%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : SEU741^1 : TPTP v9.2.1. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n007.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 09:36:49 AM UTC 2026
% Result : Theorem 4.47s 2.19s
% Output : Refutation 4.47s
% Verified :
% SZS Type : Refutation
% Derivation depth : 13
% Number of leaves : 10
% Syntax : Number of formulae : 45 ( 13 unt; 0 typ; 1 def)
% Number of atoms : 4422 (1175 equ; 0 cnn)
% Maximal formula atoms : 263 ( 98 avg)
% Number of connectives : 2414 ( 56 ~; 32 |; 830 &; 431 @)
% ( 5 <=>;1060 =>; 0 <=; 0 <~>)
% Maximal formula depth : 272 ( 45 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 460 ( 457 usr; 271 con; 0-4 aty)
% Number of variables : 113 ( 0 ^; 68 !; 45 ?; 113 :)
% 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_313,type,
sP4: $i > $i > $i > $o ).
thf(func_def_314,type,
sP5: $i > $o > $i > $i > $o ).
thf(func_def_315,type,
sP6: $i > $i > $i > $i > $o ).
thf(func_def_317,type,
sP8: $i > $i > $o ).
thf(func_def_318,type,
sP9: $i > $o ).
thf(func_def_319,type,
sP10: $i > $i > $o ).
thf(func_def_320,type,
sP11: $i > $i > $o ).
thf(func_def_330,type,
sK21: $i > $o ).
thf(func_def_332,type,
sK23: ( $i > $o ) > $i > $i ).
thf(func_def_333,type,
sK24: ( $i > $o ) > $i > $i ).
thf(func_def_364,type,
sK55: $i > $o ).
thf(func_def_367,type,
sK58: $i > $i ).
thf(func_def_369,type,
sK60: $i > $i ).
thf(func_def_380,type,
sK71: $i > $i > $o ).
thf(func_def_387,type,
sK78: ( $i > $o ) > $i ).
thf(func_def_388,type,
sK79: $i > $o ).
thf(func_def_389,type,
sK80: $i > $i ).
thf(func_def_396,type,
sK87: $i > $o ).
thf(func_def_414,type,
sK105: $i > $i > $i ).
thf(func_def_424,type,
sK115: $i > $i > $i ).
thf(func_def_425,type,
sK116: $i > $i > $i ).
thf(func_def_429,type,
sK120: $i > $i ).
thf(func_def_432,type,
sK123: $i > $i > ( $i > $i ) > $i ).
thf(func_def_444,type,
sK135: $i > $o ).
thf(func_def_446,type,
sK137: $i > ( $i > $o ) > $i ).
thf(func_def_474,type,
sK165: $i > $i ).
thf(func_def_478,type,
sK169: $i > $i > ( $i > $i ) > $i ).
thf(func_def_479,type,
sK170: $i > $o ).
thf(func_def_482,type,
sK173: $i > ( $i > $o ) > $i ).
thf(func_def_493,type,
sK184: $i > $i > $o ).
thf(func_def_494,type,
sK185: $i > $i ).
thf(func_def_495,type,
sK186: $i > $i ).
thf(func_def_496,type,
sK187: ( $i > $i > $o ) > $i > $i ).
thf(func_def_497,type,
sK188: ( $i > $i > $o ) > $i > $i ).
thf(func_def_498,type,
sK189: $i > ( $i > $i > $o ) > $i > $i ).
thf(func_def_503,type,
sK194: $i > $o ).
thf(func_def_504,type,
sK195: $i > $o ).
thf(func_def_505,type,
sK196: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_506,type,
sK197: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_509,type,
sK200: $i > $i ).
thf(func_def_513,type,
sK204: $i > $i > ( $i > $i ) > $i ).
thf(func_def_526,type,
sK217: $i > ( $i > $o ) > $i ).
thf(func_def_527,type,
sK218: $i > $o ).
thf(func_def_539,type,
sK230: $i > $i > $i > $i > $i ).
thf(func_def_542,type,
sK233: $i > $o ).
thf(func_def_545,type,
sK236: $i > $i ).
thf(func_def_551,type,
sK242: $i > $o ).
thf(func_def_554,type,
sK245: $i > ( $i > $o ) > $i ).
thf(func_def_555,type,
sK246: $i > ( $i > $o ) > $i ).
thf(func_def_556,type,
sK247: $i > $i > $i > $i > $i ).
thf(func_def_578,type,
sK269: $i > $i > $o ).
thf(func_def_586,type,
sK277: $i > $i ).
thf(func_def_601,type,
sK292: $o > $i > $i > $i ).
thf(func_def_608,type,
sK299: $i > $o ).
thf(func_def_611,type,
sK302: $i > $i ).
thf(func_def_619,type,
sK310: $i > $o ).
thf(func_def_622,type,
sK313: $i > ( $i > $o ) > $i ).
thf(func_def_635,type,
sK326: $i > $i > $o ).
thf(func_def_640,type,
sK331: $i > $i > $i ).
thf(func_def_649,type,
sK340: $i > ( $i > $o ) > $i ).
thf(func_def_650,type,
sK341: $i > $o ).
thf(func_def_665,type,
sK356: $i > $i > $o ).
thf(func_def_673,type,
sK364: $i > $i > $i ).
thf(func_def_675,type,
sK366: $i > $o ).
thf(func_def_685,type,
sK376: $i > $i > $i ).
thf(func_def_686,type,
sK377: $i > ( $i > $o ) > $i ).
thf(func_def_687,type,
sK378: $i > $o ).
thf(func_def_692,type,
sK383: $i > $i > $o ).
thf(func_def_698,type,
sK389: $i > $o ).
thf(func_def_708,type,
sK399: $i > $i > ( $i > $i ) > $i ).
thf(func_def_709,type,
sK400: $i > $i ).
thf(func_def_712,type,
sK403: ( $i > $o ) > $i > $i ).
thf(func_def_714,type,
sK405: $i > $o ).
thf(func_def_722,type,
sK413: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_723,type,
sK414: ( $i > $o ) > ( $i > $o ) > $i > $i > $i ).
thf(func_def_726,type,
sK417: $i > $o ).
thf(func_def_727,type,
sK418: $i > $o ).
thf(func_def_745,type,
sK436: $i > $i > $i > $i > $i ).
thf(func_def_761,type,
sK452: $i > $o ).
thf(func_def_763,type,
sK454: $i > $o ).
thf(func_def_774,type,
sK465: $i > $i > $i > $i ).
thf(func_def_780,type,
sK471: $i > $i > $o ).
thf(func_def_792,type,
sK483: $i > $o ).
thf(func_def_794,type,
sK485: $i > ( $i > $o ) > $i ).
thf(func_def_798,type,
sK489: $i > $i > $o ).
thf(func_def_802,type,
sK493: $i > $o ).
thf(func_def_803,type,
sK494: $i > $o ).
thf(func_def_804,type,
sK495: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_805,type,
sK496: ( $i > $o ) > ( $i > $o ) > $i ).
thf(func_def_811,type,
sK502: $i > $i > $i > $i ).
thf(func_def_812,type,
sK503: $i > $i > $i > $i ).
thf(func_def_831,type,
sK522: $i > $i > $i ).
thf(func_def_846,type,
sK537: $i > $i > $i > $i ).
thf(func_def_850,type,
sK541: $i > $i > $i > $i ).
thf(func_def_856,type,
sK547: $i > $i ).
thf(func_def_857,type,
sK548: ( $i > $i ) > $i > $i > $i ).
thf(func_def_871,type,
sK562: $i > $i > $i ).
thf(func_def_878,type,
sK569: $i > $o ).
thf(func_def_880,type,
sK571: $i > ( $i > $o ) > $i ).
thf(func_def_888,type,
sK579: $i > $o ).
thf(func_def_890,type,
sK581: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_891,type,
sK582: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_904,type,
sK595: $i > $o ).
thf(func_def_905,type,
sK596: $i > $i ).
thf(func_def_906,type,
sK597: ( $i > $o ) > $i ).
thf(func_def_915,type,
sK606: $i > $i > $i ).
thf(func_def_960,type,
sK651: $i > $i ).
thf(func_def_966,type,
sK657: $i > $i > $i > $i > $i ).
thf(func_def_967,type,
sK658: $i > $i > $i > $i > $i ).
thf(func_def_968,type,
sK659: $i > $o ).
thf(func_def_973,type,
sK664: $i > $i > $i > $i ).
thf(func_def_978,type,
sK669: $i > $i > $i > $i > $i ).
thf(func_def_979,type,
sK670: $i > $i > $i > $i > $i ).
thf(func_def_980,type,
sK671: $i > $i > $i > $i > $i ).
thf(func_def_981,type,
sK672: $i > $i > $i > $i > $i ).
thf(func_def_984,type,
sK675: $i > $o ).
thf(func_def_1003,type,
sK694: $i > $i ).
thf(func_def_1022,type,
sK713: ( $i > $o ) > $i ).
thf(func_def_1023,type,
sK714: $i > $o ).
thf(func_def_1024,type,
sK715: $i > $i > $i ).
thf(func_def_1025,type,
sK716: $i > $i > $i ).
thf(func_def_1026,type,
sK717: $i > $i > $i ).
thf(func_def_1027,type,
sK718: $i > $i > $i ).
thf(func_def_1028,type,
sK719: $i > $i > $i ).
thf(func_def_1029,type,
sK720: $i > $i > $i > $i ).
thf(func_def_1030,type,
sK721: $i > $i ).
thf(func_def_1031,type,
sK722: $i > $i ).
thf(func_def_1032,type,
sK723: $i > $i ).
thf(func_def_1033,type,
sK724: $i > $i ).
thf(func_def_1034,type,
sK725: $i > $i > $i ).
thf(func_def_1035,type,
sK726: $i > $i > $i > $i ).
thf(func_def_1036,type,
sK727: $i > $i > $i > $i ).
thf(func_def_1037,type,
sK728: $i > $i > $i ).
thf(func_def_1038,type,
sK729: $i > $i > $i ).
thf(func_def_1039,type,
sK730: $i > $i ).
thf(func_def_1041,type,
sK732: $i > $i ).
thf(func_def_1042,type,
sK733: $i > $i ).
thf(func_def_1049,type,
sK740: ( $i > $o ) > $i ).
thf(func_def_1050,type,
sK741: ( $i > $o ) > $i ).
thf(func_def_1051,type,
sK742: $i > $o ).
thf(func_def_1057,type,
sK748: $i > $o ).
thf(func_def_1073,type,
sK764: $i > $o ).
thf(func_def_1075,type,
sK766: $i > ( $i > $o ) > $i > $i ).
thf(func_def_1086,type,
sK777: $i > $i > $i ).
thf(func_def_1107,type,
sK798: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1108,type,
sK799: ( $i > $o ) > $i > $i > $i > $i ).
thf(func_def_1112,type,
sK803: $i > $o ).
thf(func_def_1116,type,
sK807: $i > $o ).
thf(func_def_1118,type,
sK809: $i > $o ).
thf(func_def_1125,type,
sK816: $i > $i ).
thf(func_def_1128,type,
sK819: $i > $i > ( $i > $i ) > $i ).
thf(func_def_1130,type,
ph821:
!>[X0: $tType] : X0 ).
thf(f4178,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f4132,f4177]) ).
thf(f4177,plain,
spl820_13,
inference(avatar_split_clause,[status(thm)],[f4057,f4087]) ).
thf(f4087,definition,
( spl820_13
<=> ( $true
= ( in @ sK521 @ sK519 ) ) ),
introduced(definition,[new_symbols(naming,[spl820_13])],[avatar_definition]) ).
thf(f4057,plain,
( $true
= ( in @ sK521 @ sK519 ) ),
inference(trivial_inequality_removal,[status(thm)],[f4056]) ).
thf(f4056,plain,
( ( $true
= ( in @ sK521 @ sK519 ) )
| ( $true != $true ) ),
inference(forward_demodulation,[status(thm)],[f4043,f3066]) ).
thf(f3066,plain,
binintersectER = $true,
inference(cnf_transformation,[status(thm)],[f1917]) ).
thf(f1917,plain,
( ( setminusI = $true )
& ( subset2powerset = $true )
& ( upairsetIL = $true )
& ( binintersectT_lem = $true )
& ( omega__Cong = $true )
& ( emptyI = $true )
& ( setadjoinIL = $true )
& ( binintersectRsub = $true )
& ( emptyinPowerset = $true )
& ( lamp = $true )
& ( dpsetconstrI = $true )
& ( emptysetE = $true )
& ( upairsetE = $true )
& ( cartprodmempair1 = $true )
& ( upairset2E = $true )
& ( powersetsubset = $true )
& ( contrasubsetT1 = $true )
& ( cartprodfstpairEq = $true )
& ( ap2apEq1 = $true )
& ( ksndsingleton = $true )
& ( beta1 = $true )
& ( setunionE2 = $true )
& ( dsetconstrI = $true )
& ( binunionEcases = $true )
& ( lam2p = $true )
& ( complementTI1 = $true )
& ( powersetTI1 = $true )
& ( symdiffIneg1 = $true )
& ( dpsetconstrEL1 = $true )
& ( setukpairinjR1 = $true )
& ( dpsetconstrEL2 = $true )
& ( symdiffI2 = $true )
& ( omegaSAx = $true )
& ( setadjoinAx = $true )
& ( notdexE = $true )
& ( cartprodpairsurjEq = $true )
& ( binintersectSubset3 = $true )
& ( brelnall1 = $true )
& ( binintersectSubset2 = $true )
& ( replAx = $true )
& ( subsetI2 = $true )
& ( setoftrueEq = $true )
& ( upairsubunion = $true )
& ( powersetTE1 = $true )
& ( descr__Cong = $true )
& ( app = $true )
& ( powerset__Cong = $true )
& ( symdiffI1 = $true )
& ( exuE1 = $true )
& ( ubforcartprodlem1 = $true )
& ( singletoninpowerset = $true )
& ( cartprodmempaircEq = $true )
& ( funcext = $true )
& ( upairequniteq = $true )
& ( subsetemptysetimpeq = $true )
& ( powersetI1 = $true )
& ( complementTnotintersectT = $true )
& ( exuE3e = $true )
& ( kpairiskpair = $true )
& ( $true
= ( in @ sK518 @ ( powerset @ sK517 ) ) )
& ( $true
!= ( in @ sK521 @ ( binunion @ sK519 @ sK520 ) ) )
& ( $true
= ( in @ sK521 @ sK517 ) )
& ( $true
= ( in @ sK521 @ ( binintersect @ sK518 @ sK519 ) ) )
& ( $true
= ( in @ sK520 @ ( powerset @ sK517 ) ) )
& ( $true
= ( in @ sK519 @ ( powerset @ sK517 ) ) )
& ( dpsetconstrSub = $true )
& ( doubleComplementSub2 = $true )
& ( secondinupair = $true )
& ( binunionE = $true )
& ( setext = $true )
& ( iftrueProp2 = $true )
& ( notinemptyset = $true )
& ( notsubsetI = $true )
& ( emptysetsubset = $true )
& ( emptyE1 = $true )
& ( setminusSubset1 = $true )
& ( dpsetconstrERa = $true )
& ( ex1E2 = $true )
& ( setunionE = $true )
& ( inPowerset = $true )
& ( omega0Ax = $true )
& ( apProp = $true )
& ( setukpairIR = $true )
& ( disjointsetsI1 = $true )
& ( subsetE2 = $true )
& ( setextAx = $true )
& ( theprop = $true )
& ( emptyInPowerset = $true )
& ( singletonprop = $true )
& ( contrasubsetT3 = $true )
& ( funcextLem = $true )
& ( lamProp = $true )
& ( emptysetimpfalse = $true )
& ( cartprodpairin = $true )
& ( binunionRsub = $true )
& ( eta1 = $true )
& ( ap2p = $true )
& ( setadjoinOr = $true )
& ( subPowSU = $true )
& ( subsetE = $true )
& ( iftrueorfalse = $true )
& ( setminusERneg = $true )
& ( singletonsswitch = $true )
& ( contraSubsetComplement = $true )
& ( subsetTI = $true )
& ( subsetTrans = $true )
& ( ex1I2 = $true )
& ( dsetconstrEL = $true )
& ( funcGraphProp1 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( iftrue = $true )
& ( emptyinunitempty = $true )
& ( dsetconstr__Cong = $true )
& ( setadjoinSub2 = $true )
& ( powersetI = $true )
& ( omegaIndAx = $true )
& ( singletonsuniq = $true )
& ( setunionI = $true )
& ( exuI1 = $true )
& ( quantDeMorgan2 = $true )
& ( iffalseProp1 = $true )
& ( binintersectLsub = $true )
& ( setukpairinjR2 = $true )
& ( notinsingleton = $true )
& ( setunionAx = $true )
& ( nonemptyI = $true )
& ( complementT_lem = $true )
& ( setukpairinjR11 = $true )
& ( funcGraphProp2 = $true )
& ( setminusILneg = $true )
& ( kpairsurjEq = $true )
& ( cartprodpairmemEL = $true )
& ( beta2 = $true )
& ( exuEu = $true )
& ( infuncsetfunc = $true )
& ( nonemptyI1 = $true )
& ( singletonsubset = $true )
& ( bs114d = $true )
& ( ubforcartprodlem3 = $true )
& ( binintersectTELcontra = $true )
& ( kfstsingleton = $true )
& ( setadjoin__Cong = $true )
& ( powersetE = $true )
& ( setextsub = $true )
& ( subsetI1 = $true )
& ( contrasubsetT = $true )
& ( wellorderingAx = $true )
& ( in__Cong = $true )
& ( eqimpsubset2 = $true )
& ( setadjoinE = $true )
& ( ifp = $true )
& ( complementImpComplementIntersect = $true )
& ( setminusSubset2 = $true )
& ( uniqinunit = $true )
& ( quantDeMorgan4 = $true )
& ( eqinunit = $true )
& ( eqbreln = $true )
& ( ifSingleton = $true )
& ( dpsetconstrER = $true )
& ( binintersectTERcontra = $true )
& ( binunionLsub = $true )
& ( doubleComplementE1 = $true )
& ( funcinfuncset = $true )
& ( powersetAx = $true )
& ( theeq = $true )
& ( complementTE1 = $true )
& ( setukpairinjR = $true )
& ( cartprodpairmemER = $true )
& ( contrasubsetT2 = $true )
& ( binunionTIRcontra = $true )
& ( notequalI1 = $true )
& ( ap2apEq2 = $true )
& ( prop2setE = $true )
& ( foundationAx = $true )
& ( setminusT_lem = $true )
& ( binintersectEL = $true )
& ( exuI3 = $true )
& ( ubforcartprodlem2 = $true )
& ( vacuousDall = $true )
& ( upairset2IR = $true )
& ( setminusELneg = $true )
& ( cartprodmempair = $true )
& ( emptyset__Cong = $true )
& ( funcGraphProp3 = $true )
& ( upairinpowunion = $true )
& ( nonemptyE1 = $true )
& ( setukpairinjL2 = $true )
& ( setbeta = $true )
& ( funcext2 = $true )
& ( doubleComplementEq = $true )
& ( upairsetIR = $true )
& ( prop2setI = $true )
& ( sepSubset = $true )
& ( quantDeMorgan1 = $true )
& ( subbreln = $true )
& ( binintersectI = $true )
& ( funcGraphProp4 = $true )
& ( exu__Cong = $true )
& ( ex1I = $true )
& ( brelnall2 = $true )
& ( setextT = $true )
& ( binunionT_lem = $true )
& ( eqimpsubset1 = $true )
& ( complementTcontraSubset = $true )
& ( noeltsimpempty = $true )
& ( cartprodsndpairEq = $true )
& ( doubleComplementI1 = $true )
& ( symdiffE = $true )
& ( setukpairIL = $true )
& ( setminusEL = $true )
& ( subsetRefl = $true )
& ( setminusLsub = $true )
& ( ex1E1 = $true )
& ( descrp = $true )
& ( binunionIL = $true )
& ( quantDeMorgan3 = $true )
& ( setadjoinSub = $true )
& ( singletoninpowunion = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjL1 = $true )
& ( funcImageSingleton = $true )
& ( eta2 = $true )
& ( binunionTILcontra = $true )
& ( binunionIR = $true )
& ( binintersectSubset5 = $true )
& ( notdallE = $true )
& ( setunionsingleton2 = $true )
& ( cartprodsndin = $true )
& ( exuE3u = $true )
& ( powersetE1 = $true )
& ( kpairp = $true )
& ( doubleComplementSub1 = $true )
& ( notequalI2 = $true )
& ( setunionsingleton = $true )
& ( dsetconstrER = $true )
& ( kfstpairEq = $true )
& ( setminusIRneg = $true )
& ( setOfPairsIsBReln = $true )
& ( iffalseProp2 = $true )
& ( setunion__Cong = $true )
& ( iftrueProp1 = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( binintersectSubset4 = $true )
& ( exuE2 = $true )
& ( powersetT_lem = $true )
& ( binintersectSubset1 = $true )
& ( prop2set2propI = $true )
& ( setminusER = $true )
& ( setunionsingleton1 = $true )
& ( inIntersectImpInUnion = $true )
& ( cartprodfstin = $true )
& ( nonemptyImpWitness = $true )
& ( inCongP = $true )
& ( binintersectER = $true )
& ( symdiffIneg2 = $true )
& ( sepInPowerset = $true )
& ( ksndpairEq = $true )
& ( lam2lamEq = $true )
& ( setadjoinIR = $true )
& ( exuI2 = $true )
& ( emptysetAx = $true )
& ( iffalse = $true )
& ( setukpairinjL = $true ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK517,sK518,sK519,sK520,sK521])],[f969,f1916,f1915,f1914,f1913]) ).
thf(f1913,plain,
( ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
& ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( $true
!= ( in @ X4 @ ( binunion @ X2 @ X3 ) ) )
& ( $true
= ( in @ X4 @ X0 ) )
& ( $true
= ( in @ X4 @ ( binintersect @ X1 @ X2 ) ) ) )
& ( $true
= ( in @ X3 @ ( powerset @ X0 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) ) )
=> ( ( $true
= ( in @ sK518 @ ( powerset @ sK517 ) ) )
& ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( $true
!= ( in @ X4 @ ( binunion @ X2 @ X3 ) ) )
& ( $true
= ( in @ X4 @ sK517 ) )
& ( $true
= ( in @ X4 @ ( binintersect @ sK518 @ X2 ) ) ) )
& ( $true
= ( in @ X3 @ ( powerset @ sK517 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ sK517 ) ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1914,plain,
( ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( $true
!= ( in @ X4 @ ( binunion @ X2 @ X3 ) ) )
& ( $true
= ( in @ X4 @ sK517 ) )
& ( $true
= ( in @ X4 @ ( binintersect @ sK518 @ X2 ) ) ) )
& ( $true
= ( in @ X3 @ ( powerset @ sK517 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ sK517 ) ) ) )
=> ( ? [X3: $i] :
( ? [X4: $i] :
( ( $true
!= ( in @ X4 @ ( binunion @ sK519 @ X3 ) ) )
& ( $true
= ( in @ X4 @ sK517 ) )
& ( $true
= ( in @ X4 @ ( binintersect @ sK518 @ sK519 ) ) ) )
& ( $true
= ( in @ X3 @ ( powerset @ sK517 ) ) ) )
& ( $true
= ( in @ sK519 @ ( powerset @ sK517 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1915,plain,
( ? [X3: $i] :
( ? [X4: $i] :
( ( $true
!= ( in @ X4 @ ( binunion @ sK519 @ X3 ) ) )
& ( $true
= ( in @ X4 @ sK517 ) )
& ( $true
= ( in @ X4 @ ( binintersect @ sK518 @ sK519 ) ) ) )
& ( $true
= ( in @ X3 @ ( powerset @ sK517 ) ) ) )
=> ( ? [X4: $i] :
( ( $true
!= ( in @ X4 @ ( binunion @ sK519 @ sK520 ) ) )
& ( $true
= ( in @ X4 @ sK517 ) )
& ( $true
= ( in @ X4 @ ( binintersect @ sK518 @ sK519 ) ) ) )
& ( $true
= ( in @ sK520 @ ( powerset @ sK517 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1916,plain,
( ? [X4: $i] :
( ( $true
!= ( in @ X4 @ ( binunion @ sK519 @ sK520 ) ) )
& ( $true
= ( in @ X4 @ sK517 ) )
& ( $true
= ( in @ X4 @ ( binintersect @ sK518 @ sK519 ) ) ) )
=> ( ( $true
!= ( in @ sK521 @ ( binunion @ sK519 @ sK520 ) ) )
& ( $true
= ( in @ sK521 @ sK517 ) )
& ( $true
= ( in @ sK521 @ ( binintersect @ sK518 @ sK519 ) ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f969,plain,
( ( setminusI = $true )
& ( subset2powerset = $true )
& ( upairsetIL = $true )
& ( binintersectT_lem = $true )
& ( omega__Cong = $true )
& ( emptyI = $true )
& ( setadjoinIL = $true )
& ( binintersectRsub = $true )
& ( emptyinPowerset = $true )
& ( lamp = $true )
& ( dpsetconstrI = $true )
& ( emptysetE = $true )
& ( upairsetE = $true )
& ( cartprodmempair1 = $true )
& ( upairset2E = $true )
& ( powersetsubset = $true )
& ( contrasubsetT1 = $true )
& ( cartprodfstpairEq = $true )
& ( ap2apEq1 = $true )
& ( ksndsingleton = $true )
& ( beta1 = $true )
& ( setunionE2 = $true )
& ( dsetconstrI = $true )
& ( binunionEcases = $true )
& ( lam2p = $true )
& ( complementTI1 = $true )
& ( powersetTI1 = $true )
& ( symdiffIneg1 = $true )
& ( dpsetconstrEL1 = $true )
& ( setukpairinjR1 = $true )
& ( dpsetconstrEL2 = $true )
& ( symdiffI2 = $true )
& ( omegaSAx = $true )
& ( setadjoinAx = $true )
& ( notdexE = $true )
& ( cartprodpairsurjEq = $true )
& ( binintersectSubset3 = $true )
& ( brelnall1 = $true )
& ( binintersectSubset2 = $true )
& ( replAx = $true )
& ( subsetI2 = $true )
& ( setoftrueEq = $true )
& ( upairsubunion = $true )
& ( powersetTE1 = $true )
& ( descr__Cong = $true )
& ( app = $true )
& ( powerset__Cong = $true )
& ( symdiffI1 = $true )
& ( exuE1 = $true )
& ( ubforcartprodlem1 = $true )
& ( singletoninpowerset = $true )
& ( cartprodmempaircEq = $true )
& ( funcext = $true )
& ( upairequniteq = $true )
& ( subsetemptysetimpeq = $true )
& ( powersetI1 = $true )
& ( complementTnotintersectT = $true )
& ( exuE3e = $true )
& ( kpairiskpair = $true )
& ? [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
& ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( $true
!= ( in @ X4 @ ( binunion @ X2 @ X3 ) ) )
& ( $true
= ( in @ X4 @ X0 ) )
& ( $true
= ( in @ X4 @ ( binintersect @ X1 @ X2 ) ) ) )
& ( $true
= ( in @ X3 @ ( powerset @ X0 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) ) )
& ( dpsetconstrSub = $true )
& ( doubleComplementSub2 = $true )
& ( secondinupair = $true )
& ( binunionE = $true )
& ( setext = $true )
& ( iftrueProp2 = $true )
& ( notinemptyset = $true )
& ( notsubsetI = $true )
& ( emptysetsubset = $true )
& ( emptyE1 = $true )
& ( setminusSubset1 = $true )
& ( dpsetconstrERa = $true )
& ( ex1E2 = $true )
& ( setunionE = $true )
& ( inPowerset = $true )
& ( omega0Ax = $true )
& ( apProp = $true )
& ( setukpairIR = $true )
& ( disjointsetsI1 = $true )
& ( subsetE2 = $true )
& ( setextAx = $true )
& ( theprop = $true )
& ( emptyInPowerset = $true )
& ( singletonprop = $true )
& ( contrasubsetT3 = $true )
& ( funcextLem = $true )
& ( lamProp = $true )
& ( emptysetimpfalse = $true )
& ( cartprodpairin = $true )
& ( binunionRsub = $true )
& ( eta1 = $true )
& ( ap2p = $true )
& ( setadjoinOr = $true )
& ( subPowSU = $true )
& ( subsetE = $true )
& ( iftrueorfalse = $true )
& ( setminusERneg = $true )
& ( singletonsswitch = $true )
& ( contraSubsetComplement = $true )
& ( subsetTI = $true )
& ( subsetTrans = $true )
& ( ex1I2 = $true )
& ( dsetconstrEL = $true )
& ( funcGraphProp1 = $true )
& ( complementSubsetComplementIntersect = $true )
& ( iftrue = $true )
& ( emptyinunitempty = $true )
& ( dsetconstr__Cong = $true )
& ( setadjoinSub2 = $true )
& ( powersetI = $true )
& ( omegaIndAx = $true )
& ( singletonsuniq = $true )
& ( setunionI = $true )
& ( exuI1 = $true )
& ( quantDeMorgan2 = $true )
& ( iffalseProp1 = $true )
& ( binintersectLsub = $true )
& ( setukpairinjR2 = $true )
& ( notinsingleton = $true )
& ( setunionAx = $true )
& ( nonemptyI = $true )
& ( complementT_lem = $true )
& ( setukpairinjR11 = $true )
& ( funcGraphProp2 = $true )
& ( setminusILneg = $true )
& ( kpairsurjEq = $true )
& ( cartprodpairmemEL = $true )
& ( beta2 = $true )
& ( exuEu = $true )
& ( infuncsetfunc = $true )
& ( nonemptyI1 = $true )
& ( singletonsubset = $true )
& ( bs114d = $true )
& ( ubforcartprodlem3 = $true )
& ( binintersectTELcontra = $true )
& ( kfstsingleton = $true )
& ( setadjoin__Cong = $true )
& ( powersetE = $true )
& ( setextsub = $true )
& ( subsetI1 = $true )
& ( contrasubsetT = $true )
& ( wellorderingAx = $true )
& ( in__Cong = $true )
& ( eqimpsubset2 = $true )
& ( setadjoinE = $true )
& ( ifp = $true )
& ( complementImpComplementIntersect = $true )
& ( setminusSubset2 = $true )
& ( uniqinunit = $true )
& ( quantDeMorgan4 = $true )
& ( eqinunit = $true )
& ( eqbreln = $true )
& ( ifSingleton = $true )
& ( dpsetconstrER = $true )
& ( binintersectTERcontra = $true )
& ( binunionLsub = $true )
& ( doubleComplementE1 = $true )
& ( funcinfuncset = $true )
& ( powersetAx = $true )
& ( theeq = $true )
& ( complementTE1 = $true )
& ( setukpairinjR = $true )
& ( cartprodpairmemER = $true )
& ( contrasubsetT2 = $true )
& ( binunionTIRcontra = $true )
& ( notequalI1 = $true )
& ( ap2apEq2 = $true )
& ( prop2setE = $true )
& ( foundationAx = $true )
& ( setminusT_lem = $true )
& ( binintersectEL = $true )
& ( exuI3 = $true )
& ( ubforcartprodlem2 = $true )
& ( vacuousDall = $true )
& ( upairset2IR = $true )
& ( setminusELneg = $true )
& ( cartprodmempair = $true )
& ( emptyset__Cong = $true )
& ( funcGraphProp3 = $true )
& ( upairinpowunion = $true )
& ( nonemptyE1 = $true )
& ( setukpairinjL2 = $true )
& ( setbeta = $true )
& ( funcext2 = $true )
& ( doubleComplementEq = $true )
& ( upairsetIR = $true )
& ( prop2setI = $true )
& ( sepSubset = $true )
& ( quantDeMorgan1 = $true )
& ( subbreln = $true )
& ( binintersectI = $true )
& ( funcGraphProp4 = $true )
& ( exu__Cong = $true )
& ( ex1I = $true )
& ( brelnall2 = $true )
& ( setextT = $true )
& ( binunionT_lem = $true )
& ( eqimpsubset1 = $true )
& ( complementTcontraSubset = $true )
& ( noeltsimpempty = $true )
& ( cartprodsndpairEq = $true )
& ( doubleComplementI1 = $true )
& ( symdiffE = $true )
& ( setukpairIL = $true )
& ( setminusEL = $true )
& ( subsetRefl = $true )
& ( setminusLsub = $true )
& ( ex1E1 = $true )
& ( descrp = $true )
& ( binunionIL = $true )
& ( quantDeMorgan3 = $true )
& ( setadjoinSub = $true )
& ( singletoninpowunion = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjL1 = $true )
& ( funcImageSingleton = $true )
& ( eta2 = $true )
& ( binunionTILcontra = $true )
& ( binunionIR = $true )
& ( binintersectSubset5 = $true )
& ( notdallE = $true )
& ( setunionsingleton2 = $true )
& ( cartprodsndin = $true )
& ( exuE3u = $true )
& ( powersetE1 = $true )
& ( kpairp = $true )
& ( doubleComplementSub1 = $true )
& ( notequalI2 = $true )
& ( setunionsingleton = $true )
& ( dsetconstrER = $true )
& ( kfstpairEq = $true )
& ( setminusIRneg = $true )
& ( setOfPairsIsBReln = $true )
& ( iffalseProp2 = $true )
& ( setunion__Cong = $true )
& ( iftrueProp1 = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( binintersectSubset4 = $true )
& ( exuE2 = $true )
& ( powersetT_lem = $true )
& ( binintersectSubset1 = $true )
& ( prop2set2propI = $true )
& ( setminusER = $true )
& ( setunionsingleton1 = $true )
& ( inIntersectImpInUnion = $true )
& ( cartprodfstin = $true )
& ( nonemptyImpWitness = $true )
& ( inCongP = $true )
& ( binintersectER = $true )
& ( symdiffIneg2 = $true )
& ( sepInPowerset = $true )
& ( ksndpairEq = $true )
& ( lam2lamEq = $true )
& ( setadjoinIR = $true )
& ( exuI2 = $true )
& ( emptysetAx = $true )
& ( iffalse = $true )
& ( setukpairinjL = $true ) ),
inference(flattening,[status(thm)],[f968]) ).
thf(f968,plain,
( ? [X0: $i,X1: $i] :
( ? [X2: $i] :
( ? [X3: $i] :
( ? [X4: $i] :
( ( $true
!= ( in @ X4 @ ( binunion @ X2 @ X3 ) ) )
& ( $true
= ( in @ X4 @ ( binintersect @ X1 @ X2 ) ) )
& ( $true
= ( in @ X4 @ X0 ) ) )
& ( $true
= ( in @ X3 @ ( powerset @ X0 ) ) ) )
& ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) ) )
& ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) ) )
& ( inIntersectImpInUnion = $true )
& ( binunionTIRcontra = $true )
& ( binunionTILcontra = $true )
& ( complementTcontraSubset = $true )
& ( contraSubsetComplement = $true )
& ( complementInPowersetComplementIntersect = $true )
& ( complementSubsetComplementIntersect = $true )
& ( complementImpComplementIntersect = $true )
& ( complementTnotintersectT = $true )
& ( doubleComplementEq = $true )
& ( doubleComplementSub2 = $true )
& ( doubleComplementSub1 = $true )
& ( doubleComplementE1 = $true )
& ( doubleComplementI1 = $true )
& ( contrasubsetT3 = $true )
& ( contrasubsetT2 = $true )
& ( contrasubsetT1 = $true )
& ( contrasubsetT = $true )
& ( binintersectTERcontra = $true )
& ( binintersectTELcontra = $true )
& ( complementTE1 = $true )
& ( complementTI1 = $true )
& ( powersetTE1 = $true )
& ( powersetTI1 = $true )
& ( subsetTI = $true )
& ( setextT = $true )
& ( complementT_lem = $true )
& ( setminusT_lem = $true )
& ( powersetT_lem = $true )
& ( binunionT_lem = $true )
& ( binintersectT_lem = $true )
& ( iftrueorfalse = $true )
& ( iffalse = $true )
& ( iftrue = $true )
& ( theeq = $true )
& ( ifp = $true )
& ( ifSingleton = $true )
& ( iftrueProp2 = $true )
& ( iftrueProp1 = $true )
& ( iffalseProp2 = $true )
& ( iffalseProp1 = $true )
& ( eta2 = $true )
& ( beta2 = $true )
& ( lam2lamEq = $true )
& ( eta1 = $true )
& ( beta1 = $true )
& ( ap2apEq2 = $true )
& ( ap2apEq1 = $true )
& ( funcext2 = $true )
& ( funcext = $true )
& ( eqbreln = $true )
& ( subbreln = $true )
& ( funcGraphProp4 = $true )
& ( funcextLem = $true )
& ( funcGraphProp2 = $true )
& ( funcGraphProp3 = $true )
& ( funcGraphProp1 = $true )
& ( ex1E2 = $true )
& ( brelnall2 = $true )
& ( brelnall1 = $true )
& ( lam2p = $true )
& ( lamp = $true )
& ( lamProp = $true )
& ( funcinfuncset = $true )
& ( ap2p = $true )
& ( infuncsetfunc = $true )
& ( app = $true )
& ( apProp = $true )
& ( funcImageSingleton = $true )
& ( dpsetconstrER = $true )
& ( dpsetconstrEL2 = $true )
& ( dpsetconstrEL1 = $true )
& ( dpsetconstrERa = $true )
& ( setOfPairsIsBReln = $true )
& ( dpsetconstrSub = $true )
& ( dpsetconstrI = $true )
& ( cartprodpairsurjEq = $true )
& ( cartprodsndpairEq = $true )
& ( cartprodfstpairEq = $true )
& ( cartprodmempaircEq = $true )
& ( cartprodpairmemER = $true )
& ( cartprodpairmemEL = $true )
& ( cartprodsndin = $true )
& ( kpairsurjEq = $true )
& ( ksndpairEq = $true )
& ( ksndsingleton = $true )
& ( setukpairinjR = $true )
& ( setukpairinjR2 = $true )
& ( upairequniteq = $true )
& ( setukpairinjR1 = $true )
& ( setukpairinjR12 = $true )
& ( setukpairinjR11 = $true )
& ( setukpairinjL = $true )
& ( setukpairinjL2 = $true )
& ( cartprodfstin = $true )
& ( kfstpairEq = $true )
& ( theprop = $true )
& ( kfstsingleton = $true )
& ( setukpairinjL1 = $true )
& ( singletonsuniq = $true )
& ( ex1I2 = $true )
& ( ex1I = $true )
& ( ex1E1 = $true )
& ( singletonprop = $true )
& ( setunionsingleton = $true )
& ( setunionsingleton2 = $true )
& ( setunionsingleton1 = $true )
& ( setunionE2 = $true )
& ( cartprodmempair = $true )
& ( cartprodmempair1 = $true )
& ( cartprodpairin = $true )
& ( ubforcartprodlem3 = $true )
& ( ubforcartprodlem2 = $true )
& ( ubforcartprodlem1 = $true )
& ( upairinpowunion = $true )
& ( upairsubunion = $true )
& ( upairset2E = $true )
& ( singletoninpowunion = $true )
& ( singletoninpowerset = $true )
& ( singletonsubset = $true )
& ( kpairp = $true )
& ( kpairiskpair = $true )
& ( setukpairIR = $true )
& ( setukpairIL = $true )
& ( secondinupair = $true )
& ( symdiffIneg2 = $true )
& ( symdiffIneg1 = $true )
& ( symdiffI2 = $true )
& ( symdiffI1 = $true )
& ( symdiffE = $true )
& ( setminusSubset1 = $true )
& ( setminusLsub = $true )
& ( setminusIRneg = $true )
& ( setminusILneg = $true )
& ( setminusELneg = $true )
& ( setminusERneg = $true )
& ( setminusSubset2 = $true )
& ( setminusER = $true )
& ( setminusEL = $true )
& ( setminusI = $true )
& ( bs114d = $true )
& ( binintersectSubset1 = $true )
& ( binintersectSubset4 = $true )
& ( binintersectRsub = $true )
& ( disjointsetsI1 = $true )
& ( binintersectER = $true )
& ( binintersectSubset3 = $true )
& ( binintersectSubset2 = $true )
& ( binintersectLsub = $true )
& ( binintersectEL = $true )
& ( binintersectSubset5 = $true )
& ( binintersectI = $true )
& ( binunionRsub = $true )
& ( binunionLsub = $true )
& ( binunionE = $true )
& ( binunionEcases = $true )
& ( binunionIR = $true )
& ( upairset2IR = $true )
& ( binunionIL = $true )
& ( sepSubset = $true )
& ( sepInPowerset = $true )
& ( powersetsubset = $true )
& ( inPowerset = $true )
& ( powersetE1 = $true )
& ( powersetI1 = $true )
& ( subsetemptysetimpeq = $true )
& ( setextsub = $true )
& ( subset2powerset = $true )
& ( setadjoinSub2 = $true )
& ( setadjoinSub = $true )
& ( subsetTrans = $true )
& ( subsetRefl = $true )
& ( notequalI2 = $true )
& ( notequalI1 = $true )
& ( notsubsetI = $true )
& ( subsetE2 = $true )
& ( subsetE = $true )
& ( emptysetsubset = $true )
& ( subsetI2 = $true )
& ( eqimpsubset1 = $true )
& ( eqimpsubset2 = $true )
& ( subsetI1 = $true )
& ( dsetconstr__Cong = $true )
& ( descr__Cong = $true )
& ( exuEu = $true )
& ( omega__Cong = $true )
& ( setunion__Cong = $true )
& ( powerset__Cong = $true )
& ( setadjoin__Cong = $true )
& ( emptyset__Cong = $true )
& ( exu__Cong = $true )
& ( exuE3u = $true )
& ( in__Cong = $true )
& ( inCongP = $true )
& ( exuI2 = $true )
& ( exuI3 = $true )
& ( exuI1 = $true )
& ( notdallE = $true )
& ( notdexE = $true )
& ( prop2set2propI = $true )
& ( prop2setI = $true )
& ( quantDeMorgan4 = $true )
& ( quantDeMorgan3 = $true )
& ( quantDeMorgan2 = $true )
& ( quantDeMorgan1 = $true )
& ( vacuousDall = $true )
& ( emptyE1 = $true )
& ( upairsetIR = $true )
& ( upairsetIL = $true )
& ( upairsetE = $true )
& ( singletonsswitch = $true )
& ( eqinunit = $true )
& ( notinsingleton = $true )
& ( uniqinunit = $true )
& ( nonemptyImpWitness = $true )
& ( exuE2 = $true )
& ( subPowSU = $true )
& ( setunionE = $true )
& ( setunionI = $true )
& ( powersetE = $true )
& ( emptyInPowerset = $true )
& ( emptyinPowerset = $true )
& ( powersetI = $true )
& ( setoftrueEq = $true )
& ( setadjoinOr = $true )
& ( setadjoinE = $true )
& ( setadjoinIR = $true )
& ( emptyinunitempty = $true )
& ( setadjoinIL = $true )
& ( nonemptyI1 = $true )
& ( nonemptyI = $true )
& ( nonemptyE1 = $true )
& ( setbeta = $true )
& ( noeltsimpempty = $true )
& ( emptyI = $true )
& ( setext = $true )
& ( exuE3e = $true )
& ( notinemptyset = $true )
& ( emptysetimpfalse = $true )
& ( emptysetE = $true )
& ( prop2setE = $true )
& ( exuE1 = $true )
& ( dsetconstrER = $true )
& ( dsetconstrEL = $true )
& ( dsetconstrI = $true )
& ( descrp = $true )
& ( wellorderingAx = $true )
& ( foundationAx = $true )
& ( replAx = $true )
& ( omegaIndAx = $true )
& ( omegaSAx = $true )
& ( omega0Ax = $true )
& ( setunionAx = $true )
& ( powersetAx = $true )
& ( setadjoinAx = $true )
& ( emptysetAx = $true )
& ( setextAx = $true ) ),
inference(ennf_transformation,[status(thm)],[f494]) ).
thf(f494,plain,
~ ( ( setextAx = $true )
=> ( ( emptysetAx = $true )
=> ( ( setadjoinAx = $true )
=> ( ( powersetAx = $true )
=> ( ( setunionAx = $true )
=> ( ( omega0Ax = $true )
=> ( ( omegaSAx = $true )
=> ( ( omegaIndAx = $true )
=> ( ( replAx = $true )
=> ( ( foundationAx = $true )
=> ( ( wellorderingAx = $true )
=> ( ( descrp = $true )
=> ( ( dsetconstrI = $true )
=> ( ( dsetconstrEL = $true )
=> ( ( dsetconstrER = $true )
=> ( ( exuE1 = $true )
=> ( ( prop2setE = $true )
=> ( ( emptysetE = $true )
=> ( ( emptysetimpfalse = $true )
=> ( ( notinemptyset = $true )
=> ( ( exuE3e = $true )
=> ( ( setext = $true )
=> ( ( emptyI = $true )
=> ( ( noeltsimpempty = $true )
=> ( ( setbeta = $true )
=> ( ( nonemptyE1 = $true )
=> ( ( nonemptyI = $true )
=> ( ( nonemptyI1 = $true )
=> ( ( setadjoinIL = $true )
=> ( ( emptyinunitempty = $true )
=> ( ( setadjoinIR = $true )
=> ( ( setadjoinE = $true )
=> ( ( setadjoinOr = $true )
=> ( ( setoftrueEq = $true )
=> ( ( powersetI = $true )
=> ( ( emptyinPowerset = $true )
=> ( ( emptyInPowerset = $true )
=> ( ( powersetE = $true )
=> ( ( setunionI = $true )
=> ( ( setunionE = $true )
=> ( ( subPowSU = $true )
=> ( ( exuE2 = $true )
=> ( ( nonemptyImpWitness = $true )
=> ( ( uniqinunit = $true )
=> ( ( notinsingleton = $true )
=> ( ( eqinunit = $true )
=> ( ( singletonsswitch = $true )
=> ( ( upairsetE = $true )
=> ( ( upairsetIL = $true )
=> ( ( upairsetIR = $true )
=> ( ( emptyE1 = $true )
=> ( ( vacuousDall = $true )
=> ( ( quantDeMorgan1 = $true )
=> ( ( quantDeMorgan2 = $true )
=> ( ( quantDeMorgan3 = $true )
=> ( ( quantDeMorgan4 = $true )
=> ( ( prop2setI = $true )
=> ( ( prop2set2propI = $true )
=> ( ( notdexE = $true )
=> ( ( notdallE = $true )
=> ( ( exuI1 = $true )
=> ( ( exuI3 = $true )
=> ( ( exuI2 = $true )
=> ( ( inCongP = $true )
=> ( ( in__Cong = $true )
=> ( ( exuE3u = $true )
=> ( ( exu__Cong = $true )
=> ( ( emptyset__Cong = $true )
=> ( ( setadjoin__Cong = $true )
=> ( ( powerset__Cong = $true )
=> ( ( setunion__Cong = $true )
=> ( ( omega__Cong = $true )
=> ( ( exuEu = $true )
=> ( ( descr__Cong = $true )
=> ( ( dsetconstr__Cong = $true )
=> ( ( subsetI1 = $true )
=> ( ( eqimpsubset2 = $true )
=> ( ( eqimpsubset1 = $true )
=> ( ( subsetI2 = $true )
=> ( ( emptysetsubset = $true )
=> ( ( subsetE = $true )
=> ( ( subsetE2 = $true )
=> ( ( notsubsetI = $true )
=> ( ( notequalI1 = $true )
=> ( ( notequalI2 = $true )
=> ( ( subsetRefl = $true )
=> ( ( subsetTrans = $true )
=> ( ( setadjoinSub = $true )
=> ( ( setadjoinSub2 = $true )
=> ( ( subset2powerset = $true )
=> ( ( setextsub = $true )
=> ( ( subsetemptysetimpeq = $true )
=> ( ( powersetI1 = $true )
=> ( ( powersetE1 = $true )
=> ( ( inPowerset = $true )
=> ( ( powersetsubset = $true )
=> ( ( sepInPowerset = $true )
=> ( ( sepSubset = $true )
=> ( ( binunionIL = $true )
=> ( ( upairset2IR = $true )
=> ( ( binunionIR = $true )
=> ( ( binunionEcases = $true )
=> ( ( binunionE = $true )
=> ( ( binunionLsub = $true )
=> ( ( binunionRsub = $true )
=> ( ( binintersectI = $true )
=> ( ( binintersectSubset5 = $true )
=> ( ( binintersectEL = $true )
=> ( ( binintersectLsub = $true )
=> ( ( binintersectSubset2 = $true )
=> ( ( binintersectSubset3 = $true )
=> ( ( binintersectER = $true )
=> ( ( disjointsetsI1 = $true )
=> ( ( binintersectRsub = $true )
=> ( ( binintersectSubset4 = $true )
=> ( ( binintersectSubset1 = $true )
=> ( ( bs114d = $true )
=> ( ( setminusI = $true )
=> ( ( setminusEL = $true )
=> ( ( setminusER = $true )
=> ( ( setminusSubset2 = $true )
=> ( ( setminusERneg = $true )
=> ( ( setminusELneg = $true )
=> ( ( setminusILneg = $true )
=> ( ( setminusIRneg = $true )
=> ( ( setminusLsub = $true )
=> ( ( setminusSubset1 = $true )
=> ( ( symdiffE = $true )
=> ( ( symdiffI1 = $true )
=> ( ( symdiffI2 = $true )
=> ( ( symdiffIneg1 = $true )
=> ( ( symdiffIneg2 = $true )
=> ( ( secondinupair = $true )
=> ( ( setukpairIL = $true )
=> ( ( setukpairIR = $true )
=> ( ( kpairiskpair = $true )
=> ( ( kpairp = $true )
=> ( ( singletonsubset = $true )
=> ( ( singletoninpowerset = $true )
=> ( ( singletoninpowunion = $true )
=> ( ( upairset2E = $true )
=> ( ( upairsubunion = $true )
=> ( ( upairinpowunion = $true )
=> ( ( ubforcartprodlem1 = $true )
=> ( ( ubforcartprodlem2 = $true )
=> ( ( ubforcartprodlem3 = $true )
=> ( ( cartprodpairin = $true )
=> ( ( cartprodmempair1 = $true )
=> ( ( cartprodmempair = $true )
=> ( ( setunionE2 = $true )
=> ( ( setunionsingleton1 = $true )
=> ( ( setunionsingleton2 = $true )
=> ( ( setunionsingleton = $true )
=> ( ( singletonprop = $true )
=> ( ( ex1E1 = $true )
=> ( ( ex1I = $true )
=> ( ( ex1I2 = $true )
=> ( ( singletonsuniq = $true )
=> ( ( setukpairinjL1 = $true )
=> ( ( kfstsingleton = $true )
=> ( ( theprop = $true )
=> ( ( kfstpairEq = $true )
=> ( ( cartprodfstin = $true )
=> ( ( setukpairinjL2 = $true )
=> ( ( setukpairinjL = $true )
=> ( ( setukpairinjR11 = $true )
=> ( ( setukpairinjR12 = $true )
=> ( ( setukpairinjR1 = $true )
=> ( ( upairequniteq = $true )
=> ( ( setukpairinjR2 = $true )
=> ( ( setukpairinjR = $true )
=> ( ( ksndsingleton = $true )
=> ( ( ksndpairEq = $true )
=> ( ( kpairsurjEq = $true )
=> ( ( cartprodsndin = $true )
=> ( ( cartprodpairmemEL = $true )
=> ( ( cartprodpairmemER = $true )
=> ( ( cartprodmempaircEq = $true )
=> ( ( cartprodfstpairEq = $true )
=> ( ( cartprodsndpairEq = $true )
=> ( ( cartprodpairsurjEq = $true )
=> ( ( dpsetconstrI = $true )
=> ( ( dpsetconstrSub = $true )
=> ( ( setOfPairsIsBReln = $true )
=> ( ( dpsetconstrERa = $true )
=> ( ( dpsetconstrEL1 = $true )
=> ( ( dpsetconstrEL2 = $true )
=> ( ( dpsetconstrER = $true )
=> ( ( funcImageSingleton = $true )
=> ( ( apProp = $true )
=> ( ( app = $true )
=> ( ( infuncsetfunc = $true )
=> ( ( ap2p = $true )
=> ( ( funcinfuncset = $true )
=> ( ( lamProp = $true )
=> ( ( lamp = $true )
=> ( ( lam2p = $true )
=> ( ( brelnall1 = $true )
=> ( ( brelnall2 = $true )
=> ( ( ex1E2 = $true )
=> ( ( funcGraphProp1 = $true )
=> ( ( funcGraphProp3 = $true )
=> ( ( funcGraphProp2 = $true )
=> ( ( funcextLem = $true )
=> ( ( funcGraphProp4 = $true )
=> ( ( subbreln = $true )
=> ( ( eqbreln = $true )
=> ( ( funcext = $true )
=> ( ( funcext2 = $true )
=> ( ( ap2apEq1 = $true )
=> ( ( ap2apEq2 = $true )
=> ( ( beta1 = $true )
=> ( ( eta1 = $true )
=> ( ( lam2lamEq = $true )
=> ( ( beta2 = $true )
=> ( ( eta2 = $true )
=> ( ( iffalseProp1 = $true )
=> ( ( iffalseProp2 = $true )
=> ( ( iftrueProp1 = $true )
=> ( ( iftrueProp2 = $true )
=> ( ( ifSingleton = $true )
=> ( ( ifp = $true )
=> ( ( theeq = $true )
=> ( ( iftrue = $true )
=> ( ( iffalse = $true )
=> ( ( iftrueorfalse = $true )
=> ( ( binintersectT_lem = $true )
=> ( ( binunionT_lem = $true )
=> ( ( powersetT_lem = $true )
=> ( ( setminusT_lem = $true )
=> ( ( complementT_lem = $true )
=> ( ( setextT = $true )
=> ( ( subsetTI = $true )
=> ( ( powersetTI1 = $true )
=> ( ( powersetTE1 = $true )
=> ( ( complementTI1 = $true )
=> ( ( complementTE1 = $true )
=> ( ( binintersectTELcontra = $true )
=> ( ( binintersectTERcontra = $true )
=> ( ( contrasubsetT = $true )
=> ( ( contrasubsetT1 = $true )
=> ( ( contrasubsetT2 = $true )
=> ( ( contrasubsetT3 = $true )
=> ( ( doubleComplementI1 = $true )
=> ( ( doubleComplementE1 = $true )
=> ( ( doubleComplementSub1 = $true )
=> ( ( doubleComplementSub2 = $true )
=> ( ( doubleComplementEq = $true )
=> ( ( complementTnotintersectT = $true )
=> ( ( complementImpComplementIntersect = $true )
=> ( ( complementSubsetComplementIntersect = $true )
=> ( ( complementInPowersetComplementIntersect = $true )
=> ( ( contraSubsetComplement = $true )
=> ( ( complementTcontraSubset = $true )
=> ( ( binunionTILcontra = $true )
=> ( ( binunionTIRcontra = $true )
=> ( ( inIntersectImpInUnion = $true )
=> ! [X0: $i,X1: $i] :
( ( $true
= ( in @ X1 @ ( powerset @ X0 ) ) )
=> ! [X2: $i] :
( ( $true
= ( in @ X2 @ ( powerset @ X0 ) ) )
=> ! [X3: $i] :
( ( $true
= ( in @ X3 @ ( powerset @ X0 ) ) )
=> ! [X4: $i] :
( ( $true
= ( in @ X4 @ X0 ) )
=> ( ( $true
= ( in @ X4 @ ( binintersect @ X1 @ X2 ) ) )
=> ( $true
= ( in @ X4 @ ( binunion @ X2 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f493]) ).
thf(f493,plain,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ! [X0: $i,X1: $i] :
( ( in @ X1 @ ( powerset @ X0 ) )
=> ! [X2: $i] :
( ( in @ X2 @ ( powerset @ X0 ) )
=> ! [X3: $i] :
( ( in @ X3 @ ( powerset @ X0 ) )
=> ! [X4: $i] :
( ( in @ X4 @ X0 )
=> ( ( in @ X4 @ ( binintersect @ X1 @ X2 ) )
=> ( in @ X4 @ ( binunion @ X2 @ X3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(rectify,[status(thm)],[f263]) ).
thf(f263,negated_conjecture,
~ ( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X17: $i] :
( ( in @ X17 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( in @ X1 @ ( binintersect @ X11 @ X16 ) )
=> ( in @ X1 @ ( binunion @ X16 @ X17 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
inference(negated_conjecture,[status(cth)],[f262]) ).
thf(f262,conjecture,
( setextAx
=> ( emptysetAx
=> ( setadjoinAx
=> ( powersetAx
=> ( setunionAx
=> ( omega0Ax
=> ( omegaSAx
=> ( omegaIndAx
=> ( replAx
=> ( foundationAx
=> ( wellorderingAx
=> ( descrp
=> ( dsetconstrI
=> ( dsetconstrEL
=> ( dsetconstrER
=> ( exuE1
=> ( prop2setE
=> ( emptysetE
=> ( emptysetimpfalse
=> ( notinemptyset
=> ( exuE3e
=> ( setext
=> ( emptyI
=> ( noeltsimpempty
=> ( setbeta
=> ( nonemptyE1
=> ( nonemptyI
=> ( nonemptyI1
=> ( setadjoinIL
=> ( emptyinunitempty
=> ( setadjoinIR
=> ( setadjoinE
=> ( setadjoinOr
=> ( setoftrueEq
=> ( powersetI
=> ( emptyinPowerset
=> ( emptyInPowerset
=> ( powersetE
=> ( setunionI
=> ( setunionE
=> ( subPowSU
=> ( exuE2
=> ( nonemptyImpWitness
=> ( uniqinunit
=> ( notinsingleton
=> ( eqinunit
=> ( singletonsswitch
=> ( upairsetE
=> ( upairsetIL
=> ( upairsetIR
=> ( emptyE1
=> ( vacuousDall
=> ( quantDeMorgan1
=> ( quantDeMorgan2
=> ( quantDeMorgan3
=> ( quantDeMorgan4
=> ( prop2setI
=> ( prop2set2propI
=> ( notdexE
=> ( notdallE
=> ( exuI1
=> ( exuI3
=> ( exuI2
=> ( inCongP
=> ( in__Cong
=> ( exuE3u
=> ( exu__Cong
=> ( emptyset__Cong
=> ( setadjoin__Cong
=> ( powerset__Cong
=> ( setunion__Cong
=> ( omega__Cong
=> ( exuEu
=> ( descr__Cong
=> ( dsetconstr__Cong
=> ( subsetI1
=> ( eqimpsubset2
=> ( eqimpsubset1
=> ( subsetI2
=> ( emptysetsubset
=> ( subsetE
=> ( subsetE2
=> ( notsubsetI
=> ( notequalI1
=> ( notequalI2
=> ( subsetRefl
=> ( subsetTrans
=> ( setadjoinSub
=> ( setadjoinSub2
=> ( subset2powerset
=> ( setextsub
=> ( subsetemptysetimpeq
=> ( powersetI1
=> ( powersetE1
=> ( inPowerset
=> ( powersetsubset
=> ( sepInPowerset
=> ( sepSubset
=> ( binunionIL
=> ( upairset2IR
=> ( binunionIR
=> ( binunionEcases
=> ( binunionE
=> ( binunionLsub
=> ( binunionRsub
=> ( binintersectI
=> ( binintersectSubset5
=> ( binintersectEL
=> ( binintersectLsub
=> ( binintersectSubset2
=> ( binintersectSubset3
=> ( binintersectER
=> ( disjointsetsI1
=> ( binintersectRsub
=> ( binintersectSubset4
=> ( binintersectSubset1
=> ( bs114d
=> ( setminusI
=> ( setminusEL
=> ( setminusER
=> ( setminusSubset2
=> ( setminusERneg
=> ( setminusELneg
=> ( setminusILneg
=> ( setminusIRneg
=> ( setminusLsub
=> ( setminusSubset1
=> ( symdiffE
=> ( symdiffI1
=> ( symdiffI2
=> ( symdiffIneg1
=> ( symdiffIneg2
=> ( secondinupair
=> ( setukpairIL
=> ( setukpairIR
=> ( kpairiskpair
=> ( kpairp
=> ( singletonsubset
=> ( singletoninpowerset
=> ( singletoninpowunion
=> ( upairset2E
=> ( upairsubunion
=> ( upairinpowunion
=> ( ubforcartprodlem1
=> ( ubforcartprodlem2
=> ( ubforcartprodlem3
=> ( cartprodpairin
=> ( cartprodmempair1
=> ( cartprodmempair
=> ( setunionE2
=> ( setunionsingleton1
=> ( setunionsingleton2
=> ( setunionsingleton
=> ( singletonprop
=> ( ex1E1
=> ( ex1I
=> ( ex1I2
=> ( singletonsuniq
=> ( setukpairinjL1
=> ( kfstsingleton
=> ( theprop
=> ( kfstpairEq
=> ( cartprodfstin
=> ( setukpairinjL2
=> ( setukpairinjL
=> ( setukpairinjR11
=> ( setukpairinjR12
=> ( setukpairinjR1
=> ( upairequniteq
=> ( setukpairinjR2
=> ( setukpairinjR
=> ( ksndsingleton
=> ( ksndpairEq
=> ( kpairsurjEq
=> ( cartprodsndin
=> ( cartprodpairmemEL
=> ( cartprodpairmemER
=> ( cartprodmempaircEq
=> ( cartprodfstpairEq
=> ( cartprodsndpairEq
=> ( cartprodpairsurjEq
=> ( dpsetconstrI
=> ( dpsetconstrSub
=> ( setOfPairsIsBReln
=> ( dpsetconstrERa
=> ( dpsetconstrEL1
=> ( dpsetconstrEL2
=> ( dpsetconstrER
=> ( funcImageSingleton
=> ( apProp
=> ( app
=> ( infuncsetfunc
=> ( ap2p
=> ( funcinfuncset
=> ( lamProp
=> ( lamp
=> ( lam2p
=> ( brelnall1
=> ( brelnall2
=> ( ex1E2
=> ( funcGraphProp1
=> ( funcGraphProp3
=> ( funcGraphProp2
=> ( funcextLem
=> ( funcGraphProp4
=> ( subbreln
=> ( eqbreln
=> ( funcext
=> ( funcext2
=> ( ap2apEq1
=> ( ap2apEq2
=> ( beta1
=> ( eta1
=> ( lam2lamEq
=> ( beta2
=> ( eta2
=> ( iffalseProp1
=> ( iffalseProp2
=> ( iftrueProp1
=> ( iftrueProp2
=> ( ifSingleton
=> ( ifp
=> ( theeq
=> ( iftrue
=> ( iffalse
=> ( iftrueorfalse
=> ( binintersectT_lem
=> ( binunionT_lem
=> ( powersetT_lem
=> ( setminusT_lem
=> ( complementT_lem
=> ( setextT
=> ( subsetTI
=> ( powersetTI1
=> ( powersetTE1
=> ( complementTI1
=> ( complementTE1
=> ( binintersectTELcontra
=> ( binintersectTERcontra
=> ( contrasubsetT
=> ( contrasubsetT1
=> ( contrasubsetT2
=> ( contrasubsetT3
=> ( doubleComplementI1
=> ( doubleComplementE1
=> ( doubleComplementSub1
=> ( doubleComplementSub2
=> ( doubleComplementEq
=> ( complementTnotintersectT
=> ( complementImpComplementIntersect
=> ( complementSubsetComplementIntersect
=> ( complementInPowersetComplementIntersect
=> ( contraSubsetComplement
=> ( complementTcontraSubset
=> ( binunionTILcontra
=> ( binunionTIRcontra
=> ( inIntersectImpInUnion
=> ! [X3: $i,X11: $i] :
( ( in @ X11 @ ( powerset @ X3 ) )
=> ! [X16: $i] :
( ( in @ X16 @ ( powerset @ X3 ) )
=> ! [X17: $i] :
( ( in @ X17 @ ( powerset @ X3 ) )
=> ! [X1: $i] :
( ( in @ X1 @ X3 )
=> ( ( in @ X1 @ ( binintersect @ X11 @ X16 ) )
=> ( in @ X1 @ ( binunion @ X16 @ X17 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.JCQicskBu4/DTF2THF_13002.p',inIntersectImpInUnion2) ).
thf(f4043,plain,
( ( $true
= ( in @ sK521 @ sK519 ) )
| ( binintersectER != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4034]) ).
thf(f4034,plain,
( ( $true != $true )
| ( binintersectER != $true )
| ( $true
= ( in @ sK521 @ sK519 ) ) ),
inference(superposition,[status(thm)],[f2719,f3257]) ).
thf(f3257,plain,
( $true
= ( in @ sK521 @ ( binintersect @ sK518 @ sK519 ) ) ),
inference(cnf_transformation,[status(thm)],[f1917]) ).
thf(f2719,plain,
! [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4 @ ( binintersect @ X5 @ X3 ) ) )
| ( ( in @ X4 @ X3 )
= $true )
| ( binintersectER != $true ) ),
inference(cnf_transformation,[status(thm)],[f1518]) ).
thf(f1518,plain,
( ( ( binintersectER = $true )
| ( ( $true
= ( in @ sK253 @ ( binintersect @ sK254 @ sK252 ) ) )
& ( $true
!= ( in @ sK253 @ sK252 ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4 @ ( binintersect @ X5 @ X3 ) ) )
| ( ( in @ X4 @ X3 )
= $true ) )
| ( binintersectER != $true ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK252,sK253,sK254])],[f1516,f1517]) ).
thf(f1517,plain,
( ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( binintersect @ X2 @ X0 ) )
= $true )
& ( $true
!= ( in @ X1 @ X0 ) ) )
=> ( ( $true
= ( in @ sK253 @ ( binintersect @ sK254 @ sK252 ) ) )
& ( $true
!= ( in @ sK253 @ sK252 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f1516,plain,
( ( ( binintersectER = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( binintersect @ X2 @ X0 ) )
= $true )
& ( $true
!= ( in @ X1 @ X0 ) ) ) )
& ( ! [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X4 @ ( binintersect @ X5 @ X3 ) ) )
| ( ( in @ X4 @ X3 )
= $true ) )
| ( binintersectER != $true ) ) ),
inference(rectify,[status(thm)],[f1515]) ).
thf(f1515,plain,
( ( ( binintersectER = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( binintersect @ X2 @ X0 ) )
= $true )
& ( $true
!= ( in @ X1 @ X0 ) ) ) )
& ( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( binintersect @ X2 @ X0 ) )
!= $true )
| ( $true
= ( in @ X1 @ X0 ) ) )
| ( binintersectER != $true ) ) ),
inference(nnf_transformation,[status(thm)],[f851]) ).
thf(f851,plain,
( ( binintersectER = $true )
<=> ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( binintersect @ X2 @ X0 ) )
!= $true )
| ( $true
= ( in @ X1 @ X0 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f738]) ).
thf(f738,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( ( in @ X1 @ ( binintersect @ X2 @ X0 ) )
= $true )
=> ( $true
= ( in @ X1 @ X0 ) ) )
<=> ( binintersectER = $true ) ),
inference(fool_elimination,[status(thm)],[f737]) ).
thf(f737,plain,
( binintersectER
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X1 @ ( binintersect @ X2 @ X0 ) )
=> ( in @ X1 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f114]) ).
thf(f114,axiom,
( binintersectER
= ( ! [X4: $i,X1: $i,X3: $i] :
( ( in @ X1 @ ( binintersect @ X3 @ X4 ) )
=> ( in @ X1 @ X4 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.JCQicskBu4/DTF2THF_13002.p',binintersectER) ).
thf(f4132,plain,
~ spl820_13,
inference(avatar_split_clause,[status(thm)],[f4131,f4087]) ).
thf(f4131,plain,
( $true
!= ( in @ sK521 @ sK519 ) ),
inference(trivial_inequality_removal,[status(thm)],[f4130]) ).
thf(f4130,plain,
( ( $true != $true )
| ( $true
!= ( in @ sK521 @ sK519 ) ) ),
inference(forward_demodulation,[status(thm)],[f4106,f3105]) ).
thf(f3105,plain,
binunionIL = $true,
inference(cnf_transformation,[status(thm)],[f1917]) ).
thf(f4106,plain,
( ( $true
!= ( in @ sK521 @ sK519 ) )
| ( binunionIL != $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f4095]) ).
thf(f4095,plain,
( ( binunionIL != $true )
| ( $true != $true )
| ( $true
!= ( in @ sK521 @ sK519 ) ) ),
inference(superposition,[status(thm)],[f3259,f3658]) ).
thf(f3658,plain,
! [X2: $i,X0: $i,X1: $i] :
( ( $true
= ( in @ X0 @ ( binunion @ X2 @ X1 ) ) )
| ( binunionIL != $true )
| ( $true
!= ( in @ X0 @ X2 ) ) ),
inference(cnf_transformation,[status(thm)],[f2295]) ).
thf(f2295,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( binunion @ X2 @ X1 ) ) )
| ( $true
!= ( in @ X0 @ X2 ) ) )
| ( binunionIL != $true ) )
& ( ( binunionIL = $true )
| ( ( $true
!= ( in @ sK755 @ ( binunion @ sK757 @ sK756 ) ) )
& ( $true
= ( in @ sK755 @ sK757 ) ) ) ) ),
inference(skolemisation,[status(esa),new_symbols(skolem,[sK755,sK756,sK757])],[f2293,f2294]) ).
thf(f2294,plain,
( ? [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X3 @ ( binunion @ X5 @ X4 ) ) )
& ( $true
= ( in @ X3 @ X5 ) ) )
=> ( ( $true
!= ( in @ sK755 @ ( binunion @ sK757 @ sK756 ) ) )
& ( $true
= ( in @ sK755 @ sK757 ) ) ) ),
introduced(definition,[],[choice_axiom]) ).
thf(f2293,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( binunion @ X2 @ X1 ) ) )
| ( $true
!= ( in @ X0 @ X2 ) ) )
| ( binunionIL != $true ) )
& ( ( binunionIL = $true )
| ? [X3: $i,X4: $i,X5: $i] :
( ( $true
!= ( in @ X3 @ ( binunion @ X5 @ X4 ) ) )
& ( $true
= ( in @ X3 @ X5 ) ) ) ) ),
inference(rectify,[status(thm)],[f2292]) ).
thf(f2292,plain,
( ( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( binunion @ X2 @ X1 ) ) )
| ( $true
!= ( in @ X0 @ X2 ) ) )
| ( binunionIL != $true ) )
& ( ( binunionIL = $true )
| ? [X0: $i,X1: $i,X2: $i] :
( ( $true
!= ( in @ X0 @ ( binunion @ X2 @ X1 ) ) )
& ( $true
= ( in @ X0 @ X2 ) ) ) ) ),
inference(nnf_transformation,[status(thm)],[f993]) ).
thf(f993,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ ( binunion @ X2 @ X1 ) ) )
| ( $true
!= ( in @ X0 @ X2 ) ) )
<=> ( binunionIL = $true ) ),
inference(ennf_transformation,[status(thm)],[f378]) ).
thf(f378,plain,
( ! [X0: $i,X1: $i,X2: $i] :
( ( $true
= ( in @ X0 @ X2 ) )
=> ( $true
= ( in @ X0 @ ( binunion @ X2 @ X1 ) ) ) )
<=> ( binunionIL = $true ) ),
inference(fool_elimination,[status(thm)],[f377]) ).
thf(f377,plain,
( binunionIL
= ( ! [X0: $i,X1: $i,X2: $i] :
( ( in @ X0 @ X2 )
=> ( in @ X0 @ ( binunion @ X2 @ X1 ) ) ) ) ),
inference(rectify,[status(thm)],[f101]) ).
thf(f101,axiom,
( binunionIL
= ( ! [X1: $i,X4: $i,X3: $i] :
( ( in @ X1 @ X3 )
=> ( in @ X1 @ ( binunion @ X3 @ X4 ) ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.JCQicskBu4/DTF2THF_13002.p',binunionIL) ).
thf(f3259,plain,
( $true
!= ( in @ sK521 @ ( binunion @ sK519 @ sK520 ) ) ),
inference(cnf_transformation,[status(thm)],[f1917]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : SEU741^1 : TPTP v9.2.1. Released v3.7.0.
% 0.13/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n007.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 12:31:53 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.32/0.48 ---- Original DTF file ---
% 0.32/0.49 thf(spec,logic,$$dhol).
% 0.32/0.49 %------------------------------------------------------------------------------
% 0.32/0.49 % File : SEU741^1 : TPTP v9.2.1. Released v3.7.0.
% 0.32/0.49 % Domain : Set Theory
% 0.32/0.49 % Problem : Typed Set Theory - Laws for Typed Sets
% 0.32/0.49 % Version : Especial.
% 0.32/0.49 % English : (! A:i.! X:i.in X (powerset A) -> (! Y:i.in Y (powerset A) ->
% 0.32/0.49 % (! Z:i.in Z (powerset A) -> (! x:i.in x A ->
% 0.32/0.49 % in x (binintersect X Y) -> in x (binunion Y Z)))))
% 0.32/0.49
% 0.32/0.49 % Refs : [Bro08] Brown (2008), Email to G. Sutcliffe
% 0.32/0.49 % Source : [Bro08]
% 0.32/0.49 % Names : ZFC243g [Bro08]
% 0.32/0.49
% 0.32/0.49 % Status : Theorem
% 0.32/0.49 % 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, 0.50 v5.4.0, 0.60 v5.2.0, 0.80 v4.1.0, 1.00 v3.7.0
% 0.32/0.49 % Syntax : Number of formulae : 557 ( 261 unt; 295 typ; 261 def)
% 0.32/0.49 % Number of atoms : 1517 ( 387 equ; 0 cnn)
% 0.32/0.49 % Maximal formula atoms : 263 ( 5 avg)
% 0.32/0.49 % Number of connectives : 3064 ( 69 ~; 12 |; 44 &;2182 @)
% 0.32/0.49 % ( 14 <=>; 743 =>; 0 <=; 0 <~>)
% 0.32/0.49 % Maximal formula depth : 271 ( 2 avg)
% 0.32/0.49 % Number of types : 2 ( 0 usr)
% 0.32/0.49 % Number of type conns : 132 ( 132 >; 0 *; 0 +; 0 <<)
% 0.32/0.49 % Number of symbols : 298 ( 295 usr; 261 con; 0-4 aty)
% 0.32/0.49 % Number of variables : 881 ( 74 ^; 772 !; 35 ?; 881 :)
% 0.32/0.49 % SPC : TH0_THM_EQU_NAR
% 0.32/0.49
% 0.32/0.49 % Comments : http://mathgate.info/detsetitem.php?id=303
% 0.32/0.49 %------------------------------------------------------------------------------
% 0.32/0.49 thf(in_type,type,
% 0.32/0.49 in: $i > $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(exu_type,type,
% 0.32/0.49 exu: ( $i > $o ) > $o ).
% 0.32/0.49
% 0.32/0.49 thf(exu,definition,
% 0.32/0.49 ( exu
% 0.32/0.49 = ( ^ [Xphi: $i > $o] :
% 0.32/0.49 ? [Xx: $i] :
% 0.32/0.49 ( ( Xphi @ Xx )
% 0.32/0.49 & ! [Xy: $i] :
% 0.32/0.49 ( ( Xphi @ Xy )
% 0.32/0.49 => ( Xx = Xy ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setextAx_type,type,
% 0.32/0.49 setextAx: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setextAx,definition,
% 0.32/0.49 ( setextAx
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 <=> ( in @ Xx @ B ) )
% 0.32/0.49 => ( A = B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(emptyset_type,type,
% 0.32/0.49 emptyset: $i ).
% 0.32/0.49
% 0.32/0.49 thf(emptysetAx_type,type,
% 0.32/0.49 emptysetAx: $o ).
% 0.32/0.49
% 0.32/0.49 thf(emptysetAx,definition,
% 0.32/0.49 ( emptysetAx
% 0.32/0.49 = ( ! [Xx: $i] :
% 0.32/0.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoin_type,type,
% 0.32/0.49 setadjoin: $i > $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoinAx_type,type,
% 0.32/0.49 setadjoinAx: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoinAx,definition,
% 0.32/0.49 ( setadjoinAx
% 0.32/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/0.49 <=> ( ( Xy = Xx )
% 0.32/0.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(powerset_type,type,
% 0.32/0.49 powerset: $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(powersetAx_type,type,
% 0.32/0.49 powersetAx: $o ).
% 0.32/0.49
% 0.32/0.49 thf(powersetAx,definition,
% 0.32/0.49 ( powersetAx
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( in @ B @ ( powerset @ A ) )
% 0.32/0.49 <=> ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ B )
% 0.32/0.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setunion_type,type,
% 0.32/0.49 setunion: $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(setunionAx_type,type,
% 0.32/0.49 setunionAx: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setunionAx,definition,
% 0.32/0.49 ( setunionAx
% 0.32/0.49 = ( ! [A: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.32/0.49 <=> ? [B: $i] :
% 0.32/0.49 ( ( in @ Xx @ B )
% 0.32/0.49 & ( in @ B @ A ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(omega_type,type,
% 0.32/0.49 omega: $i ).
% 0.32/0.49
% 0.32/0.49 thf(omega0Ax_type,type,
% 0.32/0.49 omega0Ax: $o ).
% 0.32/0.49
% 0.32/0.49 thf(omega0Ax,definition,
% 0.32/0.49 ( omega0Ax
% 0.32/0.49 = ( in @ emptyset @ omega ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(omegaSAx_type,type,
% 0.32/0.49 omegaSAx: $o ).
% 0.32/0.49
% 0.32/0.49 thf(omegaSAx,definition,
% 0.32/0.49 ( omegaSAx
% 0.32/0.49 = ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ omega )
% 0.32/0.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(omegaIndAx_type,type,
% 0.32/0.49 omegaIndAx: $o ).
% 0.32/0.49
% 0.32/0.49 thf(omegaIndAx,definition,
% 0.32/0.49 ( omegaIndAx
% 0.32/0.49 = ( ! [A: $i] :
% 0.32/0.49 ( ( ( in @ emptyset @ A )
% 0.32/0.49 & ! [Xx: $i] :
% 0.32/0.49 ( ( ( in @ Xx @ omega )
% 0.32/0.49 & ( in @ Xx @ A ) )
% 0.32/0.49 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ omega )
% 0.32/0.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(replAx_type,type,
% 0.32/0.49 replAx: $o ).
% 0.32/0.49
% 0.32/0.49 thf(replAx,definition,
% 0.32/0.49 ( replAx
% 0.32/0.49 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( exu
% 0.32/0.49 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 0.32/0.49 => ? [B: $i] :
% 0.32/0.49 ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ B )
% 0.32/0.49 <=> ? [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(foundationAx_type,type,
% 0.32/0.49 foundationAx: $o ).
% 0.32/0.49
% 0.32/0.49 thf(foundationAx,definition,
% 0.32/0.49 ( foundationAx
% 0.32/0.49 = ( ! [A: $i] :
% 0.32/0.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.32/0.49 => ? [B: $i] :
% 0.32/0.49 ( ( in @ B @ A )
% 0.32/0.49 & ~ ? [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ B )
% 0.32/0.49 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(wellorderingAx_type,type,
% 0.32/0.49 wellorderingAx: $o ).
% 0.32/0.49
% 0.32/0.49 thf(wellorderingAx,definition,
% 0.32/0.49 ( wellorderingAx
% 0.32/0.49 = ( ! [A: $i] :
% 0.32/0.49 ? [B: $i] :
% 0.32/0.49 ( ! [C: $i] :
% 0.32/0.49 ( ( in @ C @ B )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ C )
% 0.32/0.49 => ( in @ Xx @ A ) ) )
% 0.32/0.49 & ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( ( in @ Xx @ A )
% 0.32/0.49 & ( in @ Xy @ A ) )
% 0.32/0.49 => ( ! [C: $i] :
% 0.32/0.49 ( ( in @ C @ B )
% 0.32/0.49 => ( ( in @ Xx @ C )
% 0.32/0.49 <=> ( in @ Xy @ C ) ) )
% 0.32/0.49 => ( Xx = Xy ) ) )
% 0.32/0.49 & ! [C: $i,D: $i] :
% 0.32/0.49 ( ( ( in @ C @ B )
% 0.32/0.49 & ( in @ D @ B ) )
% 0.32/0.49 => ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ C )
% 0.32/0.49 => ( in @ Xx @ D ) )
% 0.32/0.49 | ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ D )
% 0.32/0.49 => ( in @ Xx @ C ) ) ) )
% 0.32/0.49 & ! [C: $i] :
% 0.32/0.49 ( ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ C )
% 0.32/0.49 => ( in @ Xx @ A ) )
% 0.32/0.49 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 0.32/0.49 => ? [D: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ D @ B )
% 0.32/0.49 & ( in @ Xx @ C )
% 0.32/0.49 & ~ ? [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ D )
% 0.32/0.49 & ( in @ Xy @ C ) )
% 0.32/0.49 & ! [E: $i] :
% 0.32/0.49 ( ( in @ E @ B )
% 0.32/0.49 => ( ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ E )
% 0.32/0.49 => ( in @ Xy @ D ) )
% 0.32/0.49 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(descr_type,type,
% 0.32/0.49 descr: ( $i > $o ) > $i ).
% 0.32/0.49
% 0.32/0.49 thf(descrp_type,type,
% 0.32/0.49 descrp: $o ).
% 0.32/0.49
% 0.32/0.49 thf(descrp,definition,
% 0.32/0.49 ( descrp
% 0.32/0.49 = ( ! [Xphi: $i > $o] :
% 0.32/0.49 ( ( exu
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.49 => ( Xphi
% 0.32/0.49 @ ( descr
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(dsetconstr_type,type,
% 0.32/0.49 dsetconstr: $i > ( $i > $o ) > $i ).
% 0.32/0.49
% 0.32/0.49 thf(dsetconstrI_type,type,
% 0.32/0.49 dsetconstrI: $o ).
% 0.32/0.49
% 0.32/0.49 thf(dsetconstrI,definition,
% 0.32/0.49 ( dsetconstrI
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( Xphi @ Xx )
% 0.32/0.49 => ( in @ Xx
% 0.32/0.49 @ ( dsetconstr @ A
% 0.32/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(dsetconstrEL_type,type,
% 0.32/0.49 dsetconstrEL: $o ).
% 0.32/0.49
% 0.32/0.49 thf(dsetconstrEL,definition,
% 0.32/0.49 ( dsetconstrEL
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx
% 0.32/0.49 @ ( dsetconstr @ A
% 0.32/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.32/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(dsetconstrER_type,type,
% 0.32/0.49 dsetconstrER: $o ).
% 0.32/0.49
% 0.32/0.49 thf(dsetconstrER,definition,
% 0.32/0.49 ( dsetconstrER
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx
% 0.32/0.49 @ ( dsetconstr @ A
% 0.32/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.32/0.49 => ( Xphi @ Xx ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(exuE1_type,type,
% 0.32/0.49 exuE1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(exuE1,definition,
% 0.32/0.49 ( exuE1
% 0.32/0.49 = ( ! [Xphi: $i > $o] :
% 0.32/0.49 ( ( exu
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.49 => ? [Xx: $i] :
% 0.32/0.49 ( ( Xphi @ Xx )
% 0.32/0.49 & ! [Xy: $i] :
% 0.32/0.49 ( ( Xphi @ Xy )
% 0.32/0.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(prop2set_type,type,
% 0.32/0.49 prop2set: $o > $i ).
% 0.32/0.49
% 0.32/0.49 thf(prop2setE_type,type,
% 0.32/0.49 prop2setE: $o ).
% 0.32/0.49
% 0.32/0.49 thf(prop2setE,definition,
% 0.32/0.49 ( prop2setE
% 0.32/0.49 = ( ! [Xphi: $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 0.32/0.49 => Xphi ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(emptysetE_type,type,
% 0.32/0.49 emptysetE: $o ).
% 0.32/0.49
% 0.32/0.49 thf(emptysetE,definition,
% 0.32/0.49 ( emptysetE
% 0.32/0.49 = ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ emptyset )
% 0.32/0.49 => ! [Xphi: $o] : Xphi ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(emptysetimpfalse_type,type,
% 0.32/0.49 emptysetimpfalse: $o ).
% 0.32/0.49
% 0.32/0.49 thf(emptysetimpfalse,definition,
% 0.32/0.49 ( emptysetimpfalse
% 0.32/0.49 = ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ emptyset )
% 0.32/0.49 => $false ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(notinemptyset_type,type,
% 0.32/0.49 notinemptyset: $o ).
% 0.32/0.49
% 0.32/0.49 thf(notinemptyset,definition,
% 0.32/0.49 ( notinemptyset
% 0.32/0.49 = ( ! [Xx: $i] :
% 0.32/0.49 ~ ( in @ Xx @ emptyset ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(exuE3e_type,type,
% 0.32/0.49 exuE3e: $o ).
% 0.32/0.49
% 0.32/0.49 thf(exuE3e,definition,
% 0.32/0.49 ( exuE3e
% 0.32/0.49 = ( ! [Xphi: $i > $o] :
% 0.32/0.49 ( ( exu
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.49 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setext_type,type,
% 0.32/0.49 setext: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setext,definition,
% 0.32/0.49 ( setext
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ Xx @ B ) )
% 0.32/0.49 => ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ B )
% 0.32/0.49 => ( in @ Xx @ A ) )
% 0.32/0.49 => ( A = B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(emptyI_type,type,
% 0.32/0.49 emptyI: $o ).
% 0.32/0.49
% 0.32/0.49 thf(emptyI,definition,
% 0.32/0.49 ( emptyI
% 0.32/0.49 = ( ! [A: $i] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ~ ( in @ Xx @ A )
% 0.32/0.49 => ( A = emptyset ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(noeltsimpempty_type,type,
% 0.32/0.49 noeltsimpempty: $o ).
% 0.32/0.49
% 0.32/0.49 thf(noeltsimpempty,definition,
% 0.32/0.49 ( noeltsimpempty
% 0.32/0.49 = ( ! [A: $i] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ~ ( in @ Xx @ A )
% 0.32/0.49 => ( A = emptyset ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setbeta_type,type,
% 0.32/0.49 setbeta: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setbeta,definition,
% 0.32/0.49 ( setbeta
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx
% 0.32/0.49 @ ( dsetconstr @ A
% 0.32/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 0.32/0.49 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(nonempty_type,type,
% 0.32/0.49 nonempty: $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(nonempty,definition,
% 0.32/0.49 ( nonempty
% 0.32/0.49 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(nonemptyE1_type,type,
% 0.32/0.49 nonemptyE1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(nonemptyE1,definition,
% 0.32/0.49 ( nonemptyE1
% 0.32/0.49 = ( ! [A: $i] :
% 0.32/0.49 ( ( nonempty @ A )
% 0.32/0.49 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(nonemptyI_type,type,
% 0.32/0.49 nonemptyI: $o ).
% 0.32/0.49
% 0.32/0.49 thf(nonemptyI,definition,
% 0.32/0.49 ( nonemptyI
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( Xphi @ Xx )
% 0.32/0.49 => ( nonempty
% 0.32/0.49 @ ( dsetconstr @ A
% 0.32/0.49 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(nonemptyI1_type,type,
% 0.32/0.49 nonemptyI1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(nonemptyI1,definition,
% 0.32/0.49 ( nonemptyI1
% 0.32/0.49 = ( ! [A: $i] :
% 0.32/0.49 ( ? [Xx: $i] : ( in @ Xx @ A )
% 0.32/0.49 => ( nonempty @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoinIL_type,type,
% 0.32/0.49 setadjoinIL: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoinIL,definition,
% 0.32/0.49 ( setadjoinIL
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(emptyinunitempty_type,type,
% 0.32/0.49 emptyinunitempty: $o ).
% 0.32/0.49
% 0.32/0.49 thf(emptyinunitempty,definition,
% 0.32/0.49 ( emptyinunitempty
% 0.32/0.49 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoinIR_type,type,
% 0.32/0.49 setadjoinIR: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoinIR,definition,
% 0.32/0.49 ( setadjoinIR
% 0.32/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoinE_type,type,
% 0.32/0.49 setadjoinE: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoinE,definition,
% 0.32/0.49 ( setadjoinE
% 0.32/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/0.49 => ! [Xphi: $o] :
% 0.32/0.49 ( ( ( Xy = Xx )
% 0.32/0.49 => Xphi )
% 0.32/0.49 => ( ( ( in @ Xy @ A )
% 0.32/0.49 => Xphi )
% 0.32/0.49 => Xphi ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoinOr_type,type,
% 0.32/0.49 setadjoinOr: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoinOr,definition,
% 0.32/0.49 ( setadjoinOr
% 0.32/0.49 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 0.32/0.49 => ( ( Xy = Xx )
% 0.32/0.49 | ( in @ Xy @ A ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setoftrueEq_type,type,
% 0.32/0.49 setoftrueEq: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setoftrueEq,definition,
% 0.32/0.49 ( setoftrueEq
% 0.32/0.49 = ( ! [A: $i] :
% 0.32/0.49 ( ( dsetconstr @ A
% 0.32/0.49 @ ^ [Xx: $i] : $true )
% 0.32/0.49 = A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(powersetI_type,type,
% 0.32/0.49 powersetI: $o ).
% 0.32/0.49
% 0.32/0.49 thf(powersetI,definition,
% 0.32/0.49 ( powersetI
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ B )
% 0.32/0.49 => ( in @ Xx @ A ) )
% 0.32/0.49 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(emptyinPowerset_type,type,
% 0.32/0.49 emptyinPowerset: $o ).
% 0.32/0.49
% 0.32/0.49 thf(emptyinPowerset,definition,
% 0.32/0.49 ( emptyinPowerset
% 0.32/0.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(emptyInPowerset_type,type,
% 0.32/0.49 emptyInPowerset: $o ).
% 0.32/0.49
% 0.32/0.49 thf(emptyInPowerset,definition,
% 0.32/0.49 ( emptyInPowerset
% 0.32/0.49 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(powersetE_type,type,
% 0.32/0.49 powersetE: $o ).
% 0.32/0.49
% 0.32/0.49 thf(powersetE,definition,
% 0.32/0.49 ( powersetE
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ B @ ( powerset @ A ) )
% 0.32/0.49 => ( ( in @ Xx @ B )
% 0.32/0.49 => ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setunionI_type,type,
% 0.32/0.49 setunionI: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setunionI,definition,
% 0.32/0.49 ( setunionI
% 0.32/0.49 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.32/0.49 ( ( in @ Xx @ B )
% 0.32/0.49 => ( ( in @ B @ A )
% 0.32/0.49 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setunionE_type,type,
% 0.32/0.49 setunionE: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setunionE,definition,
% 0.32/0.49 ( setunionE
% 0.32/0.49 = ( ! [A: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.32/0.49 => ! [Xphi: $o] :
% 0.32/0.49 ( ! [B: $i] :
% 0.32/0.49 ( ( in @ Xx @ B )
% 0.32/0.49 => ( ( in @ B @ A )
% 0.32/0.49 => Xphi ) )
% 0.32/0.49 => Xphi ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(subPowSU_type,type,
% 0.32/0.49 subPowSU: $o ).
% 0.32/0.49
% 0.32/0.49 thf(subPowSU,definition,
% 0.32/0.49 ( subPowSU
% 0.32/0.49 = ( ! [A: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(exuE2_type,type,
% 0.32/0.49 exuE2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(exuE2,definition,
% 0.32/0.49 ( exuE2
% 0.32/0.49 = ( ! [Xphi: $i > $o] :
% 0.32/0.49 ( ( exu
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.49 => ? [Xx: $i] :
% 0.32/0.49 ! [Xy: $i] :
% 0.32/0.49 ( ( Xphi @ Xy )
% 0.32/0.49 <=> ( Xy = Xx ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(nonemptyImpWitness_type,type,
% 0.32/0.49 nonemptyImpWitness: $o ).
% 0.32/0.49
% 0.32/0.49 thf(nonemptyImpWitness,definition,
% 0.32/0.49 ( nonemptyImpWitness
% 0.32/0.49 = ( ! [A: $i] :
% 0.32/0.49 ( ( nonempty @ A )
% 0.32/0.49 => ? [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 & $true ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(uniqinunit_type,type,
% 0.32/0.49 uniqinunit: $o ).
% 0.32/0.49
% 0.32/0.49 thf(uniqinunit,definition,
% 0.32/0.49 ( uniqinunit
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.49 => ( Xx = Xy ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(notinsingleton_type,type,
% 0.32/0.49 notinsingleton: $o ).
% 0.32/0.49
% 0.32/0.49 thf(notinsingleton,definition,
% 0.32/0.49 ( notinsingleton
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( Xx != Xy )
% 0.32/0.49 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(eqinunit_type,type,
% 0.32/0.49 eqinunit: $o ).
% 0.32/0.49
% 0.32/0.49 thf(eqinunit,definition,
% 0.32/0.49 ( eqinunit
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( Xx = Xy )
% 0.32/0.49 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(singletonsswitch_type,type,
% 0.32/0.49 singletonsswitch: $o ).
% 0.32/0.49
% 0.32/0.49 thf(singletonsswitch,definition,
% 0.32/0.49 ( singletonsswitch
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.49 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(upairsetE_type,type,
% 0.32/0.49 upairsetE: $o ).
% 0.32/0.49
% 0.32/0.49 thf(upairsetE,definition,
% 0.32/0.49 ( upairsetE
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.49 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.32/0.49 => ( ( Xz = Xx )
% 0.32/0.49 | ( Xz = Xy ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(upairsetIL_type,type,
% 0.32/0.49 upairsetIL: $o ).
% 0.32/0.49
% 0.32/0.49 thf(upairsetIL,definition,
% 0.32/0.49 ( upairsetIL
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(upairsetIR_type,type,
% 0.32/0.49 upairsetIR: $o ).
% 0.32/0.49
% 0.32/0.49 thf(upairsetIR,definition,
% 0.32/0.49 ( upairsetIR
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(emptyE1_type,type,
% 0.32/0.49 emptyE1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(emptyE1,definition,
% 0.32/0.49 ( emptyE1
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.49 ( ? [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 & ( Xphi @ Xx ) )
% 0.32/0.49 => ( ( ( dsetconstr @ A
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.49 = emptyset )
% 0.32/0.49 => $false ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(vacuousDall_type,type,
% 0.32/0.49 vacuousDall: $o ).
% 0.32/0.49
% 0.32/0.49 thf(vacuousDall,definition,
% 0.32/0.49 ( vacuousDall
% 0.32/0.49 = ( ! [Xphi: $i > $o,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ emptyset )
% 0.32/0.49 => ( Xphi @ Xx ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(quantDeMorgan1_type,type,
% 0.32/0.49 quantDeMorgan1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(quantDeMorgan1,definition,
% 0.32/0.49 ( quantDeMorgan1
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.49 ( ~ ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( Xphi @ Xx ) )
% 0.32/0.49 => ? [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(quantDeMorgan2_type,type,
% 0.32/0.49 quantDeMorgan2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(quantDeMorgan2,definition,
% 0.32/0.49 ( quantDeMorgan2
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ~ ( Xphi @ Xx ) )
% 0.32/0.49 => ~ ? [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(quantDeMorgan3_type,type,
% 0.32/0.49 quantDeMorgan3: $o ).
% 0.32/0.49
% 0.32/0.49 thf(quantDeMorgan3,definition,
% 0.32/0.49 ( quantDeMorgan3
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.49 ( ~ ? [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 & ( Xphi @ Xx ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(quantDeMorgan4_type,type,
% 0.32/0.49 quantDeMorgan4: $o ).
% 0.32/0.49
% 0.32/0.49 thf(quantDeMorgan4,definition,
% 0.32/0.49 ( quantDeMorgan4
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.49 ( ? [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 & ~ ( Xphi @ Xx ) )
% 0.32/0.49 => ~ ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(prop2setI_type,type,
% 0.32/0.49 prop2setI: $o ).
% 0.32/0.49
% 0.32/0.49 thf(prop2setI,definition,
% 0.32/0.49 ( prop2setI
% 0.32/0.49 = ( ! [Xphi: $o] :
% 0.32/0.49 ( Xphi
% 0.32/0.49 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(set2prop_type,type,
% 0.32/0.49 set2prop: $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(prop2set2propI_type,type,
% 0.32/0.49 prop2set2propI: $o ).
% 0.32/0.49
% 0.32/0.49 thf(prop2set2propI,definition,
% 0.32/0.49 ( prop2set2propI
% 0.32/0.49 = ( ! [Xphi: $o] :
% 0.32/0.49 ( Xphi
% 0.32/0.49 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(notdexE_type,type,
% 0.32/0.49 notdexE: $o ).
% 0.32/0.49
% 0.32/0.49 thf(notdexE,definition,
% 0.32/0.49 ( notdexE
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.49 ( ~ ? [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 & ( Xphi @ Xx ) )
% 0.32/0.49 => ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(notdallE_type,type,
% 0.32/0.49 notdallE: $o ).
% 0.32/0.49
% 0.32/0.49 thf(notdallE,definition,
% 0.32/0.49 ( notdallE
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.49 ( ~ ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( Xphi @ Xx ) )
% 0.32/0.49 => ? [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(exuI1_type,type,
% 0.32/0.49 exuI1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(exuI1,definition,
% 0.32/0.49 ( exuI1
% 0.32/0.49 = ( ! [Xphi: $i > $o] :
% 0.32/0.49 ( ? [Xx: $i] :
% 0.32/0.49 ( ( Xphi @ Xx )
% 0.32/0.49 & ! [Xy: $i] :
% 0.32/0.49 ( ( Xphi @ Xy )
% 0.32/0.49 => ( Xx = Xy ) ) )
% 0.32/0.49 => ( exu
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(exuI3_type,type,
% 0.32/0.49 exuI3: $o ).
% 0.32/0.49
% 0.32/0.49 thf(exuI3,definition,
% 0.32/0.49 ( exuI3
% 0.32/0.49 = ( ! [Xphi: $i > $o] :
% 0.32/0.49 ( ? [Xx: $i] : ( Xphi @ Xx )
% 0.32/0.49 => ( ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( Xphi @ Xx )
% 0.32/0.49 => ( ( Xphi @ Xy )
% 0.32/0.49 => ( Xx = Xy ) ) )
% 0.32/0.49 => ( exu
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(exuI2_type,type,
% 0.32/0.49 exuI2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(exuI2,definition,
% 0.32/0.49 ( exuI2
% 0.32/0.49 = ( ! [Xphi: $i > $o] :
% 0.32/0.49 ( ? [Xx: $i] :
% 0.32/0.49 ! [Xy: $i] :
% 0.32/0.49 ( ( Xphi @ Xy )
% 0.32/0.49 <=> ( Xy = Xx ) )
% 0.32/0.49 => ( exu
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(inCongP_type,type,
% 0.32/0.49 inCongP: $o ).
% 0.32/0.49
% 0.32/0.49 thf(inCongP,definition,
% 0.32/0.49 ( inCongP
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( A = B )
% 0.32/0.49 => ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( Xx = Xy )
% 0.32/0.49 => ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(in__Cong_type,type,
% 0.32/0.49 in__Cong: $o ).
% 0.32/0.49
% 0.32/0.49 thf(in__Cong,definition,
% 0.32/0.49 ( in__Cong
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( A = B )
% 0.32/0.49 => ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( Xx = Xy )
% 0.32/0.49 => ( ( in @ Xx @ A )
% 0.32/0.49 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(exuE3u_type,type,
% 0.32/0.49 exuE3u: $o ).
% 0.32/0.49
% 0.32/0.49 thf(exuE3u,definition,
% 0.32/0.49 ( exuE3u
% 0.32/0.49 = ( ! [Xphi: $i > $o] :
% 0.32/0.49 ( ( exu
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.49 => ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( Xphi @ Xx )
% 0.32/0.49 => ( ( Xphi @ Xy )
% 0.32/0.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(exu__Cong_type,type,
% 0.32/0.49 exu__Cong: $o ).
% 0.32/0.49
% 0.32/0.49 thf(exu__Cong,definition,
% 0.32/0.49 ( exu__Cong
% 0.32/0.49 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/0.49 ( ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( Xx = Xy )
% 0.32/0.49 => ( ( Xphi @ Xx )
% 0.32/0.49 <=> ( Xpsi @ Xy ) ) )
% 0.32/0.49 => ( ( exu
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.49 <=> ( exu
% 0.32/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(emptyset__Cong_type,type,
% 0.32/0.49 emptyset__Cong: $o ).
% 0.32/0.49
% 0.32/0.49 thf(emptyset__Cong,definition,
% 0.32/0.49 ( emptyset__Cong
% 0.32/0.49 = ( emptyset = emptyset ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoin__Cong_type,type,
% 0.32/0.49 setadjoin__Cong: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoin__Cong,definition,
% 0.32/0.49 ( setadjoin__Cong
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( Xx = Xy )
% 0.32/0.49 => ! [Xz: $i,Xu: $i] :
% 0.32/0.49 ( ( Xz = Xu )
% 0.32/0.49 => ( ( setadjoin @ Xx @ Xz )
% 0.32/0.49 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(powerset__Cong_type,type,
% 0.32/0.49 powerset__Cong: $o ).
% 0.32/0.49
% 0.32/0.49 thf(powerset__Cong,definition,
% 0.32/0.49 ( powerset__Cong
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( A = B )
% 0.32/0.49 => ( ( powerset @ A )
% 0.32/0.49 = ( powerset @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setunion__Cong_type,type,
% 0.32/0.49 setunion__Cong: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setunion__Cong,definition,
% 0.32/0.49 ( setunion__Cong
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( A = B )
% 0.32/0.49 => ( ( setunion @ A )
% 0.32/0.49 = ( setunion @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(omega__Cong_type,type,
% 0.32/0.49 omega__Cong: $o ).
% 0.32/0.49
% 0.32/0.49 thf(omega__Cong,definition,
% 0.32/0.49 ( omega__Cong
% 0.32/0.49 = ( omega = omega ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(exuEu_type,type,
% 0.32/0.49 exuEu: $o ).
% 0.32/0.49
% 0.32/0.49 thf(exuEu,definition,
% 0.32/0.49 ( exuEu
% 0.32/0.49 = ( ! [Xphi: $i > $o] :
% 0.32/0.49 ( ( exu
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.49 => ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( Xphi @ Xx )
% 0.32/0.49 => ( ( Xphi @ Xy )
% 0.32/0.49 => ( Xx = Xy ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(descr__Cong_type,type,
% 0.32/0.49 descr__Cong: $o ).
% 0.32/0.49
% 0.32/0.49 thf(descr__Cong,definition,
% 0.32/0.49 ( descr__Cong
% 0.32/0.49 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/0.49 ( ! [Xx: $i,Xy: $i] :
% 0.32/0.49 ( ( Xx = Xy )
% 0.32/0.49 => ( ( Xphi @ Xx )
% 0.32/0.49 <=> ( Xpsi @ Xy ) ) )
% 0.32/0.49 => ( ( exu
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.49 => ( ( exu
% 0.32/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 0.32/0.49 => ( ( descr
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.49 = ( descr
% 0.32/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(dsetconstr__Cong_type,type,
% 0.32/0.49 dsetconstr__Cong: $o ).
% 0.32/0.49
% 0.32/0.49 thf(dsetconstr__Cong,definition,
% 0.32/0.49 ( dsetconstr__Cong
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( A = B )
% 0.32/0.49 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( ( Xx = Xy )
% 0.32/0.49 => ( ( Xphi @ Xx )
% 0.32/0.49 <=> ( Xpsi @ Xy ) ) ) ) )
% 0.32/0.49 => ( ( dsetconstr @ A
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.49 = ( dsetconstr @ B
% 0.32/0.49 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(subset_type,type,
% 0.32/0.49 subset: $i > $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(disjoint_type,type,
% 0.32/0.49 disjoint: $i > $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(setsmeet_type,type,
% 0.32/0.49 setsmeet: $i > $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(subsetI1_type,type,
% 0.32/0.49 subsetI1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(subsetI1,definition,
% 0.32/0.49 ( subsetI1
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ Xx @ B ) )
% 0.32/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(eqimpsubset2_type,type,
% 0.32/0.49 eqimpsubset2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(eqimpsubset2,definition,
% 0.32/0.49 ( eqimpsubset2
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( A = B )
% 0.32/0.49 => ( subset @ B @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(eqimpsubset1_type,type,
% 0.32/0.49 eqimpsubset1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(eqimpsubset1,definition,
% 0.32/0.49 ( eqimpsubset1
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( A = B )
% 0.32/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(subsetI2_type,type,
% 0.32/0.49 subsetI2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(subsetI2,definition,
% 0.32/0.49 ( subsetI2
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ Xx @ B ) )
% 0.32/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(emptysetsubset_type,type,
% 0.32/0.49 emptysetsubset: $o ).
% 0.32/0.49
% 0.32/0.49 thf(emptysetsubset,definition,
% 0.32/0.49 ( emptysetsubset
% 0.32/0.49 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(subsetE_type,type,
% 0.32/0.49 subsetE: $o ).
% 0.32/0.49
% 0.32/0.49 thf(subsetE,definition,
% 0.32/0.49 ( subsetE
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( subset @ A @ B )
% 0.32/0.49 => ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(subsetE2_type,type,
% 0.32/0.49 subsetE2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(subsetE2,definition,
% 0.32/0.49 ( subsetE2
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( subset @ A @ B )
% 0.32/0.49 => ( ~ ( in @ Xx @ B )
% 0.32/0.49 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(notsubsetI_type,type,
% 0.32/0.49 notsubsetI: $o ).
% 0.32/0.49
% 0.32/0.49 thf(notsubsetI,definition,
% 0.32/0.49 ( notsubsetI
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ~ ( in @ Xx @ B )
% 0.32/0.49 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(notequalI1_type,type,
% 0.32/0.49 notequalI1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(notequalI1,definition,
% 0.32/0.49 ( notequalI1
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ~ ( subset @ A @ B )
% 0.32/0.49 => ( A != B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(notequalI2_type,type,
% 0.32/0.49 notequalI2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(notequalI2,definition,
% 0.32/0.49 ( notequalI2
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ~ ( in @ Xx @ B )
% 0.32/0.49 => ( A != B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(subsetRefl_type,type,
% 0.32/0.49 subsetRefl: $o ).
% 0.32/0.49
% 0.32/0.49 thf(subsetRefl,definition,
% 0.32/0.49 ( subsetRefl
% 0.32/0.49 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(subsetTrans_type,type,
% 0.32/0.49 subsetTrans: $o ).
% 0.32/0.49
% 0.32/0.49 thf(subsetTrans,definition,
% 0.32/0.49 ( subsetTrans
% 0.32/0.49 = ( ! [A: $i,B: $i,C: $i] :
% 0.32/0.49 ( ( subset @ A @ B )
% 0.32/0.49 => ( ( subset @ B @ C )
% 0.32/0.49 => ( subset @ A @ C ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoinSub_type,type,
% 0.32/0.49 setadjoinSub: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoinSub,definition,
% 0.32/0.49 ( setadjoinSub
% 0.32/0.49 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoinSub2_type,type,
% 0.32/0.49 setadjoinSub2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setadjoinSub2,definition,
% 0.32/0.49 ( setadjoinSub2
% 0.32/0.49 = ( ! [A: $i,Xx: $i,B: $i] :
% 0.32/0.49 ( ( subset @ A @ B )
% 0.32/0.49 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(subset2powerset_type,type,
% 0.32/0.49 subset2powerset: $o ).
% 0.32/0.49
% 0.32/0.49 thf(subset2powerset,definition,
% 0.32/0.49 ( subset2powerset
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( subset @ A @ B )
% 0.32/0.49 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setextsub_type,type,
% 0.32/0.49 setextsub: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setextsub,definition,
% 0.32/0.49 ( setextsub
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( subset @ A @ B )
% 0.32/0.49 => ( ( subset @ B @ A )
% 0.32/0.49 => ( A = B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(subsetemptysetimpeq_type,type,
% 0.32/0.49 subsetemptysetimpeq: $o ).
% 0.32/0.49
% 0.32/0.49 thf(subsetemptysetimpeq,definition,
% 0.32/0.49 ( subsetemptysetimpeq
% 0.32/0.49 = ( ! [A: $i] :
% 0.32/0.49 ( ( subset @ A @ emptyset )
% 0.32/0.49 => ( A = emptyset ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(powersetI1_type,type,
% 0.32/0.49 powersetI1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(powersetI1,definition,
% 0.32/0.49 ( powersetI1
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( subset @ B @ A )
% 0.32/0.49 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(powersetE1_type,type,
% 0.32/0.49 powersetE1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(powersetE1,definition,
% 0.32/0.49 ( powersetE1
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( in @ B @ ( powerset @ A ) )
% 0.32/0.49 => ( subset @ B @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(inPowerset_type,type,
% 0.32/0.49 inPowerset: $o ).
% 0.32/0.49
% 0.32/0.49 thf(inPowerset,definition,
% 0.32/0.49 ( inPowerset
% 0.32/0.49 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(powersetsubset_type,type,
% 0.32/0.49 powersetsubset: $o ).
% 0.32/0.49
% 0.32/0.49 thf(powersetsubset,definition,
% 0.32/0.49 ( powersetsubset
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( subset @ A @ B )
% 0.32/0.49 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(sepInPowerset_type,type,
% 0.32/0.49 sepInPowerset: $o ).
% 0.32/0.49
% 0.32/0.49 thf(sepInPowerset,definition,
% 0.32/0.49 ( sepInPowerset
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.49 ( in
% 0.32/0.49 @ ( dsetconstr @ A
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.49 @ ( powerset @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(sepSubset_type,type,
% 0.32/0.49 sepSubset: $o ).
% 0.32/0.49
% 0.32/0.49 thf(sepSubset,definition,
% 0.32/0.49 ( sepSubset
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.49 ( subset
% 0.32/0.49 @ ( dsetconstr @ A
% 0.32/0.49 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.49 @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binunion_type,type,
% 0.32/0.49 binunion: $i > $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(binunionIL_type,type,
% 0.32/0.49 binunionIL: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binunionIL,definition,
% 0.32/0.49 ( binunionIL
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(upairset2IR_type,type,
% 0.32/0.49 upairset2IR: $o ).
% 0.32/0.49
% 0.32/0.49 thf(upairset2IR,definition,
% 0.32/0.49 ( upairset2IR
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binunionIR_type,type,
% 0.32/0.49 binunionIR: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binunionIR,definition,
% 0.32/0.49 ( binunionIR
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ B )
% 0.32/0.49 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binunionEcases_type,type,
% 0.32/0.49 binunionEcases: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binunionEcases,definition,
% 0.32/0.49 ( binunionEcases
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 0.32/0.49 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.32/0.49 => ( ( ( in @ Xx @ A )
% 0.32/0.49 => Xphi )
% 0.32/0.49 => ( ( ( in @ Xx @ B )
% 0.32/0.49 => Xphi )
% 0.32/0.49 => Xphi ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binunionE_type,type,
% 0.32/0.49 binunionE: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binunionE,definition,
% 0.32/0.49 ( binunionE
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 0.32/0.49 => ( ( in @ Xx @ A )
% 0.32/0.49 | ( in @ Xx @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binunionLsub_type,type,
% 0.32/0.49 binunionLsub: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binunionLsub,definition,
% 0.32/0.49 ( binunionLsub
% 0.32/0.49 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binunionRsub_type,type,
% 0.32/0.49 binunionRsub: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binunionRsub,definition,
% 0.32/0.49 ( binunionRsub
% 0.32/0.49 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binintersect_type,type,
% 0.32/0.49 binintersect: $i > $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectI_type,type,
% 0.32/0.49 binintersectI: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectI,definition,
% 0.32/0.49 ( binintersectI
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ B )
% 0.32/0.49 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectSubset5_type,type,
% 0.32/0.49 binintersectSubset5: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectSubset5,definition,
% 0.32/0.49 ( binintersectSubset5
% 0.32/0.49 = ( ! [A: $i,B: $i,C: $i] :
% 0.32/0.49 ( ( subset @ C @ A )
% 0.32/0.49 => ( ( subset @ C @ B )
% 0.32/0.49 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectEL_type,type,
% 0.32/0.49 binintersectEL: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectEL,definition,
% 0.32/0.49 ( binintersectEL
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.32/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectLsub_type,type,
% 0.32/0.49 binintersectLsub: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectLsub,definition,
% 0.32/0.49 ( binintersectLsub
% 0.32/0.49 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectSubset2_type,type,
% 0.32/0.49 binintersectSubset2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectSubset2,definition,
% 0.32/0.49 ( binintersectSubset2
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( subset @ A @ B )
% 0.32/0.49 => ( ( binintersect @ A @ B )
% 0.32/0.49 = A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectSubset3_type,type,
% 0.32/0.49 binintersectSubset3: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectSubset3,definition,
% 0.32/0.49 ( binintersectSubset3
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( ( binintersect @ A @ B )
% 0.32/0.49 = B )
% 0.32/0.49 => ( subset @ B @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectER_type,type,
% 0.32/0.49 binintersectER: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectER,definition,
% 0.32/0.49 ( binintersectER
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 0.32/0.49 => ( in @ Xx @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(disjointsetsI1_type,type,
% 0.32/0.49 disjointsetsI1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(disjointsetsI1,definition,
% 0.32/0.49 ( disjointsetsI1
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ~ ? [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 & ( in @ Xx @ B ) )
% 0.32/0.49 => ( ( binintersect @ A @ B )
% 0.32/0.49 = emptyset ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectRsub_type,type,
% 0.32/0.49 binintersectRsub: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectRsub,definition,
% 0.32/0.49 ( binintersectRsub
% 0.32/0.49 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectSubset4_type,type,
% 0.32/0.49 binintersectSubset4: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectSubset4,definition,
% 0.32/0.49 ( binintersectSubset4
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( subset @ B @ A )
% 0.32/0.49 => ( ( binintersect @ A @ B )
% 0.32/0.49 = B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectSubset1_type,type,
% 0.32/0.49 binintersectSubset1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(binintersectSubset1,definition,
% 0.32/0.49 ( binintersectSubset1
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( ( binintersect @ A @ B )
% 0.32/0.49 = A )
% 0.32/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(bs114d_type,type,
% 0.32/0.49 bs114d: $o ).
% 0.32/0.49
% 0.32/0.49 thf(bs114d,definition,
% 0.32/0.49 ( bs114d
% 0.32/0.49 = ( ! [A: $i,B: $i,C: $i] :
% 0.32/0.49 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 0.32/0.49 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(regular_type,type,
% 0.32/0.49 regular: $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(setminus_type,type,
% 0.32/0.49 setminus: $i > $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(setminusI_type,type,
% 0.32/0.49 setminusI: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setminusI,definition,
% 0.32/0.49 ( setminusI
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ~ ( in @ Xx @ B )
% 0.32/0.49 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setminusEL_type,type,
% 0.32/0.49 setminusEL: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setminusEL,definition,
% 0.32/0.49 ( setminusEL
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/0.49 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setminusER_type,type,
% 0.32/0.49 setminusER: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setminusER,definition,
% 0.32/0.49 ( setminusER
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/0.49 => ~ ( in @ Xx @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setminusSubset2_type,type,
% 0.32/0.49 setminusSubset2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setminusSubset2,definition,
% 0.32/0.49 ( setminusSubset2
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( subset @ A @ B )
% 0.32/0.49 => ( ( setminus @ A @ B )
% 0.32/0.49 = emptyset ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setminusERneg_type,type,
% 0.32/0.49 setminusERneg: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setminusERneg,definition,
% 0.32/0.49 ( setminusERneg
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/0.49 => ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ Xx @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setminusELneg_type,type,
% 0.32/0.49 setminusELneg: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setminusELneg,definition,
% 0.32/0.49 ( setminusELneg
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 0.32/0.49 => ( ~ ( in @ Xx @ B )
% 0.32/0.49 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setminusILneg_type,type,
% 0.32/0.49 setminusILneg: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setminusILneg,definition,
% 0.32/0.49 ( setminusILneg
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ~ ( in @ Xx @ A )
% 0.32/0.49 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setminusIRneg_type,type,
% 0.32/0.49 setminusIRneg: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setminusIRneg,definition,
% 0.32/0.49 ( setminusIRneg
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ B )
% 0.32/0.49 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setminusLsub_type,type,
% 0.32/0.49 setminusLsub: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setminusLsub,definition,
% 0.32/0.49 ( setminusLsub
% 0.32/0.49 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setminusSubset1_type,type,
% 0.32/0.49 setminusSubset1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setminusSubset1,definition,
% 0.32/0.49 ( setminusSubset1
% 0.32/0.49 = ( ! [A: $i,B: $i] :
% 0.32/0.49 ( ( ( setminus @ A @ B )
% 0.32/0.49 = emptyset )
% 0.32/0.49 => ( subset @ A @ B ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(symdiff_type,type,
% 0.32/0.49 symdiff: $i > $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(symdiffE_type,type,
% 0.32/0.49 symdiffE: $o ).
% 0.32/0.49
% 0.32/0.49 thf(symdiffE,definition,
% 0.32/0.49 ( symdiffE
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 0.32/0.49 => ! [Xphi: $o] :
% 0.32/0.49 ( ( ( in @ Xx @ A )
% 0.32/0.49 => ( ~ ( in @ Xx @ B )
% 0.32/0.49 => Xphi ) )
% 0.32/0.49 => ( ( ~ ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ B )
% 0.32/0.49 => Xphi ) )
% 0.32/0.49 => Xphi ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(symdiffI1_type,type,
% 0.32/0.49 symdiffI1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(symdiffI1,definition,
% 0.32/0.49 ( symdiffI1
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ~ ( in @ Xx @ B )
% 0.32/0.49 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(symdiffI2_type,type,
% 0.32/0.49 symdiffI2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(symdiffI2,definition,
% 0.32/0.49 ( symdiffI2
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ~ ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ B )
% 0.32/0.49 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(symdiffIneg1_type,type,
% 0.32/0.49 symdiffIneg1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(symdiffIneg1,definition,
% 0.32/0.49 ( symdiffIneg1
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( ( in @ Xx @ B )
% 0.32/0.49 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(symdiffIneg2_type,type,
% 0.32/0.49 symdiffIneg2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(symdiffIneg2,definition,
% 0.32/0.49 ( symdiffIneg2
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ~ ( in @ Xx @ A )
% 0.32/0.49 => ( ~ ( in @ Xx @ B )
% 0.32/0.49 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(iskpair_type,type,
% 0.32/0.49 iskpair: $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(secondinupair_type,type,
% 0.32/0.49 secondinupair: $o ).
% 0.32/0.49
% 0.32/0.49 thf(secondinupair,definition,
% 0.32/0.49 ( secondinupair
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairIL_type,type,
% 0.32/0.49 setukpairIL: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairIL,definition,
% 0.32/0.49 ( setukpairIL
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairIR_type,type,
% 0.32/0.49 setukpairIR: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setukpairIR,definition,
% 0.32/0.49 ( setukpairIR
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(kpairiskpair_type,type,
% 0.32/0.49 kpairiskpair: $o ).
% 0.32/0.49
% 0.32/0.49 thf(kpairiskpair,definition,
% 0.32/0.49 ( kpairiskpair
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(kpair_type,type,
% 0.32/0.49 kpair: $i > $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(kpairp_type,type,
% 0.32/0.49 kpairp: $o ).
% 0.32/0.49
% 0.32/0.49 thf(kpairp,definition,
% 0.32/0.49 ( kpairp
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(cartprod_type,type,
% 0.32/0.49 cartprod: $i > $i > $i ).
% 0.32/0.49
% 0.32/0.49 thf(singletonsubset_type,type,
% 0.32/0.49 singletonsubset: $o ).
% 0.32/0.49
% 0.32/0.49 thf(singletonsubset,definition,
% 0.32/0.49 ( singletonsubset
% 0.32/0.49 = ( ! [A: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(singletoninpowerset_type,type,
% 0.32/0.49 singletoninpowerset: $o ).
% 0.32/0.49
% 0.32/0.49 thf(singletoninpowerset,definition,
% 0.32/0.49 ( singletoninpowerset
% 0.32/0.49 = ( ! [A: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(singletoninpowunion_type,type,
% 0.32/0.49 singletoninpowunion: $o ).
% 0.32/0.49
% 0.32/0.49 thf(singletoninpowunion,definition,
% 0.32/0.49 ( singletoninpowunion
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(upairset2E_type,type,
% 0.32/0.49 upairset2E: $o ).
% 0.32/0.49
% 0.32/0.49 thf(upairset2E,definition,
% 0.32/0.49 ( upairset2E
% 0.32/0.49 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.49 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 0.32/0.49 => ( ( Xz = Xx )
% 0.32/0.49 | ( Xz = Xy ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(upairsubunion_type,type,
% 0.32/0.49 upairsubunion: $o ).
% 0.32/0.49
% 0.32/0.49 thf(upairsubunion,definition,
% 0.32/0.49 ( upairsubunion
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(upairinpowunion_type,type,
% 0.32/0.49 upairinpowunion: $o ).
% 0.32/0.49
% 0.32/0.49 thf(upairinpowunion,definition,
% 0.32/0.49 ( upairinpowunion
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ubforcartprodlem1_type,type,
% 0.32/0.49 ubforcartprodlem1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ubforcartprodlem1,definition,
% 0.32/0.49 ( ubforcartprodlem1
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ubforcartprodlem2_type,type,
% 0.32/0.49 ubforcartprodlem2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ubforcartprodlem2,definition,
% 0.32/0.49 ( ubforcartprodlem2
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(ubforcartprodlem3_type,type,
% 0.32/0.49 ubforcartprodlem3: $o ).
% 0.32/0.49
% 0.32/0.49 thf(ubforcartprodlem3,definition,
% 0.32/0.49 ( ubforcartprodlem3
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodpairin_type,type,
% 0.32/0.49 cartprodpairin: $o ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodpairin,definition,
% 0.32/0.49 ( cartprodpairin
% 0.32/0.49 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodmempair1_type,type,
% 0.32/0.49 cartprodmempair1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodmempair1,definition,
% 0.32/0.49 ( cartprodmempair1
% 0.32/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.49 => ? [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 & ? [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ B )
% 0.32/0.49 & ( Xu
% 0.32/0.49 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodmempair_type,type,
% 0.32/0.49 cartprodmempair: $o ).
% 0.32/0.49
% 0.32/0.49 thf(cartprodmempair,definition,
% 0.32/0.49 ( cartprodmempair
% 0.32/0.49 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.49 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.49 => ( iskpair @ Xu ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setunionE2_type,type,
% 0.32/0.49 setunionE2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setunionE2,definition,
% 0.32/0.49 ( setunionE2
% 0.32/0.49 = ( ! [A: $i,Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ ( setunion @ A ) )
% 0.32/0.49 => ? [X: $i] :
% 0.32/0.49 ( ( in @ X @ A )
% 0.32/0.49 & ( in @ Xx @ X ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setunionsingleton1_type,type,
% 0.32/0.49 setunionsingleton1: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setunionsingleton1,definition,
% 0.32/0.49 ( setunionsingleton1
% 0.32/0.49 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setunionsingleton2_type,type,
% 0.32/0.49 setunionsingleton2: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setunionsingleton2,definition,
% 0.32/0.49 ( setunionsingleton2
% 0.32/0.49 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(setunionsingleton_type,type,
% 0.32/0.49 setunionsingleton: $o ).
% 0.32/0.49
% 0.32/0.49 thf(setunionsingleton,definition,
% 0.32/0.49 ( setunionsingleton
% 0.32/0.49 = ( ! [Xx: $i] :
% 0.32/0.49 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 0.32/0.49 = Xx ) ) ) ).
% 0.32/0.49
% 0.32/0.49 thf(singleton_type,type,
% 0.32/0.49 singleton: $i > $o ).
% 0.32/0.49
% 0.32/0.49 thf(singletonprop_type,type,
% 0.32/0.49 singletonprop: $o ).
% 0.32/0.49
% 0.32/0.49 thf(singletonprop,definition,
% 0.32/0.49 ( singletonprop
% 0.32/0.49 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.49 ( ! [Xx: $i] :
% 0.32/0.49 ( ( in @ Xx @ A )
% 0.32/0.49 => ! [Xy: $i] :
% 0.32/0.49 ( ( in @ Xy @ A )
% 0.32/0.49 => ( ( Xphi @ Xx )
% 0.32/0.49 => ( ( Xphi @ Xy )
% 0.32/0.49 => ( Xx = Xy ) ) ) ) )
% 0.32/0.50 => ( ? [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 & ( Xphi @ Xx ) )
% 0.32/0.50 => ( singleton
% 0.32/0.50 @ ( dsetconstr @ A
% 0.32/0.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(ex1_type,type,
% 0.32/0.50 ex1: $i > ( $i > $o ) > $o ).
% 0.32/0.50
% 0.32/0.50 thf(ex1,definition,
% 0.32/0.50 ( ex1
% 0.32/0.50 = ( ^ [A: $i,Xphi: $i > $o] :
% 0.32/0.50 ( singleton
% 0.32/0.50 @ ( dsetconstr @ A
% 0.32/0.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(ex1E1_type,type,
% 0.32/0.50 ex1E1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(ex1E1,definition,
% 0.32/0.50 ( ex1E1
% 0.32/0.50 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.50 ( ( ex1 @ A
% 0.32/0.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.50 => ? [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 & ( Xphi @ Xx ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(ex1I_type,type,
% 0.32/0.50 ex1I: $o ).
% 0.32/0.50
% 0.32/0.50 thf(ex1I,definition,
% 0.32/0.50 ( ex1I
% 0.32/0.50 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( Xphi @ Xx )
% 0.32/0.50 => ( ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ A )
% 0.32/0.50 => ( ( Xphi @ Xy )
% 0.32/0.50 => ( Xy = Xx ) ) )
% 0.32/0.50 => ( ex1 @ A
% 0.32/0.50 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(ex1I2_type,type,
% 0.32/0.50 ex1I2: $o ).
% 0.32/0.50
% 0.32/0.50 thf(ex1I2,definition,
% 0.32/0.50 ( ex1I2
% 0.32/0.50 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.50 ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ A )
% 0.32/0.50 => ( ( Xphi @ Xx )
% 0.32/0.50 => ( ( Xphi @ Xy )
% 0.32/0.50 => ( Xx = Xy ) ) ) ) )
% 0.32/0.50 => ( ? [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 & ( Xphi @ Xx ) )
% 0.32/0.50 => ( ex1 @ A
% 0.32/0.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(singletonsuniq_type,type,
% 0.32/0.50 singletonsuniq: $o ).
% 0.32/0.50
% 0.32/0.50 thf(singletonsuniq,definition,
% 0.32/0.50 ( singletonsuniq
% 0.32/0.50 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.50 ( ( ( setadjoin @ Xx @ emptyset )
% 0.32/0.50 = ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.50 => ( Xx = Xy ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(atmost1p_type,type,
% 0.32/0.50 atmost1p: $i > $o ).
% 0.32/0.50
% 0.32/0.50 thf(atleast2p_type,type,
% 0.32/0.50 atleast2p: $i > $o ).
% 0.32/0.50
% 0.32/0.50 thf(atmost2p_type,type,
% 0.32/0.50 atmost2p: $i > $o ).
% 0.32/0.50
% 0.32/0.50 thf(upairsetp_type,type,
% 0.32/0.50 upairsetp: $i > $o ).
% 0.32/0.50
% 0.32/0.50 thf(setukpairinjL1_type,type,
% 0.32/0.50 setukpairinjL1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(setukpairinjL1,definition,
% 0.32/0.50 ( setukpairinjL1
% 0.32/0.50 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.50 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 0.32/0.50 => ( Xx = Xz ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(kfstsingleton_type,type,
% 0.32/0.50 kfstsingleton: $o ).
% 0.32/0.50
% 0.32/0.50 thf(kfstsingleton,definition,
% 0.32/0.50 ( kfstsingleton
% 0.32/0.50 = ( ! [Xu: $i] :
% 0.32/0.50 ( ( iskpair @ Xu )
% 0.32/0.50 => ( singleton
% 0.32/0.50 @ ( dsetconstr @ ( setunion @ Xu )
% 0.32/0.50 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(theprop_type,type,
% 0.32/0.50 theprop: $o ).
% 0.32/0.50
% 0.32/0.50 thf(theprop,definition,
% 0.32/0.50 ( theprop
% 0.32/0.50 = ( ! [X: $i] :
% 0.32/0.50 ( ( singleton @ X )
% 0.32/0.50 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(kfst_type,type,
% 0.32/0.50 kfst: $i > $i ).
% 0.32/0.50
% 0.32/0.50 thf(kfstpairEq_type,type,
% 0.32/0.50 kfstpairEq: $o ).
% 0.32/0.50
% 0.32/0.50 thf(kfstpairEq,definition,
% 0.32/0.50 ( kfstpairEq
% 0.32/0.50 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.50 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.32/0.50 = Xx ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(cartprodfstin_type,type,
% 0.32/0.50 cartprodfstin: $o ).
% 0.32/0.50
% 0.32/0.50 thf(cartprodfstin,definition,
% 0.32/0.50 ( cartprodfstin
% 0.32/0.50 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.50 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.50 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(setukpairinjL2_type,type,
% 0.32/0.50 setukpairinjL2: $o ).
% 0.32/0.50
% 0.32/0.50 thf(setukpairinjL2,definition,
% 0.32/0.50 ( setukpairinjL2
% 0.32/0.50 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.50 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/0.50 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.32/0.50 => ( Xx = Xz ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(setukpairinjL_type,type,
% 0.32/0.50 setukpairinjL: $o ).
% 0.32/0.50
% 0.32/0.50 thf(setukpairinjL,definition,
% 0.32/0.50 ( setukpairinjL
% 0.32/0.50 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.50 ( ( ( kpair @ Xx @ Xy )
% 0.32/0.50 = ( kpair @ Xz @ Xu ) )
% 0.32/0.50 => ( Xx = Xz ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(setukpairinjR11_type,type,
% 0.32/0.50 setukpairinjR11: $o ).
% 0.32/0.50
% 0.32/0.50 thf(setukpairinjR11,definition,
% 0.32/0.50 ( setukpairinjR11
% 0.32/0.50 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.50 ( ( Xx = Xy )
% 0.32/0.50 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.50 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(setukpairinjR12_type,type,
% 0.32/0.50 setukpairinjR12: $o ).
% 0.32/0.50
% 0.32/0.50 thf(setukpairinjR12,definition,
% 0.32/0.50 ( setukpairinjR12
% 0.32/0.50 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.50 ( ( Xx = Xy )
% 0.32/0.50 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/0.50 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(setukpairinjR1_type,type,
% 0.32/0.50 setukpairinjR1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(setukpairinjR1,definition,
% 0.32/0.50 ( setukpairinjR1
% 0.32/0.50 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.50 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/0.50 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.32/0.50 => ( ( Xz = Xu )
% 0.32/0.50 => ( Xy = Xu ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(upairequniteq_type,type,
% 0.32/0.50 upairequniteq: $o ).
% 0.32/0.50
% 0.32/0.50 thf(upairequniteq,definition,
% 0.32/0.50 ( upairequniteq
% 0.32/0.50 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 0.32/0.50 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 0.32/0.50 = ( setadjoin @ Xz @ emptyset ) )
% 0.32/0.50 => ( Xx = Xy ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(setukpairinjR2_type,type,
% 0.32/0.50 setukpairinjR2: $o ).
% 0.32/0.50
% 0.32/0.50 thf(setukpairinjR2,definition,
% 0.32/0.50 ( setukpairinjR2
% 0.32/0.50 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.50 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 0.32/0.50 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 0.32/0.50 => ( Xy = Xu ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(setukpairinjR_type,type,
% 0.32/0.50 setukpairinjR: $o ).
% 0.32/0.50
% 0.32/0.50 thf(setukpairinjR,definition,
% 0.32/0.50 ( setukpairinjR
% 0.32/0.50 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 0.32/0.50 ( ( ( kpair @ Xx @ Xy )
% 0.32/0.50 = ( kpair @ Xz @ Xu ) )
% 0.32/0.50 => ( Xy = Xu ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(ksndsingleton_type,type,
% 0.32/0.50 ksndsingleton: $o ).
% 0.32/0.50
% 0.32/0.50 thf(ksndsingleton,definition,
% 0.32/0.50 ( ksndsingleton
% 0.32/0.50 = ( ! [Xu: $i] :
% 0.32/0.50 ( ( iskpair @ Xu )
% 0.32/0.50 => ( singleton
% 0.32/0.50 @ ( dsetconstr @ ( setunion @ Xu )
% 0.32/0.50 @ ^ [Xx: $i] :
% 0.32/0.50 ( Xu
% 0.32/0.50 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(ksnd_type,type,
% 0.32/0.50 ksnd: $i > $i ).
% 0.32/0.50
% 0.32/0.50 thf(ksndpairEq_type,type,
% 0.32/0.50 ksndpairEq: $o ).
% 0.32/0.50
% 0.32/0.50 thf(ksndpairEq,definition,
% 0.32/0.50 ( ksndpairEq
% 0.32/0.50 = ( ! [Xx: $i,Xy: $i] :
% 0.32/0.50 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.32/0.50 = Xy ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(kpairsurjEq_type,type,
% 0.32/0.50 kpairsurjEq: $o ).
% 0.32/0.50
% 0.32/0.50 thf(kpairsurjEq,definition,
% 0.32/0.50 ( kpairsurjEq
% 0.32/0.50 = ( ! [Xu: $i] :
% 0.32/0.50 ( ( iskpair @ Xu )
% 0.32/0.50 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.32/0.50 = Xu ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(cartprodsndin_type,type,
% 0.32/0.50 cartprodsndin: $o ).
% 0.32/0.50
% 0.32/0.50 thf(cartprodsndin,definition,
% 0.32/0.50 ( cartprodsndin
% 0.32/0.50 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.50 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.50 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(cartprodpairmemEL_type,type,
% 0.32/0.50 cartprodpairmemEL: $o ).
% 0.32/0.50
% 0.32/0.50 thf(cartprodpairmemEL,definition,
% 0.32/0.50 ( cartprodpairmemEL
% 0.32/0.50 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.32/0.50 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.32/0.50 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(cartprodpairmemER_type,type,
% 0.32/0.50 cartprodpairmemER: $o ).
% 0.32/0.50
% 0.32/0.50 thf(cartprodpairmemER,definition,
% 0.32/0.50 ( cartprodpairmemER
% 0.32/0.50 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 0.32/0.50 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 0.32/0.50 => ( in @ Xy @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(cartprodmempaircEq_type,type,
% 0.32/0.50 cartprodmempaircEq: $o ).
% 0.32/0.50
% 0.32/0.50 thf(cartprodmempaircEq,definition,
% 0.32/0.50 ( cartprodmempaircEq
% 0.32/0.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ B )
% 0.32/0.50 => ( ( kpair @ Xx @ Xy )
% 0.32/0.50 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(cartprodfstpairEq_type,type,
% 0.32/0.50 cartprodfstpairEq: $o ).
% 0.32/0.50
% 0.32/0.50 thf(cartprodfstpairEq,definition,
% 0.32/0.50 ( cartprodfstpairEq
% 0.32/0.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ B )
% 0.32/0.50 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 0.32/0.50 = Xx ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(cartprodsndpairEq_type,type,
% 0.32/0.50 cartprodsndpairEq: $o ).
% 0.32/0.50
% 0.32/0.50 thf(cartprodsndpairEq,definition,
% 0.32/0.50 ( cartprodsndpairEq
% 0.32/0.50 = ( ! [A: $i,B: $i,Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ B )
% 0.32/0.50 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 0.32/0.50 = Xy ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(cartprodpairsurjEq_type,type,
% 0.32/0.50 cartprodpairsurjEq: $o ).
% 0.32/0.50
% 0.32/0.50 thf(cartprodpairsurjEq,definition,
% 0.32/0.50 ( cartprodpairsurjEq
% 0.32/0.50 = ( ! [A: $i,B: $i,Xu: $i] :
% 0.32/0.50 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 0.32/0.50 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 0.32/0.50 = Xu ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(breln_type,type,
% 0.32/0.50 breln: $i > $i > $i > $o ).
% 0.32/0.50
% 0.32/0.50 thf(breln,definition,
% 0.32/0.50 ( breln
% 0.32/0.50 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(dpsetconstr_type,type,
% 0.32/0.50 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 0.32/0.50
% 0.32/0.50 thf(dpsetconstrI_type,type,
% 0.32/0.50 dpsetconstrI: $o ).
% 0.32/0.50
% 0.32/0.50 thf(dpsetconstrI,definition,
% 0.32/0.50 ( dpsetconstrI
% 0.32/0.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ B )
% 0.32/0.50 => ( ( Xphi @ Xx @ Xy )
% 0.32/0.50 => ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.50 @ ( dpsetconstr @ A @ B
% 0.32/0.50 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(dpsetconstrSub_type,type,
% 0.32/0.50 dpsetconstrSub: $o ).
% 0.32/0.50
% 0.32/0.50 thf(dpsetconstrSub,definition,
% 0.32/0.50 ( dpsetconstrSub
% 0.32/0.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.32/0.50 ( subset
% 0.32/0.50 @ ( dpsetconstr @ A @ B
% 0.32/0.50 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 0.32/0.50 @ ( cartprod @ A @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(setOfPairsIsBReln_type,type,
% 0.32/0.50 setOfPairsIsBReln: $o ).
% 0.32/0.50
% 0.32/0.50 thf(setOfPairsIsBReln,definition,
% 0.32/0.50 ( setOfPairsIsBReln
% 0.32/0.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 0.32/0.50 ( breln @ A @ B
% 0.32/0.50 @ ( dpsetconstr @ A @ B
% 0.32/0.50 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(dpsetconstrERa_type,type,
% 0.32/0.50 dpsetconstrERa: $o ).
% 0.32/0.50
% 0.32/0.50 thf(dpsetconstrERa,definition,
% 0.32/0.50 ( dpsetconstrERa
% 0.32/0.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ B )
% 0.32/0.50 => ( ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.50 @ ( dpsetconstr @ A @ B
% 0.32/0.50 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.32/0.50 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(dpsetconstrEL1_type,type,
% 0.32/0.50 dpsetconstrEL1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(dpsetconstrEL1,definition,
% 0.32/0.50 ( dpsetconstrEL1
% 0.32/0.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.32/0.50 ( ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.50 @ ( dpsetconstr @ A @ B
% 0.32/0.50 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.32/0.50 => ( in @ Xx @ A ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(dpsetconstrEL2_type,type,
% 0.32/0.50 dpsetconstrEL2: $o ).
% 0.32/0.50
% 0.32/0.50 thf(dpsetconstrEL2,definition,
% 0.32/0.50 ( dpsetconstrEL2
% 0.32/0.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.32/0.50 ( ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.50 @ ( dpsetconstr @ A @ B
% 0.32/0.50 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.32/0.50 => ( in @ Xy @ B ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(dpsetconstrER_type,type,
% 0.32/0.50 dpsetconstrER: $o ).
% 0.32/0.50
% 0.32/0.50 thf(dpsetconstrER,definition,
% 0.32/0.50 ( dpsetconstrER
% 0.32/0.50 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 0.32/0.50 ( ( in @ ( kpair @ Xx @ Xy )
% 0.32/0.50 @ ( dpsetconstr @ A @ B
% 0.32/0.50 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 0.32/0.50 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(func_type,type,
% 0.32/0.50 func: $i > $i > $i > $o ).
% 0.32/0.50
% 0.32/0.50 thf(funcSet_type,type,
% 0.32/0.50 funcSet: $i > $i > $i ).
% 0.32/0.50
% 0.32/0.50 thf(funcImageSingleton_type,type,
% 0.32/0.50 funcImageSingleton: $o ).
% 0.32/0.50
% 0.32/0.50 thf(funcImageSingleton,definition,
% 0.32/0.50 ( funcImageSingleton
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( func @ A @ B @ Xf )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( singleton
% 0.32/0.50 @ ( dsetconstr @ B
% 0.32/0.50 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(apProp_type,type,
% 0.32/0.50 apProp: $o ).
% 0.32/0.50
% 0.32/0.50 thf(apProp,definition,
% 0.32/0.50 ( apProp
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( func @ A @ B @ Xf )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( in
% 0.32/0.50 @ ( setunion
% 0.32/0.50 @ ( dsetconstr @ B
% 0.32/0.50 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 0.32/0.50 @ B ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(ap_type,type,
% 0.32/0.50 ap: $i > $i > $i > $i > $i ).
% 0.32/0.50
% 0.32/0.50 thf(app_type,type,
% 0.32/0.50 app: $o ).
% 0.32/0.50
% 0.32/0.50 thf(app,definition,
% 0.32/0.50 ( app
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( func @ A @ B @ Xf )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(infuncsetfunc_type,type,
% 0.32/0.50 infuncsetfunc: $o ).
% 0.32/0.50
% 0.32/0.50 thf(infuncsetfunc,definition,
% 0.32/0.50 ( infuncsetfunc
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.50 => ( func @ A @ B @ Xf ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(ap2p_type,type,
% 0.32/0.50 ap2p: $o ).
% 0.32/0.50
% 0.32/0.50 thf(ap2p,definition,
% 0.32/0.50 ( ap2p
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(funcinfuncset_type,type,
% 0.32/0.50 funcinfuncset: $o ).
% 0.32/0.50
% 0.32/0.50 thf(funcinfuncset,definition,
% 0.32/0.50 ( funcinfuncset
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( func @ A @ B @ Xf )
% 0.32/0.50 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(lamProp_type,type,
% 0.32/0.50 lamProp: $o ).
% 0.32/0.50
% 0.32/0.50 thf(lamProp,definition,
% 0.32/0.50 ( lamProp
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.50 ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.50 => ( func @ A @ B
% 0.32/0.50 @ ( dpsetconstr @ A @ B
% 0.32/0.50 @ ^ [Xx: $i,Xy: $i] :
% 0.32/0.50 ( ( Xf @ Xx )
% 0.32/0.50 = Xy ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(lam_type,type,
% 0.32/0.50 lam: $i > $i > ( $i > $i ) > $i ).
% 0.32/0.50
% 0.32/0.50 thf(lamp_type,type,
% 0.32/0.50 lamp: $o ).
% 0.32/0.50
% 0.32/0.50 thf(lamp,definition,
% 0.32/0.50 ( lamp
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.50 ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.50 => ( func @ A @ B
% 0.32/0.50 @ ( lam @ A @ B
% 0.32/0.50 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(lam2p_type,type,
% 0.32/0.50 lam2p: $o ).
% 0.32/0.50
% 0.32/0.50 thf(lam2p,definition,
% 0.32/0.50 ( lam2p
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.50 ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.50 => ( in
% 0.32/0.50 @ ( lam @ A @ B
% 0.32/0.50 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.32/0.50 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(brelnall1_type,type,
% 0.32/0.50 brelnall1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(brelnall1,definition,
% 0.32/0.50 ( brelnall1
% 0.32/0.50 = ( ! [A: $i,B: $i,R: $i] :
% 0.32/0.50 ( ( breln @ A @ B @ R )
% 0.32/0.50 => ! [Xphi: $i > $o] :
% 0.32/0.50 ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ B )
% 0.32/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.50 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ R )
% 0.32/0.50 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(brelnall2_type,type,
% 0.32/0.50 brelnall2: $o ).
% 0.32/0.50
% 0.32/0.50 thf(brelnall2,definition,
% 0.32/0.50 ( brelnall2
% 0.32/0.50 = ( ! [A: $i,B: $i,R: $i] :
% 0.32/0.50 ( ( breln @ A @ B @ R )
% 0.32/0.50 => ! [Xphi: $i > $o] :
% 0.32/0.50 ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ B )
% 0.32/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.50 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ R )
% 0.32/0.50 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(ex1E2_type,type,
% 0.32/0.50 ex1E2: $o ).
% 0.32/0.50
% 0.32/0.50 thf(ex1E2,definition,
% 0.32/0.50 ( ex1E2
% 0.32/0.50 = ( ! [A: $i,Xphi: $i > $o] :
% 0.32/0.50 ( ( ex1 @ A
% 0.32/0.50 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ A )
% 0.32/0.50 => ( ( Xphi @ Xx )
% 0.32/0.50 => ( ( Xphi @ Xy )
% 0.32/0.50 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(funcGraphProp1_type,type,
% 0.32/0.50 funcGraphProp1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(funcGraphProp1,definition,
% 0.32/0.50 ( funcGraphProp1
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( func @ A @ B @ Xf )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(funcGraphProp3_type,type,
% 0.32/0.50 funcGraphProp3: $o ).
% 0.32/0.50
% 0.32/0.50 thf(funcGraphProp3,definition,
% 0.32/0.50 ( funcGraphProp3
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(funcGraphProp2_type,type,
% 0.32/0.50 funcGraphProp2: $o ).
% 0.32/0.50
% 0.32/0.50 thf(funcGraphProp2,definition,
% 0.32/0.50 ( funcGraphProp2
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( func @ A @ B @ Xf )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ B )
% 0.32/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.32/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.50 = Xy ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(funcextLem_type,type,
% 0.32/0.50 funcextLem: $o ).
% 0.32/0.50
% 0.32/0.50 thf(funcextLem,definition,
% 0.32/0.50 ( funcextLem
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( func @ A @ B @ Xf )
% 0.32/0.50 => ! [Xg: $i] :
% 0.32/0.50 ( ( func @ A @ B @ Xg )
% 0.32/0.50 => ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.50 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ B )
% 0.32/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 0.32/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(funcGraphProp4_type,type,
% 0.32/0.50 funcGraphProp4: $o ).
% 0.32/0.50
% 0.32/0.50 thf(funcGraphProp4,definition,
% 0.32/0.50 ( funcGraphProp4
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ B )
% 0.32/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 0.32/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.50 = Xy ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(subbreln_type,type,
% 0.32/0.50 subbreln: $o ).
% 0.32/0.50
% 0.32/0.50 thf(subbreln,definition,
% 0.32/0.50 ( subbreln
% 0.32/0.50 = ( ! [A: $i,B: $i,R: $i] :
% 0.32/0.50 ( ( breln @ A @ B @ R )
% 0.32/0.50 => ! [S: $i] :
% 0.32/0.50 ( ( breln @ A @ B @ S )
% 0.32/0.50 => ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ B )
% 0.32/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.32/0.50 => ( subset @ R @ S ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(eqbreln_type,type,
% 0.32/0.50 eqbreln: $o ).
% 0.32/0.50
% 0.32/0.50 thf(eqbreln,definition,
% 0.32/0.50 ( eqbreln
% 0.32/0.50 = ( ! [A: $i,B: $i,R: $i] :
% 0.32/0.50 ( ( breln @ A @ B @ R )
% 0.32/0.50 => ! [S: $i] :
% 0.32/0.50 ( ( breln @ A @ B @ S )
% 0.32/0.50 => ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ B )
% 0.32/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 0.32/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 0.32/0.50 => ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ B )
% 0.32/0.50 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 0.32/0.50 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 0.32/0.50 => ( R = S ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(funcext_type,type,
% 0.32/0.50 funcext: $o ).
% 0.32/0.50
% 0.32/0.50 thf(funcext,definition,
% 0.32/0.50 ( funcext
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( func @ A @ B @ Xf )
% 0.32/0.50 => ! [Xg: $i] :
% 0.32/0.50 ( ( func @ A @ B @ Xg )
% 0.32/0.50 => ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.50 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.32/0.50 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(funcext2_type,type,
% 0.32/0.50 funcext2: $o ).
% 0.32/0.50
% 0.32/0.50 thf(funcext2,definition,
% 0.32/0.50 ( funcext2
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.50 => ! [Xg: $i] :
% 0.32/0.50 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 0.32/0.50 => ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.50 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 0.32/0.50 => ( Xf = Xg ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(ap2apEq1_type,type,
% 0.32/0.50 ap2apEq1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(ap2apEq1,definition,
% 0.32/0.50 ( ap2apEq1
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.50 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(ap2apEq2_type,type,
% 0.32/0.50 ap2apEq2: $o ).
% 0.32/0.50
% 0.32/0.50 thf(ap2apEq2,definition,
% 0.32/0.50 ( ap2apEq2
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( func @ A @ B @ Xf )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( ap @ A @ B @ Xf @ Xx )
% 0.32/0.50 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(beta1_type,type,
% 0.32/0.50 beta1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(beta1,definition,
% 0.32/0.50 ( beta1
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.50 ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( ap @ A @ B
% 0.32/0.50 @ ( lam @ A @ B
% 0.32/0.50 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.32/0.50 @ Xx )
% 0.32/0.50 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(eta1_type,type,
% 0.32/0.50 eta1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(eta1,definition,
% 0.32/0.50 ( eta1
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( func @ A @ B @ Xf )
% 0.32/0.50 => ( ( lam @ A @ B
% 0.32/0.50 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.32/0.50 = Xf ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(lam2lamEq_type,type,
% 0.32/0.50 lam2lamEq: $o ).
% 0.32/0.50
% 0.32/0.50 thf(lam2lamEq,definition,
% 0.32/0.50 ( lam2lamEq
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.50 ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.50 => ( ( lam @ A @ B
% 0.32/0.50 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 0.32/0.50 = ( lam @ A @ B
% 0.32/0.50 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(beta2_type,type,
% 0.32/0.50 beta2: $o ).
% 0.32/0.50
% 0.32/0.50 thf(beta2,definition,
% 0.32/0.50 ( beta2
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 0.32/0.50 ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( in @ ( Xf @ Xx ) @ B ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( ap @ A @ B
% 0.32/0.50 @ ( lam @ A @ B
% 0.32/0.50 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 0.32/0.50 @ Xx )
% 0.32/0.50 = ( Xf @ Xx ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(eta2_type,type,
% 0.32/0.50 eta2: $o ).
% 0.32/0.50
% 0.32/0.50 thf(eta2,definition,
% 0.32/0.50 ( eta2
% 0.32/0.50 = ( ! [A: $i,B: $i,Xf: $i] :
% 0.32/0.50 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 0.32/0.50 => ( ( lam @ A @ B
% 0.32/0.50 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 0.32/0.50 = Xf ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(iffalseProp1_type,type,
% 0.32/0.50 iffalseProp1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(iffalseProp1,definition,
% 0.32/0.50 ( iffalseProp1
% 0.32/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ A )
% 0.32/0.50 => ( ~ Xphi
% 0.32/0.50 => ( in @ Xy
% 0.32/0.50 @ ( dsetconstr @ A
% 0.32/0.50 @ ^ [Xz: $i] :
% 0.32/0.50 ( ( Xphi
% 0.32/0.50 & ( Xz = Xx ) )
% 0.32/0.50 | ( ~ Xphi
% 0.32/0.50 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(iffalseProp2_type,type,
% 0.32/0.50 iffalseProp2: $o ).
% 0.32/0.50
% 0.32/0.50 thf(iffalseProp2,definition,
% 0.32/0.50 ( iffalseProp2
% 0.32/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ A )
% 0.32/0.50 => ( ~ Xphi
% 0.32/0.50 => ( ( dsetconstr @ A
% 0.32/0.50 @ ^ [Xz: $i] :
% 0.32/0.50 ( ( Xphi
% 0.32/0.50 & ( Xz = Xx ) )
% 0.32/0.50 | ( ~ Xphi
% 0.32/0.50 & ( Xz = Xy ) ) ) )
% 0.32/0.50 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(iftrueProp1_type,type,
% 0.32/0.50 iftrueProp1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(iftrueProp1,definition,
% 0.32/0.50 ( iftrueProp1
% 0.32/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ A )
% 0.32/0.50 => ( Xphi
% 0.32/0.50 => ( in @ Xx
% 0.32/0.50 @ ( dsetconstr @ A
% 0.32/0.50 @ ^ [Xz: $i] :
% 0.32/0.50 ( ( Xphi
% 0.32/0.50 & ( Xz = Xx ) )
% 0.32/0.50 | ( ~ Xphi
% 0.32/0.50 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(iftrueProp2_type,type,
% 0.32/0.50 iftrueProp2: $o ).
% 0.32/0.50
% 0.32/0.50 thf(iftrueProp2,definition,
% 0.32/0.50 ( iftrueProp2
% 0.32/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ A )
% 0.32/0.50 => ( Xphi
% 0.32/0.50 => ( ( dsetconstr @ A
% 0.32/0.50 @ ^ [Xz: $i] :
% 0.32/0.50 ( ( Xphi
% 0.32/0.50 & ( Xz = Xx ) )
% 0.32/0.50 | ( ~ Xphi
% 0.32/0.50 & ( Xz = Xy ) ) ) )
% 0.32/0.50 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(ifSingleton_type,type,
% 0.32/0.50 ifSingleton: $o ).
% 0.32/0.50
% 0.32/0.50 thf(ifSingleton,definition,
% 0.32/0.50 ( ifSingleton
% 0.32/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ A )
% 0.32/0.50 => ( singleton
% 0.32/0.50 @ ( dsetconstr @ A
% 0.32/0.50 @ ^ [Xz: $i] :
% 0.32/0.50 ( ( Xphi
% 0.32/0.50 & ( Xz = Xx ) )
% 0.32/0.50 | ( ~ Xphi
% 0.32/0.50 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(if_type,type,
% 0.32/0.50 if: $i > $o > $i > $i > $i ).
% 0.32/0.50
% 0.32/0.50 thf(ifp_type,type,
% 0.32/0.50 ifp: $o ).
% 0.32/0.50
% 0.32/0.50 thf(ifp,definition,
% 0.32/0.50 ( ifp
% 0.32/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ A )
% 0.32/0.50 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(theeq_type,type,
% 0.32/0.50 theeq: $o ).
% 0.32/0.50
% 0.32/0.50 thf(theeq,definition,
% 0.32/0.50 ( theeq
% 0.32/0.50 = ( ! [X: $i] :
% 0.32/0.50 ( ( singleton @ X )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ X )
% 0.32/0.50 => ( ( setunion @ X )
% 0.32/0.50 = Xx ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(iftrue_type,type,
% 0.32/0.50 iftrue: $o ).
% 0.32/0.50
% 0.32/0.50 thf(iftrue,definition,
% 0.32/0.50 ( iftrue
% 0.32/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ A )
% 0.32/0.50 => ( Xphi
% 0.32/0.50 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.32/0.50 = Xx ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(iffalse_type,type,
% 0.32/0.50 iffalse: $o ).
% 0.32/0.50
% 0.32/0.50 thf(iffalse,definition,
% 0.32/0.50 ( iffalse
% 0.32/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ A )
% 0.32/0.50 => ( ~ Xphi
% 0.32/0.50 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 0.32/0.50 = Xy ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(iftrueorfalse_type,type,
% 0.32/0.50 iftrueorfalse: $o ).
% 0.32/0.50
% 0.32/0.50 thf(iftrueorfalse,definition,
% 0.32/0.50 ( iftrueorfalse
% 0.32/0.50 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ! [Xy: $i] :
% 0.32/0.50 ( ( in @ Xy @ A )
% 0.32/0.50 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(binintersectT_lem_type,type,
% 0.32/0.50 binintersectT_lem: $o ).
% 0.32/0.50
% 0.32/0.50 thf(binintersectT_lem,definition,
% 0.32/0.50 ( binintersectT_lem
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(binunionT_lem_type,type,
% 0.32/0.50 binunionT_lem: $o ).
% 0.32/0.50
% 0.32/0.50 thf(binunionT_lem,definition,
% 0.32/0.50 ( binunionT_lem
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(powersetT_lem_type,type,
% 0.32/0.50 powersetT_lem: $o ).
% 0.32/0.50
% 0.32/0.50 thf(powersetT_lem,definition,
% 0.32/0.50 ( powersetT_lem
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(setminusT_lem_type,type,
% 0.32/0.50 setminusT_lem: $o ).
% 0.32/0.50
% 0.32/0.50 thf(setminusT_lem,definition,
% 0.32/0.50 ( setminusT_lem
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(complementT_lem_type,type,
% 0.32/0.50 complementT_lem: $o ).
% 0.32/0.50
% 0.32/0.50 thf(complementT_lem,definition,
% 0.32/0.50 ( complementT_lem
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(setextT_type,type,
% 0.32/0.50 setextT: $o ).
% 0.32/0.50
% 0.32/0.50 thf(setextT,definition,
% 0.32/0.50 ( setextT
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( in @ Xx @ X )
% 0.32/0.50 => ( in @ Xx @ Y ) ) )
% 0.32/0.50 => ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( in @ Xx @ Y )
% 0.32/0.50 => ( in @ Xx @ X ) ) )
% 0.32/0.50 => ( X = Y ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(subsetTI_type,type,
% 0.32/0.50 subsetTI: $o ).
% 0.32/0.50
% 0.32/0.50 thf(subsetTI,definition,
% 0.32/0.50 ( subsetTI
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( in @ Xx @ X )
% 0.32/0.50 => ( in @ Xx @ Y ) ) )
% 0.32/0.50 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(powersetTI1_type,type,
% 0.32/0.50 powersetTI1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(powersetTI1,definition,
% 0.32/0.50 ( powersetTI1
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ( ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( in @ Xx @ X )
% 0.32/0.50 => ( in @ Xx @ Y ) ) )
% 0.32/0.50 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(powersetTE1_type,type,
% 0.32/0.50 powersetTE1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(powersetTE1,definition,
% 0.32/0.50 ( powersetTE1
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( in @ X @ ( powerset @ Y ) )
% 0.32/0.50 => ( ( in @ Xx @ X )
% 0.32/0.50 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(complementTI1_type,type,
% 0.32/0.50 complementTI1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(complementTI1,definition,
% 0.32/0.50 ( complementTI1
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( in @ Xx @ X )
% 0.32/0.50 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(complementTE1_type,type,
% 0.32/0.50 complementTE1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(complementTE1,definition,
% 0.32/0.50 ( complementTE1
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 0.32/0.50 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(binintersectTELcontra_type,type,
% 0.32/0.50 binintersectTELcontra: $o ).
% 0.32/0.50
% 0.32/0.50 thf(binintersectTELcontra,definition,
% 0.32/0.50 ( binintersectTELcontra
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ~ ( in @ Xx @ X )
% 0.32/0.50 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(binintersectTERcontra_type,type,
% 0.32/0.50 binintersectTERcontra: $o ).
% 0.32/0.50
% 0.32/0.50 thf(binintersectTERcontra,definition,
% 0.32/0.50 ( binintersectTERcontra
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ~ ( in @ Xx @ Y )
% 0.32/0.50 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(contrasubsetT_type,type,
% 0.32/0.50 contrasubsetT: $o ).
% 0.32/0.50
% 0.32/0.50 thf(contrasubsetT,definition,
% 0.32/0.50 ( contrasubsetT
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.32/0.50 => ( ( in @ Xx @ Y )
% 0.32/0.50 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(contrasubsetT1_type,type,
% 0.32/0.50 contrasubsetT1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(contrasubsetT1,definition,
% 0.32/0.50 ( contrasubsetT1
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( subset @ X @ Y )
% 0.32/0.50 => ( ~ ( in @ Xx @ Y )
% 0.32/0.50 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(contrasubsetT2_type,type,
% 0.32/0.50 contrasubsetT2: $o ).
% 0.32/0.50
% 0.32/0.50 thf(contrasubsetT2,definition,
% 0.32/0.50 ( contrasubsetT2
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ( ( subset @ X @ Y )
% 0.32/0.50 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(contrasubsetT3_type,type,
% 0.32/0.50 contrasubsetT3: $o ).
% 0.32/0.50
% 0.32/0.50 thf(contrasubsetT3,definition,
% 0.32/0.50 ( contrasubsetT3
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 0.32/0.50 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(doubleComplementI1_type,type,
% 0.32/0.50 doubleComplementI1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(doubleComplementI1,definition,
% 0.32/0.50 ( doubleComplementI1
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( in @ Xx @ X )
% 0.32/0.50 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(doubleComplementE1_type,type,
% 0.32/0.50 doubleComplementE1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(doubleComplementE1,definition,
% 0.32/0.50 ( doubleComplementE1
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 0.32/0.50 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(doubleComplementSub1_type,type,
% 0.32/0.50 doubleComplementSub1: $o ).
% 0.32/0.50
% 0.32/0.50 thf(doubleComplementSub1,definition,
% 0.32/0.50 ( doubleComplementSub1
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(doubleComplementSub2_type,type,
% 0.32/0.50 doubleComplementSub2: $o ).
% 0.32/0.50
% 0.32/0.50 thf(doubleComplementSub2,definition,
% 0.32/0.50 ( doubleComplementSub2
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(doubleComplementEq_type,type,
% 0.32/0.50 doubleComplementEq: $o ).
% 0.32/0.50
% 0.32/0.50 thf(doubleComplementEq,definition,
% 0.32/0.50 ( doubleComplementEq
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ( X
% 0.32/0.50 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(complementTnotintersectT_type,type,
% 0.32/0.50 complementTnotintersectT: $o ).
% 0.32/0.50
% 0.32/0.50 thf(complementTnotintersectT,definition,
% 0.32/0.50 ( complementTnotintersectT
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.32/0.50 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(complementImpComplementIntersect_type,type,
% 0.32/0.50 complementImpComplementIntersect: $o ).
% 0.32/0.50
% 0.32/0.50 thf(complementImpComplementIntersect,definition,
% 0.32/0.50 ( complementImpComplementIntersect
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 0.32/0.50 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(complementSubsetComplementIntersect_type,type,
% 0.32/0.50 complementSubsetComplementIntersect: $o ).
% 0.32/0.50
% 0.32/0.50 thf(complementSubsetComplementIntersect,definition,
% 0.32/0.50 ( complementSubsetComplementIntersect
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(complementInPowersetComplementIntersect_type,type,
% 0.32/0.50 complementInPowersetComplementIntersect: $o ).
% 0.32/0.50
% 0.32/0.50 thf(complementInPowersetComplementIntersect,definition,
% 0.32/0.50 ( complementInPowersetComplementIntersect
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(contraSubsetComplement_type,type,
% 0.32/0.50 contraSubsetComplement: $o ).
% 0.32/0.50
% 0.32/0.50 thf(contraSubsetComplement,definition,
% 0.32/0.50 ( contraSubsetComplement
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( in @ Xx @ Y )
% 0.32/0.50 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(complementTcontraSubset_type,type,
% 0.32/0.50 complementTcontraSubset: $o ).
% 0.32/0.50
% 0.32/0.50 thf(complementTcontraSubset,definition,
% 0.32/0.50 ( complementTcontraSubset
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 0.32/0.50 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(binunionTILcontra_type,type,
% 0.32/0.50 binunionTILcontra: $o ).
% 0.32/0.50
% 0.32/0.50 thf(binunionTILcontra,definition,
% 0.32/0.50 ( binunionTILcontra
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.32/0.50 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(binunionTIRcontra_type,type,
% 0.32/0.50 binunionTIRcontra: $o ).
% 0.32/0.50
% 0.32/0.50 thf(binunionTIRcontra,definition,
% 0.32/0.50 ( binunionTIRcontra
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 0.32/0.50 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(inIntersectImpInUnion_type,type,
% 0.32/0.50 inIntersectImpInUnion: $o ).
% 0.32/0.50
% 0.32/0.50 thf(inIntersectImpInUnion,definition,
% 0.32/0.50 ( inIntersectImpInUnion
% 0.32/0.50 = ( ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ! [Z: $i] :
% 0.32/0.50 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.32/0.50 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 thf(inIntersectImpInUnion2,conjecture,
% 0.32/0.50 ( setextAx
% 0.32/0.50 => ( emptysetAx
% 0.32/0.50 => ( setadjoinAx
% 0.32/0.50 => ( powersetAx
% 0.32/0.50 => ( setunionAx
% 0.32/0.50 => ( omega0Ax
% 0.32/0.50 => ( omegaSAx
% 0.32/0.50 => ( omegaIndAx
% 0.32/0.50 => ( replAx
% 0.32/0.50 => ( foundationAx
% 0.32/0.50 => ( wellorderingAx
% 0.32/0.50 => ( descrp
% 0.32/0.50 => ( dsetconstrI
% 0.32/0.50 => ( dsetconstrEL
% 0.32/0.50 => ( dsetconstrER
% 0.32/0.50 => ( exuE1
% 0.32/0.50 => ( prop2setE
% 0.32/0.50 => ( emptysetE
% 0.32/0.50 => ( emptysetimpfalse
% 0.32/0.50 => ( notinemptyset
% 0.32/0.50 => ( exuE3e
% 0.32/0.50 => ( setext
% 0.32/0.50 => ( emptyI
% 0.32/0.50 => ( noeltsimpempty
% 0.32/0.50 => ( setbeta
% 0.32/0.50 => ( nonemptyE1
% 0.32/0.50 => ( nonemptyI
% 0.32/0.50 => ( nonemptyI1
% 0.32/0.50 => ( setadjoinIL
% 0.32/0.50 => ( emptyinunitempty
% 0.32/0.50 => ( setadjoinIR
% 0.32/0.50 => ( setadjoinE
% 0.32/0.50 => ( setadjoinOr
% 0.32/0.50 => ( setoftrueEq
% 0.32/0.50 => ( powersetI
% 0.32/0.50 => ( emptyinPowerset
% 0.32/0.50 => ( emptyInPowerset
% 0.32/0.50 => ( powersetE
% 0.32/0.50 => ( setunionI
% 0.32/0.50 => ( setunionE
% 0.32/0.50 => ( subPowSU
% 0.32/0.50 => ( exuE2
% 0.32/0.50 => ( nonemptyImpWitness
% 0.32/0.50 => ( uniqinunit
% 0.32/0.50 => ( notinsingleton
% 0.32/0.50 => ( eqinunit
% 0.32/0.50 => ( singletonsswitch
% 0.32/0.50 => ( upairsetE
% 0.32/0.50 => ( upairsetIL
% 0.32/0.50 => ( upairsetIR
% 0.32/0.50 => ( emptyE1
% 0.32/0.50 => ( vacuousDall
% 0.32/0.50 => ( quantDeMorgan1
% 0.32/0.50 => ( quantDeMorgan2
% 0.32/0.50 => ( quantDeMorgan3
% 0.32/0.50 => ( quantDeMorgan4
% 0.32/0.50 => ( prop2setI
% 0.32/0.50 => ( prop2set2propI
% 0.32/0.50 => ( notdexE
% 0.32/0.50 => ( notdallE
% 0.32/0.50 => ( exuI1
% 0.32/0.50 => ( exuI3
% 0.32/0.50 => ( exuI2
% 0.32/0.50 => ( inCongP
% 0.32/0.50 => ( in__Cong
% 0.32/0.50 => ( exuE3u
% 0.32/0.50 => ( exu__Cong
% 0.32/0.50 => ( emptyset__Cong
% 0.32/0.50 => ( setadjoin__Cong
% 0.32/0.50 => ( powerset__Cong
% 0.32/0.50 => ( setunion__Cong
% 0.32/0.50 => ( omega__Cong
% 0.32/0.50 => ( exuEu
% 0.32/0.50 => ( descr__Cong
% 0.32/0.50 => ( dsetconstr__Cong
% 0.32/0.50 => ( subsetI1
% 0.32/0.50 => ( eqimpsubset2
% 0.32/0.50 => ( eqimpsubset1
% 0.32/0.50 => ( subsetI2
% 0.32/0.50 => ( emptysetsubset
% 0.32/0.50 => ( subsetE
% 0.32/0.50 => ( subsetE2
% 0.32/0.50 => ( notsubsetI
% 0.32/0.50 => ( notequalI1
% 0.32/0.50 => ( notequalI2
% 0.32/0.50 => ( subsetRefl
% 0.32/0.50 => ( subsetTrans
% 0.32/0.50 => ( setadjoinSub
% 0.32/0.50 => ( setadjoinSub2
% 0.32/0.50 => ( subset2powerset
% 0.32/0.50 => ( setextsub
% 0.32/0.50 => ( subsetemptysetimpeq
% 0.32/0.50 => ( powersetI1
% 0.32/0.50 => ( powersetE1
% 0.32/0.50 => ( inPowerset
% 0.32/0.50 => ( powersetsubset
% 0.32/0.50 => ( sepInPowerset
% 0.32/0.50 => ( sepSubset
% 0.32/0.50 => ( binunionIL
% 0.32/0.50 => ( upairset2IR
% 0.32/0.50 => ( binunionIR
% 0.32/0.50 => ( binunionEcases
% 0.32/0.50 => ( binunionE
% 0.32/0.50 => ( binunionLsub
% 0.32/0.50 => ( binunionRsub
% 0.32/0.50 => ( binintersectI
% 0.32/0.50 => ( binintersectSubset5
% 0.32/0.50 => ( binintersectEL
% 0.32/0.50 => ( binintersectLsub
% 0.32/0.50 => ( binintersectSubset2
% 0.32/0.50 => ( binintersectSubset3
% 0.32/0.50 => ( binintersectER
% 0.32/0.50 => ( disjointsetsI1
% 0.32/0.50 => ( binintersectRsub
% 0.32/0.50 => ( binintersectSubset4
% 0.32/0.50 => ( binintersectSubset1
% 0.32/0.50 => ( bs114d
% 0.32/0.50 => ( setminusI
% 0.32/0.50 => ( setminusEL
% 0.32/0.50 => ( setminusER
% 0.32/0.50 => ( setminusSubset2
% 0.32/0.50 => ( setminusERneg
% 0.32/0.50 => ( setminusELneg
% 0.32/0.50 => ( setminusILneg
% 0.32/0.50 => ( setminusIRneg
% 0.32/0.50 => ( setminusLsub
% 0.32/0.50 => ( setminusSubset1
% 0.32/0.50 => ( symdiffE
% 0.32/0.50 => ( symdiffI1
% 0.32/0.50 => ( symdiffI2
% 0.32/0.50 => ( symdiffIneg1
% 0.32/0.50 => ( symdiffIneg2
% 0.32/0.50 => ( secondinupair
% 0.32/0.50 => ( setukpairIL
% 0.32/0.50 => ( setukpairIR
% 0.32/0.50 => ( kpairiskpair
% 0.32/0.50 => ( kpairp
% 0.32/0.50 => ( singletonsubset
% 0.32/0.50 => ( singletoninpowerset
% 0.32/0.50 => ( singletoninpowunion
% 0.32/0.50 => ( upairset2E
% 0.32/0.50 => ( upairsubunion
% 0.32/0.50 => ( upairinpowunion
% 0.32/0.50 => ( ubforcartprodlem1
% 0.32/0.50 => ( ubforcartprodlem2
% 0.32/0.50 => ( ubforcartprodlem3
% 0.32/0.50 => ( cartprodpairin
% 0.32/0.50 => ( cartprodmempair1
% 0.32/0.50 => ( cartprodmempair
% 0.32/0.50 => ( setunionE2
% 0.32/0.50 => ( setunionsingleton1
% 0.32/0.50 => ( setunionsingleton2
% 0.32/0.50 => ( setunionsingleton
% 0.32/0.50 => ( singletonprop
% 0.32/0.50 => ( ex1E1
% 0.32/0.50 => ( ex1I
% 0.32/0.50 => ( ex1I2
% 0.32/0.50 => ( singletonsuniq
% 0.32/0.50 => ( setukpairinjL1
% 0.32/0.50 => ( kfstsingleton
% 0.32/0.50 => ( theprop
% 0.32/0.50 => ( kfstpairEq
% 0.32/0.50 => ( cartprodfstin
% 0.32/0.50 => ( setukpairinjL2
% 0.32/0.50 => ( setukpairinjL
% 0.32/0.50 => ( setukpairinjR11
% 0.32/0.50 => ( setukpairinjR12
% 0.32/0.50 => ( setukpairinjR1
% 0.32/0.50 => ( upairequniteq
% 0.32/0.50 => ( setukpairinjR2
% 0.32/0.50 => ( setukpairinjR
% 0.32/0.50 => ( ksndsingleton
% 0.32/0.50 => ( ksndpairEq
% 0.32/0.50 => ( kpairsurjEq
% 0.32/0.50 => ( cartprodsndin
% 0.32/0.50 => ( cartprodpairmemEL
% 0.32/0.50 => ( cartprodpairmemER
% 0.32/0.50 => ( cartprodmempaircEq
% 0.32/0.50 => ( cartprodfstpairEq
% 0.32/0.50 => ( cartprodsndpairEq
% 0.32/0.50 => ( cartprodpairsurjEq
% 0.32/0.50 => ( dpsetconstrI
% 0.32/0.50 => ( dpsetconstrSub
% 0.32/0.50 => ( setOfPairsIsBReln
% 0.32/0.50 => ( dpsetconstrERa
% 0.32/0.50 => ( dpsetconstrEL1
% 0.32/0.50 => ( dpsetconstrEL2
% 0.32/0.50 => ( dpsetconstrER
% 0.32/0.50 => ( funcImageSingleton
% 0.32/0.50 => ( apProp
% 0.32/0.50 => ( app
% 0.32/0.50 => ( infuncsetfunc
% 0.32/0.50 => ( ap2p
% 0.32/0.50 => ( funcinfuncset
% 0.32/0.50 => ( lamProp
% 0.32/0.50 => ( lamp
% 0.32/0.50 => ( lam2p
% 0.32/0.50 => ( brelnall1
% 0.32/0.50 => ( brelnall2
% 0.32/0.50 => ( ex1E2
% 0.32/0.50 => ( funcGraphProp1
% 0.32/0.50 => ( funcGraphProp3
% 0.32/0.50 => ( funcGraphProp2
% 0.32/0.50 => ( funcextLem
% 0.32/0.50 => ( funcGraphProp4
% 0.32/0.50 => ( subbreln
% 0.32/0.50 => ( eqbreln
% 0.32/0.50 => ( funcext
% 0.32/0.50 => ( funcext2
% 0.32/0.50 => ( ap2apEq1
% 0.32/0.50 => ( ap2apEq2
% 0.32/0.50 => ( beta1
% 0.32/0.50 => ( eta1
% 0.32/0.50 => ( lam2lamEq
% 0.32/0.50 => ( beta2
% 0.32/0.50 => ( eta2
% 0.32/0.50 => ( iffalseProp1
% 0.32/0.50 => ( iffalseProp2
% 0.32/0.50 => ( iftrueProp1
% 0.32/0.50 => ( iftrueProp2
% 0.32/0.50 => ( ifSingleton
% 0.32/0.50 => ( ifp
% 0.32/0.50 => ( theeq
% 0.32/0.50 => ( iftrue
% 0.32/0.50 => ( iffalse
% 0.32/0.50 => ( iftrueorfalse
% 0.32/0.50 => ( binintersectT_lem
% 0.32/0.50 => ( binunionT_lem
% 0.32/0.50 => ( powersetT_lem
% 0.32/0.50 => ( setminusT_lem
% 0.32/0.50 => ( complementT_lem
% 0.32/0.50 => ( setextT
% 0.32/0.50 => ( subsetTI
% 0.32/0.50 => ( powersetTI1
% 0.32/0.50 => ( powersetTE1
% 0.32/0.50 => ( complementTI1
% 0.32/0.50 => ( complementTE1
% 0.32/0.50 => ( binintersectTELcontra
% 0.32/0.50 => ( binintersectTERcontra
% 0.32/0.50 => ( contrasubsetT
% 0.32/0.50 => ( contrasubsetT1
% 0.32/0.50 => ( contrasubsetT2
% 0.32/0.50 => ( contrasubsetT3
% 0.32/0.50 => ( doubleComplementI1
% 0.32/0.50 => ( doubleComplementE1
% 0.32/0.50 => ( doubleComplementSub1
% 0.32/0.50 => ( doubleComplementSub2
% 0.32/0.50 => ( doubleComplementEq
% 0.32/0.50 => ( complementTnotintersectT
% 0.32/0.50 => ( complementImpComplementIntersect
% 0.32/0.50 => ( complementSubsetComplementIntersect
% 0.32/0.50 => ( complementInPowersetComplementIntersect
% 0.32/0.50 => ( contraSubsetComplement
% 0.32/0.50 => ( complementTcontraSubset
% 0.32/0.50 => ( binunionTILcontra
% 0.32/0.50 => ( binunionTIRcontra
% 0.32/0.50 => ( inIntersectImpInUnion
% 0.32/0.50 => ! [A: $i,X: $i] :
% 0.32/0.50 ( ( in @ X @ ( powerset @ A ) )
% 0.32/0.50 => ! [Y: $i] :
% 0.32/0.50 ( ( in @ Y @ ( powerset @ A ) )
% 0.32/0.50 => ! [Z: $i] :
% 0.32/0.50 ( ( in @ Z @ ( powerset @ A ) )
% 0.32/0.50 => ! [Xx: $i] :
% 0.32/0.50 ( ( in @ Xx @ A )
% 0.32/0.50 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 0.32/0.50 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.32/0.50
% 0.32/0.50 %------------------------------------------------------------------------------
% 0.32/0.50 ------------------------
% 3.17/1.81 ---- Embedded in THF ---
% 3.17/1.81 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 3.17/1.81 %%% Generated on Tue Feb 24 12:31:54 EST 2026
% 3.17/1.81 %%% using '$$dhol' embedding, version 1.3.0.
% 3.17/1.81 %%% Logic specification used:
% 3.17/1.81 %%% thf(spec, logic, $$dhol).
% 3.17/1.81
% 3.17/1.81 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.JCQicskBu4/DTF2DTF13002.p
% See solution above
% 3.17/1.81 ------------------------
% 3.17/1.87 ---- Cleaned THF ---
% 3.17/1.87 thf(in_type,type,
% 3.17/1.87 in: $i > $i > $o ).
% 3.17/1.87
% 3.17/1.87 thf(exu_type,type,
% 3.17/1.87 exu: ( $i > $o ) > $o ).
% 3.17/1.87
% 3.17/1.87 thf(exu,definition,
% 3.17/1.87 ( exu
% 3.17/1.87 = ( ^ [Xphi: $i > $o] :
% 3.17/1.87 ? [Xx: $i] :
% 3.17/1.87 ( ( Xphi @ Xx )
% 3.17/1.87 & ! [Xy: $i] :
% 3.17/1.87 ( ( Xphi @ Xy )
% 3.17/1.87 => ( Xx = Xy ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setextAx_type,type,
% 3.17/1.87 setextAx: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setextAx,definition,
% 3.17/1.87 ( setextAx
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 <=> ( in @ Xx @ B ) )
% 3.17/1.87 => ( A = B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(emptyset_type,type,
% 3.17/1.87 emptyset: $i ).
% 3.17/1.87
% 3.17/1.87 thf(emptysetAx_type,type,
% 3.17/1.87 emptysetAx: $o ).
% 3.17/1.87
% 3.17/1.87 thf(emptysetAx,definition,
% 3.17/1.87 ( emptysetAx
% 3.17/1.87 = ( ! [Xx: $i] :
% 3.17/1.87 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoin_type,type,
% 3.17/1.87 setadjoin: $i > $i > $i ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoinAx_type,type,
% 3.17/1.87 setadjoinAx: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoinAx,definition,
% 3.17/1.87 ( setadjoinAx
% 3.17/1.87 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.17/1.87 <=> ( ( Xy = Xx )
% 3.17/1.87 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(powerset_type,type,
% 3.17/1.87 powerset: $i > $i ).
% 3.17/1.87
% 3.17/1.87 thf(powersetAx_type,type,
% 3.17/1.87 powersetAx: $o ).
% 3.17/1.87
% 3.17/1.87 thf(powersetAx,definition,
% 3.17/1.87 ( powersetAx
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( in @ B @ ( powerset @ A ) )
% 3.17/1.87 <=> ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ B )
% 3.17/1.87 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setunion_type,type,
% 3.17/1.87 setunion: $i > $i ).
% 3.17/1.87
% 3.17/1.87 thf(setunionAx_type,type,
% 3.17/1.87 setunionAx: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setunionAx,definition,
% 3.17/1.87 ( setunionAx
% 3.17/1.87 = ( ! [A: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ ( setunion @ A ) )
% 3.17/1.87 <=> ? [B: $i] :
% 3.17/1.87 ( ( in @ Xx @ B )
% 3.17/1.87 & ( in @ B @ A ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(omega_type,type,
% 3.17/1.87 omega: $i ).
% 3.17/1.87
% 3.17/1.87 thf(omega0Ax_type,type,
% 3.17/1.87 omega0Ax: $o ).
% 3.17/1.87
% 3.17/1.87 thf(omega0Ax,definition,
% 3.17/1.87 ( omega0Ax
% 3.17/1.87 = ( in @ emptyset @ omega ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(omegaSAx_type,type,
% 3.17/1.87 omegaSAx: $o ).
% 3.17/1.87
% 3.17/1.87 thf(omegaSAx,definition,
% 3.17/1.87 ( omegaSAx
% 3.17/1.87 = ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ omega )
% 3.17/1.87 => ( in @ ( setadjoin @ Xx @ Xx ) @ omega ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(omegaIndAx_type,type,
% 3.17/1.87 omegaIndAx: $o ).
% 3.17/1.87
% 3.17/1.87 thf(omegaIndAx,definition,
% 3.17/1.87 ( omegaIndAx
% 3.17/1.87 = ( ! [A: $i] :
% 3.17/1.87 ( ( ( in @ emptyset @ A )
% 3.17/1.87 & ! [Xx: $i] :
% 3.17/1.87 ( ( ( in @ Xx @ omega )
% 3.17/1.87 & ( in @ Xx @ A ) )
% 3.17/1.87 => ( in @ ( setadjoin @ Xx @ Xx ) @ A ) ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ omega )
% 3.17/1.87 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(replAx_type,type,
% 3.17/1.87 replAx: $o ).
% 3.17/1.87
% 3.17/1.87 thf(replAx,definition,
% 3.17/1.87 ( replAx
% 3.17/1.87 = ( ! [Xphi: $i > $i > $o,A: $i] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( exu
% 3.17/1.87 @ ^ [Xy: $i] : ( Xphi @ Xx @ Xy ) ) )
% 3.17/1.87 => ? [B: $i] :
% 3.17/1.87 ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ B )
% 3.17/1.87 <=> ? [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ A )
% 3.17/1.87 & ( Xphi @ Xy @ Xx ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(foundationAx_type,type,
% 3.17/1.87 foundationAx: $o ).
% 3.17/1.87
% 3.17/1.87 thf(foundationAx,definition,
% 3.17/1.87 ( foundationAx
% 3.17/1.87 = ( ! [A: $i] :
% 3.17/1.87 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.17/1.87 => ? [B: $i] :
% 3.17/1.87 ( ( in @ B @ A )
% 3.17/1.87 & ~ ? [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ B )
% 3.17/1.87 & ( in @ Xx @ A ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(wellorderingAx_type,type,
% 3.17/1.87 wellorderingAx: $o ).
% 3.17/1.87
% 3.17/1.87 thf(wellorderingAx,definition,
% 3.17/1.87 ( wellorderingAx
% 3.17/1.87 = ( ! [A: $i] :
% 3.17/1.87 ? [B: $i] :
% 3.17/1.87 ( ! [C: $i] :
% 3.17/1.87 ( ( in @ C @ B )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ C )
% 3.17/1.87 => ( in @ Xx @ A ) ) )
% 3.17/1.87 & ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( ( in @ Xx @ A )
% 3.17/1.87 & ( in @ Xy @ A ) )
% 3.17/1.87 => ( ! [C: $i] :
% 3.17/1.87 ( ( in @ C @ B )
% 3.17/1.87 => ( ( in @ Xx @ C )
% 3.17/1.87 <=> ( in @ Xy @ C ) ) )
% 3.17/1.87 => ( Xx = Xy ) ) )
% 3.17/1.87 & ! [C: $i,D: $i] :
% 3.17/1.87 ( ( ( in @ C @ B )
% 3.17/1.87 & ( in @ D @ B ) )
% 3.17/1.87 => ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ C )
% 3.17/1.87 => ( in @ Xx @ D ) )
% 3.17/1.87 | ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ D )
% 3.17/1.87 => ( in @ Xx @ C ) ) ) )
% 3.17/1.87 & ! [C: $i] :
% 3.17/1.87 ( ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ C )
% 3.17/1.87 => ( in @ Xx @ A ) )
% 3.17/1.87 & ? [Xx: $i] : ( in @ Xx @ C ) )
% 3.17/1.87 => ? [D: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ D @ B )
% 3.17/1.87 & ( in @ Xx @ C )
% 3.17/1.87 & ~ ? [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ D )
% 3.17/1.87 & ( in @ Xy @ C ) )
% 3.17/1.87 & ! [E: $i] :
% 3.17/1.87 ( ( in @ E @ B )
% 3.17/1.87 => ( ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ E )
% 3.17/1.87 => ( in @ Xy @ D ) )
% 3.17/1.87 | ( in @ Xx @ E ) ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(descr_type,type,
% 3.17/1.87 descr: ( $i > $o ) > $i ).
% 3.17/1.87
% 3.17/1.87 thf(descrp_type,type,
% 3.17/1.87 descrp: $o ).
% 3.17/1.87
% 3.17/1.87 thf(descrp,definition,
% 3.17/1.87 ( descrp
% 3.17/1.87 = ( ! [Xphi: $i > $o] :
% 3.17/1.87 ( ( exu
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.87 => ( Xphi
% 3.17/1.87 @ ( descr
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(dsetconstr_type,type,
% 3.17/1.87 dsetconstr: $i > ( $i > $o ) > $i ).
% 3.17/1.87
% 3.17/1.87 thf(dsetconstrI_type,type,
% 3.17/1.87 dsetconstrI: $o ).
% 3.17/1.87
% 3.17/1.87 thf(dsetconstrI,definition,
% 3.17/1.87 ( dsetconstrI
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( Xphi @ Xx )
% 3.17/1.87 => ( in @ Xx
% 3.17/1.87 @ ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(dsetconstrEL_type,type,
% 3.17/1.87 dsetconstrEL: $o ).
% 3.17/1.87
% 3.17/1.87 thf(dsetconstrEL,definition,
% 3.17/1.87 ( dsetconstrEL
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx
% 3.17/1.87 @ ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.17/1.87 => ( in @ Xx @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(dsetconstrER_type,type,
% 3.17/1.87 dsetconstrER: $o ).
% 3.17/1.87
% 3.17/1.87 thf(dsetconstrER,definition,
% 3.17/1.87 ( dsetconstrER
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx
% 3.17/1.87 @ ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.17/1.87 => ( Xphi @ Xx ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(exuE1_type,type,
% 3.17/1.87 exuE1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(exuE1,definition,
% 3.17/1.87 ( exuE1
% 3.17/1.87 = ( ! [Xphi: $i > $o] :
% 3.17/1.87 ( ( exu
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.87 => ? [Xx: $i] :
% 3.17/1.87 ( ( Xphi @ Xx )
% 3.17/1.87 & ! [Xy: $i] :
% 3.17/1.87 ( ( Xphi @ Xy )
% 3.17/1.87 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(prop2set_type,type,
% 3.17/1.87 prop2set: $o > $i ).
% 3.17/1.87
% 3.17/1.87 thf(prop2setE_type,type,
% 3.17/1.87 prop2setE: $o ).
% 3.17/1.87
% 3.17/1.87 thf(prop2setE,definition,
% 3.17/1.87 ( prop2setE
% 3.17/1.87 = ( ! [Xphi: $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ ( prop2set @ Xphi ) )
% 3.17/1.87 => Xphi ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(emptysetE_type,type,
% 3.17/1.87 emptysetE: $o ).
% 3.17/1.87
% 3.17/1.87 thf(emptysetE,definition,
% 3.17/1.87 ( emptysetE
% 3.17/1.87 = ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ emptyset )
% 3.17/1.87 => ! [Xphi: $o] : Xphi ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(emptysetimpfalse_type,type,
% 3.17/1.87 emptysetimpfalse: $o ).
% 3.17/1.87
% 3.17/1.87 thf(emptysetimpfalse,definition,
% 3.17/1.87 ( emptysetimpfalse
% 3.17/1.87 = ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ emptyset )
% 3.17/1.87 => $false ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(notinemptyset_type,type,
% 3.17/1.87 notinemptyset: $o ).
% 3.17/1.87
% 3.17/1.87 thf(notinemptyset,definition,
% 3.17/1.87 ( notinemptyset
% 3.17/1.87 = ( ! [Xx: $i] :
% 3.17/1.87 ~ ( in @ Xx @ emptyset ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(exuE3e_type,type,
% 3.17/1.87 exuE3e: $o ).
% 3.17/1.87
% 3.17/1.87 thf(exuE3e,definition,
% 3.17/1.87 ( exuE3e
% 3.17/1.87 = ( ! [Xphi: $i > $o] :
% 3.17/1.87 ( ( exu
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.87 => ? [Xx: $i] : ( Xphi @ Xx ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setext_type,type,
% 3.17/1.87 setext: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setext,definition,
% 3.17/1.87 ( setext
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ Xx @ B ) )
% 3.17/1.87 => ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ B )
% 3.17/1.87 => ( in @ Xx @ A ) )
% 3.17/1.87 => ( A = B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(emptyI_type,type,
% 3.17/1.87 emptyI: $o ).
% 3.17/1.87
% 3.17/1.87 thf(emptyI,definition,
% 3.17/1.87 ( emptyI
% 3.17/1.87 = ( ! [A: $i] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ~ ( in @ Xx @ A )
% 3.17/1.87 => ( A = emptyset ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(noeltsimpempty_type,type,
% 3.17/1.87 noeltsimpempty: $o ).
% 3.17/1.87
% 3.17/1.87 thf(noeltsimpempty,definition,
% 3.17/1.87 ( noeltsimpempty
% 3.17/1.87 = ( ! [A: $i] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ~ ( in @ Xx @ A )
% 3.17/1.87 => ( A = emptyset ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setbeta_type,type,
% 3.17/1.87 setbeta: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setbeta,definition,
% 3.17/1.87 ( setbeta
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( in @ Xx
% 3.17/1.87 @ ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) )
% 3.17/1.87 <=> ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(nonempty_type,type,
% 3.17/1.87 nonempty: $i > $o ).
% 3.17/1.87
% 3.17/1.87 thf(nonempty,definition,
% 3.17/1.87 ( nonempty
% 3.17/1.87 = ( ^ [Xx: $i] : ( Xx != emptyset ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(nonemptyE1_type,type,
% 3.17/1.87 nonemptyE1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(nonemptyE1,definition,
% 3.17/1.87 ( nonemptyE1
% 3.17/1.87 = ( ! [A: $i] :
% 3.17/1.87 ( ( nonempty @ A )
% 3.17/1.87 => ? [Xx: $i] : ( in @ Xx @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(nonemptyI_type,type,
% 3.17/1.87 nonemptyI: $o ).
% 3.17/1.87
% 3.17/1.87 thf(nonemptyI,definition,
% 3.17/1.87 ( nonemptyI
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( Xphi @ Xx )
% 3.17/1.87 => ( nonempty
% 3.17/1.87 @ ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(nonemptyI1_type,type,
% 3.17/1.87 nonemptyI1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(nonemptyI1,definition,
% 3.17/1.87 ( nonemptyI1
% 3.17/1.87 = ( ! [A: $i] :
% 3.17/1.87 ( ? [Xx: $i] : ( in @ Xx @ A )
% 3.17/1.87 => ( nonempty @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoinIL_type,type,
% 3.17/1.87 setadjoinIL: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoinIL,definition,
% 3.17/1.87 ( setadjoinIL
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ Xy ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(emptyinunitempty_type,type,
% 3.17/1.87 emptyinunitempty: $o ).
% 3.17/1.87
% 3.17/1.87 thf(emptyinunitempty,definition,
% 3.17/1.87 ( emptyinunitempty
% 3.17/1.87 = ( in @ emptyset @ ( setadjoin @ emptyset @ emptyset ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoinIR_type,type,
% 3.17/1.87 setadjoinIR: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoinIR,definition,
% 3.17/1.87 ( setadjoinIR
% 3.17/1.87 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ A )
% 3.17/1.87 => ( in @ Xy @ ( setadjoin @ Xx @ A ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoinE_type,type,
% 3.17/1.87 setadjoinE: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoinE,definition,
% 3.17/1.87 ( setadjoinE
% 3.17/1.87 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.17/1.87 => ! [Xphi: $o] :
% 3.17/1.87 ( ( ( Xy = Xx )
% 3.17/1.87 => Xphi )
% 3.17/1.87 => ( ( ( in @ Xy @ A )
% 3.17/1.87 => Xphi )
% 3.17/1.87 => Xphi ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoinOr_type,type,
% 3.17/1.87 setadjoinOr: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoinOr,definition,
% 3.17/1.87 ( setadjoinOr
% 3.17/1.87 = ( ! [Xx: $i,A: $i,Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ ( setadjoin @ Xx @ A ) )
% 3.17/1.87 => ( ( Xy = Xx )
% 3.17/1.87 | ( in @ Xy @ A ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setoftrueEq_type,type,
% 3.17/1.87 setoftrueEq: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setoftrueEq,definition,
% 3.17/1.87 ( setoftrueEq
% 3.17/1.87 = ( ! [A: $i] :
% 3.17/1.87 ( ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xx: $i] : $true )
% 3.17/1.87 = A ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(powersetI_type,type,
% 3.17/1.87 powersetI: $o ).
% 3.17/1.87
% 3.17/1.87 thf(powersetI,definition,
% 3.17/1.87 ( powersetI
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ B )
% 3.17/1.87 => ( in @ Xx @ A ) )
% 3.17/1.87 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(emptyinPowerset_type,type,
% 3.17/1.87 emptyinPowerset: $o ).
% 3.17/1.87
% 3.17/1.87 thf(emptyinPowerset,definition,
% 3.17/1.87 ( emptyinPowerset
% 3.17/1.87 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(emptyInPowerset_type,type,
% 3.17/1.87 emptyInPowerset: $o ).
% 3.17/1.87
% 3.17/1.87 thf(emptyInPowerset,definition,
% 3.17/1.87 ( emptyInPowerset
% 3.17/1.87 = ( ! [A: $i] : ( in @ emptyset @ ( powerset @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(powersetE_type,type,
% 3.17/1.87 powersetE: $o ).
% 3.17/1.87
% 3.17/1.87 thf(powersetE,definition,
% 3.17/1.87 ( powersetE
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ B @ ( powerset @ A ) )
% 3.17/1.87 => ( ( in @ Xx @ B )
% 3.17/1.87 => ( in @ Xx @ A ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setunionI_type,type,
% 3.17/1.87 setunionI: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setunionI,definition,
% 3.17/1.87 ( setunionI
% 3.17/1.87 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.17/1.87 ( ( in @ Xx @ B )
% 3.17/1.87 => ( ( in @ B @ A )
% 3.17/1.87 => ( in @ Xx @ ( setunion @ A ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setunionE_type,type,
% 3.17/1.87 setunionE: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setunionE,definition,
% 3.17/1.87 ( setunionE
% 3.17/1.87 = ( ! [A: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ ( setunion @ A ) )
% 3.17/1.87 => ! [Xphi: $o] :
% 3.17/1.87 ( ! [B: $i] :
% 3.17/1.87 ( ( in @ Xx @ B )
% 3.17/1.87 => ( ( in @ B @ A )
% 3.17/1.87 => Xphi ) )
% 3.17/1.87 => Xphi ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(subPowSU_type,type,
% 3.17/1.87 subPowSU: $o ).
% 3.17/1.87
% 3.17/1.87 thf(subPowSU,definition,
% 3.17/1.87 ( subPowSU
% 3.17/1.87 = ( ! [A: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ Xx @ ( powerset @ ( setunion @ A ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(exuE2_type,type,
% 3.17/1.87 exuE2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(exuE2,definition,
% 3.17/1.87 ( exuE2
% 3.17/1.87 = ( ! [Xphi: $i > $o] :
% 3.17/1.87 ( ( exu
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.87 => ? [Xx: $i] :
% 3.17/1.87 ! [Xy: $i] :
% 3.17/1.87 ( ( Xphi @ Xy )
% 3.17/1.87 <=> ( Xy = Xx ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(nonemptyImpWitness_type,type,
% 3.17/1.87 nonemptyImpWitness: $o ).
% 3.17/1.87
% 3.17/1.87 thf(nonemptyImpWitness,definition,
% 3.17/1.87 ( nonemptyImpWitness
% 3.17/1.87 = ( ! [A: $i] :
% 3.17/1.87 ( ( nonempty @ A )
% 3.17/1.87 => ? [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 & $true ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(uniqinunit_type,type,
% 3.17/1.87 uniqinunit: $o ).
% 3.17/1.87
% 3.17/1.87 thf(uniqinunit,definition,
% 3.17/1.87 ( uniqinunit
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.17/1.87 => ( Xx = Xy ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(notinsingleton_type,type,
% 3.17/1.87 notinsingleton: $o ).
% 3.17/1.87
% 3.17/1.87 thf(notinsingleton,definition,
% 3.17/1.87 ( notinsingleton
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( Xx != Xy )
% 3.17/1.87 => ~ ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(eqinunit_type,type,
% 3.17/1.87 eqinunit: $o ).
% 3.17/1.87
% 3.17/1.87 thf(eqinunit,definition,
% 3.17/1.87 ( eqinunit
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( Xx = Xy )
% 3.17/1.87 => ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(singletonsswitch_type,type,
% 3.17/1.87 singletonsswitch: $o ).
% 3.17/1.87
% 3.17/1.87 thf(singletonsswitch,definition,
% 3.17/1.87 ( singletonsswitch
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( in @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.17/1.87 => ( in @ Xy @ ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(upairsetE_type,type,
% 3.17/1.87 upairsetE: $o ).
% 3.17/1.87
% 3.17/1.87 thf(upairsetE,definition,
% 3.17/1.87 ( upairsetE
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.17/1.87 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.17/1.87 => ( ( Xz = Xx )
% 3.17/1.87 | ( Xz = Xy ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(upairsetIL_type,type,
% 3.17/1.87 upairsetIL: $o ).
% 3.17/1.87
% 3.17/1.87 thf(upairsetIL,definition,
% 3.17/1.87 ( upairsetIL
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(upairsetIR_type,type,
% 3.17/1.87 upairsetIR: $o ).
% 3.17/1.87
% 3.17/1.87 thf(upairsetIR,definition,
% 3.17/1.87 ( upairsetIR
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(emptyE1_type,type,
% 3.17/1.87 emptyE1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(emptyE1,definition,
% 3.17/1.87 ( emptyE1
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.87 ( ? [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 & ( Xphi @ Xx ) )
% 3.17/1.87 => ( ( ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.87 = emptyset )
% 3.17/1.87 => $false ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(vacuousDall_type,type,
% 3.17/1.87 vacuousDall: $o ).
% 3.17/1.87
% 3.17/1.87 thf(vacuousDall,definition,
% 3.17/1.87 ( vacuousDall
% 3.17/1.87 = ( ! [Xphi: $i > $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ emptyset )
% 3.17/1.87 => ( Xphi @ Xx ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(quantDeMorgan1_type,type,
% 3.17/1.87 quantDeMorgan1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(quantDeMorgan1,definition,
% 3.17/1.87 ( quantDeMorgan1
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.87 ( ~ ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( Xphi @ Xx ) )
% 3.17/1.87 => ? [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(quantDeMorgan2_type,type,
% 3.17/1.87 quantDeMorgan2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(quantDeMorgan2,definition,
% 3.17/1.87 ( quantDeMorgan2
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ~ ( Xphi @ Xx ) )
% 3.17/1.87 => ~ ? [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(quantDeMorgan3_type,type,
% 3.17/1.87 quantDeMorgan3: $o ).
% 3.17/1.87
% 3.17/1.87 thf(quantDeMorgan3,definition,
% 3.17/1.87 ( quantDeMorgan3
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.87 ( ~ ? [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 & ( Xphi @ Xx ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(quantDeMorgan4_type,type,
% 3.17/1.87 quantDeMorgan4: $o ).
% 3.17/1.87
% 3.17/1.87 thf(quantDeMorgan4,definition,
% 3.17/1.87 ( quantDeMorgan4
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.87 ( ? [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 & ~ ( Xphi @ Xx ) )
% 3.17/1.87 => ~ ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(prop2setI_type,type,
% 3.17/1.87 prop2setI: $o ).
% 3.17/1.87
% 3.17/1.87 thf(prop2setI,definition,
% 3.17/1.87 ( prop2setI
% 3.17/1.87 = ( ! [Xphi: $o] :
% 3.17/1.87 ( Xphi
% 3.17/1.87 => ( in @ emptyset @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(set2prop_type,type,
% 3.17/1.87 set2prop: $i > $o ).
% 3.17/1.87
% 3.17/1.87 thf(prop2set2propI_type,type,
% 3.17/1.87 prop2set2propI: $o ).
% 3.17/1.87
% 3.17/1.87 thf(prop2set2propI,definition,
% 3.17/1.87 ( prop2set2propI
% 3.17/1.87 = ( ! [Xphi: $o] :
% 3.17/1.87 ( Xphi
% 3.17/1.87 => ( set2prop @ ( prop2set @ Xphi ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(notdexE_type,type,
% 3.17/1.87 notdexE: $o ).
% 3.17/1.87
% 3.17/1.87 thf(notdexE,definition,
% 3.17/1.87 ( notdexE
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.87 ( ~ ? [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 & ( Xphi @ Xx ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(notdallE_type,type,
% 3.17/1.87 notdallE: $o ).
% 3.17/1.87
% 3.17/1.87 thf(notdallE,definition,
% 3.17/1.87 ( notdallE
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.87 ( ~ ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( Xphi @ Xx ) )
% 3.17/1.87 => ? [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 & ~ ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(exuI1_type,type,
% 3.17/1.87 exuI1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(exuI1,definition,
% 3.17/1.87 ( exuI1
% 3.17/1.87 = ( ! [Xphi: $i > $o] :
% 3.17/1.87 ( ? [Xx: $i] :
% 3.17/1.87 ( ( Xphi @ Xx )
% 3.17/1.87 & ! [Xy: $i] :
% 3.17/1.87 ( ( Xphi @ Xy )
% 3.17/1.87 => ( Xx = Xy ) ) )
% 3.17/1.87 => ( exu
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(exuI3_type,type,
% 3.17/1.87 exuI3: $o ).
% 3.17/1.87
% 3.17/1.87 thf(exuI3,definition,
% 3.17/1.87 ( exuI3
% 3.17/1.87 = ( ! [Xphi: $i > $o] :
% 3.17/1.87 ( ? [Xx: $i] : ( Xphi @ Xx )
% 3.17/1.87 => ( ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( Xphi @ Xx )
% 3.17/1.87 => ( ( Xphi @ Xy )
% 3.17/1.87 => ( Xx = Xy ) ) )
% 3.17/1.87 => ( exu
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(exuI2_type,type,
% 3.17/1.87 exuI2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(exuI2,definition,
% 3.17/1.87 ( exuI2
% 3.17/1.87 = ( ! [Xphi: $i > $o] :
% 3.17/1.87 ( ? [Xx: $i] :
% 3.17/1.87 ! [Xy: $i] :
% 3.17/1.87 ( ( Xphi @ Xy )
% 3.17/1.87 <=> ( Xy = Xx ) )
% 3.17/1.87 => ( exu
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(inCongP_type,type,
% 3.17/1.87 inCongP: $o ).
% 3.17/1.87
% 3.17/1.87 thf(inCongP,definition,
% 3.17/1.87 ( inCongP
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( A = B )
% 3.17/1.87 => ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( Xx = Xy )
% 3.17/1.87 => ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(in__Cong_type,type,
% 3.17/1.87 in__Cong: $o ).
% 3.17/1.87
% 3.17/1.87 thf(in__Cong,definition,
% 3.17/1.87 ( in__Cong
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( A = B )
% 3.17/1.87 => ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( Xx = Xy )
% 3.17/1.87 => ( ( in @ Xx @ A )
% 3.17/1.87 <=> ( in @ Xy @ B ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(exuE3u_type,type,
% 3.17/1.87 exuE3u: $o ).
% 3.17/1.87
% 3.17/1.87 thf(exuE3u,definition,
% 3.17/1.87 ( exuE3u
% 3.17/1.87 = ( ! [Xphi: $i > $o] :
% 3.17/1.87 ( ( exu
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.87 => ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( Xphi @ Xx )
% 3.17/1.87 => ( ( Xphi @ Xy )
% 3.17/1.87 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(exu__Cong_type,type,
% 3.17/1.87 exu__Cong: $o ).
% 3.17/1.87
% 3.17/1.87 thf(exu__Cong,definition,
% 3.17/1.87 ( exu__Cong
% 3.17/1.87 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.17/1.87 ( ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( Xx = Xy )
% 3.17/1.87 => ( ( Xphi @ Xx )
% 3.17/1.87 <=> ( Xpsi @ Xy ) ) )
% 3.17/1.87 => ( ( exu
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.87 <=> ( exu
% 3.17/1.87 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(emptyset__Cong_type,type,
% 3.17/1.87 emptyset__Cong: $o ).
% 3.17/1.87
% 3.17/1.87 thf(emptyset__Cong,definition,
% 3.17/1.87 ( emptyset__Cong
% 3.17/1.87 = ( emptyset = emptyset ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoin__Cong_type,type,
% 3.17/1.87 setadjoin__Cong: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoin__Cong,definition,
% 3.17/1.87 ( setadjoin__Cong
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( Xx = Xy )
% 3.17/1.87 => ! [Xz: $i,Xu: $i] :
% 3.17/1.87 ( ( Xz = Xu )
% 3.17/1.87 => ( ( setadjoin @ Xx @ Xz )
% 3.17/1.87 = ( setadjoin @ Xy @ Xu ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(powerset__Cong_type,type,
% 3.17/1.87 powerset__Cong: $o ).
% 3.17/1.87
% 3.17/1.87 thf(powerset__Cong,definition,
% 3.17/1.87 ( powerset__Cong
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( A = B )
% 3.17/1.87 => ( ( powerset @ A )
% 3.17/1.87 = ( powerset @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setunion__Cong_type,type,
% 3.17/1.87 setunion__Cong: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setunion__Cong,definition,
% 3.17/1.87 ( setunion__Cong
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( A = B )
% 3.17/1.87 => ( ( setunion @ A )
% 3.17/1.87 = ( setunion @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(omega__Cong_type,type,
% 3.17/1.87 omega__Cong: $o ).
% 3.17/1.87
% 3.17/1.87 thf(omega__Cong,definition,
% 3.17/1.87 ( omega__Cong
% 3.17/1.87 = ( omega = omega ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(exuEu_type,type,
% 3.17/1.87 exuEu: $o ).
% 3.17/1.87
% 3.17/1.87 thf(exuEu,definition,
% 3.17/1.87 ( exuEu
% 3.17/1.87 = ( ! [Xphi: $i > $o] :
% 3.17/1.87 ( ( exu
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.87 => ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( Xphi @ Xx )
% 3.17/1.87 => ( ( Xphi @ Xy )
% 3.17/1.87 => ( Xx = Xy ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(descr__Cong_type,type,
% 3.17/1.87 descr__Cong: $o ).
% 3.17/1.87
% 3.17/1.87 thf(descr__Cong,definition,
% 3.17/1.87 ( descr__Cong
% 3.17/1.87 = ( ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.17/1.87 ( ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( Xx = Xy )
% 3.17/1.87 => ( ( Xphi @ Xx )
% 3.17/1.87 <=> ( Xpsi @ Xy ) ) )
% 3.17/1.87 => ( ( exu
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.87 => ( ( exu
% 3.17/1.87 @ ^ [Xx: $i] : ( Xpsi @ Xx ) )
% 3.17/1.87 => ( ( descr
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.87 = ( descr
% 3.17/1.87 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(dsetconstr__Cong_type,type,
% 3.17/1.87 dsetconstr__Cong: $o ).
% 3.17/1.87
% 3.17/1.87 thf(dsetconstr__Cong,definition,
% 3.17/1.87 ( dsetconstr__Cong
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( A = B )
% 3.17/1.87 => ! [Xphi: $i > $o,Xpsi: $i > $o] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( ( Xx = Xy )
% 3.17/1.87 => ( ( Xphi @ Xx )
% 3.17/1.87 <=> ( Xpsi @ Xy ) ) ) ) )
% 3.17/1.87 => ( ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.87 = ( dsetconstr @ B
% 3.17/1.87 @ ^ [Xx: $i] : ( Xpsi @ Xx ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(subset_type,type,
% 3.17/1.87 subset: $i > $i > $o ).
% 3.17/1.87
% 3.17/1.87 thf(disjoint_type,type,
% 3.17/1.87 disjoint: $i > $i > $o ).
% 3.17/1.87
% 3.17/1.87 thf(setsmeet_type,type,
% 3.17/1.87 setsmeet: $i > $i > $o ).
% 3.17/1.87
% 3.17/1.87 thf(subsetI1_type,type,
% 3.17/1.87 subsetI1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(subsetI1,definition,
% 3.17/1.87 ( subsetI1
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ Xx @ B ) )
% 3.17/1.87 => ( subset @ A @ B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(eqimpsubset2_type,type,
% 3.17/1.87 eqimpsubset2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(eqimpsubset2,definition,
% 3.17/1.87 ( eqimpsubset2
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( A = B )
% 3.17/1.87 => ( subset @ B @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(eqimpsubset1_type,type,
% 3.17/1.87 eqimpsubset1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(eqimpsubset1,definition,
% 3.17/1.87 ( eqimpsubset1
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( A = B )
% 3.17/1.87 => ( subset @ A @ B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(subsetI2_type,type,
% 3.17/1.87 subsetI2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(subsetI2,definition,
% 3.17/1.87 ( subsetI2
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ Xx @ B ) )
% 3.17/1.87 => ( subset @ A @ B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(emptysetsubset_type,type,
% 3.17/1.87 emptysetsubset: $o ).
% 3.17/1.87
% 3.17/1.87 thf(emptysetsubset,definition,
% 3.17/1.87 ( emptysetsubset
% 3.17/1.87 = ( ! [A: $i] : ( subset @ emptyset @ A ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(subsetE_type,type,
% 3.17/1.87 subsetE: $o ).
% 3.17/1.87
% 3.17/1.87 thf(subsetE,definition,
% 3.17/1.87 ( subsetE
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( subset @ A @ B )
% 3.17/1.87 => ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(subsetE2_type,type,
% 3.17/1.87 subsetE2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(subsetE2,definition,
% 3.17/1.87 ( subsetE2
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( subset @ A @ B )
% 3.17/1.87 => ( ~ ( in @ Xx @ B )
% 3.17/1.87 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(notsubsetI_type,type,
% 3.17/1.87 notsubsetI: $o ).
% 3.17/1.87
% 3.17/1.87 thf(notsubsetI,definition,
% 3.17/1.87 ( notsubsetI
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ~ ( in @ Xx @ B )
% 3.17/1.87 => ~ ( subset @ A @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(notequalI1_type,type,
% 3.17/1.87 notequalI1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(notequalI1,definition,
% 3.17/1.87 ( notequalI1
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ~ ( subset @ A @ B )
% 3.17/1.87 => ( A != B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(notequalI2_type,type,
% 3.17/1.87 notequalI2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(notequalI2,definition,
% 3.17/1.87 ( notequalI2
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ~ ( in @ Xx @ B )
% 3.17/1.87 => ( A != B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(subsetRefl_type,type,
% 3.17/1.87 subsetRefl: $o ).
% 3.17/1.87
% 3.17/1.87 thf(subsetRefl,definition,
% 3.17/1.87 ( subsetRefl
% 3.17/1.87 = ( ! [A: $i] : ( subset @ A @ A ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(subsetTrans_type,type,
% 3.17/1.87 subsetTrans: $o ).
% 3.17/1.87
% 3.17/1.87 thf(subsetTrans,definition,
% 3.17/1.87 ( subsetTrans
% 3.17/1.87 = ( ! [A: $i,B: $i,C: $i] :
% 3.17/1.87 ( ( subset @ A @ B )
% 3.17/1.87 => ( ( subset @ B @ C )
% 3.17/1.87 => ( subset @ A @ C ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoinSub_type,type,
% 3.17/1.87 setadjoinSub: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoinSub,definition,
% 3.17/1.87 ( setadjoinSub
% 3.17/1.87 = ( ! [Xx: $i,A: $i] : ( subset @ A @ ( setadjoin @ Xx @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoinSub2_type,type,
% 3.17/1.87 setadjoinSub2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setadjoinSub2,definition,
% 3.17/1.87 ( setadjoinSub2
% 3.17/1.87 = ( ! [A: $i,Xx: $i,B: $i] :
% 3.17/1.87 ( ( subset @ A @ B )
% 3.17/1.87 => ( subset @ A @ ( setadjoin @ Xx @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(subset2powerset_type,type,
% 3.17/1.87 subset2powerset: $o ).
% 3.17/1.87
% 3.17/1.87 thf(subset2powerset,definition,
% 3.17/1.87 ( subset2powerset
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( subset @ A @ B )
% 3.17/1.87 => ( in @ A @ ( powerset @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setextsub_type,type,
% 3.17/1.87 setextsub: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setextsub,definition,
% 3.17/1.87 ( setextsub
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( subset @ A @ B )
% 3.17/1.87 => ( ( subset @ B @ A )
% 3.17/1.87 => ( A = B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(subsetemptysetimpeq_type,type,
% 3.17/1.87 subsetemptysetimpeq: $o ).
% 3.17/1.87
% 3.17/1.87 thf(subsetemptysetimpeq,definition,
% 3.17/1.87 ( subsetemptysetimpeq
% 3.17/1.87 = ( ! [A: $i] :
% 3.17/1.87 ( ( subset @ A @ emptyset )
% 3.17/1.87 => ( A = emptyset ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(powersetI1_type,type,
% 3.17/1.87 powersetI1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(powersetI1,definition,
% 3.17/1.87 ( powersetI1
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( subset @ B @ A )
% 3.17/1.87 => ( in @ B @ ( powerset @ A ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(powersetE1_type,type,
% 3.17/1.87 powersetE1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(powersetE1,definition,
% 3.17/1.87 ( powersetE1
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( in @ B @ ( powerset @ A ) )
% 3.17/1.87 => ( subset @ B @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(inPowerset_type,type,
% 3.17/1.87 inPowerset: $o ).
% 3.17/1.87
% 3.17/1.87 thf(inPowerset,definition,
% 3.17/1.87 ( inPowerset
% 3.17/1.87 = ( ! [A: $i] : ( in @ A @ ( powerset @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(powersetsubset_type,type,
% 3.17/1.87 powersetsubset: $o ).
% 3.17/1.87
% 3.17/1.87 thf(powersetsubset,definition,
% 3.17/1.87 ( powersetsubset
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( subset @ A @ B )
% 3.17/1.87 => ( subset @ ( powerset @ A ) @ ( powerset @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(sepInPowerset_type,type,
% 3.17/1.87 sepInPowerset: $o ).
% 3.17/1.87
% 3.17/1.87 thf(sepInPowerset,definition,
% 3.17/1.87 ( sepInPowerset
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.87 ( in
% 3.17/1.87 @ ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.87 @ ( powerset @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(sepSubset_type,type,
% 3.17/1.87 sepSubset: $o ).
% 3.17/1.87
% 3.17/1.87 thf(sepSubset,definition,
% 3.17/1.87 ( sepSubset
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.87 ( subset
% 3.17/1.87 @ ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.87 @ A ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binunion_type,type,
% 3.17/1.87 binunion: $i > $i > $i ).
% 3.17/1.87
% 3.17/1.87 thf(binunionIL_type,type,
% 3.17/1.87 binunionIL: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binunionIL,definition,
% 3.17/1.87 ( binunionIL
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(upairset2IR_type,type,
% 3.17/1.87 upairset2IR: $o ).
% 3.17/1.87
% 3.17/1.87 thf(upairset2IR,definition,
% 3.17/1.87 ( upairset2IR
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binunionIR_type,type,
% 3.17/1.87 binunionIR: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binunionIR,definition,
% 3.17/1.87 ( binunionIR
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ B )
% 3.17/1.87 => ( in @ Xx @ ( binunion @ A @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binunionEcases_type,type,
% 3.17/1.87 binunionEcases: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binunionEcases,definition,
% 3.17/1.87 ( binunionEcases
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i,Xphi: $o] :
% 3.17/1.87 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.17/1.87 => ( ( ( in @ Xx @ A )
% 3.17/1.87 => Xphi )
% 3.17/1.87 => ( ( ( in @ Xx @ B )
% 3.17/1.87 => Xphi )
% 3.17/1.87 => Xphi ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binunionE_type,type,
% 3.17/1.87 binunionE: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binunionE,definition,
% 3.17/1.87 ( binunionE
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ ( binunion @ A @ B ) )
% 3.17/1.87 => ( ( in @ Xx @ A )
% 3.17/1.87 | ( in @ Xx @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binunionLsub_type,type,
% 3.17/1.87 binunionLsub: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binunionLsub,definition,
% 3.17/1.87 ( binunionLsub
% 3.17/1.87 = ( ! [A: $i,B: $i] : ( subset @ A @ ( binunion @ A @ B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binunionRsub_type,type,
% 3.17/1.87 binunionRsub: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binunionRsub,definition,
% 3.17/1.87 ( binunionRsub
% 3.17/1.87 = ( ! [A: $i,B: $i] : ( subset @ B @ ( binunion @ A @ B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binintersect_type,type,
% 3.17/1.87 binintersect: $i > $i > $i ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectI_type,type,
% 3.17/1.87 binintersectI: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectI,definition,
% 3.17/1.87 ( binintersectI
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( in @ Xx @ B )
% 3.17/1.87 => ( in @ Xx @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectSubset5_type,type,
% 3.17/1.87 binintersectSubset5: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectSubset5,definition,
% 3.17/1.87 ( binintersectSubset5
% 3.17/1.87 = ( ! [A: $i,B: $i,C: $i] :
% 3.17/1.87 ( ( subset @ C @ A )
% 3.17/1.87 => ( ( subset @ C @ B )
% 3.17/1.87 => ( subset @ C @ ( binintersect @ A @ B ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectEL_type,type,
% 3.17/1.87 binintersectEL: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectEL,definition,
% 3.17/1.87 ( binintersectEL
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.17/1.87 => ( in @ Xx @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectLsub_type,type,
% 3.17/1.87 binintersectLsub: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectLsub,definition,
% 3.17/1.87 ( binintersectLsub
% 3.17/1.87 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectSubset2_type,type,
% 3.17/1.87 binintersectSubset2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectSubset2,definition,
% 3.17/1.87 ( binintersectSubset2
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( subset @ A @ B )
% 3.17/1.87 => ( ( binintersect @ A @ B )
% 3.17/1.87 = A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectSubset3_type,type,
% 3.17/1.87 binintersectSubset3: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectSubset3,definition,
% 3.17/1.87 ( binintersectSubset3
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( ( binintersect @ A @ B )
% 3.17/1.87 = B )
% 3.17/1.87 => ( subset @ B @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectER_type,type,
% 3.17/1.87 binintersectER: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectER,definition,
% 3.17/1.87 ( binintersectER
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ ( binintersect @ A @ B ) )
% 3.17/1.87 => ( in @ Xx @ B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(disjointsetsI1_type,type,
% 3.17/1.87 disjointsetsI1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(disjointsetsI1,definition,
% 3.17/1.87 ( disjointsetsI1
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ~ ? [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 & ( in @ Xx @ B ) )
% 3.17/1.87 => ( ( binintersect @ A @ B )
% 3.17/1.87 = emptyset ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectRsub_type,type,
% 3.17/1.87 binintersectRsub: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectRsub,definition,
% 3.17/1.87 ( binintersectRsub
% 3.17/1.87 = ( ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ B ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectSubset4_type,type,
% 3.17/1.87 binintersectSubset4: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectSubset4,definition,
% 3.17/1.87 ( binintersectSubset4
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( subset @ B @ A )
% 3.17/1.87 => ( ( binintersect @ A @ B )
% 3.17/1.87 = B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectSubset1_type,type,
% 3.17/1.87 binintersectSubset1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectSubset1,definition,
% 3.17/1.87 ( binintersectSubset1
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( ( binintersect @ A @ B )
% 3.17/1.87 = A )
% 3.17/1.87 => ( subset @ A @ B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(bs114d_type,type,
% 3.17/1.87 bs114d: $o ).
% 3.17/1.87
% 3.17/1.87 thf(bs114d,definition,
% 3.17/1.87 ( bs114d
% 3.17/1.87 = ( ! [A: $i,B: $i,C: $i] :
% 3.17/1.87 ( ( binintersect @ A @ ( binunion @ B @ C ) )
% 3.17/1.87 = ( binunion @ ( binintersect @ A @ B ) @ ( binintersect @ A @ C ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(regular_type,type,
% 3.17/1.87 regular: $i > $o ).
% 3.17/1.87
% 3.17/1.87 thf(setminus_type,type,
% 3.17/1.87 setminus: $i > $i > $i ).
% 3.17/1.87
% 3.17/1.87 thf(setminusI_type,type,
% 3.17/1.87 setminusI: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setminusI,definition,
% 3.17/1.87 ( setminusI
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ~ ( in @ Xx @ B )
% 3.17/1.87 => ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setminusEL_type,type,
% 3.17/1.87 setminusEL: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setminusEL,definition,
% 3.17/1.87 ( setminusEL
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.17/1.87 => ( in @ Xx @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setminusER_type,type,
% 3.17/1.87 setminusER: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setminusER,definition,
% 3.17/1.87 ( setminusER
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ ( setminus @ A @ B ) )
% 3.17/1.87 => ~ ( in @ Xx @ B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setminusSubset2_type,type,
% 3.17/1.87 setminusSubset2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setminusSubset2,definition,
% 3.17/1.87 ( setminusSubset2
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( subset @ A @ B )
% 3.17/1.87 => ( ( setminus @ A @ B )
% 3.17/1.87 = emptyset ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setminusERneg_type,type,
% 3.17/1.87 setminusERneg: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setminusERneg,definition,
% 3.17/1.87 ( setminusERneg
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.17/1.87 => ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ Xx @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setminusELneg_type,type,
% 3.17/1.87 setminusELneg: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setminusELneg,definition,
% 3.17/1.87 ( setminusELneg
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ~ ( in @ Xx @ ( setminus @ A @ B ) )
% 3.17/1.87 => ( ~ ( in @ Xx @ B )
% 3.17/1.87 => ~ ( in @ Xx @ A ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setminusILneg_type,type,
% 3.17/1.87 setminusILneg: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setminusILneg,definition,
% 3.17/1.87 ( setminusILneg
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ~ ( in @ Xx @ A )
% 3.17/1.87 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setminusIRneg_type,type,
% 3.17/1.87 setminusIRneg: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setminusIRneg,definition,
% 3.17/1.87 ( setminusIRneg
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ B )
% 3.17/1.87 => ~ ( in @ Xx @ ( setminus @ A @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setminusLsub_type,type,
% 3.17/1.87 setminusLsub: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setminusLsub,definition,
% 3.17/1.87 ( setminusLsub
% 3.17/1.87 = ( ! [A: $i,B: $i] : ( subset @ ( setminus @ A @ B ) @ A ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setminusSubset1_type,type,
% 3.17/1.87 setminusSubset1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setminusSubset1,definition,
% 3.17/1.87 ( setminusSubset1
% 3.17/1.87 = ( ! [A: $i,B: $i] :
% 3.17/1.87 ( ( ( setminus @ A @ B )
% 3.17/1.87 = emptyset )
% 3.17/1.87 => ( subset @ A @ B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(symdiff_type,type,
% 3.17/1.87 symdiff: $i > $i > $i ).
% 3.17/1.87
% 3.17/1.87 thf(symdiffE_type,type,
% 3.17/1.87 symdiffE: $o ).
% 3.17/1.87
% 3.17/1.87 thf(symdiffE,definition,
% 3.17/1.87 ( symdiffE
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ ( symdiff @ A @ B ) )
% 3.17/1.87 => ! [Xphi: $o] :
% 3.17/1.87 ( ( ( in @ Xx @ A )
% 3.17/1.87 => ( ~ ( in @ Xx @ B )
% 3.17/1.87 => Xphi ) )
% 3.17/1.87 => ( ( ~ ( in @ Xx @ A )
% 3.17/1.87 => ( ( in @ Xx @ B )
% 3.17/1.87 => Xphi ) )
% 3.17/1.87 => Xphi ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(symdiffI1_type,type,
% 3.17/1.87 symdiffI1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(symdiffI1,definition,
% 3.17/1.87 ( symdiffI1
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ~ ( in @ Xx @ B )
% 3.17/1.87 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(symdiffI2_type,type,
% 3.17/1.87 symdiffI2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(symdiffI2,definition,
% 3.17/1.87 ( symdiffI2
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ~ ( in @ Xx @ A )
% 3.17/1.87 => ( ( in @ Xx @ B )
% 3.17/1.87 => ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(symdiffIneg1_type,type,
% 3.17/1.87 symdiffIneg1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(symdiffIneg1,definition,
% 3.17/1.87 ( symdiffIneg1
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( in @ Xx @ B )
% 3.17/1.87 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(symdiffIneg2_type,type,
% 3.17/1.87 symdiffIneg2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(symdiffIneg2,definition,
% 3.17/1.87 ( symdiffIneg2
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ~ ( in @ Xx @ A )
% 3.17/1.87 => ( ~ ( in @ Xx @ B )
% 3.17/1.87 => ~ ( in @ Xx @ ( symdiff @ A @ B ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(iskpair_type,type,
% 3.17/1.87 iskpair: $i > $o ).
% 3.17/1.87
% 3.17/1.87 thf(secondinupair_type,type,
% 3.17/1.87 secondinupair: $o ).
% 3.17/1.87
% 3.17/1.87 thf(secondinupair,definition,
% 3.17/1.87 ( secondinupair
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairIL_type,type,
% 3.17/1.87 setukpairIL: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairIL,definition,
% 3.17/1.87 ( setukpairIL
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] : ( in @ Xx @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairIR_type,type,
% 3.17/1.87 setukpairIR: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairIR,definition,
% 3.17/1.87 ( setukpairIR
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] : ( in @ Xy @ ( setunion @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(kpairiskpair_type,type,
% 3.17/1.87 kpairiskpair: $o ).
% 3.17/1.87
% 3.17/1.87 thf(kpairiskpair,definition,
% 3.17/1.87 ( kpairiskpair
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(kpair_type,type,
% 3.17/1.87 kpair: $i > $i > $i ).
% 3.17/1.87
% 3.17/1.87 thf(kpairp_type,type,
% 3.17/1.87 kpairp: $o ).
% 3.17/1.87
% 3.17/1.87 thf(kpairp,definition,
% 3.17/1.87 ( kpairp
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] : ( iskpair @ ( kpair @ Xx @ Xy ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(cartprod_type,type,
% 3.17/1.87 cartprod: $i > $i > $i ).
% 3.17/1.87
% 3.17/1.87 thf(singletonsubset_type,type,
% 3.17/1.87 singletonsubset: $o ).
% 3.17/1.87
% 3.17/1.87 thf(singletonsubset,definition,
% 3.17/1.87 ( singletonsubset
% 3.17/1.87 = ( ! [A: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( subset @ ( setadjoin @ Xx @ emptyset ) @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(singletoninpowerset_type,type,
% 3.17/1.87 singletoninpowerset: $o ).
% 3.17/1.87
% 3.17/1.87 thf(singletoninpowerset,definition,
% 3.17/1.87 ( singletoninpowerset
% 3.17/1.87 = ( ! [A: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ A ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(singletoninpowunion_type,type,
% 3.17/1.87 singletoninpowunion: $o ).
% 3.17/1.87
% 3.17/1.87 thf(singletoninpowunion,definition,
% 3.17/1.87 ( singletoninpowunion
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ ( setadjoin @ Xx @ emptyset ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(upairset2E_type,type,
% 3.17/1.87 upairset2E: $o ).
% 3.17/1.87
% 3.17/1.87 thf(upairset2E,definition,
% 3.17/1.87 ( upairset2E
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.17/1.87 ( ( in @ Xz @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) )
% 3.17/1.87 => ( ( Xz = Xx )
% 3.17/1.87 | ( Xz = Xy ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(upairsubunion_type,type,
% 3.17/1.87 upairsubunion: $o ).
% 3.17/1.87
% 3.17/1.87 thf(upairsubunion,definition,
% 3.17/1.87 ( upairsubunion
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( subset @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( binunion @ A @ B ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(upairinpowunion_type,type,
% 3.17/1.87 upairinpowunion: $o ).
% 3.17/1.87
% 3.17/1.87 thf(upairinpowunion,definition,
% 3.17/1.87 ( upairinpowunion
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( in @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(ubforcartprodlem1_type,type,
% 3.17/1.87 ubforcartprodlem1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(ubforcartprodlem1,definition,
% 3.17/1.87 ( ubforcartprodlem1
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( subset @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(ubforcartprodlem2_type,type,
% 3.17/1.87 ubforcartprodlem2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(ubforcartprodlem2,definition,
% 3.17/1.87 ( ubforcartprodlem2
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( in @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(ubforcartprodlem3_type,type,
% 3.17/1.87 ubforcartprodlem3: $o ).
% 3.17/1.87
% 3.17/1.87 thf(ubforcartprodlem3,definition,
% 3.17/1.87 ( ubforcartprodlem3
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( in @ ( kpair @ Xx @ Xy ) @ ( powerset @ ( powerset @ ( binunion @ A @ B ) ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodpairin_type,type,
% 3.17/1.87 cartprodpairin: $o ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodpairin,definition,
% 3.17/1.87 ( cartprodpairin
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodmempair1_type,type,
% 3.17/1.87 cartprodmempair1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodmempair1,definition,
% 3.17/1.87 ( cartprodmempair1
% 3.17/1.87 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.17/1.87 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.17/1.87 => ? [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 & ? [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 & ( Xu
% 3.17/1.87 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodmempair_type,type,
% 3.17/1.87 cartprodmempair: $o ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodmempair,definition,
% 3.17/1.87 ( cartprodmempair
% 3.17/1.87 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.17/1.87 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.17/1.87 => ( iskpair @ Xu ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setunionE2_type,type,
% 3.17/1.87 setunionE2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setunionE2,definition,
% 3.17/1.87 ( setunionE2
% 3.17/1.87 = ( ! [A: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ ( setunion @ A ) )
% 3.17/1.87 => ? [X: $i] :
% 3.17/1.87 ( ( in @ X @ A )
% 3.17/1.87 & ( in @ Xx @ X ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setunionsingleton1_type,type,
% 3.17/1.87 setunionsingleton1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setunionsingleton1,definition,
% 3.17/1.87 ( setunionsingleton1
% 3.17/1.87 = ( ! [A: $i] : ( subset @ ( setunion @ ( setadjoin @ A @ emptyset ) ) @ A ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setunionsingleton2_type,type,
% 3.17/1.87 setunionsingleton2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setunionsingleton2,definition,
% 3.17/1.87 ( setunionsingleton2
% 3.17/1.87 = ( ! [A: $i] : ( subset @ A @ ( setunion @ ( setadjoin @ A @ emptyset ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setunionsingleton_type,type,
% 3.17/1.87 setunionsingleton: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setunionsingleton,definition,
% 3.17/1.87 ( setunionsingleton
% 3.17/1.87 = ( ! [Xx: $i] :
% 3.17/1.87 ( ( setunion @ ( setadjoin @ Xx @ emptyset ) )
% 3.17/1.87 = Xx ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(singleton_type,type,
% 3.17/1.87 singleton: $i > $o ).
% 3.17/1.87
% 3.17/1.87 thf(singletonprop_type,type,
% 3.17/1.87 singletonprop: $o ).
% 3.17/1.87
% 3.17/1.87 thf(singletonprop,definition,
% 3.17/1.87 ( singletonprop
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ A )
% 3.17/1.87 => ( ( Xphi @ Xx )
% 3.17/1.87 => ( ( Xphi @ Xy )
% 3.17/1.87 => ( Xx = Xy ) ) ) ) )
% 3.17/1.87 => ( ? [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 & ( Xphi @ Xx ) )
% 3.17/1.87 => ( singleton
% 3.17/1.87 @ ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(ex1_type,type,
% 3.17/1.87 ex1: $i > ( $i > $o ) > $o ).
% 3.17/1.87
% 3.17/1.87 thf(ex1,definition,
% 3.17/1.87 ( ex1
% 3.17/1.87 = ( ^ [A: $i,Xphi: $i > $o] :
% 3.17/1.87 ( singleton
% 3.17/1.87 @ ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(ex1E1_type,type,
% 3.17/1.87 ex1E1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(ex1E1,definition,
% 3.17/1.87 ( ex1E1
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.87 ( ( ex1 @ A
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.87 => ? [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 & ( Xphi @ Xx ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(ex1I_type,type,
% 3.17/1.87 ex1I: $o ).
% 3.17/1.87
% 3.17/1.87 thf(ex1I,definition,
% 3.17/1.87 ( ex1I
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( Xphi @ Xx )
% 3.17/1.87 => ( ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ A )
% 3.17/1.87 => ( ( Xphi @ Xy )
% 3.17/1.87 => ( Xy = Xx ) ) )
% 3.17/1.87 => ( ex1 @ A
% 3.17/1.87 @ ^ [Xy: $i] : ( Xphi @ Xy ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(ex1I2_type,type,
% 3.17/1.87 ex1I2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(ex1I2,definition,
% 3.17/1.87 ( ex1I2
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ A )
% 3.17/1.87 => ( ( Xphi @ Xx )
% 3.17/1.87 => ( ( Xphi @ Xy )
% 3.17/1.87 => ( Xx = Xy ) ) ) ) )
% 3.17/1.87 => ( ? [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 & ( Xphi @ Xx ) )
% 3.17/1.87 => ( ex1 @ A
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(singletonsuniq_type,type,
% 3.17/1.87 singletonsuniq: $o ).
% 3.17/1.87
% 3.17/1.87 thf(singletonsuniq,definition,
% 3.17/1.87 ( singletonsuniq
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( ( setadjoin @ Xx @ emptyset )
% 3.17/1.87 = ( setadjoin @ Xy @ emptyset ) )
% 3.17/1.87 => ( Xx = Xy ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(atmost1p_type,type,
% 3.17/1.87 atmost1p: $i > $o ).
% 3.17/1.87
% 3.17/1.87 thf(atleast2p_type,type,
% 3.17/1.87 atleast2p: $i > $o ).
% 3.17/1.87
% 3.17/1.87 thf(atmost2p_type,type,
% 3.17/1.87 atmost2p: $i > $o ).
% 3.17/1.87
% 3.17/1.87 thf(upairsetp_type,type,
% 3.17/1.87 upairsetp: $i > $o ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairinjL1_type,type,
% 3.17/1.87 setukpairinjL1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairinjL1,definition,
% 3.17/1.87 ( setukpairinjL1
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.17/1.87 ( ( in @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) ) )
% 3.17/1.87 => ( Xx = Xz ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(kfstsingleton_type,type,
% 3.17/1.87 kfstsingleton: $o ).
% 3.17/1.87
% 3.17/1.87 thf(kfstsingleton,definition,
% 3.17/1.87 ( kfstsingleton
% 3.17/1.87 = ( ! [Xu: $i] :
% 3.17/1.87 ( ( iskpair @ Xu )
% 3.17/1.87 => ( singleton
% 3.17/1.87 @ ( dsetconstr @ ( setunion @ Xu )
% 3.17/1.87 @ ^ [Xx: $i] : ( in @ ( setadjoin @ Xx @ emptyset ) @ Xu ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(theprop_type,type,
% 3.17/1.87 theprop: $o ).
% 3.17/1.87
% 3.17/1.87 thf(theprop,definition,
% 3.17/1.87 ( theprop
% 3.17/1.87 = ( ! [X: $i] :
% 3.17/1.87 ( ( singleton @ X )
% 3.17/1.87 => ( in @ ( setunion @ X ) @ X ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(kfst_type,type,
% 3.17/1.87 kfst: $i > $i ).
% 3.17/1.87
% 3.17/1.87 thf(kfstpairEq_type,type,
% 3.17/1.87 kfstpairEq: $o ).
% 3.17/1.87
% 3.17/1.87 thf(kfstpairEq,definition,
% 3.17/1.87 ( kfstpairEq
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.17/1.87 = Xx ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodfstin_type,type,
% 3.17/1.87 cartprodfstin: $o ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodfstin,definition,
% 3.17/1.87 ( cartprodfstin
% 3.17/1.87 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.17/1.87 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.17/1.87 => ( in @ ( kfst @ Xu ) @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairinjL2_type,type,
% 3.17/1.87 setukpairinjL2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairinjL2,definition,
% 3.17/1.87 ( setukpairinjL2
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.17/1.87 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.17/1.87 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.17/1.87 => ( Xx = Xz ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairinjL_type,type,
% 3.17/1.87 setukpairinjL: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairinjL,definition,
% 3.17/1.87 ( setukpairinjL
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.17/1.87 ( ( ( kpair @ Xx @ Xy )
% 3.17/1.87 = ( kpair @ Xz @ Xu ) )
% 3.17/1.87 => ( Xx = Xz ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairinjR11_type,type,
% 3.17/1.87 setukpairinjR11: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairinjR11,definition,
% 3.17/1.87 ( setukpairinjR11
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( Xx = Xy )
% 3.17/1.87 => ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.17/1.87 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairinjR12_type,type,
% 3.17/1.87 setukpairinjR12: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairinjR12,definition,
% 3.17/1.87 ( setukpairinjR12
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( Xx = Xy )
% 3.17/1.87 => ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.17/1.87 = ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ emptyset ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairinjR1_type,type,
% 3.17/1.87 setukpairinjR1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairinjR1,definition,
% 3.17/1.87 ( setukpairinjR1
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.17/1.87 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.17/1.87 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.17/1.87 => ( ( Xz = Xu )
% 3.17/1.87 => ( Xy = Xu ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(upairequniteq_type,type,
% 3.17/1.87 upairequniteq: $o ).
% 3.17/1.87
% 3.17/1.87 thf(upairequniteq,definition,
% 3.17/1.87 ( upairequniteq
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i] :
% 3.17/1.87 ( ( ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) )
% 3.17/1.87 = ( setadjoin @ Xz @ emptyset ) )
% 3.17/1.87 => ( Xx = Xy ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairinjR2_type,type,
% 3.17/1.87 setukpairinjR2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairinjR2,definition,
% 3.17/1.87 ( setukpairinjR2
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.17/1.87 ( ( ( setadjoin @ ( setadjoin @ Xx @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) @ emptyset ) )
% 3.17/1.87 = ( setadjoin @ ( setadjoin @ Xz @ emptyset ) @ ( setadjoin @ ( setadjoin @ Xz @ ( setadjoin @ Xu @ emptyset ) ) @ emptyset ) ) )
% 3.17/1.87 => ( Xy = Xu ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairinjR_type,type,
% 3.17/1.87 setukpairinjR: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setukpairinjR,definition,
% 3.17/1.87 ( setukpairinjR
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i,Xz: $i,Xu: $i] :
% 3.17/1.87 ( ( ( kpair @ Xx @ Xy )
% 3.17/1.87 = ( kpair @ Xz @ Xu ) )
% 3.17/1.87 => ( Xy = Xu ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(ksndsingleton_type,type,
% 3.17/1.87 ksndsingleton: $o ).
% 3.17/1.87
% 3.17/1.87 thf(ksndsingleton,definition,
% 3.17/1.87 ( ksndsingleton
% 3.17/1.87 = ( ! [Xu: $i] :
% 3.17/1.87 ( ( iskpair @ Xu )
% 3.17/1.87 => ( singleton
% 3.17/1.87 @ ( dsetconstr @ ( setunion @ Xu )
% 3.17/1.87 @ ^ [Xx: $i] :
% 3.17/1.87 ( Xu
% 3.17/1.87 = ( kpair @ ( kfst @ Xu ) @ Xx ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(ksnd_type,type,
% 3.17/1.87 ksnd: $i > $i ).
% 3.17/1.87
% 3.17/1.87 thf(ksndpairEq_type,type,
% 3.17/1.87 ksndpairEq: $o ).
% 3.17/1.87
% 3.17/1.87 thf(ksndpairEq,definition,
% 3.17/1.87 ( ksndpairEq
% 3.17/1.87 = ( ! [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.17/1.87 = Xy ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(kpairsurjEq_type,type,
% 3.17/1.87 kpairsurjEq: $o ).
% 3.17/1.87
% 3.17/1.87 thf(kpairsurjEq,definition,
% 3.17/1.87 ( kpairsurjEq
% 3.17/1.87 = ( ! [Xu: $i] :
% 3.17/1.87 ( ( iskpair @ Xu )
% 3.17/1.87 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.17/1.87 = Xu ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodsndin_type,type,
% 3.17/1.87 cartprodsndin: $o ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodsndin,definition,
% 3.17/1.87 ( cartprodsndin
% 3.17/1.87 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.17/1.87 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.17/1.87 => ( in @ ( ksnd @ Xu ) @ B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodpairmemEL_type,type,
% 3.17/1.87 cartprodpairmemEL: $o ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodpairmemEL,definition,
% 3.17/1.87 ( cartprodpairmemEL
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.17/1.87 => ( in @ Xx @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodpairmemER_type,type,
% 3.17/1.87 cartprodpairmemER: $o ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodpairmemER,definition,
% 3.17/1.87 ( cartprodpairmemER
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( in @ ( kpair @ Xx @ Xy ) @ ( cartprod @ A @ B ) )
% 3.17/1.87 => ( in @ Xy @ B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodmempaircEq_type,type,
% 3.17/1.87 cartprodmempaircEq: $o ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodmempaircEq,definition,
% 3.17/1.87 ( cartprodmempaircEq
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( ( kpair @ Xx @ Xy )
% 3.17/1.87 = ( kpair @ Xx @ Xy ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodfstpairEq_type,type,
% 3.17/1.87 cartprodfstpairEq: $o ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodfstpairEq,definition,
% 3.17/1.87 ( cartprodfstpairEq
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( ( kfst @ ( kpair @ Xx @ Xy ) )
% 3.17/1.87 = Xx ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodsndpairEq_type,type,
% 3.17/1.87 cartprodsndpairEq: $o ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodsndpairEq,definition,
% 3.17/1.87 ( cartprodsndpairEq
% 3.17/1.87 = ( ! [A: $i,B: $i,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( ( ksnd @ ( kpair @ Xx @ Xy ) )
% 3.17/1.87 = Xy ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodpairsurjEq_type,type,
% 3.17/1.87 cartprodpairsurjEq: $o ).
% 3.17/1.87
% 3.17/1.87 thf(cartprodpairsurjEq,definition,
% 3.17/1.87 ( cartprodpairsurjEq
% 3.17/1.87 = ( ! [A: $i,B: $i,Xu: $i] :
% 3.17/1.87 ( ( in @ Xu @ ( cartprod @ A @ B ) )
% 3.17/1.87 => ( ( kpair @ ( kfst @ Xu ) @ ( ksnd @ Xu ) )
% 3.17/1.87 = Xu ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(breln_type,type,
% 3.17/1.87 breln: $i > $i > $i > $o ).
% 3.17/1.87
% 3.17/1.87 thf(breln,definition,
% 3.17/1.87 ( breln
% 3.17/1.87 = ( ^ [A: $i,B: $i,C: $i] : ( subset @ C @ ( cartprod @ A @ B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(dpsetconstr_type,type,
% 3.17/1.87 dpsetconstr: $i > $i > ( $i > $i > $o ) > $i ).
% 3.17/1.87
% 3.17/1.87 thf(dpsetconstrI_type,type,
% 3.17/1.87 dpsetconstrI: $o ).
% 3.17/1.87
% 3.17/1.87 thf(dpsetconstrI,definition,
% 3.17/1.87 ( dpsetconstrI
% 3.17/1.87 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( ( Xphi @ Xx @ Xy )
% 3.17/1.87 => ( in @ ( kpair @ Xx @ Xy )
% 3.17/1.87 @ ( dpsetconstr @ A @ B
% 3.17/1.87 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(dpsetconstrSub_type,type,
% 3.17/1.87 dpsetconstrSub: $o ).
% 3.17/1.87
% 3.17/1.87 thf(dpsetconstrSub,definition,
% 3.17/1.87 ( dpsetconstrSub
% 3.17/1.87 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.17/1.87 ( subset
% 3.17/1.87 @ ( dpsetconstr @ A @ B
% 3.17/1.87 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) )
% 3.17/1.87 @ ( cartprod @ A @ B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setOfPairsIsBReln_type,type,
% 3.17/1.87 setOfPairsIsBReln: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setOfPairsIsBReln,definition,
% 3.17/1.87 ( setOfPairsIsBReln
% 3.17/1.87 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o] :
% 3.17/1.87 ( breln @ A @ B
% 3.17/1.87 @ ( dpsetconstr @ A @ B
% 3.17/1.87 @ ^ [Xx: $i,Xy: $i] : ( Xphi @ Xx @ Xy ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(dpsetconstrERa_type,type,
% 3.17/1.87 dpsetconstrERa: $o ).
% 3.17/1.87
% 3.17/1.87 thf(dpsetconstrERa,definition,
% 3.17/1.87 ( dpsetconstrERa
% 3.17/1.87 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( ( in @ ( kpair @ Xx @ Xy )
% 3.17/1.87 @ ( dpsetconstr @ A @ B
% 3.17/1.87 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.17/1.87 => ( Xphi @ Xx @ Xy ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(dpsetconstrEL1_type,type,
% 3.17/1.87 dpsetconstrEL1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(dpsetconstrEL1,definition,
% 3.17/1.87 ( dpsetconstrEL1
% 3.17/1.87 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( in @ ( kpair @ Xx @ Xy )
% 3.17/1.87 @ ( dpsetconstr @ A @ B
% 3.17/1.87 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.17/1.87 => ( in @ Xx @ A ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(dpsetconstrEL2_type,type,
% 3.17/1.87 dpsetconstrEL2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(dpsetconstrEL2,definition,
% 3.17/1.87 ( dpsetconstrEL2
% 3.17/1.87 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( in @ ( kpair @ Xx @ Xy )
% 3.17/1.87 @ ( dpsetconstr @ A @ B
% 3.17/1.87 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.17/1.87 => ( in @ Xy @ B ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(dpsetconstrER_type,type,
% 3.17/1.87 dpsetconstrER: $o ).
% 3.17/1.87
% 3.17/1.87 thf(dpsetconstrER,definition,
% 3.17/1.87 ( dpsetconstrER
% 3.17/1.87 = ( ! [A: $i,B: $i,Xphi: $i > $i > $o,Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( in @ ( kpair @ Xx @ Xy )
% 3.17/1.87 @ ( dpsetconstr @ A @ B
% 3.17/1.87 @ ^ [Xz: $i,Xu: $i] : ( Xphi @ Xz @ Xu ) ) )
% 3.17/1.87 => ( Xphi @ Xx @ Xy ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(func_type,type,
% 3.17/1.87 func: $i > $i > $i > $o ).
% 3.17/1.87
% 3.17/1.87 thf(funcSet_type,type,
% 3.17/1.87 funcSet: $i > $i > $i ).
% 3.17/1.87
% 3.17/1.87 thf(funcImageSingleton_type,type,
% 3.17/1.87 funcImageSingleton: $o ).
% 3.17/1.87
% 3.17/1.87 thf(funcImageSingleton,definition,
% 3.17/1.87 ( funcImageSingleton
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( func @ A @ B @ Xf )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( singleton
% 3.17/1.87 @ ( dsetconstr @ B
% 3.17/1.87 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(apProp_type,type,
% 3.17/1.87 apProp: $o ).
% 3.17/1.87
% 3.17/1.87 thf(apProp,definition,
% 3.17/1.87 ( apProp
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( func @ A @ B @ Xf )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in
% 3.17/1.87 @ ( setunion
% 3.17/1.87 @ ( dsetconstr @ B
% 3.17/1.87 @ ^ [Xy: $i] : ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) )
% 3.17/1.87 @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(ap_type,type,
% 3.17/1.87 ap: $i > $i > $i > $i > $i ).
% 3.17/1.87
% 3.17/1.87 thf(app_type,type,
% 3.17/1.87 app: $o ).
% 3.17/1.87
% 3.17/1.87 thf(app,definition,
% 3.17/1.87 ( app
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( func @ A @ B @ Xf )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(infuncsetfunc_type,type,
% 3.17/1.87 infuncsetfunc: $o ).
% 3.17/1.87
% 3.17/1.87 thf(infuncsetfunc,definition,
% 3.17/1.87 ( infuncsetfunc
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.87 => ( func @ A @ B @ Xf ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(ap2p_type,type,
% 3.17/1.87 ap2p: $o ).
% 3.17/1.87
% 3.17/1.87 thf(ap2p,definition,
% 3.17/1.87 ( ap2p
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ ( ap @ A @ B @ Xf @ Xx ) @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(funcinfuncset_type,type,
% 3.17/1.87 funcinfuncset: $o ).
% 3.17/1.87
% 3.17/1.87 thf(funcinfuncset,definition,
% 3.17/1.87 ( funcinfuncset
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( func @ A @ B @ Xf )
% 3.17/1.87 => ( in @ Xf @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(lamProp_type,type,
% 3.17/1.87 lamProp: $o ).
% 3.17/1.87
% 3.17/1.87 thf(lamProp,definition,
% 3.17/1.87 ( lamProp
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.87 => ( func @ A @ B
% 3.17/1.87 @ ( dpsetconstr @ A @ B
% 3.17/1.87 @ ^ [Xx: $i,Xy: $i] :
% 3.17/1.87 ( ( Xf @ Xx )
% 3.17/1.87 = Xy ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(lam_type,type,
% 3.17/1.87 lam: $i > $i > ( $i > $i ) > $i ).
% 3.17/1.87
% 3.17/1.87 thf(lamp_type,type,
% 3.17/1.87 lamp: $o ).
% 3.17/1.87
% 3.17/1.87 thf(lamp,definition,
% 3.17/1.87 ( lamp
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.87 => ( func @ A @ B
% 3.17/1.87 @ ( lam @ A @ B
% 3.17/1.87 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(lam2p_type,type,
% 3.17/1.87 lam2p: $o ).
% 3.17/1.87
% 3.17/1.87 thf(lam2p,definition,
% 3.17/1.87 ( lam2p
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.87 => ( in
% 3.17/1.87 @ ( lam @ A @ B
% 3.17/1.87 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.17/1.87 @ ( funcSet @ A @ B ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(brelnall1_type,type,
% 3.17/1.87 brelnall1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(brelnall1,definition,
% 3.17/1.87 ( brelnall1
% 3.17/1.87 = ( ! [A: $i,B: $i,R: $i] :
% 3.17/1.87 ( ( breln @ A @ B @ R )
% 3.17/1.87 => ! [Xphi: $i > $o] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.87 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ R )
% 3.17/1.87 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(brelnall2_type,type,
% 3.17/1.87 brelnall2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(brelnall2,definition,
% 3.17/1.87 ( brelnall2
% 3.17/1.87 = ( ! [A: $i,B: $i,R: $i] :
% 3.17/1.87 ( ( breln @ A @ B @ R )
% 3.17/1.87 => ! [Xphi: $i > $o] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.87 => ( Xphi @ ( kpair @ Xx @ Xy ) ) ) ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ R )
% 3.17/1.87 => ( Xphi @ Xx ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(ex1E2_type,type,
% 3.17/1.87 ex1E2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(ex1E2,definition,
% 3.17/1.87 ( ex1E2
% 3.17/1.87 = ( ! [A: $i,Xphi: $i > $o] :
% 3.17/1.87 ( ( ex1 @ A
% 3.17/1.87 @ ^ [Xx: $i] : ( Xphi @ Xx ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ A )
% 3.17/1.87 => ( ( Xphi @ Xx )
% 3.17/1.87 => ( ( Xphi @ Xy )
% 3.17/1.87 => ( Xx = Xy ) ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(funcGraphProp1_type,type,
% 3.17/1.87 funcGraphProp1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(funcGraphProp1,definition,
% 3.17/1.87 ( funcGraphProp1
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( func @ A @ B @ Xf )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(funcGraphProp3_type,type,
% 3.17/1.87 funcGraphProp3: $o ).
% 3.17/1.87
% 3.17/1.87 thf(funcGraphProp3,definition,
% 3.17/1.87 ( funcGraphProp3
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ ( kpair @ Xx @ ( ap @ A @ B @ Xf @ Xx ) ) @ Xf ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(funcGraphProp2_type,type,
% 3.17/1.87 funcGraphProp2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(funcGraphProp2,definition,
% 3.17/1.87 ( funcGraphProp2
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( func @ A @ B @ Xf )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.17/1.87 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.87 = Xy ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(funcextLem_type,type,
% 3.17/1.87 funcextLem: $o ).
% 3.17/1.87
% 3.17/1.87 thf(funcextLem,definition,
% 3.17/1.87 ( funcextLem
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( func @ A @ B @ Xf )
% 3.17/1.87 => ! [Xg: $i] :
% 3.17/1.87 ( ( func @ A @ B @ Xg )
% 3.17/1.87 => ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.87 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xg )
% 3.17/1.87 => ( in @ ( kpair @ Xx @ Xy ) @ Xf ) ) ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(funcGraphProp4_type,type,
% 3.17/1.87 funcGraphProp4: $o ).
% 3.17/1.87
% 3.17/1.87 thf(funcGraphProp4,definition,
% 3.17/1.87 ( funcGraphProp4
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( ( in @ ( kpair @ Xx @ Xy ) @ Xf )
% 3.17/1.87 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.87 = Xy ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(subbreln_type,type,
% 3.17/1.87 subbreln: $o ).
% 3.17/1.87
% 3.17/1.87 thf(subbreln,definition,
% 3.17/1.87 ( subbreln
% 3.17/1.87 = ( ! [A: $i,B: $i,R: $i] :
% 3.17/1.87 ( ( breln @ A @ B @ R )
% 3.17/1.87 => ! [S: $i] :
% 3.17/1.87 ( ( breln @ A @ B @ S )
% 3.17/1.87 => ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.87 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.17/1.87 => ( subset @ R @ S ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(eqbreln_type,type,
% 3.17/1.87 eqbreln: $o ).
% 3.17/1.87
% 3.17/1.87 thf(eqbreln,definition,
% 3.17/1.87 ( eqbreln
% 3.17/1.87 = ( ! [A: $i,B: $i,R: $i] :
% 3.17/1.87 ( ( breln @ A @ B @ R )
% 3.17/1.87 => ! [S: $i] :
% 3.17/1.87 ( ( breln @ A @ B @ S )
% 3.17/1.87 => ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( ( in @ ( kpair @ Xx @ Xy ) @ R )
% 3.17/1.87 => ( in @ ( kpair @ Xx @ Xy ) @ S ) ) ) )
% 3.17/1.87 => ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ B )
% 3.17/1.87 => ( ( in @ ( kpair @ Xx @ Xy ) @ S )
% 3.17/1.87 => ( in @ ( kpair @ Xx @ Xy ) @ R ) ) ) )
% 3.17/1.87 => ( R = S ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(funcext_type,type,
% 3.17/1.87 funcext: $o ).
% 3.17/1.87
% 3.17/1.87 thf(funcext,definition,
% 3.17/1.87 ( funcext
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( func @ A @ B @ Xf )
% 3.17/1.87 => ! [Xg: $i] :
% 3.17/1.87 ( ( func @ A @ B @ Xg )
% 3.17/1.87 => ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.87 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.17/1.87 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(funcext2_type,type,
% 3.17/1.87 funcext2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(funcext2,definition,
% 3.17/1.87 ( funcext2
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.87 => ! [Xg: $i] :
% 3.17/1.87 ( ( in @ Xg @ ( funcSet @ A @ B ) )
% 3.17/1.87 => ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.87 = ( ap @ A @ B @ Xg @ Xx ) ) )
% 3.17/1.87 => ( Xf = Xg ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(ap2apEq1_type,type,
% 3.17/1.87 ap2apEq1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(ap2apEq1,definition,
% 3.17/1.87 ( ap2apEq1
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.87 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(ap2apEq2_type,type,
% 3.17/1.87 ap2apEq2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(ap2apEq2,definition,
% 3.17/1.87 ( ap2apEq2
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( func @ A @ B @ Xf )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( ap @ A @ B @ Xf @ Xx )
% 3.17/1.87 = ( ap @ A @ B @ Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(beta1_type,type,
% 3.17/1.87 beta1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(beta1,definition,
% 3.17/1.87 ( beta1
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( ap @ A @ B
% 3.17/1.87 @ ( lam @ A @ B
% 3.17/1.87 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.17/1.87 @ Xx )
% 3.17/1.87 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(eta1_type,type,
% 3.17/1.87 eta1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(eta1,definition,
% 3.17/1.87 ( eta1
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( func @ A @ B @ Xf )
% 3.17/1.87 => ( ( lam @ A @ B
% 3.17/1.87 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.17/1.87 = Xf ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(lam2lamEq_type,type,
% 3.17/1.87 lam2lamEq: $o ).
% 3.17/1.87
% 3.17/1.87 thf(lam2lamEq,definition,
% 3.17/1.87 ( lam2lamEq
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.87 => ( ( lam @ A @ B
% 3.17/1.87 @ ^ [Xx: $i] : ( Xf @ Xx ) )
% 3.17/1.87 = ( lam @ A @ B
% 3.17/1.87 @ ^ [Xx: $i] : ( Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(beta2_type,type,
% 3.17/1.87 beta2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(beta2,definition,
% 3.17/1.87 ( beta2
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i > $i] :
% 3.17/1.87 ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( in @ ( Xf @ Xx ) @ B ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( ap @ A @ B
% 3.17/1.87 @ ( lam @ A @ B
% 3.17/1.87 @ ^ [Xy: $i] : ( Xf @ Xy ) )
% 3.17/1.87 @ Xx )
% 3.17/1.87 = ( Xf @ Xx ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(eta2_type,type,
% 3.17/1.87 eta2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(eta2,definition,
% 3.17/1.87 ( eta2
% 3.17/1.87 = ( ! [A: $i,B: $i,Xf: $i] :
% 3.17/1.87 ( ( in @ Xf @ ( funcSet @ A @ B ) )
% 3.17/1.87 => ( ( lam @ A @ B
% 3.17/1.87 @ ^ [Xx: $i] : ( ap @ A @ B @ Xf @ Xx ) )
% 3.17/1.87 = Xf ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(iffalseProp1_type,type,
% 3.17/1.87 iffalseProp1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(iffalseProp1,definition,
% 3.17/1.87 ( iffalseProp1
% 3.17/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ A )
% 3.17/1.87 => ( ~ Xphi
% 3.17/1.87 => ( in @ Xy
% 3.17/1.87 @ ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xz: $i] :
% 3.17/1.87 ( ( Xphi
% 3.17/1.87 & ( Xz = Xx ) )
% 3.17/1.87 | ( ~ Xphi
% 3.17/1.87 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(iffalseProp2_type,type,
% 3.17/1.87 iffalseProp2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(iffalseProp2,definition,
% 3.17/1.87 ( iffalseProp2
% 3.17/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ A )
% 3.17/1.87 => ( ~ Xphi
% 3.17/1.87 => ( ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xz: $i] :
% 3.17/1.87 ( ( Xphi
% 3.17/1.87 & ( Xz = Xx ) )
% 3.17/1.87 | ( ~ Xphi
% 3.17/1.87 & ( Xz = Xy ) ) ) )
% 3.17/1.87 = ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(iftrueProp1_type,type,
% 3.17/1.87 iftrueProp1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(iftrueProp1,definition,
% 3.17/1.87 ( iftrueProp1
% 3.17/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ A )
% 3.17/1.87 => ( Xphi
% 3.17/1.87 => ( in @ Xx
% 3.17/1.87 @ ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xz: $i] :
% 3.17/1.87 ( ( Xphi
% 3.17/1.87 & ( Xz = Xx ) )
% 3.17/1.87 | ( ~ Xphi
% 3.17/1.87 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(iftrueProp2_type,type,
% 3.17/1.87 iftrueProp2: $o ).
% 3.17/1.87
% 3.17/1.87 thf(iftrueProp2,definition,
% 3.17/1.87 ( iftrueProp2
% 3.17/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ A )
% 3.17/1.87 => ( Xphi
% 3.17/1.87 => ( ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xz: $i] :
% 3.17/1.87 ( ( Xphi
% 3.17/1.87 & ( Xz = Xx ) )
% 3.17/1.87 | ( ~ Xphi
% 3.17/1.87 & ( Xz = Xy ) ) ) )
% 3.17/1.87 = ( setadjoin @ Xx @ emptyset ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(ifSingleton_type,type,
% 3.17/1.87 ifSingleton: $o ).
% 3.17/1.87
% 3.17/1.87 thf(ifSingleton,definition,
% 3.17/1.87 ( ifSingleton
% 3.17/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ A )
% 3.17/1.87 => ( singleton
% 3.17/1.87 @ ( dsetconstr @ A
% 3.17/1.87 @ ^ [Xz: $i] :
% 3.17/1.87 ( ( Xphi
% 3.17/1.87 & ( Xz = Xx ) )
% 3.17/1.87 | ( ~ Xphi
% 3.17/1.87 & ( Xz = Xy ) ) ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(if_type,type,
% 3.17/1.87 if: $i > $o > $i > $i > $i ).
% 3.17/1.87
% 3.17/1.87 thf(ifp_type,type,
% 3.17/1.87 ifp: $o ).
% 3.17/1.87
% 3.17/1.87 thf(ifp,definition,
% 3.17/1.87 ( ifp
% 3.17/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ A )
% 3.17/1.87 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ A ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(theeq_type,type,
% 3.17/1.87 theeq: $o ).
% 3.17/1.87
% 3.17/1.87 thf(theeq,definition,
% 3.17/1.87 ( theeq
% 3.17/1.87 = ( ! [X: $i] :
% 3.17/1.87 ( ( singleton @ X )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ X )
% 3.17/1.87 => ( ( setunion @ X )
% 3.17/1.87 = Xx ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(iftrue_type,type,
% 3.17/1.87 iftrue: $o ).
% 3.17/1.87
% 3.17/1.87 thf(iftrue,definition,
% 3.17/1.87 ( iftrue
% 3.17/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ A )
% 3.17/1.87 => ( Xphi
% 3.17/1.87 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.17/1.87 = Xx ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(iffalse_type,type,
% 3.17/1.87 iffalse: $o ).
% 3.17/1.87
% 3.17/1.87 thf(iffalse,definition,
% 3.17/1.87 ( iffalse
% 3.17/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ A )
% 3.17/1.87 => ( ~ Xphi
% 3.17/1.87 => ( ( if @ A @ Xphi @ Xx @ Xy )
% 3.17/1.87 = Xy ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(iftrueorfalse_type,type,
% 3.17/1.87 iftrueorfalse: $o ).
% 3.17/1.87
% 3.17/1.87 thf(iftrueorfalse,definition,
% 3.17/1.87 ( iftrueorfalse
% 3.17/1.87 = ( ! [A: $i,Xphi: $o,Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ! [Xy: $i] :
% 3.17/1.87 ( ( in @ Xy @ A )
% 3.17/1.87 => ( in @ ( if @ A @ Xphi @ Xx @ Xy ) @ ( setadjoin @ Xx @ ( setadjoin @ Xy @ emptyset ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectT_lem_type,type,
% 3.17/1.87 binintersectT_lem: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectT_lem,definition,
% 3.17/1.87 ( binintersectT_lem
% 3.17/1.87 = ( ! [A: $i,X: $i] :
% 3.17/1.87 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.87 => ! [Y: $i] :
% 3.17/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.87 => ( in @ ( binintersect @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binunionT_lem_type,type,
% 3.17/1.87 binunionT_lem: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binunionT_lem,definition,
% 3.17/1.87 ( binunionT_lem
% 3.17/1.87 = ( ! [A: $i,X: $i] :
% 3.17/1.87 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.87 => ! [Y: $i] :
% 3.17/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.87 => ( in @ ( binunion @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(powersetT_lem_type,type,
% 3.17/1.87 powersetT_lem: $o ).
% 3.17/1.87
% 3.17/1.87 thf(powersetT_lem,definition,
% 3.17/1.87 ( powersetT_lem
% 3.17/1.87 = ( ! [A: $i,X: $i] :
% 3.17/1.87 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.87 => ( in @ ( powerset @ X ) @ ( powerset @ ( powerset @ A ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setminusT_lem_type,type,
% 3.17/1.87 setminusT_lem: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setminusT_lem,definition,
% 3.17/1.87 ( setminusT_lem
% 3.17/1.87 = ( ! [A: $i,X: $i] :
% 3.17/1.87 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.87 => ! [Y: $i] :
% 3.17/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.87 => ( in @ ( setminus @ X @ Y ) @ ( powerset @ A ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(complementT_lem_type,type,
% 3.17/1.87 complementT_lem: $o ).
% 3.17/1.87
% 3.17/1.87 thf(complementT_lem,definition,
% 3.17/1.87 ( complementT_lem
% 3.17/1.87 = ( ! [A: $i,X: $i] :
% 3.17/1.87 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.87 => ( in @ ( setminus @ A @ X ) @ ( powerset @ A ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(setextT_type,type,
% 3.17/1.87 setextT: $o ).
% 3.17/1.87
% 3.17/1.87 thf(setextT,definition,
% 3.17/1.87 ( setextT
% 3.17/1.87 = ( ! [A: $i,X: $i] :
% 3.17/1.87 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.87 => ! [Y: $i] :
% 3.17/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.87 => ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( in @ Xx @ X )
% 3.17/1.87 => ( in @ Xx @ Y ) ) )
% 3.17/1.87 => ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( in @ Xx @ Y )
% 3.17/1.87 => ( in @ Xx @ X ) ) )
% 3.17/1.87 => ( X = Y ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(subsetTI_type,type,
% 3.17/1.87 subsetTI: $o ).
% 3.17/1.87
% 3.17/1.87 thf(subsetTI,definition,
% 3.17/1.87 ( subsetTI
% 3.17/1.87 = ( ! [A: $i,X: $i] :
% 3.17/1.87 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.87 => ! [Y: $i] :
% 3.17/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.87 => ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( in @ Xx @ X )
% 3.17/1.87 => ( in @ Xx @ Y ) ) )
% 3.17/1.87 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(powersetTI1_type,type,
% 3.17/1.87 powersetTI1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(powersetTI1,definition,
% 3.17/1.87 ( powersetTI1
% 3.17/1.87 = ( ! [A: $i,X: $i] :
% 3.17/1.87 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.87 => ! [Y: $i] :
% 3.17/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.87 => ( ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( in @ Xx @ X )
% 3.17/1.87 => ( in @ Xx @ Y ) ) )
% 3.17/1.87 => ( in @ X @ ( powerset @ Y ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(powersetTE1_type,type,
% 3.17/1.87 powersetTE1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(powersetTE1,definition,
% 3.17/1.87 ( powersetTE1
% 3.17/1.87 = ( ! [A: $i,X: $i] :
% 3.17/1.87 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.87 => ! [Y: $i] :
% 3.17/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( in @ X @ ( powerset @ Y ) )
% 3.17/1.87 => ( ( in @ Xx @ X )
% 3.17/1.87 => ( in @ Xx @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(complementTI1_type,type,
% 3.17/1.87 complementTI1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(complementTI1,definition,
% 3.17/1.87 ( complementTI1
% 3.17/1.87 = ( ! [A: $i,X: $i] :
% 3.17/1.87 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( in @ Xx @ X )
% 3.17/1.87 => ~ ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(complementTE1_type,type,
% 3.17/1.87 complementTE1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(complementTE1,definition,
% 3.17/1.87 ( complementTE1
% 3.17/1.87 = ( ! [A: $i,X: $i] :
% 3.17/1.87 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ~ ( in @ Xx @ ( setminus @ A @ X ) )
% 3.17/1.87 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectTELcontra_type,type,
% 3.17/1.87 binintersectTELcontra: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectTELcontra,definition,
% 3.17/1.87 ( binintersectTELcontra
% 3.17/1.87 = ( ! [A: $i,X: $i] :
% 3.17/1.87 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.87 => ! [Y: $i] :
% 3.17/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ~ ( in @ Xx @ X )
% 3.17/1.87 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectTERcontra_type,type,
% 3.17/1.87 binintersectTERcontra: $o ).
% 3.17/1.87
% 3.17/1.87 thf(binintersectTERcontra,definition,
% 3.17/1.87 ( binintersectTERcontra
% 3.17/1.87 = ( ! [A: $i,X: $i] :
% 3.17/1.87 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.87 => ! [Y: $i] :
% 3.17/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ~ ( in @ Xx @ Y )
% 3.17/1.87 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(contrasubsetT_type,type,
% 3.17/1.87 contrasubsetT: $o ).
% 3.17/1.87
% 3.17/1.87 thf(contrasubsetT,definition,
% 3.17/1.87 ( contrasubsetT
% 3.17/1.87 = ( ! [A: $i,X: $i] :
% 3.17/1.87 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.87 => ! [Y: $i] :
% 3.17/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.17/1.87 => ( ( in @ Xx @ Y )
% 3.17/1.87 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(contrasubsetT1_type,type,
% 3.17/1.87 contrasubsetT1: $o ).
% 3.17/1.87
% 3.17/1.87 thf(contrasubsetT1,definition,
% 3.17/1.87 ( contrasubsetT1
% 3.17/1.87 = ( ! [A: $i,X: $i] :
% 3.17/1.87 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.87 => ! [Y: $i] :
% 3.17/1.87 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.87 => ! [Xx: $i] :
% 3.17/1.87 ( ( in @ Xx @ A )
% 3.17/1.87 => ( ( subset @ X @ Y )
% 3.17/1.87 => ( ~ ( in @ Xx @ Y )
% 3.17/1.87 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ) ).
% 3.17/1.87
% 3.17/1.87 thf(contrasubsetT2_type,type,
% 3.17/1.88 contrasubsetT2: $o ).
% 3.17/1.88
% 3.17/1.88 thf(contrasubsetT2,definition,
% 3.17/1.88 ( contrasubsetT2
% 3.17/1.88 = ( ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ! [Y: $i] :
% 3.17/1.88 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.88 => ( ( subset @ X @ Y )
% 3.17/1.88 => ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.17/1.88
% 3.17/1.88 thf(contrasubsetT3_type,type,
% 3.17/1.88 contrasubsetT3: $o ).
% 3.17/1.88
% 3.17/1.88 thf(contrasubsetT3,definition,
% 3.17/1.88 ( contrasubsetT3
% 3.17/1.88 = ( ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ! [Y: $i] :
% 3.17/1.88 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.88 => ( ( subset @ ( setminus @ A @ Y ) @ ( setminus @ A @ X ) )
% 3.17/1.88 => ( subset @ X @ Y ) ) ) ) ) ) ).
% 3.17/1.88
% 3.17/1.88 thf(doubleComplementI1_type,type,
% 3.17/1.88 doubleComplementI1: $o ).
% 3.17/1.88
% 3.17/1.88 thf(doubleComplementI1,definition,
% 3.17/1.88 ( doubleComplementI1
% 3.17/1.88 = ( ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ! [Xx: $i] :
% 3.17/1.88 ( ( in @ Xx @ A )
% 3.17/1.88 => ( ( in @ Xx @ X )
% 3.17/1.88 => ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ).
% 3.17/1.88
% 3.17/1.88 thf(doubleComplementE1_type,type,
% 3.17/1.88 doubleComplementE1: $o ).
% 3.17/1.88
% 3.17/1.88 thf(doubleComplementE1,definition,
% 3.17/1.88 ( doubleComplementE1
% 3.17/1.88 = ( ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ! [Xx: $i] :
% 3.17/1.88 ( ( in @ Xx @ A )
% 3.17/1.88 => ( ( in @ Xx @ ( setminus @ A @ ( setminus @ A @ X ) ) )
% 3.17/1.88 => ( in @ Xx @ X ) ) ) ) ) ) ).
% 3.17/1.88
% 3.17/1.88 thf(doubleComplementSub1_type,type,
% 3.17/1.88 doubleComplementSub1: $o ).
% 3.17/1.88
% 3.17/1.88 thf(doubleComplementSub1,definition,
% 3.17/1.88 ( doubleComplementSub1
% 3.17/1.88 = ( ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ( subset @ X @ ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.17/1.88
% 3.17/1.88 thf(doubleComplementSub2_type,type,
% 3.17/1.88 doubleComplementSub2: $o ).
% 3.17/1.88
% 3.17/1.88 thf(doubleComplementSub2,definition,
% 3.17/1.88 ( doubleComplementSub2
% 3.17/1.88 = ( ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ( subset @ ( setminus @ A @ ( setminus @ A @ X ) ) @ X ) ) ) ) ).
% 3.17/1.88
% 3.17/1.88 thf(doubleComplementEq_type,type,
% 3.17/1.88 doubleComplementEq: $o ).
% 3.17/1.88
% 3.17/1.88 thf(doubleComplementEq,definition,
% 3.17/1.88 ( doubleComplementEq
% 3.17/1.88 = ( ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ( X
% 3.17/1.88 = ( setminus @ A @ ( setminus @ A @ X ) ) ) ) ) ) ).
% 3.17/1.88
% 3.17/1.88 thf(complementTnotintersectT_type,type,
% 3.17/1.88 complementTnotintersectT: $o ).
% 3.17/1.88
% 3.17/1.88 thf(complementTnotintersectT,definition,
% 3.17/1.88 ( complementTnotintersectT
% 3.17/1.88 = ( ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ! [Y: $i] :
% 3.17/1.88 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.88 => ! [Xx: $i] :
% 3.17/1.88 ( ( in @ Xx @ A )
% 3.17/1.88 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.17/1.88 => ~ ( in @ Xx @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.88
% 3.17/1.88 thf(complementImpComplementIntersect_type,type,
% 3.17/1.88 complementImpComplementIntersect: $o ).
% 3.17/1.88
% 3.17/1.88 thf(complementImpComplementIntersect,definition,
% 3.17/1.88 ( complementImpComplementIntersect
% 3.17/1.88 = ( ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ! [Y: $i] :
% 3.17/1.88 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.88 => ! [Xx: $i] :
% 3.17/1.88 ( ( in @ Xx @ A )
% 3.17/1.88 => ( ( in @ Xx @ ( setminus @ A @ X ) )
% 3.17/1.88 => ( in @ Xx @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ) ).
% 3.17/1.88
% 3.17/1.88 thf(complementSubsetComplementIntersect_type,type,
% 3.17/1.88 complementSubsetComplementIntersect: $o ).
% 3.17/1.88
% 3.17/1.88 thf(complementSubsetComplementIntersect,definition,
% 3.17/1.88 ( complementSubsetComplementIntersect
% 3.17/1.88 = ( ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ! [Y: $i] :
% 3.17/1.88 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.88 => ( subset @ ( setminus @ A @ X ) @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ).
% 3.17/1.88
% 3.17/1.88 thf(complementInPowersetComplementIntersect_type,type,
% 3.17/1.88 complementInPowersetComplementIntersect: $o ).
% 3.17/1.88
% 3.17/1.88 thf(complementInPowersetComplementIntersect,definition,
% 3.17/1.88 ( complementInPowersetComplementIntersect
% 3.17/1.88 = ( ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ! [Y: $i] :
% 3.17/1.88 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.88 => ( in @ ( setminus @ A @ X ) @ ( powerset @ ( setminus @ A @ ( binintersect @ X @ Y ) ) ) ) ) ) ) ) ).
% 3.17/1.88
% 3.17/1.88 thf(contraSubsetComplement_type,type,
% 3.17/1.88 contraSubsetComplement: $o ).
% 3.17/1.88
% 3.17/1.88 thf(contraSubsetComplement,definition,
% 3.17/1.88 ( contraSubsetComplement
% 3.17/1.88 = ( ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ! [Y: $i] :
% 3.17/1.88 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.88 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.17/1.88 => ! [Xx: $i] :
% 3.17/1.88 ( ( in @ Xx @ A )
% 3.17/1.88 => ( ( in @ Xx @ Y )
% 3.17/1.88 => ( in @ Xx @ ( setminus @ A @ X ) ) ) ) ) ) ) ) ) ).
% 3.17/1.88
% 3.17/1.88 thf(complementTcontraSubset_type,type,
% 3.17/1.88 complementTcontraSubset: $o ).
% 3.17/1.88
% 3.17/1.88 thf(complementTcontraSubset,definition,
% 3.17/1.88 ( complementTcontraSubset
% 3.17/1.88 = ( ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ! [Y: $i] :
% 3.17/1.88 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.88 => ( ( subset @ X @ ( setminus @ A @ Y ) )
% 3.17/1.88 => ( subset @ Y @ ( setminus @ A @ X ) ) ) ) ) ) ) ).
% 3.17/1.88
% 3.17/1.88 thf(binunionTILcontra_type,type,
% 3.17/1.88 binunionTILcontra: $o ).
% 3.17/1.88
% 3.17/1.88 thf(binunionTILcontra,definition,
% 3.17/1.88 ( binunionTILcontra
% 3.17/1.88 = ( ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ! [Y: $i] :
% 3.17/1.88 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.88 => ! [Xx: $i] :
% 3.17/1.88 ( ( in @ Xx @ A )
% 3.17/1.88 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.17/1.88 => ~ ( in @ Xx @ X ) ) ) ) ) ) ) ).
% 3.17/1.88
% 3.17/1.88 thf(binunionTIRcontra_type,type,
% 3.17/1.88 binunionTIRcontra: $o ).
% 3.17/1.88
% 3.17/1.88 thf(binunionTIRcontra,definition,
% 3.17/1.88 ( binunionTIRcontra
% 3.17/1.88 = ( ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ! [Y: $i] :
% 3.17/1.88 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.88 => ! [Xx: $i] :
% 3.17/1.88 ( ( in @ Xx @ A )
% 3.17/1.88 => ( ~ ( in @ Xx @ ( binunion @ X @ Y ) )
% 3.17/1.88 => ~ ( in @ Xx @ Y ) ) ) ) ) ) ) ).
% 3.17/1.88
% 3.17/1.88 thf(inIntersectImpInUnion_type,type,
% 3.17/1.88 inIntersectImpInUnion: $o ).
% 3.17/1.88
% 3.17/1.88 thf(inIntersectImpInUnion,definition,
% 3.17/1.88 ( inIntersectImpInUnion
% 3.17/1.88 = ( ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ! [Y: $i] :
% 3.17/1.88 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.88 => ! [Z: $i] :
% 3.17/1.88 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.88 => ! [Xx: $i] :
% 3.17/1.88 ( ( in @ Xx @ A )
% 3.17/1.88 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.17/1.88 => ( in @ Xx @ ( binunion @ X @ Z ) ) ) ) ) ) ) ) ) ).
% 3.17/1.88
% 3.17/1.88 thf(inIntersectImpInUnion2,conjecture,
% 3.17/1.88 ( setextAx
% 3.17/1.88 => ( emptysetAx
% 3.17/1.88 => ( setadjoinAx
% 3.17/1.88 => ( powersetAx
% 3.17/1.88 => ( setunionAx
% 3.17/1.88 => ( omega0Ax
% 3.17/1.88 => ( omegaSAx
% 3.17/1.88 => ( omegaIndAx
% 3.17/1.88 => ( replAx
% 3.17/1.88 => ( foundationAx
% 3.17/1.88 => ( wellorderingAx
% 3.17/1.88 => ( descrp
% 3.17/1.88 => ( dsetconstrI
% 3.17/1.88 => ( dsetconstrEL
% 3.17/1.88 => ( dsetconstrER
% 3.17/1.88 => ( exuE1
% 3.17/1.88 => ( prop2setE
% 3.17/1.88 => ( emptysetE
% 3.17/1.88 => ( emptysetimpfalse
% 3.17/1.88 => ( notinemptyset
% 3.17/1.88 => ( exuE3e
% 3.17/1.88 => ( setext
% 3.17/1.88 => ( emptyI
% 3.17/1.88 => ( noeltsimpempty
% 3.17/1.88 => ( setbeta
% 3.17/1.88 => ( nonemptyE1
% 3.17/1.88 => ( nonemptyI
% 3.17/1.88 => ( nonemptyI1
% 3.17/1.88 => ( setadjoinIL
% 3.17/1.88 => ( emptyinunitempty
% 3.17/1.88 => ( setadjoinIR
% 3.17/1.88 => ( setadjoinE
% 3.17/1.88 => ( setadjoinOr
% 3.17/1.88 => ( setoftrueEq
% 3.17/1.88 => ( powersetI
% 3.17/1.88 => ( emptyinPowerset
% 3.17/1.88 => ( emptyInPowerset
% 3.17/1.88 => ( powersetE
% 3.17/1.88 => ( setunionI
% 3.17/1.88 => ( setunionE
% 3.17/1.88 => ( subPowSU
% 3.17/1.88 => ( exuE2
% 3.17/1.88 => ( nonemptyImpWitness
% 3.17/1.88 => ( uniqinunit
% 3.17/1.88 => ( notinsingleton
% 3.17/1.88 => ( eqinunit
% 3.17/1.88 => ( singletonsswitch
% 3.17/1.88 => ( upairsetE
% 3.17/1.88 => ( upairsetIL
% 3.17/1.88 => ( upairsetIR
% 3.17/1.88 => ( emptyE1
% 3.17/1.88 => ( vacuousDall
% 3.17/1.88 => ( quantDeMorgan1
% 3.17/1.88 => ( quantDeMorgan2
% 3.17/1.88 => ( quantDeMorgan3
% 3.17/1.88 => ( quantDeMorgan4
% 3.17/1.88 => ( prop2setI
% 3.17/1.88 => ( prop2set2propI
% 3.17/1.88 => ( notdexE
% 3.17/1.88 => ( notdallE
% 3.17/1.88 => ( exuI1
% 3.17/1.88 => ( exuI3
% 3.17/1.88 => ( exuI2
% 3.17/1.88 => ( inCongP
% 3.17/1.88 => ( in__Cong
% 3.17/1.88 => ( exuE3u
% 3.17/1.88 => ( exu__Cong
% 3.17/1.88 => ( emptyset__Cong
% 3.17/1.88 => ( setadjoin__Cong
% 3.17/1.88 => ( powerset__Cong
% 3.17/1.88 => ( setunion__Cong
% 3.17/1.88 => ( omega__Cong
% 3.17/1.88 => ( exuEu
% 3.17/1.88 => ( descr__Cong
% 3.17/1.88 => ( dsetconstr__Cong
% 3.17/1.88 => ( subsetI1
% 3.17/1.88 => ( eqimpsubset2
% 3.17/1.88 => ( eqimpsubset1
% 3.17/1.88 => ( subsetI2
% 3.17/1.88 => ( emptysetsubset
% 3.17/1.88 => ( subsetE
% 3.17/1.88 => ( subsetE2
% 3.17/1.88 => ( notsubsetI
% 3.17/1.88 => ( notequalI1
% 3.17/1.88 => ( notequalI2
% 3.17/1.88 => ( subsetRefl
% 3.17/1.88 => ( subsetTrans
% 3.17/1.88 => ( setadjoinSub
% 3.17/1.88 => ( setadjoinSub2
% 3.17/1.88 => ( subset2powerset
% 3.17/1.88 => ( setextsub
% 3.17/1.88 => ( subsetemptysetimpeq
% 3.17/1.88 => ( powersetI1
% 3.17/1.88 => ( powersetE1
% 3.17/1.88 => ( inPowerset
% 3.17/1.88 => ( powersetsubset
% 3.17/1.88 => ( sepInPowerset
% 3.17/1.88 => ( sepSubset
% 3.17/1.88 => ( binunionIL
% 3.17/1.88 => ( upairset2IR
% 3.17/1.88 => ( binunionIR
% 3.17/1.88 => ( binunionEcases
% 3.17/1.88 => ( binunionE
% 3.17/1.88 => ( binunionLsub
% 3.17/1.88 => ( binunionRsub
% 3.17/1.88 => ( binintersectI
% 3.17/1.88 => ( binintersectSubset5
% 3.17/1.88 => ( binintersectEL
% 3.17/1.88 => ( binintersectLsub
% 3.17/1.88 => ( binintersectSubset2
% 3.17/1.88 => ( binintersectSubset3
% 3.17/1.88 => ( binintersectER
% 3.17/1.88 => ( disjointsetsI1
% 3.17/1.88 => ( binintersectRsub
% 3.17/1.88 => ( binintersectSubset4
% 3.17/1.88 => ( binintersectSubset1
% 3.17/1.88 => ( bs114d
% 3.17/1.88 => ( setminusI
% 3.17/1.88 => ( setminusEL
% 3.17/1.88 => ( setminusER
% 3.17/1.88 => ( setminusSubset2
% 3.17/1.88 => ( setminusERneg
% 3.17/1.88 => ( setminusELneg
% 3.17/1.88 => ( setminusILneg
% 3.17/1.88 => ( setminusIRneg
% 3.17/1.88 => ( setminusLsub
% 3.17/1.88 => ( setminusSubset1
% 3.17/1.88 => ( symdiffE
% 3.17/1.88 => ( symdiffI1
% 3.17/1.88 => ( symdiffI2
% 3.17/1.88 => ( symdiffIneg1
% 3.17/1.88 => ( symdiffIneg2
% 3.17/1.88 => ( secondinupair
% 3.17/1.88 => ( setukpairIL
% 3.17/1.88 => ( setukpairIR
% 3.17/1.88 => ( kpairiskpair
% 3.17/1.88 => ( kpairp
% 3.17/1.88 => ( singletonsubset
% 3.17/1.88 => ( singletoninpowerset
% 3.17/1.88 => ( singletoninpowunion
% 3.17/1.88 => ( upairset2E
% 3.17/1.88 => ( upairsubunion
% 3.17/1.88 => ( upairinpowunion
% 3.17/1.88 => ( ubforcartprodlem1
% 3.17/1.88 => ( ubforcartprodlem2
% 3.17/1.88 => ( ubforcartprodlem3
% 3.17/1.88 => ( cartprodpairin
% 3.17/1.88 => ( cartprodmempair1
% 3.17/1.88 => ( cartprodmempair
% 3.17/1.88 => ( setunionE2
% 3.17/1.88 => ( setunionsingleton1
% 3.17/1.88 => ( setunionsingleton2
% 3.17/1.88 => ( setunionsingleton
% 3.17/1.88 => ( singletonprop
% 3.17/1.88 => ( ex1E1
% 3.17/1.88 => ( ex1I
% 3.17/1.88 => ( ex1I2
% 3.17/1.88 => ( singletonsuniq
% 3.17/1.88 => ( setukpairinjL1
% 3.17/1.88 => ( kfstsingleton
% 3.17/1.88 => ( theprop
% 3.17/1.88 => ( kfstpairEq
% 3.17/1.88 => ( cartprodfstin
% 3.17/1.88 => ( setukpairinjL2
% 3.17/1.88 => ( setukpairinjL
% 3.17/1.88 => ( setukpairinjR11
% 3.17/1.88 => ( setukpairinjR12
% 3.17/1.88 => ( setukpairinjR1
% 3.17/1.88 => ( upairequniteq
% 3.17/1.88 => ( setukpairinjR2
% 3.17/1.88 => ( setukpairinjR
% 3.17/1.88 => ( ksndsingleton
% 3.17/1.88 => ( ksndpairEq
% 3.17/1.88 => ( kpairsurjEq
% 3.17/1.88 => ( cartprodsndin
% 3.17/1.88 => ( cartprodpairmemEL
% 3.17/1.88 => ( cartprodpairmemER
% 3.17/1.88 => ( cartprodmempaircEq
% 3.17/1.88 => ( cartprodfstpairEq
% 3.17/1.88 => ( cartprodsndpairEq
% 3.17/1.88 => ( cartprodpairsurjEq
% 3.17/1.88 => ( dpsetconstrI
% 3.17/1.88 => ( dpsetconstrSub
% 3.17/1.88 => ( setOfPairsIsBReln
% 3.17/1.88 => ( dpsetconstrERa
% 3.17/1.88 => ( dpsetconstrEL1
% 3.17/1.88 => ( dpsetconstrEL2
% 3.17/1.88 => ( dpsetconstrER
% 3.17/1.88 => ( funcImageSingleton
% 3.17/1.88 => ( apProp
% 3.17/1.88 => ( app
% 3.17/1.88 => ( infuncsetfunc
% 3.17/1.88 => ( ap2p
% 3.17/1.88 => ( funcinfuncset
% 3.17/1.88 => ( lamProp
% 3.17/1.88 => ( lamp
% 3.17/1.88 => ( lam2p
% 3.17/1.88 => ( brelnall1
% 3.17/1.88 => ( brelnall2
% 3.17/1.88 => ( ex1E2
% 3.17/1.88 => ( funcGraphProp1
% 3.17/1.88 => ( funcGraphProp3
% 3.17/1.88 => ( funcGraphProp2
% 3.17/1.88 => ( funcextLem
% 3.17/1.88 => ( funcGraphProp4
% 3.17/1.88 => ( subbreln
% 3.17/1.88 => ( eqbreln
% 3.17/1.88 => ( funcext
% 3.17/1.88 => ( funcext2
% 3.17/1.88 => ( ap2apEq1
% 3.17/1.88 => ( ap2apEq2
% 3.17/1.88 => ( beta1
% 3.17/1.88 => ( eta1
% 3.17/1.88 => ( lam2lamEq
% 3.17/1.88 => ( beta2
% 3.17/1.88 => ( eta2
% 3.17/1.88 => ( iffalseProp1
% 3.17/1.88 => ( iffalseProp2
% 3.17/1.88 => ( iftrueProp1
% 3.17/1.88 => ( iftrueProp2
% 3.17/1.88 => ( ifSingleton
% 3.17/1.88 => ( ifp
% 3.17/1.88 => ( theeq
% 3.17/1.88 => ( iftrue
% 3.17/1.88 => ( iffalse
% 3.17/1.88 => ( iftrueorfalse
% 3.17/1.88 => ( binintersectT_lem
% 3.17/1.88 => ( binunionT_lem
% 3.17/1.88 => ( powersetT_lem
% 3.17/1.88 => ( setminusT_lem
% 3.17/1.88 => ( complementT_lem
% 3.17/1.88 => ( setextT
% 3.17/1.88 => ( subsetTI
% 3.17/1.88 => ( powersetTI1
% 3.17/1.88 => ( powersetTE1
% 3.17/1.88 => ( complementTI1
% 3.17/1.88 => ( complementTE1
% 3.17/1.88 => ( binintersectTELcontra
% 3.17/1.88 => ( binintersectTERcontra
% 3.17/1.88 => ( contrasubsetT
% 3.17/1.88 => ( contrasubsetT1
% 3.17/1.88 => ( contrasubsetT2
% 3.17/1.88 => ( contrasubsetT3
% 3.17/1.88 => ( doubleComplementI1
% 3.17/1.88 => ( doubleComplementE1
% 3.17/1.88 => ( doubleComplementSub1
% 3.17/1.88 => ( doubleComplementSub2
% 3.17/1.88 => ( doubleComplementEq
% 3.17/1.88 => ( complementTnotintersectT
% 3.17/1.88 => ( complementImpComplementIntersect
% 3.17/1.88 => ( complementSubsetComplementIntersect
% 3.17/1.88 => ( complementInPowersetComplementIntersect
% 3.17/1.88 => ( contraSubsetComplement
% 3.17/1.88 => ( complementTcontraSubset
% 3.17/1.88 => ( binunionTILcontra
% 3.17/1.88 => ( binunionTIRcontra
% 3.17/1.88 => ( inIntersectImpInUnion
% 3.17/1.88 => ! [A: $i,X: $i] :
% 3.17/1.88 ( ( in @ X @ ( powerset @ A ) )
% 3.17/1.88 => ! [Y: $i] :
% 3.17/1.88 ( ( in @ Y @ ( powerset @ A ) )
% 3.17/1.88 => ! [Z: $i] :
% 3.17/1.88 ( ( in @ Z @ ( powerset @ A ) )
% 3.17/1.88 => ! [Xx: $i] :
% 3.17/1.88 ( ( in @ Xx @ A )
% 3.17/1.88 => ( ( in @ Xx @ ( binintersect @ X @ Y ) )
% 3.17/1.88 => ( in @ Xx @ ( binunion @ Y @ Z ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 3.17/1.88 ------------------------
% 3.24/1.92 % (13138)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_13002 for (2999ds/4Mi)
% 3.24/1.92 % (13137)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_13002 for (2999ds/183Mi)
% 3.24/1.92 % (13140)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_13002 for (2999ds/2Mi)
% 3.24/1.92 % (13142)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_13002 for (2999ds/275Mi)
% 3.24/1.92 % (13141)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_13002 for (2999ds/2Mi)
% 3.24/1.92 % (13139)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_13002 for (2999ds/27Mi)
% 3.24/1.92 % (13143)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_13002 for (2999ds/18Mi)
% 3.24/1.92 % (13140)Instruction limit reached!
% 3.24/1.92 % (13140)------------------------------
% 3.24/1.92 % (13140)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.92 % (13140)Termination reason: Unknown
% 3.24/1.92 % (13141)Instruction limit reached!
% 3.24/1.92 % (13141)------------------------------
% 3.24/1.92 % (13141)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.92 % (13140)Termination phase: shuffling
% 3.24/1.92
% 3.24/1.92 % (13140)Memory used [KB]: 1407
% 3.24/1.92 % (13140)Time elapsed: 0.003 s
% 3.24/1.92 % (13140)Instructions burned: 2 (million)
% 3.24/1.92 % (13140)------------------------------
% 3.24/1.92 % (13140)------------------------------
% 3.24/1.92 % (13141)Termination reason: Unknown
% 3.24/1.92 % (13141)Termination phase: shuffling
% 3.24/1.92
% 3.24/1.92 % (13141)Memory used [KB]: 1407
% 3.24/1.92 % (13141)Time elapsed: 0.003 s
% 3.24/1.92 % (13141)Instructions burned: 2 (million)
% 3.24/1.92 % (13141)------------------------------
% 3.24/1.92 % (13141)------------------------------
% 3.24/1.92 % (13138)Instruction limit reached!
% 3.24/1.92 % (13138)------------------------------
% 3.24/1.92 % (13138)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.92 % (13138)Termination reason: Unknown
% 3.24/1.92 % (13138)Termination phase: shuffling
% 3.24/1.92
% 3.24/1.92 % (13138)Memory used [KB]: 1535
% 3.24/1.92 % (13138)Time elapsed: 0.005 s
% 3.24/1.92 % (13138)Instructions burned: 6 (million)
% 3.24/1.92 % (13138)------------------------------
% 3.24/1.92 % (13138)------------------------------
% 3.24/1.93 % (13143)Instruction limit reached!
% 3.24/1.93 % (13143)------------------------------
% 3.24/1.93 % (13143)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.93 % (13143)Termination reason: Unknown
% 3.24/1.93 % (13143)Termination phase: shuffling
% 3.24/1.93
% 3.24/1.93 % (13143)Memory used [KB]: 1791
% 3.24/1.93 % (13143)Time elapsed: 0.011 s
% 3.24/1.93 % (13143)Instructions burned: 18 (million)
% 3.24/1.93 % (13143)------------------------------
% 3.24/1.93 % (13143)------------------------------
% 3.24/1.93 % (13139)Instruction limit reached!
% 3.24/1.93 % (13139)------------------------------
% 3.24/1.93 % (13139)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.93 % (13139)Termination reason: Unknown
% 3.24/1.93 % (13139)Termination phase: shuffling
% 3.24/1.93
% 3.24/1.93 % (13139)Memory used [KB]: 1918
% 3.24/1.93 % (13139)Time elapsed: 0.015 s
% 3.24/1.93 % (13139)Instructions burned: 27 (million)
% 3.24/1.93 % (13139)------------------------------
% 3.24/1.93 % (13139)------------------------------
% 3.24/1.93 % (13144)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_13002 for (2999ds/3Mi)
% 3.24/1.93 % (13145)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_13002 for (2999ds/37Mi)
% 3.24/1.94 % (13146)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_13002 for (2999ds/15Mi)
% 3.24/1.94 % (13144)Instruction limit reached!
% 3.24/1.94 % (13144)------------------------------
% 3.24/1.94 % (13144)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.94 % (13144)Termination reason: Unknown
% 3.24/1.94 % (13144)Termination phase: shuffling
% 3.24/1.94
% 3.24/1.94 % (13144)Memory used [KB]: 1535
% 3.24/1.94 % (13144)Time elapsed: 0.004 s
% 3.24/1.94 % (13144)Instructions burned: 4 (million)
% 3.24/1.94 % (13144)------------------------------
% 3.24/1.94 % (13144)------------------------------
% 3.24/1.94 % (13147)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_13002 for (2999ds/3Mi)
% 3.24/1.94 % (13146)Instruction limit reached!
% 3.24/1.94 % (13146)------------------------------
% 3.24/1.94 % (13146)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.94 % (13146)Termination reason: Unknown
% 3.24/1.94 % (13146)Termination phase: shuffling
% 3.24/1.94
% 3.24/1.94 % (13146)Memory used [KB]: 1791
% 3.24/1.94 % (13146)Time elapsed: 0.009 s
% 3.24/1.94 % (13146)Instructions burned: 15 (million)
% 3.24/1.94 % (13146)------------------------------
% 3.24/1.94 % (13146)------------------------------
% 3.24/1.94 % (13147)Instruction limit reached!
% 3.24/1.94 % (13147)------------------------------
% 3.24/1.94 % (13147)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.94 % (13147)Termination reason: Unknown
% 3.24/1.94 % (13147)Termination phase: shuffling
% 3.24/1.94
% 3.24/1.94 % (13147)Memory used [KB]: 1407
% 3.24/1.94 % (13147)Time elapsed: 0.003 s
% 3.24/1.94 % (13147)Instructions burned: 3 (million)
% 3.24/1.94 % (13147)------------------------------
% 3.24/1.94 % (13147)------------------------------
% 3.24/1.95 % (13148)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_13002 for (2999ds/1041Mi)
% 3.24/1.95 % (13145)Instruction limit reached!
% 3.24/1.95 % (13145)------------------------------
% 3.24/1.95 % (13145)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.95 % (13145)Termination reason: Unknown
% 3.24/1.95 % (13145)Termination phase: shuffling
% 3.24/1.95
% 3.24/1.95 % (13145)Memory used [KB]: 2302
% 3.24/1.95 % (13145)Time elapsed: 0.019 s
% 3.24/1.95 % (13145)Instructions burned: 37 (million)
% 3.24/1.95 % (13145)------------------------------
% 3.24/1.95 % (13145)------------------------------
% 3.24/1.95 % (13149)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_13002 for (2999ds/7Mi)
% 3.24/1.96 % (13149)Instruction limit reached!
% 3.24/1.96 % (13149)------------------------------
% 3.24/1.96 % (13149)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.96 % (13149)Termination reason: Unknown
% 3.24/1.96 % (13149)Termination phase: shuffling
% 3.24/1.96
% 3.24/1.96 % (13149)Memory used [KB]: 1535
% 3.24/1.96 % (13149)Time elapsed: 0.005 s
% 3.24/1.96 % (13149)Instructions burned: 7 (million)
% 3.24/1.96 % (13149)------------------------------
% 3.24/1.96 % (13149)------------------------------
% 3.24/1.96 % (13150)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_13002 for (2999ds/16Mi)
% 3.24/1.96 % (13151)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_13002 for (2999ds/3Mi)
% 3.24/1.96 % (13151)Instruction limit reached!
% 3.24/1.96 % (13151)------------------------------
% 3.24/1.96 % (13151)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.96 % (13151)Termination reason: Unknown
% 3.24/1.96 % (13151)Termination phase: shuffling
% 3.24/1.96
% 3.24/1.96 % (13151)Memory used [KB]: 1407
% 3.24/1.96 % (13151)Time elapsed: 0.003 s
% 3.24/1.96 % (13151)Instructions burned: 3 (million)
% 3.24/1.96 % (13151)------------------------------
% 3.24/1.96 % (13151)------------------------------
% 3.24/1.97 % (13150)Instruction limit reached!
% 3.24/1.97 % (13150)------------------------------
% 3.24/1.97 % (13150)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.97 % (13150)Termination reason: Unknown
% 3.24/1.97 % (13150)Termination phase: shuffling
% 3.24/1.97
% 3.24/1.97 % (13150)Memory used [KB]: 1791
% 3.24/1.97 % (13150)Time elapsed: 0.010 s
% 3.24/1.97 % (13150)Instructions burned: 18 (million)
% 3.24/1.97 % (13150)------------------------------
% 3.24/1.97 % (13150)------------------------------
% 3.24/1.97 % (13152)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_13002 for (2999ds/3Mi)
% 3.24/1.97 % (13152)Instruction limit reached!
% 3.24/1.97 % (13152)------------------------------
% 3.24/1.97 % (13152)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.97 % (13152)Termination reason: Unknown
% 3.24/1.97 % (13152)Termination phase: shuffling
% 3.24/1.97
% 3.24/1.97 % (13152)Memory used [KB]: 1407
% 3.24/1.97 % (13152)Time elapsed: 0.003 s
% 3.24/1.97 % (13152)Instructions burned: 3 (million)
% 3.24/1.97 % (13152)------------------------------
% 3.24/1.97 % (13152)------------------------------
% 3.24/1.97 % (13153)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_13002 for (2999ds/7Mi)
% 3.24/1.98 % (13153)Instruction limit reached!
% 3.24/1.98 % (13153)------------------------------
% 3.24/1.98 % (13153)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.98 % (13153)Termination reason: Unknown
% 3.24/1.98 % (13153)Termination phase: shuffling
% 3.24/1.98
% 3.24/1.98 % (13153)Memory used [KB]: 1535
% 3.24/1.98 % (13153)Time elapsed: 0.005 s
% 3.24/1.98 % (13153)Instructions burned: 7 (million)
% 3.24/1.98 % (13153)------------------------------
% 3.24/1.98 % (13153)------------------------------
% 3.24/1.98 % (13154)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_13002 for (2999ds/3Mi)
% 3.24/1.98 % (13154)Instruction limit reached!
% 3.24/1.98 % (13154)------------------------------
% 3.24/1.98 % (13154)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.98 % (13154)Termination reason: Unknown
% 3.24/1.98 % (13154)Termination phase: shuffling
% 3.24/1.98
% 3.24/1.98 % (13154)Memory used [KB]: 1407
% 3.24/1.98 % (13154)Time elapsed: 0.003 s
% 3.24/1.98 % (13154)Instructions burned: 3 (million)
% 3.24/1.98 % (13154)------------------------------
% 3.24/1.98 % (13154)------------------------------
% 3.24/1.98 % (13155)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_13002 for (2999ds/4Mi)
% 3.24/1.99 % (13156)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_13002 for (2999ds/18Mi)
% 3.24/1.99 % (13155)Instruction limit reached!
% 3.24/1.99 % (13155)------------------------------
% 3.24/1.99 % (13155)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.99 % (13155)Termination reason: Unknown
% 3.24/1.99 % (13155)Termination phase: shuffling
% 3.24/1.99
% 3.24/1.99 % (13155)Memory used [KB]: 1535
% 3.24/1.99 % (13155)Time elapsed: 0.004 s
% 3.24/1.99 % (13155)Instructions burned: 5 (million)
% 3.24/1.99 % (13155)------------------------------
% 3.24/1.99 % (13155)------------------------------
% 3.24/1.99 % (13157)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_13002 for (2999ds/710Mi)
% 3.24/1.99 % (13156)Instruction limit reached!
% 3.24/1.99 % (13156)------------------------------
% 3.24/1.99 % (13156)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/1.99 % (13156)Termination reason: Unknown
% 3.24/1.99 % (13156)Termination phase: shuffling
% 3.24/1.99
% 3.24/1.99 % (13156)Memory used [KB]: 1791
% 3.24/1.99 % (13156)Time elapsed: 0.010 s
% 3.24/1.99 % (13156)Instructions burned: 18 (million)
% 3.24/1.99 % (13156)------------------------------
% 3.24/1.99 % (13156)------------------------------
% 3.24/2.00 % (13158)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_13002 for (2999ds/6Mi)
% 3.24/2.00 % (13137)Instruction limit reached!
% 3.24/2.00 % (13137)------------------------------
% 3.24/2.00 % (13137)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/2.00 % (13137)Termination reason: Unknown
% 3.24/2.00 % (13137)Termination phase: Saturation
% 3.24/2.00
% 3.24/2.00 % (13137)Memory used [KB]: 7419
% 3.24/2.00 % (13137)Time elapsed: 0.083 s
% 3.24/2.00 % (13137)Instructions burned: 184 (million)
% 3.24/2.00 % (13137)------------------------------
% 3.24/2.00 % (13137)------------------------------
% 3.24/2.00 % (13158)Instruction limit reached!
% 3.24/2.00 % (13158)------------------------------
% 3.24/2.00 % (13158)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/2.00 % (13158)Termination reason: Unknown
% 3.24/2.00 % (13158)Termination phase: shuffling
% 3.24/2.00
% 3.24/2.00 % (13158)Memory used [KB]: 1535
% 3.24/2.00 % (13158)Time elapsed: 0.006 s
% 3.24/2.00 % (13158)Instructions burned: 7 (million)
% 3.24/2.00 % (13158)------------------------------
% 3.24/2.00 % (13158)------------------------------
% 3.24/2.00 % (13159)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_13002 for (2999ds/902Mi)
% 3.24/2.01 % (13160)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_13002 for (2998ds/21Mi)
% 3.24/2.02 % (13161)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_13002 for (2998ds/5Mi)
% 3.24/2.02 % (13162)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_13002 for (2998ds/6Mi)
% 3.24/2.02 % (13161)Instruction limit reached!
% 3.24/2.02 % (13161)------------------------------
% 3.24/2.02 % (13161)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/2.02 % (13161)Termination reason: Unknown
% 3.24/2.02 % (13161)Termination phase: shuffling
% 3.24/2.02
% 3.24/2.02 % (13161)Memory used [KB]: 1535
% 3.24/2.02 % (13161)Time elapsed: 0.004 s
% 3.24/2.02 % (13161)Instructions burned: 5 (million)
% 3.24/2.02 % (13161)------------------------------
% 3.24/2.02 % (13161)------------------------------
% 3.24/2.02 % (13162)Instruction limit reached!
% 3.24/2.02 % (13162)------------------------------
% 3.24/2.02 % (13162)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/2.02 % (13162)Termination reason: Unknown
% 3.24/2.02 % (13162)Termination phase: shuffling
% 3.24/2.02
% 3.24/2.02 % (13162)Memory used [KB]: 1535
% 3.24/2.02 % (13162)Time elapsed: 0.005 s
% 3.24/2.02 % (13162)Instructions burned: 6 (million)
% 3.24/2.02 % (13162)------------------------------
% 3.24/2.02 % (13162)------------------------------
% 3.24/2.02 % (13160)Instruction limit reached!
% 3.24/2.02 % (13160)------------------------------
% 3.24/2.02 % (13160)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.24/2.02 % (13160)Termination reason: Unknown
% 3.24/2.02 % (13160)Termination phase: shuffling
% 3.24/2.02
% 3.24/2.02 % (13160)Memory used [KB]: 1791
% 3.24/2.02 % (13160)Time elapsed: 0.012 s
% 3.24/2.02 % (13160)Instructions burned: 22 (million)
% 3.24/2.02 % (13160)------------------------------
% 3.24/2.02 % (13160)------------------------------
% 4.05/2.03 % (13163)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_13002 for (2998ds/377Mi)
% 4.05/2.04 % (13164)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_13002 for (2998ds/779Mi)
% 4.05/2.04 % (13165)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_13002 for (2998ds/19Mi)
% 4.05/2.05 % (13165)Instruction limit reached!
% 4.05/2.05 % (13165)------------------------------
% 4.05/2.05 % (13165)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.05 % (13165)Termination reason: Unknown
% 4.05/2.05 % (13165)Termination phase: shuffling
% 4.05/2.05
% 4.05/2.05 % (13165)Memory used [KB]: 1791
% 4.05/2.05 % (13165)Time elapsed: 0.011 s
% 4.05/2.05 % (13165)Instructions burned: 20 (million)
% 4.05/2.05 % (13165)------------------------------
% 4.05/2.05 % (13165)------------------------------
% 4.05/2.05 % (13142)Instruction limit reached!
% 4.05/2.05 % (13142)------------------------------
% 4.05/2.05 % (13142)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.05 % (13142)Termination reason: Unknown
% 4.05/2.05 % (13142)Termination phase: Saturation
% 4.05/2.05
% 4.05/2.05 % (13142)Memory used [KB]: 9722
% 4.05/2.05 % (13142)Time elapsed: 0.137 s
% 4.05/2.05 % (13142)Instructions burned: 276 (million)
% 4.05/2.05 % (13142)------------------------------
% 4.05/2.05 % (13142)------------------------------
% 4.05/2.06 % (13166)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_13002 for (2998ds/879Mi)
% 4.05/2.07 % (13167)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_13002 for (2998ds/17Mi)
% 4.05/2.08 % (13167)Instruction limit reached!
% 4.05/2.08 % (13167)------------------------------
% 4.05/2.08 % (13167)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.08 % (13167)Termination reason: Unknown
% 4.05/2.08 % (13167)Termination phase: shuffling
% 4.05/2.08
% 4.05/2.08 % (13167)Memory used [KB]: 1791
% 4.05/2.08 % (13167)Time elapsed: 0.010 s
% 4.05/2.08 % (13167)Instructions burned: 17 (million)
% 4.05/2.08 % (13167)------------------------------
% 4.05/2.08 % (13167)------------------------------
% 4.05/2.09 % (13168)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_13002 for (2998ds/3Mi)
% 4.05/2.10 % (13168)Instruction limit reached!
% 4.05/2.10 % (13168)------------------------------
% 4.05/2.10 % (13168)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.10 % (13168)Termination reason: Unknown
% 4.05/2.10 % (13168)Termination phase: shuffling
% 4.05/2.10
% 4.05/2.10 % (13168)Memory used [KB]: 1535
% 4.05/2.10 % (13168)Time elapsed: 0.004 s
% 4.05/2.10 % (13168)Instructions burned: 3 (million)
% 4.05/2.10 % (13168)------------------------------
% 4.05/2.10 % (13168)------------------------------
% 4.05/2.11 % (13169)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_13002 for (2998ds/30Mi)
% 4.05/2.13 % (13169)Instruction limit reached!
% 4.05/2.13 % (13169)------------------------------
% 4.05/2.13 % (13169)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.05/2.13 % (13169)Termination reason: Unknown
% 4.05/2.13 % (13169)Termination phase: shuffling
% 4.05/2.13
% 4.05/2.13 % (13169)Memory used [KB]: 2046
% 4.05/2.13 % (13169)Time elapsed: 0.016 s
% 4.05/2.13 % (13169)Instructions burned: 30 (million)
% 4.05/2.13 % (13169)------------------------------
% 4.05/2.13 % (13169)------------------------------
% 4.47/2.14 % (13170)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_13002 for (2997ds/127Mi)
% 4.47/2.18 % (13159)First to succeed.
% 4.47/2.19 % (13159)Refutation found. Thanks to Tanya!
% 4.47/2.19 % SZS status Theorem for DTF2THF_13002
% 4.47/2.19 % SZS output start Proof for DTF2THF_13002
% 4.47/2.19 thf(func_def_0, type, in: $i > $i > $o).
% 4.47/2.19 thf(func_def_1, type, exu: ($i > $o) > $o).
% 4.47/2.19 thf(func_def_6, type, setadjoin: $i > $i > $i).
% 4.47/2.19 thf(func_def_8, type, powerset: $i > $i).
% 4.47/2.19 thf(func_def_10, type, setunion: $i > $i).
% 4.47/2.19 thf(func_def_19, type, descr: ($i > $o) > $i).
% 4.47/2.19 thf(func_def_21, type, dsetconstr: $i > ($i > $o) > $i).
% 4.47/2.19 thf(func_def_26, type, prop2set: $o > $i).
% 4.47/2.19 thf(func_def_36, type, nonempty: $i > $o).
% 4.47/2.19 thf(func_def_69, type, set2prop: $i > $o).
% 4.47/2.19 thf(func_def_88, type, subset: $i > $i > $o).
% 4.47/2.19 thf(func_def_89, type, disjoint: $i > $i > $o).
% 4.47/2.19 thf(func_def_90, type, setsmeet: $i > $i > $o).
% 4.47/2.19 thf(func_def_114, type, binunion: $i > $i > $i).
% 4.47/2.19 thf(func_def_122, type, binintersect: $i > $i > $i).
% 4.47/2.19 thf(func_def_135, type, regular: $i > $o).
% 4.47/2.19 thf(func_def_136, type, setminus: $i > $i > $i).
% 4.47/2.19 thf(func_def_147, type, symdiff: $i > $i > $i).
% 4.47/2.19 thf(func_def_153, type, iskpair: $i > $o).
% 4.47/2.19 thf(func_def_158, type, kpair: $i > $i > $i).
% 4.47/2.19 thf(func_def_160, type, cartprod: $i > $i > $i).
% 4.47/2.19 thf(func_def_177, type, singleton: $i > $o).
% 4.47/2.19 thf(func_def_179, type, ex1: $i > ($i > $o) > $o).
% 4.47/2.19 thf(func_def_184, type, atmost1p: $i > $o).
% 4.47/2.19 thf(func_def_185, type, atleast2p: $i > $o).
% 4.47/2.19 thf(func_def_186, type, atmost2p: $i > $o).
% 4.47/2.19 thf(func_def_187, type, upairsetp: $i > $o).
% 4.47/2.19 thf(func_def_191, type, kfst: $i > $i).
% 4.47/2.19 thf(func_def_203, type, ksnd: $i > $i).
% 4.47/2.19 thf(func_def_213, type, breln: $i > $i > $i > $o).
% 4.47/2.19 thf(func_def_214, type, dpsetconstr: $i > $i > ($i > $i > $o) > $i).
% 4.47/2.19 thf(func_def_222, type, func: $i > $i > $i > $o).
% 4.47/2.19 thf(func_def_223, type, funcSet: $i > $i > $i).
% 4.47/2.19 thf(func_def_226, type, ap: $i > $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_232, type, lam: $i > $i > ($i > $i) > $i).
% 4.47/2.19 thf(func_def_259, type, if: $i > $o > $i > $i > $i).
% 4.47/2.19 thf(func_def_313, type, sP4: $i > $i > $i > $o).
% 4.47/2.19 thf(func_def_314, type, sP5: $i > $o > $i > $i > $o).
% 4.47/2.19 thf(func_def_315, type, sP6: $i > $i > $i > $i > $o).
% 4.47/2.19 thf(func_def_317, type, sP8: $i > $i > $o).
% 4.47/2.19 thf(func_def_318, type, sP9: $i > $o).
% 4.47/2.19 thf(func_def_319, type, sP10: $i > $i > $o).
% 4.47/2.19 thf(func_def_320, type, sP11: $i > $i > $o).
% 4.47/2.19 thf(func_def_330, type, sK21: $i > $o).
% 4.47/2.19 thf(func_def_332, type, sK23: ($i > $o) > $i > $i).
% 4.47/2.19 thf(func_def_333, type, sK24: ($i > $o) > $i > $i).
% 4.47/2.19 thf(func_def_364, type, sK55: $i > $o).
% 4.47/2.19 thf(func_def_367, type, sK58: $i > $i).
% 4.47/2.19 thf(func_def_369, type, sK60: $i > $i).
% 4.47/2.19 thf(func_def_380, type, sK71: $i > $i > $o).
% 4.47/2.19 thf(func_def_387, type, sK78: ($i > $o) > $i).
% 4.47/2.19 thf(func_def_388, type, sK79: $i > $o).
% 4.47/2.19 thf(func_def_389, type, sK80: $i > $i).
% 4.47/2.19 thf(func_def_396, type, sK87: $i > $o).
% 4.47/2.19 thf(func_def_414, type, sK105: $i > $i > $i).
% 4.47/2.19 thf(func_def_424, type, sK115: $i > $i > $i).
% 4.47/2.19 thf(func_def_425, type, sK116: $i > $i > $i).
% 4.47/2.19 thf(func_def_429, type, sK120: $i > $i).
% 4.47/2.19 thf(func_def_432, type, sK123: $i > $i > ($i > $i) > $i).
% 4.47/2.19 thf(func_def_444, type, sK135: $i > $o).
% 4.47/2.19 thf(func_def_446, type, sK137: $i > ($i > $o) > $i).
% 4.47/2.19 thf(func_def_474, type, sK165: $i > $i).
% 4.47/2.19 thf(func_def_478, type, sK169: $i > $i > ($i > $i) > $i).
% 4.47/2.19 thf(func_def_479, type, sK170: $i > $o).
% 4.47/2.19 thf(func_def_482, type, sK173: $i > ($i > $o) > $i).
% 4.47/2.19 thf(func_def_493, type, sK184: $i > $i > $o).
% 4.47/2.19 thf(func_def_494, type, sK185: $i > $i).
% 4.47/2.19 thf(func_def_495, type, sK186: $i > $i).
% 4.47/2.19 thf(func_def_496, type, sK187: ($i > $i > $o) > $i > $i).
% 4.47/2.19 thf(func_def_497, type, sK188: ($i > $i > $o) > $i > $i).
% 4.47/2.19 thf(func_def_498, type, sK189: $i > ($i > $i > $o) > $i > $i).
% 4.47/2.19 thf(func_def_503, type, sK194: $i > $o).
% 4.47/2.19 thf(func_def_504, type, sK195: $i > $o).
% 4.47/2.19 thf(func_def_505, type, sK196: ($i > $o) > ($i > $o) > $i).
% 4.47/2.19 thf(func_def_506, type, sK197: ($i > $o) > ($i > $o) > $i).
% 4.47/2.19 thf(func_def_509, type, sK200: $i > $i).
% 4.47/2.19 thf(func_def_513, type, sK204: $i > $i > ($i > $i) > $i).
% 4.47/2.19 thf(func_def_526, type, sK217: $i > ($i > $o) > $i).
% 4.47/2.19 thf(func_def_527, type, sK218: $i > $o).
% 4.47/2.19 thf(func_def_539, type, sK230: $i > $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_542, type, sK233: $i > $o).
% 4.47/2.19 thf(func_def_545, type, sK236: $i > $i).
% 4.47/2.19 thf(func_def_551, type, sK242: $i > $o).
% 4.47/2.19 thf(func_def_554, type, sK245: $i > ($i > $o) > $i).
% 4.47/2.19 thf(func_def_555, type, sK246: $i > ($i > $o) > $i).
% 4.47/2.19 thf(func_def_556, type, sK247: $i > $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_578, type, sK269: $i > $i > $o).
% 4.47/2.19 thf(func_def_586, type, sK277: $i > $i).
% 4.47/2.19 thf(func_def_601, type, sK292: $o > $i > $i > $i).
% 4.47/2.19 thf(func_def_608, type, sK299: $i > $o).
% 4.47/2.19 thf(func_def_611, type, sK302: $i > $i).
% 4.47/2.19 thf(func_def_619, type, sK310: $i > $o).
% 4.47/2.19 thf(func_def_622, type, sK313: $i > ($i > $o) > $i).
% 4.47/2.19 thf(func_def_635, type, sK326: $i > $i > $o).
% 4.47/2.19 thf(func_def_640, type, sK331: $i > $i > $i).
% 4.47/2.19 thf(func_def_649, type, sK340: $i > ($i > $o) > $i).
% 4.47/2.19 thf(func_def_650, type, sK341: $i > $o).
% 4.47/2.19 thf(func_def_665, type, sK356: $i > $i > $o).
% 4.47/2.19 thf(func_def_673, type, sK364: $i > $i > $i).
% 4.47/2.19 thf(func_def_675, type, sK366: $i > $o).
% 4.47/2.19 thf(func_def_685, type, sK376: $i > $i > $i).
% 4.47/2.19 thf(func_def_686, type, sK377: $i > ($i > $o) > $i).
% 4.47/2.19 thf(func_def_687, type, sK378: $i > $o).
% 4.47/2.19 thf(func_def_692, type, sK383: $i > $i > $o).
% 4.47/2.19 thf(func_def_698, type, sK389: $i > $o).
% 4.47/2.19 thf(func_def_708, type, sK399: $i > $i > ($i > $i) > $i).
% 4.47/2.19 thf(func_def_709, type, sK400: $i > $i).
% 4.47/2.19 thf(func_def_712, type, sK403: ($i > $o) > $i > $i).
% 4.47/2.19 thf(func_def_714, type, sK405: $i > $o).
% 4.47/2.19 thf(func_def_722, type, sK413: ($i > $o) > ($i > $o) > $i > $i > $i).
% 4.47/2.19 thf(func_def_723, type, sK414: ($i > $o) > ($i > $o) > $i > $i > $i).
% 4.47/2.19 thf(func_def_726, type, sK417: $i > $o).
% 4.47/2.19 thf(func_def_727, type, sK418: $i > $o).
% 4.47/2.19 thf(func_def_745, type, sK436: $i > $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_761, type, sK452: $i > $o).
% 4.47/2.19 thf(func_def_763, type, sK454: $i > $o).
% 4.47/2.19 thf(func_def_774, type, sK465: $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_780, type, sK471: $i > $i > $o).
% 4.47/2.19 thf(func_def_792, type, sK483: $i > $o).
% 4.47/2.19 thf(func_def_794, type, sK485: $i > ($i > $o) > $i).
% 4.47/2.19 thf(func_def_798, type, sK489: $i > $i > $o).
% 4.47/2.19 thf(func_def_802, type, sK493: $i > $o).
% 4.47/2.19 thf(func_def_803, type, sK494: $i > $o).
% 4.47/2.19 thf(func_def_804, type, sK495: ($i > $o) > ($i > $o) > $i).
% 4.47/2.19 thf(func_def_805, type, sK496: ($i > $o) > ($i > $o) > $i).
% 4.47/2.19 thf(func_def_811, type, sK502: $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_812, type, sK503: $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_831, type, sK522: $i > $i > $i).
% 4.47/2.19 thf(func_def_846, type, sK537: $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_850, type, sK541: $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_856, type, sK547: $i > $i).
% 4.47/2.19 thf(func_def_857, type, sK548: ($i > $i) > $i > $i > $i).
% 4.47/2.19 thf(func_def_871, type, sK562: $i > $i > $i).
% 4.47/2.19 thf(func_def_878, type, sK569: $i > $o).
% 4.47/2.19 thf(func_def_880, type, sK571: $i > ($i > $o) > $i).
% 4.47/2.19 thf(func_def_888, type, sK579: $i > $o).
% 4.47/2.19 thf(func_def_890, type, sK581: ($i > $o) > $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_891, type, sK582: ($i > $o) > $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_904, type, sK595: $i > $o).
% 4.47/2.19 thf(func_def_905, type, sK596: $i > $i).
% 4.47/2.19 thf(func_def_906, type, sK597: ($i > $o) > $i).
% 4.47/2.19 thf(func_def_915, type, sK606: $i > $i > $i).
% 4.47/2.19 thf(func_def_960, type, sK651: $i > $i).
% 4.47/2.19 thf(func_def_966, type, sK657: $i > $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_967, type, sK658: $i > $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_968, type, sK659: $i > $o).
% 4.47/2.19 thf(func_def_973, type, sK664: $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_978, type, sK669: $i > $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_979, type, sK670: $i > $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_980, type, sK671: $i > $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_981, type, sK672: $i > $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_984, type, sK675: $i > $o).
% 4.47/2.19 thf(func_def_1003, type, sK694: $i > $i).
% 4.47/2.19 thf(func_def_1022, type, sK713: ($i > $o) > $i).
% 4.47/2.19 thf(func_def_1023, type, sK714: $i > $o).
% 4.47/2.19 thf(func_def_1024, type, sK715: $i > $i > $i).
% 4.47/2.19 thf(func_def_1025, type, sK716: $i > $i > $i).
% 4.47/2.19 thf(func_def_1026, type, sK717: $i > $i > $i).
% 4.47/2.19 thf(func_def_1027, type, sK718: $i > $i > $i).
% 4.47/2.19 thf(func_def_1028, type, sK719: $i > $i > $i).
% 4.47/2.19 thf(func_def_1029, type, sK720: $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_1030, type, sK721: $i > $i).
% 4.47/2.19 thf(func_def_1031, type, sK722: $i > $i).
% 4.47/2.19 thf(func_def_1032, type, sK723: $i > $i).
% 4.47/2.19 thf(func_def_1033, type, sK724: $i > $i).
% 4.47/2.19 thf(func_def_1034, type, sK725: $i > $i > $i).
% 4.47/2.19 thf(func_def_1035, type, sK726: $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_1036, type, sK727: $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_1037, type, sK728: $i > $i > $i).
% 4.47/2.19 thf(func_def_1038, type, sK729: $i > $i > $i).
% 4.47/2.19 thf(func_def_1039, type, sK730: $i > $i).
% 4.47/2.19 thf(func_def_1041, type, sK732: $i > $i).
% 4.47/2.19 thf(func_def_1042, type, sK733: $i > $i).
% 4.47/2.19 thf(func_def_1049, type, sK740: ($i > $o) > $i).
% 4.47/2.19 thf(func_def_1050, type, sK741: ($i > $o) > $i).
% 4.47/2.19 thf(func_def_1051, type, sK742: $i > $o).
% 4.47/2.19 thf(func_def_1057, type, sK748: $i > $o).
% 4.47/2.19 thf(func_def_1073, type, sK764: $i > $o).
% 4.47/2.19 thf(func_def_1075, type, sK766: $i > ($i > $o) > $i > $i).
% 4.47/2.19 thf(func_def_1086, type, sK777: $i > $i > $i).
% 4.47/2.19 thf(func_def_1107, type, sK798: ($i > $o) > $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_1108, type, sK799: ($i > $o) > $i > $i > $i > $i).
% 4.47/2.19 thf(func_def_1112, type, sK803: $i > $o).
% 4.47/2.19 thf(func_def_1116, type, sK807: $i > $o).
% 4.47/2.19 thf(func_def_1118, type, sK809: $i > $o).
% 4.47/2.19 thf(func_def_1125, type, sK816: $i > $i).
% 4.47/2.19 thf(func_def_1128, type, sK819: $i > $i > ($i > $i) > $i).
% 4.47/2.19 thf(func_def_1130, type, ph821: !>[X0: $tType]:(X0)).
% 4.47/2.19 thf(f4178,plain,(
% 4.47/2.19 $false),
% 4.47/2.19 inference(avatar_sat_refutation,[status(thm)],[f4132,f4177])).
% 4.47/2.19 thf(f4177,plain,(
% 4.47/2.19 spl820_13),
% 4.47/2.19 inference(avatar_split_clause,[status(thm)],[f4057,f4087])).
% 4.47/2.19 thf(f4087,definition,(
% 4.47/2.19 spl820_13 <=> ($true = (in @ sK521 @ sK519))),
% 4.47/2.19 introduced(definition,[new_symbols(naming,[spl820_13])],[avatar_definition])).
% 4.47/2.19 thf(f4057,plain,(
% 4.47/2.19 ($true = (in @ sK521 @ sK519))),
% 4.47/2.19 inference(trivial_inequality_removal,[status(thm)],[f4056])).
% 4.47/2.19 thf(f4056,plain,(
% 4.47/2.19 ($true = (in @ sK521 @ sK519)) | ($true != $true)),
% 4.47/2.19 inference(forward_demodulation,[status(thm)],[f4043,f3066])).
% 4.47/2.19 thf(f3066,plain,(
% 4.47/2.19 (binintersectER = $true)),
% 4.47/2.19 inference(cnf_transformation,[status(thm)],[f1917])).
% 4.47/2.19 thf(f1917,plain,(
% 4.47/2.19 (setminusI = $true) & (subset2powerset = $true) & (upairsetIL = $true) & (binintersectT_lem = $true) & (omega__Cong = $true) & (emptyI = $true) & (setadjoinIL = $true) & (binintersectRsub = $true) & (emptyinPowerset = $true) & (lamp = $true) & (dpsetconstrI = $true) & (emptysetE = $true) & (upairsetE = $true) & (cartprodmempair1 = $true) & (upairset2E = $true) & (powersetsubset = $true) & (contrasubsetT1 = $true) & (cartprodfstpairEq = $true) & (ap2apEq1 = $true) & (ksndsingleton = $true) & (beta1 = $true) & (setunionE2 = $true) & (dsetconstrI = $true) & (binunionEcases = $true) & (lam2p = $true) & (complementTI1 = $true) & (powersetTI1 = $true) & (symdiffIneg1 = $true) & (dpsetconstrEL1 = $true) & (setukpairinjR1 = $true) & (dpsetconstrEL2 = $true) & (symdiffI2 = $true) & (omegaSAx = $true) & (setadjoinAx = $true) & (notdexE = $true) & (cartprodpairsurjEq = $true) & (binintersectSubset3 = $true) & (brelnall1 = $true) & (binintersectSubset2 = $true) & (replAx = $true) & (subsetI2 = $true) & (setoftrueEq = $true) & (upairsubunion = $true) & (powersetTE1 = $true) & (descr__Cong = $true) & (app = $true) & (powerset__Cong = $true) & (symdiffI1 = $true) & (exuE1 = $true) & (ubforcartprodlem1 = $true) & (singletoninpowerset = $true) & (cartprodmempaircEq = $true) & (funcext = $true) & (upairequniteq = $true) & (subsetemptysetimpeq = $true) & (powersetI1 = $true) & (complementTnotintersectT = $true) & (exuE3e = $true) & (kpairiskpair = $true) & (($true = (in @ sK518 @ (powerset @ sK517))) & (((($true != (in @ sK521 @ (binunion @ sK519 @ sK520))) & ($true = (in @ sK521 @ sK517)) & ($true = (in @ sK521 @ (binintersect @ sK518 @ sK519)))) & ($true = (in @ sK520 @ (powerset @ sK517)))) & ($true = (in @ sK519 @ (powerset @ sK517))))) & (dpsetconstrSub = $true) & (doubleComplementSub2 = $true) & (secondinupair = $true) & (binunionE = $true) & (setext = $true) & (iftrueProp2 = $true) & (notinemptyset = $true) & (notsubsetI = $true) & (emptysetsubset = $true) & (emptyE1 = $true) & (setminusSubset1 = $true) & (dpsetconstrERa = $true) & (ex1E2 = $true) & (setunionE = $true) & (inPowerset = $true) & (omega0Ax = $true) & (apProp = $true) & (setukpairIR = $true) & (disjointsetsI1 = $true) & (subsetE2 = $true) & (setextAx = $true) & (theprop = $true) & (emptyInPowerset = $true) & (singletonprop = $true) & (contrasubsetT3 = $true) & (funcextLem = $true) & (lamProp = $true) & (emptysetimpfalse = $true) & (cartprodpairin = $true) & (binunionRsub = $true) & (eta1 = $true) & (ap2p = $true) & (setadjoinOr = $true) & (subPowSU = $true) & (subsetE = $true) & (iftrueorfalse = $true) & (setminusERneg = $true) & (singletonsswitch = $true) & (contraSubsetComplement = $true) & (subsetTI = $true) & (subsetTrans = $true) & (ex1I2 = $true) & (dsetconstrEL = $true) & (funcGraphProp1 = $true) & (complementSubsetComplementIntersect = $true) & (iftrue = $true) & (emptyinunitempty = $true) & (dsetconstr__Cong = $true) & (setadjoinSub2 = $true) & (powersetI = $true) & (omegaIndAx = $true) & (singletonsuniq = $true) & (setunionI = $true) & (exuI1 = $true) & (quantDeMorgan2 = $true) & (iffalseProp1 = $true) & (binintersectLsub = $true) & (setukpairinjR2 = $true) & (notinsingleton = $true) & (setunionAx = $true) & (nonemptyI = $true) & (complementT_lem = $true) & (setukpairinjR11 = $true) & (funcGraphProp2 = $true) & (setminusILneg = $true) & (kpairsurjEq = $true) & (cartprodpairmemEL = $true) & (beta2 = $true) & (exuEu = $true) & (infuncsetfunc = $true) & (nonemptyI1 = $true) & (singletonsubset = $true) & (bs114d = $true) & (ubforcartprodlem3 = $true) & (binintersectTELcontra = $true) & (kfstsingleton = $true) & (setadjoin__Cong = $true) & (powersetE = $true) & (setextsub = $true) & (subsetI1 = $true) & (contrasubsetT = $true) & (wellorderingAx = $true) & (in__Cong = $true) & (eqimpsubset2 = $true) & (setadjoinE = $true) & (ifp = $true) & (complementImpComplementIntersect = $true) & (setminusSubset2 = $true) & (uniqinunit = $true) & (quantDeMorgan4 = $true) & (eqinunit = $true) & (eqbreln = $true) & (ifSingleton = $true) & (dpsetconstrER = $true) & (binintersectTERcontra = $true) & (binunionLsub = $true) & (doubleComplementE1 = $true) & (funcinfuncset = $true) & (powersetAx = $true) & (theeq = $true) & (complementTE1 = $true) & (setukpairinjR = $true) & (cartprodpairmemER = $true) & (contrasubsetT2 = $true) & (binunionTIRcontra = $true) & (notequalI1 = $true) & (ap2apEq2 = $true) & (prop2setE = $true) & (foundationAx = $true) & (setminusT_lem = $true) & (binintersectEL = $true) & (exuI3 = $true) & (ubforcartprodlem2 = $true) & (vacuousDall = $true) & (upairset2IR = $true) & (setminusELneg = $true) & (cartprodmempair = $true) & (emptyset__Cong = $true) & (funcGraphProp3 = $true) & (upairinpowunion = $true) & (nonemptyE1 = $true) & (setukpairinjL2 = $true) & (setbeta = $true) & (funcext2 = $true) & (doubleComplementEq = $true) & (upairsetIR = $true) & (prop2setI = $true) & (sepSubset = $true) & (quantDeMorgan1 = $true) & (subbreln = $true) & (binintersectI = $true) & (funcGraphProp4 = $true) & (exu__Cong = $true) & (ex1I = $true) & (brelnall2 = $true) & (setextT = $true) & (binunionT_lem = $true) & (eqimpsubset1 = $true) & (complementTcontraSubset = $true) & (noeltsimpempty = $true) & (cartprodsndpairEq = $true) & (doubleComplementI1 = $true) & (symdiffE = $true) & (setukpairIL = $true) & (setminusEL = $true) & (subsetRefl = $true) & (setminusLsub = $true) & (ex1E1 = $true) & (descrp = $true) & (binunionIL = $true) & (quantDeMorgan3 = $true) & (setadjoinSub = $true) & (singletoninpowunion = $true) & (setukpairinjR12 = $true) & (setukpairinjL1 = $true) & (funcImageSingleton = $true) & (eta2 = $true) & (binunionTILcontra = $true) & (binunionIR = $true) & (binintersectSubset5 = $true) & (notdallE = $true) & (setunionsingleton2 = $true) & (cartprodsndin = $true) & (exuE3u = $true) & (powersetE1 = $true) & (kpairp = $true) & (doubleComplementSub1 = $true) & (notequalI2 = $true) & (setunionsingleton = $true) & (dsetconstrER = $true) & (kfstpairEq = $true) & (setminusIRneg = $true) & (setOfPairsIsBReln = $true) & (iffalseProp2 = $true) & (setunion__Cong = $true) & (iftrueProp1 = $true) & (complementInPowersetComplementIntersect = $true) & (binintersectSubset4 = $true) & (exuE2 = $true) & (powersetT_lem = $true) & (binintersectSubset1 = $true) & (prop2set2propI = $true) & (setminusER = $true) & (setunionsingleton1 = $true) & (inIntersectImpInUnion = $true) & (cartprodfstin = $true) & (nonemptyImpWitness = $true) & (inCongP = $true) & (binintersectER = $true) & (symdiffIneg2 = $true) & (sepInPowerset = $true) & (ksndpairEq = $true) & (lam2lamEq = $true) & (setadjoinIR = $true) & (exuI2 = $true) & (emptysetAx = $true) & (iffalse = $true) & (setukpairinjL = $true)),
% 4.47/2.19 inference(skolemisation,[status(esa),new_symbols(skolem,[sK517,sK518,sK519,sK520,sK521])],[f969,f1916,f1915,f1914,f1913])).
% 4.47/2.19 thf(f1913,plain,(
% 4.47/2.19 ? [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) & ? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : (($true != (in @ X4 @ (binunion @ X2 @ X3))) & ($true = (in @ X4 @ X0)) & ($true = (in @ X4 @ (binintersect @ X1 @ X2)))) & ($true = (in @ X3 @ (powerset @ X0)))) & ($true = (in @ X2 @ (powerset @ X0))))) => (($true = (in @ sK518 @ (powerset @ sK517))) & ? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : (($true != (in @ X4 @ (binunion @ X2 @ X3))) & ($true = (in @ X4 @ sK517)) & ($true = (in @ X4 @ (binintersect @ sK518 @ X2)))) & ($true = (in @ X3 @ (powerset @ sK517)))) & ($true = (in @ X2 @ (powerset @ sK517)))))),
% 4.47/2.19 introduced(definition,[],[choice_axiom])).
% 4.47/2.19 thf(f1914,plain,(
% 4.47/2.19 ? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : (($true != (in @ X4 @ (binunion @ X2 @ X3))) & ($true = (in @ X4 @ sK517)) & ($true = (in @ X4 @ (binintersect @ sK518 @ X2)))) & ($true = (in @ X3 @ (powerset @ sK517)))) & ($true = (in @ X2 @ (powerset @ sK517)))) => (? [X3 : $i] : (? [X4 : $i] : (($true != (in @ X4 @ (binunion @ sK519 @ X3))) & ($true = (in @ X4 @ sK517)) & ($true = (in @ X4 @ (binintersect @ sK518 @ sK519)))) & ($true = (in @ X3 @ (powerset @ sK517)))) & ($true = (in @ sK519 @ (powerset @ sK517))))),
% 4.47/2.19 introduced(definition,[],[choice_axiom])).
% 4.47/2.19 thf(f1915,plain,(
% 4.47/2.19 ? [X3 : $i] : (? [X4 : $i] : (($true != (in @ X4 @ (binunion @ sK519 @ X3))) & ($true = (in @ X4 @ sK517)) & ($true = (in @ X4 @ (binintersect @ sK518 @ sK519)))) & ($true = (in @ X3 @ (powerset @ sK517)))) => (? [X4 : $i] : (($true != (in @ X4 @ (binunion @ sK519 @ sK520))) & ($true = (in @ X4 @ sK517)) & ($true = (in @ X4 @ (binintersect @ sK518 @ sK519)))) & ($true = (in @ sK520 @ (powerset @ sK517))))),
% 4.47/2.19 introduced(definition,[],[choice_axiom])).
% 4.47/2.19 thf(f1916,plain,(
% 4.47/2.19 ? [X4 : $i] : (($true != (in @ X4 @ (binunion @ sK519 @ sK520))) & ($true = (in @ X4 @ sK517)) & ($true = (in @ X4 @ (binintersect @ sK518 @ sK519)))) => (($true != (in @ sK521 @ (binunion @ sK519 @ sK520))) & ($true = (in @ sK521 @ sK517)) & ($true = (in @ sK521 @ (binintersect @ sK518 @ sK519))))),
% 4.47/2.19 introduced(definition,[],[choice_axiom])).
% 4.47/2.19 thf(f969,plain,(
% 4.47/2.19 (setminusI = $true) & (subset2powerset = $true) & (upairsetIL = $true) & (binintersectT_lem = $true) & (omega__Cong = $true) & (emptyI = $true) & (setadjoinIL = $true) & (binintersectRsub = $true) & (emptyinPowerset = $true) & (lamp = $true) & (dpsetconstrI = $true) & (emptysetE = $true) & (upairsetE = $true) & (cartprodmempair1 = $true) & (upairset2E = $true) & (powersetsubset = $true) & (contrasubsetT1 = $true) & (cartprodfstpairEq = $true) & (ap2apEq1 = $true) & (ksndsingleton = $true) & (beta1 = $true) & (setunionE2 = $true) & (dsetconstrI = $true) & (binunionEcases = $true) & (lam2p = $true) & (complementTI1 = $true) & (powersetTI1 = $true) & (symdiffIneg1 = $true) & (dpsetconstrEL1 = $true) & (setukpairinjR1 = $true) & (dpsetconstrEL2 = $true) & (symdiffI2 = $true) & (omegaSAx = $true) & (setadjoinAx = $true) & (notdexE = $true) & (cartprodpairsurjEq = $true) & (binintersectSubset3 = $true) & (brelnall1 = $true) & (binintersectSubset2 = $true) & (replAx = $true) & (subsetI2 = $true) & (setoftrueEq = $true) & (upairsubunion = $true) & (powersetTE1 = $true) & (descr__Cong = $true) & (app = $true) & (powerset__Cong = $true) & (symdiffI1 = $true) & (exuE1 = $true) & (ubforcartprodlem1 = $true) & (singletoninpowerset = $true) & (cartprodmempaircEq = $true) & (funcext = $true) & (upairequniteq = $true) & (subsetemptysetimpeq = $true) & (powersetI1 = $true) & (complementTnotintersectT = $true) & (exuE3e = $true) & (kpairiskpair = $true) & ? [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) & ? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : (($true != (in @ X4 @ (binunion @ X2 @ X3))) & ($true = (in @ X4 @ X0)) & ($true = (in @ X4 @ (binintersect @ X1 @ X2)))) & ($true = (in @ X3 @ (powerset @ X0)))) & ($true = (in @ X2 @ (powerset @ X0))))) & (dpsetconstrSub = $true) & (doubleComplementSub2 = $true) & (secondinupair = $true) & (binunionE = $true) & (setext = $true) & (iftrueProp2 = $true) & (notinemptyset = $true) & (notsubsetI = $true) & (emptysetsubset = $true) & (emptyE1 = $true) & (setminusSubset1 = $true) & (dpsetconstrERa = $true) & (ex1E2 = $true) & (setunionE = $true) & (inPowerset = $true) & (omega0Ax = $true) & (apProp = $true) & (setukpairIR = $true) & (disjointsetsI1 = $true) & (subsetE2 = $true) & (setextAx = $true) & (theprop = $true) & (emptyInPowerset = $true) & (singletonprop = $true) & (contrasubsetT3 = $true) & (funcextLem = $true) & (lamProp = $true) & (emptysetimpfalse = $true) & (cartprodpairin = $true) & (binunionRsub = $true) & (eta1 = $true) & (ap2p = $true) & (setadjoinOr = $true) & (subPowSU = $true) & (subsetE = $true) & (iftrueorfalse = $true) & (setminusERneg = $true) & (singletonsswitch = $true) & (contraSubsetComplement = $true) & (subsetTI = $true) & (subsetTrans = $true) & (ex1I2 = $true) & (dsetconstrEL = $true) & (funcGraphProp1 = $true) & (complementSubsetComplementIntersect = $true) & (iftrue = $true) & (emptyinunitempty = $true) & (dsetconstr__Cong = $true) & (setadjoinSub2 = $true) & (powersetI = $true) & (omegaIndAx = $true) & (singletonsuniq = $true) & (setunionI = $true) & (exuI1 = $true) & (quantDeMorgan2 = $true) & (iffalseProp1 = $true) & (binintersectLsub = $true) & (setukpairinjR2 = $true) & (notinsingleton = $true) & (setunionAx = $true) & (nonemptyI = $true) & (complementT_lem = $true) & (setukpairinjR11 = $true) & (funcGraphProp2 = $true) & (setminusILneg = $true) & (kpairsurjEq = $true) & (cartprodpairmemEL = $true) & (beta2 = $true) & (exuEu = $true) & (infuncsetfunc = $true) & (nonemptyI1 = $true) & (singletonsubset = $true) & (bs114d = $true) & (ubforcartprodlem3 = $true) & (binintersectTELcontra = $true) & (kfstsingleton = $true) & (setadjoin__Cong = $true) & (powersetE = $true) & (setextsub = $true) & (subsetI1 = $true) & (contrasubsetT = $true) & (wellorderingAx = $true) & (in__Cong = $true) & (eqimpsubset2 = $true) & (setadjoinE = $true) & (ifp = $true) & (complementImpComplementIntersect = $true) & (setminusSubset2 = $true) & (uniqinunit = $true) & (quantDeMorgan4 = $true) & (eqinunit = $true) & (eqbreln = $true) & (ifSingleton = $true) & (dpsetconstrER = $true) & (binintersectTERcontra = $true) & (binunionLsub = $true) & (doubleComplementE1 = $true) & (funcinfuncset = $true) & (powersetAx = $true) & (theeq = $true) & (complementTE1 = $true) & (setukpairinjR = $true) & (cartprodpairmemER = $true) & (contrasubsetT2 = $true) & (binunionTIRcontra = $true) & (notequalI1 = $true) & (ap2apEq2 = $true) & (prop2setE = $true) & (foundationAx = $true) & (setminusT_lem = $true) & (binintersectEL = $true) & (exuI3 = $true) & (ubforcartprodlem2 = $true) & (vacuousDall = $true) & (upairset2IR = $true) & (setminusELneg = $true) & (cartprodmempair = $true) & (emptyset__Cong = $true) & (funcGraphProp3 = $true) & (upairinpowunion = $true) & (nonemptyE1 = $true) & (setukpairinjL2 = $true) & (setbeta = $true) & (funcext2 = $true) & (doubleComplementEq = $true) & (upairsetIR = $true) & (prop2setI = $true) & (sepSubset = $true) & (quantDeMorgan1 = $true) & (subbreln = $true) & (binintersectI = $true) & (funcGraphProp4 = $true) & (exu__Cong = $true) & (ex1I = $true) & (brelnall2 = $true) & (setextT = $true) & (binunionT_lem = $true) & (eqimpsubset1 = $true) & (complementTcontraSubset = $true) & (noeltsimpempty = $true) & (cartprodsndpairEq = $true) & (doubleComplementI1 = $true) & (symdiffE = $true) & (setukpairIL = $true) & (setminusEL = $true) & (subsetRefl = $true) & (setminusLsub = $true) & (ex1E1 = $true) & (descrp = $true) & (binunionIL = $true) & (quantDeMorgan3 = $true) & (setadjoinSub = $true) & (singletoninpowunion = $true) & (setukpairinjR12 = $true) & (setukpairinjL1 = $true) & (funcImageSingleton = $true) & (eta2 = $true) & (binunionTILcontra = $true) & (binunionIR = $true) & (binintersectSubset5 = $true) & (notdallE = $true) & (setunionsingleton2 = $true) & (cartprodsndin = $true) & (exuE3u = $true) & (powersetE1 = $true) & (kpairp = $true) & (doubleComplementSub1 = $true) & (notequalI2 = $true) & (setunionsingleton = $true) & (dsetconstrER = $true) & (kfstpairEq = $true) & (setminusIRneg = $true) & (setOfPairsIsBReln = $true) & (iffalseProp2 = $true) & (setunion__Cong = $true) & (iftrueProp1 = $true) & (complementInPowersetComplementIntersect = $true) & (binintersectSubset4 = $true) & (exuE2 = $true) & (powersetT_lem = $true) & (binintersectSubset1 = $true) & (prop2set2propI = $true) & (setminusER = $true) & (setunionsingleton1 = $true) & (inIntersectImpInUnion = $true) & (cartprodfstin = $true) & (nonemptyImpWitness = $true) & (inCongP = $true) & (binintersectER = $true) & (symdiffIneg2 = $true) & (sepInPowerset = $true) & (ksndpairEq = $true) & (lam2lamEq = $true) & (setadjoinIR = $true) & (exuI2 = $true) & (emptysetAx = $true) & (iffalse = $true) & (setukpairinjL = $true)),
% 4.47/2.19 inference(flattening,[status(thm)],[f968])).
% 4.47/2.19 thf(f968,plain,(
% 4.47/2.19 ((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((((? [X0 : $i,X1 : $i] : (? [X2 : $i] : (? [X3 : $i] : (? [X4 : $i] : ((($true != (in @ X4 @ (binunion @ X2 @ X3))) & ($true = (in @ X4 @ (binintersect @ X1 @ X2)))) & ($true = (in @ X4 @ X0))) & ($true = (in @ X3 @ (powerset @ X0)))) & ($true = (in @ X2 @ (powerset @ X0)))) & ($true = (in @ X1 @ (powerset @ X0)))) & (inIntersectImpInUnion = $true)) & (binunionTIRcontra = $true)) & (binunionTILcontra = $true)) & (complementTcontraSubset = $true)) & (contraSubsetComplement = $true)) & (complementInPowersetComplementIntersect = $true)) & (complementSubsetComplementIntersect = $true)) & (complementImpComplementIntersect = $true)) & (complementTnotintersectT = $true)) & (doubleComplementEq = $true)) & (doubleComplementSub2 = $true)) & (doubleComplementSub1 = $true)) & (doubleComplementE1 = $true)) & (doubleComplementI1 = $true)) & (contrasubsetT3 = $true)) & (contrasubsetT2 = $true)) & (contrasubsetT1 = $true)) & (contrasubsetT = $true)) & (binintersectTERcontra = $true)) & (binintersectTELcontra = $true)) & (complementTE1 = $true)) & (complementTI1 = $true)) & (powersetTE1 = $true)) & (powersetTI1 = $true)) & (subsetTI = $true)) & (setextT = $true)) & (complementT_lem = $true)) & (setminusT_lem = $true)) & (powersetT_lem = $true)) & (binunionT_lem = $true)) & (binintersectT_lem = $true)) & (iftrueorfalse = $true)) & (iffalse = $true)) & (iftrue = $true)) & (theeq = $true)) & (ifp = $true)) & (ifSingleton = $true)) & (iftrueProp2 = $true)) & (iftrueProp1 = $true)) & (iffalseProp2 = $true)) & (iffalseProp1 = $true)) & (eta2 = $true)) & (beta2 = $true)) & (lam2lamEq = $true)) & (eta1 = $true)) & (beta1 = $true)) & (ap2apEq2 = $true)) & (ap2apEq1 = $true)) & (funcext2 = $true)) & (funcext = $true)) & (eqbreln = $true)) & (subbreln = $true)) & (funcGraphProp4 = $true)) & (funcextLem = $true)) & (funcGraphProp2 = $true)) & (funcGraphProp3 = $true)) & (funcGraphProp1 = $true)) & (ex1E2 = $true)) & (brelnall2 = $true)) & (brelnall1 = $true)) & (lam2p = $true)) & (lamp = $true)) & (lamProp = $true)) & (funcinfuncset = $true)) & (ap2p = $true)) & (infuncsetfunc = $true)) & (app = $true)) & (apProp = $true)) & (funcImageSingleton = $true)) & (dpsetconstrER = $true)) & (dpsetconstrEL2 = $true)) & (dpsetconstrEL1 = $true)) & (dpsetconstrERa = $true)) & (setOfPairsIsBReln = $true)) & (dpsetconstrSub = $true)) & (dpsetconstrI = $true)) & (cartprodpairsurjEq = $true)) & (cartprodsndpairEq = $true)) & (cartprodfstpairEq = $true)) & (cartprodmempaircEq = $true)) & (cartprodpairmemER = $true)) & (cartprodpairmemEL = $true)) & (cartprodsndin = $true)) & (kpairsurjEq = $true)) & (ksndpairEq = $true)) & (ksndsingleton = $true)) & (setukpairinjR = $true)) & (setukpairinjR2 = $true)) & (upairequniteq = $true)) & (setukpairinjR1 = $true)) & (setukpairinjR12 = $true)) & (setukpairinjR11 = $true)) & (setukpairinjL = $true)) & (setukpairinjL2 = $true)) & (cartprodfstin = $true)) & (kfstpairEq = $true)) & (theprop = $true)) & (kfstsingleton = $true)) & (setukpairinjL1 = $true)) & (singletonsuniq = $true)) & (ex1I2 = $true)) & (ex1I = $true)) & (ex1E1 = $true)) & (singletonprop = $true)) & (setunionsingleton = $true)) & (setunionsingleton2 = $true)) & (setunionsingleton1 = $true)) & (setunionE2 = $true)) & (cartprodmempair = $true)) & (cartprodmempair1 = $true)) & (cartprodpairin = $true)) & (ubforcartprodlem3 = $true)) & (ubforcartprodlem2 = $true)) & (ubforcartprodlem1 = $true)) & (upairinpowunion = $true)) & (upairsubunion = $true)) & (upairset2E = $true)) & (singletoninpowunion = $true)) & (singletoninpowerset = $true)) & (singletonsubset = $true)) & (kpairp = $true)) & (kpairiskpair = $true)) & (setukpairIR = $true)) & (setukpairIL = $true)) & (secondinupair = $true)) & (symdiffIneg2 = $true)) & (symdiffIneg1 = $true)) & (symdiffI2 = $true)) & (symdiffI1 = $true)) & (symdiffE = $true)) & (setminusSubset1 = $true)) & (setminusLsub = $true)) & (setminusIRneg = $true)) & (setminusILneg = $true)) & (setminusELneg = $true)) & (setminusERneg = $true)) & (setminusSubset2 = $true)) & (setminusER = $true)) & (setminusEL = $true)) & (setminusI = $true)) & (bs114d = $true)) & (binintersectSubset1 = $true)) & (binintersectSubset4 = $true)) & (binintersectRsub = $true)) & (disjointsetsI1 = $true)) & (binintersectER = $true)) & (binintersectSubset3 = $true)) & (binintersectSubset2 = $true)) & (binintersectLsub = $true)) & (binintersectEL = $true)) & (binintersectSubset5 = $true)) & (binintersectI = $true)) & (binunionRsub = $true)) & (binunionLsub = $true)) & (binunionE = $true)) & (binunionEcases = $true)) & (binunionIR = $true)) & (upairset2IR = $true)) & (binunionIL = $true)) & (sepSubset = $true)) & (sepInPowerset = $true)) & (powersetsubset = $true)) & (inPowerset = $true)) & (powersetE1 = $true)) & (powersetI1 = $true)) & (subsetemptysetimpeq = $true)) & (setextsub = $true)) & (subset2powerset = $true)) & (setadjoinSub2 = $true)) & (setadjoinSub = $true)) & (subsetTrans = $true)) & (subsetRefl = $true)) & (notequalI2 = $true)) & (notequalI1 = $true)) & (notsubsetI = $true)) & (subsetE2 = $true)) & (subsetE = $true)) & (emptysetsubset = $true)) & (subsetI2 = $true)) & (eqimpsubset1 = $true)) & (eqimpsubset2 = $true)) & (subsetI1 = $true)) & (dsetconstr__Cong = $true)) & (descr__Cong = $true)) & (exuEu = $true)) & (omega__Cong = $true)) & (setunion__Cong = $true)) & (powerset__Cong = $true)) & (setadjoin__Cong = $true)) & (emptyset__Cong = $true)) & (exu__Cong = $true)) & (exuE3u = $true)) & (in__Cong = $true)) & (inCongP = $true)) & (exuI2 = $true)) & (exuI3 = $true)) & (exuI1 = $true)) & (notdallE = $true)) & (notdexE = $true)) & (prop2set2propI = $true)) & (prop2setI = $true)) & (quantDeMorgan4 = $true)) & (quantDeMorgan3 = $true)) & (quantDeMorgan2 = $true)) & (quantDeMorgan1 = $true)) & (vacuousDall = $true)) & (emptyE1 = $true)) & (upairsetIR = $true)) & (upairsetIL = $true)) & (upairsetE = $true)) & (singletonsswitch = $true)) & (eqinunit = $true)) & (notinsingleton = $true)) & (uniqinunit = $true)) & (nonemptyImpWitness = $true)) & (exuE2 = $true)) & (subPowSU = $true)) & (setunionE = $true)) & (setunionI = $true)) & (powersetE = $true)) & (emptyInPowerset = $true)) & (emptyinPowerset = $true)) & (powersetI = $true)) & (setoftrueEq = $true)) & (setadjoinOr = $true)) & (setadjoinE = $true)) & (setadjoinIR = $true)) & (emptyinunitempty = $true)) & (setadjoinIL = $true)) & (nonemptyI1 = $true)) & (nonemptyI = $true)) & (nonemptyE1 = $true)) & (setbeta = $true)) & (noeltsimpempty = $true)) & (emptyI = $true)) & (setext = $true)) & (exuE3e = $true)) & (notinemptyset = $true)) & (emptysetimpfalse = $true)) & (emptysetE = $true)) & (prop2setE = $true)) & (exuE1 = $true)) & (dsetconstrER = $true)) & (dsetconstrEL = $true)) & (dsetconstrI = $true)) & (descrp = $true)) & (wellorderingAx = $true)) & (foundationAx = $true)) & (replAx = $true)) & (omegaIndAx = $true)) & (omegaSAx = $true)) & (omega0Ax = $true)) & (setunionAx = $true)) & (powersetAx = $true)) & (setadjoinAx = $true)) & (emptysetAx = $true)) & (setextAx = $true)),
% 4.47/2.19 inference(ennf_transformation,[status(thm)],[f494])).
% 4.47/2.19 thf(f494,plain,(
% 4.47/2.19 ~((setextAx = $true) => ((emptysetAx = $true) => ((setadjoinAx = $true) => ((powersetAx = $true) => ((setunionAx = $true) => ((omega0Ax = $true) => ((omegaSAx = $true) => ((omegaIndAx = $true) => ((replAx = $true) => ((foundationAx = $true) => ((wellorderingAx = $true) => ((descrp = $true) => ((dsetconstrI = $true) => ((dsetconstrEL = $true) => ((dsetconstrER = $true) => ((exuE1 = $true) => ((prop2setE = $true) => ((emptysetE = $true) => ((emptysetimpfalse = $true) => ((notinemptyset = $true) => ((exuE3e = $true) => ((setext = $true) => ((emptyI = $true) => ((noeltsimpempty = $true) => ((setbeta = $true) => ((nonemptyE1 = $true) => ((nonemptyI = $true) => ((nonemptyI1 = $true) => ((setadjoinIL = $true) => ((emptyinunitempty = $true) => ((setadjoinIR = $true) => ((setadjoinE = $true) => ((setadjoinOr = $true) => ((setoftrueEq = $true) => ((powersetI = $true) => ((emptyinPowerset = $true) => ((emptyInPowerset = $true) => ((powersetE = $true) => ((setunionI = $true) => ((setunionE = $true) => ((subPowSU = $true) => ((exuE2 = $true) => ((nonemptyImpWitness = $true) => ((uniqinunit = $true) => ((notinsingleton = $true) => ((eqinunit = $true) => ((singletonsswitch = $true) => ((upairsetE = $true) => ((upairsetIL = $true) => ((upairsetIR = $true) => ((emptyE1 = $true) => ((vacuousDall = $true) => ((quantDeMorgan1 = $true) => ((quantDeMorgan2 = $true) => ((quantDeMorgan3 = $true) => ((quantDeMorgan4 = $true) => ((prop2setI = $true) => ((prop2set2propI = $true) => ((notdexE = $true) => ((notdallE = $true) => ((exuI1 = $true) => ((exuI3 = $true) => ((exuI2 = $true) => ((inCongP = $true) => ((in__Cong = $true) => ((exuE3u = $true) => ((exu__Cong = $true) => ((emptyset__Cong = $true) => ((setadjoin__Cong = $true) => ((powerset__Cong = $true) => ((setunion__Cong = $true) => ((omega__Cong = $true) => ((exuEu = $true) => ((descr__Cong = $true) => ((dsetconstr__Cong = $true) => ((subsetI1 = $true) => ((eqimpsubset2 = $true) => ((eqimpsubset1 = $true) => ((subsetI2 = $true) => ((emptysetsubset = $true) => ((subsetE = $true) => ((subsetE2 = $true) => ((notsubsetI = $true) => ((notequalI1 = $true) => ((notequalI2 = $true) => ((subsetRefl = $true) => ((subsetTrans = $true) => ((setadjoinSub = $true) => ((setadjoinSub2 = $true) => ((subset2powerset = $true) => ((setextsub = $true) => ((subsetemptysetimpeq = $true) => ((powersetI1 = $true) => ((powersetE1 = $true) => ((inPowerset = $true) => ((powersetsubset = $true) => ((sepInPowerset = $true) => ((sepSubset = $true) => ((binunionIL = $true) => ((upairset2IR = $true) => ((binunionIR = $true) => ((binunionEcases = $true) => ((binunionE = $true) => ((binunionLsub = $true) => ((binunionRsub = $true) => ((binintersectI = $true) => ((binintersectSubset5 = $true) => ((binintersectEL = $true) => ((binintersectLsub = $true) => ((binintersectSubset2 = $true) => ((binintersectSubset3 = $true) => ((binintersectER = $true) => ((disjointsetsI1 = $true) => ((binintersectRsub = $true) => ((binintersectSubset4 = $true) => ((binintersectSubset1 = $true) => ((bs114d = $true) => ((setminusI = $true) => ((setminusEL = $true) => ((setminusER = $true) => ((setminusSubset2 = $true) => ((setminusERneg = $true) => ((setminusELneg = $true) => ((setminusILneg = $true) => ((setminusIRneg = $true) => ((setminusLsub = $true) => ((setminusSubset1 = $true) => ((symdiffE = $true) => ((symdiffI1 = $true) => ((symdiffI2 = $true) => ((symdiffIneg1 = $true) => ((symdiffIneg2 = $true) => ((secondinupair = $true) => ((setukpairIL = $true) => ((setukpairIR = $true) => ((kpairiskpair = $true) => ((kpairp = $true) => ((singletonsubset = $true) => ((singletoninpowerset = $true) => ((singletoninpowunion = $true) => ((upairset2E = $true) => ((upairsubunion = $true) => ((upairinpowunion = $true) => ((ubforcartprodlem1 = $true) => ((ubforcartprodlem2 = $true) => ((ubforcartprodlem3 = $true) => ((cartprodpairin = $true) => ((cartprodmempair1 = $true) => ((cartprodmempair = $true) => ((setunionE2 = $true) => ((setunionsingleton1 = $true) => ((setunionsingleton2 = $true) => ((setunionsingleton = $true) => ((singletonprop = $true) => ((ex1E1 = $true) => ((ex1I = $true) => ((ex1I2 = $true) => ((singletonsuniq = $true) => ((setukpairinjL1 = $true) => ((kfstsingleton = $true) => ((theprop = $true) => ((kfstpairEq = $true) => ((cartprodfstin = $true) => ((setukpairinjL2 = $true) => ((setukpairinjL = $true) => ((setukpairinjR11 = $true) => ((setukpairinjR12 = $true) => ((setukpairinjR1 = $true) => ((upairequniteq = $true) => ((setukpairinjR2 = $true) => ((setukpairinjR = $true) => ((ksndsingleton = $true) => ((ksndpairEq = $true) => ((kpairsurjEq = $true) => ((cartprodsndin = $true) => ((cartprodpairmemEL = $true) => ((cartprodpairmemER = $true) => ((cartprodmempaircEq = $true) => ((cartprodfstpairEq = $true) => ((cartprodsndpairEq = $true) => ((cartprodpairsurjEq = $true) => ((dpsetconstrI = $true) => ((dpsetconstrSub = $true) => ((setOfPairsIsBReln = $true) => ((dpsetconstrERa = $true) => ((dpsetconstrEL1 = $true) => ((dpsetconstrEL2 = $true) => ((dpsetconstrER = $true) => ((funcImageSingleton = $true) => ((apProp = $true) => ((app = $true) => ((infuncsetfunc = $true) => ((ap2p = $true) => ((funcinfuncset = $true) => ((lamProp = $true) => ((lamp = $true) => ((lam2p = $true) => ((brelnall1 = $true) => ((brelnall2 = $true) => ((ex1E2 = $true) => ((funcGraphProp1 = $true) => ((funcGraphProp3 = $true) => ((funcGraphProp2 = $true) => ((funcextLem = $true) => ((funcGraphProp4 = $true) => ((subbreln = $true) => ((eqbreln = $true) => ((funcext = $true) => ((funcext2 = $true) => ((ap2apEq1 = $true) => ((ap2apEq2 = $true) => ((beta1 = $true) => ((eta1 = $true) => ((lam2lamEq = $true) => ((beta2 = $true) => ((eta2 = $true) => ((iffalseProp1 = $true) => ((iffalseProp2 = $true) => ((iftrueProp1 = $true) => ((iftrueProp2 = $true) => ((ifSingleton = $true) => ((ifp = $true) => ((theeq = $true) => ((iftrue = $true) => ((iffalse = $true) => ((iftrueorfalse = $true) => ((binintersectT_lem = $true) => ((binunionT_lem = $true) => ((powersetT_lem = $true) => ((setminusT_lem = $true) => ((complementT_lem = $true) => ((setextT = $true) => ((subsetTI = $true) => ((powersetTI1 = $true) => ((powersetTE1 = $true) => ((complementTI1 = $true) => ((complementTE1 = $true) => ((binintersectTELcontra = $true) => ((binintersectTERcontra = $true) => ((contrasubsetT = $true) => ((contrasubsetT1 = $true) => ((contrasubsetT2 = $true) => ((contrasubsetT3 = $true) => ((doubleComplementI1 = $true) => ((doubleComplementE1 = $true) => ((doubleComplementSub1 = $true) => ((doubleComplementSub2 = $true) => ((doubleComplementEq = $true) => ((complementTnotintersectT = $true) => ((complementImpComplementIntersect = $true) => ((complementSubsetComplementIntersect = $true) => ((complementInPowersetComplementIntersect = $true) => ((contraSubsetComplement = $true) => ((complementTcontraSubset = $true) => ((binunionTILcontra = $true) => ((binunionTIRcontra = $true) => ((inIntersectImpInUnion = $true) => ! [X0 : $i,X1 : $i] : (($true = (in @ X1 @ (powerset @ X0))) => ! [X2 : $i] : (($true = (in @ X2 @ (powerset @ X0))) => ! [X3 : $i] : (($true = (in @ X3 @ (powerset @ X0))) => ! [X4 : $i] : (($true = (in @ X4 @ X0)) => (($true = (in @ X4 @ (binintersect @ X1 @ X2))) => ($true = (in @ X4 @ (binunion @ X2 @ X3)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 4.47/2.19 inference(fool_elimination,[status(thm)],[f493])).
% 4.47/2.19 thf(f493,plain,(
% 4.47/2.19 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => ! [X0 : $i,X1 : $i] : ((in @ X1 @ (powerset @ X0)) => ! [X2 : $i] : ((in @ X2 @ (powerset @ X0)) => ! [X3 : $i] : ((in @ X3 @ (powerset @ X0)) => ! [X4 : $i] : ((in @ X4 @ X0) => ((in @ X4 @ (binintersect @ X1 @ X2)) => (in @ X4 @ (binunion @ X2 @ X3))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 4.47/2.19 inference(rectify,[status(thm)],[f263])).
% 4.47/2.19 thf(f263,negated_conjecture,(
% 4.47/2.19 ~(setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X17 : $i] : ((in @ X17 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((in @ X1 @ (binintersect @ X11 @ X16)) => (in @ X1 @ (binunion @ X16 @ X17))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 4.47/2.19 inference(negated_conjecture,[status(cth)],[f262])).
% 4.47/2.19 thf(f262,conjecture,(
% 4.47/2.19 setextAx => (emptysetAx => (setadjoinAx => (powersetAx => (setunionAx => (omega0Ax => (omegaSAx => (omegaIndAx => (replAx => (foundationAx => (wellorderingAx => (descrp => (dsetconstrI => (dsetconstrEL => (dsetconstrER => (exuE1 => (prop2setE => (emptysetE => (emptysetimpfalse => (notinemptyset => (exuE3e => (setext => (emptyI => (noeltsimpempty => (setbeta => (nonemptyE1 => (nonemptyI => (nonemptyI1 => (setadjoinIL => (emptyinunitempty => (setadjoinIR => (setadjoinE => (setadjoinOr => (setoftrueEq => (powersetI => (emptyinPowerset => (emptyInPowerset => (powersetE => (setunionI => (setunionE => (subPowSU => (exuE2 => (nonemptyImpWitness => (uniqinunit => (notinsingleton => (eqinunit => (singletonsswitch => (upairsetE => (upairsetIL => (upairsetIR => (emptyE1 => (vacuousDall => (quantDeMorgan1 => (quantDeMorgan2 => (quantDeMorgan3 => (quantDeMorgan4 => (prop2setI => (prop2set2propI => (notdexE => (notdallE => (exuI1 => (exuI3 => (exuI2 => (inCongP => (in__Cong => (exuE3u => (exu__Cong => (emptyset__Cong => (setadjoin__Cong => (powerset__Cong => (setunion__Cong => (omega__Cong => (exuEu => (descr__Cong => (dsetconstr__Cong => (subsetI1 => (eqimpsubset2 => (eqimpsubset1 => (subsetI2 => (emptysetsubset => (subsetE => (subsetE2 => (notsubsetI => (notequalI1 => (notequalI2 => (subsetRefl => (subsetTrans => (setadjoinSub => (setadjoinSub2 => (subset2powerset => (setextsub => (subsetemptysetimpeq => (powersetI1 => (powersetE1 => (inPowerset => (powersetsubset => (sepInPowerset => (sepSubset => (binunionIL => (upairset2IR => (binunionIR => (binunionEcases => (binunionE => (binunionLsub => (binunionRsub => (binintersectI => (binintersectSubset5 => (binintersectEL => (binintersectLsub => (binintersectSubset2 => (binintersectSubset3 => (binintersectER => (disjointsetsI1 => (binintersectRsub => (binintersectSubset4 => (binintersectSubset1 => (bs114d => (setminusI => (setminusEL => (setminusER => (setminusSubset2 => (setminusERneg => (setminusELneg => (setminusILneg => (setminusIRneg => (setminusLsub => (setminusSubset1 => (symdiffE => (symdiffI1 => (symdiffI2 => (symdiffIneg1 => (symdiffIneg2 => (secondinupair => (setukpairIL => (setukpairIR => (kpairiskpair => (kpairp => (singletonsubset => (singletoninpowerset => (singletoninpowunion => (upairset2E => (upairsubunion => (upairinpowunion => (ubforcartprodlem1 => (ubforcartprodlem2 => (ubforcartprodlem3 => (cartprodpairin => (cartprodmempair1 => (cartprodmempair => (setunionE2 => (setunionsingleton1 => (setunionsingleton2 => (setunionsingleton => (singletonprop => (ex1E1 => (ex1I => (ex1I2 => (singletonsuniq => (setukpairinjL1 => (kfstsingleton => (theprop => (kfstpairEq => (cartprodfstin => (setukpairinjL2 => (setukpairinjL => (setukpairinjR11 => (setukpairinjR12 => (setukpairinjR1 => (upairequniteq => (setukpairinjR2 => (setukpairinjR => (ksndsingleton => (ksndpairEq => (kpairsurjEq => (cartprodsndin => (cartprodpairmemEL => (cartprodpairmemER => (cartprodmempaircEq => (cartprodfstpairEq => (cartprodsndpairEq => (cartprodpairsurjEq => (dpsetconstrI => (dpsetconstrSub => (setOfPairsIsBReln => (dpsetconstrERa => (dpsetconstrEL1 => (dpsetconstrEL2 => (dpsetconstrER => (funcImageSingleton => (apProp => (app => (infuncsetfunc => (ap2p => (funcinfuncset => (lamProp => (lamp => (lam2p => (brelnall1 => (brelnall2 => (ex1E2 => (funcGraphProp1 => (funcGraphProp3 => (funcGraphProp2 => (funcextLem => (funcGraphProp4 => (subbreln => (eqbreln => (funcext => (funcext2 => (ap2apEq1 => (ap2apEq2 => (beta1 => (eta1 => (lam2lamEq => (beta2 => (eta2 => (iffalseProp1 => (iffalseProp2 => (iftrueProp1 => (iftrueProp2 => (ifSingleton => (ifp => (theeq => (iftrue => (iffalse => (iftrueorfalse => (binintersectT_lem => (binunionT_lem => (powersetT_lem => (setminusT_lem => (complementT_lem => (setextT => (subsetTI => (powersetTI1 => (powersetTE1 => (complementTI1 => (complementTE1 => (binintersectTELcontra => (binintersectTERcontra => (contrasubsetT => (contrasubsetT1 => (contrasubsetT2 => (contrasubsetT3 => (doubleComplementI1 => (doubleComplementE1 => (doubleComplementSub1 => (doubleComplementSub2 => (doubleComplementEq => (complementTnotintersectT => (complementImpComplementIntersect => (complementSubsetComplementIntersect => (complementInPowersetComplementIntersect => (contraSubsetComplement => (complementTcontraSubset => (binunionTILcontra => (binunionTIRcontra => (inIntersectImpInUnion => ! [X3 : $i,X11 : $i] : ((in @ X11 @ (powerset @ X3)) => ! [X16 : $i] : ((in @ X16 @ (powerset @ X3)) => ! [X17 : $i] : ((in @ X17 @ (powerset @ X3)) => ! [X1 : $i] : ((in @ X1 @ X3) => ((in @ X1 @ (binintersect @ X11 @ X16)) => (in @ X1 @ (binunion @ X16 @ X17)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))),
% 4.47/2.19 file('/export/starexec/sandbox/tmp/tmp.JCQicskBu4/DTF2THF_13002.p',inIntersectImpInUnion2)).
% 4.47/2.19 thf(f4043,plain,(
% 4.47/2.19 ($true = (in @ sK521 @ sK519)) | (binintersectER != $true)),
% 4.47/2.19 inference(trivial_inequality_removal,[status(thm)],[f4034])).
% 4.47/2.19 thf(f4034,plain,(
% 4.47/2.19 ($true != $true) | (binintersectER != $true) | ($true = (in @ sK521 @ sK519))),
% 4.47/2.19 inference(superposition,[status(thm)],[f2719,f3257])).
% 4.47/2.19 thf(f3257,plain,(
% 4.47/2.19 ($true = (in @ sK521 @ (binintersect @ sK518 @ sK519)))),
% 4.47/2.19 inference(cnf_transformation,[status(thm)],[f1917])).
% 4.47/2.19 thf(f2719,plain,(
% 4.47/2.19 ( ! [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X4 @ (binintersect @ X5 @ X3))) | ((in @ X4 @ X3) = $true) | (binintersectER != $true)) )),
% 4.47/2.19 inference(cnf_transformation,[status(thm)],[f1518])).
% 4.47/2.19 thf(f1518,plain,(
% 4.47/2.19 ((binintersectER = $true) | (($true = (in @ sK253 @ (binintersect @ sK254 @ sK252))) & ($true != (in @ sK253 @ sK252)))) & (! [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X4 @ (binintersect @ X5 @ X3))) | ((in @ X4 @ X3) = $true)) | (binintersectER != $true))),
% 4.47/2.19 inference(skolemisation,[status(esa),new_symbols(skolem,[sK252,sK253,sK254])],[f1516,f1517])).
% 4.47/2.19 thf(f1517,plain,(
% 4.47/2.19 ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (binintersect @ X2 @ X0)) = $true) & ($true != (in @ X1 @ X0))) => (($true = (in @ sK253 @ (binintersect @ sK254 @ sK252))) & ($true != (in @ sK253 @ sK252)))),
% 4.47/2.19 introduced(definition,[],[choice_axiom])).
% 4.47/2.19 thf(f1516,plain,(
% 4.47/2.19 ((binintersectER = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (binintersect @ X2 @ X0)) = $true) & ($true != (in @ X1 @ X0)))) & (! [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X4 @ (binintersect @ X5 @ X3))) | ((in @ X4 @ X3) = $true)) | (binintersectER != $true))),
% 4.47/2.19 inference(rectify,[status(thm)],[f1515])).
% 4.47/2.19 thf(f1515,plain,(
% 4.47/2.19 ((binintersectER = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (binintersect @ X2 @ X0)) = $true) & ($true != (in @ X1 @ X0)))) & (! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (binintersect @ X2 @ X0)) != $true) | ($true = (in @ X1 @ X0))) | (binintersectER != $true))),
% 4.47/2.19 inference(nnf_transformation,[status(thm)],[f851])).
% 4.47/2.19 thf(f851,plain,(
% 4.47/2.19 (binintersectER = $true) <=> ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (binintersect @ X2 @ X0)) != $true) | ($true = (in @ X1 @ X0)))),
% 4.47/2.19 inference(ennf_transformation,[status(thm)],[f738])).
% 4.47/2.19 thf(f738,plain,(
% 4.47/2.19 ! [X0 : $i,X1 : $i,X2 : $i] : (((in @ X1 @ (binintersect @ X2 @ X0)) = $true) => ($true = (in @ X1 @ X0))) <=> (binintersectER = $true)),
% 4.47/2.19 inference(fool_elimination,[status(thm)],[f737])).
% 4.47/2.19 thf(f737,plain,(
% 4.47/2.19 (binintersectER = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X1 @ (binintersect @ X2 @ X0)) => (in @ X1 @ X0)))),
% 4.47/2.19 inference(rectify,[status(thm)],[f114])).
% 4.47/2.19 thf(f114,axiom,(
% 4.47/2.19 (binintersectER = ! [X4 : $i,X1 : $i,X3 : $i] : ((in @ X1 @ (binintersect @ X3 @ X4)) => (in @ X1 @ X4)))),
% 4.47/2.19 file('/export/starexec/sandbox/tmp/tmp.JCQicskBu4/DTF2THF_13002.p',binintersectER)).
% 4.47/2.19 thf(f4132,plain,(
% 4.47/2.19 ~spl820_13),
% 4.47/2.19 inference(avatar_split_clause,[status(thm)],[f4131,f4087])).
% 4.47/2.19 thf(f4131,plain,(
% 4.47/2.19 ($true != (in @ sK521 @ sK519))),
% 4.47/2.19 inference(trivial_inequality_removal,[status(thm)],[f4130])).
% 4.47/2.19 thf(f4130,plain,(
% 4.47/2.19 ($true != $true) | ($true != (in @ sK521 @ sK519))),
% 4.47/2.19 inference(forward_demodulation,[status(thm)],[f4106,f3105])).
% 4.47/2.19 thf(f3105,plain,(
% 4.47/2.19 (binunionIL = $true)),
% 4.47/2.19 inference(cnf_transformation,[status(thm)],[f1917])).
% 4.47/2.19 thf(f4106,plain,(
% 4.47/2.19 ($true != (in @ sK521 @ sK519)) | (binunionIL != $true)),
% 4.47/2.19 inference(trivial_inequality_removal,[status(thm)],[f4095])).
% 4.47/2.19 thf(f4095,plain,(
% 4.47/2.19 (binunionIL != $true) | ($true != $true) | ($true != (in @ sK521 @ sK519))),
% 4.47/2.19 inference(superposition,[status(thm)],[f3259,f3658])).
% 4.47/2.19 thf(f3658,plain,(
% 4.47/2.19 ( ! [X2 : $i,X0 : $i,X1 : $i] : (($true = (in @ X0 @ (binunion @ X2 @ X1))) | (binunionIL != $true) | ($true != (in @ X0 @ X2))) )),
% 4.47/2.19 inference(cnf_transformation,[status(thm)],[f2295])).
% 4.47/2.19 thf(f2295,plain,(
% 4.47/2.19 (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (binunion @ X2 @ X1))) | ($true != (in @ X0 @ X2))) | (binunionIL != $true)) & ((binunionIL = $true) | (($true != (in @ sK755 @ (binunion @ sK757 @ sK756))) & ($true = (in @ sK755 @ sK757))))),
% 4.47/2.19 inference(skolemisation,[status(esa),new_symbols(skolem,[sK755,sK756,sK757])],[f2293,f2294])).
% 4.47/2.19 thf(f2294,plain,(
% 4.47/2.19 ? [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X3 @ (binunion @ X5 @ X4))) & ($true = (in @ X3 @ X5))) => (($true != (in @ sK755 @ (binunion @ sK757 @ sK756))) & ($true = (in @ sK755 @ sK757)))),
% 4.47/2.19 introduced(definition,[],[choice_axiom])).
% 4.47/2.19 thf(f2293,plain,(
% 4.47/2.19 (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (binunion @ X2 @ X1))) | ($true != (in @ X0 @ X2))) | (binunionIL != $true)) & ((binunionIL = $true) | ? [X3 : $i,X4 : $i,X5 : $i] : (($true != (in @ X3 @ (binunion @ X5 @ X4))) & ($true = (in @ X3 @ X5))))),
% 4.47/2.19 inference(rectify,[status(thm)],[f2292])).
% 4.47/2.19 thf(f2292,plain,(
% 4.47/2.19 (! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (binunion @ X2 @ X1))) | ($true != (in @ X0 @ X2))) | (binunionIL != $true)) & ((binunionIL = $true) | ? [X0 : $i,X1 : $i,X2 : $i] : (($true != (in @ X0 @ (binunion @ X2 @ X1))) & ($true = (in @ X0 @ X2))))),
% 4.47/2.19 inference(nnf_transformation,[status(thm)],[f993])).
% 4.47/2.19 thf(f993,plain,(
% 4.47/2.19 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ (binunion @ X2 @ X1))) | ($true != (in @ X0 @ X2))) <=> (binunionIL = $true)),
% 4.47/2.19 inference(ennf_transformation,[status(thm)],[f378])).
% 4.47/2.19 thf(f378,plain,(
% 4.47/2.19 ! [X0 : $i,X1 : $i,X2 : $i] : (($true = (in @ X0 @ X2)) => ($true = (in @ X0 @ (binunion @ X2 @ X1)))) <=> (binunionIL = $true)),
% 4.47/2.19 inference(fool_elimination,[status(thm)],[f377])).
% 4.47/2.19 thf(f377,plain,(
% 4.47/2.19 (binunionIL = ! [X0 : $i,X1 : $i,X2 : $i] : ((in @ X0 @ X2) => (in @ X0 @ (binunion @ X2 @ X1))))),
% 4.47/2.19 inference(rectify,[status(thm)],[f101])).
% 4.47/2.19 thf(f101,axiom,(
% 4.47/2.19 (binunionIL = ! [X1 : $i,X4 : $i,X3 : $i] : ((in @ X1 @ X3) => (in @ X1 @ (binunion @ X3 @ X4))))),
% 4.47/2.19 file('/export/starexec/sandbox/tmp/tmp.JCQicskBu4/DTF2THF_13002.p',binunionIL)).
% 4.47/2.19 thf(f3259,plain,(
% 4.47/2.19 ($true != (in @ sK521 @ (binunion @ sK519 @ sK520)))),
% 4.47/2.19 inference(cnf_transformation,[status(thm)],[f1917])).
% 4.47/2.19 % SZS output end Proof for DTF2THF_13002
% 4.47/2.19 % (13159)------------------------------
% 4.47/2.19 % (13159)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 4.47/2.19 % (13159)Termination reason: Refutation
% 4.47/2.19
% 4.47/2.19 % (13159)Memory used [KB]: 10106
% 4.47/2.19 % (13159)Time elapsed: 0.182 s
% 4.47/2.19 % (13159)Instructions burned: 357 (million)
% 4.47/2.19 % (13159)------------------------------
% 4.47/2.19 % (13159)------------------------------
% 4.47/2.19 % (13136)Success in time 0.288 s
%------------------------------------------------------------------------------