (24668)Time elapsed: 0.004 s
% 2.20/1.55 % (24668)Instructions burned: 4 (million)
% 2.20/1.55 % (24668)------------------------------
% 2.20/1.55 % (24668)------------------------------
% 2.20/1.55 % (24671)Instruction limit reached!
% 2.20/1.55 % (24671)------------------------------
% 2.20/1.55 % (24671)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.55 % (24671)Termination reason: Unknown
% 2.20/1.55 % (24671)Termination phase: shuffling
% 2.20/1.55
% 2.20/1.55 % (24671)Memory used [KB]: 1791
% 2.20/1.55 % (24671)Time elapsed: 0.006 s
% 2.20/1.55 % (24671)Instructions burned: 18 (million)
% 2.20/1.55 % (24671)------------------------------
% 2.20/1.55 % (24671)------------------------------
% 2.20/1.55 % (24670)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_24371 for (2999ds/4Mi)
% 2.20/1.56 % (24674)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_24371 for (2999ds/710Mi)
% 2.20/1.56 % (24670)Instruction limit reached!
% 2.20/1.56 % (24670)------------------------------
% 2.20/1.56 % (24670)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.56 % (24670)Termination reason: Unknown
% 2.20/1.56 % (24670)Termination phase: shuffling
% 2.20/1.56
% 2.20/1.56 % (24670)Memory used [KB]: 1535
% 2.20/1.56 % (24670)Time elapsed: 0.005 s
% 2.20/1.56 % (24670)Instructions burned: 5 (million)
% 2.20/1.56 % (24670)------------------------------
% 2.20/1.56 % (24670)------------------------------
% 2.20/1.56 % (24632)Instruction limit reached!
% 2.20/1.56 % (24632)------------------------------
% 2.20/1.56 % (24632)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.56 % (24632)Termination reason: Unknown
% 2.20/1.56 % (24632)Termination phase: Saturation
% 2.20/1.56
% 2.20/1.56 % (24632)Memory used [KB]: 7547
% 2.20/1.56 % (24632)Time elapsed: 0.068 s
% 2.20/1.56 % (24632)Instructions burned: 186 (million)
% 2.20/1.56 % (24632)------------------------------
% 2.20/1.56 % (24632)------------------------------
% 2.20/1.56 % (24677)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_24371 for (2999ds/902Mi)
% 2.20/1.57 % (24637)Instruction limit reached!
% 2.20/1.57 % (24637)------------------------------
% 2.20/1.57 % (24637)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.57 % (24637)Termination reason: Unknown
% 2.20/1.57 % (24637)Termination phase: Saturation
% 2.20/1.57
% 2.20/1.57 % (24637)Memory used [KB]: 9466
% 2.20/1.57 % (24637)Time elapsed: 0.082 s
% 2.20/1.57 % (24637)Instructions burned: 276 (million)
% 2.20/1.57 % (24637)------------------------------
% 2.20/1.57 % (24637)------------------------------
% 2.20/1.57 % (24676)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_24371 for (2999ds/6Mi)
% 2.20/1.57 % (24676)Instruction limit reached!
% 2.20/1.57 % (24676)------------------------------
% 2.20/1.57 % (24676)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.57 % (24676)Termination reason: Unknown
% 2.20/1.57 % (24676)Termination phase: shuffling
% 2.20/1.57
% 2.20/1.57 % (24676)Memory used [KB]: 1535
% 2.20/1.57 % (24676)Time elapsed: 0.005 s
% 2.20/1.57 % (24676)Instructions burned: 6 (million)
% 2.20/1.57 % (24676)------------------------------
% 2.20/1.57 % (24676)------------------------------
% 2.20/1.57 % (24684)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_24371 for (2999ds/5Mi)
% 2.20/1.57 % (24680)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_24371 for (2999ds/21Mi)
% 2.20/1.57 % (24684)Instruction limit reached!
% 2.20/1.57 % (24684)------------------------------
% 2.20/1.57 % (24684)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.57 % (24684)Termination reason: Unknown
% 2.20/1.57 % (24684)Termination phase: shuffling
% 2.20/1.57
% 2.20/1.57 % (24684)Memory used [KB]: 1535
% 2.20/1.57 % (24684)Time elapsed: 0.003 s
% 2.20/1.57 % (24684)Instructions burned: 6 (million)
% 2.20/1.57 % (24684)------------------------------
% 2.20/1.57 % (24684)------------------------------
% 2.20/1.58 % (24687)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_24371 for (2999ds/6Mi)
% 2.20/1.58 % (24687)Instruction limit reached!
% 2.20/1.58 % (24687)------------------------------
% 2.20/1.58 % (24687)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.58 % (24687)Termination reason: Unknown
% 2.20/1.58 % (24687)Termination phase: shuffling
% 2.20/1.58
% 2.20/1.58 % (24687)Memory used [KB]: 1535
% 2.20/1.58 % (24687)Time elapsed: 0.003 s
% 2.20/1.58 % (24687)Instructions burned: 6 (million)
% 2.20/1.58 % (24687)------------------------------
% 2.20/1.58 % (24687)------------------------------
% 2.20/1.58 % (24680)Instruction limit reached!
% 2.20/1.58 % (24680)------------------------------
% 2.20/1.58 % (24680)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.58 % (24680)Termination reason: Unknown
% 2.20/1.58 % (24680)Termination phase: shuffling
% 2.20/1.58
% 2.20/1.58 % (24680)Memory used [KB]: 1791
% 2.20/1.58 % (24680)Time elapsed: 0.009 s
% 2.20/1.58 % (24680)Instructions burned: 24 (million)
% 2.20/1.58 % (24680)------------------------------
% 2.20/1.58 % (24680)------------------------------
% 2.20/1.58 % (24690)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_24371 for (2999ds/779Mi)
% 2.20/1.58 % (24688)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_24371 for (2999ds/377Mi)
% 2.20/1.59 % (24695)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_24371 for (2999ds/19Mi)
% 2.20/1.59 % (24696)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_24371 for (2998ds/879Mi)
% 2.20/1.59 % (24695)Instruction limit reached!
% 2.20/1.59 % (24695)------------------------------
% 2.20/1.59 % (24695)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.59 % (24695)Termination reason: Unknown
% 2.20/1.59 % (24695)Termination phase: shuffling
% 2.20/1.59
% 2.20/1.59 % (24695)Memory used [KB]: 1791
% 2.20/1.59 % (24695)Time elapsed: 0.007 s
% 2.20/1.59 % (24695)Instructions burned: 22 (million)
% 2.20/1.59 % (24695)------------------------------
% 2.20/1.59 % (24695)------------------------------
% 2.20/1.60 % (24701)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_24371 for (2998ds/17Mi)
% 2.20/1.61 % (24701)Instruction limit reached!
% 2.20/1.61 % (24701)------------------------------
% 2.20/1.61 % (24701)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.61 % (24701)Termination reason: Unknown
% 2.20/1.61 % (24701)Termination phase: shuffling
% 2.20/1.61
% 2.20/1.61 % (24701)Memory used [KB]: 1791
% 2.20/1.61 % (24701)Time elapsed: 0.007 s
% 2.20/1.61 % (24701)Instructions burned: 19 (million)
% 2.20/1.61 % (24701)------------------------------
% 2.20/1.61 % (24701)------------------------------
% 2.20/1.62 % (24707)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_24371 for (2998ds/3Mi)
% 2.20/1.62 % (24707)Instruction limit reached!
% 2.20/1.62 % (24707)------------------------------
% 2.20/1.62 % (24707)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.62 % (24707)Termination reason: Unknown
% 2.20/1.62 % (24707)Termination phase: shuffling
% 2.20/1.62
% 2.20/1.62 % (24707)Memory used [KB]: 1407
% 2.20/1.62 % (24707)Time elapsed: 0.002 s
% 2.20/1.62 % (24707)Instructions burned: 3 (million)
% 2.20/1.62 % (24707)------------------------------
% 2.20/1.62 % (24707)------------------------------
% 2.20/1.63 % (24714)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_24371 for (2998ds/30Mi)
% 2.20/1.64 % (24714)Instruction limit reached!
% 2.20/1.64 % (24714)------------------------------
% 2.20/1.64 % (24714)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.64 % (24714)Termination reason: Unknown
% 2.20/1.64 % (24714)Termination phase: shuffling
% 2.20/1.64
% 2.20/1.64 % (24714)Memory used [KB]: 2046
% 2.20/1.64 % (24714)Time elapsed: 0.010 s
% 2.20/1.64 % (24714)Instructions burned: 33 (million)
% 2.20/1.64 % (24714)------------------------------
% 2.20/1.64 % (24714)------------------------------
% 2.20/1.65 % (24727)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_24371 for (2998ds/127Mi)
% 2.20/1.66 % (24677)First to succeed.
% 2.20/1.66 % (24677)Refutation found. Thanks to Tanya!
% 2.20/1.66 % SZS status Theorem for DTF2THF_24371
% 2.20/1.66 % SZS output start Proof for DTF2THF_24371
% 2.20/1.66 thf(func_def_0, type, in: $i > $i > $o).
% 2.20/1.66 thf(func_def_1, type, exu: ($i > $o) > $o).
% 2.20/1.66 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 2.20/1.66 thf(func_def_8, type, powerset: $i > $i).
% 2.20/1.66 thf(func_def_10, type, setunion: $i > $i).
% 2.20/1.66 thf(func_def_19, type, descr: ($i > $o) > $i).
% 2.20/1.66 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 2.20/1.66 thf(func_def_26, type, prop2set: $o > $i).
% 2.20/1.66 thf(func_def_36, type, nonempty: $i > $o).
% 2.20/1.66 thf(func_def_69, type, set2prop: $i > $o).
% 2.20/1.66 thf(func_def_88, type, subset: $i > $i > $o).
% 2.20/1.66 thf(func_def_89, type, disjoint: $i > $i > $o).
% 2.20/1.66 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 2.20/1.66 thf(func_def_114, type, binunion: $i > $i > $i).
% 2.20/1.66 thf(func_def_122, type, binintersect: $i > $i > $i).
% 2.20/1.66 thf(func_def_135, type, regular: $i > $o).
% 2.20/1.66 thf(func_def_136, type, setminus: $i > $i > $i).
% 2.20/1.66 thf(func_def_147, type, symdiff: $i > $i > $i).
% 2.20/1.66 thf(func_def_153, type, iskpair: $i > $o).
% 2.20/1.66 thf(func_def_158, type, kpair: $i > $i > $i).
% 2.20/1.66 thf(func_def_160, type, cartprod: $i > $i > $i).
% 2.20/1.66 thf(func_def_177, type, singleton: $i > $o).
% 2.20/1.66 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 2.20/1.66 thf(func_def_184, type, atmost1p: $i > $o).
% 2.20/1.66 thf(func_def_185, type, atleast2p: $i > $o).
% 2.20/1.66 thf(func_def_186, type, atmost2p: $i > $o).
% 2.20/1.66 thf(func_def_187, type, upairsetp: $i > $o).
% 2.20/1.66 thf(func_def_191, type, kfst: $i > $i).
% 2.20/1.66 thf(func_def_203, type, ksnd: $i > $i).
% 2.20/1.66 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 2.20/1.66 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 2.20/1.66 thf(func_def_222, type, func: $i > $i > $i > $o).
% 2.20/1.66 thf(func_def_223, type, funcSet: $i > $i > $i).
% 2.20/1.66 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 2.20/1.66 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 2.20/1.66 thf(func_def_290, type, sP2: $o > $i > $i > $i > $o).
% 2.20/1.66 thf(func_def_291, type, sP3: $i > $i > $o).
% 2.20/1.66 thf(func_def_292, type, sP4: $i > $o).
% 2.20/1.66 thf(func_def_293, type, sP5: $i > $i > $o).
% 2.20/1.66 thf(func_def_294, type, sP6: $i > $i > $o).
% 2.20/1.66 thf(func_def_296, type, sP8: $i > $i > $i > $i > $o).
% 2.20/1.66 thf(func_def_299, type, sP11: $i > $i > $i > $o).
% 2.20/1.66 thf(func_def_304, type, sK16: $i > $o).
% 2.20/1.66 thf(func_def_305, type, sK17: $i > $o).
% 2.20/1.66 thf(func_def_315, type, sK27: $i > $i).
% 2.20/1.66 thf(func_def_318, type, sK30: $i > ($i > $i) > $i > $i).
% 2.20/1.66 thf(func_def_340, type, sK52: $i > $i > $o).
% 2.20/1.66 thf(func_def_370, type, sK82: $i > $i > $o).
% 2.20/1.66 thf(func_def_378, type, sK90: $i > $i).
% 2.20/1.66 thf(func_def_382, type, sK94: $i > $i > $o).
% 2.20/1.66 thf(func_def_389, type, sK101: $i > $o).
% 2.20/1.66 thf(func_def_413, type, sK125: $i > $o).
% 2.20/1.66 thf(func_def_415, type, sK127: ($i > $o) > $i > $i).
% 2.20/1.66 thf(func_def_419, type, sK131: $i > $o).
% 2.20/1.66 thf(func_def_422, type, sK134: $i > ($i > $o) > $i).
% 2.20/1.66 thf(func_def_423, type, sK135: $i > ($i > $o) > $i).
% 2.20/1.66 thf(func_def_432, type, sK144: $i > $i > ($i > $i) > $i).
% 2.20/1.66 thf(func_def_433, type, sK145: $i > $i).
% 2.20/1.66 thf(func_def_439, type, sK151: $i > $o).
% 2.20/1.66 thf(func_def_451, type, sK163: $i > $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_452, type, sK164: $i > $i > $i).
% 2.20/1.66 thf(func_def_455, type, sK167: $i > $i).
% 2.20/1.66 thf(func_def_457, type, sK169: $i > $i).
% 2.20/1.66 thf(func_def_466, type, sK178: $i > $o).
% 2.20/1.66 thf(func_def_468, type, sK180: ($i > $o) > $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_469, type, sK181: ($i > $o) > $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_470, type, sK182: $i > $i > $i).
% 2.20/1.66 thf(func_def_471, type, sK183: $i > $i > $i).
% 2.20/1.66 thf(func_def_479, type, sK191: $i > $i).
% 2.20/1.66 thf(func_def_482, type, sK194: $i > $i > $i).
% 2.20/1.66 thf(func_def_485, type, sK197: ($i > $o) > $i).
% 2.20/1.66 thf(func_def_486, type, sK198: $i > $o).
% 2.20/1.66 thf(func_def_487, type, sK199: $i > $i).
% 2.20/1.66 thf(func_def_496, type, sK208: $i > $o).
% 2.20/1.66 thf(func_def_500, type, sK212: $i > $o).
% 2.20/1.66 thf(func_def_503, type, sK215: $i > ($i > $o) > $i > $i).
% 2.20/1.66 thf(func_def_505, type, sK217: $i > $o).
% 2.20/1.66 thf(func_def_520, type, sK232: ($i > $i) > $i > $i > $i).
% 2.20/1.66 thf(func_def_523, type, sK235: $i > $i).
% 2.20/1.66 thf(func_def_527, type, sK239: $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_552, type, sK264: $i > $o).
% 2.20/1.66 thf(func_def_572, type, sK284: $i > $o).
% 2.20/1.66 thf(func_def_574, type, sK286: ($i > $o) > $i > $i).
% 2.20/1.66 thf(func_def_584, type, sK296: $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_591, type, sK303: $i > ($i > $o) > $i).
% 2.20/1.66 thf(func_def_592, type, sK304: $i > $o).
% 2.20/1.66 thf(func_def_603, type, sK315: $i > $i > $i).
% 2.20/1.66 thf(func_def_610, type, sK322: $i > $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_615, type, sK327: $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_616, type, sK328: $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_639, type, sK351: $i > $o).
% 2.20/1.66 thf(func_def_660, type, sK372: ($i > $o) > $i).
% 2.20/1.66 thf(func_def_661, type, sK373: $i > $o).
% 2.20/1.66 thf(func_def_662, type, sK374: $i > $i).
% 2.20/1.66 thf(func_def_669, type, sK381: $i > $i).
% 2.20/1.66 thf(func_def_670, type, sK382: ($i > $i) > $i > $i > $i).
% 2.20/1.66 thf(func_def_684, type, sK396: $i > $o).
% 2.20/1.66 thf(func_def_687, type, sK399: $i > $i).
% 2.20/1.66 thf(func_def_703, type, sK415: ($i > $o) > ($i > $o) > $i).
% 2.20/1.66 thf(func_def_704, type, sK416: ($i > $o) > ($i > $o) > $i).
% 2.20/1.66 thf(func_def_705, type, sK417: $i > $o).
% 2.20/1.66 thf(func_def_706, type, sK418: $i > $o).
% 2.20/1.66 thf(func_def_707, type, sK419: ($i > $o) > $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_708, type, sK420: ($i > $o) > $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_712, type, sK424: $i > $o).
% 2.20/1.66 thf(func_def_717, type, sK429: $i > $i > $i).
% 2.20/1.66 thf(func_def_718, type, sK430: $i > $i > $i).
% 2.20/1.66 thf(func_def_719, type, sK431: $i > $i > $i).
% 2.20/1.66 thf(func_def_720, type, sK432: $i > $i > $i).
% 2.20/1.66 thf(func_def_721, type, sK433: $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_722, type, sK434: $i > $i).
% 2.20/1.66 thf(func_def_723, type, sK435: $i > $i).
% 2.20/1.66 thf(func_def_724, type, sK436: $i > $i).
% 2.20/1.66 thf(func_def_725, type, sK437: $i > $i).
% 2.20/1.66 thf(func_def_726, type, sK438: $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_727, type, sK439: $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_728, type, sK440: $i > $i > $i).
% 2.20/1.66 thf(func_def_729, type, sK441: $i > $i > $i).
% 2.20/1.66 thf(func_def_730, type, sK442: $i > $i > $i).
% 2.20/1.66 thf(func_def_732, type, sK444: $i > $i).
% 2.20/1.66 thf(func_def_733, type, sK445: $i > $i).
% 2.20/1.66 thf(func_def_734, type, sK446: $i > $i).
% 2.20/1.66 thf(func_def_735, type, sK447: $i > $i > $i).
% 2.20/1.66 thf(func_def_736, type, sK448: $i > $i > $i).
% 2.20/1.66 thf(func_def_741, type, sK453: $i > $o).
% 2.20/1.66 thf(func_def_745, type, sK457: $i > $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_746, type, sK458: $i > $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_757, type, sK469: $i > $o).
% 2.20/1.66 thf(func_def_760, type, sK472: $i > ($i > $o) > $i).
% 2.20/1.66 thf(func_def_761, type, sK473: $i > $o).
% 2.20/1.66 thf(func_def_763, type, sK475: $i > ($i > $o) > $i).
% 2.20/1.66 thf(func_def_764, type, sK476: $i > $o).
% 2.20/1.66 thf(func_def_765, type, sK477: ($i > $o) > $i).
% 2.20/1.66 thf(func_def_778, type, sK490: $i > $i > $i).
% 2.20/1.66 thf(func_def_796, type, sK508: $i > $i > ($i > $i) > $i).
% 2.20/1.66 thf(func_def_797, type, sK509: $i > $i).
% 2.20/1.66 thf(func_def_813, type, sK525: $i > $i > $o).
% 2.20/1.66 thf(func_def_819, type, sK531: $i > ($i > $o) > $i).
% 2.20/1.66 thf(func_def_820, type, sK532: $i > $o).
% 2.20/1.66 thf(func_def_829, type, sK541: $i > $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_856, type, sK568: $i > $i).
% 2.20/1.66 thf(func_def_860, type, sK572: $o > $i > $i > $i).
% 2.20/1.66 thf(func_def_871, type, sK583: $i > $o).
% 2.20/1.66 thf(func_def_886, type, sK598: $i > $i).
% 2.20/1.66 thf(func_def_887, type, sK599: ($i > $i) > $i > $i > $i).
% 2.20/1.66 thf(func_def_890, type, sK602: $i > $i > $i).
% 2.20/1.66 thf(func_def_897, type, sK609: $i > $o).
% 2.20/1.66 thf(func_def_900, type, sK612: $i > ($i > $o) > $i).
% 2.20/1.66 thf(func_def_901, type, sK613: $i > ($i > $o) > $i).
% 2.20/1.66 thf(func_def_916, type, sK628: $i > $o).
% 2.20/1.66 thf(func_def_921, type, sK633: $i > $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_922, type, sK634: $i > $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_923, type, sK635: $i > $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_924, type, sK636: $i > $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_931, type, sK643: $i > $i > $i).
% 2.20/1.66 thf(func_def_934, type, sK646: $i > $i > $o).
% 2.20/1.66 thf(func_def_937, type, sK649: $i > $o).
% 2.20/1.66 thf(func_def_939, type, sK651: $i > ($i > $o) > $i).
% 2.20/1.66 thf(func_def_944, type, sK656: $i > $o).
% 2.20/1.66 thf(func_def_948, type, sK660: $i > $i > $o).
% 2.20/1.66 thf(func_def_950, type, sK662: $i > $i).
% 2.20/1.66 thf(func_def_951, type, sK663: $i > $i).
% 2.20/1.66 thf(func_def_952, type, sK664: $i > ($i > $i > $o) > $i).
% 2.20/1.66 thf(func_def_953, type, sK665: $i > ($i > $i > $o) > $i).
% 2.20/1.66 thf(func_def_954, type, sK666: $i > $i > ($i > $i > $o) > $i).
% 2.20/1.66 thf(func_def_971, type, sK683: $i > $i > $i).
% 2.20/1.66 thf(func_def_972, type, sK684: $i > $i > $o).
% 2.20/1.66 thf(func_def_986, type, sK698: $i > $i > $o).
% 2.20/1.66 thf(func_def_990, type, sK702: $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_994, type, sK706: $i > $i > $i > $i).
% 2.20/1.66 thf(func_def_995, type, sK707: ($i > $o) > $i).
% 2.20/1.66 thf(func_def_996, type, sK708: ($i > $o) > $i).
% 2.20/1.66 thf(func_def_997, type, sK709: $i > $o).
% 2.20/1.66 thf(func_def_1001, type, sK713: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.20/1.66 thf(func_def_1002, type, sK714: ($i > $o) > ($i > $o) > $i > $i > $i).
% 2.20/1.66 thf(func_def_1005, type, sK717: $i > $o).
% 2.20/1.66 thf(func_def_1006, type, sK718: $i > $o).
% 2.20/1.66 thf(func_def_1007, type, sK719: $i > ($i > $o) > $i).
% 2.20/1.66 thf(func_def_1008, type, sK720: $i > $o).
% 2.20/1.66 thf(func_def_1011, type, sK723: $i > $i).
% 2.20/1.66 thf(func_def_1014, type, sK726: $i > ($i > $o) > $i).
% 2.20/1.66 thf(func_def_1015, type, sK727: $i > $o).
% 2.20/1.66 thf(func_def_1031, type, sK743: $i > $o).
% 2.20/1.66 thf(func_def_1032, type, sK744: $i > $o).
% 2.20/1.66 thf(func_def_1033, type, sK745: ($i > $o) > ($i > $o) > $i).
% 2.20/1.66 thf(func_def_1034, type, sK746: ($i > $o) > ($i > $o) > $i).
% 2.20/1.66 thf(func_def_1036, type, ph748: !>[X0: $tType]:(X0)).
% 2.20/1.66 thf(f3729,plain,(
% 2.20/1.66 $false),
% 2.20/1.66 inference(avatar_sat_refutation,[status(thm)],[f3667,f3692,f3715])).
% 2.20/1.66 thf(f3715,plain,(
% 2.20/1.66 ~spl747_6),
% 2.20/1.66 inference(avatar_contradiction_clause,[status(thm)],[f3714])).
% 2.20/1.66 thf(f3714,plain,(
% 2.20/1.66 $false | ~spl747_6),
% 2.20/1.66 inference(trivial_inequality_removal,[status(thm)],[f3709])).
% 2.20/1.66 thf(f3709,plain,(
% 2.20/1.66 ($true != $true) | ~spl747_6),
% 2.20/1.66 inference(superposition,[status(thm)],[f2382,f3647])).
% 2.20/1.66 thf(f3647,plain,(
% 2.20/1.66 ((in @ sK130 @ sK128) = $true) | ~spl747_6),
% 2.20/1.66 inference(avatar_component_clause,[status(thm)],[f3645])).
% 2.20/1.66 thf(f3645,definition,(
% 2.20/1.66 spl747_6 <=> ((in @ sK130 @ sK128) = $true)),
% 2.20/1.66 introduced(definition,[new_symbols(naming,[spl747_6])],[avatar_definition])).
% 2.20/1.66 thf(f2382,plain,(
% 2.20/1.66 ((in @ sK130 @ sK128) != $true)),
% 2.20/1.66 inference(cnf_transformation,[status(thm)],[f1229])).
% 2.20/1.66 thf(f1229,plain,(
% 2.20/1.66 (exuI2 = $true) & (app = $true) & (prop2setE = $true) & (setoftrueEq = $true) & (powersetTE1 = $true) & (exuI1 = $true) & (notinemptyset = $true) & (funcext = $true) & (cartprodpairmemEL = $true) & (binintersectSubset2 = $true) & (setadjoinIR = $true) & (iftrueProp1 = $true) & (iftrueProp2 = $true) & (omega0Ax = $true) & (emptysetimpfalse = $true) & (binintersectI = $true) & (setunionsingleton1 = $true) & (cartprodmempair1 = $true) & (cartprodsndpairEq = $true) & (exuE3e = $true) & (prop2setI = $true) & (kfstsingleton = $true) & (singletoninpowunion = $true) & (binintersectLsub = $true) & (nonemptyI1 = $true) & (omegaIndAx = $true) & (funcImageSingleton = $true) & (powersetE1 = $true) & (nonemptyE1 = $true) & (cartprodmempair = $true) & (setadjoin__Cong = $true) & (setextAx = $true) & (ex1E1 = $true) & (funcGraphProp4 = $true) & (notdallE = $true) & (noeltsimpempty = $true) & (setminusIRneg = $true) & (upairsetIR = $true) & (funcextLem = $true) & (subbreln = $true) & (funcGraphProp1 = $true) & (setunionE2 = $true) & (quantDeMorgan1 = $true) & (setukpairinjL1 = $true) & (dpsetconstrEL2 = $true) & (emptyset__Cong = $true) & (subsetTrans = $true) & (ex1I2 = $true) & (powersetTI1 = $true) & (singletonsubset = $true) & (notequalI1 = $true) & (theeq = $true) & (setukpairinjR12 = $true) & (ifp = $true) & (setunionAx = $true) & (cartprodfstin = $true) & (setbeta = $true) & (apProp = $true) & (subset2powerset = $true) & (setminusI = $true) & (lam2lamEq = $true) & (setminusELneg = $true) & (powersetI = $true) & (nonemptyI = $true) & (iftrue = $true) & (singletoninpowerset = $true) & (singletonprop = $true) & (notsubsetI = $true) & (eqbreln = $true) & (funcGraphProp2 = $true) & (binintersectSubset3 = $true) & (exuI3 = $true) & (emptyI = $true) & (powersetAx = $true) & (symdiffE = $true) & (wellorderingAx = $true) & (ap2apEq1 = $true) & (setminusERneg = $true) & (descrp = $true) & (exuE1 = $true) & (setadjoinE = $true) & (notdexE = $true) & (setminusILneg = $true) & (powersetsubset = $true) & (nonemptyImpWitness = $true) & (subsetTI = $true) & (setadjoinAx = $true) & (setext = $true) & (funcinfuncset = $true) & (setminusSubset1 = $true) & (singletonsuniq = $true) & (emptyInPowerset = $true) & (lamp = $true) & (disjointsetsI1 = $true) & (exuE2 = $true) & (binintersectEL = $true) & (subsetE = $true) & (iffalseProp1 = $true) & (sepInPowerset = $true) & (binunionE = $true) & (setminusLsub = $true) & (setunionsingleton = $true) & (binunionIL = $true) & (inCongP = $true) & (dsetconstrER = $true) & (binintersectER = $true) & (subPowSU = $true) & (sepSubset = $true) & (setextT = $true) & (prop2set2propI = $true) & (theprop = $true) & (binunionLsub = $true) & (eta2 = $true) & (ap2apEq2 = $true) & (dpsetconstrEL1 = $true) & (setadjoinIL = $true) & (setukpairinjL2 = $true) & (emptyinPowerset = $true) & (cartprodpairin = $true) & (ap2p = $true) & (omega__Cong = $true) & (setminusT_lem = $true) & (iffalse = $true) & (uniqinunit = $true) & (exu__Cong = $true) & (infuncsetfunc = $true) & (cartprodpairsurjEq = $true) & (upairinpowunion = $true) & (quantDeMorgan2 = $true) & (dpsetconstrERa = $true) & (setukpairinjR11 = $true) & (ksndsingleton = $true) & (omegaSAx = $true) & (symdiffIneg2 = $true) & (inPowerset = $true) & (setadjoinSub = $true) & (eta1 = $true) & (symdiffI1 = $true) & (beta1 = $true) & (setadjoinSub2 = $true) & (eqimpsubset2 = $true) & (funcGraphProp3 = $true) & (foundationAx = $true) & (setukpairinjR = $true) & (cartprodsndin = $true) & (upairsetE = $true) & (upairequniteq = $true) & (dpsetconstrER = $true) & (ubforcartprodlem3 = $true) & (setunionI = $true) & (eqimpsubset1 = $true) & (funcext2 = $true) & (cartprodmempaircEq = $true) & (iffalseProp2 = $true) & (notinsingleton = $true) & (setukpairIL = $true) & (kpairsurjEq = $true) & (kpairp = $true) & (subsetRefl = $true) & (binintersectSubset5 = $true) & (setukpairIR = $true) & (emptyinunitempty = $true) & (in__Cong = $true) & (secondinupair = $true) & (setminusSubset2 = $true) & (quantDeMorgan3 = $true) & (binintersectT_lem = $true) & (binintersectSubset1 = $true) & (upairsetIL = $true) & (dsetconstr__Cong = $true) & (setminusER = $true) & (eqinunit = $true) & (beta2 = $true) & (setunion__Cong = $true) & (setukpairinjR2 = $true) & (setunionE = $true) & (ubforcartprodlem2 = $true) & (powersetI1 = $true) & (ifSingleton = $true) & ((((in @ sK130 @ (setminus @ sK129 @ sK128)) != $true) & ((in @ sK130 @ sK128) != $true) & ((in @ sK130 @ sK129) = $true)) & ((in @ sK128 @ (powerset @ sK129)) = $true)) & (upairset2IR = $true) & (upairsubunion = $true) & (cartprodfstpairEq = $true) & (subsetE2 = $true) & (brelnall2 = $true) & (setminusEL = $true) & (symdiffI2 = $true) & (subsetI2 = $true) & (subsetI1 = $true) & (brelnall1 = $true) & (complementTI1 = $true) & (emptysetAx = $true) & (setunionsingleton2 = $true) & (lam2p = $true) & (ex1E2 = $true) & (bs114d = $true) & (binunionIR = $true) & (kpairiskpair = $true) & (ksndpairEq = $true) & (binunionEcases = $true) & (dpsetconstrI = $true) & (binintersectSubset4 = $true) & (notequalI2 = $true) & (vacuousDall = $true) & (powersetE = $true) & (iftrueorfalse = $true) & (setextsub = $true) & (ubforcartprodlem1 = $true) & (emptysetE = $true) & (cartprodpairmemER = $true) & (binunionRsub = $true) & (powerset__Cong = $true) & (quantDeMorgan4 = $true) & (setadjoinOr = $true) & (dpsetconstrSub = $true) & (powersetT_lem = $true) & (kfstpairEq = $true) & (complementT_lem = $true) & (binintersectRsub = $true) & (binunionT_lem = $true) & (symdiffIneg1 = $true) & (setOfPairsIsBReln = $true) & (singletonsswitch = $true) & (exuE3u = $true) & (lamProp = $true) & (dsetconstrI = $true) & (subsetemptysetimpeq = $true) & (upairset2E = $true) & (exuEu = $true) & (emptysetsubset = $true) & (descr__Cong = $true) & (ex1I = $true) & (replAx = $true) & (emptyE1 = $true) & (dsetconstrEL = $true) & (setukpairinjR1 = $true) & (setukpairinjL = $true)),
% 2.20/1.66 inference(skolemisation,[status(esa),new_symbols(skolem,[sK128,sK129,sK130])],[f862,f1228,f1227])).
% 2.20/1.66 thf(f1227,plain,(
% 2.20/1.66 ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true != (in @ X2 @ (setminus @ X1 @ X0))) & ((in @ X2 @ X0) != $true) & ((in @ X2 @ X1) = $true)) & ((in @ X0 @ (powerset @ X1)) = $true)) => (? [X2 : $i] : (((in @ X2 @ (setminus @ sK129 @ sK128)) != $true) & ((in @ X2 @ sK128) != $true) & ((in @ X2 @ sK129) = $true)) & ((in @ sK128 @ (powerset @ sK129)) = $true))),
% 2.20/1.66 introduced(definition,[],[choice_axiom])).
% 2.20/1.66 thf(f1228,plain,(
% 2.20/1.66 ? [X2 : $i] : (((in @ X2 @ (setminus @ sK129 @ sK128)) != $true) & ((in @ X2 @ sK128) != $true) & ((in @ X2 @ sK129) = $true)) => (((in @ sK130 @ (setminus @ sK129 @ sK128)) != $true) & ((in @ sK130 @ sK128) != $true) & ((in @ sK130 @ sK129) = $true))),
% 2.20/1.66 introduced(definition,[],[choice_axiom])).
% 2.20/1.66 thf(f862,plain,(
% 2.20/1.66 (exuI2 = $true) & (app = $true) & (prop2setE = $true) & (setoftrueEq = $true) & (powersetTE1 = $true) & (exuI1 = $true) & (notinemptyset = $true) & (funcext = $true) & (cartprodpairmemEL = $true) & (binintersectSubset2 = $true) & (setadjoinIR = $true) & (iftrueProp1 = $true) & (iftrueProp2 = $true) & (omega0Ax = $true) & (emptysetimpfalse = $true) & (binintersectI = $true) & (setunionsingleton1 = $true) & (cartprodmempair1 = $true) & (cartprodsndpairEq = $true) & (exuE3e = $true) & (prop2setI = $true) & (kfstsingleton = $true) & (singletoninpowunion = $true) & (binintersectLsub = $true) & (nonemptyI1 = $true) & (omegaIndAx = $true) & (funcImageSingleton = $true) & (powersetE1 = $true) & (nonemptyE1 = $true) & (cartprodmempair = $true) & (setadjoin__Cong = $true) & (setextAx = $true) & (ex1E1 = $true) & (funcGraphProp4 = $true) & (notdallE = $true) & (noeltsimpempty = $true) & (setminusIRneg = $true) & (upairsetIR = $true) & (funcextLem = $true) & (subbreln = $true) & (funcGraphProp1 = $true) & (setunionE2 = $true) & (quantDeMorgan1 = $true) & (setukpairinjL1 = $true) & (dpsetconstrEL2 = $true) & (emptyset__Cong = $true) & (subsetTrans = $true) & (ex1I2 = $true) & (powersetTI1 = $true) & (singletonsubset = $true) & (notequalI1 = $true) & (theeq = $true) & (setukpairinjR12 = $true) & (ifp = $true) & (setunionAx = $true) & (cartprodfstin = $true) & (setbeta = $true) & (apProp = $true) & (subset2powerset = $true) & (setminusI = $true) & (lam2lamEq = $true) & (setminusELneg = $true) & (powersetI = $true) & (nonemptyI = $true) & (iftrue = $true) & (singletoninpowerset = $true) & (singletonprop = $true) & (notsubsetI = $true) & (eqbreln = $true) & (funcGraphProp2 = $true) & (binintersectSubset3 = $true) & (exuI3 = $true) & (emptyI = $true) & (powersetAx = $true) & (symdiffE = $true) & (wellorderingAx = $true) & (ap2apEq1 = $true) & (setminusERneg = $true) & (descrp = $true) & (exuE1 = $true) & (setadjoinE = $true) & (notdexE = $true) & (setminusILneg = $true) & (powersetsubset = $true) & (nonemptyImpWitness = $true) & (subsetTI = $true) & (setadjoinAx = $true) & (setext = $true) & (funcinfuncset = $true) & (setminusSubset1 = $true) & (singletonsuniq = $true) & (emptyInPowerset = $true) & (lamp = $true) & (disjointsetsI1 = $true) & (exuE2 = $true) & (binintersectEL = $true) & (subsetE = $true) & (iffalseProp1 = $true) & (sepInPowerset = $true) & (binunionE = $true) & (setminusLsub = $true) & (setunionsingleton = $true) & (binunionIL = $true) & (inCongP = $true) & (dsetconstrER = $true) & (binintersectER = $true) & (subPowSU = $true) & (sepSubset = $true) & (setextT = $true) & (prop2set2propI = $true) & (theprop = $true) & (binunionLsub = $true) & (eta2 = $true) & (ap2apEq2 = $true) & (dpsetconstrEL1 = $true) & (setadjoinIL = $true) & (setukpairinjL2 = $true) & (emptyinPowerset = $true) & (cartprodpairin = $true) & (ap2p = $true) & (omega__Cong = $true) & (setminusT_lem = $true) & (iffalse = $true) & (uniqinunit = $true) & (exu__Cong = $true) & (infuncsetfunc = $true) & (cartprodpairsurjEq = $true) & (upairinpowunion = $true) & (quantDeMorgan2 = $true) & (dpsetconstrERa = $true) & (setukpairinjR11 = $true) & (ksndsingleton = $true) & (omegaSAx = $true) & (symdiffIneg2 = $true) & (inPowerset = $true) & (setadjoinSub = $true) & (eta1 = $true) & (symdiffI1 = $true) & (beta1 = $true) & (setadjoinSub2 = $true) & (eqimpsubset2 = $true) & (funcGraphProp3 = $true) & (foundationAx = $true) & (setukpairinjR = $true) & (cartprodsndin = $true) & (upairsetE = $true) & (upairequniteq = $true) & (dpsetconstrER = $true) & (ubforcartprodlem3 = $true) & (setunionI = $true) & (eqimpsubset1 = $true) & (funcext2 = $true) & (cartprodmempaircEq = $true) & (iffalseProp2 = $true) & (notinsingleton = $true) & (setukpairIL = $true) & (kpairsurjEq = $true) & (kpairp = $true) & (subsetRefl = $true) & (binintersectSubset5 = $true) & (setukpairIR = $true) & (emptyinunitempty = $true) & (in__Cong = $true) & (secondinupair = $true) & (setminusSubset2 = $true) & (quantDeMorgan3 = $true) & (binintersectT_lem = $true) & (binintersectSubset1 = $true) & (upairsetIL = $true) & (dsetconstr__Cong = $true) & (setminusER = $true) & (eqinunit = $true) & (beta2 = $true) & (setunion__Cong = $true) & (setukpairinjR2 = $true) & (setunionE = $true) & (ubforcartprodlem2 = $true) & (powersetI1 = $true) & (ifSingleton = $true) & ? [X0 : $i,X1 : $i] : (? [X2 : $i] : (($true != (in @ X2 @ (setminus @ X1 @ X0))) & ((in @ X2 @ X0) != $true) & ((in @ X2 @ X1) = $true)) & ((in @ X0 @ (powerset @ X1)) = $true)) & (upairset2IR = $true) & (upairsubunion = $true) & (cartprodfstpairEq = $true) & (subsetE2 = $true) & (brelnall2 = $true) & (setminusEL = $true) & (symdiffI2 = $true) & (subsetI2 = $true) & (subsetI1 = $true) & (brelnall1 = $true) & (complementTI1 = $true) & (emptysetAx = $true) & (setunionsingleton2 = $true) & (lam2p = $true) & (ex1E2 = $true) & (bs114d = $true) & (binunionIR = $true) & (kpairiskpair = $true) & (ksndpairEq = $true) & (binunionEcases = $true) & (dpsetconstrI = $true) & (binintersectSubset4 = $true) & (notequalI2 = $true) & (vacuousDall = $true) & (powersetE = $true) & (iftrueorfalse = $true) & (setextsub = $true) & (ubforcartprodlem1 = $true) & (emptysetE = $true) & (cartprodpairmemER = $true) & (binunionRsub = $true) & (powerset__Cong = $true) & (quantDeMorgan4 = $true) & (setadjoinOr = $true) & (dpsetconstrSub = $true) & (powersetT_lem = $true) & (kfstpairEq = $true) & (complementT_lem = $true) & (binintersectRsub = $true) & (binunionT_lem = $true) & (symdiffIneg1 = $true) & (setOfPairsIsBReln = $true) & (singletonsswitch = $true) & (exuE3u = $true) & (lamProp = $true) & (dsetconstrI = $true) & (subsetemptysetimpeq = $true) & (upairset2E = $true) & (exuEu = $true) & (emptysetsubset = $true) & (descr__Cong = $true) & (ex1I = $true) & (replAx = $true) & (emptyE1 = $true) & (dsetconstrEL = $true) & (setukpairinjR1 = $true) & (setukpairinjL = $true)),
% 2.20/1.66 inference(flattening,[status(thm)],[f861])).
% 2.20/1.66 thf(f861,plain,(
% 2.20/1.66 (((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : ((((in @ X2 @ X0) != $true) & ($true != (in @ X2 @ (setminus @ X1 @ X0)))) & ((in @ X2 @ X1) = $true)) & ((in @ X0 @ (powerset @ X1)) = $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)),
% 2.20/1.66 inference(ennf_transformation,[status(thm)],[f742])).
% 2.20/1.66 thf(f742,plain,(
% 2.20/1.66 ~((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) => ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) => ! [X2 : $i] : (((in @ X2 @ X1) = $true) => (($true != (in @ X2 @ (setminus @ X1 @ X0))) => ((in @ X2 @ X0) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.20/1.66 inference(flattening,[status(thm)],[f527])).
% 2.20/1.66 thf(f527,plain,(
% 2.20/1.66 ~((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) => ! [X0 : $i,X1 : $i] : (((in @ X0 @ (powerset @ X1)) = $true) => ! [X2 : $i] : (((in @ X2 @ X1) = $true) => (~($true = (in @ X2 @ (setminus @ X1 @ X0))) => ((in @ X2 @ X0) = $true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.20/1.66 inference(fool_elimination,[status(thm)],[f526])).
% 2.20/1.66 thf(f526,plain,(
% 2.20/1.66 ~(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 => ! [X0 : $i,X1 : $i] : ((in @ X0 @ (powerset @ X1)) => ! [X2 : $i] : ((in @ X2 @ X1) => (~(in @ X2 @ (setminus @ X1 @ X0)) => (in @ X2 @ X0))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.20/1.66 inference(rectify,[status(thm)],[f242])).
% 2.20/1.66 thf(f242,negated_conjecture,(
% 2.20/1.66 ~(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 => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => (~(in @ X1 @ (setminus @ X3 @ X11)) => (in @ X1 @ X11))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.20/1.66 inference(negated_conjecture,[status(cth)],[f241])).
% 2.20/1.66 thf(f241,conjecture,(
% 2.20/1.66 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 => ! [X11 : $i,X3 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => (~(in @ X1 @ (setminus @ X3 @ X11)) => (in @ X1 @ X11)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 2.20/1.66 file('/export/starexec/sandbox2/tmp/tmp.SCLFyHJTHZ/DTF2THF_24371.p',complementTE1)).
% 2.20/1.66 thf(f3692,plain,(
% 2.20/1.66 spl747_5),
% 2.20/1.66 inference(avatar_contradiction_clause,[status(thm)],[f3691])).
% 2.20/1.66 thf(f3691,plain,(
% 2.20/1.66 $false | spl747_5),
% 2.20/1.66 inference(trivial_inequality_removal,[status(thm)],[f3681])).
% 2.20/1.66 thf(f3681,plain,(
% 2.20/1.66 ($true != $true) | spl747_5),
% 2.20/1.66 inference(superposition,[status(thm)],[f3643,f2381])).
% 2.20/1.66 thf(f2381,plain,(
% 2.20/1.66 ((in @ sK130 @ sK129) = $true)),
% 2.20/1.66 inference(cnf_transformation,[status(thm)],[f1229])).
% 2.20/1.66 thf(f3643,plain,(
% 2.20/1.66 ((in @ sK130 @ sK129) != $true) | spl747_5),
% 2.20/1.66 inference(avatar_component_clause,[status(thm)],[f3641])).
% 2.20/1.66 thf(f3641,definition,(
% 2.20/1.66 spl747_5 <=> ((in @ sK130 @ sK129) = $true)),
% 2.20/1.66 introduced(definition,[new_symbols(naming,[spl747_5])],[avatar_definition])).
% 2.20/1.66 thf(f3667,plain,(
% 2.20/1.66 spl747_6 | ~spl747_5),
% 2.20/1.66 inference(avatar_split_clause,[status(thm)],[f3666,f3641,f3645])).
% 2.20/1.66 thf(f3666,plain,(
% 2.20/1.66 ((in @ sK130 @ sK128) = $true) | ((in @ sK130 @ sK129) != $true)),
% 2.20/1.66 inference(trivial_inequality_removal,[status(thm)],[f3665])).
% 2.20/1.66 thf(f3665,plain,(
% 2.20/1.66 ((in @ sK130 @ sK128) = $true) | ((in @ sK130 @ sK129) != $true) | ($true != $true)),
% 2.20/1.66 inference(forward_demodulation,[status(thm)],[f3633,f2503])).
% 2.20/1.66 thf(f2503,plain,(
% 2.20/1.66 (setminusI = $true)),
% 2.20/1.66 inference(cnf_transformation,[status(thm)],[f1229])).
% 2.20/1.66 thf(f3633,plain,(
% 2.20/1.66 (setminusI != $true) | ((in @ sK130 @ sK128) = $true) | ((in @ sK130 @ sK129) != $true)),
% 2.20/1.66 inference(trivial_inequality_removal,[status(thm)],[f3624])).
% 2.20/1.66 thf(f3624,plain,(
% 2.20/1.66 (setminusI != $true) | ($true != $true) | ((in @ sK130 @ sK128) = $true) | ((in @ sK130 @ sK129) != $true)),
% 2.20/1.66 inference(superposition,[status(thm)],[f2383,f3132])).
% 2.20/1.66 thf(f3132,plain,(
% 2.20/1.66 ( ! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ (setminus @ X5 @ X3)) = $true) | (setminusI != $true) | ((in @ X4 @ X3) = $true) | ((in @ X4 @ X5) != $true)) )),
% 2.20/1.66 inference(cnf_transformation,[status(thm)],[f1883])).
% 2.20/1.66 thf(f1883,plain,(
% 2.20/1.66 ((setminusI = $true) | (((in @ sK561 @ sK560) != $true) & ($true != (in @ sK561 @ (setminus @ sK562 @ sK560))) & ((in @ sK561 @ sK562) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ X3) = $true) | ((in @ X4 @ (setminus @ X5 @ X3)) = $true) | ((in @ X4 @ X5) != $true)) | (setminusI != $true))),
% 2.20/1.66 inference(skolemisation,[status(esa),new_symbols(skolem,[sK560,sK561,sK562])],[f1881,f1882])).
% 2.20/1.66 thf(f1882,plain,(
% 2.20/1.66 ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ X0)) & ((in @ X1 @ (setminus @ X2 @ X0)) != $true) & ((in @ X1 @ X2) = $true)) => (((in @ sK561 @ sK560) != $true) & ($true != (in @ sK561 @ (setminus @ sK562 @ sK560))) & ((in @ sK561 @ sK562) = $true))),
% 2.20/1.66 introduced(definition,[],[choice_axiom])).
% 2.20/1.66 thf(f1881,plain,(
% 2.20/1.66 ((setminusI = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ X0)) & ((in @ X1 @ (setminus @ X2 @ X0)) != $true) & ((in @ X1 @ X2) = $true))) & (! [X3 : $i,X4 : $i,X5 : $i] : (((in @ X4 @ X3) = $true) | ((in @ X4 @ (setminus @ X5 @ X3)) = $true) | ((in @ X4 @ X5) != $true)) | (setminusI != $true))),
% 2.20/1.66 inference(rectify,[status(thm)],[f1880])).
% 2.20/1.66 thf(f1880,plain,(
% 2.20/1.66 ((setminusI = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X1 @ X0)) & ((in @ X1 @ (setminus @ X2 @ X0)) != $true) & ((in @ X1 @ X2) = $true))) & (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ X0)) | ((in @ X1 @ (setminus @ X2 @ X0)) = $true) | ((in @ X1 @ X2) != $true)) | (setminusI != $true))),
% 2.20/1.66 inference(nnf_transformation,[status(thm)],[f956])).
% 2.20/1.66 thf(f956,plain,(
% 2.20/1.66 (setminusI = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X1 @ X0)) | ((in @ X1 @ (setminus @ X2 @ X0)) = $true) | ((in @ X1 @ X2) != $true))),
% 2.20/1.66 inference(flattening,[status(thm)],[f955])).
% 2.20/1.66 thf(f955,plain,(
% 2.20/1.66 (setminusI = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : ((((in @ X1 @ (setminus @ X2 @ X0)) = $true) | ($true = (in @ X1 @ X0))) | ((in @ X1 @ X2) != $true))),
% 2.20/1.66 inference(ennf_transformation,[status(thm)],[f744])).
% 2.20/1.66 thf(f744,plain,(
% 2.20/1.66 (setminusI = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) => (($true != (in @ X1 @ X0)) => ((in @ X1 @ (setminus @ X2 @ X0)) = $true)))),
% 2.20/1.66 inference(flattening,[status(thm)],[f535])).
% 2.20/1.66 thf(f535,plain,(
% 2.20/1.66 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ X2) = $true) => (~($true = (in @ X1 @ X0)) => ((in @ X1 @ (setminus @ X2 @ X0)) = $true))) <=> (setminusI = $true)),
% 2.20/1.66 inference(fool_elimination,[status(thm)],[f534])).
% 2.20/1.66 thf(f534,plain,(
% 2.20/1.66 (! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ X2) => (~(in @ X1 @ X0) => (in @ X1 @ (setminus @ X2 @ X0)))) = setminusI)),
% 2.20/1.66 inference(rectify,[status(thm)],[f120])).
% 2.20/1.66 thf(f120,axiom,(
% 2.20/1.66 (! [X4 : $i,X1 : $i,X3 : $i] : ((in @ X1 @ X3) => (~(in @ X1 @ X4) => (in @ X1 @ (setminus @ X3 @ X4)))) = setminusI)),
% 2.20/1.66 file('/export/starexec/sandbox2/tmp/tmp.SCLFyHJTHZ/DTF2THF_24371.p',setminusI)).
% 2.20/1.66 thf(f2383,plain,(
% 2.20/1.66 ((in @ sK130 @ (setminus @ sK129 @ sK128)) != $true)),
% 2.20/1.66 inference(cnf_transformation,[status(thm)],[f1229])).
% 2.20/1.66 % SZS output end Proof for DTF2THF_24371
% 2.20/1.66 % (24677)------------------------------
% 2.20/1.66 % (24677)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.20/1.66 % (24677)Termination reason: Refutation
% 2.20/1.66
% 2.20/1.66 % (24677)Memory used [KB]: 9850
% 2.20/1.66 % (24677)Time elapsed: 0.101 s
% 2.20/1.66 % (24677)Instructions burned: 325 (million)
% 2.20/1.66 % (24677)------------------------------
% 2.20/1.66 % (24677)------------------------------
% 2.20/1.66 % (24628)Success in time 0.201 s
%------------------------------------------------------------------------------